The Correspondence of John Wallis: Volume II (1660 - September 1668)

The Correspondence of John Wallis, Volume II

PHILIP BEELEY CHRISTOPH J. SCRIBA, Editors

OXFORD UNIVERSITY PRESS

John Wallis c.1668 engraving by William Faithorne (16167-91)

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The Correspondence of John Wallis Volume II (1660-September 1668) Editors PHILIP BEELEY CHRISTOPH J. SCRIBA With the Assistance of Uwe Mayer

OXPORD UNIVERSITY PRESS

OXPORD UNIVERSITY PRESS

Great Clarendon Street Oxford OX2 6DP Oxford University Press is a department of the University of Oxford. Tt furthers the University's objective of excellence in research, scholarship, and education by publishing worldwide in Oxford New York Auckland Cape Town Dar es Salaam Honk Kong Karachi Kuala Lumpur Madrid Melbourne Mexico City Nairobi New Delhi Shanghai Taipei Toroanto With offices in Argentina Austria Brazil Chile Czech Republic France Greece Guatemala Hungray Italy Japan South Korea Poland Portugal Singapore Switzerland Thailand Turkey Ukraine Vietnam Published in the United States by Oxford University Press Inc., New York © Oxford University Pi-ess, 2005 The moral rights of the author have been asserted Database right Oxford University Press (maker) First published 2005 All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, without the prior permission in writing of Oxford University Press, or as expressly permitted by law, or under terms agreed with the appropriate reprographics rights organization. Enquiries concerning reproduction outside the scope of the above should be sent to the Rights Department, Oxford University Press, at the address above You must not circulate this book in any other binding or cover and you must impose this same condition on any acquirer British Library Cataloguing in Publication data Data available Library of Congress Cataloging in Publication Data Data available ISBN 0-19-856601-8 1 3 5 7 9 1 0 8 6 4 2 Typeset by the Editors Printed in Great Britain on acid-free paper by Bukkes Ltd., King's Hynn, Norfolk

The Correspondence of John Wallis Volume II (1660 - September 1668)

PHILIP BEELEY CHRISTOPH J. SCRIBA Editors With the Assistance of Uwe Mayer

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For Inge Scriba

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PREFACE

The second volume of the Correspondence of John Wallis (1616-1703) covers a decisive period of political and scientific reorganization in England following the eleven years of the Interregnum. It begins just a matter of weeks before the triumphal return of Charles II from the Netherlands and his accession to the throne on 29 May 1660 and sees the emergence of the Royal Society, soon to become the leading organization of its kind in Europe, from two main scientific circles which had flourished in London and Oxford during the revolutionary years. Wallis, who had been appointed Savilian professor of geometry at the University of Oxford largely in recognition of his services to parliament as a decipherer during the Civil Wars was in this period establishing himself as one of the central figures in the growth of modern science in England. Fittingly, his name stands alongside those of Robert Boyle, Kenelm Digby, Robert Moray, John Wilkins, and Christopher Wren as one of the original members of council, appointed on 15 July 1662, when the first charter 'for the incorporation of the society under the title of the Royal Society' was passed. The years encompassed by the present volume witness the Savilian professor consolidate his reputation both in London and in Oxford. At the Royal Society he was a close associate of its secretary Henry Oldenburg, with whom he corresponded regularly and in whose journal, the The Philosophical Transactions of the Royal Society, he published articles on a wide range of themes from the fields of mathematics and the physical sciences. Within the walls of the University of Oxford he continued to fulfil his professorial duties and at the same time succeeded in acquiring considerable authority and influence through the office of keeper of the archives, to which he had been elected under somewhat questionable circumstances in 1657/8. Although not a trained lawyer, Wallis displayed considerable legal acumen and an unswayable commitment to defending the University's rights in the face of numerous attempts to infringe these on the part of city authorities. The volume contains a total of 232 existent or identified letters exchanged between Wallis and over 30 correspondents at home and abroad Vll

between February 1660 and September 1668. More than sixty of these letters have never appeared in print before, while many of the rest are only to be found in publications which are rare or not widely accessible. Eighteen of the letters printed are part of the former Macclesfield Collection, now in the possession of Cambridge University Library. Although these letters have been previously available in S. J. Rigaud's Correspondence of Scientific Men (1841), they are published here completely in the form of a critical edition for the first time. As in the first volume, additional letters and material have been included where these form an essential part of exchanges between Wallis and other scholars or where they are able to throw light on correspondence for which no other documentary evidence has survived. In this way it has been aimed to provide as complete a picture as possible of the literary activity of the Savilian professor during the nine years covered. Work on the edition has been made possible through funding from the Deutsche Forschungsgemeinschaft. The editors should like to express their sincere thanks to this organization for its continued generosity. An enterprise such as the Wallis correspondence project can only succeed through close international cooperation of scholars working in the field. The editors are therefore once more glad to record their debt to numerous friends and colleagues who through advice or assistance have contributed significantly to realizing the volume in the present form. The editors are once again grateful to Robert Hatch for providing information on manuscript sources in France and the United States. David Cram and Jackie Stedall are to be thanked not only for sharing their profound knowledge of Wallis and his contemporaries but also for their willingness to provide help in almost every conceivable way. Further, the editors should like to express their gratitude to Jordan Avramov for useful discussions on questions concerning Henry Oldenburg and Robert Moray. They are grateful also to Jason Rampelt for kindly allowing them to read parts of his as yet unfinished Cambridge dissertation on the life and work of Wallis. Special thanks go to Mordechai Feingold who read the manuscript at an early stage of completion and who as always provided many helpful suggestions on commentaries and general presentation. On the technical side the editors have once again had the good fortune to be able to fall back on the sound advice of Menso Folkerts and his colleagues in Munich. They are particularly grateful to Albert Krayer for giving vital assistance on questions concerning the ET^X system used for editing. vin

The editors have continued to enjoy the generous support of the Institute for History of Science, Mathematics, and Technology at the University of Hamburg. They should like to express their deep gratitude in particular to Karin Reich in her capacity as director of the Institute for ensuring that the correspondence project continues to nourish. The University of Hamburg has generously provided rooms to accommodate the project, and the Department of Mathematics has made sure that those employed on the project receive all the necessary technical assistance. Without this help much of the work on the edition would not have been possible. A personal note of thanks goes to Siegmund Probst who has continued to share generously in his knowledge of seventeenth-century mathematics. The editors are also grateful for the dedication shown by Ines Harrie and Oliver Leistert while working on the project in the course of their studies. The editors' greatest debt goes to Uwe Mayer who has taken on a substantial part of the workload over the last years. His editorial efforts have been enormous. Without his energy and commitment the second volume could not have been realized in the short length of time it has taken. Staff at numerous libraries and archives have provided often invaluable help in preparing the present volume. The editors should like to express their particular gratitude to Adam Perkins of Cambridge University Library, Rupert Baker of the Library of the Royal Society, Frances Harris of the British Library, William Hodges of Duke Humfrey, Bodleian Library, and Peter Rau and staff of the Staatsbibliothek Hamburg. Once again, the editors are especially indebted to Simon Bailey, the Keeper of the Archives of the University of Oxford, and to Alice B. Millea, the Assistant Keeper. The editors are grateful to the following persons and institutions for granting permission to publish copyright material held in their possession: The Syndics of Cambridge University Library; the Librarian of the Bibliotheek der Rijksuniversiteit, Leiden; the British Library Board; the Librarian of the Bodleian Library, Oxford; the Keeper of the Archives, University of Oxford; the Koninklijke Bibliotheek, The Hague; the Bibliotheque Nationale de France; the Royal Society; the National Archives, Kew; and the Governors and Guardians of Marsh's Library, Dublin. Quotations from the Devonshire Manuscripts, Chatsworth House, are published by permission of the Chatsworth Settlement Trustees. IX

Finally, the editors should like to thank editorial staff at Oxford University Press for their continued help in realizing the edition, their care in proof reading, and their patience in the light of often considerable delays in the submission of material. Philip Beeley Christoph J. Scriba Miinster (Westf.) and Hamburg, June 2004

x

CONTENTS

Introduction Restoration politics and scientific practice Wallis, Oldenburg, and the Royal Society Correspondence with Boyle Mathematical correspondence Correspondence with Collins Wallis, Gregory, and Huygens Cryptanalysis University affairs Personal affairs Theological affairs Editorial principles and abbreviations

xix XX

xxii XXV

xxvi xxix XXX

xxxi xxxi xxxiii xxxiii xxxv

Correspondence 1. HUYGENS to C ARC AVI, [16] /26 February [1659] / 1660 . . . . 2. HUYGENS to CARCAVI, [17]/27 March [1659]/1660 3. HUYGENS to WALLIS, [21]/31 March [1659]/1660 4. CARCAVI to WALLIS, June? 1660 5. CARCAVI to HUYGENS, [15]/25 June 1660 6. HUYGENS to WALLIS, [5]/15 July 1660 7. WALLIS to DIGBY, 24 August/[3 September] 1660 8. WALLIS to HUYGENS, 31 August/[10 September] 1660 . . . . 9. FELL to YATE, 5/[15] February [1660/1 ?] 10. WALLIS to DILLINGHAM, 16/[26] March 1660/1 11. WALLIS: Humble Petition to Charles II, [March? 1661] . . . 12. DILLINGHAM to WALLIS, 1/[11] April 1661 13. COOPER to WALLIS, 23 April/ [3 May] 1661 14. [HOBBES]: La duplication du cube, [June? 1661] 15. WALLIS to [BROUNCKER?], 23 June/[3 July] 1661 16. FRENICLE to WALLIS, [October/November ? 1661] XI

1 1 7 11 13 13 20 22 27 30 32 35 36 37 38 41 45

Contents 17. WALLIS to FRENICLE, [November/December ? 1661] . . . . 18. FRENICLE to WALLIS, [[10]/20 December 1661] 19. WALLIS to BOYLE, 30 December 1661/[9 January 1662] . . 20. BOYLE to WALLIS, 4 or 5/[14 or 15] January 1661/[1662] . 21. WALLIS to BOYLE, 20 February/[2 March] 1661/2 22. BOYLE to WALLIS, 26 February/[8 March] 1661/[1662] . . . 23. WALLIS to BOYLE, 14/[24] March 1661/2 (i) 24. [WALLIS:] Note on letter to Boyle, 11/[21] July 1670 . . . . 25. WALLIS to BOYLE, 14/[24] March 1661/2 (ii) 26. BOYLE to WALLIS, 5/[15] April 1662 27. WALLIS to MORAY, 7/[17] April 1662 28. MORAY to WALLIS, April/May 1662 29. WALLIS to MORAY, 6/[16] May 1662 30. MORAY? ET AL. to WALLIS, 8/[18] May 1662 31. WALLIS to TITUS, 12/[22] June 1662 32. WHARTON to WALLIS, 30 September/[10 October] 1662 . . 33. BROWNE to CHYLINSKI, 3/[13] October 1662 34. CHYLINSKI to BROWNE, 4/[14] October 1662 35. CHYLINSKI to WALLIS, 4/[14] October 1662 36. WALLIS to DILLINGHAM, 21/[31] October 1662 37. DILLINGHAM to WALLIS, 27 October/[6 November] 1662 . . 38. HOBBES to WALLIS, end of 1662 39. WALLIS to HEVELIUS, 9/[19] February 1662/3 40. WALLIS to BOYLE, 27 March/[6 April] 1663 41. WALLIS to HEVELIUS, 30 March/[9 April] 1663 42. BOYLE to WALLIS, August/September 1663 43. WALLIS to BOYLE, 10/[20] September 1663 44. WALLIS to BOYLE, 10/[20] September 1663, enclosure (i) . . 45. WALLIS to BOYLE, 10/[20] September 1663, enclosure (ii) . 46. WALLIS for the ROYAL SOCIETY, 24 September/[4 October] 1663 47. HEVELIUS to WALLIS, [25 December 1663]/4 January 1664 48. WALLIS to BROUNCKER?, ? March 1664 (i) 49. WALLIS to BROUNCKER?, ? March 1664 (ii) 50. WALLIS to HEVELIUS, 5/[15] April 1664 51. WALLIS to OLDENBURG, 6/[16] April 1664 52. WALLIS to OLDENBURG, 7/[17] April 1664 53. OLDENBURG to WALLIS, 21 April/[l May] 1664 54. WALLIS to OLDENBURG, 30 April/[10 May] 1664 Xll

45 46 47 50 50 51 51 60 63 68 69 71 71 72 73 73 74 75 76 77 77 79 79 80 82 86 87 88 91

99 100 102 102 103 106 110 Ill Ill

Contents 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92.

OLDENBURG to WALLIS, 4/[14] May 1664 WALLIS to OLDENBURG, 7/[17] May 1664 WALLIS to OLDENBURG, 14/[24] May 1664 WALLIS to OLDENBURG?, 16/[26] May 1664 WALLIS to OLDENBURG, 25 May/[4 June] 1664 WALLIS to BLANDFORD, 13/[23] June 1664 JENKINS to WALLIS, 15/[25] June 1664 WALLIS to BLANDFORD, 15/[25] ? June 1664 BLANDFORD to WALLIS, 16/[26] June 1664 WALLIS to BLANDFORD, 18/[28] June 1664 WALLIS to BLANDFORD, 20/[30] June 1664 JENKINS to WALLIS, 21 June/[l July] 1664 BLANDFORD to WALLIS, 21 June/[l July] 1664 WALLIS to BLANDFORD, 23 June/[3 July] 1664 BLANDFORD to WALLIS, 27 June/[7 July] 1664 WALLIS to BLANDFORD, 28 June/[8 July] 1664 JENKINS to WALLIS, early-mid July 1664 WALLIS to BLANDFORD, 18/[28] July 1664 HEVELIUS to WALLIS, c.31 August/[10 September] 1664 . . CONSTANTUN HuYGENS for WALLIS, 7/[17] September 1664 WALLIS to OLDENBURG, 21 September/]! October] 1664 . . OLDENBURG to WALLIS, 29 September/[9 October] 1664 . . CONSTANTUN HUYGENS for WALLIS, 6/[16] October 1664 . OLDENBURG to WALLIS, 13/[23] October 1664 OLDENBURG to WALLIS, 21/[31] October 1664 WALLIS to OLDENBURG, 29 October/[8 November] 1664 . . WALLIS to CHYLINSKI, c.!7/[27] November 1664 CHYLINSKI to WALLIS, 19/[29] November 1664 WALLIS to BURSCOUGH, ? November 1664 BURSCOUGH to WALLIS, 24 November/[4 December] 1664 . OLDENBURG to WALLIS, mid-December 1664 WALLIS to OLDENBURG?, 24 December 1664/[3 January 1665] WALLIS to OLDENBURG, 21/[31] January 1664/5 H. L. to WALLIS, 10/[20] April 1665 BOYLE to WALLIS, early 1665 WALLIS to BOYLE, 24 April/[4 May] 1665 WALLIS to BOYLE, 29 April/[9 May] 1665 WALLIS to OLDENBURG?, 8/[18] May 1665 Xlll

114 114 118 137 137 139 142 144 144 145 146 147 148 149 154 155 157 157 158 159 160 162 163 164 164 164 166 166 168 168 170 171 174 179 182 182 182 184

Contents 93. MATTHEW WREN to WALLIS, 30 May/[9 June] 1665 . . . . 185 94. WALLIS to MATTHEW WREN, 4/[14] June 1665 186 95. OLDENBURG to WALLIS, 28 September/[8 October] 1665 . . 186 96. OLDENBURG to WALLIS, 26 December 1665/[5 January 1666] 187 97. OLDENBURG to WALLIS, 30 December 1665/[9 January 1666] 187 98. COLLINS to WALLIS, 2/[12] January 1665/[1666] 188 99. WALLIS to COLLINS, January/February 1665/6 191 100. COLLINS to WALLIS, 28 February/[10 March] [1665/6] . . 191 101. OLDENBURG to WALLIS, early 1666 195 102. WALLIS to OLDENBURG, 3/[13] April 1666 195 103. WALLIS to OLDENBURG, 23 April/[3 May] 1666 197 104. WALLIS to BOYLE, 25 April/[5 May] 1666 200 105. OLDENBURG to WALLIS, 5/[15] May 1666 223 106. WALLIS to OLDENBURG, 7/[17] May 1666 223 107. WALLIS to OLDENBURG, 12/[22] May 1666 226 108. OLDENBURG to WALLIS, 17/[27] May 1666 231 109. WALLIS to OLDENBURG, 19/[29] May 1666 232 110. WALLIS to OLDENBURG, 24 May/[3 June] 1666 235 111. OLDENBURG to WALLIS, 29 May/[8 June] 1666 238 112. OLDENBURG to WALLIS, 31 May/[10 June] 1666 240 113. WALLIS to OLDENBURG, 2/[12] June 1666 240 114. WALLIS to OLDENBURG, 8/[18] June 1666 246 115. OLDENBURG: Objections against Dr Wallis's Hypothesis of Tides, [June? 1666] 250 116. WALLIS to OLDENBURG, 18/[28] July 1666 251 117. WALLIS to [OLDENBURG], 24 July/[3 August] 1666 . . . . 263 118. OLDENBURG to WALLIS, 26 July/[5 August] 1666 270 119. OLDENBURG to WALLIS, 31 July/[10 August] 1666 . . . . 270 120. WALLIS to OLDENBURG, 2/[12] August 1666 271 121. COLLINS to WALLIS, 2/[12] August [1666] 275 122. OLDENBURG to WALLIS, 4/[14] August 1666 278 123. WALLIS to COLLINS, 7/[17] August 1666 278 124. WALLIS to OLDENBURG, 11/[21] August 1666 281 125. OLDENBURG to WALLIS, mid-August 1666 283 126. WALLIS to OLDENBURG, 18/[28] August 1666 284 127. WALLIS to [OLDENBURG?], 23 October/[2 November] 1666 287 128. FELL to WALLIS, October/November 1666 (i) 288 129. FELL to WALLIS, October/November 1666 (ii) 288 130. FELL to WALLIS, 18/[28] November [1666] 288 XIV

Contents 131. 132. 133. 134. 135. 136. 137. 138. 139.

FELL to WALLIS, 5/[15] December [1666 ?] OLDENBURG to WALLIS, c.8/[18] January 1666/7 OLDENBURG to WALLIS, 15/[25] January 1666/7 WALLIS to OLDENBURG, 19/[29] January 1666/7 OLDENBURG to WALLIS, 24 January/[3 February] 1666/7 . WALLIS to OLDENBURG, 31 January/[10 February] 1666/7 COLLINS to WALLIS, [January? 1666/7] WALLIS to COLLINS, January? 1666/7 COLLINS to WALLIS, 2/[12] February 1666/7

290 291 291 291 295 296 298 301 301

140. COLLINS to WALLIS, 2/[12] February 1666/7, enclosure . . 307

141. 142. 143. 144. 145. 146. 147. 148. 149. 150. 151. 152. 153. 154. 155. 156. 157. 158. 159. 160. 161. 162. 163. 164. 165. 166.

WALLIS to COLLINS, 5/[15] February 1666/7 COLLINS to WALLIS, [c. 10/[20] February 1666/7] WALLIS to OLDENBURG, 12/[22] February 1666/7 FELL to WALLIS, 19 February/[l March] [1666/7?] . . . . OLDENBURG to WALLIS, 19/[29] March 1666/7 WALLIS to OLDENBURG, 21/[31] March 1666/7 WALLIS to CLENDON, 5/[15] May 1667 OLDENBURG to BOYLE, 17/[27] September 1667 WALLIS to JENKINS, 26 October/[5 November] 1667 . . . . JENKINS to WALLIS, 26 October/[5 November] 1667 . . . . WALLIS to JENKINS, 27 October/[6 November] 1667 . . . . WALLIS to JENKINS, 31 October/[10 November] 1667 . . . JENKINS to WALLIS, 31 October/[10 November] 1667 . . . WALLIS to HOPKINS, 1/[11] November 1667 WALLIS to JENKINS, 2/[12] November 1667 WALLIS to JENKINS, 2/[12] November 1667, enclosure . . . WALLIS: Notes on Martivell's case, [1667 ?] JENKINS to WALLIS, 5/[15] November [1667] WALLIS to JENKINS, 11/[21] November 1667 WALLIS to OLDENBURG, 16/[26] November 1667 JENKINS to WALLIS, 16/[26] November 1667 OLDENBURG to WALLIS, 26 November/[6 December] 1667 WALLIS to OLDENBURG, 30 November/[10 December] 1667 OLDENBURG to WALLIS, 10/[20] December 1667 WALLIS to OLDENBURG, 13/[23] December 1667 OLDENBURG to WALLIS, 24 December 1667/[3 January 1668] 167. JENKINS to [WALLIS?], [January 1667/8?] 168. WALLIS to JENKINS, 21/[31] January 1667/8 XV

309 311 318 321 322 322 325 325 330 332 332 334 337 339 339 346 350 352 353 354 359 360 361 365 368 375 378 379

Contents

169. WALLIS to JENKINS 21/[31] January 1667/8, enclosure . . 380 170. WALLIS to JENKINS, 28 January/[7 February] 1667/8 . . . 382

171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure 384 172. BROUNCKER: Solution to Dulaurens's Problem, [December 1667/January 1667/8]

394

173. 174. 175. 176.

395 396 400 400

JENKINS to WALLIS, 30 January/[9 February] 1667/[1668] WALLIS to OLDENBURG, I/[11] February 1667/[1668] . . . OLDENBURG to WALLIS, 4/[14] February 1667/8 WALLIS to OLDENBURG, 8/[18] February 1667/8

177. WALLIS: Solution to Dulaurens's Problem, 8/[18] February

178. 179. 180. 181. 182. 183. 184. 185. 186. 187. 188. 189. 190. 191. 192. 193. 194. 195. 196. 197. 198. 199.

1667/8 WALLIS to JENKINS, 11/[21] February 1667/8 OLDENBURG to WALLIS, 11/[21] February 1667/8 JENKINS to WALLIS, 13/[23] February 1667/[1668] WALLIS to COLLINS, early 1668? WALLIS to COLLINS, 15/[25] February 1667/8 WALLIS to LEOTAUD, 17/[27] February 1667/[1668] . . . . OLDENBURG to WALLIS, 25 February/[6 March] 1667/8 . COLLINS to WALLIS, 25 February/[6 March] 1667/8 . . . . WALLIS to COLLINS, 27 February/[8 March] 1667/8 . . . . WALLIS to OLDENBURG, 29 February/[10 March] ? 1667/8 MORAY? to WALLIS, ? February 1667/8 WALLIS to MORAY?, ? February 1667/8 WALLIS to OLDENBURG, 7/[17] March 1667/8 (i) WALLIS to OLDENBURG, 7/[17] March 1667/8 (ii) OLDENBURG to WALLIS, 10/[20] March 1667/8 WALLIS to OLDENBURG, 17/[27] March 1667/8 (i) WALLIS to OLDENBURG, 17/[27] March 1667/8 (ii) . . . . OLDENBURG to WALLIS, 26 March/[5 April] 1668 WALLIS to OLDENBURG, 30 March/[9 April] 1668 BROUNCKER to OLDENBURG, March? 1667/8 [WALLIS?] to OLDENBURG, March/April? 1668 PHILIPS to WALLIS, 6/[16] April 1668

402 405 406 408 409 409 412 428 429 429 432 432 432 434 435 440 440 441 446 446 454 461 462

200. CONSTANTUN HuYGENS to WALLIS, [21 June]/I July 1668 467

201. 202. 203. 204.

OLDENBURG to WALLIS, 30 June/[10 July] 1668 WALLIS to COLLINS, early July 1668 WALLIS to OLDENBURG, 2/[12] July 1668 (i) WALLIS to OLDENBURG, 2/[12] July 1668 (ii) xvi

469 469 470 471

Contents 205. 206. 207. 208. 209. 210. 211. 212. 213. 214. 215. 216. 217. 218. 219. 220. 221. 222. 223. 224. 225. 226. 227. 228. 229. 230. 231. 232. 233. 234. 235. 236. 237. 238. 239. 240. 241. 242.

WALLIS to OLDENBURG, 4/[14] July 1668 OLDENBURG to WALLIS, 4/[14] ? July 1668 WALLIS to OLDENBURG, 6/[16] July 1668 WALLIS to OLDENBURG, 8/[18] July 1668 WALLIS to OLDENBURG, 11/[21] July 1668 COLLINS to BRERETON, 11/[21] July 1668 OLDENBURG to WALLIS, 13/[23] July 1668 WILKINS to WALLIS, 13/[23] July 1668 WALLIS to MORAY, 14/J24] July 1668 COLLINS to WALLIS, 14/[24] July 1668 WALLIS to WILKINS, 16/[26] July 1668 WALLIS to OLDENBURG, 16/[26] July 1668 WALLIS to OLDENBURG, 18/[28] July 1668 OLDENBURG to WALLIS, 18/[28] July 1668 COLLINS to PELL, 18/[28] July 1668 COLLINS to PELL, 18/[28] July 1668, enclosure WALLIS to OLDENBURG, 20/[30] July 1668 WALLIS to COLLINS, 21/[31] July 1668 PELL to BRANCKER, 21/[31] July 1668 BRANCKER to PELL, 25 July/[4 August] 1668 WALLIS to BRAMSTON, 27 July/[6 August] 1668 WALLIS and CHRISTOPHER WREN to the ESSEX COMMISSIONERS, 27 July/[6 August] 1668 WALLIS to BROUNCKER, July/August 1668 OLDENBURG to WALLIS, July/August 1668 WALLIS to OLDENBURG, 3/[13] August 1668 WALLIS to BROUNCKER, 3/[13] August 1668 WALLIS to BROUNCKER, 3/[13] August 1668, enclosure . . BROUNCKER to WALLIS, 3/[13] August 1668 WALLIS to BROUNCKER, 5/[15] August 1668 WALLIS to BROUNCKER, 6/[16] August 1668 WALLIS to BROUNCKER, 8/[18] August 1668 WALLIS to BROUNCKER, 8/[18] August 1668, enclosure . . COLLINS to WALLIS, 15/[25] August 1668 WALLIS to COLLINS, 25 August/[4 September] 1668 . . . . WALLIS to OLDENBURG, 25 August/[4 September] 1668 . OLDENBURG to WALLIS, 28 August/[7 September] 1668 . PELL to COLLINS, 29 August/[8 September] 1668 WALLIS to HUYGENS, 31 August/[10 September] 1668 . . XVll

477 479 479 481 489 491 492 492 494 495 496 497 498 525 525 527 530 531 533 534 536 538 541 542 542 545 545 554 554 558 559 559 561 562 564 565 566 568

Contents 243. 244. 245. 246. 247. 248. 249. 250. 251. 252.

WALLIS to BROUNCKER, [August ? 1668] OLDENBURG to WALLIS, I/[II] September 1668 WALLIS to OLDENBURG, 3/[13] September 1668 PELL to COLLINS, 6/[16] September 1668 WALLIS to COLLINS, 8/[18] September 1668 WALLIS to COLLINS, 10/[20] September 1668 WALLIS to LALOUBERE, c.lO/[20] September 1668 COLLINS to WALLIS, 22 September/[2 October] 1668 . . . WALLIS to COLLINS, 26 September/[6 October] 1668 . . . WALLIS to COLLINS, 26 September/[6 October] 1668, enclosure 253. GREGORY to WALLIS, September/October 1668 254. OLDENBURG to WALLIS, September/early October 1668 .

573 592 592 594 596 601 603 603 604 610 614 615

Biographies of correspondents

617

List of manuscripts

629

Bibliography

631

List of letters

651

Index: persons and subjects

661

XVlll

INTRODUCTION

The restoration of the monarchy in England in May 1660, following almost a year of political instability and uncertainty after Richard Cromwell's (1626-1712) resignation, meant not only the return to some form of constitutional order, but also provided conditions in which the creation of a central scientific organization at last became possible. Although a group of some of the key figures had begun to meet regularly in London in the mid-1640s, and later more formally also in Oxford, it was only now that the efforts of these two centres of activity were able to be merged into one fully fledged institution dedicated to the promotion of mathematical and experimental knowledge. This transformation took place remarkably quickly. While the official beginning of the Royal Society coincides with the granting of the first royal charter in July 1662, its predecessor in all but name began to take shape in London already towards the end of 1660. At regular meetings at Gresham College, which at this time typically took place after the lecture of the astronomy professor Christopher Wren (1632-1723), concrete plans were drawn up by William Brouncker (16207-84), Paul Neile (1613-86), William Petty (1623-87) and others to create a college whose aim would be to promote learning in the mathematical and experimental sciences. Particularly prominent from the outset in these moves to provide an institutional focal point for scientific activity in England and Scotland was Robert Moray (1608-73), who had spent much of the Interregnum in exile and was able to combine his interest in experimental philosophy with influence at the royal court. It was he who promoted the constitution of the society into a corporation. But others, too, played a decisive role in the creation of what was soon to become the most important institution of its kind in Europe. Many of these were men who had pursued their scientific endeavours in the intervening years almost entirely in Oxford and had been active in the philosophical association which had formed there around the pivotal figures of John Wilkins (1614-72) and Robert Boyle (1627-91). Wallis was among the most notable of these together xix

Introduction with Jonathan Goddard (1616-75), Seth Ward (1617-89), and Christopher Wren.

Restoration politics and scientific practice The Restoration was in many respects, at least in the early days, a time of reconciliation; only in this way could the political and religious divides which had driven the country into civil war be overcome. However, goodwill on the royalist side extended usually only to those who had not aligned themselves too openly with the Independents during the revolutionary years. Despite having widely different political leanings, many of the most accomplished mathematicians and experimental philosophers who gathered in the future Royal Society from 1660 onwards had made their careers during the Commonwealth and the Protectorate. Wallis had been appointed Savilian professor of geometry in place of Peter Turner (1586-1652), just a matter of weeks after the proclamation of the republic in 1649. There he had joined Wilkins who had replaced John Pitt (1584?c.1648) as warden of Wadham College, while his colleague, the Savilian professor of astronomy, Seth Ward moved into the place forceably vacated by John Greaves (1602-52). Another of Wallis's long-standing associates, the physician Jonathan Goddard was made warden of Merton College in place of Nathaniel Brent (1573?-1652) in 1651. Fates at the Restoration were largely decided by political sympathies of old. Wallis came out of the Cromwellian period largely unscathed, whereas Goddard and Wilkins, who had in the meantime been appointed master of Trinity College, Cambridge, witnessed a reversal of earlier events and were ejected from their posts. In effect, both men were punished on account of having sided too openly with Cromwell—although Wilkins, with the help of influential friends at court, was soon thereafter able to embark upon a successful career in the church. Undoubtedly, one of the main reasons for Wallis's survival at the University was his moderation in both political and religious affairs. Not only had he vigorously opposed the execution of Charles I in 1649, but also, as he was later careful to point out in his autobiography, he had consistently followed a Presbyterian path throughout the Cromwellian period.1 But this was not all. At least two further factors counted heavily in his favour: his renown as Europe's greatest living decipherer and his proven skill in serving the interests of the University of Oxford as keeper 1

ScaiBA, 'Autobiography of John Wallis', 35, 42-3. XX

Introduction of the archives. In these areas of activity Wallis had succeeded in making himself absolutely indispensable. Both of Wallis's posts at the University, the Savilian professorship in geometry and the office of Gustos archivorum, were confirmed at the Restoration and his efforts in each of them during the period covered by the present volume played a part in extending still further his already considerable scholarly reputation. On the one side he continued to produce important contributions to the mathematical and physical sciences, particularly through the medium of papers or letters read at meetings of the Royal Society and articles published in Henry Oldenburg's (1618-77) Philosophical Transactions. On the other side he was actively engaged by successive vice-chancellors in defending University rights in the face of repeated moves on the part of the city of Oxford to have the University's privileges removed or at least significantly reduced. In his publication practice Wallis made full use of the modern means available for the communication of scientific results. Some of the papers presented at meetings of the Royal Society were later incorporated into his monumental Mechanica sive de motu tractatus geometricus (1670-1) which served to generate new interest in a mathematical approach to physical problems and which had immense impact on the scientific world. His Treatise of Algebra (1685), which like the Mechanica was printed in London, also contains pieces largely written or even completed before the end of 1668. Among these is his Cono-Cuneus, a tract on the shipwright's circular wedge, which he wrote at the request of Robert Moray, in 1662 (No. 27). In many ways Cono-Cuneus reflected the interests and the utilitarian focus of the Royal Society. Not only did it arise out of a whole series of discussions on the nature of various solids, but also the demonstrations in piano which Wallis produced of sections of the double-coned wedge by means of analytical geometry were conceived as being potentially useful for the construction of ships. Other works appeared as publications in their own right. In continuation of the dispute with Thomas Hobbes (1588-1679), which in the long term did justice to neither of the two men, Wallis published Hobbius Heauton-timorumenos (1662) by way of reply to the philosopher's Examinatio et emendatio mathematicae hodiernae (1660) and his more recent Dialogus physicus de natura aeris (1661). In the Dialogus physicus Hobbes had attacked not only the statutes and the scientific practice of the Royal Society, but also Boyle's New Experiments Physico-Mechanicall (1660) and not least for this reason Wallis chose the form of an open letxxi

Introduction ter to Boyle for the publication of his reply (No. 21). As an appendix to the Dialogus physicus Hobbes ambitiously published his solution to one of the three great classical mathematical problems, that of constructing a cube double in volume to a given cube. This solution had already been published by him anonymously in the summer of 1661 as a printed paper purportedly originating from France (No. 14). It was subsequently refuted by Laurence Rooke (1622-62), professor of geometry at Gresham College, as well as by Wallis (No. 15), and Hobbes used the opportunity of Dialogus physicus not only to reprint his solution but also to reply to the objections raised by Rooke and Wallis. Not to be outdone by Wallis's latest attempt to humiliate him, Hobbes replied to Hobbius Heauton-timorumenos by means of an epistolary tract entitled Mr Hobbes considered in his loyalty, religion, reputation, and manners (No. 38). A more substantial publication, which however did not appear separately until 1684, reflects the close cooperation which existed between Wallis and Oldenburg. In a series of papers presented to the Royal Society, the Savilian professor of geometry put forward a new hypothesis to explain apparent irregularities of the tides (Nos. 104, 106, 109, 111, 11314). Galileo (1564-1642) had attributed this phenomenon to the different velocities of different parts of the earth. The new hypothesis consisted in assuming the earth and the moon to be a single body with a common centre of gravity. The variation of the tides from spring to neap was correspondingly to be accounted for by movements in the centre of gravity. In perfecting his hypothesis, Wallis was able to profit from numerous discussions of parts of it at meetings of the Royal Society as well as from observations of the tides which Oldenburg enlisted from some of his correspondents living near to the sea (Nos. 108, 115-16, 134, 136, 146, 179, 191, 199).

Wallis, Oldenburg, and the Royal Society Like Boyle, who remained in Oxford until 1668, Wallis relied on correspondence not only to keep abreast of developments in the scientific world, but also to play an active part in discussions taking place a considerable distance away in London. Oldenburg was the decisive figure in this respect. In return for communicating Wallis's papers to the Royal Society, and for acting as an intermediary in subsequent discussions, Oldenburg supplied the Savilian professor with news from the Republic of Letters and occasionally sent him material for perusal and criticism. He also set up the xxu

Introduction direct correspondence between Wallis and the Liege mathematician Rene Frangois de Sluse (1622-85). Although regrettably only a few of Oldenburg's letters have survived, the correspondence between the two men is by far the most extensive individual collection in the present volume. Any disadvantage which Wallis might have felt by being in Oxford rather than London vanished in the second half of 1665 on account of the outbreak of what became known as the Great Plague. With the exception of Brouncker and Oldenburg, most leading members of the Royal Society normally resident in London sought refuge in the university city where they were soon joined by the king and parliament. By September, there was almost a return to the old state of affairs when a group including Moray, Neile, Petty, and Wallis decided to meet regularly in the rooms of Robert Boyle. In view of increasing disruption to normal life in the capital, Oldenburg also placed the publication of Philosophical Transactions temporarily in the hands of Boyle, Moray, and Wallis in Oxford (No. 96). At the same time, the secretary of the Royal Society expressed the hope that all this scientific activity might have some influence on academic life, suggesting in his letter to Boyle of 5 October 1665 that the Oxonians might come to prefer 'that solidity of knowledge' associated with the Society rather than their usual 'scholasticall contentions'.2 Apart from this, Wallis usually went at least once a year to London and as a matter of course used the opportunity of being there to attend meetings of the Royal Society. Occasionally, his stays in the capital occured en route to and from his home town of Ashford. However, the most common reason for his travelling to London was to deal with legal proceedings involving the University, often being required to stay there for many weeks on end until a satisfactory solution had been reached. Sometimes both were combined. During one visit to Kent in September and October 1666, following the death of his brother Henry, he devoted time to making observations on the tides which he subsequently communicated to the Royal Society (No. 127). On arriving in London on his way back to Oxford he was called upon to look after the University's interests in the latest legal dispute (Nos. 128-30). During his stay in the capital, a large part of which had been devastated by the Great Fire in September, he was able to be present at least at one meeting of the Royal Society and to give members a personal account of the observations he had sent earlier based on his new hypothesis. 2

OLDENBURG-BOYLE 5/[15].X.1665, The Correspondence of Henry Oldenburg, ed. A. R. Hall and M. B. Hall, II, 543-5, 545. XXlll

Introduction But Wallis's being in Oxford was at times advantageous for the Royal Society, too. When Henri Justel (1620-93), who was assisting in the publication of Peter of Blois's (c.H35-c.l204) Opera omnia in Paris, requested Oldenburg's help in obtaining transcripts of the French author's manuscripts contained in English collections, the secretary engaged Wallis to investigate the holdings of libraries in Oxford and have copies made of those which were required (Nos. 102). Wallis in fact transcribed some of the shorter pieces himself (Nos. 103, 110, 134). This was in many ways typical of his sense of scientific cooperation and reflects at the same time his general attitude to the importance of the accurate transmission of texts, which in later years finds its expression in numerous exemplary editions of classical works such as Ptolemy's Harmonics. In precisely this spirit, Wallis conveyed the wishes of the Royal Society to the oriental scholar and Bodley librarian Thomas Hyde (1636-1703) that a translation of Ulug Beg's (1394-1449) catalogue of the fixed stars be made for Johannes Hevelius (1611-87). In order to minimize any possibility of error in the numerical tables, the Savilian professor made a transcript of the corresponding part of the translation and collated this against the Persian originals. He subsequently enclosed this transcript in a letter to the Danzig astronomer (Nos. 50-1). This continued a pattern of cooperation which by then was already well established. A number of years beforehand, Wallis had sent to Hevelius the account of the transit of Venus which the English astronomer, Jeremiah Horrox (16177-41), had written on the basis of observations he had made in November 1639. Hevelius subsequently published Horrox's Venus in Sole visa as an appendix to his own account of the transit of Mercury, Mercurius in Sole visus. Appropriately, he sent copies of this book, printed in Danzig in 1662, both to Wallis and to Seth Ward, who had only recently been succeeded as Savilian professor of astronomy by Christopher Wren (No. 39). Wallis, who had been a contemporary of Horrox at Emmanuel College, was in 1664 assigned the task by the Royal Society of publishing a selection or composition of his papers and in numerous letters to Oldenburg he reports on the progress of this project (Nos. 51, 75-6, 80). The result of these efforts, Horrox's Opera posthuma, was eventually published in Oxford in 1673. Wallis also pursued astronomical investigations of his own. In 1664-5 he collaborated with Christopher Wren and Robert Hooke (1635-1703) in observing a new comet and reported to Oldenburg on the successes and failures of his undertakings made from the Schools' Tower in Oxford XXIV

Introduction (Nos. 86-7). With Robert Moray operating as an intermediary, Adrien Auzout (1622-91) and Christiaan Huygens (1629-95) in Paris worked together with the English side in collecting and comparing observational data. When subsequently a controversy over results broke out between Auzout and Hevelius, who had carried out his investigations from his own observatory, Wallis was among the members of the Society who were called upon to act as impartial judges. Ultimately, it was concluded that Auzout's observations were more reliable than those of Hevelius and Oldenburg tactfully communicated this verdict to his friend in Danzig.3 Just as astronomy, so too was music closely tied in with Wallis's mathematical interests. When Oldenburg reported to him that John Birchensha (fl. 1664-72) had written to the Royal Society on the question of perfecting the art of music, the Savilian professor responded by writing an extensive paper on the theory of music in which he points out defects in ancient and modern accounts (Nos. 56-7). Only after he had sent the paper to London did Wallis find time to consult Marin Mersenne's (15881648) work on harmonics and was thereupon forced to concede that his proposals for improvement had already been anticipated by others (No. 59). Further letters to Oldenburg reflect the interest of members of the Royal Society in gathering news of unusual natural phenomena or innovations in the field of technology. For example, Wallis reports observations he made on the hatching of pigeons (No. 46), on a man killed in a thunderstorm in Oxford (No. 107), and the use of an otacousticon to relieve defects in hearing (No. 245). As in the case of his astronomical or tidal observations these, too, were presented at meetings of the Society in London for discussion. Correspondence with Boyle Wallis's correspondence with Boyle likewise covers a variety of topics and provides evidence of the two men's close professional ties. Thus Wallis gladly obliged the Irish scholar's request to comment on his book entitled Some Considerations Touching the Usefulnesse of Experimental Natural Philosophy (No. 43) and at the same time sought his advice as chemist on possible treatment for an ailment affecting his sister in law, Rebecca Wallis (d. 1677). To this end he sent Boyle a copy of a report on her illness which 3 OLDENBURG-HEVELius 24.1/[3.II]. 1665/6, The Correspondence of Henry Oldenburg, ed. A. R. Hall and M. B. Hall, III, 29-30.

XXV

Introduction was almost certainly written by her brother, the physician John Nowell (d. 1701) (No. 44). Wallis also cooperated with Boyle in supporting the Lithuanian scholar and translator Samuel Chylihski (c. 1634-68), who with their help spent a number of years in Oxford before moving on to London (Nos. 25, 33-5, 82). Without doubt the most remarkable topic dealt with in the letters exchanged by Wallis and Boyle concerns their mutual interest in language. In January 1662, Wallis undertook the task of teaching Daniel Whalley, a young man who had been born deaf-mute, to speak distinctly. Just two months later he wrote an extensive letter to Boyle, explaining his motives for taking on the task and giving details of the approach he was using in order to achieve his goal (Nos. 19, 23-4, 29). Shortly afterwards he presented Whalley at a meeting of the Royal Society in order to demonstrate his initial results. Such was his success that he was later also employed by Anne Wharton (d. 1692) to teach her deaf-mute son, Alexander Popham (No. 32). This particular case, however, ultimately brought Wallis into dispute with another member of the Royal Society, William Holder (1616-98). When, in 1670, Wallis gave the Society his account of the cure of Popham's handicap, he neglected to mention Holder's name, despite the fact he had been responsible for the young man's earlier language tuition. In his long letter to Boyle on language training, Wallis considers the possibility of developing and introducing a universal character which would signify things independently of words and sounds. The seventeenth century witnessed numerous projects of this nature to overcome linguistic divides. At the same time, an artifical language based on characters or signs promised a means to improving the acquisition of knowledge. Wallis likens universal character to the role of numbers and symbols in algebra. Although he did not doubt that a philosophical language such as that conceived by John Wilkins could be developed, he felt its universal implementation was unlikely (No. 207). Nevertheless, when Wilkins wrote him a note in his real character (No. 212), Wallis replied in kind (No. 215).

Mathematical correspondence A substantial part of the correspondence in the present volume concerns mathematical themes. At the beginning of 1660, Pierre de Fermat's (1607/8-65) challenges on number theory and Blaise Pascal's (1623-62) XXVI

Introduction prize questions on the cycloid continued to influence discussions. In a letter to Pierre de Carcavi (c. 1600-84), Huygens announces the arrival of Wallis's newly published Tractatus duo in which he provides an account of the history of the cycloid quite different from that of Pascal. In many ways anticipating later academic squabbles on questions of priority, Huygens suggests that the Savilian professor in his latest book had sought at all costs to maintain the honour of his nation (No. 2). Carcavi for his part gives Huygens a clear impression of the prevailing attitude towards Wallis in the circles around Pascal and Roberval (1602-75) at this time (No. 4). Likewise, when Kenelm Digby (1603-65) sent the Savilian professor a copy of Fermat's De linearum curvarum cum lineis rectis comparatione in the summer of the same year, Wallis replied within two days, arguing that the rectification of the semicubic parabola contained therein had already been carried out by the English mathematician William Neile (1637-70) in 1657 (No. 7). By this time Huygens had also made the same claim for his fellow countryman Hendrik van Heuraet (1633-60?) (No. 6). More than once in the following years Anglo-Dutch rivalry in the field of mathematics came to the surface over this particular question of priority. One of the theorems on which Fermat's challenges in 1657/8 had been based was concerned with the indeterminate quadratic equation nx2 + 1 = y 2 , generally known as Pell's equation. Wallis returned to this topic in a paper which he enclosed in a letter sent to Brouncker in August 1668, where he also investigates the French mathematician's so-called last theorem for n = 3, which holds that a rational cube cannot be divided into two further rational cubes, that is, that a;3 + y3 = z3 has no nonzero integer solutions for a;, y, and z (Nos. 230, 231). He later reworked the two parts of this paper with the probable intention of publishing the piece in the Philosophical Transactions (No. 243). Wallis's demonstration of Fermat's other negative theorem, which holds that there is no rightangled triangle in numbers whose area is a square appears likewise to have been intended for publication (No. 236). In the reworked letter for Brouncker, Wallis also addresses once more some of the problems proposed by Bernard Frenicle de Bessy (1605-75) in the context of the discussion with him and Fermat over questions of number theory in the 1650s, including that of finding a number n such that 6n + 1 is a cube. Since the 1650s there had been at least one more exchange between the two men, rendered possible by their long-standing intermediary Kenelm Digby. Towards the end of 1661, Frenicle proposed a new problem on triangles to Wallis. The Savilian professor obliged in subXXVll

Introduction mitting a solution to which Frenicle subsequently wrote a short comment (Nos. 16-18). In the seventeenth century the practice of publishing reviews of the latest books in journals was becoming an increasingly important part of scientific discourse. Wallis was no exception in this respect. Typically, Oldenburg would ask for his comments on a new publication in the field of mathematics or experimental philosophy and Wallis would duly submit his views for possible inclusion in the Philosophical Transactions. Such was the case with Nicolaus Mercator's (1620-87) Logarithmotechnia (1668), in which the German-born mathematician demonstrated his method for finding the sums of logarithms and published his well-known series for the area of the hyperbola based on this. Already by this time there was fierce rivalry in the Republic of Letters on most questions concerning quadratures. After the claim had been made in French circles that Mercator had based his quadrature of the hyperbola on the efforts of Laloubere (160064), Wallis was quick to point out that it owed even more to his own Arithmetica infinitorum (1656) (No. 247). Another work on which Wallis was called upon by Oldenburg to provide an assessment was the Specimina mathematica (1667) of Frangois Dulaurens (d. c.1675). A largely unknown figure at the time, Dulaurens managed to create a stir already in advance of sending copies of his book to London by setting English mathematicians a problem on the properties of the circle (No. 164). After the problem had been read at a meeting of the Royal Society, John Collins (1625-83), who often advised Oldenburg on mathematical topics, Brouncker, and Wallis were asked to consider it and to produce a solution. All three did so, but not without beforehand finding fault with the way in which it had been expressed (Nos. 165, 172, 174, 1767). When the book finally arrived, Wallis was dismayed to find that in it Dulaurens ascribed to him the prize question which had been posed in the 1650s by a certain 'Jean de Montfert'. Apart from attacking this ascription as being without foundation, Wallis claimed that much of the first part of Dulauren's book had been taken from Oughtred (1575-1660) and himself, while much of the second part was openly based on Schooten and Viete (Nos. 182, 186, 196, 204). Initially, Oldenburg published only the first half of Wallis's review of Specimina mathematica. But after Dulaurens had replied to the Savilian professor's account by publishing his Responsio ad epistolam Wallisii, Oldenburg lifted the diplomatic restriction he had imposed on himself and printed the rest. XXVlll

Introduction Correspondence with Collins Occasionally, Collins too asked Wallis for his views on new publications. Alongside Honore Fabri's (1607-88) Synopsis optica (1667), of which Wallis wrote only a summary of contents (Nos. 186, 190), the most prominent example is undoubtedly that of Vincent Leotaud's (1595-1672) Cyclomathia (1663). With this work the French Jesuit mathematician entered a long-standing dispute over the nature of the angle of contact which had been re-ignited some seven years earlier when Wallis published his De angulo contactus (1656). There, he provided arguments in support of the view originally put forward by Jacques Peletier (Peletarius) (151782) against Christoph Clavius (1537-1612) that the angle of contact is not only less than any possible right-lined angle, but also cannot even be classified as a true angle at all. After receiving from Collins a copy of the Cydomathia (No. 182), in which Leotaud maintains against Wallis tha the angle of contact does have a magnitude, the Savilian professor wrote an extensive reply to him on the topic, showing what he considered to be the irreconcilability of Clavius's concept of angulus contactus with Euclid's definition of a plane angle (No. 183). Oldenburg sent Wallis's letter to France for conveyance to Leotaud in February/March 1668, but as fa as can be established no reply was ever written. Wallis's correspondence with Collins covers almost as equally large and varied a number of themes as that with Oldenburg. Many of the letters contain details of recent scientific publications on the Continent, particularly in the field of mathematics. Sometimes Collins presented Wallis with copies of books he had obtained from abroad in repayment for favours he had received (No. 98). Not a few of his letters betray his in terest in sea charts and navigation originating from the time spent at sea earlier in his life (Nos. 99, 100, 137). On one occasion he gives th Savilian professor a short account of some of his own publications (No. 100). In addition, Collins kept Wallis informed on progress in the printing of Thomas Brancker's (1633-76) English translation of Rahn's (1622-76) Teutsche Algebra, which had been significantly reworked and expanded by John Pell (1611-85). Moreover, he ensured that the printer, Moses Pitt (fl. 1654-96), sent Wallis those parts of the book which had already been printed for proofreading and correction (No. 137). In the course of doing this Wallis was able to produce a list of errors in Brancker's table of incomposite numbers, which he subsequently conveyed to Pell (Nos. 220, 222-4). Later on, Collins similarly supervised the printing of Wallis's xxix

Introduction Mechanica, the task of which was also taken on by Pitt (Nos. 247-8, 251). When Wallis drew up plans for reprinting William Oughtred's Clavis mathematicae early in 1667, Collins assisted by presenting the proposal to the same London printer (No. 139). However, Pitt was evidently not to be convinced of the viability of the project and the book was instead reprinted by Lichfield in Oxford later the same year. Although Collins's assistance led to no success with the Clavis project in London, it is nevertheless indicative of a spirit of cooperation which pervades his correspondence with fellow mathematicians. What is particularly remarkable in this case is that he too was clearly doubtful of there being a need for a new edition of Oughtred's book, citing contemporary authors critical of its form and content (No. 142). Wallis, by contrast, had an unswayable respect for Oughtred right up to the end of his life and not seldomly accused other authors such as Dulaurens of having appropriated parts of his work for their own (Nos. 141 and 196). Wallis, Gregory, and Huygens Disputes based on accusations of plagiarism or concerning questions of priority increasingly dominated scientific discourse from the mid-1650s onwards. As one of the leading mathematicians of his day, Wallis was often—indeed, more often than most—involved in such academic quarrels. But sometimes, as with Auzout and Hevelius, the Savilian professor was called upon to adjudicate in controversies involving others. The same task fell upon him in a dispute between James Gregory (1638-75) and Christiaan Huygens which came about towards the end of the period covered by the present volume. As before, tact was required, as the adversaries had close ties with the Royal Society—Huygens had been a fellow since 1663 and Gregory was elected in 1668. The Scottish mathematician had sent Huygens a copy of his Vera circuli et hyperbolae quadratura (1667), in which he set out his discovery of the infinitely converging series for the area of the circle and of the hyperbola, soon after its publication in September 1667. Huygens subsequently wrote a review of the book for the Journal des Sgavans. While recognizing its importance, Huygens pointed out a number of faults and suggested that some of the propositions, including the derivation of logarithms from hyperbolic areas, had already appeared in his own work. Moreover, he attacked the book's central proposition, according to which the circle cannot be squared 'analytically'. By this Gregory meant that TT could not be obtained by a finite sequence of XXX

Introduction the five basic operations: addition, subtraction, multiplication, division, and root extraction. Wallis's initial reaction to Gregory's work, based only on a superficial reading, was one of approval (No. 186). As the dispute emerged and on the basis of a more detailed study he modified his opinion, but nevertheless he remained even-handed towards the two opponents. Thus although he expressed surprise at the ferocity of Huygen's attack, he sided with the Dutch mathematician on a number of points, noting, for example, that Gregory had failed to prove conclusively that there is no 'analytical' means of squaring the circle other than that considered by him (No. 229). As to the derivation of logarithms from hyperbolic areas, Wallis finds that he is able to correct both authors: this had in his view already been set out by Gregoire de Saint-Vincent (1584-1667) in his Opus geometricum (1647) and later by himself in letters published in his Commercium epistolicum (1658) (Nos. 229, 251). Cryptanalysis In comparison to later years of his life, there was apparently little call upon Wallis's deciphering skills during the period covered by the present volume. The only case evident in the correspondence appears to have arisen from fears on the part of the lord chancellor, Edward Hyde (1609-74), that the repressive legislation against dissenters introduced by parliament in the course of re-establishing the authority of the Church of England might lead to a conspiracy against the ruling powers. After two letters written in cipher had been intercepted at Banbury, the lord chancellor, who clearly valued Wallis's skills, commanded his secretary, Matthew Wren (162972), to convey them to the Savilian professor to be deciphered (No. 93). Just a few days later, Wallis was able to send back the results of his labours (No. 94). The encoded letters were, as it turned out, perfectly harmless.

University affairs By contrast, enormous demands were made on Wallis in his capacity as Keeper of the Archives. Legal disputes involving the University abounded, partly as a result of efforts to restore discipline after years of moral decline (No. 9), but mainly because its historical rights and privileges were increasingly called into question by the civic authorities. Relations between the city of Oxford and the University became particularly strained XXXI

Introduction towards the end of 1660, when the mayor refused to take the oath obliging him 'to observe and keep all manner of lawfull liberties and customes of the University of Oxford'.4 Among the privileges which the city subsequently disregarded were the University's rights to licence alehouses and victuallers as well as printers and apothecaries, to govern the market, and to take possession of felons' goods. In the disputes which followed, Wallis was required to assist in preparing the University's case at court by investigating legal precedents and by studying the exact wording of diverse royal charters. On occasion he also sought the collaboration of a colleague in the University of Cambridge, Theophilus Dillingham (1613-78), who had been his contemporary at Emmanuel College (Nos. 10-12, 36-7). Two disputes between 1660 and 1668 were especially time consuming for Wallis. In 1664 an attorney called William Thackwell was accused of libel by a manciple at Balliol College and then imprisoned by the vice-chancellor on refusing to put in a security following his arrest. Since Thackwell was able to provide a writ of privilege and a Habeas corpus from the Court of Common Pleas, the focus of attention switched to London, where it was necessary for Wallis to spend a number of weeks attending the interests of the University. In the end, he was only able to achieve moderate success, despite meetings with the lord chancellor and the lord chief justice (Nos. 60-72). The second major dispute also concerned an attorney. In 1667 Fish Lyne was apprehended in an alehouse by University proctors conducting their night walk. He subsequently ignored a summons to attend the vicechancellor's court and later pleaded himself free from the University's jurisdiction at the Court of Common Pleas. In a series of letters sent to Llewelyn Jenkins (1623-85) in London, Wallis provided material from the Archives in Oxford with which the principal of Jesus College sought to defend the rights and privileges of the University (Nos. 149-59, 161, 167-71, 173, 178). However, the University was not always on the defensive in this way. As one of Wallis's letters to Jenkins makes clear, the city was at least on one occasion concerned about the effects continuing legal disputes might have and sought to bring an end to its most recent conflict with the University. Quite simply, townsmen feared they might lose the University's trade and custom to outsiders (No. 170). l

The Life and Times of Anthony Wood, ed. A. Clark, I, 370-2. xxxu

Introduction

Personal affairs Only rarely does Wallis's surviving correspondence allow us insight into the private side of his life. This makes the few instances there are exceedingly valuable. Among the more personal letters is one from John Burscough (c.l629-c.!707), formerly a fellow of Brasenose College and since 1662 rector of Stoke by Guildford. Having recently married, Burscough evidently arranged a loan from Wallis in order to overcome financial difficulties he was currently experiencing (No. 84). Help of a rather different kind from Wallis is sought in a letter from a mother whose daughter had been lodging in the household of the Oxford printer Ann Lichfield. On learning of her daughter's unsuitable behaviour in that environment, she asks Wallis and his wife to take her into their care and to prevent her from having access to potentially corrupting influences in the future (No. 88). From a number of letters we discover that Wallis's house in Oxford was broken into and plundered by thieves at the beginning of 1667. Even his books were not spared. As he reports to Oldenburg, he found his copy of Hevelius's Descriptio cometae (1666) torn apart after the thieves had departed (Nos. 134, 137, 143). Finally, two pieces of correspondence document in different ways the uncertainties of the time. The stipend of the Savilian professors was raised entirely from the rents of the properties donated by Sir Henry Savile (1549-1622) to the University of Oxford. The imposition of taxes of whatever nature could therefore—and often did—have a direct influence on the income received by Wallis and his counterpart, the professor of astronomy, who from 1661 onwards was Christopher Wren. In a joint letter to the Essex commissioners of monthly assessments, the two men apply for redress for moneys already deducted from the rent of their tenant in Rettendon, and thereby cite a proviso in parliamentary and other acts, according to which the stipends of readers at both universities are exempt from such charges (No. 225-6).

Theological affairs One of Wallis's ways of responding to financial and professional uncertainty of this kind was to take on additional responsibilities. Having succeeded in supplementing his Savilian professorship by the office of Gustos archivorum in 1658, he sought in 1661 to gain an additional stipend, by petitioning the king for the post of prebendary at Christ Church, OxXXXlll

Introduction ford (No. 11). Many dignities, canonries, and prebends had fallen vacant by 1660 and cathedral chapters needed to be restored. This attempt at gaining a prebendaryship reflects at the same time his re-emergence as a prominent theologian at the beginning of the Restoration. Having spent the years following the Westminster Assembly largely in the background, Wallis was one of a dozen or so Presbyterian divines, including Richard Baxter (1615-91) and Edmund Calamy (1600-66), who were appointed chaplains-in-ordinary to the king in the course of 1660 in recognition of the important part leading Presbyterians had played in bringing about the return of the monarchy.5 Moreover, he was a key participant in negotiations which took place well into 1661, aimed at overcoming the significant constitutional and liturgical differences which existed between moderate Presbyterians on the one side and Anglican episcopalians on the other side.6 Although these talks got off to a promising start in the presence of the king at Worcester House, the residence of the lord chancellor, they finally ended in deadlock and acrimony at the Savoy Conference. This failure eventually led to the decision by the Cavalier parliament to take religious settlement into its own hands. Against the background of the resulting Act of Uniformity (1662), the following years were marked by rumours of Presbyterian plots, such as was probably thought to be behind the encoded letters the Savilian professor was asked to decipher. The success of the Royal Society in the 1660s, to which Wallis as one of the leading scientists of his day made a decisive contribution, came about despite the absence of the religious unity he and others, particulary on the Presbyterian side, had sought to achieve.

5 6

R. BAXTER, Reliquiae Baxterianae, ed. M. Sylvester, II, 229. Ibid., 230, 276-7; E. CARDWELL, A History of Conferences, third ed., Oxford 1849,

257. XXXIV

EDITORIAL PRINCIPLES AND ABBREVIATIONS

All letters are preceded by an account (Transmission) of the various manuscript and printed forms in which they have been handed down. In the case of those letters whose text has not survived, the reasons for assuming that they did exist at some time are given. The Transmission section also puts each letter in context, records, when known, how it was conveyed to the addressee, and supplies additional information such as postmarks, details of notes appended to manuscripts, enclosures, and so on. Manuscript and printed sources are denoted according to the following scheme: W w C c E

original manuscript in Wallis's hand copy of Wallis manuscript in scribal (or identified) hand original manuscript in correspondent's hand copy of correspondents manuscript in scribal (or identified) hand contemporary edition

Where there is more than one source in a particular category, these are numbered successively W1, W2, . . . , w1, w2, . . . , and so on. All letters contained in the volume are dated according to both the old style or Julian calendar employed in England until 1752 and the new style or Gregorian calendar widely used on the Continent, with the form not given in a particular letter placed in square brackets. In the period covered by the present volume the difference between the two calendars was ten days. To accommodate the English year, which began on Annunciation or Lady Day (25 March) and which permitted a new style date such as 16 February 1663 to be expressed in a number of ways old style— 6 February 1662, 6 February 1662/3 or even 6 February 1663—the most common form (1662/3) has been used in brackets where a correspondent on the Continent employing new style has not supplied the date old style XXXV

Editorial principles and abbreviations himself. (For reasons of legibility, only the Gregorian calendar has been used in creating the Index of Letters.) Where the place at which a letter was written can only be surmised, this also is set in square brackets. The spelling, capitalization, and punctuation of manuscript and printed sources has been retained throughout. Contractions have been silently expanded, except where they are still in common use today, and thorn has been altered to 'th'. The use of i/j and u/v has been modernized. All symbols, the ampersand, and use of superscripts to denote pounds, shillings, and pence have likewise been kept. Full stops have been diplomatically added at the end of sentences where their absence could lead to misunderstandings. As a general rule, sentences begin with a capital letter even when a miniscule is used in the handed-down text. All underlining in manuscripts is reproduced as italics. The reproduction of italics from printed sources has been treated diplomatically. In mathematical passages, letters used to indicate points or places in figures, and likewise all algebraic formulae, have been italicized where the writer or printer has not already done this.

Editorial signs (text) { }

[paper torn]

11 add. alt. corr. | text del. ed. ins. suppl.

uncertain reading illegible words (the number of dashes indicates the number of illegible words to a maximum of three) words omitted Editor's remarks (N.B. upright square brackets contained in text or in variant readings of the critical apparatus are always either employed by the author himself or represent a contemporary addition, as indicated) new paragraph within a variant reading added contemporary alteration to text by someone other than the author corrected word or words deleted editor inserted supplied

The critical apparatus shows the development of text through its various stages. Each successive stage replaces the preceding one. Thus XXXVI

Editorial principles and abbreviations stage (1) is superseded by stage (2} and this in turn by stage (3). Further subdivisions are indicated by letters: (a) is replaced by (6) and then by (c), (aa) is replaced by (66), (aaa) by (666), and so on. As in the case of the critical apparatus, but placed above this, marginal annotations to texts are referenced by means of line numbers. Editorial comments (footnotes) are indicated by numerical superscripts. Astronomical and mathematical symbols T a)b(c H 6 $ S>

Aries, vernal equinox division Gemini Mars Mercury Moon

a.b:: c.d I I,D t? D O A

xxxvii

proportion rectangle Saturn, Saturday square Sun triangle

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CORRESPONDENCE

1. CHRISTIAN HUYGENS to PIERRE DE CARCAVI [16J/26 February [1659J/1660 Transmission:

C Heavily revised and corrected draft of missing letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 722, 3 pp, (our source).—printed: HUYGENS, (Euvres completes III, 26-8. c Copy of draft: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 3611, f. 151r-152v. Reply to: CARCAVi-HiJYGENS [3]/13.IX.1659 (HUYGENS, (Euvres completes II, 534-6). Answered by: CARCAVI-HUYGENS [25.II]/6.III.1659/60 (HUYGENS, (Euvres completes III, 38-9).

Carcavy. 26 feb. 1660.

Monsieur. Depuis que j'ay receu vostre derniere7 qui a este au mois de decembre, j'ay escrit8 a Mr. Wallis mon correspondant en Angleterre, sur le suject des Livres de Mr. d'Etonville9, et en attendant tousjours sa response 4 [In left margin in Huygens's hand:] perpet. mob. Pascalin. horologe de Mr de Boismorand.

4 receu (1) vostr breaks off (2) vostre 4 qui (1) fust (2) a este 7

vostre derniere: i.e. CARCAVI-HUYGENS [3]/13.IX.1659; HUYGENS, (Euvres completes II, 534-6. 8 j'ay escrit: i.e. HuYGENS-WALLis XII.1659-II.1659/60. 9 Livres de Mr. d'Etonville: i.e. [PASCAL], Lettres de A. Dettonv^lle, Paris 1658. 1

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pour vous pouvoir satisfaire je suis demeure plus long temps qu'il ne faloit a vous en faire moymesme, de quoy je vous demande pardon. Je ne puis deviner la raison pourquoy il ne m'escrit point depuis tant de mois, scachant bien pourtant que les dits livres avec ceux10 de Mr. Sluse et depuis encore ceux de mon Systeme11 ont este delivrez au libraire12 a Londres a qui j'addresse tous mes pacquets. Peut estre que les siens se sont esgarez. Je luy escriray13 encore une fois, et lorsqu'il me viendra response14 je ne manqueray pas de vous en faire part. Je vous remercie beaucoup des extraits15 qu'il vous a pleu m'envoyer des lettres de Mr. de Fermat. Pource qui est de la demonstration de la spirale et parabole, je vous ay escrit16 que j'y trouvois de la difficulte, et que Monsr. Sluse non plus que moy ne la pouvoit resoudre, c'est a dire

1 satisfaire (1) j'ay t breaks off (2) je 1 qu'il (1) {breaks off (2) ne 2 moymesme, (1) et dont je vous en demande (2) de quoy je vous demande pardon. Je ne (a) scay (b) puis 4 les (1) Livres (2) dits livres avec ceux de Mr. Sluse add] et 6 j'addresse (1) toutes mes lettres (2) tous mes pacquets. (a) Lors (b) Peut 7 et (1) s'il me vient (2) lorsqu'il me viendra 9 remercie (1) des (2) beaucoup 10 de (1) sa (2) la demonstration (a) de Mr. Dettonville (b) de la spirale et parabole, je (oa) ne scay (66) vous 11-3, 2 difficulte, (1) a scavoir pour 1' (2) et que (a) ny Mr. (b) Monsr. Sluse non plus que moy ne la (oa) pouvoit (bb) pouvions resoudre, c'est a dire (aoa) en retenant les paroles (aaaa) de (bbbb) de Mr. dettonville. Car (bbb) que selon nostre jugement il y avoit de la faute en cette demonstration I, comme il y en a en effect add. \ . Mais 10

ceux: i.e. SLUSE, Mesolabum, Liege 1659. mon Systeme: i.e. HUYGENS, Systema saturnium, The Hague 1659. 12 libraire: i.e. Samuel Thompson (d. 1668), London bookseller at the 'White Horse' in St Paul's Churchyard and (from 1664 onwards) at the 'Bishop's Head' in Duck Lane. Cf. WALLIS-HUYGENS 24.XI/[4.XII].1659. 13 Je . . . escriray: Huygens in fact wrote to Wallis at the end of March (HUYGENSWALLIS [21]/31.III.1659/60) and again in July (HuYGENS-WALLis [5]/15.VII.1660). 14 lorsqu'il . . . response: Wallis did not in fact reply to Huygens until August: WALLISHUYGENS 31.VIII/[10.IX].1660. 15 extraits: i.e. excerpts of letters from Fermat to Carcavi concerning the comparison of spiral and parabolic lines, as dealt with in Pascal's Lettres de A. Dettonville. See HUYGENS, (Euvres completes II, 536-8 and 538-40. 16 je ... escrit: i.e. HUYGENS-CARCAVI [25.VIIIJ/4.IX.1659; HUYGENS, (Euvres completes II, 474. Cf. HUYGENS-CARCAVI [12J/22.V.1659; HUYGENS, (Euvres completes II, 411-12. 11

2

1. HUYGENS to CARCAVI, [16]/26 February [1659]/1660 que selon nostre jugement il y avoit de la faute en cette demonstration, comme il y en a en effect. Mais j'ay bien veu d'abbord qu'en la changeant, Ton y pouvoit remedier. Et voycy comme je 1'avois conciie, en gardant de plus pres ce me semble 1'intention de Mr. dettonville que n'a fait 17 Mr. de Fermat. Si elles ne sont pas egales, soit X la difference et soit Z la cinquiesme partie de X, et soient inscrites et circonscrites les figures ainsi que dit 1'autheur. Maintenant puis que la difference entre 1'inscrite en la spirale et 1'inscrite en la parabole est moindre que Z\ et que aussi la difference entre 1'inscrite en la parabole et la circonscrite a la mesnie parabole est moindre que Z; done la diff. entre 1'inscr. en la spirale et la circonscr. a la parabole est moindre que deux Z. Mais la diff. entre la circonscr. a la parabole et la circonscr. a la spirale est aussi moindre que Z: done la difference entre 1'inscr. a la spirale et la circonscr. a la mesme spirale est moindre que 3Z. Et a plus forte raison la diff. entre la spirale mesme et le tour de sa figure inscrite sera moindre que 3Z. Mais la diff. entre 1'inscr. en la spirale, et 1'inscrite en la parabole est moindre que Z, done la diff. entre la spirale et 1'inscrite en la parabole sera moindre que 4:Z. Enfin la diff. entre 1'inscrite en la parabole et la parabole mesme est aussi moindre que Z. done la diff. de la spirale et de la parabole sera moindre que 5Z,

2 qu'en (1) y (2) la changeant, (a) quelque (b) 1'on 3 Et (1) voyla (2) voycy comme je 1'avois conciie, (a) sans (6) sans la reforme entierement comme a fait Mr. de Fermat (c) en 4 ce me semble add. 4 dettonville (1) au lieu qu'il dit, Et soit Z le tiers de X, je (a) mets (6) dis, et soit Z une cinquiesme de X, c'est tout ce (2) que n'a fait Mr. de Fermat. 6 et (1) puis (2) soient 9 que Z; et ( 1 ) dereclief (2) que aussi 12 circonscr. (1) en la spirale (2) a 14 1'inscr. (1) en la et en (2) a 17-4,1 est jderechef del] jmoindre que Z, done la diff. entre la spirale et (1) la (2) 1'inscrite en la parabole sera moindre que 4Z. (a) Mais (6) Enfin la diff. . . . supposition &c. in left margin] 17 fait: i.e. in FERMAT-CARCAVi VIII.1659; HUYGENS, (Euvres completes II, 536-8; FERMAT, (Euvres (1891-1912/22) II, 438-40.

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c'est a dire que .X", centre la supposition &c.| [2] La comparaison des autres sortes de spirales avec les lignes paraboloides que donne18 Mr. de Fermat est veritable mais non pas fort difficile a trouver apres que la premiere est conniie. Et je m'estonne qu'il prend plaisir a inventer des lignes nouvelles, qui n'ont pas autrement de proprietez dignes de consideration. Les propositions touchant les surfaces des Conoides et Sphaeroides comme aussi de la ligne parabolique sont les mesmes que je vous ay cydevant communiquees19, et a plusieurs autres de mes amis. Je croy bien pourtant que Mr. de Fermat n'en avoit veu aucune puis qu'il 1'assure, mais d'autres peutestre seront plus incredules, si en les donnant au public il n'allegue celuy a qui il les ait fait veoir auparavant. La mesure de la superficie du conoide que fait la parabole autour de 1'appliquee la quelle

2 (1) II y (a) aura (6) a (oa) done (oao) cinq (666) ces 6 (bb) premierement ces 4 quantitez dont chacune differe de la suivante (aaaa) d'une moindre (bbbb) que (cccc) de nioins que n'est Z; a scavoir 1'inscrite (aaaaa) a (bbbbb) en la spirale, 1'inscrite en la parabole, la circonscrite a la parabole, la circonscrite a la spirale. (aaaaaa) Done la (bbbbbb) Done la difference entre la premiere et la derniere, a scavoir entre 1'inscrite et la circonscrite a la spirale sera moindre que 3Z. Et a plus forte raison, sera la difference entre la spirale mesme et sa figure inscrite moindre que 3Z. \\ Mais la d breaks off (2) La comparaison (aaaaaaa) que (bbbbbbb) des 2-3 paraboloides (1) est (2) que donne Mr. de Fermat est (a) assez subtile mais pour moy je ne suis pas (b) veritable (oa) mais non p breaks off (bb) mais non add. pas 6 proprietez (1) fort (2) fort remarquables. (3) dignes de (a) remar breaks off (b) consideration. 7 Les (1) propositions (2) inventions (3) propositions (a) qu'il d breaks off (b) touchant 8 comme aussi de la ligne parabolique add. 10 n'en (1) a (2) avoit 11 mais (1) il (2) d'autres 11-5, 2 si (1) ce n'est qu'en (2) en les donnant au public il (a) ne (ao) puisse alleguer 18

donne: i.e. FERMAT-CARCAVI IX. 1659; HUYGENS, (Euvres completes II, 538-49; FERMAT, (Euvres (1891-1912/22) II, 441-4. 19 je . . . communiquees: i.e. HuYGENS-CARCAVi [6]/16.1.1658/9; HUYGENS, (Euvres completes II, 315-17. 4

1. HUYGENS to CARCAVI, [16]/26 February [1659]/1660 il promet en supposant la quadrature de 1'hyperbole sera quelque chose de nouveau si elle est vraye. Vous m'auriez fait grand plaisir si lors que Mr. de Boismorand20 vous parla de son horologe a pendule, vous 1'eussiez demande plus particulierement de quelle fagon ce pendule est applique, et s'il fait un bon effect. II y a de 1'apparence que non, parce qu'il n'auroit pas ainsi laisse se perdre une invention qu'il eust juge utile. C'est une chose estrange que personne devant moy n'ait parle de ces horologes, et qu'a cette heure il s'en de[3] couvre tant d'autres autheurs.| J'espere de vous faire voir bien tost ce que j'y ay adjouste de nouveau, qui est une invention que les Geometres

(66) scache dire (6) n'allegue celuy (aoa) a qui il (bbb) qu'il (ccc) a qui il les ait (aaaa) communique ci devant que les mienes paravent. (bbbb) fait veoir auparavant. La |mesure de la add.\ superficie jpourtant add. and del] du conoide que fait la parabole (aaaaa) estant tournee sur (bbbbb) autour de 1'appliquee (aaaaaa) est quelque qu'il d breaks off (bbbbbb) la quelle il (aaaaaaa) mesure (bbbbbbb) dit mesurer (ccccccc) promet en supposant la quadrature de 1'hyperbole (aaaaaaaa) est (bbbbbbbb) sera quelque chose de nouveau (aaaaaaaaa) et de bien joly a (666666666) la quelle je ne (aaaaaaaaaa) croyois pas (6666666666) scay pas encore mais je doute si elle vraye. (ccccccccc) pourveu qu'elle soit (ddddddddd) si elle est vraye. 3 (1) Vous (2) Je suis marry que vous (—) (3) Lors (4) Vous 5 s'il (1) faisoit (2) fait un bon effect, (a) Ce (6) II 6-9 perdre (1) cette (2) une invention qu'il (a) 1'eust (b) eust juge utile. (ao) II y en a (aoa) plus de (666) 5 ou (aaaa) 6 (bbbb) six qui maintienent d'avoir eu cette (ccc) encore plusieurs qui disent avoir eu cett invention en teste devant moy, et pour ce qui est de Galilee j'ay veu dernierement le dessein du modelle que Le Prince Leopold de Toscane assure avoir trouve apres sa morte. (aaaaa) A tout cela je ne puis respondre, que (bbbbb) Mais tout cela ne fait rien contre moy si ce n'est que l'(—} de sorte que (aaaaaa) cette (bbbbbb) la pensee semble avoir este assez commune, mais mon modelle a jbien add] succede le p(—} et je (aaaaaaa) ne veoy (bbbbbbb) scay bien qu'on ne trouve point, que j'en aye rien emprunte de personne. (66) |C'est une . . . d'autres autheurs. in left margin\ 7 choye corr. ed. 9 autheurs. (1) L'on ne (—) (2) Vous verrez (3) J'espere de vous faire voir bien tost (a) une nouvelle invention (6) ce que j'y ay adjouste (oa) a ces horologes (aaa) pour (666) pour leur derniere perfection |et justesse add. , la quelle [invention add. (66) de nouveau, qui est une invention que 20

Mr. de Boismorand: a magistrate in Anguleme, who was in possession of a pendulum clock made in 1615 or 1616; see CARCAVI-HUYGENS [3J/13.IX.1659 HUYGENS, (Euvres completes II, 534-6, 535.

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estimeront infiniment plus que tout le reste de la fabrique. Celle de la machine de Mr. Pascal21 pour supputer, que 1'on m'a envoyee de Paris, est certainement digne d'admiration, et contient plusieurs belles pensees. Je luy en escriray bientost puis que j'apprens qu'il commence a se porter mieux. Mais toutes nos inventions Monsieur vont estre peu considerables si celle de 1'allemand Johannes Joachimus Becherus22 s'effectue, ou s'il ne nous trompe pas, car il m'a assure, m'ayant este veoir icy, qu'il a construit23 un mouvement perpetuel a Mayence, qui continue d'aller depuis six mois. Et hier il m'en envoya les figures qu'il a fait graver en deux grandes planches a Amsterdam. L'on ne peut pas pourtant comprendre le secret de 1'invention par ces figures, devant que de voir la description qu'il en promet, ayant par tout adjouste de lettres et des nombres. Seulement 1'inscription tient et 1'on le voit a peu pres que 1'une des machines (car il a deux inventions diverses du perpetuum mobile) est purement mechanique et 1'autre physico-mechanique. Pour celle cy la chose ne me paroit pas tout a fait impossible, mais de 1'autre j'advoue qu'elle passe ma croyance. Je suis Mons.

2 (1) J'ay icy (2) Celle de 3 Paris, (1) qui certainement est (2) est certainement digne d'admiration, (a) p breaks off (b) et 4-5 pensees. (1) Je luy en escr breaks off (2) |Je luy ... mieux. add] Mais 5 inventions jmechaniques add. and del] Monsieur (1) doivent ceder (2) vont 6 1'allemand add. 7 m'ayant este veoir icy add. 8 un (1) perpetuum (2) mouvement perpetuel a Mayence, qui (a) va en dessin (b) continue d'aller 11 que (1) d'avoir (2) de voir 13 des (1) chiffres (2) nombres. Seulement (a) 1'invent breaks off (b) 1'inscription tient et 1'on le voit a peu pres 14 (car il a deux inventions diverses |du perpetuum mobile add.\ ) add. 16-18 1'autre (1) je n'en croy rien. (2) j'advoue qu'elle passe ma croyance (a) Je suis (b) Je 21 machine de Mr. Pascal: i.e. Pascal's arithmetical machine; cf. BELLAIR-HUYGENS [24.VI]/4.VII.1659 (HUYGENS, (Euvres completes II, 426-9). 22 Becherus: i.e. Johann Joachim Becher (1635-82), German alchemist and inventor. 23 construit: see SCHOOTEN-HuYGENS [7]/17.1.1659/60; HUYGENS, (Euvres completes III, 10-11.

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2. HUYGENS to CARCAVI, [17]/27 March [1659]/1660 Le livre24 de Mr. de Wit n'est pas encore acheve d'imprimer. Celuy25 de Wallis que vous desirez de veoir ne se trouve pas icy chez les libraires, mais a la premiere occasion qui s'offre je vous envoyeray mon Exemplaire et seray tousjours

2.

CHRISTIAN HUYGENS to PIERRE DE CARCAVI [17J/27 March [1659J/1660 Transmission:

Cl Preliminary draft of missing letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 735, 1 p.—printed: HUYGENS, CEuvres completes III, 56. C2 Heavily revised and corrected draft of missing letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 735, 2 pp. (our source).—printed: HUYGENS, (Euvre completes III, 56-7. c Copy of C2: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 3611, f. 147r-148r. Date: given in C1 and c. Reply to: CARCAVl-HuYGENS [25.IIJ/6.III.1659/60 (HUYGENS, (Euvres completes III, 38-9). Answered by: CARCAVl-HuYGENS [15J/25.VI.1660.

Carcavy. Monsieur II en est arrive ce que je me estois imagine du silence de Mrs. les Anglois, car 2 ou trois jours apres que je vous eus envoye ma precedente26 je receus

2 Wallis (1) que v breaks off (2) que 2 icy (1) avec (2) chez 3 Exemplaire (1) . Je suis (2) et seray 7 (1) Deux ou trois jours apres vous avoir escrit ma derniere, je receus nouvelles de Mr. Wallis a s§avoir une lettre (2) II en est arrive (a) cornme je me 1'estois (&) ce que je me estois 7 Anglois, (1) car il y a longtemps que (2) et (3) car 24

livre: i.e. WlTT, Elementa curvarum linearum, ed. F. van Schooten, Amsterdam 1659. 25 Celuy: i.e. WALLIS, Commercium epistolicum, Oxford 1658. 26 ma precedente: i.e. HUYGENS-CARCAVI [16]/26.II.1659/60. 7

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une lettre27 de Mr. Wallis avec un nouveau livre28 qu'il a fait imprimer dont le titre est Joh. Wallisii &c. tractatus duo Prior de Cycloide et corporibus inde genitis, Posterior Epistolaris in qua agitur de Cissoide et corporibus inde genitis. Et de curvarum turn linearum ev'&vvoei^ turn superficierum -K\arva\j1u). Je m'estonne que vous ne 1'ayez pas encore vu lors que vous me fites 1'honneur de m'escrire la derniere fois29, car sans doute il en aura envoye aussi des exemplaires a Paris pour Mr. Pascal et pour vous Mr. qui y estes nomine si souvent. Je ne vous escriray que sommairement ce qu'il contient, parce que vous le verrez vous mesme ou peutestre 1'aurez veu desia. Dans le premier traicte il resout par ses methodes les problemes que Mr. Pascal a proposez et resolus. Et il y adjouste la demonstration de Mr. Wren de la dimension de la Cycloide more veterum, qui est ingenieuse. L'autre traicte est la lettre que j'ay attendue si long temps, au commencement de la quelle il dit avoir receu et distribue les exemplaires du livre30 de Mr. dettonville que je luy avois envoye de vostre part. II donne en suite ses specul. touchant la Cissoide a 1'occasion d'un theoreme que cy devant je luy avois communique pour la dimension de 1'espace infini qui est entre cette courbe et son asymptote. Au reste il y a

(1) Pent estre (2) Je m'estonne que vous ne (a) 1'avez (6) 1'ayez pas 5 encore vu add] lors 8 nomine (1) tant de fois. (2) si souvent. 8 escriray (1) pas le detail de ce qu'il traite dans ce livre (2) que sommairement ce qu'il (a) est contenu dans (6) contient 9 verrez (1) ou ... veu desia vous mesme. (2) vous mesme ... veu desia. (a) Au premier (6) Dans (ao) la premiere partie (66) le premier traicte 11 les jmesmes add. and del.\ problemes 11 proposez et add. 12 dimension (1) des Cycloides (2) de la Cycloide 14 il (1) (avoit) (2) dit avoir 16 part. (1) Pour apres (2) Les speculations touchant la cycloide (a) son breaks off (6) qu'il traite au long sont (3) II (ao) traite en suite d breaks off (bb) donne en suite ses specul. touchant la Cissoide (aaa) auxquelles (bbb) a 17 communique (1) pour (2) concernant (3) pour 18 asymptote. (1) Dans la preface (2) (—) (3) II (4) Au reste il y a (a) (aussi) (b) et 27

lettre: i.e. WALLis-HuYGENS 24.XI/[4.XII].1659. nouveau livre: i.e. WALLIS, Tractatus duo, Oxford 1659. 29 derniere fois: i.e. CARCAVI-HUYGENS [25.II]/6.III.1659/60; HUYGENS, (Euvres completes III, 38-9. 30 livre: i.e. [PASCAL], Lettres de A. Dettonv^lle, Paris 1659. 28

8

2. HUYGENS to CARCAVI, [17]/27 March [1659]/1660 et dans cette lettre et dans la preface du traite de la Cycloide beaucoup de choses centre Mrs. Dettonville et de Roberval et centre leur histoire31 de la Roulette aux quelles peut estre ils ne demeureront pas sans replique. Ce Mr. Wallis tesmoigne certes d'avoir 1'esprit prompt et il y a du plaisir a veoir comme il tasche a toute force de maintenir 1'honneur de sa nation. Je vous rends graces tresh. de 1'extrait32 de la lettre de Mr. de Fermat [2] de qui je trouve cette derniere speculation touchant| les roulettes proportionelles beaucoup plus belle que la precedente33 des spirales. Pour ce qui est de la proposition dont j'estois aucunement en doute, je ne voudrois pas qu'il prit la peine d'en escrire la demonstration separement, pour me la faire veoir, mais plustost Mr. ayant le traite entier et tant d'autres excellens ouvrages de ce grand geometre dont vous pourrez obliger le public lors qu'il vous aura tout envoye, ainsi qu'il a promis. II ne se plaint pas

1 de | la add] Cycloide 3-5 Roulette |aux quelles . . . sans replique add] . (1) (—} (2) Vous le verrez Monsieur et peut estre (a) (ces) (&) les dits Messieurs. II (aa) ne manque point d'esprit (bb) a 'lesprit prompt, ce Mr. Wallis, et dans (aaa) de (bbb) les controverses il tasche (3) Ce Mr. Wallis tesmoigne certes d'avoir 1'esprit prompt, et (aaaa) c'est chose (bbbb) il y a du plaisir a veoir (aaaaa) il (bbbbb) comme il tasche a toute force add. de 6 graces |tresh. add.\ (1) des nouveaux (2) de 6 Fermat (1) qui {—} (2) de 8-13 que (1) celle des spirales la (2) la precedente des spirales. (a) II (b) Ne luy donnons pas la peine de (c) ne luy donnez pas la peine Monsieur d'en (d) Pour ce qui est de la (aa) superficie (66) proposition dont j'estois aucunement en doute je ne voudrois pas qu'il prit la peine d'en escrire la demonstration separement, (aaa) mais insistez tousjours (aaaa) qu breaks off (bbbb) a fin qu'il (bbb) pour me la faire veoir, mais (aaaaa) insistez (bbbbb) ayons plustost quelque (ccccc) plustost Mr. ayant le traite entier |et tant d'autres excellens (aaaaaa) de ce (aaaaaaa) (grand) (bbbbbbb) bel esprit (bbbbbb) ouvrages de ce grand geometre add] dont vous pourrez obliger le public (aaaaaaaa) Mr puis (bbbbbbbb) lors qu'il 13-10,14 promis. (1) \(a) Cette proposition touchant la superficie du conoide 31

histoire: i.e. [PASCAL], Historia trochoidis sive cycloidis, Paris 1658. This work was generally ascribed both to Pascal and to Roberval. 32 extrait: i.e. FERMAT-CARCAVI 11.1659/60; HUYGENS, CEuvres completes III, 39-40; FERMAT, (Euvres (1891-1912/22) II, 445-6. This was quoted by Carcavi in CARCAVIHUYGENS [25.II]/6.III.1659/60. 33 precedente: cf. FERMAT-CARCAVI VIII.1659; HUYGENS, (Euvres completes II, 5368; FERMAT, CEuvres (1891-1912/22) II, 438-40. This was quoted by Carcavi in CARCAVI-HUYGENS [3]/13.IX.1659.

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2. HUYGENS to CARCAVI, [17]/27 March [1659]/1660

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tout a fait sans raison de Monsieur Schooten de ce qu'il n'a pas fait mention de luy en publiant ses lieux plans.34 Car encore qu'il n'ayt jamais veu ce que Mr. de Fermat en avoit escrit comme il m'a assure tousjours, il 1'a bien sceu et pourtant il n'auroit pas deu le dissimuler. Si je n'estois apres a faire la description de mon horologe avec ce que j'y ay adjouste nouvellement je vous en ferois part des a cet heure, mais puis que j'espere de 1'achever bientost et de la faire imprimer je vous prie de ne 1'exiger pas auparavant. Je voy plus que jamais par ce dernier escrit de Wallisius les inconvenients et disputes qui en peuvent naistre lors que des inventions de quelque consequence vont de main en main devant que d'estre publiees. Apres que j'auray acheve cela je vous feray aussi veoir ma methode de mesurer les lignes courbes la quelle je ne croy pas que personne encore s'est imaginee. Le traite35 de Mr. de Wit est imprime et je chercheray quelque occasion pour vous 1'envoyer comme toutes celles par ou je pourray tesmoigner

parabolique (6) Je suis bien loin de craindre (oa) qu'il en la dedans (66) qu'il y puisse avoir en cela quelque chose qui soit (aoa) en (bbb) a mon prejudice, ou outre que nos propositions sont desia connues de plusieurs, (aaaa) je (bbbb) j'y croy (en) (cccc) je scay que Mr. de Fermat ( ) des belles clioses luy mesnie. 11 Si je m'estois apres del.\ || Si je n'estois apres . . . s'est imaginee. || Monsieur de Fermat ne se plaint pas tout a fait sans raison de Monsieur Schooten . . . il n'auroit pas deu le dissimuler. 11 Le traite de Mr. de Wit . . . (2) II ne se plaint pas . . . 1 tout a fait add. 3 assure |tousjours add. , il (1) me semble que la candeur requiert (a) que (b) qu'il (c) que 1'autrement (2) 1'a bien sceu (ao) neanmoins (bb) et pourtant (aaa) ce qu'il n'auroit pas deu (bbb) il n'auroit pas deu le 5 horologe (1) et de (2) avec ce que j'y ay adjouste (a) de (b) nouvellement 8 auparavant. (1) J'ay (2) Je 9 de | Mr. add. and del] Wallisius 9 en (1) peuvent (2) naissent (3) peuvent naistre 12 je (1) scay bien (2) ne 14 est (1) desia sorty (2) imprime 15 1'envoyer (1) et (2) comme (a) toute autre par la quelle (b) toutes celles par ou je pourray |vous del. tesmoigner 34

en publiant . . . plans: i.e. SCHOOTEN, Exercitationum mathematicarum, liber III. Continens Apollonii Pergaei loco, plana restituta, Leiden 1656. 35

Le traite: i.e. WITT, Elementa curvarum linearum, ed. Frans van Schooten, Amsterdam 1659. 10

3. HUYGENS to WALLIS, [21]/31 March [1659]/1660 combien j 'estime I'honneur d'estre Monsieur &c.

3. CHRISTIAN HUYGENS to WALLIS [21J/31 March [1659]/1660 Transmission:

C Preliminary draft of missing letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 736, 1 p. (our source).—printed: HUYGENS, (Euvres completes III, 58. c Copy of draft: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 3611, f. 149r. Reply to: WALLis-HuYGENS 24.XI/[4.XII].1659. Answered by: WALLIS-HUYGENS 31.VIII/[10.IX].1660. Wallis's letter of November 1659 to which this was a reply only reached Huygens on [10]/20 March 1660.

Wallis. 31 Mart. 60. 20 martio accepi.36

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Dat ick wel gedacht had dat hy niet gesien moet hebben 1'examen &c. dat daer sijn fauten in staen. Sendt het hem. Wrens demonstratie elegans, ingeniosa, sed clarior potuisset esse in prop, bedanck voor d'eer die hy my doet. mittat in Galliam. Heuratius. In Cissoide apparet vis methodi. Horologii mutatio quam dicit nova non est. talia multa hinc in Angliam missa38. Additio mea tota constructione melior. Methodus curvarum. Sa-

2 Monsieur &c. 27.° Mars 1660. c 6 ick (1) g breaks off (2) wel 8 sed (1) obscuri breaks off (2) clarior 9 Galliam. \ ( 1 ) Robervall (2) de Robervalli suspicione {—} quod insinuas { } fatum est ego expertus in horologio {—} hoc me invenisse (ut) et de (a) Saturno (6) Saturni annulo del} Heuratius 36

accepi: i.e. WALLIS-HUYGENS 24.XI/[4.XII].1659. l'examen: i.e. [PASCAL], Recit de 1'examen, Paris 1658. 38 Horologii mutatio . . . missa: on this error on Wallis's part cf. HUYGENS, (Euvres compleies XVII, 27, n. 3. 37

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turni Systema39 me misisse. Balii40 observationes frater41 ejus mihi communicavit42 ad Jun. 17. 1659. Quomodo balteum f? pingat. Exemplum ad Schotenium ipse Broun43 misit44. Nelio45 de telescopiis. 6 diam. non eo modo qui est in systemate46. Dicat et ipse de suis telescopiis. Contigua vitra sunt ocularia. nulla mine diaphragmata. dicuntur optima habere, nominatim Boilius47. Guisonius48 retulerit. Vlack49 spreecken vande exempl. Examinare Wall, de Cycloide.50 De Cissoide more veterum demonstratio51.

2 pingat. |Error in propositione de Conoide del] Exemplum 5 habere, (1) prae (2) nominatim 6 Guisonius (1) dix breaks off (2) retulit. (3) retulerit. 39

Saturni Systema: i.e. HUYGENS, Systema Saturnium, The Hague 1659. Balii: i.e. William Ball (c.1627-90), astronomer and member of Gresham College, later fellow and treasurer of the Royal Society. 41 frater: i.e. Sir Peter Ball (d. 1680), secretary of state to the Queen in the reigns of Charles I and Charles II, DNB. 42 communicavit: i.e. BALL-HuYGENS 17/[27].VI.1659. This apparently detailed description of William Ball's observations of Saturn has not survived. Cf. WALLISHUYGENS 22.XII.1658/[1.I.1659] and HUYGENS, Brevis assertio systematis Saturnn sui, The Hague 1660, 7; HUYGENS, CEuvres completes XV, 446-7. 43 Broun: i.e. Samuel Brown (d. 1665), English bookseller, who established his business in The Hague near to the Anglican church in 1647. Cf. HuYGENS-ScHOOTEN [18]/28.VL1656; HUYGENS, CEuvres completes I, 440. 44 Exemplum . . . misit: i.e. the copy of Wallis's Tractatus duo destined for Schooten. Cf. WALLIS-HUYGENS 24.XI/[4.XII].1659 and HUYGENS-SCHOOTEN [9]/19.III.1660; HUYGENS, CEuvres completes III, 44. 45 Nelio: i.e. Sir Paul Neil(e) (1613-86), courtier and astronomer, founding member of the Royal Society. Cf. WALLIS-HUYGENS 24.XI/[4.XII].1659. 46 in systemate: see HUYGENS, Systema Saturnium, The Hague 1659, 79; HUYGENS, CEuvres completes XV, 345-6. 47 Boilius: i.e. Robert Boyle, q.v. Cf. HuYGENS-MoRAY [14J/24.VI.1661; HUYGENS, CEuvres completes III, 283-4. 48 Guisonius: i.e. Pierre Guisony, medical practitioner in Avignon. 49 Vlack: i.e. Adriaen Vlacq (16007-67). Bookseller, first in London, then in Paris and finally The Hague. From 1626 to 1633 he worked in London on the calculation and edition of a table of logarithms. 50 Wall. de Cycloide: i.e. WALLIS, Tractatus duo, Oxford 1659. 51 De Cissoide . . . demonstratio: Huygens sent Wallis his proof of the quadrature of the cissoid (HUYGENS, CEuvres completes II, 170-3), which he had completed already in April 1658, some time before the appearance of the third part of Mechanica (London 1671), in which it was subsequently printed. In the second edition it was inserted in part two of this work; see WALLIS, Opera mathematica I, 906. 40

12

4. CARCAVI to WALLIS, June? 1660 4.

PIERRE DE CARCAVI to WALLIS June ? 1660 Transmission:

Manuscript missing. Existence and date: Mentioned in CARCAVi-HuYGENS [15J/25.VI.1660. Carcavi apparently sent this letter via Huygens in reply to the letter from Wallis, which reached him at the same time as a copy of his Tractatus duo, Oxford 1659. Reply to: WALLis-CARCAVi 1659?. Enclosure to: HUYGENS-WALLIS [5J/15.VII.1660.

5.

PIERRE DE CARCAVI to CHRISTIAN HUYGENS Paris, [15]/25 June 1660 Transmission:

C Letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 754, 6 pp. (our source).—printed: HENRY, 'Pierre de Carcavy', Bullettino di bibliografia XVII (1884), 317-79, 346-50; HUYGENS, (Euvres completes III, 85-90. Reply to: HuYGENS-CARCAVi [17J/27.III.1659/60. Answered by: HUYGENS-CARCAVI [5]/15.VII. 1660 (HUYGENS, (Euvres completes III, 97-8). Huygens transmitted a copy of this letter to Wallis as enclosure to HUYGENS-WALLIS [5]/15.VIL1660.

de Paris ce 25e. Juin 1660. Monsieur a mon retour de la campagne ou J'ay este oblige de demeurer Environ trois moys pour des affaires qui ne m'ont laisse aucun loisir de me donner 1'honneur de vous escrire J'ay trouve un petit livre52 de Mr. Fermat qu'il vous envoye, il m'a aussy fait tenir pendant ce terns un autre pe52

livre: i.e. [FERMAT], De linearum curvarum cum lineis rectis comparatione dissertatio geometrica, autore M. P. E. A. S., Toulouse 1660; FERMAT, (Euvres (1891-1912/22) I, 211-54; Varia opera mathematica, 89-103.

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tit traicte53 de solutions, problematum geometricorum per curvas simplicissimas et unicuique problematum generi proprie convenientes. que Je vous feray coppier si vous le desirez, II y fait voir plusieurs fautes de Mr. Descartes dans sa geometrie54 dont Mr. Schooten n'a dit mot55. Voicy Encore Trois de ses propositions. 1. Data quadratura hyperboles, datur circulus aequalis superficiei curvae paraboles circa applicatam rotatae.

Sit data parabola AD, cujus axis AE, applicata seu semibasis DE, rectum latus ABC, quaeritur circulus aequalis superficiei curvae solidi quod ex rotatione figurae ADE, circa applicatam DE tanquam immobilem circumductae conficitur. Bisecetur latus rectum AC, in B, et axi AE, ponatur in directum recta EF aequalis rectae AB, seu dimidio recti lateris, et jungatur DF, Exponatur separatim recta IQ aequalis axi AE, cujus dupla sit recta IR, fiat ut FE, sive AB, ad DF ita recta QI, ad rectam QH, et a puncto H, ducatur HG, perpendicularis ad HIR, et fiat HG, aequalis rectae DE, per punctum

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5 Encore (1) deux (2) Trois 53

traicte: i.e. FERMAT, De solutions problematum geometricorum per curvas simplicissimas et unicuique problematum generi proprie convenientes dissertatio tripartita, Manuscript 1657/58; FERMAT, CEuvres (1891-1912/22) I, 118-31; Varia opera mathematica, 110-15. 54 Descartes dans sa geometrie: i.e. DESCARTES, La geometrie, Leiden 1637. 55 Schooten n'a dit mot: i.e. in Geometria a R. Des Cartes, ed. Frans van Schooten, Leiden 1649. 14

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/, tanquam verticem describatur Hyperbole cujus transversum latus sit recta IR, centrum Q et transeat hyperbole per punctum G, et sit IG.\ Describatur item alia hyperbole separatim cujus transversum latus MN, sit aequale quartae parti

recti paraboles lateris, hoc est quartae parti rectae AC, centrum vero sit V, rectum latus OVP, aequale transverse lateri, sit autem hyperbole ita descripta MK, cujus vertex M, axis ML, qui continuetur donee recta ML, sit aequalis axi 15

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5. CARCAVI to HUYGENS, [15]/25 June 1660 paraboles AE, et ducatur perpendicularis seu applicata LK, a rectangulo sub QH, in HG, deducantur duo spatia hyperbolica IGH, MKL, quorum quadraturae supponuntur, et quod supererit aequetur quadrato, Diagonia istius quadrat! erit radius circuli superficiei curvae, cujus dimensionem quaerimus, aequalis. 2. Esto cyclois primaria ANIF, cujus axis AD, semibasis DF, Et ab ea formentur aliae curvae vel extra ipsam vel intra, quarum applicatae sint semper in eadem ratione data, ad applicatas primariae cycloidis. Ex. grat. in curva exteriori AMHG, ducantur applicatae GFD, HIC, MNB, ratio autem GD, ad DF, sit data et sit semper eadem quae HC,

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ad CI, et MB, ad BN. In curva autem interiori AROE, ratio FD, ad DE, sit data, et sit semper eadem quae 1C, ad CO, et NB, ad RB. Dico contingere ut curvae Exteriores qualis est AMHG, sint semper aequales aggregate lineae circularis et lineae rectae. Curvae autem Interiores, qualis est AROE, sint semper aequales parabolis primariis sive archimedeis. Theorematis generalis enuntiationem, quando volueris exhibebo, immo et demonstrationem.56

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10 ad applicatas primariae cycloidis. add. in margin 12 semper (1) et s breaks off (2) eadem 56

1. Data quadratura . . . demonstrationem: i.e. FERMAT-CARCAVI 7.VI.1660; FER-

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[3]

5. CARCAVI to HUYGENS, [15J/25 June 1660 3. Voicy 1'extrait d'une sienne lettre.

Pour me sauver un peu de 1'accusation57 de Mr. de Zulychem, qui dit que mes spirales n'ont pas des proprietez qui soyent autrement considerables vous pourrez si vous voulez luy proposer celle qui suit. Soit le cercle BCDM, duquel le centre A, et le Rayon AB, et soit la spirale BOZA, de laquelle la propriete soit telle, BA, est a AO, comme le quarre de toute la circonferance BCDMB, au quarre de la portion de la mesme circonferance CDMB, cette spirale par mon Theoreme general est esgale a une parabole en laquelle les cubes des appliquees sont en mesme raison que les quarres des portions de 1'axe, laquelle parabole est esgalle a une ligne droitte, J'espere que cette propriete suffira pour me reconcilier avec Mr. de Zulychem. Et puisque je luy cede touts mes droits sur les surfaces courbes des spheroides et conoides, Je souheterois qu'en revenche il m'indiquat s'il scait aucune surface courbe esgale a un quarre par voye purement geometrique et pareille a celle dont je me suis servi en donnant des droittes esgales a des courbes.58 MAT, CEuvres (1891-1912/22) II, 446-8. 57 l'accusation: i.e. in HuYGENS-CARCAVi [16]/26.II.1659/60; HUYGENS, CEuvres completes III, 26-8; 27. Cf. FERMAT-CARCAVI IX. 1659; FERMAT, CEuvres (18911912/22) II, 441-4. The latter was quoted by Carcavi in CARCAVI-HUYGENS [3J/13.IX.1659; HUYGENS, CEuvres completes II, 538-40. 58 3. Pour ... courbes: i.e. FERMAT-CARCAVI 7.VI.1660; FERMAT, CEuvres (1891-

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Voila Monsieur ce que J'ay receu de Mr. de Fermat depuis que Je n'ay eu 1'honneur de vous escrire59, Aprez quoy je vous supplie tres humblement agreer que je vous dise quelque chose de Mr. Walls, et vous en userez avec luy de la sorte qu'il vous plaira, n'ayant pas cru que je deusse respondre autre chose ny a son livre60 ni a une lettre61 qu'il m'escrivit au mesme terns qu'il m'a este rendu, qui fut le Jour avant mon depart de cette ville, que ce qu'il vous plaira de voir par la lettre62 que je luy escris, II est vray que J'ay este surpris de son precede, et que Je n'eusse pas attandu qu'il en diit user| de la sorte Envers Mr. D'Ettonville et par consequent En mon endroit ne luy en ayant Jamais donne aucun sujet, que seroit ce si nous avions fait imprimer non seulement les lettres qu'il m'a escrit, mais encore celles qui sont entre les mains de Mr. de Roberval qui Justifient et ses paralogismes, et son aveuglement, pour ne pas dire davantage, a ne s'en point corriger, II ne faudroit point d'autre responce a toutes ses inpertinences, Et vous verriez Monsieur qu'en ce qui concerne les problemes du dit Seigneur d'Ettonville il n'a pas seulement failly mais encore a advoue qu'il ne pouvoit pas y donner davantage de satisfaction, aprez cela le livre estant imprime il veut qu'on croye qu'il ne luy a servi de rien pour se corriger, et ce qui est le plus outrageux qu'on a pris de luy ou d'autruy ce qu'il n'a Jamais sceu, il faut avoir bien peu de sincerite, pour faire paroistre aux yeux de tout le monde des bagatelles et des bassesses de cette nature, pour moy Je ne scaurois concevoir les raisons qui 1'ont porte a cela, Mr. de Roberval m'a bien dit avoir escrit a un de ses amis quelque chose sur les fautes qui sont tant dans son livre Intitule Elenchus geomet. hobbian.^ que dans son arithmetique des infinis64, mais il n'a rien dit sinon qu'il y avoit telle et telle faute, et II ne 1'a point fait imprimer, Et quand cela seroit qu'est ce qu'il y auroit de commun avec le livre de Mr. Dettonville et la maniere toute genereuse dont il en a use, car il ne s'est pas contente de donner seulement le terns porte dans son deffi, 4 deusse |luy del.\ respondre 11 imprimer (1) ses (2) non 1912/22) II, 448-9. 59 Je . . . escrire: see CARCAVI-HUYGENS [25.IIJ/6.III.1659/60; HUYGENS, (Euvres completes III, 38-9. 60 livre: i.e. WALLIS, Tractatus duo, Oxford 1659. 61 lettre: i.e. WALLIS-CARCAVI 1659?. 62 lettre: i.e. CARCAVI-WALLIS VI?. 1660. 63 Elenchus geomet. hobbian.: i.e. WALLIS, Elenchus geometriae Hobbianae, Oxford 1655. 64 arithmetique des infinis: i.e. WALLIS, Arithmetica infinitorum, Oxford 1656. 18

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5. CARCAVI to HUYGENS, [15J/25 June 1660 mais encore trois moys davantage, durant lesquels Mr. Vallis ny personne [5] autre n'ont fait rien paroistre de ce qui| avoit este demande, aprez quoy il a donne Jusques a ses principes et a ses methodes, Et pour tout cela ce Brave professeur traitte en pedant des personnes de condition, et cherche a leur dire des Injures sur chaque mot qu'il tourne a sa fantesie. II impute a crime d'avoir propose un prix, ad pompam facere visum est. II chicane sur des clauses que nous avons mis, qui ne dependent neanmoins que de nostre volonte, et veut que nous y ayons mis de 1'equivoque, affigat (dit il) quam velit mentem verbis suis, qu'il nous connoit mal! Pourquoy cette longue et inutile apologie de Toricelly repetee en tant d'endroits, que nous pouvons facilement convaincre de faux et de ridicule par les lettres mesme originales de Toricelly que nous avons Entre les mains, Et pouvoit on dire ce qui s'est passe dans la recherche de la ligne dont il estoit question qu'en rapportant fidellement ce qui est icy connu de touts les geometres, Mr. Walis vouloit il qu'on mentit comm'il a fait en tant d'endroits de son livre. Quand II trouvera quelque chose nous ne dirons pas qu'il ne 1'a pas trouve, mais quand nous en aurons veu les demonstrations donnees par un autre, nous dirons librement et en verite qu'il n'en est pas 1'inventeur. Je vous serois trop inportun si Je vous disois tout ce qui ne devroit point estre dans ce livre, Je n'ay fait que le parcourir, et ce que J'y trouve encore de plus beau en achevant de le lire c'est qu'il veut que Mr. D'Ettonville ayt pris de luy ce qu'il a de meilleur, avec quel front me peut il dire cela ayant eu toutes ces demonstrations avant que d'avoir receu aucune nouvelle d'Angleterre. En voyla assez s'il vous plaist et mesme trop, dont| vous me per[6] mettrez de vous dire encore une fois que vous userez comm'il vous plaira, mais je crois que la chose ne vaut la peyne d'en parler davantage, la verite n'ayant point besoin d'autre deffence que d'elle mesme, Je suis de tout mon coeur.

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Monsieur Vostre tres humble et tres obeissant serviteur Decarcavy 9 neanmoins add. 20 librement (1) en (2) et 25 demonstrations avoir que d'avoir corr. ed.

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6. HUYGENS to WALLIS, [5]/15 July 1660 Mr. Boulliaud65 qui ma promis de vous faire tenir cette lettre me presse si fort que je n'ay eu loisir de la relire. Apres avoir Escrit cette lettre J'ay trouve un imprime66 de 1'annee 1640. qui Justine que Mr. de Roberval a pense le premier a la cycloide.

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CHRISTIAAN HUYGENS to WALLIS [5]/15 July 1660 Transmission:

C Preliminary draft of missing letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 759, 1 p. (our source).—printed: HUYGENS, (Euvres completes III, 96; FERMAT (Euvres (1891-1912/1922) IV, 130. c Copy of draft: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 36 II, f. 157r-157v. W Excerpt from missing letter sent (quoted in WALLIS-OLDENBURG 4/[14].X.1673): LONDON Royal Society Early Letters W2, No. 14, p. 2 of 4 pp. (our source).—printed: Philosophical Transactions No. 98 (17 November 1673), 6148; HUYGENS, CEuvrcs completes VII, 342; OLDENBURG, Correspondence X, 278 (Latin original), 281 (English translation). Answered by: WALLis-HuYGENS 31.VIII/[10.IX].1660. Enclosures: CARCAVI-WALLIS VI?. 1660 and (a copy/part copy of) CARCAVi-HuYGENS [15]/25.VI.1660.

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Wallis. ic send de apologie67 van M. de Carcavy, en den brief68 ut videas quid

7 send (1) den brief (2) de apologie 7 Monsieur |de add] Carcavy 7 brief \ ( 1 ) quorum (2) ut videas .. . reprehendant add. . dat 65

Mr. Boulliaud: i.e. Ismael Boulliau (1605-94). imprime: not identified. 67 apologie: probably CARCAVI-WALLIS VI?.1660. Cf. WALLIS-HUYGENS 31.VIII/ [10.IX].1660. 68 den brief: presumably CARCAVl-HuYGENS [15J/25.VI.1660. Cf. WALLlS-HliYGENS 31.VIII/[10.IX].1660. 66

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6. HUYGENS to WALLIS, [5]/15 July 1660 in tuo libro69 reprehendant. dat ick uyt Brouncker verstaen heb70 dat hij mijn laetste71 heeft ontfangen. Fermat send my een boeck72, forte etiam ad vos, de Curvarum cum rectis comparatione.precipuum in eo vande parabolois die Heuraet73 hier en Nelius74 apud vos rectae aequavit. quorum scripta mirum est ilium non vidisse. veruntamen et alias curvas ex ilia paraboloide enatas subtili admodum ratione rectificare docet. scribe quid novi apud vestrates ubi jam studia si unquam florere incipient, restituta patriae tranquillitate, de qua tibi plurimum gratulor. Exemplaren van 't systema75 alhoewel het laet is, waer daer nae te vernemen. Schoten doot.76 (W)

Fermatii libellum novum simul ad me misit Carcavius, de Curvarum linearum cum rectis comparatione; in quo praecipue agitur de Paraboloide

2 ontfangen. (1) Caxcavy send my een boeck van Fermat, (2) Fermat send my een boeck, 3 comparatione. |sed eundem del. precipuum 3 eo (1) dat (2) vande 4 aequavit. (1) quanquam (2) quorum scripta (a) non oportet (6) mirum est ilium 6 scribe add. 7 studia (1) reflor breaks off (2) magis quam (3) si 8 tibi (1) max breaks off (2) plurimum 69

tuo libro: i.e. WALLIS, Tractatus duo, Oxford 1659. dat . . . heb: This letter from Brouncker to Huygens is missing. n mijn laetste: i.e. HuYGENS-WALLis [21J/31.III.1659/60. 72 een boeck: i.e. [FERMAT], De linearum curvarum cum lineis rectis comparatione dissertatio geometrica, Toulouse 1660. This tract was sent to Wallis by Kenelm Digby. See WALLIS, Algebra, 293; Opera mathematica II, 319. 73 Heuraet: on Heuraet's rectification of the semicubic parabola see HEURAET, Epistola de transmutatione curvarum, linearum, in rectas, in: Geometric, a Renato Des Cartes, ed. Fr. v. Schooten, Amsterdam 1659, 517-20. 74 Nelius: on William Neile's rectification of the semicubic parabola see WALLIS, Tractatus duo, Oxford 1659, 92; Opera mathematica I, 551-2. According to Wallis, Neile achieved this result already in 1657 and could therefore claim priority in this over Heuraet: Tractatus duo, 91; Opera mathematica I, 551. See also his Algebra, 293; Opera mathematica II, 319. 75 systema: i.e. HUYGENS, Systema Saturnium, The Hague 1659. 76 Schoten doot: Frans van Schooten the younger died in Leiden [19J/29.V.1660. 70

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7. WALLIS to DIGBY, 24 August/[3 September] 1660 ilia, quam jam ante apud nos Heuratius, apud vos Gu. Nelius rectae lineae adaequavit.

7. WALLIS to KENELM DIGBY London, 24 August/[3 September] 1660 Transmission:

E1 First edition of missing copy of missing letter sent: WALLIS, Algebra, 294-6 (our source). E2 Second edition: WALLIS, Opera mathematica II, 320-2. Probably in summer 1660, Kenelm Digby sent Wallis a copy of Fermat's tract De linearum curvarum cum lineis rectis comparatione. Within two days of its receipt, Wallis wrote the present letter to Digby in Paris, arguing that the rectification of the semicubic parabola had already been carried out by William Neile in 1657. Cf. WALLIS-OLDENBURG 4/[14].X.1673.

Illustrissimo Nobilissimoque Viro, D. Kenelmo Digby, Equiti Anglo. 5

August! 24. 1660. Londini. Illustrissime Vir,

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Videbam ego nudius tertius Fermatii quod miseras acutum opus77; quo Curvam Paraboloeidem, quam ego Semicubicalem appello, (cujus Ordinatim-applicatae sint in Diametrorum ratione Subtriplicatae-duplicata,) aequalem Rectae ostendit. Quod acute quidem & Geometrice (ut sua solet) peragit. Unum autem est aut alterum, quod monendum duxi. Primo quidem, Eandem ipsam Curvam Rectae aequalem, primus

9 sunt E2 13 Primum E2 77 Fermatii . . . opus: i.e. [FERMAT], De linearum curvarum cum lineis rectis comparatione dissertatio geometrica, autore M. P. E. A. S., Toulouse 1660.

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7. WALLIS to DIGBY, 24 August/[3 September] 1660 (credo) omnium ostenderat78 Gulielmus Nelius, Equitis Pauli films; suamque hujus Demonstrationem jam Anno 1657 divulgaverat: quod & a pluribus apud nos post ilium demonstratum est, & passim notum. Id Ipsum deinde, post annum circiter, ab Heuratio Batavo peractum est,79 quod (nescius, puto, quid apud nos factum fuerat) iteratae suae Cartesiani Operis Editioni subjunxit Schootenius. Eandemque rem in Epistola80, quam Tractatui de Cycloide (Anno praeterito a me edito) subjunxi, fusius prosecutus sum. Quae tamen omnia cum Fermatio, credo, minime innotuerint, non mirum est si ipse se primum invenisse putaverit. Alterum est, Quod, cum (praeter primas illas,) Secundas, Tertias, Quartas, aliasque in infinitum a primis derivatas, in Dissertatione sua memoratas, quas item a primis specie differentes appellat, rectis aequales dederit; non videtur Vir acutissimus animadvertisse, non alias illas esse Curvas, a primis diversas, sed earundem partes ab aliis aliisque punctis inchoatas. Quod sic brevi demonstro. Esto, in ipsius Fig. 11. paraboloides sua Semicubicalis, cujus vertex A, latus rectum AD, quod sit, verbi gratia, 9, (quae nempe recta, in quadratum interceptae diametri ducta, solidum efficiat Cubo ordinatimapplicatae aequale,) sitque Semibasis EF. Formenturque ad mentem suam ES, ER, EL, Secunda, Tertia, & Quarta, ab ilia Prima derivatae. Exponatur autem Parabola G\, cujus Latusrectum GH sit 4, (nempe | rectae AD.} Sumptisque (in Diametro) GK aequali lateri-recto, & GY ejusdem quadrupla, continuentur KQ, & Yd, quarum utraque sit semibasi EF aequalis, & ordinatim-applicentur KI, QP, YT, OX. I ostendit E2 19 aequalem,) E2 78

ostenderat: Neile's demonstration was described by Wallis in Tractatus duo, Oxford 1659, 92; Opera mathematica I, 551-2. This priority dispute came to a head again in 1673 when Huygens in his Horologium oscillatorium denied that Neile had fully succeeded in rectifying the curve. Wallis thereupon published a defence of Neile's claim along with further papers to this end from William Brouncker and Christopher Wren. See WALLIS, 'Primarn inventionem et dernonstrationern aequalitatis lineae curvae paraboloidis cum recta, anno 1657 . . . ' , in: Philosophical Transactions No. 98 (17 November 1673), 6146-9. 79 ab ... peractum est: HEURAET, Epistola de transmutations, curvarum linearum in rectas, in: Gcometria a Renato DCS Cartes, ed. Frans van Schooten, Amsterdam 1659, 517-20. Although the Epistola is dated 13 January 1659 (new style), we know that Heuraet in fact made his discovery in the summer of 1658. 80 Epistola: i.e. Wallis's Tractatus cpistolaris de cissoide et corporibus inde genitis, in: Tractatus duo, Oxford 1659, 75-121; Opera mathematica I, 542-69.

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7. WALLIS to DIGBY, 24 August/[3 September] 1660 His ita ad mentem suam constructis; Assume, tanquam ab ipso demonstrata, Curvae AE particulas, quantumvis minutas, (vel potius harum tangentes,) rectis Diametro Parallelis abscissas, respectivis in trunco Parabolico KIPQ ordinatim-applicatis proportionales esse; (nempe, curvae 5 particulas, sive harum Tangentes, ad correspondentes particulas basis, ita esse ut sunt respectivae ordinatim-applicatae in Parabola ad suum Latus-rectum:) Item, Curvae LE particulas, respectivis in Trunco Yr\9 ordinatim-applicatis similiter proportionales. His positis; AE Curva eousque deorsum continuetur donee basin ^v 10 aequalem habeat toti K6. Eaque ita in po~ divisa, ut in QY dividitur KO, erigantur inde perpendiculares; quarum altera occurret Curvae in E; occurrat altera in T.

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Manifestum est, ex suis demonstratis, ut AE Curva Trunco KIPQ, sic Curvam Av Trunco KIXO correspondere, & partes partibus: Adeoque ET Curvam Trunco QPTY, & Curvam TV Trunco YTX9-, & partes partibus respective. Sed, eidem YTX9 trunco similiter correspondet LE Curva, (quod ex 1 demonstratum E2 24

7. WALLIS to DIGBY, 24 August/[3 September] 1660 illo supra ostensum est,) & partes partibus. Ergo (per ipsius concessa & demonstrata) Curva LE eadem est atque TV. Et similiter ostendetur; si sumeretur [io dupla rectae GH (& av ut prius aequalis rectae EF,) esset TV Curva eadem atque RE. Sin /j,a rectae GH aequalis, esset TV eadem atque SE. Et in reliquis similiter. Non sunt igitur £"5, ER, EL, aliae ab AE Curvae, specie distinctae; sed, ejusdem continuatae, aliae atque aliae partes. Atque haec sunt, Vir Illustrissime, quae impraesentiarum monenda duxi. Caeterum Vale, Vir Illustrissime, Tuoque faveas, Observantissimo & Devinctissimo, Joh. Wallis. Ad meam Circuli Quadraturam quod spectat, quam (ex mea Arithmetica Infinitorum petitam) sub finem Epist. XXIII, sic designaveram81: Ut fac[296] tum ex quadratis numerorum imparium 3, 5, 7, 9, &c, in infinitum, ad factum ex iisdem quadratis unitate minutis: Sic Quadratum Diametri, ad aream Circuli. Puta, ut 9 x 25 x 49 x 81 x 121, &c, in infinitum; ad 8 x 24 x 48 x 80 x 120, &c, in infinitum. (Quae quadraturae meae nonnisi pars est; quatenus nempe ad numeros absolutes reduci possit.) Quod reponit82 D. Freniclius, Hanc aliam non esse quam Methodum approximandi, qualis est ilia Archimedis per inscriptas & circumscriptas; & ut nunquam perventuri sumus ad illud infinitum, ita nee ad perfectam Circuli Quadraturam hac via pertingemus: Omnino verum est, prout hie per numeros absolutos designatur. Sicut nee potest numerus surdus, puta Y/2, aliter designari in numeris absolutis, quam simili approximatione in infinitum; puta, per Unitatem cum annexis partibus decimalibus, ut 1.41421356 &c, (continuando radicis quadraticae extractionem in infinitum.) Nee tamen culpandus ille erit qui valorem numeri Surdi ^/2, numeris absolutis sic designandum dixerit: Quoniam ut numeris absolutis perfecte designetur (aliter quam per approximationem) numerorum natura non patitur; quique illud fieri postulet, postulat aSvvaTOV. Idemque & hie obtinet. Demonstraveram83 enim (Arithm. Infin. prop. 189, 27 radices E1 corr. ed. 31 fieri postulat, E2 81

designaveram: i.e. WALLIS-DIGBY 4/[14].III.1657/8 (Epist. 23). reponit: not ascertained. 83 Demonstraveram: i.e. WALLIS, Arithmetica infinitorum, 169-78; Opera mathemati82

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190, hujusque Scholio,) primus credo omnium, fortasse & solus; Rationem Quadrati ad Circulum Inscriptum, talem esse, ut nee numeris absolutis exprimi possit, nee etiam Radicibus Surdis (puta Quadraticis, Cubicis, Biquadraticis, &c.) sed neque ulla adhuc recepta Aequationum formula: Quippe ad hoc requiritur, ut numerus Impar in duos integros aequales dividatur; atque ut Aequationis formula reperiatur Laterali & Quadraticae intermedia; adeoque quae radices habeat plures quam unam, sed pauciores quam duas. Quorum utrumque est impossibile. Quod autem in radice Surda designanda fit; nempe, ut, quod exacte fieri non possit, nota aliqua insinuetur quasi factum; puta ^/2, vel -y/1 x 2; quo significetur terminus intermedius inter 1 & 2 in serie continue proportionalium 1, 2, 4, 8, &c, quae fit continua multiplicatione per communem Multiplicatorem 2; puta 1 x 2 x 2 x 2 &c: Idem hie faciendum ostendimus; nempe cum demonstratum sit, rationem Circuli ad Quadratum Diametri esse ut 1 ad d terminum intermedium inter 1 & | in serie 1, |, ^, ^, &c, quaefitex continua Multiplicatione (non quidem per eundem communem Multiplicatorem, ut in continue-Proportionalibus, sed) numerorum 1 x | x | x | &c, poterit ille (ad formam medii Proportionalis, inter 1 & 2, puta \/l x 2,) sic utcunque designari; mr11|| (vel alia forma simili.) Et propterea (prout latus ad diagonium quadrati est ut 1 ad -y/1 x 2, sic) Circulus ad Quadratum Diametri, ut 1 ad jnT 1 |. Quae vera est Circuli quadratura in numeris, quatenus ipsa numerorum natura patitur. Quam ad numeros absolutes (per continuam approximationem) sic reduci posse ut supra dictum est, ibidem demonstravimus84 Prop. 191. Quomodo autem in lineis exhibeatur, ostensum est85 ibidem Prop. 192, 193, 194. Quas autem memorat D. Fermatii rectas Curvis aequales, jam consideravimus.

10 v/2 aut /L x 2 E2 12 Multiplicatorem 2: Idem hie faciendum E2 16 eundem continuum Multiplicatorem E2 23 ut dictum est E2

ca I, 462-7. 84 demonstravimus: i.e. WALLIS, Arithmetica infinitorum, 178-93; Opera mathematica I, 467-76. 85 ostensum est: i.e. WALLIS, Arithmetica infinitorum, 193-8; Opera mathematica I, 476-8.

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8. WALLIS to HUYGENS, 31 August/[10 September] 1660 8.

WALLIS to CHRISTIAAN HUYGENS London, 31 August/fit) September] 1660 Transmission: W Letter sent: LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 779, 4pp. (p. 3 blank) (our source).—printed: HUYGENS, (Euvres completes III, 126-8.

Reply to: HUYGENS-WALLIS [21J/31.III.1659/60 and HUYGENS-WALLIS [5J/15.VII.1660.

Accepi, Vir Nobilissime, binas a Te literas86 (alteras prid. Cal. Apr. alteras 15 Julii datas) quibus hactenus nihil responsi dedi: partim ob varias apud nos indies mutationes, quae animum alio avocarunt; partim quia nihil habui, nee dum habeo, quod Te dignum judicem, impertiendum. Ne tamen ingratus videar, aut neglectus arguar, rescribendum utcunque duxi. Priores quod spectat, quasque inibi memoras literas tardius ad Te quam vellem datas87; id hinc evenisse intelligo, quod quas ego Oxoniae eo qui indicatur die scripserim, Londini apud Bibliopolam aliquandiu haerebant; sed nee inde etiam ad Te ea quam sperabam diligentia perferebantur. Judicium vero quod de me meisque feceris, quanto candidius sit quam quod tulerint Galli, posteriores quas misisti literae notum faciunt. Quam autem memoras Narrationem Examinis88, non illam prius videram, aut de ilia inaudiveram quicquam, (necque ex nostris quern scio aliquis) quam Tu illam miseris. Literas vero quas ea memorat, non aliae sunt quam quarum ego in libro edito89 mentionem feceram; (si saltern unam excipias, quae hoc saltern petebat, ut indicare vellent num priores accepissent.) Ad quas omnes quum mihi ne tribus quidem verbis quicquam responsi dederint, (sed neque Examen illud, quod me spectabat maxime, transmitti curabant;) mirabar equidem; nee imputabam tamen (aliter, quam nuda facti narratione) me magis fortasse quam par erat fuisse neglectum. Quum vero 4 Te add. 15 edito add. 86

binas a Te literas: i.e. HUYGENS-WALLIS [21J/31.III.1659/60 and HUYGENS-WALLIS [5J/15.VII.1660. 87 literas ... datas: i.e. WALLIS-HUYGENS 24.XI/[4.XII].1659. This letter did not reach Huygens until [10]/20 March 1659/60. 88

Narrationem Examinis: i.e. [PASCAL], Recit de I'examen et du jugement des ecrits envoyes pour les prix proposes publique.me.nt sur le sujet de la, Roulette . . ., Paris 1658. 89 libro edito: i.e. WALLIS, Tractatus duo, Oxford 1659.

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in literis tandem D. Carcavii turn ad Te90, turn ad me91, (in posterioribus tuis hue transmissis92) tantas tragoedias excitari video, quamque sit irae impotens Vir Nobilissimus; mirari subit, quas sibi solent Nobiles Galli loquendi formulas indulgere, quamque a nostris diversas! (nisi forsan triobularem me nebulonem putaverint, quern eapropter corripiant quod Gallos ausim compellare.) Quicquid sit, negligenda certe mihi videntur isthaec omnia. Non moror enim quod ex Dettonvillio me desumsisse insinuet, quae de Cycloide scripseram;93 quum jamdudum Illustrissimum Brounkerum testem advocaverim94, qui per aliquot menses ante legerat isthaec omnia prout nunc extant scripta, (quodque amplius est, integrum ille calculum a capite ad calcem severe examinaverat,) et, magnam etiam (si memini) partem, prius impressa, quam prodiit illud Dettonvillii opus. Neque suum illud ex adverse mentiri me, me quicquam movet. Non novit me, Vir Nobilissimus, nee mores meos, qui hoc impingit. Quinam autem sint Errores illi, quos in meo sive Hobbii Elencho95, sive Infinitorum Arithmetical^, vel me vel amicorum meorum aliquem (utrum enim vult, non satis assequor) fassum esse asserat97 Robervallius, ego non intelligo; aut, quis sit ille amicorum. (Sed nee intelligo, quid hoc ad Rhombum; Quid ad Cycloidem ista?) Nondum enim adhuc mihi conscius sum, errorem ullum Geometricum me admisisse in utrovis libro, nedum fassum esse. (Sed nee in aliis a me editis; unum illud si excipias quod in Epistola98 ad Illustrissimum Brounkerum quae scriptis in Meibomium praefigitur revocavi.) Sed neque memini me ad Robervallium scripsisse unquam, (aut ad me ilium,) utut de illo nonnunquam| ad alios scripserim: [1] (unicam saltern si excipias Epistolam quam olim Gassendo, eove absente 11 calculum (1) severe (2) a 19 hoc | ad add.\ Rhombum 90

literis . . . ad Te: i.e. CARCAVI-HUYGENS [15]/25.VI.1660. literis ... ad me: i.e. CARCAVI-WALLIS VI?.1660. 92 in . . . transmissis: i.e. HuYGENS-WALLis [5]/15.VII.1660. 93 Non moror . . . scripseram: see CARCAVI-HUYGENS [15J/25.VI.1660. 94 advocaverim: Wallis claims that he sent Brouncker the first tract of his Tractatus duo, i.e. De cydoide, for examination in March 1659. See WALLIS, Tractatus duo, Praefatio, Sig. a4r; Opera mathematica I, 495, and WALLIS, Mechanicorum, sive tractatus de motu, pars secunda, Oxford 1670, 458-9; Opera mathematica I, 858-9. 95 Hobbii Elencho: i.e. WALLIS, Elenchus geometriae Hobbianae, Oxford 1655. 96 Infinitorum Arithmetica: i.e. WALLIS, Arithmetica infinitorum, Oxford 1656. 97 asserat: see CARCAVI-HUYGENS [15J/25.VI.1660: 'Monsieur de Roberval m'a bien dit avoir . . . II ne 1'a point fait imprimer.' 98 Epistola: i.e. WALLIS-BROUNCKER 5/[15].XII. 1656. This is the epistle dedicatory 91

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8. WALLIS to HUYGENS, 31 August/[10 September] 1660 Robervallio, indifferenter inscripseram"; quam periisse autumo100; vel, si exstet, nihil sane horum continet, aut etiam continere potuit;) ut mirum sit, quid ille ex literis meis ad Robervallium scriptis101 potis sit depromere quod hue spectet. Sed mittamus ista: et, quae de Torricellio dicta sunt: aliaque quae sequuntur. Quorum nihil, credo, est quod Te moveat. Ad hasce vero Carcavii literas nihil hactenus respondendum putavi, aut etiamnum puto. Non, quod Nobilissimum Virum neglexerim: sed, quoniam, quum eo stilo scriptae sint qui non admodum deceat ingenuos viros, cuique vix responderi posse videatur quin in aperta jurgia delabamur; condonare malim Nobiliss. Viro, ultra quam par est effervescenti, quam irritare. (quod et literis ad Digbaeum Equitem Parisios scriptis insinuavi). Quid interim sit, cur Pascalium a Dettonvillio videatur distinguere, non intelligo: Hos enim pro eodem (Te primum indicante) hactenus habui. Sin erratum sit, erroris fontem habes. Fermatii quern memoras libellum novum102 nuper vidi; quo eandem, quam prius turn nostri turn vestri etiam curvam aequaverant rectae, contemplatur. Quas autem in Dissertatione sua curvas alias inde derivatas et rectis comparatas, specie diversas existimat; non aliae sunt (aut ego admodum fallor) quam ejusdem curvae aliae atque aliae partes. In primaria siquidem, deorsum continuata, reperientur secunda tertia aliaeque in infinitum. Recta utique axi primariae parallela, quae inde distat | lateris recti, designat punctum quo incipit secundaria (deorsum in infinitum continuanda:) quaeque ab hac tantundem distat, tertiam ostendit; quaeque tantundem ab hac, quartam: et sic deinceps. Quod ubi examinaveris, facile deprehendes. Quod et literis ad Digbaeum Equitem scriptis demonstravi. Schotenii nostri mortem doleo. Quae de Cissoide ipse scripseris, lubenter videro. Quid Slusius praestitit103, ignoramus; neque enim quos miseris libros 17 alias specie diversas del.\ inde 27 lubenter (1) d breaks off (2) videro. to WALLIS, Adversus Marci Meibomii ... tractatus elenchticus, Oxford 1657. "Epistolam ... inscripseram: i.e. WALLIS-GASSENDI 31.VIII/[10.IX].1655. Cf. WALLIS-BROUNCKER 16/[26].X.1656 and WALLIS-BROUNCKER 20/[30].X.1656. 100 quam periisse autumo: Wallis was wrong in believing that this letter to Gassendi had gone missing. 101 ex literis . . . scriptis: Carcavi was possibly referring to the letters WALLISBROUNCKER 16/[26].X.1656 and WALLIS-BROUNCKER 20/[30].X.1656. 102 libellum novum: i.e. [FERMAT], De linearum curvarum ... dissertatio geometrica, Toulouse 1660. 103 praestitit: presumably in Sluse's Mesolabum, Liege 1659. 29

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9. FELL to YATE, 5/[15] February [1660/1 ?] nacti sumus. Quae de D. Pauli Nelii tubis sciscitaris104; partim, credo, Illustr. Brounkerus tibi corani indicavit; partim, spero, ubi in Angliam trajeceris, videbis. Interim Vale, Vir Nobilissime, Tuoque faveas 5

Londini, Aug. 31. 1660.

Observantissimo, deditissimoque Joh. Wallis.

Quae posthac ad me mittere dignaberis, inscribas licet D. Samueli Thomson105, Bibliopolae Londinensi in Caemiterio Paulino: Nam D. Underbill106 mortuus est. Dominum Dicas107, (cui dicis libros missos fuisse) quaerebam, sed 10 Parisiis jam agere intelligo; Collega suus Martinius108, negat (saltern non agnoscit) se accepisse libros ullos mini inscriptos.

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Eruditissimo Nobilissimoque Viro, D. Christiano Hugenio de Zulichem, Hagae Comitis.

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9. JOHN FELL to THOMAS YATE 5/[15] February [1660/1?] Transmission:

C Letter sent: OXFORD University Archives WPj/16/1, f. 92r-92v. On f. 92V in unknown hand: 'Letter of Bsp: Fell concerning Encroachments by the Town.'

7 D. (1) Th breaks off (2) Samueli 104

sciscitaris: cf. HUYGENS-WALLIS [21J/31.III.1659/60; WALLIS-OLDENBURG 4/[14]. X.1673. 105 Thomson: i.e. Samuel Thompson. 106 Underhill: i.e. Thomas Underhill (d. 1660), London bookseller at the 'Bible' in Wood Street and at the 'Anchor and Bible' in St Paul's Church Street. 107 Dicas: i.e. Thomas Dicas (d. 1669), London bookseller established at the 'Bell' and at the 'Hen & chickens' in St Paul's Churchyard. In 1660, he joined James Allestree (d. 1670) and John Martin. 108 Martinius: i.e. John Martin (fl. 1649-80), publisher and bookseller established at the 'Bell' in St Paul's Churchyard, publisher to the Royal Society. 30

9. FELL to YATE, 5/[15] February [1660/1 ?] The background to this and subsequent letters are numerous grievances of the University in respect of the city of Oxford, which apparently emerged following the failure of the mayor to take the oath 'to observe and keep all manner of lawfull liberties and customes of the University of Oxford'. During Convocation on 15 March 1660/1 ten grievances of the University of Oxford to the king were read, including the refusal of city officials to make the yearly oblation on St Scholastica's day, which was 10 February. See WOOD, Life and Times I, 370-2. In the course of pursuing the University's petition to the king, Cooper, Wallis, and Yate were required to spend considerable periods of time in London. See WOOD, Life and Times IV, 65.

Sir

Since my last I have considerd of several other greivances, occasiond by the towns very insolent and apparent encroachments on our priviledges, besides those you have breviats of. For instance the denying the homage on St. Scholasticas day which will be on the 10 of this moneth, in reference to which I have desird the vice-chancellor to give the maior109 an admonition before hand, that he may not pretend ignorance as he did in the case of the Oath. Secondly the Commission for making the river of thames navigable wherein we are grossely abusd, the profit in a manner coming wholy to the town and the Charges to us. Sly the licensing of Vintners which not only the wine office but the town have now usurpt. 41y the licensing of ale houses which they have engrost' contrary to our Charter & at this day have licensed about 700 as I am credibly enformed. Sly the proclaiming of the Market which they first assumed in 49 and think now they have prescription for. 61y the placing of the Market. This you know grounds a present contest between us & them. 71y the corporation of the brewers should depend on the university & take an oath to them, which is now left off. Sly licensing of Apothecarys &; Printers is usurpt by them. As lastly the royalty of having felons goods. I have spoken to Dr Wallis, to draw up with all speed, our pleadings to each of the fore mentioned breviats; which I suppose are all material & worth our insisting on. He tells me a memorable case of one Painter110 a Bayliffe of this town who in Queen Eliz: time took upon him to walk in the night: and being brought into the vicechancellors court & there sentenc't to pay twenty pounds (having it seems walkt ten times) he refusing to pay was put in prison, and having 24 pounds (1) it being (2) (having 109

maior: i.e. Sampson White. case of one Painter: for the case of the bailiff Thomas Painter cf. Wallis's Notes relating to the allowance of the claim in the Exchequer enclosed in WALLIS-JENKINS 21/[31].I.1667/8. 110

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10. WALLIS to DILLINGHAM, 16/[26] March 1660/1

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taken out his habeas corpus cum causa and removed himself, upon the universitys defence was cast, and fain to pay his fine. The particular of this I shall send you speedily, & suppose it may be a leading case, in all our other contests; and a better way then to insist upon a councel table order. As to the businesse of Justices of peace; the vicechancellor has ever since the late Act bin our, and the present Vice-chancellor111 owns himself as our. But the businesse of all our priviledgd persons, and other incidental occurrents cannot be dispatcht by any single man. By the Charter of Hen: 8112 which was confirmd by act of Parliament in Queen Eliz: time we are to have two besides him; named by our Chancellor, and this I think we should stand upon: I shall by my next give you a more particular account, be pleasd to excuse this very rude & hasty one, from Sir

Feb: 5.

Your assured friend & faithful servant J. Fell.

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For the reverend his honourd Friend Dr Yate, at Mr George Hills house in Hemlock Court in Sheer lane These D.D. London.

[92V]

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WALLIS to THEOPHILUS DILLINGHAM Oxford, 16/[26] March 1660/1 Transmission:

W Letter sent: OXFORD University Archives WP//3/15/9/30, 4 pp. (pp. 2 and 3 blank). On p. 4 in Wallis's hand: 'Letter to Cambridge. Mar. 16. 1660./I. To Dr Theophilus 1 himself (1) to the Kings bench (2) , upon 111

present Vice-chancellor: i.e. Paul Hood. Charter of Hen: 8: i.e. the charter of Henry VIII of 1 April 1523 ('Wolsey's charter'); cf. the Extracts from the University Archives enclosed in WALLIS-JENKINS 28.I/[7.II]. 1667/8. 112

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10. WALLIS to DILLINGHAM, 16/[26] March 1660/1 Dilingliam.' At top of p. 1 in Wallis's hand: 'To Dr Tlieopliilus Dillingham Master of Clare-hall Cambridge.' Answered by: DlLLINGHAM-WALLIS 1/[11].IV.1661.

Sir

I am desired by Mr Vicechancellor of Oxford113, in the behalf of this University, to request the favour from some of your University, of a particular information concerning the practise about the businesse of Alehouses & Victuallers. If I am not mistaken; the Government of Bread Wine Bear & other Victualls, having been anciently in the Mayor & Bailifs, & afterwards (perhaps) in a mixt jurisdiction of the Chancellor & Mayor jointly; about the beginning of Rich. 2. by a Grant in Parliament (upon the seisure of the Town liberties) that you should have Custodiam Assisae Panis Vini et Cervisiae, ac etiam plenam potestatem inquirendi et cognoscendi de Fore-Stallatoribus et Regratariis omnibus et super his punitionem debitam faciendi, ac etiam gubernationem correctionem et punitionem praemissorum et aliorum victualium quorumcunque, simul cum finibus forisfacturis et amerciamentis ex eis provenientibus, sicut Cancellarius et Scholares Universitatis Oxon ipsa habent in sua Villa Oxon et suburbiis ejusdem. (and by a Charter of 5° Ric. 2. in pursuance of that Parliamentary grant,) the Government of Victualls were settled on the University; & that ever since then have without interruption injoyed the licensing of all Taverns, Innes, Ale-houses, & Victuallers; & (I think) take

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12 [In left margin in Wallis's hand:] Rot. Parl. de anno 5° Ric. 2. memb. 49. in Tur. Lond.

2 Oxford, (1) to write to some of your (2) in 3 from some of your University, add. 4 practise |in your University del] about 5 the (1) power of the (2) Government of Bread Wine Bear fe other Victualls, (a) being (b) having been anciently in the (oa) Town, & then (66) Mayor 7 the (1) M breaks off (2) Chancellor & Mayor jointly; |it was del] about 16 by add. 113

Mr Vicechancellor of Oxford: i.e. Paul Hood (c.1585-1668), B.D. Lincoln College, Oxford 1617, D.D. 1623. Rector of Lincoln College 1620-68. Vice-chancellor of the University of Oxford 1660. 33

10. WALLIS to DILLINGHAM, 16/[26] March 1660/1

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the Recognizances of Butchers & Victuallers not to kill or dresse flesh in Lent, (but do not return those or the Recognizances of Alehouse-keepers, to the Quarter Sessions;) & grant licenses to sick persons yet for eating flesh, & to buchers to kill for such. If it be thus (as I presume it is) we should desire the favour of a Certificate thereof (or so much of it as you find so to bee) particularly & clearly set forth, (under the hands of Mr Vicechancellor or your Register or both;) in reference to a hearing at the Councell Table (betwixt this University & the City of Oxford,) which we expect very suddenly (within a fortnight or thereabouts;) Which being a businesse of common concernment to both Universities (to strengthen the hands each of other) hath, the rather induced mee to give you this trouble at present (having no acquaintance with any to whom I might more fitly apply myself in this particular; & having had experience formerly of your readynesse in this kind,) assuring you of a like readynesse from hence upon any the like occasion, so far as shall bee in the power of Sir

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From my house in Cat-street Oxon March. 16. 1660./1.

Your affectionate friend & humble servant John Wallis.

The shortnesse of the time makes mee request your answer as soon as can bee, directed either to Dr Hood our Vicechancellor or myself here; or at

1 Victuallers (1) not to dressin breaks off (2) for (3) not 2-3 (but do ... Sessions;) add. 5 should |{—} del.\ desire 5 a (1) Testimoniall to this purpose (or at lest (2) Certificate thereof (a) so (6) or so 6 find (1) thus (2) so 8 Table | which we add. and del.\ (betwixt 9 Oxford,) (1) within a (2) which we expect 12 rather (1) persuaded (2) induced 12 present (1) (not knowing to whom better to apply myself, as more acquainted & having had (2) bee breaks off (3) (not being acquainted with (4) (having no ... having had 14 your (1) forwardnesse in this (2) readynesse 15 upon any the like occasion, add.

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11. WALLIS: Humble Petition to Charles II, [March? 1661] lest, if it admit any delay, to be left for mee at Mr Clendons114 house in the Strand over against the New Exchange London.

11. WALLIS: Humble Petition to Charles II for a Grant of the Next Vacant Prebend in Christ Church, Oxford [March? 1661] Transmission:

W Paper sent: KEW The National Archives PRO SP 29/33, No. 66, 1 p. Wallis was appointed chaplain-in-ordinary to the king at the end of 1660 and was involved in negotiations between Presbyterians and Anglicans in London in early 1661. It is probable that he applied for the post of prebendary at the height of this activity around March.

To The Kings Most Excellent Majestic; The humble Petition of John Wallis Dr in Divinity, and Professor of Geometry in the University of Oxford,

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Sheweth, That your Petitioner, incouraged thereunto by your Majesties gracious favour, doth humbly crave, A Grant under the Great Seal, of the next Prebends place in Christchurch Oxon which shall be voyd; as an Addition to his place of Geometry Professor. And your Petitioner shall ever pray &c. 114

Mr Clendons: i.e. John Clendon, q.v.

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THEOPHILUS DILLINGHAM to WALLIS Cambridge, I/[II] April 1661 Transmission:

C Letter sent: OXFORD University Archives WP//9/R/7b/6, 2 pp. On p. 2 in unknown hand: 'Apr. 1. 1661. Dr Theoph. Dillinghams Letters to Dr Wallis. No. 6.' Reply to: WALLis-DiLLiNGHAM 16/[26].III.1660/1. The present letter enclosed a paper by Dillingham concerning the University of Cambridge's practice in respect of alehouses and victuallers. This was evidently in response to a request made by Wallis in his letter of 16 March 1660/1.

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I have sent you here inclosed a paper which I scribbled over this morning in answer to yours115 concerning our practise about the business of Alehouses, & Victualers. I am very sorry that I have been incumbred with so much business as I could not answer yours before this time, & should bee much troubled if your hearing bee past before these come to you. I have according to your desire added the Vicechancellers116 hand to it & have allso sent you a copy of a letter concerning Lent which it may bee your concernment at this time if you shall need anything els which I can procure for you as to this concernment or any other. I shall with much chearfulness bee very ready to serve your university or your self, remaining Your very loving friend Theoph. Dillingham

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8 may bee |may bee del. ed.\ your 115

yours: i.e. WALLIS-DILLINGHAM 16/[26].III.1660/1. Vicechancellers: i.e. Henry Feme (1602-62), who was vice-chancellor of the University of Cambridge in both 1660 and 1661; DNB. 116

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13. COOPER to WALLIS, 23 April/[3 May] 1661

13.

BENJAMIN COOPER to WALLIS Oxford, 23 April/[3 May] 1661 Transmission:

C Letter sent: OXFORD University Archives WPj/16/1, f. 70r-71av, lower half of f. 71ar-71av missing (f. 70V and 71ar blank). On f. 71av beneath address in Wallis's hand: '23 Apr 1661 Mr Cooper the Register letter'.

Sir

I conceive their will be noe necessity of my attending upon the University buisinesse at Whitehall the next Saturday; and therefore I sent up to you by Dr Lamplugh117 the Originall Order under Secretary Nicholas118 hand, least otherwise it might not have come safe to you. I left the Act of Convocation with Dr Yates119, which, I thinke agrees allmost in every particular with this which is enclosed. I made what enquiry I could to find out what estates the Bayliffs seized on of Stevenson120 and Clarkes121 who murdered themselves, concerning the first I could get noe information, though his friends were asked about it; of the last you have a particular; and allsoe the time when the present Bayliffs undertooke to place the Markett, which as I remember is what you left me in trust withall. I am

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Your very humble servant Ben: Cooper Oxon April the 23: 1661

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This morning your Children were very well.

10 about (1) , (2) it; 117

Dr Lamplugh: i.e. Thomas Lamplugh (1615-91), fellow of the Queen's College, Oxford. Created D.D. in 1660, he was later made bishop of Exeter and finally archbishop of York; DNB. 118 Nicholas: i.e. Sir Edward Nicholas (1593-1669), secretary of state to Charles I and Charles II; DNB. 119 Dr Yates: i.e. Thomas Yate, q.v. 120 Stevenson: not identified. 121 Clarkes: not identified. 37

14. [HOBBESJ: La duplication du cube, [June? 1661]

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These For the Reverend Doctor John Wallis at Mr Clendon's house a Tallow Chandler next door to the signe of the Horse-Shoe over against the New-Exchange London

[71av]

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[THOMAS HOBBES]: La duplication du cube [June? 1661] Transmission:

E1 Printed paper. Exemplar (with, handwritten corrections): CHATSWORTH The Devonshire Collection Hobbes MSS, letter 85, 2 pp. (our source). E2 WALLIS, Hobbius Heauton-timorumenos, Oxford 1662, 129-32 (English translation), figure on p. 130 (our source for the figure). Hobbes published La duplication du cube anonymously axound June 1661. He claime that it was printed in Paris, although Wallis describes this as pretence (Hobbius Heautontimorumenos, 127-8). It was reprinted with replies to objections by Wallis and Rooke as an appendix to Dialogue physicus (1661). Amended versions appeared in an appendix to Problemata, physica, London 1662 (Opera, philosophica IV, 378-82), and in Seven Philosophical Problems, printed in London 1682 (English Works VII, 59-62). The acronym is solved by Hobbes in the Dialogus physicus, London 1661, as 'Un Autre Que Roberval' (Opera philosophica IV, 295). As he explains in Seven Philosophical Problems, he had the paper printed in Paris 'on purpose to see what objections would be made to it by our professors of algebra here' (English Works VII, 59). As confirmed in WALLis-BaouNCKER? 23.VI/[3.VII].1661, the printed paper contained a diagram, supplemented here from the English translation in Wallis's Hobbius Heauton-timorumenos. The handwritten corrections to the present exemplar probably originate from Hobbes himself, who is known to have subjected the demonstration to numerous revisions. At the foot of the reverse is a refutation (in Hobbes's hand) entitled 'La Refufation monstrante que la droit YZ produite ne tombe sur P: c'est a dire que en la Triangle DYP la coste PD n'est pas egale a DC, mais plus court.' This is possibly the refutation referred to by Hobbes in Seven Philosophical Problems (English Works VII, 59).

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14. [HOBBESJ: La duplication du cube, [June? 1661]

(E1} La duplication du cube Par V. A. Q. R. Une ligne droite etant donnee, trouver entre elle & sa moitie deux moyennes proportionelles. Soit donnee la droite AB, dont le quarre ABCD, soit coupe en quatre quarres egaus par les deux droites EF, GH, qui s'entrecoupent au centre du quarre ABCD, au point I. Ainsi que les quatre cotes soient divises en deux parties egales aus quatre points E, F, G, H. II faut done trouver deux moyennes proportionelles entre DC & DF. Je tire les diagonales AC, BD, & decris les quatre quarts de cercle ABD, BCA, CDB, DAC, dont les arcs coupent lesdites diagonales en K, L, M, N. Ausquels points les arcs sont coupes, chacun en deux parties egales, ce qui est asses concu. Je produis BA, CD, aus points O & P jusques a tant qu'elles soient egales a AB, DC, chacune a chacune, Et ayant decrit le quart de cercle ADO, & tire la diagonale AP (qui coupera 1'arc DO en deux parties egales au point Q,} Et etant produite de 1'autre part en R, marquera BR egale au sinus droit de 45 degres, c'est a dire a la semidiagonale BI. Et par consequent SD est 1'exces de la plus grande extreme AD au dessus de la semidiagonale AS. Je coupe cette SD en deux parties egales en T. En AD produite, je pren DV egale a DF, & faisant T le centre & TV semidiametre, je decris le cercle VXYZ, coupant DC en X, DA en Y, & la droite RS produite en Z, Et dis que les deux droites DY, DX, sont les deux moyennes proportionnelles demandees entre DP, egale a DC, & DV, egale a sa moitie, DF. Car tirant les lignes droites VX, XY, Tangle VXY (dans le demicercle) sera droit; Et la droite XT, tiree & produite jusqu'a la concavite du cercle VXYZ tombera sur Z, parce que ST, TD, sont egales, & par [2] consequent) SZ, egale a DX, Et XZ sera le diametre du cercle VXYZ.

26 egale a AB corr. 26-27 , DF add.

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L'angle done XYZ dans le demicercle est droit; Et en tirant les droites YZ, VX, on fait VXYZ rectangle, & ses cotes VX, YZ paralleles. Or, si la droite YZ produite, tombe sur P, touts la ligne P, Z, Y, sera droite & parallele a VX, & les angles alternes YPX, VXP seront egaus. Aussi les angles YPX, & XYD, seront egaus, & les trois triangles rectangles PDY, YDX, XDV, seront semblables. Et par consequent, les quatre droites PD, DY, DX, DV, seront en la mesme raison continuelles. II faut done demonstrer que la droite YZ produite, tombe sur P. Soit tiree PV & coupee en deux parties egales en a. Soit tiree aussi la droite ab parallele a AV coupant PD en c. Soit encore tiree, Td, parallele a PD, coupant ab en d; & divisee, dc, en deux parties egales en g; sur le centre g, a la distance, ga, soit decrit le demicercle ahb, coupant PD en h, & ab en b. Cela fait, les deux droites tirees, ah, bh, feront un angle droit en h. Or ac est la moitie de DV, Et parce que dg, gc, sont egales, db, sera aussi egale a la moitie de DV, & ab sera la moitie de YV. Comme est done PD a DY, c'est a dire a la composee de DS & SY, ainsi est PC (la moitie de PD) a cb la composee des moities de DS & 5T, & par consequent Pb etant produite tombera sur Y. Et les droites hb, ha seront les moities des droites, XY, XV, Et XY etant divisee en deux parties egales en i, la figure Yihb sera rectangle, & Yb sera parallele a XV. Or yZ, est parallele a XV. Done, YZ, produite tombera sur P. Et (par ce qui a este demonstre) les quatre droites PD, DY, DX, DV, sont en une mesme raison continuelle. J'ay done entre une ligne droite donnee & sa moitie, trouve deux moyennes proportionelles. Ce qu'il falloit faire. Consect. Un Cube qui a pour cote la plus grande de ces deux moyennes, est le double du Cube qui a pour cote la moitie de la plus grande extreme. Car la raison de Cube a Cube est tripliquee de la raison de cote a cote; Et la raison de PD a DV est tripliquee de la raison de PD a DY.

1 tirant la (1) droite VZ (2) droites YZ, VX corr. ed. 40

15. WALLIS to [BROUNCKER?], 23 June/[3 July] 1661

(E2, figure)

15. WALLIS to [WILLIAM BROUNCKER ?] Oxford, 23 June/[3 July] 1661 Transmission:

W Letter sent: CHATSWORTH The Devonshire Collection Hobbes MSS, letter 51, 2 pp. (our source).—partly printed: ScHUHMANN, Hobbes. Une chronique, Paris 1998, p. 172. El First edition: Thomas Hobbes, Dialogue physicus, London 1661, 32-3; Opera philosophica IV, 288-91 (partly). 41

15. WALLIS to [BROUNCKER?], 23 June/[3 July] 1661 E2 Second edition: Thomas Hobbes, Problemata physica, London 1662, 119-20; Opera philosophica IV, 379-80 (partly). It would appear that after Wallis had received a copy of Hobbes's printed paper La duplication du cube he sent Brouncker the present refutation for his consideration. Further refutations were produced by Brouncker himself—see WALLIS, Hobbius Heautontimorumenos, Oxford 1662, 144—and by the professor of geometry at Gresham College, Lawrence Rooke, whose calculation is quoted by Hobbes in Problemata, physica, London 1662, 124; Opera philosophica IV, 382. Cf. Walter Pope, The life o f . . . Seth, Lord Bishop of Salisbury, London 1697, 118-19.

Oxoniae. Junii 23. 1661. Illustrissime Vir,

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Tradebatur mihi hesterna nocte, sub horam vespertinam, impressa pagina, ex Gallia (ut ferebatur) allata, et quidem idiomate Gallicano; Cui Titulus La Duplication du Cube, par V. A. Q. R.122 cum adjuncta Scheniate lineoso. Linearum vero magnam partem, et constructionis non exiguam, superfluam esse plane, nee ad problematis vel Constructionem vel Demonstrationem quicquam spectare, statim deprehendi: Rem ipsam autem infectam esse; et Demonstrationem paralogismo laborare.123 Quae visum est subjecta pagina breviter indicare; Meque interim profiteri Dominationis vestrae Observantissimum, Joh: Wallis. Summa dictorum in Pseudodiplasiasmo Cubi nupero, (rescissis quae turn in Schemate turn in Constructionibus sunt superflua,) haec est. Exposita AD recta, continuetur ad V, ut sit DV semissi rectae AD aequalis. Centro A, distantia AD, scribatur DO circuli quadrans, bisectus in Q: et QS, rectae AD perpendicularis. Bisecta vero SD, in T; centre T, ducatur per V circulus VXYZ: Cui occurrat DX (rectae AD perpendicularis) in XAD, in Y; et QS in Z.

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Affirmat, Rectas DY, DX, medias esse proportionales inter AD et DV. 122

La Duplication ... V. A. Q. R.: i.e. [HOBBES], La duplication du cube, [June? 1661]. Rem . . . laborare: cf. HuYGENS-MoRAY [25.XJ/4.XI.1661; HUYGENS, (Euvres completes III, 384. 123

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15. WALLIS to [BROUNCKER?], 23 June/[3 July] 1661

Quod sic conatur demonstrare. Ductis rectis VX, XY; erit angulus VXY in semicirculo rectus. Ductaque XT et continuata, (propter bisectam SD in T) occurret circulo in Z. Adeoque ducta Y Z] erit XYZ angulus in semicirculo rectus; ipsaque YZ rectae XY parallela. Producatur XD ad P, ut sit DP rectae AD aequalis. Si itaque YZ producta rectae PD occurrat in puncto P; erunt (propter similia triangula PDY', YDX, XDV,) rectae DP, DY, DX, DV, continue proportionales. Et consequenter (propter Cubos in ratione laterum triplicata) Cubus lateris DY subduplus [sic enim dicendum erat, non Duplus] cubi lateris DP, sive DA. Atque hactenus recte. Rectam vero YZ continuatam, puncto P occurrere; sic probare (frustra) contendit. Ducta PV, et bisecta in o; ducatur ab rectae DY parallela, rectam DP occurrens in c. Rectaeque ab perpendicularis Td. Bisecta vero dc in g; centro g ducatur per a, semicirculus ahb, rectae CD occurrens in h, rectaeque ab in b. Ideoque, propter turn ca aequalem semissi rectae DV turn eg semissi TD, erit ab semissi VY aequalis. Adeoque juncta Pb et continuata, occurret puncto Y. Ductisque bh, ba, rectis; erit an43

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15. WALLIS to [BROUNCKER?], 23 June/[3 July] 1661 gulus bha rectus; ipsaeque bh, ha, semissibus rectarum YX, XV, aequales et parallelae. Quae quidem vera sunt. Sed non item et sequentia. Nempe 5

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Bisecta itaque YX in i, junctaque ih: erit Yihb rectangulum; et Yb rectae XV parallela. Sed et YZ eidem XV parallela est. Ergo et YZ (sicut ipsa Yb) producta occurret puncto P. Haec ille. Sed male. Sequitur utique ex praedictis Yihb Parallelogrammum esse; sed non item, Rectangulum; (adeoque nee Yb rectae XV parallelam.) Non enim (quod ipsi fraudi fuit) quia bha angulum rectum esse ostenderat, ideo bhi rectum esse sequitur; nisi et simul demonstrarat, Rectam ah continuatam, ad punctum i pertingere. Quos neque demonstrat ille, neque verum est. Et quidem tantum abest ut ex argumento quicquam de mediis proportionalibus concludatur, ut, quantacunque sumatur DP (sive ipsi AD aequalis, sive major sive minor in quacunque ratione,) reliquis manentibus, eadem demonstratione non minus concludatur, easdem DX, DY', medias esse proportionates, inter eandem DV atque hanc DP quamlibet. Quod autem non modo indemonstratum sit, sed neque verum, quod affirmat, nempe DX, DY, medias esse proportionales inter DV, et DP rectae AD aequalem,) sic Demonstro.| Ponamus DV — 1; adeoque DA vel DP — 2. Cum itaque sint ejusdem circuli, turn AD radius, turn AS sinus graduum 45; erit AS = ^2; et SD = 2-^/2; et TD = 1-^2; adeoque TV = 2- ^2; et DY = 3-^2; et DX = •/: 3 — -/2. Ideoque tribus DV, DX, DY, quarta proportionalis (quam quidem abscindet YZ recta ad rectam DP continuata) erit 3 — ^/2 in \f: 3 — ^2 :, hoc est, 1,997 fere; minor quam DP = 2. Adeoque YZ producta occurret rectae DP, non quidem in puncto P, sed in puncto aliquot inter P et D. Et consequenter, cum sit XYZ angulus rectus, erit XYP recto major. Verum itaque non est, vel rectam YZ continuatam, ad punctum P pertingere, vel PYX aut bhi angulum rectum esse, vel hi eandem esse rectam atque ah continuatam, aut rectae VX parallelam, vel denique rectas DX, DY, medias esse proportionales inter DV, et DP rectae DA aequalem. Quod demonstrandum suscepi. 23 AS (1) radius (2) sinus 24 TV jhoc est TX del] = 2 - |^2;

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[2]

16. FRENICLE to WALLIS, [October/November ? 1661]

16.

BERNARD FRENICLE DE BESSY to WALLIS [October/November? 1661] Transmission: C Part copy of missing letter sent (quoted in FRENlCLE-DlGBY [10J/20.XII.1661): LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 971, 1 p. (our source).—printed: HUYGENS, (Euvres completes IV, 44; HENRY, 'Recherches sur les manuscrits de Fermat', 695.

Answered by: WALLIS-FRENICLE XI/XII7.1661. Frenicle probably proposed this problem to Wallis in October or November 1661. Wallis subsequently sent Frenicle a solution (WALLIS-FRENICLE XI/XII7.1661), to which the latter then replied (FRENICLE-WALLIS [10J/20.XII.1661). Cf. FERMAT-[CARCAVI] XII?.1661; HUYGENS, (Euvres completes IV, 2-3; HENRY, 'Pierre de Carcavy', 351. Fermat's short account of the exchanges between Frenicle and Wallis was among various pieces which Carcavi sent enclosed with CARCAVI-HUYGENS [22.XII.1661]/!.1.1662; HUYGENS, (Euvres completes IV, 1.

Problema. Invenire duo Triangula Rectangula in numeris ita constituta, ut laterum circa angulum rectum differentia sit in utroque eadem; & quod in altero est majus duorum laterum circa angulum rectum, sit in reliquo Hypothenusa.

17. WALLIS to BERNARD FRENICLE DE BESSY [November/December? 1661] Transmission: w Part copy of missing letter sent (quoted in FRENlCLE-DlGBY [10J/20.XII. 1661): LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 971, 1 p. (our source).—printed: HUYGENS, (Euvres completes IV, 44-5; HENRY, 'Recherches sur les manuscrits de Fermat', 695. Reply to: FRENICLE-WALLIS X/XI7.1661. Answered by: FRENICLE-WALLIS [10]/20.XII.1661. This letter contained Wallis's reply to the problem Frenicle had posed him probably in October or November 1661. It was no doubt sent through the hands of Digby as was Frenicle's subsequent reply. The title was clearly supplied by Frenicle.

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18. FRENICLE to WALLIS, [[10]/20 December 1661] Solutio Clarissimi D. Wallisii.

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Esto duorum TViangulorum alterum BAG. alterum BCE. Sitque BC = 5 + x. BA — 5 — x. (ut sit 5 semisumma a;, semidifferentia laterum BC. BA.) Adeoque BCq. = 52 + 2.5x + x2, BAq. = 52 - 2.5x + x2 & horum differentia ACq. = 4.5ar, qui cum numerus quadratus esse debeat, oportet 5.x esse inter se, ut numeri quadrati.

Esto igitur 5 = a2, x = e2. Saltern 5 = ba2, x = be2. Ergo BC = ba + be2. BA = ba2 - be2. BCq = &V + 2.b2a2e2 + &2e4. BAq = b2a4 2.b2a2e2 + 62e4. Adeoque ACq. = 4.62a2e2. & AC = 2bae = AD. BD = ba2 - be2 - 2bae = BS. 6C = 2be2 + 2bae = CE. & CEq = 4.62e4 + 862ae3 + 462a2e2. Adeoque BEq = &V + 562e4 + 6.62a2e2 + 8.62ae3. Qui quum numerus quadratus esse debeat (etiam per b2 divisus) Quaerendum restat Quomodo investigandi erunt duo numeri a.e. ita constituti ut a4 + 5.e4 + 6.a2e2 + 8.a.e3 sit numerus quadratus. Interim suspicor (propter 8.ae3) num non casus sit impossibilis. Sed Nihil pronuncio. 2

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BERNARD FRENICLE DE BESSY to WALLIS [[10]/20 December 1661] Transmission:

C Part copy of missing letter sent (quoted in FRENlCLE-DlGBY [10J/20.XII.1661): LEIDEN Bibliotheek der Rijksuniversiteit Hug. 45, No. 972, 2 pp. (our source).—printed: HUYGENS, (Euvres completes IV, 45; HENRY, 'Recherches sur les manuscrits de Fermat', 696. 10-11 CEq = 4.62e4 + 862ae3 + & 2 aV corr. ed.

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19. WALLIS to BOYLE, 30 December 1661/[9 January 1662] Reply to: WALLIS-FRENICLE XI/XII7.1661. Frenicle sent the present response to Wallis's solution to the problem he had set as an enclosure to FRENICLE-DIGBY [10J/20.XII.1661; HUYGENS, CEuvres completes IV, 379. Digby probably forwarded it directly to Wallis. Cf. FERMAT-[CARCAVI] XII?.1661; HUYGENS, CEuvres completes IV, 2-3. A copy of the exchanges was made for Robert Moray who then sent them to Huygens with MoRAY-HuYGENS 30.1/[9.II]. 1661/2; HUYGENS, CEuvres completes IV, 34-6. See also MORAY-HUYGENS 24!/[3!I]. 1661/2; HUYGENS, CEuvres completes IV, 25-8.

Ad Clarissimi D. Wallisii Solutionem Responsum proponentis. Si absque alia conditione proponantur investigandi duo numeri a. e. ita constituti ut a4 + 5.e4 + Q.a2e2 + 8.ae3 sit numerus quadratus; facillima erit hujus Problematis Solutio. Sit namque a. quilibet numerus puta 2. e — 2.a. nempe 4. Erit a4 + 5.e4 + 6.a2e2 + 8.ae3 — 2704. numero quadrato [2] cujus radix 52. | Attamen non sufficit quaestioni ad quam solvendam numerus a. excedere deberet numerum e. in quo casu non ita faciles sunt inventu hi duo numeri a. e. In his autem perquirendis stat omnis quaestionis nodus.

19. WALLIS to ROBERT BOYLE Oxford, 30 December 1661/[9 January 1662] Transmission:

E1 First edition of missing letter sent: BOYLE, Works (1744), V, 511-12.—reprinted: BOYLE, Correspondence I, 473-5. E'2 Second edition: BOYLE, Works (1772), VI, 453-5 (our source). The present letter enclosed part of Wallis's Hobbius Heauton-timorumenos, the printing of which had not been completed at the time. This work takes the form of an epistolary tract addressed to Boyle.

Oxon. Dec. 30, 1661.

Sir, This, I suppose, may be at you time enough to wish you a happy new 47

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19. WALLIS to BOYLE, 30 December 1661/[9 January 1662]

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year; and many more to ensue. It brings with it a part of that in print124, which you had before an account of in writing125 (for it was thought fit by friends that it should also appear in this dress.) You might have had the whole ere this, had the press been as much at leisure, and as diligent (or rather not so much at leisure, but as diligent) as I could wish. I am now upon another work; as hard almost as to make Mr. Hobbes understand a demonstration. It is to teach a person126 dumb and deaf to speak, and to understand a language127: of which if he could do either, the other would be more easy; but knowjing neither, makes both the harder. [454] And though the former may be thought the more difficult, the latter may perhaps require as much of time. For, if a considerable time be requisite for him, that can speak one, to learn a second language; much more for him, that knows none, nor hath so much as the advantage of speech. You may think, perhaps, that it is not a piece of mathematics, to teach either speech, or language; as Mr. Hobbes, that the attendants of Jupiter were not found out by algebra. But though I am in this fully of his opinion, yet I must in that except to yours, at least thus far, that I find it therein no small advantage to have been versed in mathematics. I was the more willing to attempt it, because the person was represented as very ingenious and apprehensive (and at least so much of a mathematician, as to limn very well, being taught it by some of the best masters in London.) And I was the more confident that the defect was not in the organs of speech, (though possibly not so pliable as in a child, to the forming of unacquainted sounds) not only upon the common presumption, that the defect of speech in deaf persons is but an accidental consequent of their want of hearing: but also because he could once speak (though so long since that he does not remember it) till that about five years of age, having by accident lost his hearing, he thereupon lost his speech also: not at once, but gradually; that is, he was about half a year in losing it. He had, before he came, learned to write, I mean, as an English scrivener writes Greek; of which he knows neither the sound nor sense; and thereby hath saved me so much labour as the teaching him an alphabet: but hears either so little, or not at all, that I cannot, as I hoped, make any 124

tliat in print: i.e. WALLIS, Hobbius Heauton-timorumenos, Oxford 1662. account of in writing: probably in a now missing letter to Boyle. 126 person: i.e. Daniel Whalley. 12r lt is to teach. . . . language: see Wallis's account of this in SCRIBA, 'Autobiography', 41, and in WALLIS, Defence of the Royal Society, London 1678, passim. 125

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19. WALLIS to BOYLE, 30 December 1661/[9 January 1662] advantage of it. He cannot, from the back-side of my house, (which is a little more than a stone's cast, and no obstacle between) hear St. Mary's great bell ring; nor, in Christ Church choir, hear the organs. Close at his ear he can hear a sound, but not a voice, (unless I should rather say he feels it:) I mean, he discerns a noise, but not the articulation; and of a smart sound, rather than a loud. When a coach at night rushing close by the window, I perceived he discerned it: asking whether he heard it, he signified no, but he felt it, by the shaking of the ground. He hath now been with me a fortnight, and somewhat more. In which time, as to the language, he hath already learned many words, and somewhat of the Syntax. And, as to speech, hath pronounced all the sounds of our language (even those of L and R, and those of th in thy and thigh, which the French and others complain of as most difficult, and can hardly attain unto) which secures me, that the organs of speech are not indisposed to the forming of articulate sounds: and at some of these he is very ready, though he cannot at pleasure command them all. If you ask what my conjecture is as to the issue, or what I do design in it; I must confess, that as to the matter of speech, though I doubt not but he may come to speak any thing, yet I do not expect that he shall make the like advantage of it as those that hear: because, that neither hearing himself nor others, he will be subject to forget or mistake in forming sounds; and not to correct those mistakes, because he hears them not. For as one, who knows very well how to write, and hath a good command of his hand, yet if he want either sight or light, will hardly write well; the like must be expected in a deaf man's speaking; for, as then the eye guides the hand, so here, the ear the tongue. But as to the language, I know not but that he may, by writing, both express his own, and understand the conceptions of others, as well as other men; and so converse with men, as we do with the ancients, or persons distant, which is no small advantage [455] in human affairs: and may very much supply the defect of speech. | You may please to acquaint Sir Robert Moray128 with this adventure, who is himself so much a virtuoso, that he is the more inquisitive what others are doing; and will not allow me to be unemployed: but whether he will infer that I am busy, or was much at leisure, I cannot tell. Nor that I further trespass on your present affairs, than to say, that I am,

128

Moray: i.e. Robert Moray, q.v. 49

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21. WALLIS to BOYLE, 20 February/[2 March] 1661/2 your honour's affectionate and very humble servant, John Wallis.

20. ROBERT BOYLE to WALLIS 4 or 5/[14 or 15] January 1661/[1662] Transmission:

Manuscript missing. Existence and date: Mentioned in WALLIS, Defence of the Royal Society, London 1678, 10 (where the date given is 5 January) and 27 (where the date given is 4 January). Answered by: WALLIS-BOYLE 14/[24]!II.1661/2 (i). In this letter, as in BOYLE-WALLIS 26!I/[8!II]. 1661/2, Boyle requested an account of Wallis's progress in teaching Daniel Whalley how to speak.

21. WALLIS to ROBERT BOYLE Oxford, 20 February/[2 March] 1661/2 Transmission:

E WALLIS, Hobbius Heauton-timorumenos. Or a consideration of Mr Hobbes his Dialogues in an epistolary discourse, addressed, to the Honourable Robert Boyle, Esq., Oxford 1662. Written in the form of an epistolary tract addressed to Boyle, Wallis's Hobbius Heautontimorumenos is ostensibly a reply not only to Hobbes's ExamAnatio et emendatio mathematicae hodiernae (London 1660), but also to his recent Dialogue physicus, sive de natura aeris (London 1661). In the latter, Hobbes had attacked the statutes and scientific practice of the Royal Society as well as Boyle's New Experiments PhysicoMechanicall (Oxford 1660). In addition, he had sought to reply to Wallis's and Rooke's refutations of his duplication of the cube, publishing his solution as an appendix. 50

22. BOYLE to WALLIS, 26 February/[8 March] 1661/[1662]

22. ROBERT BOYLE to WALLIS 26 February/[8 March] 1661/[1662] Transmission:

Manuscript missing. Existence and date: Mentioned in WALLIS, Defence of the Royal Society, London 1678, 10 and 27. Answered by: WALLIS-BOYLE 14/[24].III.1661/2 (i).

23. WALLIS to ROBERT BOYLE Oxford, 14/[24] March 1661/2 (i) Transmission:

El First edition of missing letter sent: Philosophical Transactions No. 61 (18 July 1670), 1087-97 ('A Letter of Dr. John Wallis to Robert Boyle Esq, concerning the said Doctor's Essay of Teaching a person Dumb and Deaf to speak, and to Understand a Language; together with the success thereof: Which Letter though written many years since, was but lately obtain'd to be inserted here, it being esteemed very well worth to be preserv'd and communicated for Publick Use') (our source).—reprinted: BOYLE, Correspondence II, 11-18. E2 Latin translation of E1: Miscellanea curiosa medico-physica Academiae naturae curiosorum sive ephemeridum medico-physicarum germanicarum curiosarum annus primus, Leipzig 1670; Appendix sen addenda curiosa omissorum ad annum primum miscellanorum medico-physicorum, Breslau 1670, 11-20. E3 Richard BOULTON, The Theological Works of the Honourable Robert Boyle, Esq; epitomiz'd, London 1715, I, 292-9. E* German translation of E2: AMMAN, Abhandlung von der Sprache, und wie Taubstumme darin zu unterrichten sind. Nebst zwei Briefen des Dr. Joh. Wallis ... vom Unterrichte der Taubstummen. Aus dem Lateinischen iibersetzt . . . von Dr. L. Graflhoff ..., Berlin 1828, 103-18. Reply to: BoYLE-WALLis 4 or 5/[14 or 15].1.1661/2 and BOYLE-WALLIS 26.II/[8.III]. 1661/2.

Sir,

1 did acquaint you a while since,129 That (beside the consideration of 2 the consideration I had in hand E3 129

I did ... since: i.e. in WALLIS-BOYLE 30.XII.1661/[9.I.1662]. 51

23. WALLIS to BOYLE, 14/[24] March 1661/2 (i) 13

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°, which I had in hand;) I had undertaken another Task, (almost as 131 hard as to make Mr understand Reason,) to Teach a per son Dumb [1088] and Deaf, to speak and to Understand a Language. Of which if he could do either, the other would be more easy: But his knowing neither, makes both harder. And though the Former may be thought the more difficult; the Latter may perhaps require as much of Time. For if a considerable Time be requisite, for him that can speak One, to learn a Second Language; much more for him that knows None, to learn the First. I told you, in my last, that my Mute was now at least Semivocalis, whereof because you desire a more particular Information, I thought my self obliged to give you this brief Account of that whole Affair: that you may at once perceive, as well, upon what considerations I was induced to Attempt that Work, and what I did propose to my self as Fesible therein, as what Success hath hitherto attended that Essay. The Task it self consists of Two very different parts; each of which doth render the other more difficult. For, beside that which appears upon the first view, To teach a person who cannot Hear, to Pronounce the Sound of Words: There is that other, of teaching him to Understand a Language, and know the signification of those words, whether spoken or written, whereby he may both express his own sense, and understand the Thoughts of others: without which latter, that former were only to speak like a Parrot; or to write like a Scrivener, who understanding no Language but English, transcribes a piece of Latin, Welsh, or Irish; or like a Printer of Greek or Arabick, who knows neither the sound nor signification of what he printeth. Now, though I did not apprehend Either of these impossible; yet, that each of them doth render the other more hard, was so obvious as that I could not be Ignorant of it. For, how easily the understanding of a Language is attain'd by the benefit of Discourse, we see every day; not onely in those, who knowing one Language already, are now to learn a second; but (which doth more resemble the present case) in Children, who as yet knowing none, are now to learn their First Language. | [1089] For it is very certain, that no Two Languages can be so much different the one from the other, but that the knowledge of the one will be subservient to the gaining of the other: not only because there is now a 1 (almost . . . Reason,) omitted in E3 130

of : i.e. of the reply to Hobbes's Dialogus physicus, sive de natura aeris, which Wallis published under the title Hobbius heauton-timorumenos in 1662. 131 Mr : i.e. Thomas Hobbes.

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23. WALLIS to BOYLE, 14/[24] March 1661/2 (i) common Language, wherein the Teacher may Interpret to the Learner the signification of those Words and Notions which he knows not, and express his own Thoughts to him; but likewise (which is very considerable,) because the common Notions of Language, wherein all or most Languages do agree, and also so many of the Particularities thereof as are common to the Language he knows already, and that which he is to learn, (which will be very many,) are already known; and therefore a very considerable part already dispatched, of that work which will be necessary for the teaching of a First Language, to him who as yet knows none. But to this disadvantage (of teaching a First Language,) when that of Deafness is super-added, it must needs augment the difficulty: since it is manifestly evident from Experience, That the most advantageous way of teaching a Child his First Language, is that of perpetual Discourse; not onely what is particularly addressed to himself, as well in pleasing divertisements, or delightful sportings, (and therefore insinuates it self without any irksom or tedious labour,) as what is directly intended for his more serious Information: But that discourse also which passeth between others; where, without pains or study, he takes notice of what Actions in the Speaker do accompany such words, and what Effects they do produce in those to whom they are directed; which doth, by degrees, insinuate the intendments of those words. And, as that Deafness makes it the more difficult to teach him a Language: so on the other hand, that want of Language, makes it more hard to teach him how to speak or pronounce the Sounds. For there being no other way to direct his Speech, than by teaching him how the Tongue, the Lips, the Palate, and other Organs of speech, are to be applyed and moved in the Forming of such sounds as are required; to the end that [1090] he may, by Art, pronounce those| Sounds, which others do by Custome, they know not how; it may be thought hard enough to express in writing, even to one who understands it very well, those very nice Curiosities and Delicacies of motion, which must be observed (though we heed it not,) by him, who without help of his Ear to guide his Tongue, shall form that variety of Sounds we use in speaking: Many of which Curiosities are so nice and delicate, and the difference in forming those Sounds so very subtile, that most of our selves, who pronounce them every day, are not able, without a very serious consideration, to give an account, by what Art or Motion our-selves form them; much less to teach another how it is to be done. And if, by writing to one who understands a Language, it be thus difficult to give Instruction, how, without the help of Hearing, he 53

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23. WALLIS to BOYLE, 14/[24] March 1661/2 (i)

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may utter those Sounds, it must needs increase the Difficulty, when there is no other Language to express it in, but that of Dumb signs. These Difficulties (of which I was well aware) did not yet so far discourage me from that undertaking, but that I did still conceive it possible that both parts of this Task might be effected. As to the First of them; Though I did not doubt but that the Ear doth as much guide the Tongue in speaking, as the Eye doth the Hand in writing, or playing on the Lute: and therefore those who by accident do wholly lose their Hearing, lose also their Speech, and consequently become Dumb as well as Deaf; (for it is in a manner the same difficulty, for one that Hears not, to speak well; as for him that is blind, to write a fair hand:) yet, since we see that 'tis possible for a Lady to attain so great a Dexterity, as, in the dark, to play on a Lute, though to that variety of nimble motions, the Eyes direction, as well as the Judgment of the Ear, might seem necessary to guide the Hand; I did not think it impossible, but that the Organs of Speech might be taught to observe their due Postures, though neither the Eye behold their Motion, nor the Ear discern the Sound they make. And as to the other; That of Language might seem yet| more possible. [1091] For, since that in Children, every day, the Knowledge of words, with their various Constructions and Significations, is by degrees attained by the Ear; so that, in a few years, they arrive to a competent ability of Expressing themselves in their first Language, at least as to the more usual Parts and Notions of it; Why should it be thought impossible, that the Eye (though with some disadvantage) might as well apply such Complication of Letters or other Characters, to represent the various Conceptions of the mind; as the Ear, a like Complication of Sounds? For though, as things now are, it be very true that Letters are, with us, the immediate Characters of Sounds, as those Sounds are of Conceptions: yet is there nothing, in the nature of the Thing it self, why Letters and Characters might not as properly be applyed to represent Immediately, as by the Intervention of Sounds, what our Conceptions are. Which is so great a Truth, (though not so generally taken notice of,) that 'tis Practiced every day; not onely by the Chineses, whose whole Language is said to be made up of such Characters as to represent Things and Notions, independent on the Sound of words; and is therefore differently spoken, by those who differ not in the Writing of it: (like as what, in Figures, we Write, 1, 2, 3, for One, Two, Three- a Frenchman, for example, reads Un, Deux, Trois:) But, in part, also amongst our selves; as 54

23. WALLIS to BOYLE, 14/[24] March 1661/2 (i) in the Numeral Figures now mentioned, and many other Characters of Weights and Metals, used indifferently by divers Nations to signifie the same Conceptions, though expressed by a different Sound of words: And, more frequently, in the practice of Specious Arithmetick, and operations of Algebra, expressed in such Symbols, as so little need the Intervention of Words to make known their meaning, that, when different persons come to express, in Words, the sense of those Characters, they will as little agree upon the same Words, though all express the same sense, as two Translators of one and the same Book into another Language. [1092] And, though I will not dispute the Practical possibility) of introducing an Universal Character, in which all Nations, though of different speech, shall express their common Conceptions; yet, that some Two or Three (or more) persons may, by consent, agree upon such Characters, whereby to express each to other their sense in Writing, without attending the Sound of words; is so far from an impossibility, that it must needs be allowed to be very Fesible, if not Facile. And, if it may be done by new-invented Characters; why not as well by those already in use? Which though to those who know their common use, may signifie Sounds; yet to those that know it not, or do not attend it, may be as immediately applied to signifie Things or Notions, as if they signified nothing else: And consequently, so long as it is purely Arbitrary, by what Character to express such a Thing or Notion; we may as well make use of that Character or Collection of Letters, to express the Thing to the Eyes of him that is Deaf; by which others express the Sound or Name of it to those that Hear. So that, indeed, that shall be, to Him, a Real Character, which expresseth to Another a vocal Sound; but signifieth, to Both, the same Conception: Which is, To understand the Language. To these Fundamental Grounds of Possibility in Nature, I am next to add a Consideration which made me think it Morally-possible; that is, not impossible to succeed in Practice. And, because I am now giving an Account to one who is so good a Friend to Mathematicks, and Proficient therein, I shall not doubt but this Consideration will have the force of a great swasive. Considering therefore, from how few and despicable Principles the whole Body of Geometry, by continual consequence, is inforced; if so fair a Pile, and curious Structure may be raised, and stand fast upon so small a Bottom; I could not think it incredible, that we might attain some considerable success in this Design, how little soever we had at first to begin upon: and, from those little Actions and Gestures, which have a kind of Natural significancy, we might, if well managed, proceed gradually 55

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to the Explication of a compleat Language, and withal, direct to those curiosities of Motion and Po|sture in the Organs of speech, requisite to [1093] the Formation of a Sound desired; and, so to effect both parts of what we intend. My next Inducement to undertake it, was a consideration of the Person132, (which, in a work of this nature, is of no small concernment;) who was represented to me as very Ingenious and Apprehensive, (and therefore a fit subject to make an Essay upon;) and so far at least a Mathematician as to Draw Pictures; wherein, I was told, he had attained some good ability, which did induce me to believe that he was not uncapable of the Patience, which will be necessary to attend the Curiosity of those little varieties in the Articulation of Sounds; being already accustomed to observe and imitate those little Niceties in a Face, without which it is not possible to Draw a Picture well. I shall add this also, That, once, he could have spoken, though so long ago, that (I think) he doth scarce remember it. But having, by accident, when about five years of age lost his Hearing, he consequently lost his speech also; not all at once, but by degrees, in about half a years time: which though it do confirm what I was saying but now, How needful it is for the Ear to guide the Tongue in Speaking, (since that Habit of Speaking, which was attained by Hearing, was also lost with it,) and might therefore discourage the undertaking; yet I was thereby very much secured, that his want of Speech was but a consequent of his want of Hearing, and did not proceed originally from an Indisposition in the Organs of Speech to form those Sounds. And though the neglect of it in his younger years, when the Organs of Speech, being yet tender, were more pliable, might now render them less capable of that Accurateness which those of Children attain unto: (whereof we have daily experience; it being found very difficult, if not impossible, to teach a Forraigner well in years, the Accurate pronouncing of that Sound or Language, which in his tender years he had not learned:) yet, if he can attain to speak but so well as a Forraigner, at his years, may learn to speak English; what shall be farther wanting to that Accurateness which a| Native from his Childhood attains unto, may, to an indifferent [1094] estimate, be very well dispensed with. Having thus acquainted you with those Considerations which did induce me to attempt it; least you may think I build too confidently thereupon, and judge me guilty of too much vanity, in promising my self a 132

Person: i.e. Daniel Whalley. 56

23. WALLIS to BOYLE, 14/[24] March 1661/2 (i) greater success than can, in reason, be hoped for; It will next be necessary to give you some account, what measure of Success I might propose to my self as probable, in such an undertaking. And as to the first part of it, (that of Speaking;) Though I did believe, that much more is to be effected than is commonly thought Fesible, and that it was possible for him so to speak as to be understood; yet I cannot promise my self, that he shall speak so Accurately, but that a Critical ear may easily discern some Failures, or little differences from the ordinary tone or pronunciation of other men; (since that we see the like every day, when not Forraigners only, but those of our own Nation in the remoter parts of it, can hardly speak so Accurately, as not to discover a considerable difference from what is the common Dialect or Tone at London.) And this not onely upon the consideration last mentioned; (concerning the Organs of Speech less pliable to those Sounds to which they were not from the first accustomed;) but especially upon that other consideration, concerning the Ears usefulness to guide and correct the Tongue. For as I doubt not, but that a Person, who knows well how to Write, may attain, by custome, such a Dexterity, as to write in the Dark tolerably well; yet it could not be expected, that he should perform it with the like Elegancy, as if he saw the Motions of his hand: so neither is it reasonably to be expected, that he who cannot Hear, though he may know how to speak Truly, should yet perform it so Accurately, as if he had the advantage of his Ear also. Nor can I promise, nor indeed hope, that how Accurately soever he may learn to speak, he shoud be able to make so great Use of it as others do. For since that he cannot hear what others say to Him, as well as [1095] express his own Thoughts | to Them; he cannot make such use of it in Discourse as others may. And though it may be thought possible, that he may, in time, discern, by the motion of the Lips, visible to the Eye, what is said to him; (of which I am loth to deliver a positive judgment, since much may be said conjecturally both ways;) yet this cannot be expected, till at least he be so perfectly Master of the Language, as that, by a few Letters known, he may be able to Supply the rest of the Word; and by a few Words, the rest of the Sentence, or at least the sense of it, by a probable conjecture, (as when we Decipher Letters written in Cipher:) For, that the Eye can actually discern all the varieties of Motion in the Organs of Speech, and see what Sounds are made by those Motions, (of which many are Inward, and are not exposed to the Eye at all,) is not Imaginable. 57

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But as to the other Branch of our Design, concerning the Understanding of a Language: I see no reason at all to doubt, but that he may attain This, as perfectly as those that Hear; and that, allowing the like Time and Exercise, as to other men is requisite to attain the Perfection of a Language, and the Elegance of it, he may Understand as well, and Write as Good Language as other men; and (abating onely what doth directly depend upon Sound, as Tones, Cadencies, and such Punctilio's,) no whit inferiour to what he might Attain to, if he had his Hearing as others have. And what I speak of him in particular, I mean as well of any other Ingenious person in his Condition; who, I believe, might be taught to use their Book and Pen as well as others, if a right Course were taken to that purpose. To tell you next, What Course I have hitherto used towards this Design, it will not be so necessary. For should I descend to Particulars, it would be too Tedious; especially since they are to be used very differently, and varied as the present Case and Circumstances do require. And, as to the General way, it is sufficiently Intimated already. As to that of Speech; I must first, by the most significant signs I can, make him to understand, in what Posture and Motion I would have him apply his Tongue, Lips, and| other Organs of Speech, to the Forming of [1096] such a Sound as I direct. Which if I hit right, I confirm him in it: If he miss, I signifie to him, in what he differed from my Direction; and, to what Circumstances he must attend to mend it. By which means, with some Trials, and a little Patience, he learns first one, then another Sound; and, by frequent Repetitions, is confirmed in it; or (if he chance to forget) Recovers it again. And for this Work, I was so far prepared before hand, that I had heretofore, upon another occasion, (in my Treatise De Loquela1^3, prefixed to my Grammar for the English Tongue134",) considered very exactly (what few Attend to) the Accurate Formation of all Sounds in Speaking, (at least as to our own Language, and those I knew:) without which, it were in vain to set upon this Task. For, if we do not know, or not consider, how we Apply our own Organs in Forming those Sounds we Speak, it is not likely, that we shall, this way, Teach another. As to that of Teaching him the Language: I must (as Mathematicians 133

Treatise De Loquela: i.e. WALLIS, De loquela sive sonorum formatione tractatus grammatico-physicus, [Oxford 1653]. 134 Grammar for the English Tongue: i.e. WALLIS, Grammatica linguae anglicanae, Oxford 1653.

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23. WALLIS to BOYLE, 14/[24] March 1661/2 (i) do from a Few Principles first granted,) from that little stock (that we have to Begin upon) of such Actions and Gestures as have a kind of Natural significancy, or some Few Signs, which himself had before taken up to express his Thoughts as well as he could, Proceed to Teach him, what I mean by somewhat else; and so, by steps, to more and more: And this, so far as well I can, in such Method, as that what he Knows already, may be a step to what he is next to Learn; as, in Mathematicks, we make use, not of Principles only, but Propositions already demonstrated, in the Demonstration of that which follows. It remains now, for the Perfecting the Account which at present you desire of me, only to tell you, what Progress we have already made. Which, had not your Desires commanded from me, I should have respited a while longer, till I might have made it somewhat Fuller. He hath been already with me somewhat more than Two Moneths. In which time, though I cannot be thought to have Finished such a Work; [1097] yet the Success is not so little, | as to Discourage the Undertaking: but as much as I could hope for in so short a time; and more than I did Expect. So that I may say, the Greatest difficulty of Both Parts being almost over; what Remains, is little more than the work of Time and Exercise. There is hardly any Word, which (with deliberation) he cannot Speak; but, to do it Accurately, and with Expedition we must allow him the Practice of some considerable Time, to make it familiar to him. And, as to the Language; though it were very Indifferent to him who Knew none, which to begin withal; yet, since it is out of Question, that English, to him, is like to be the most Useful and Necessary; it was not adviseable to Begin with any other. For though he can Pronounce the Latine with much more Ease, (as being less perplexed with a multitude of concurring Consonants;) yet this is a Consideration of much less concernment than the other. To this therefore having applyed himself, he hath already Learned a great many Words, and, I may say, a considerable part of the English, as to Words of most Frequent use: But the whole Language being so Copious, though otherwise Easy, will require a longer Time to Perfect what he hath Begun. And this, Sir, is the full History of our Progress hitherto. If you shall hereafter esteem our Future Success, worthy your taking notice of, You may Command that, or what else is within the power of 4 teach him somewhat else; E3

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24. [WALLIS:] Note on letter to Boyle, 11/[21] July 1670

Oxford, March 14. 1661/2.

Sir, Your Honours very humble Servant, John Wallis.

24.

[WALLIS:] Note on letter to Boyle Oxford, 11/[21] July 1670 Transmission:

W Draft note: LONDON Royal Society Early Letters Wl, No. 110, 1 p. (our source). On reverse in Wallis's hand: 'To Mr Oldenburg. July 11. 1670. Oxford. Sir' (deleted). El First edition of draft note: Philosophical Transactions No. 61 (18 July 1670), 1098-9 ('A Letter of Dr. John Wallis to Robert Boyle Esq, concerning the said Doctor's Essay of Teaching a person Dumb and Deaf to speak, and to understand a Language; together with the success thereof: Which Letter though written many years since, was but lately obtain'd to be inserted here, it being esteemed very well worth to be preserv'd and communicated for Publick Use').—reprinted: BOYLE, Correspondence II, 18-19. E2 Latin translation of El: Miscellanea, curiosa medico-physica Academiae naturae curiosorum sive ephemeridum medico-physicarum germanicarum curiosarum annus primus, Leipzig 1670; Appendix sen addenda curiosa omissorum ad annum primum miscellanorum medico-physicorum, Breslau 1670, 20-2. E^ German translation of E2: AMMAN, Abhandlung von der Sprache, und wie Taubstumme darin zu unterrichten sind. Nebst zwei Briefen des Dr. Joh. Wallis ... vora Unterrichte der Taubstummen. Aus dem Lateinischen ilbersetzt ... von Dr. L. Grafihoff . . . , Berlin 1828, 118-19. The present note represents a postscript to WALLIS-BOYLE 14/[24].III.1661/2 (i). It was written by Wallis in form of an editor's note to be appended to the version of that letter which was printed in the July 1670 issue of Philosophical Transactions. He probably handed the manuscript to Oldenburg together with a copy of the letter itself.

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The Person to whom the foregoing Letter135 doth referre, is Mr Daniel Whalley, (son of Mr Whaley136, late of Northampton, & Mayor of 135

foregoing Letter: i.e. WALLIS-BOYLE 14/[24]!II.1661/2 (i). Mr Whaley: presumably Peter Whalley (d. 1656), mayor of Northampton

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24. [WALLIS:] Note on letter to Boyle, 11/[21] July 1670 that Town.) Hee was, (soon after the Date of this Letter) on the 21th day of May, 1662, present at a meeting of the Royal Society137, (of which the Register of that days proceedings takes particular notice,) and did in their presence (to the great satisfaction of the Company) pronounce distinctly enough such words, as by the Company were proposed to him; and, though not altogether with the usual Tone or Accent, yet so as easyly to be understood. About the same time allso (his Majestic having heard of it, & being willing to see him,) hee did the like several times at Whitehal, in the presence of his Majestic138, & Prince Rupert139, & divers others of the Nobility; though hee had then imployed but a small time in acquiring this ability. In the space of One Year, which was the whole time of his stay with Dr Wallis; hee had read over a great part of the English Bible, & had attained so much skill as to expresse himself intelligibly in ordinary affairs; to read and understand letters written to him, & to

4 presence, to their great satisfaction pronounce E1 4 pronounce (1) very distinctly (2) distinctly enough W 7 understood (1) , (—} did it appear to the company that (2) . \[here you may adde, if you please, what the Register says of it.] del] (a) In the space of One year (which was the whole time of his stay with Dr Wallis) hee had (6) About W understood: Whereupon also the said Doctor was, by the same Assembly, encouraged to pursue what he had so ingeniously and so successfully begun. About E1 8 did (1) it (2) the like several times (a) in his Ma breaks off (b) at Whitehal, W 9 Majestie, (1) of th breaks off (2) of his Highness the Duke of York, (3) & Prince W Majestie, His Highness Prince E1 12 Dr Wallis; (1) hee (2) hee had read over a great part of the English Bible, & W 13 himself (1) intelligly, in most (2) intelligibly in W 14 to |read and add.\ understand W to understand E1 (1646 and 1655) and member of parliament for burgesse of Northampton (1654-5). 137 Hee . . . Society: cf. BIRCH, History of the Royal Society, I, 84: 'Dr Wallis brought with him the young man born deaf and dumb, and made him pronounce several words; and was desired to continue his practice upon him'. 138 his Majestie: i.e. King Charles II. 139 Prince Rupert: i.e. Prince Rupert (1619-82), son of Elizabeth, Queen of Bohemia, and Frederick V, Elector Palatine. He fought on the Royalist side during the civil wars and was appointed Commander-in-Chief in 1644. After the Restoration, he rose to first lord of the Admiralty (1673-9). Elected fellow of the Royal Society in March 1664. DNB. 61

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write answers to them; though not elegantly, yet so at lest as to be understood: and in the presence of many Forraigners (who out of curiosity have come to see him) hath oft times not onely read English & Latine to them, but pronounced the most difficult words of their languages (even the Polish itself) which they could propose to him. Since that time, though he have not had the opportunity of making much further improvement, for want of an Instructer; hee doth yet retain what hee had attained to: or, wherein hee may have forgot the niceness requisite in the pronunciation of some sounds, doth easyly recover it with a little help. Nor is this the onely person on whom hee hath shewed this effect of his skill; But he hath since done the like for another140, (a Young Gentleman of a very good Family & a fair Estate,) who did, from his Birth, want his Hearing. On this occasion, I thought it very sutable to give notice of a small Latine Treatise141 of this same Author, first published in the year 1653, intituled De Loquela (of Speech;) prefixed to his Grammaire142 of the English Tongue (written allso in Latine.) In which treatise of Speech (to which he refers in this discourse, & on confidence hee durst undertake that difficult task) hee doth very distinctly lay down the manner of forming all Sounds of Letters usual in Speech, as well of the English as of other Languages. Which is, I think, the first book ever published in that kind; (For though some writers formerly have here and there occasionally sayd something of the formation of some particular Letters; yet none, that I know of, had before him undertaken to give an account of all:) Whether any, since him, have with more judgement and accurateness performed

1 though, not elegantly, yet add. W 1 yet so as to be E1 1-2 understood: (1) And since th breaks off (2) and in the presence of (a) divers (b) many W 6 making (1) any great (2) much further W 9 recover it (1) at (2) with W 15 first published . .. Speech;) add. W 20 all (manner of del] Sounds W 22 some (1) have (2) writers formerly have (a) now an breaks off (b) here W 24 all:) (1) And I question add] whether (3) Whether W 140

another: i.e. Alexander Popham, son of Anne Wharton by her first marriage. Latine Treatise: i.e. WALLIS, De loquela sive sonorum formatione tractatus grammatico-physicus, [Oxford 1653]. 142 his Grammaire: i.e. WALLIS, Grammatica linguae anglicanae, Oxford 1653. 141

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25. WALLIS to BOYLE, 14/[24] March 1661/2 (ii) the same, I will not take upon mee to determine. In his Grammaire of the English Tongue (to which this, of Speech, is prefixed;) hee hath so briefly & clearly given a true account of this language, as may be very advantageous not onely to strangers for the easy attainment thereof, but even to the English themselves for the clear discovering (which few take notice of) the true genius of their own language.

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25. WALLIS to ROBERT BOYLE [Oxford], 14/[24] March 1661/2 (ii) Transmission:

W Letter sent: LONDON Royal Society Boyle Letters 6, f. llr-12v (our source). At bottom of f. 12V at 90° in Boyle's hand: 'The Businesse about the Bible.'—printed: BOYLE, Correspondence II, 7-10.

Sir

I have, as you desired, considered of Mr Chilinski's143 businesse144, & perused the Papers & Letters by him produced, (many of the particulars being allso otherwise known to myself to be true as he relates them.) And the case, according to the best estimate I can make of it, stands thus. 1654

Mr Samuel Boguslaus Chylinski, being sent out of his own country (of Lithuania) for that purpose, studied Divinity in the University of Franeker in order to fit him the better for the service of the church at home, for the space of Two Years, being so long supplyed by means allowed him from those Churches.

1 upon me at all to determine. El 1 In (1) that (2) his W 2 so (1) cl breaks off (2) briefly W 3 given an account E1 4 attainment (1) of (2) thereof W 5 discovering (1) (of what (2) (which few W 143

Chilinski's: i.e. Samuel Boguslav Chylinski, q.v. businesse: i.e. Chylinski's efforts to publish the Bible in Lithuanian. See MADDISON, Life of the Honourable Robert Boyle, 111. 144

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The Wars145 then breaking forth & wasting those Churches, and hindering all supplies from thence, he was afterward forced to subsist on the Benevolence of such as pittyed his condition. Hee came to Oxford with Letters of Recommendation from the University of Franeker, and there for some years hee lived piously & studiously, on like Benevolence of divers there. In this condition, considering how hee might best imploy his study for the service of those Churches, he thought of Translating the Bible into the Lithuanian Language; into which (though the Gospel have been there received for about 300 years) it was never yet translated. In June 1657, he began the Translation of the New Testament; & finished it the year following; But, having nothing but Benevolence to live on, was not able to get it Printed146. In September 1658, hee began the Translation of the Old Testament, & finished it about the end of the next year. His Translatinge being finished, divers in Oxford, of the most eminent persons there, approving this design, gave him ample Testimonialls, as well of his life & conversation, as of approving so good a work, recommending the promoting of it: And many of them subscribed towards the Impression of it divers summes of many (which were afterwards payd, & imployed therein,) but far short of what were necessary for the compleating of it. In March following, hee began the Impression at London, (hoping that other Benefactors would afterwards be found, for the perfecting of it;) and carried it on, as far as the monies collected, and his own credit would reach. | In September following, the monies collected being all imployed in that work & there being farther due to the Printer above 100', (about one

1 then (1) beginning (2) breaking 2 on (1) such (2) the 16 His (1) being wellnigh (2) Translatinge being 17 Testimonialls, (1) of his (2) as well 20 were |most of them del] afterwards 23 following, |(the Translation being before that time compleated,) del] hee 28 100(, (1) p breaks off (2) (about one (a) third (b) halfe 145

Wars: i.e. the Russo-Polish war (1654-6). Printed: Chylinski's Lithuanian translation of the New Testament was not printed until 1958 as Biblia litewska Chyliriskiego. Nowy testament, Poznan 1958. 146

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[IF] 1660

25. WALLIS to BOYLE, 14/[24] March 1661/2 (ii) halfe part of the Bible being then printed:) The Printer was not willing to proceed further; but required Bond of him for the rest of the mony due for what was allready done; which was given him. In this condition, applying himself to divers of the London Ministers 1660 soliciting farther contributions for that work, they Recommended him to the Lord Mayor147 & Court of Aldermen; who so far resented his condition, & that pious work, that they were willing to Recommend his case to the King for a Publike Collection & made an Order of the Court to that purpose. 1660 Having thus far promoted this affair upon his private account, & the interest of those friends that were willing to favour him in it: When the Petition to his Majestie was drawn, & ready to be signed; as the Desire of the Lord Mayor, the Court of Aldermen, & the Ministers of the City, & the day appointed for that purpose; Mr John de Kraino Krainski148, coming over as an Agent from those Churches to desire Relief for them, occasioned a stop therein for the present. 1660 This Agent, suspecting that it might prove disadvantageous to his desired supplies if this collection for the Translation should proceed; & yet understanding how far it was promoted, & that so many considerable persons were allready ingaged in it, that he could not well hinder it; pretended a readynesse to promote it (hoping thereby the better to carry on his own businesse;) suggesting (though unknown to Mr Chylinski) that those Churches had procured this Translation, & caused it to be thus far printed; & desiring a Collection for the carrying it on, and for the further Relief of those distested Churches.

1 then add. 2 to (1) pro breaks off (2) proceed 14 John |de add.\ Craino Crainski corr. 17 might add. 19 was (1) proceed, & how (2) promoted, fe that so 20 he add. 24 desiring (1) supply (2) a Collection 24 on, (1) as we (2) and 147

Lord Mayor: i.e. Thomas Alleyn (d. 1690), lord mayor of London 1659-60. John de Kraino Krainski: i.e. Jan Krainski (1625-85), Calvinist priest. He lived in exile in London 1660-6, and raised funds there to aid the Lithuanian Calvinist church. 148

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25. WALLIS to BOYLE, 14/[24] March 1661/2 (ii)

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Meanwhile, Mr Chylinski not finding a supply of means here to carry on his Impression, did, in March following, go over into his own Country, & acquainted the Synod there, what he had done, & how far hee had proceeded, who approved the work & incouraged him to goe on; as appears by the order of the Synode to that purpose, bearing date Sept. 1. 1661. In his Absence the Agent (whose design was, rather to hinder than promote the Work, as appears by his own Letters, & the whole sequele, though he pretended otherwise) had upon suggestion that this was the Churches Work (who as yet knew nothing of it) and the Translation procured by them obtained Letters Patents for a publike Collection, for carrying on this Impression, & for the Relief of those Churches; bearing Date July 12. 1661. In November following, Mr Chylinski returning to London (having heard of a Collection granted for that work,) and willing to go on with the Work; the Printer upon credit of those Letters Patents, was willing to go forward with| the Impression; & proceeded therein as far as the Psalmes. The Agent understanding it; in stead of promoting the Work, or taking care to pay what was due to the Printer for what was allready done; gives expresse order to the Printer not to proceed. Whereupon the work is at a stand. The Printer, thus Inhibited, writes a Letter to Mr Chylinski, demanding his mony due, & threatening to Arrest him, if he did not by the next day put in Bail to Answer an Action commenced against him: The Agent having disclaimed the Work. Mr Chilinski repairing to the Agent with this Letter, told him that hee must be forced to complain to his Majestie & the Counsell for relief, if he did not otherwise take care of it.

1 carry add. 4 on add. 6 rether corr. ed. 7 sequele, (1) ) had upon sugg breaks off (2) though 9 the (1) Translated (2) Translation 13 following, (1) when Mr Chylinski returned (2) Mr Chylinski returning 14 work, |but not knowing upon what suggestion, del.\ and 14 go add. 18 pay what add] was 22 not add. 26 & the Counsell add.

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1660./ 1.

1661.

1661.

[12r]

25. WALLIS to BOYLE, 14/[24] March 1661/2 (ii) The Agent, though first he told him that before he should be able to make friends there, to have it represented to the Counsel, it would cost him as much time as himself had allready spent in getting the Letters Patents; Yet, thinking it necessary to prevent such Complaint; & being to Petition the Counsell for other Monies, obtained an Order (amongst other Payments) for 50^ to be payd to the Printer in part of his debt, (whereas he might as well have obtained an Order for the whole, if he had desired it;) & for allowance of 4f per mensem to the Translator for his Subsistence. The Printer having received this 50', (& 20' more upon another Petition of his own,) being not yet satisfied for the whole of what is allready due, (nor the Agent willing to Undertake or Own the Work,) Mr Chilinski can neither have his Bond delivered; nor can the Work of the Impression go on (being still inhibited by the Agent;) nor can he get any of that 4' per mensem for his Subsistence; the Treasurers refusing to pay any, without an Acquittance signed by the Agent, which is not to be had; And the Agent now gone out of Town, to solicite Collections in other places, is not likely to return for divers Monthes. And the Agent, more effectually to retard the Work; would now have Mr Chylinski go back again into his own Country, to have the Translation examined & reviewed there: Sometimes pretending, that he was not imployed by those Churches in this work, and therefore not the Translator intended in the Letters Patents; Sometimes, that he will send or hath sent himself for other Translators; Sometimes, that there is more necessity of relieving their poor, than printing the Bible; Sometime, wishing, that he may perish, & his work with him: Notwithstanding that he had before suggested to his Majestie, (& thereupon obtained the Letters Patents,) that those Churches had procured this Translation, &, as a Testimony

1(1) The Agent, thinking it necessary though with opprobrious words |he add.\ told (2) The Agent, though 1 to (1) find (2) make 2 Counsel, (1) he (2) it 11 what (1) was (2) is (a) due (6) allready 13 nor |nor del. ed.\ can 20 Translation (1) re breaks off (2) examined 21 not |not del. ed.\ imployed 24 that (1) the (2) there 28 that (1) the (2) those

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26. BOYLE to WALLIS, 5/[15] April 1662

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thereof, a Copy of so much thereof as| was then Printed, was presented [12-] to his Majestie. Mean while, the Person who upon his own charge, with the supply of his private friends, & his own Study & Credit, hath hitherto carried on that work, wants subsistence to live on, (having been forced to pawn what he hath;) & the work of Printing the Bible in that Language, which seemes to be the first Intention of the Letters Patents, (the Surplusage onely of that Collection being ordered for the relief of the Poor in those Churches) is actually hindred. Having thus given you account of what I take to be the true State of the businesse; I know not well, (considering this Disagrement beween the Agent & him,) what is more advisable for perfecting the Intended Edition of the Bible in that Language, (which the Letters Patents do expressely direct,) than that by Order of his Majestie & Counsel, or by such ways as they may think fit so much out of the monies collected for that purpose, be payed to Mr Chilinski or the Printer (upon the Acquittance of both of them) by the Treasurers immediately (without dependence on the Agent,) as shall be necessary for that work. I have nothing farther to adde to it, than that I am Your Honours very humble servant J. W.

20

March. 14. 1661./2.

26.

ROBERT BOYLE to WALLIS 5/[15] April 1662 Transmission:

Manuscript missing. Existence and date: Mentioned in WALLIS, Defence of the Royal Society, London 1678, 10 and 28.

5 (having (1) pawned (2) been 11 well, (1) what in this (2) (considering 13 do (1) di breaks off (2) expressely direct,) (a) that (b) than 16 the (1) T breaks off (2) Printer

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27. WALLIS to MORAY, 7/[17] April 1662 In this letter Boyle thanked Wallis for the account of his progress in teaching Daniel Whalley to speak which he had provided in WALLIS-BOYLE 14/[24].III.1661/2 (i). It would appear that he also expressed the wish to see this demonstrated. Cf. MORAY ? et al.-WALLis 8/[18].V.1662.

27.

WALLIS to ROBERT MORAY Oxford, 7/[17] April 1662 Transmission:

El First edition of missing letter sent: WALLIS, Cono-Cuneus: or, the Shipwright's circular wedge, London 1684, sig. A2 r -A2 v , Epistle dedicatory; printed as additional treatise I to WALLIS, Algebra, London 1685 (our source). E2 Latin translation of El: WALLIS, Opera mathematica II, 681. Answered by: MORAY-WALLIS IV/V.1662. The present letter enclosed Wallis's theoretical exposition of geometrical sections of the shipwright's circular wedge, suggested to him by models of these which Peter Pett had demonstrated to Brouncker and Moray in his presence in London.

To the honourable Sir Robert Moray, Kt.

Sir, Since I came home from London, I have taken some time to consider of those Solids and Lines made by the Sections thereof; proposed to Consideration (to my Lord Brouncker and your self, at your Lodgings, where I was also present) by Mr. Pett149, one of His Majesties Commissioners for the Navy, and an excellent Shipwright. The Bodies proposed to consideration were all of this form. On a plain Base, which was the Quadrant of a Circle, (like that of a Quadrantal Cone or Cylinder) stood an erect Solid, whose Altitude (being arbitrary) was there double to the Radius of that Quadrant; and from every Point of its Perimeter, streight Lines drawn to the Vertex, met there, not in a Point (as is the Apex of a Cone), nor in a parallel Quadrant (as in a Quadrantal Cylinder), but in a streight Line or sharp Edge, like that of a Wedge or Cuneus. On which consideration, I thought fit to give it the 149 Pett: i.e. Peter Pett (1610-70?), Navy Commissioner, fellow of the Royal Society from September 1662 onwards, DNB. Cf. BIRCH, History of the Royal Society, I, 85.

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27. WALLIS to MORAY, 7/[17] April 1662

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name of Cono-Cuneus, as having the Base of a Cone, and the Vertex of a Cuneus. By the various Sections of this Solid, in several Positions, he did (rightly) conceive, that divers new Lines must arise, in great variety, different from those arising from the Section of a Cone. Some of which he supposed might be of good use in the Building of Ships; in order to which it was, that he proposed them to Consideration. Now because he judged it troublesom (as indeed it would be) first to form such Solids, and then cut them by Plains in such Positions as he desired; he had (for avoiding that trouble) ingeniously contrived this Expedient. He caused divers Boards, of a good solid Wood, to be exactly planed, some of an equal thickness, some meeting in a sharp edge; those of the former, he caused to be glewed together in a parallel Position; those of the latter sort, he caused so to be glewed together, as that their sharp edges met in one common Angle. And having thus formed several Solids, of Boards thus glewed together, he then caused them to be wrought into such a form as that before described: Which being done, he then caused the Glew| to be dissolved in warm Water, whereby the several Boards, falling [A2V] asunder, did exhibit, in their several faces, the respective Sections of those Solids. And such were those he shewed us; which being put together, made up such Solids; and taken asunder, shewed the several Sections of them. I do not intend at all to disparage the ingenuity of that Contrivance, which was indeed very handsom, and neatly performed, but do withall suppose, that it would not be unpleasing to your self, or him, to see those Lines described in Piano, which would arise by such Section of the Solid. That therefore is the work of these Papers, to represent the true nature of such Lines, and the ways to draw them, without the actual Section of a Solid. Which I have the rather undertaken, because this is a Solid which I do not know that any other have before considered. And because this may be a Pattern; according to which, other Solids of like nature may be in like manner considered if there shall be occasion. If beside these Sections which he hath already considered, there be any other Sections of this or other the like Solids which he shall conceive useful to his purpose; the same may in like manner be represented (without the actual Section of such Solids) by Lines thus described in a Plain. But which of them may be most advantageous to his design, I do not pretend to understand so well, nor can with so much certainty affirm; as, 70

28. MORAY to WALLIS, April/May 1662 that I am, Sir,

Oxon, Apr. 7. 1662.

Your very humble Servant, John Wallis.

28.

ROBERT MORAY to WALLIS April/May 1662 Transmission:

Manuscript missing. Existence: Mentioned in and answered by WALLis-MoRAY 6/[16].V.1662. Reply to: WALLIS-MORAY 7/[17].IV.1662.

29. WALLIS to ROBERT MORAY Oxford, 6/[16] May 1662 Transmission:

W Letter sent: LONDON Royal Society Boyle Letters 5, f. 170r-171v (f. 170V and 171r blank) (our source). Postmark on f. 171V: 'MA/7'.—printed: BOYLE, Correspondence VI, 428-9. El First edition of letter sent: BOYLE, Works (1744), V, 512. E2 Second edition: BOYLE, Works (1772), VI, 455. Reply to: MoRAY-WALLis IV/V.1662.

Sir,

5

You will excuse mee for not returning you my thanks for yours150, (so full of civility & respect,) when you understand that I had nothing else to send with it that might bee worth giving you that trouble: And yet pardon this at present, which waits on you with this onely addition, that 6 not add. 9 which (1) onely (2) waits 150

yours: i.e. MoRAY-WALLis IV/V.1662. 71

30. MORAY? ET AL. to WALLIS, 8/[18] May 1662

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I have thoughts of waiting on you myself next week. At which time I may possibly present you with somewhat as acceptable as the last151; but in another kind. For I shall bring with niee (I suppose) a person152 on whom I have made an attempt lately of teaching him to Speak, though he cannot Hear; wherein your Curiosity may possibly make you desirous of understanding what progresse I have made. Which being of such a nature as not to be so satisfactory in a representation to the Eye as to the Ear, I shall forbear the Narrative, & refer it till you may receive a personal account thereof from himselfe, & from, Sir

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Oxon. May. 6.1662.

15

Your very humble servant John Wallis. [171V]

For my honoured Noble Friend, Sir Robert Moray, at his lodgings in the Privy Garden at White-hall.

30.

ROBERT MORAY ? ET AL. to WALLIS 8/[18] May 1662 Transmission:

Manuscript missing. Existence and date: Mentioned in WALLIS, Defence of the Royal Society, London 1678, 28. According to Wallis, members of the Gresham Society, i.e. the future Royal Society, requested that he provide them with direct knowledge of his language instruction to a deaf-mute, of which they had previously received a report. This possibly took the form of a reply to WALLIS-MORAY 6/[16].V.1662. On 14 May 1662, Wallis gave members of the Society an account of his progress in teaching Daniel Whalley how to speak. At their request he presented the young man to them at the next meeting on 21 May 1662. 8 may add. 151

last: probably WALLIS-MORAY 7/[17].IV.1662, in which Wallis enclosed his theoretical exposition of geometrical sections of a shipwright's circular wedge. 152 person: i.e. Daniel Whalley.

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31. WALLIS to TITUS, 12/[22] June 1662

31.

WALLIS to SILAS TITUS 12/[22] June 1662 Transmission: Manuscript missing. Existence and date: Referred to in WALLIS, Algebra, 225-7; WALLIS, Opera mathematical, II, 245-6. In this letter, Wallis sent his proposed solution to a problem which evidently originated from Pell, and which Titus, 'a very Ingenious Person . . . and very well accomplished in Mathematical and other Learning' (WALLIS, Algebra, 225), had communicated to him. The task concerned was to find the numbers a, &, c, given that aa + be — 16, bb + ac = 17, and cc + ab = 18. Having been informed by Titus of the origin of the problem, Wallis drew up his solution in general terms according to Pell's method. The mathematical content of Wallis's solution is to be found in the pages of the Algebra referred to above.

32. ANNE WHARTON to WALLIS 30 September/[10 October] 1662 Transmission: C Letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 130ar-130bv (f. 130av and f. 130br blank). The address is in another hand.

Sir

Your undertakeinge of the trouble of my soone153 I must needs ackolidge154 as a great obligation, since you make such waighty obrecti(ons)155 as the lose of your time and study besides the incovenence of your attendance. But the high caracter I have receaved of you, from severall persons that amonght many of your comendations this is not the smalest that you make a consience of what you undertake hath maid me more importunate 153 Your undertakeing . . . soone: i.e. Wallis's attempt to teach Anne Wharton's deafmute son Alexander Popham how to speak. Alexander was an offspring of her previous marriage to Edward Popham (16107-51). Cf. SCRIBA, 'Autobiography', 42. 154 ackolidge: i.e. acknowledge. 155 obrecti(ons): i.e. objections.

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33. BROWNE to CHYLINSKI, 3/[13] October 1662

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with you then I should have beene with any one else the sume of mony that you name for his di(et) and his mans with the reward for your instructions and care of him I suppos to be a 100 pound a yeare; I confese it was 40 pound more then was given Dr Holder156 upon the same account; I shall desire you would eithere send an abaitment of the sume by this messenger or refere it to Dr Bathurst157 who will be as impartiall to you as to her as is You sarvant Anne wharton

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Sepe 30. If you thinke fitt I shall waite one you the beginning of the next weeke

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[130bv]

These for the Reverend Dr Wallis in Catte-Street in Oxford.

33.

RICHARD BROWNE to SAMUEL BOGUSLAV CHYLINSKI [London], 3/[13] October 1662 Transmission:

c Copy of missing letter sent in Chyliiiski's hand: OXFORD Bodleian Library MS. Add. D. 105, f. 12V (noted on CHYLINSKI-WALLIS 4/[14].X.1662 and superscribed with 'Copia Literarum Innotescentiarum'). 4 account; (1) but becos (2) I 156

Holder: i.e. William Holder (1616-98), divine and writer, fellow of the Royal Society (from January 1661 onwards), DNB. He published Elements of Speech: An Essay of Inquiry into the Natural Production of Letters; together with an Appendix to instruct Persons Deaf and Dumb, London 1669, and later accused Wallis and Oldenburg of failing to do justice to his own attempt at teaching speech to Alexander Popham. Wallis, according to whom Holder's efforts were largely unsuccessfull, wrote his Defence of the Royal Society, and the Philosophical Transactions, Particularly those of July, 1670. In Answer to the Cavils of Dr. William, Holder, London 1678, in reply. 157 Bathurst: i.e. Ralph Bathurst (1620-1704) physician and divine. Fellow and (from 1664 onwards) President of Trinity College, Oxford. Elected fellow of the Royal Society in August 1663. DNB. 74

34. CHYLINSKI to BROWNE, 4/[14] October 1662 Answered by: CHYLINSKI-BROWNE 4/[14].X.1662.

Dominus Chylinski. Translator of the Lithuanian Bible is to attend the Lords of the Provie Councell Satterday the 4.th of October at the Councell Chamber at Whitehall at 3. of the clocke after dinner. 3d October 1662.

Richard Browne.

34. SAMUEL BOGUSLAV CHYLINSKI to RICHARD BROWNE [London], 4/[14] October 1662 Transmission:

C Copy of missing letter sent in unknown hand: OXFORD Bodleian Library MS. Add. D. 105, f. 13r (written on CHYLINSKI-WALLIS 4/[14].X.1662 and superscribed by Chylinski with 'Copia Responsi mei'). The signature is in Chyliriski's hand. Reply to: BaowNE-CHYLiNSKi 3/[13].X.1662. Worthy Sir,

5

Receiving a note158 or surnons this very morning from your Worship; whereby I was required to give my attendance on the Lords of his Majesties Privy Councill this present Satterday afternoone. And not being fitted for the same: Humbly make bold to crave and intreate the suspending of my said Attendance, untill Satterday next, being the Eleventh day of this instant October: Not only for that my warning was not a full day; But also in that I had not the benefit of advising with such of my friends as I hold both grave and fit to direct in so weighty an Affair. This being accquainted and made known to his Majesties said Lords, and the suspending thereof endeavoured by your Worship I crave leave to subscribe my self, October the 4th 1662.

Your Worships most faithfull servant Samuel Boguslaus Chylinsky. Translator of the Lithuanian Bible.

158

note: i.e. BROWNE-CHYLINSKI 3/[13].X.1662.

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35. CHYLINSKI to WALLIS, 4/[14] October 1662

35. SAMUEL BOGUSLAV CHYLINSKI to WALLIS London, 4/[14] October 1662 Transmission:

C Letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 12r-13v. On lower half off. 12V Chyliriski's copy of BROWNE-CHYLINSKI 3/[13].X.1662, and on f. 13r copy in unknown hand of CHYLINSKI-BROWNE 4/[14].X.1662.

Reverendissime in Christo Jesu vir, Fautor gratiosissime.

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Ex quo R. T. vidi, coram consilio S. R. Majestatis adhuc non comparui, neque enim consultum fuit e re mea et operis mei, quin illud fecissem hactenus, nisi immediate ab ipsomet Consilio ad praeteritam Petitionem meam prosequendam fuissem vocatus, a quo quia nonnisi hodie fui accitus comparere, ad probandum, quod nuper in Petitis meis, contra Adversarium meum159, proposui. Quia autem illud idonee absque interventu eorum, qui me noverant sex ab his annis et ultra in Anglia, praestare non possum, (non quod se res secius habeat,) sed quia ad negotium intelligendum multuni conducet operis quod ago, et Serenissimae R. Majestatis Consilii Honoratissimi, si aliquem habeam, qui interpretem meum agat. Distuli hodiernam comparitionem ad proximum diem Sabbathi160, eo fine potissimum, ut oculatum testem haberem, qui melius eis negotium meum exponat, quam ego ipsemet, aut adversarius meus Delegatus. Ideoque R. T. per omnia sacra, per amorem in me, et opus meum, rogatum volo, ut in gratiam mei illud oneris in se suscipere velit, venire nempe circa illud tempus et mihi in laboranti negotio assistere, quodsi confio, quod expeto R. T. ut proram et puppim negotii et fortunae meae agnoscam non solum ipse, sed etiam sequentia tempora, ni illud fiat, hoc sciat pro certo, ultimum incurret totum hoc Biblicum negotium, pericujlum, quod quia nee [12-] ullus pius, nee R. T. optaret et aptat, ideo rogo ut illam molestiam in se insumat et sumptus, quo me et opus meum eripiat ex charybdi. Hoc cum a Te Pientissimo Viro DEi rogo, ni hoc felici omine finio, DEum inditurum et R T. animum et Consilio S. R. Majestatis saluberrimum Consilium, ne 7 ad (1) respondendum (2) probandum 159 160

Adversarium meum: i.e. Jan Krainski. Cf. WALLIS-BOYLE 14/[24].III.1661/2 (ii). Distuli . . . Sabbathi: see CHYLINSKI-BROWNE 4/[14].X.1662 76

36. WALLIS to DILLINGHAM, 21/[31] October 1662 opus intercidat, quod concernit promotionem gloriae DEi ex Regni in his terris, Cujus providentiae nutu Te committo et maneo Dabam Londini 4. Sbris. 1662

Reverendae Tuae dignitatis humillimus cultor cum opere meo periclitante S. Boguslaus Chylinski.

5

[13V] These

To the Right Reverend Dr John Wallis. in Oxford. Whith care and speed I pray.

10

36.

WALLIS to THEOPHILUS DILLINGHAM 21/[31] October 1662 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by DlLLINGHAM-WALLIS 27.X/[6. XI].1662. In this letter, Wallis requested that Dillingham supply information on cases where privileges of the University of Cambridge were allowed in the Exchequer. Such cases would serve as precedents to which the University of Oxford could also refer.

37. THEOPHILUS DILLINGHAM to WALLIS Cambridge, 27 October/[6 November] 1662 Transmission:

C Letter sent: OXFORD University Archives WP7/16/1, f. 71br-71cv (f. 71bv and 71cr blank). On bottom off. 71br Wallis has noted: 'This is Dr Dillingham (Vice Chancellor 34. (1) 7bris (2) Sbris.

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37. DlLLlNGHAM to WALLIS, 27 October/[6 November] 1662 of Cambridge) his answere to one of mine desiring an account of what precedents they had, |had del] of Privileges allowed in Exchequer; it being denyed us in the case of Mr Wilkins161 the Bedle, for want of a precedent.', and underneath this: 'We had it afterwards allowed in Lichfields162 case, upon a solemn argumen: Judge Hales163 being then Lord Chief Baron.' Reply to: WALLis-DiLLiNGHAM 21/[31].X.1662.

Sir

I received yours164 of the 21 instant in answer to which I have sent you the inclosed paper, which is as much as I could in so short a time finde out for your purpose. I conceive the words of our charters will as well 5 beare us out against the proceeding of the Exchequer as of any other of the Westminster Courts. Nullus judex se intromittat justiciarii ad placita, et alii judices quicunque me thinks should serve for all. Sir though I am very full of business at this time yet I thought my self obliged to make some speedy returne to those than last bee: as I 10 remember I received letters from you about half a yeare since, which I could not then presently usherre an answer to, and afterwards thought it too late. I now crave your pardon for that neglect, & remaine Your loving friend to serve you Theoph: Dillingham 15

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from Clare hall in Camb. Oct: 27. 62. [71cv]

To his much respected friend Dr John Wallis Geometry professor in the University of Oxford these with speed

5 as Iwell as del. of 161

Wilkins: i.e. Timothy Wilkins (1617-71), esquire bedell of divinity at Oxford (from 1657 onwards) and brother of John Wilkins, q.v. 162 Lichfields: i.e. Solodell Lichfield (d. 1671), yeoman bedell of law at Oxford (from 1635 onwards) and esquire bedell of arts (from 1666 onwards). 163 Hales: i.e. Sir Matthew Hale (1609-76), justice of common pleas (1654), lord chief baron of the Exchequer (1660), lord chief justice of the King's Bench (1671), DNB. 164 yours: i.e. WALLis-DiLLiNGHAM 21/[31].X.1662.

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38. HOBBES to WALLIS, end of 1662 38.

THOMAS HOBBES to WALLIS end of 1662 Transmission:

El First edition: [HOBBES], Mr Hobbs considered in his loyalty, religion, reputation, and manners. By way of a letter to Dr Wallis, London 1662. E2 Second edition: HOBBES, English Works IV, 413-40. Written in the form of an epistolary tract addressed to Wallis, Mr Hobbs considered replies to Hobbius Heauton-timorumenos, in which Wallis had accused Hobbes of having written his Leviathan in support of Oliver Cromwell.

39. WALLIS to JOHANNES HEVELIUS Oxford, 9/[19] February 1662/3 Transmission:

W Letter sent: PARIS Bibliotheque Nationale Nouv. acq. latines 1641, f. 110r (verso blank). Answered by: HEVELIUS-WALLIS [25.XII.1663J/4.I.1664.

Oxonii Febr. 9. 1662./3. Praeclarissime Vir, Accepi hodie, dono tuo, gratissimum opus165, De Mercurio in Sole viso: et grates rependo nostras. Et quidem eo magis gratulor successus tuos166, quod Tibi faverit hac in re clementius caelum quam Nostratibus. Utut

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opus: i.e. HEVELIUS, Mercurius in Sole visus Gedani, Anno Christiana MDCLXI. d. III. Maj, St. n. Cum aliis quibusdam rerum Coelestium observationibus, rarisque phaenomenis. Cui annexa est Venus in Sole pariter visa, Anno 1639, d. 24 Nov. St. V. Liverpoliae, A Jeremia Horoxio, nunc primum edita, notisque illustrata. Quibus accedit succincta novae Historiola illius, ac mime Stellae in collo Ceti, certis anni temporibus dare admodum affulgentis, rursus omnino evanescentis. Nee non Genuina Delineatio, Paraselenarum et Paraliorum quorundam rarissimorum, Danzig 1662. 166 successus tuos: cf. HUYGENS, CEuvres completes XXI, 872. 79

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40. WALLIS to BOYLE, 27 March/[6 April] 1663

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enim ex Nostris aliqui conati fuerint id ipsum observare; vix tamen, ob caelum nubilum, id datum erat. Sub hora tamen prima post meridiem eodem die, (nempe Apr. 23. 1661, stylo nostro; qui vobis est Maii 3.) D. Hugenius (qui turn Londini fuit167) et cum illo aliqui, jam turn Mercurium Solis discum intrasse notaverunt168; sed, supervenientibus nubibus, statim occultari et Mercurium et Solem etiam. (Quod eo ipso tempore contingebat, quo, in celeberrima Magnatum corona, Corona Regia Serenissimi Caroli II. capiti turn primum imponebatur.169) Sed et horrenda Tonitrua vesperi sequebantur. Verum non dubito, quin ab ipso D. Hugenio Tibi distinctius, quid ipse observaverit, indicatum fuerit. Numqui autem ex Nostratibus alii quam qui cum illo erant, observaverint, non audio. Video interim et Horroxii170 nostri Veneris observationem, Tuae Mercurii subjunctam esse. Spero hinc utriusque Syderis motum accuratius imposterum restitutum iri. Deum Opt. Max. precor, ut longam tibi et vitam et valetudinem concedat; quo diu, bono publico, his invigilare possis observationibus. Vale. Tui observantissimus, Job. Wallis.

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WALLIS to ROBERT BOYLE Oxford, 27 March/[6 April] 1663 Transmission:

E1 First edition of missing letter sent: BOYLE, Works (1744), V, 512.—printed: BOYLE, Correspondence II, pp. 71-2. E2 Second edition of missing letter sent: BOYLE, Works (1772), VI, 455-6 (our source).

6 ipso add. 15-18 his invigilare ... Joh. Wallis. at 9(f in left margin 16r

Londini fuit: Huygens was in London from 23 March 1660/1 until about 17 May 1661. 168 notaverunt: Huygens was with the instrument-maker Richard Reeves (fl. 1641-79) when he made the observation of the transit of Mercury across the sun. See HUYGENS, (Euvres completes XV, 72-3, XVI, 307, and XXII, 575. 169 imponebatur: On the coronation of Charles II on 23 April 1661 see The Diary of John Evelyn, ed. DE BEER, III, 278-84; HUYGENS, (Euvres completes XXII, 575. lr °Horroxii: i.e. the astronomer Jeremiah Horrox (16177-41), DNB. 80

40. WALLIS to BOYLE, 27 March/[6 April] 1663 Oxon, March 27, 1663. Sir,

The inclosed171 is in obedience to your commands laid upon me, when I waited on you at London.172 If it be too large, you may extract out of it as short a one as you please; and if it may seem in ought too short, there is scope of enlarging it as far as you shall think fit. The design is so ordered as to obviate the inconveniencies, and to reserve as much of convenience as well may be. The other command173, which I received in order to Dr. Pocock174,1 have endeavoured to observe so far as was in me. I acquainted him with your desire; but he tells me, that to give account of all the longitudes and latitudes in Abulfeda, is, in a manner, to transcribe the whole book; for it contains little else but the longitude and latitude of places, with some very brief descriptions of them in two or three lines, and not digested into distinct tables, but to be collected out of the text. But he tells me, that Mr. Clark175 is designing somewhat out of him and other geographers compared; which perhaps may better satisfy the desire of the gentleman, than a bare account of Abulfeda alone. I hoped to have given a speedier account of both, had not somewhat else so often interposed. Your goodness, I trust, will excuse the delay, and accept the endeavours of,

Sir, your honour's very humble servant, John Wallis. Since I had written thus much, Mr. Hyde176 (the under library1

The inclosed: The enclosure is missing. when .. . London: Wallis was in London during the first half of March. 173 other command: Apparently, Wallis was to request of Edward Pococke that he provide an account of the longitudes and latitudes in Abulfeda's Geography. 174 Pocock: i.e. Edward Pococke (1604-91), regius professor of Hebrew and Laudian professor of Arabic at Oxford. DNB. 175 Clark: i.e. Samuel Clarke (1625-69), orientalist, M.A. in 1648 at Merton College, Oxford, since 1658 architypographus and upper bedell of the civil law at the University of Oxford. His surviving papers indicate that he was contemplating a new edition of Abulfeda's Geography. See WOOD, Athenae Oxonienses III, 882-5; TYACKE (ed.), History of the University of Oxford IV, 490; DNB. 176 Hyde: i.e. Thomas Hyde (1636-1703), since 1659 underkeeper and from 1665 to 1701 chief librarian of the Bodleian library, Laudian professor of Arabic (from 1691 onwards) and regius professor of Hebrew (from 1697 onwards). In 1665, he published an edition of Ulug Beg's catalogue of fixed stars. DNB. m

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keeper) tells me, that if you please, he will collect out of Abulfeda the longitudes and latitudes, and| transmit to you (I suppose he will expect [456] some gratuity for his pains.) He hath lately, for the bishop of Exeter177, transcribed out of Uleg Beig1^8 the longitude and latitude of all the fixed stars, according to his observations. If you think it tanti, and gave me such order, I will desire him to undertake it.

41. WALLIS to JOHANNES HEVELIUS Oxford, 30 March/[9 April] 1663 Transmission:

W1 Draft of letter sent: LONDON Royal Society Eaxly Letters Wl, No. 2, 4 pp. At top right of p. 1: 'read May 4: 64. entered LB. 1. 131.' At bottom of p. 4 Oldenburg has noted at 90°: 'A Copy of Dr Wallis's letter sent to M. Hevelius and mention'd in the letter next foregoing, about the New Staxr in Cete'. W'2 Letter sent (?): OXFORD Bodleian Library MS. Add. D. 105, f. 14r-15v (our source). w1 Copy of W1: LONDON Royal Society Letter Book Original 1, pp. 131-2. w2 LONDON Royal Society Letter Book Copy 1, pp. 153-4. Answered by: HEVELius-WALLis [25.XII.1663]/4.I.1664. A year after the present letter was sent, Wallis forwarded to Oldenburg the original draft as an enclosure to WALLIS-OLDENBURG 30!V/[10.V].1664, after the Royal Society had expressed an interest in John Palmer's observations of the Star of Cetus. This interest was apparently generated by Wallis's letter to Hevelius of 5/[15].IV.1664, which had been read at the meeting of the Society on 13 April 1664. The endorsement on p. 1 of the draft W1 is possibly incorrect: According to Birch, the letter read at the meeting on 4 May 1664 was WALLIS-OLDENBURG 30.IV/[10.V].1664 (BIRCH, History of the Royal Society I, 422). However, Birch is clearly mistaken when he dates the present letter 30 March 1664, and indicates that it was read at the meeting of the Society on 13 April 1664. As the summary he gives makes clear, the letter read at that meeting was actually WALLIS-HEVELIUS 5/[15].IV.1664 (BIRCH, History of the Royal Society I, 414). W2, which is a fair copy of the draft W1, has been folded and sealed; it is possibly the letter actually sent or may have been substituted by a second letter incorporating new or other information. lrr

bishop of Exeter: i.e. Seth Ward. Uleg Beig: i.e. Ulug Beg (1394-1449), Usbek astronomer, grandson of Tamerlane and himself Great Khan from 1447 to 1449. In 1420, he founded an observatory near Samarkand. 178

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41. WALLIS to HEVELIUS, 30 March/[9 April] 1663 Praeclarissime Vir, Post perlustratum, quern ad me misisti, Tuum, de Mercurio a Te in Sole conspecto, Tractatum179 perquam gratum; una cum eis quae subjunxisti; (inter quae Horroxii nostratis de Venere tractatum180, quern nimium diu publico negatum saepius conquestus sum, tandem comparere gaudeo:) Duo saltern sunt quae Te monendum duxi. Alterum, de mira Ceti stella; de Fixis reliquis, alterum. Nempe Stellam illam, ab Anno Dom. 1639, et deinceps; a Nostratibus (quod Te nesciisse video) multoties observatam fuisse; nunc comparere quidem, nunc disparere. Et primo quidem (quantum scio) a D. Joh. Palmero181 (Viro doctissimo, et Astronomicis Observationibus, ut per alia licet negotia, intento; munere tamen Ecclesiastico occupato:) deinde ab aliis; praesertim D. Samuele Fostero182 Astronomiae olim Professore in Collegio Greshamensi Londini; qui me, dum vixit, ea de re compellavit aliquoties jam ante multos annos. Quid autem Palmerus obiter, in libro183 ab ipso ante aliquot annos edito, de hac stella monuit184; exscriptum huic subjungam. Sed et propediem missurus sum ad ilium, tractatum quern a Te accepi tuum, ut, quum ilium perlegerit, possit, si videbitur, fusius, quae observavit, ipse aliquando impertire. Stellarum vero Fixarum cum videam Te Longitudines et Latitudines denuo instaurandas tantum non polliceri; Monendum duxi, extare apud nos, in Manuscriptis Persicis, Tabulas Astronomicas, ab Uleg-Beig, Rege

10 Joh. add. 13 Samuele add. 13 olim add. 15 jam ante multos annos add. 19 ipse Tibi aliquando Wl 179

Tractatum: i.e. HEVELIUS, Mercurius in Sole visus, Danzig 1662. See WALLISHEVELIUS 9/[19].II.1662/3. 180 tractatum: i.e. Horrox's Venus in Sole visa, which Hevelius had printed in his Mercurius in Sole visus. 181 Palmero: i.e. John Palmer (1612-79), rector of Ecton in Northamptonshire 1641-79, archdeacon of Northampton 1665-79, pupil of Samuel Foster; conducted astronomical observations. 182 Fostero: i.e. Samuel Foster (c.1600-52), mathematician, professor of astronomy at Gresham College in 1637 and from 1641 onwards, DNB. 183 libro: i.e. PALMER, The Catholique planisphaer. Which Mr. Blagrave calleth the mathematical Jewel ..., London 1658. 184 monuit: i.e. PALMER, Catholique planisphaer, 100-1. 83

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olim et Astronomo insigni, institutas; et, inter illas, Fixarum loca, prout ab ipso suisque qui sibi aderant Astronomis fuerant observata; in urbe Samarkanda, quae turn sibi turn avo proavove suo Tamerlano natalis erat: Neque enim Ptolemaei aliorumve Tabulis (quas a vero non raro deviare notaverat) satis fidendum esse duxit. Aptantur illae Fixarum Tabulae, ad initium Anni Hegirae 841; hoc est, ad mensem Julium Anni Aerae Christianae 1437. Quas quia fieri potest ut non videris, curabo, si id desideraveris, ex Persico in Latinum versas, ad Te transmitti185. Interim Vale, Praeclarissime Vir, studiaque, quod facis, Astronomica in utilitatem publicam promovere perge. Oxonii; Martii 30. 1663.

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Tui observantissimus, Joh. Wallis.

In libro186 quodam, Anglico sermone, a Dno. Johanni Palmer, Ecclesiasta [141 Ectonensi in agro Northamptoniae Angliae, edito Londini, anno 1658; cui Titulus; Planisphaerium Catholicum, (or Mr Blagrave's Mathematical Jewel, described, &c;) lib. 4. cap. 34. pag. 100, 101. inter alia haec occurrunt; quae latine versa sic sonant. Novembris 14, 1639 (stilo veteri;) Stellam observavi, tertiae Magnitudinis in Corde Ceti: Quam inde stellam non vulgarem esse cognovi, quod earn ante nunquam conspexeram, nee illam potui in Tabulis Ptolemaei, Tichonis, aut aliorum invenire. Erat ejus Ascensio Recta, grad. 30, 13'. Declinatio Australis gr. 4. 50'. prout observavi modo post tradendo, cap. 44. Quum illam primo observaveram, anno dicto, 1639; nee ullam potui illius mentionem invenire in Tabulis Ptolemaei,

1 olim in India et W1 3 Samarkanda, (1) qui (2) quae 4 Tabulis ( (1) quae (2) quas 20 nunquam notaveram, W1 185si ... transmitti: Hevelius did in fact request a copy of Ulug Beg's catalogue of fixed stars; this was sent by Oldenburg together with a diploma testifying his election to the Royal Society and a letter from Wallis (WALLIS-HEVELIUS 5/[15].IV.1664) on 21 May 1664. See OLDENBURG-HEVELIUS 27.V/[6.VI].1664; OLDENBURG, Correspondence II, 204-5. 186 libro: i.e. PALMER, Catholique planisphaer.

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41. WALLIS to HEVELIUS, 30 March/[9 April] 1663 Copernici, Tichonis, aut Magini188; hoc indicabam amicis meis non vulgaribus, Doctor! Johanni Twisden189; tune in Kantio; et Do. Samueli Foster, Astronomiae Professori in Collegio Greshamensi, turn Londini; qui eandem quam ego, quoad stellae locum, observationem fecerunt: Et convenimus omnes, lucem ipsius turn continuo crescere, ut Mense Decembri 1639, tertiam magnitudinem superaverit; nullam autem sensibilem habere parallaxin. Cumque in animo habuerim, brevem aliquam ejus insinuationem, Latino sermone, in lucem mittere; quo Astronomi transmarini, juxta atque nos, illius observationi attenderent: Literis indicavit D. Foster, se stellam hanc in imaginibus Bayeri, Anno 1616 editis190, depictam invenisse. Anno sequente, 1640, in manus meas pervenit, a Dno. Twysden transmissus, Tractatus de Stella hac turn nuper editus a Phocylide191 quodam, Logices Professore Franaquerae; cujus observationes nostris fuerunt consentaneae. Existimavit autem ille, Stellam hanc ex Ecclipsi niagna Lunari, quae Decemb. 10. 1638. in anterior! pede H contigerat, originem habuisse: Verum in hoc ab illo dissentientes fuimus. Sententiam ejus videre est, in libri sui pag. 197. Haec stella, apparet saepe, iterumque disparet. Est aliquando tertiae magnitudinis; aliquando, quartae. Saepe illam observavi in hemisphaerio Orientali; in Occidentali raro. Dis1 Copernici, Stadii187, Tichonis Wl 1 aut (1) aliorum invenire; (2) Magini; 4 quam ego, (1) observati breaks off (2) quoad 9 ejusdem insinuationem Wl 12 indicabat Wl 187

Stadii: i.e. Jan Stade (1527-79), Belgian astronomer and mathematician. Magini: i.e. Giovanni Antonio Magini (1555-1617), Italian astronomer and mathematician. 189 Twisden: i.e. John Twysden (1607-88), physician, fellow of the College of Physicians in London from 1664 onwards. In 1659 he published some of the papers left by Samuel Foster as Miscellaries or Mathematical Lucubrations. DNB. 190 imaginibus Bayeri, Anno 1616 editis: possibly a reference to the Uranometria of the German lawyer and astronomer Johann Bayer (1572-1625), which first appeared in Augsburg in 1603. 191 Tractatus . . . a Phocylide: i.e. Jan Fokkes HOLWARDA (Johannes Phocylides) (1618-51), ITaz/ (which rank of additions is so easyly made, as that after two or three numbers, it runs on as fast as you can write it; & the same order of 19 places, is in all cases the same, onely beginning at severall places; & so being once done serves for all:) 4. Now because I do not here meet with 13, till the 19th place; I conclude that 13 in the Moons Cycle, doth not concur with 22 in the Suns, till the 19th Revolution; that is, not till after 18 intire revolutions (or 28 x 18 = 504 years,) & 22 yeares onward of the 19th Revolution. For which Reason, in my second Precept, I bid multiply 28 by 18, & then adde 22: which makes, 526; which is therefore the first year of the Julian

I mutandis, (1) fit (2) be 3 year (1) of (2) either 7 the (1) Number (2) Cycle 73, (1) (which is in our case is 13.) fe (2) not 13, as in our case: II number add. 13 And (1) therefore (2) therefore 14 the (1) next (2) second 15 the (1) moo breaks off (2) Moones 25 after add. 271(1) ad breaks off (2) bid

362

163. WALLIS to OLDENBURG, 30 November/flO December] 1667 Period, when these two Numbers (O 22, S> 13,) can meet. And it is the number of the Dionysian Period (compounded of the Suns & Moons cycle) in which it allways so meets. 5. Now if, at the same time, the Indiction had been 3 (as is proposed) that year had answered the Case: But, dividing 526 by 15, I find (by the [2] Remainder) the Indiction then to be 1. And therefore I am to seek for| some other such conjunction of O 22, S> 13, which may concur with the Indiction 3. 6. Now because this doth not come again till after 532 years (which is the Revolution of the Dionysian Period, compounded of the two Cycles of Sun, & moon; that is 532 — 28 x 18,) which contains 35 intire revolutions of the Cycle of Indictions (because of 15 x 35 = 525) & 7 years onward: therefore, the next time these meet, the Indiction will be promoted by 7. 7. To find therefore, in what Revolution of the Dionisian period, the number 526 (in which O 22, & 3) 13, do meet) will meet 3 for the Indiction; To the Number 1, (which I find to agree with 526 in the first Dionysian Period,) I adde continually 7, (still casting away 15 as there is occasion,) And so I find, that to 526 in the second revolution, answeres 8; in the third, 15; & so onward. For which Reason I bid, in my Third precept, to make this continuall addition of 7 &c; till I come to 3. 1.8.15.7.14.6.13.5.12.4.11.3. Cycl: Indict. (which is so easyly continued as you see, & being once continued to 15 places, serves for all cases, onely beginning as there is occasion, at several parts of it.) 8. Now because I do not here find 3, till the 12th place; I conclude that the Indiction 3, doth not concur with the Dionysian 526, (wherein O 22, & 3 13, do allways & onely meet,) till the 12th Revolution of the Dionysian period: Therefore according to my 4th Precept; for 11 intire

1-3 And it is ... so meets, add. 7 may (1) be accomodate to (2) concur with 9 come add. 12 onward: |And del] therefore 13 be (1) forward (2) promoted 15 526 ( (1) when (2) in which 18 that (1) with (2) to 24-25 of it.) (1) Whe breaks off (2) \\ 8. Now 27 fe onely add. 28 period: (1) There (2) Therefore

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163. WALLIS to OLDENBURG, 30 November/[10 December] 1667

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revolutions, I multiply 532 (the Revolution of that period) by 11; & to the product 5852 (= 532 x 11) I adde 526 (the number in the 12th period current,) which gives the number 6378, the number current in the Julian period. So that the whole operation (the numbers given being O 22, S> 13, Indict. 3.) is but this Cycl. O, 22 (= 19 + 3.) 3.12.2.11.1.10.19.9.18.8.17.7.16.6.15.5.14.4.13. Cycl. 5) . 28 x 18 = 504 +22 11.8.15.7.14.6.13.5.12.4.11.3. Indict. Per: Dionys: 526 532 x 11 = 5852 +526 6378 Period. Jul. Where though I have been fain, in many words to set down the Notion, the operation is short & easy. And I know not well how it can be made more facile. The like methods (mutatis mutandis) may be assigned; in what order soever you place the three Cycles: whether you begin with the Sunnes, & then the Moones, & lastly the Indictions, (as here:) Or first the Suns, then the Indictions, then the Moons: Or first the Moons, then the Suns, then the Indictions: Or first the Moons, then the Indictions, then the Suns: Or, first the Indictions, then the Suns, then the Moons: Or (lastly) first the Indictions, then the Moons, then the Suns. But of this sufficient. | [3] The Construction of Slusius729, for resolving a Biquadratick Equation by a Parabola & a Circle; I have examined: & find it to effect what it undertakes. The methode from Paris730, for depressing a Biquadratick Equation,

18 order (1) the (2) soever 27 I have (1) & (2) examined 27 it (1) affirmed. (2) undertakes. 729

Construction of Slusius: see SLUSE-OLDENBURG [14J/24.XI.1667; OLDENBURG, Correspondence III, 594-6, 594-5. 730 methode from Paris: possibly the Regula nova Deprimendi aequationes quatuor Dimensionum ad Sum gradum (OLDENBURG, Correspondence III, 433-4), which had been sent to Oldenburg from Paris in summer 1667. 364

164. OLDENBURG to WALLIS, 10/[20] December 1667 to a Cubick: I have not yet considered, sufficiently to give you an account of it by this Post. Resting Yours &c. John Wallis. [4]

These For my honoured freind Mr Henry Oldenburg; at his house in the Old Pelmell near St James's London.

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164. HENRY OLDENBURG to WALLIS London, 10/[20] December 1667 Transmission:

C Letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 25r-26v (f. 26r blank) (our source). Postmark on f. 26V: 'DE/10'.—printed: OLDENBURG, Correspondence IV, 23-4. Answered by: WALLIS-OLDENBURG 13/[23].XIL1667.

London Decemb. 10. 67. Sir,

In stead of giving you by this MyLord Brounkers thoughts upon your last (which I could not yet doe, he being still absent731) I shall acquaint you with two particulars, you will, I think, be well pleased with: One is, the notice of a Book732, lately publisht at Paris, entitled,

731

he being still absent: Brouncker was at the time staying in Chatham. His absence from London was also the reason why Oldenburg had not written a week earlier. Cf. OLDENBURG-BOYLE 3/[13].XII.1667; OLDENBURG, Correspondence IV, 4-7, 7. 732 Book: i.e. DULAURENS, Specimina mathematica duobus libris comprehensa, Paris 1667. After receiving a copy in March 1668, Wallis commented on this book in WALLISOLDENBURG 30.III/[9.IV].1668.

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Francisci Du Laurens Specimina Mathematica, duobus libris comprehensa, quorum Primus, Syntheticus, agit de Genuinis Matheseos principiis in genere, in specie autem de veris Geometriae Elementis hucusque nondum traditis. Secundus, Analyticus, de methodo Compositionis atque Resolutionis fuse disserit, et multa nova complectitur, quae subtilissimam Analyseos artem mirum in modum promovent. I am promised733 to have this book sent me by the first opportunity, together with two others, 1. De vi Percussionis734, Borelli Itali. 2. De motu Musculorum735, Stenonis. The other particular is a Probleme, sent from the Author736 of this Book, which he recommends not for the difficulty of its Solution, but for the Inferences, that may be thence made, of some very fine proprieties of the Circle,| by him affirmed not to have been discovered hitherto. [251 Problema.

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Numero datis, circuli cbd radio eg, et inscripta bd extra circulum porrecta ad libitum, ita tamen, ut ad ductam cf acta parallela by, circuli circumferentiam secet in puncto y; Verum (non autem vero proximum, quod quidem per Sinuum Canones facile fieri potest) valorem exhibere lineae by.

6 fuse (1) diffe breaks off (2) disserit, 9 1. add.

12 Solution, (1) as (2) but for 14 by him add. 733

I am promised: evidently by Justel; see JusTEL-OLDENBURG [6J/16.XI.1667 (OLDENBURG, Correspondence III, 577-9, 578). Although Justel had dispatched two copies of this book not later than December 1667 (cf. JUSTEL-OLDENBURG [25.XII.1667J/4.I.1668; OLDENBURG, Correspondence IV, 84-5), Oldenburg did not receive them until February 1668; cf. OLDENBURG-SLUSE 26.II/[5.III].1667/8 (OLDENBURG, Correspondence IV, 209-10, 209). 734 De vi Percussionis: i.e. BORELLI, De vi percussionis liber, Bologna 1667. 735 De motu Musculorum: i.e. STENSEN, Elementorum myologiae specimen, sen musculi descriptio geometrica, Florence 1667. 736 Author: i.e. Francois Dulaurens. Oldenburg read this problem at the meeting of the Royal Society on 12 December 1667, where the problem 'was ordered to be communicated for solution to Mr. Collins'; BIRCH, History of the Royal Society II, 226. 366

164. OLDENBURG to WALLIS, 10/[20] December 1667

I pray, Sir, if you have leasure, consider it, and what is intimated above of the proprieties of the Circle to be drawn from thence. I am now printing, in the Transactions of this month, Mr John Collin's method and demonstration737 for finding the Julian Period, which a pretty while agoe he hath been ready with, and mentioned at the Society with applause, altogether differing of yours, I think. At the latter end of this, or the beginning of the next week, the book will be printed off; and your judgement, upon that part especially, is also desired by Sir Your faithfull humble servant H. O.

[26V] For his much honor'd friend Dr John Wallis Savilian Professor of Geometry in Oxford

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9 part (1) is (2) especially 73r

Collin's method and demonstration: This method was based on de Billy's rule pubslihed in Journal des Scavans No. 36 (6 September 1666), an account of which was printed in Philosophical Transactions No. 18 (22 October 1666); see WALLISOLDENBURG 16/[26].XL1667. Collins's method and its demonstration were printed in Philosophical Transactions No. 30 (9 December 1667), 568-75 ('A Method For finding the Number of the Julian Period for any Year assign'd, the Number of the Cycle of the Sun, the Cycle of the Moon, and of the Indictions for the same year, being given; together with the Demonstration of that Method'). 367

165. WALLIS to OLDENBURG, 13/[23] December 1667

165.

WALLIS to HENRY OLDENBURG Oxford, 13/[23] December 1667 Transmission:

W1 Draft of letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 28r-28v. On f. 28V three additional figures, not inserted in the letter sent. W2 Letter sent (fair copy of draft, of which some passages and longer marginal note are omitted): LONDON Royal Society Early Letters Wl, No. 35, 2 pp. (our source). At top of p. 2 above address at 90° note by Oldenburg (apparently arising from a meeting with Brouncker and written in preparation of Oldenburg's reply): 'The sense is That DF being ad libitum, is to be chosen of such a (./) length, (2) quantity, that BY may be rational. || As to Collins, he makes it a rational quantity in this instance, but hath not given a rule for it, for A (a) may (b) or GF may (ao) be (66) prove irrational by his methode. 11 Give always such a progression of numbers, that thereby lengthening the corde the parallel-line BY, may prove a rational quantity.' Beneath address in Oldenburg's hand: 'Rec. Dec. 16. 1667.' Postmark on p. 2: 'DE/16'.—printed: OLDENBURG, Correspondence IV, 36-8 (Latin original); 38-9 (English translation). W3 Enlarged and elaborated version of the mathematical section of the letter: OXFORD Bodleian Library MS. Add. D. 105, f. 27r-27v (our source). E First edition of letter sent: WALLIS, Opera mathematica II, 597-8. Reply to: OLDENBURG-WALLis 10/[20].XII.1667. Answered by: OLDENBURG-WALLIS 24.XII.1667/[3.I.1668]. The text of W3 represents a revised and extended version of the solution to Dulaurens's problem, as it was contained in the letter to Oldenburg. It takes up the issue raised by Brouncker to which Oldenburg refers in his reply and which he also notes in the endorsement. It can be assumed that Wallis wrote most if not all of this revised version after having received that letter. Cf. Wallis's Solution to Dulaurens's Problem 8/[18].II. 1667/8, which also discussed the case of rationality of BY taking up the reasoning of W3'.

(W)2 Oxford Decemb. 13. 1667.

Sir, The Books mentioned in yours738 of Dec. 10. I shall be willing inough to 738

yours: i.e. OLDENBURG-WALLIS 10/[20].XIL1667. 368

165. WALLIS to OLDENBURG, 13/[23] December 1667 see. Especially if the first739 of them do answere the title to which it doth pretend. Problema vero quod spectat, ibidem his verbis propositum, Problema. Numero datis, Circuli CBD radio CG, et inscripta BD, extra circulum porrecta ad libitum, ita tamen, ut ad ductam CF acta parallela BY, circuli circumferentiam secet in puncto Y; Verum (non autem vero proximum, quod quidem per Sinuum Canones facile fieri potest,) valorem exhibere lineae BY.

Hoc est (si ego ipsius mentem satis assequor;) Datis magnitudine (numeris designata) tribus hisce rectis; Nempe, Circuli BDG, turn radio CG, turn inscripta BD, turn hujus (ad libitum protractae) continuatione DF; cui occurrat in F, recta a Centro ducta CF; et huic parallela intelligatur inscripta BY: Quaeritur hujusce BY verus valor, numeris exhibendus. Quod sic solvo. Dico; Datis numeris, r = CG, 26 = BD, et / = DF: Erit Quadratum semissis rectae BY. 7 circuli (1) peripheriam (2) circuniferentiani 17 rectae BY. Hujusque radix quadratica r39

first: i.e. DULAURENS, Specimina mathematica duobus libris comprehensa, Paris 1667.

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165. WALLIS to OLDENBURG, 13/[23] December 1667 Dernonstratio.

Sint, CG = CD = CY = r.

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BE = ED = \BD = b.

DF = f .

BA = AY = \BY = CH = y.

Erunt EF = b + f.

BF = 26 + /.

EFq = b2 + 2bf + f2. BFq = 462 + 4&/ + f2. 2 CEq = CDq - EDq = r - b2. CFq = CEq + EFq = r2 + 2bf + f2. Item, in similibus triangulis, CFE, BFH, CF . CE :: BF . BH = CA. Adeoque CFq . CEq :: BFq . BHq = CAq. Hoc est r 2 + 2bf + f2 . r2 - b2 :: 462 + 46/ + f2

Ergo, 15

Hoc est, Ideoque

Quod erat demonstrandum.

17 Marginal note and additional figures in Wl: Nota. Si H sit citra C (puta, remotius ab F,) figura prima740 convenit. Si inter C et C?, vide figuram secundam (in versa 740

figura prima: i.e. a figure identical to the first figure in W2. The additional figures 2 and 3 in W1 are drawn on the reverse ('versa pagina'). 370

165. WALLIS to OLDENBURG, 13/[23] December 1667 Habes itaque Problema, eo saltern sensu quo (utut satis obscure traditum) intelligendum existimo, constructum et demonstratum. Assumo autem tres rectas, CG, BD, DF, datas esse: Quoniam, nisi hoc velit problema, aut saltern quod huic sit iaoSvvanov, poterit esse BY cujusvis longitudinis quae totam diametrum non excedit. Quaenam autem sint illae Circuli proprietates novae, hinc detegendae, quae non aliunde haberi possunt, nondum assequor. Vale.

5

Tuus Joh: Wallis. [2] These

For my honoured friend Mr Henry Oldenburg, at his house in the Pal-mal near St James's London.

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15

(W3)

Problema. Datis magnitudine (numeris designata) tribus hisce rectis; Nempe, Circuli BDG, radio CG, inscripta BD, hujusque (ad libitum protracta) continuatione DF; cui, in F, occurrat recta a Centre CF; et huic parallela

pagina.) Si in ipso C; erit BY — 0; ut in fig. 3. (quippe quae per B dianietro parallela ducitur, circuluni in B tangit.) ubique eadeni.

4 huic add. 19 Circuli (1) BY (2) BYD (3) BDG

371

20

165. WALLIS to OLDENBURG, 13/[23] December 1667 intelligatur inscripta BY: Quaeritur hujusce BY longitude, numeris item exhibenda.

Solutio. 5

Dico; Datis numeris r =CG,2b-BD, et f=DF erit Quadratum semissis rectae BY.

Dico porro; Si punctum F, quod ultra D supponitur, contingat citra D (ut in figuris quinque sequentibus,) erit Quadratum semissis BY. 7 quinque add.

372

165. WALLIS to OLDENBURG, 13/[23] December 1667

Demonstratio. Sint

CG = CB = r. BE = \BD = b. Datae. DF = ±f. (Nempe; +/, si F sit ultra D: -/, si citra;) 5A = i^y = y, Quaesita. Erunt. EF = b±f. EFq = b2 ± 2bf + f2. CEq = CBq - BEq = r2 - b2. CFq = EFq + CEq = r2 ± 2bf + f2. BF = 2b± f. BFq = 462 ± 46/ + f2. BHq = CAq = CBq - BAq = r2 - y2. Item (in similibus triangulis CFE, BFH,} CF. CE :: BF . BH. Et CFq.CEq :: BFq.BHq = CAq. Hoc est; r2 + f 2 ± 2bf .r2 -b2 :: 4fe2 + f 2 ± 46/ .

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Adeoque y

2

CJuod erat demonstrandum. Vel etiam, posito K ubi CA, BD, (productae si opus) se mutuo intersecant; propter similia Triangula BFH, RFC, CFE, KCE, KB A. Erit (in fig. prima, quod caeteris accommodabitur, mutatis ut opus erit + et -,) EF = b+ f .EC = :r2 -b2 :: EC.EK = et KB = BE-EK = b-

Et KF = FE + EK = b + f +

9-10 4&2 + /2±46f. (1) r2 -y2 (2) 15 KFC, add. 18 W («) Et

corr. ed.

373

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165. WALLIS to OLDENBURG, 13/[23] December 1667 Item KF

FC

KB=

seu y

BA=y=

[27^] Tandem petitur, 5

Ut BY sit rationalis. Respondeo.

10

1. Si illud vellet problema ut primo propositum; nimis imperfecte propositum fuit; Quippe illud quod petitur faciendum, nempe verum valorem rectae BY exhibere, nihil tale insinuet. Dixisset igitur numero rationail exhibere- vel saltern numero exhibere] et non simpliciter exhibere. 2. Illud tamen sic solvo. Quo fiat 11 Cum sit (ut demonstratum est) rationalis (positis r, 6,ex hypothesi rationalibus) requiritur ut / rationalis

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(quo sit 2b2 — r2 + bf rationalis) ita sumatur ut sit etiam ^/ : r 2 + / 2 + 26/ : rationalis: Hoc est, ut sit r2 + / 2 + 2bf quadratus. Quod sic habetur. erit Cum sit b < r, adeoque r 2 +/ 2 +26/ 2bf :< r + f. Esto r + f — c, (adeoque c rational!, propter rationales r, /.) Hujusque propterea quadratus r 2 +/ 2 +c 2 +2/r—2cr—2c/ = r 2 +/ 2 +2fe/: Adeoque c2 + 2/r - 2cr - 2cf = 2bf: Et 2/r - 2bf - 2cf = 2cr- c2; adeoque / (adeoque rationalis c < r —6, nisi velimus ut

2r — 2b — 2c sit negativus.) Ponatur (quo fractionis Denominator fiat singularis) 2r — 26 — 2c — 2x; adeoque r — b — c — x, qui itaque est minor quam r — b, (et quidem negativus, si c > r — 6,) et r — 6 — x = c; et 25

propterea 2cr = 2r2 - 26r - 2xr, et c2 = r2 + 62 + a;2 - 2r6 - 2rx + 2bx, et 2cr - c2 = r2 - b2 - x2 - 2bx, et

/, (sumpto pro x, quovis rational! qui minor sit quam r — b.) Adeoque bf Adeoque et 2b2

12 est) (l)y= (2) \BY = 16 r2 + f2 + 2rf; (i) quadrat! r 2 + (g) erit 17 r + / — c, ( (i) sumpto (2) adeoque 18 propterea add. 19 = 2bf: (1) Et (a) 2rf (6) c2 - 2cr = 2&f + 2c/ - 2r/ (g) Et 21 sit (1) quantitas (2) negativus 23 si (1) c> 6 &reafc« off (2) Or-b 25 2cr - c2 = (i) {—} (2) r 2 - &2 - x2 - 2bx 374

166. OLDENBURG to WALLIS, 24 December 1667/[3 January 1668] (propter •J : r2 + f2 + 2bf :=r + f-c

= f + b+ x =

\ erit

Hoc est, posito Radio r inscripta BD — 26, rationalibus; sumptoque x rational! quovis modo minor sit quam r—6; erit b— rationalis. Quod erat faciendum. Eademque facile accommodantur (mutatis mutandis) aliis casibus; puta cum F cadit intra circulum. Vel etiam, (hoc est / — DF) quantisi ita sumatur or, ut proveniat tas negativa; indicium est punctum F quod supponitur ultra D, cadere citra D- hoc est, BD secabit CG intra circulum; aut etiam ex parte C productum. Similiter, si pro y = BA, proveniat b — x negativa quantitas; indicio est BY non prorsum (ut in schemate principal!) sed retrorsum sumendum: Quod fieri potest si BD minor sit quam subfuerit negativa quantitas; tensa quadrantis; non secus. Item, si erit BY major quam BD, et centre propius jacebit. Similiter, si pro EK proveniat quantitas negativa; indicio est punctum K quod supponitur citra E, jacere ultra E. Item, si pro KB, proveniat negativa quantitas; erit K, quod supponitur ultra B versus D, revera citra B, in DB ex parte B producta. Atque alibi similiter. Quorum omnium determinationes non sunt difficiles: Nempe, qua ratione ponenda erit 6, vel sumenda x, ut hoc aut illud contingat.

166. HENRY OLDENBURG to WALLIS London, 24 December 1667/[3 January 1668] Transmission: C Letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 29r-30v (our source). At bottom of f. 30r part draft of Wallis's reply. Postmark on f. 30V: 'DE/24'.—printed:

1 erit (1) 2&2 + 4 breaks off (2) 2b2x - 4r2 breaks off (3) y = (4) 5 Quod erat faciendum, add. 7 proveniat (1) pro / = DF quant breaks off (2) corr. ed. (a) qu breaks off (b) (hoc est / (aa) ; (bb) = 10 pro (1) BA, proveniat (2) y = BA, proveniat corr. ed.

12 si (1) BY (2) BD 17 K, (1) citr breaks off (2) quod 18 similiter. (1) Quod (2) Quorum

375

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166. OLDENBURG to WALLIS, 24 December 1667/[3 January 1668] OLDENBURG, Correspondence IV, 82-4. Reply to: WALLS-OLDENBURG 30.XI/[10.XII].1667 and WALLIS-OLDENBURG 13/[23]. XII. 1667. Answered by: WALLIS-OLDENBURG l/[ll].II.1667/8.

London Dec. 24. 67.

Sir,

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Since MyLord Brounker's returne I exhibited to him your two last741, concerning the Demonstration of the numbers for finding out the Julian Period, and that other of the Parisian Probleme. His Lordship is satisfied with the former; and as to the latter, he understands the Proposers meaning and demand to be in short this, That DF, being ad libitum, is to be chosen of such a quantity, that BY may be rational. A certain person742 here, makes it a rational quantity in this Instance, but hath not given a Rule for it, because, as MyLord Brounker observed in his paper, by his Method, GF may prove irrational. This I thought necessary to acquaint you with. Mr Boyle will doubtlesse give you part in the news, I send him743 by this post from abroad, and particularly of a considerable Experiment744, if not mis-represented, of Transfusion, practised very lately at Paris upon a Mad man in the presence of 8. able physitians there. | [29V] 745 Dr John Palmer , now Arch-deacon of Northampton, hath begun to correspond with me Philosophically. He presses746, that, since Tanger 5 other add. 7 demand (1) is, (2) to be in short this, 8 being |drawn del] ad libitum 741

your two last: i.e. WALLIS-OLDENBURG 30.XI/[10.XII].1667 and WALLIS-OLDENBURG 13/[23].XIL1667. 742 A certain person: probably Collins, to whom the problem was communicated by the Royal Society with the request for its solution; BIRCH, History of the Royal Society II, 226. 743 I send him: i.e. OLDENBURG-BOYLE 24.XII.1667/[3.I.1668] (OLDENBURG, Correspondence IV, 78-80), which was sent together with the present letter. 744 Experiment: i.e. the experiment on blood transfusion by Jean-Baptiste Denis, an account of which was contained in JuSTEL-OLDENBURG 18/[28].XII. 1667 (see OLDENBURG, Correspondence IV, 61). 745 Palmer: i.e. John Palmer (1612-79). 746 presses: see PALMER-OLDENBURG 12/[22].XII. 1667 (OLDENBURG, Correspondence IV, 34-5). 376

166. OLDENBURG to WALLIS, 24 December 1667/[3 January 1668] is ours747, a Correspondence may be setled for Observations to be made of the Moons Meridian Altitude both there and at Edinburg, Aberdeen, or Catnes748, as well as at London, on the same days; to the end, that by taking a Chord of 17. 20. or 25. deg. (in stead of the semidiameter of the Earth) to subtend the parallactick angle, the distance of the S> might be solidly demonstrated. The like correspondence he would have held for observing the Eclipses of the Moon, with Smyrna, Aleppo or Bantam749 in the East, and with Bermuda, Barbados or Jamaica in the West, for the correcting of our Maps. We shall take this into Consideration. Since this, upon an information, I receaved, that this Learned man has a way of resolving all Equations and hard problems of Arithmetick by Regula Falsi, I have writ to him again750, and desired him to communicate it, if the information be true. Besides, I have engaged him, to procure for us from good Husbandmen the practice of Agriculture in Northamtonsh. and to take notice of all the Observables in that shire, for communication. Sir, I intend, if God vouchsafe me Life and health, to put the Transactions into Latin between this and Whitsontide, there being several particulars in it, communicated by able persons, and new, which ought to r [30 ] be kept from usurpation, by this way. You would | oblige me not a litle, and somewhat ease me, by taking yourself the pains, to turne your owne excellent Hypothesis about Tides751 into your owne Latin, as soon as conveniently you can. I am

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Sir

Your faithful humble servant H. Old. I have desired752 Mr Boyles favor, to see, whether the sub-bibliotheca14 from (1) a, good Husbandman (2) good Husbandmen 20 by add. 747

Tanger is ours: Tangier was occupied by England from 1662 to 1684. Catnes: i.e. Caithness. 749 Bantam: i.e. Bandung on Java. 750 writ to him again: i.e. OLDENBURG-PALMER 21/[31].XII. 1667 (OLDENBURG, Correspondence IV, 71-2). This request was also contained in the opening letter of the correspondence with Palmer, OLDENBURG-PALMER 3/[13].XII. 1667 (OLDENBURG, Correspondence IV, 3-4). 751 Hypothesis about Tides: i.e. WALLIS-BOYLE 25.IV/[5.V].1666, which had been printed in Philosophical Transactions No. 16 (6 August 1666). 752 I have desired: see OLDENBURG-BOYLE 24.XII.1667/[3.I.1668] (OLDENBURG, Cor748

377

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167. JENKINS to [WALLIS?], [January 1667/8?]

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15

rius of Oxford can give a good account and catalogue of the good books printed in England, these 10. or 12. years last: Monsr Carcavy having earnestly desired me753, to procure him such an one, in order to furnish the French kings Library with such Books accordingly. Pray, be pleas'd to Joyn with M. Boyle herein. A gentleman754 living in the Franche-Comte, that has been a great Traveller in the Levant, desires to know, what able men England has at present in the Oriental Tongues, especially in Arabick, Persian, Turkish. If at Oxford there be others, that ar skilld in them, besides Dr Pokock755 and Mr Hide756, I pray, signify it to me: And it may be, those two persons can give you and me notice of all others in the nation. For his much honor'd friend Dr John Wallis, Savilian Professor of Geometry in Oxford.

[30V]

167. LLEWELYN JENKINS to [WALLIS ?] [London, January 1667/8 ?] Transmission:

C Letter sent: OXFORD University Archives WP7/16/1, f. 90r-90v (f. 90V blank). On f. 90r beneath, text at 180° calculation probably in Wallis's hand. On f. 90V in unknown hand: 'Dr Jenkins letter'. It would appear that the present request was directed to Wallis at the University archives in connection with the Fish Lyne case. Wallis possibly decided to make the transcript himself and sent it with his letter to Jenkins of 28 January 1667/8.

1 the |good add] books 3 one, (1) for (2) in order respondence IV, 78-80, 79-80). 753 Monsr Carcavy having earnestly desired me: via JuSTEL-OLDENBURG ll/ [21]?.XIL1667 (OLDENBURG, Correspondence IV, 29-31, 30). 754 gentleman: not identified. 755 Dr Pokock: i.e. Edward Pococke. 756 Mr Hide: i.e. Thomas Hyde. 378

168. WALLIS to JENKINS, 21/[31] January 1667/8 Thursday Morn. Sir

I perceive there is a pluries habeas Corpus, and an Attachment coming down against me. I must therefore beg the use of King Charles his great Patent757 this morning, that I may transcribe the whole. I build upon, & send it up to some Friends for

5

I am

Your most humble servant L. Jenkins

168. WALLIS to LLEWELYN JENKINS 21/[31] January 1667/8 Transmission: W Letter sent: KEW The National Archives PRO SP 29/233, No. 4(ii), 2 pp. (p. 2 blank). Enclosure: Notes relating to the allowance of the claim in the Exchequer.

Jan. 21. 1667./8.

10

Sir

Be pleased, in my breviat758; to mend one particular, in the reciting of Painter's: where instead of according to Dr Chaces case' put according to the case of 9 Hen. 6. (viz. Term. Mich. 9 Hen. 6. cas. 24.) which is not Dr Chace's759 case; but another; which allso is to our purpose, as well as the printed case of Dr Chase. For Dr. Jenkins.

Yours. J.W.

Dr chace's case is 8° Hen. 6. 757

King Charles his great Patent: i.e. the 'great charter' of Charles I of 3 March 1636; cf. Wallis's Extracts from the University Archives enclosed in WALLIS-JENKINS 28.I/[7.II].1667/8. 758 breviat: i.e. the enclosed Notes relating to the allowance of the claim in the Exchequer. 759 Chace's: see WALLIS-JENKINS 26.X/[5.XI].1667.

379

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169. WALLIS to JENKINS 21/[31] January 1667/8, enclosure

169.

WALLIS to JENKINS 21/[31] January 1667/8, enclosure: Notes relating to the allowance of the claim in the Exchequer Transmission:

W Paper sent: KEW The National Archives PRO SP 29/233, No. 4(i), Ip. Endorsed on verso: 'Instructions from Dr. Wallis, out of the Archives, touching Prince's the Townserjeant his suit against the Bedles in the Exchequer for executing the V. Chancellors warrant. 21. Jan. 67.' Enclosure to: WALLIS-JENKINS 21/[31].1.1667/8.

For the Allowance of our claim in the Exchequer in the Bedles case arrested for executing the Vicechancellors760 warrant. 5

That the Priviledge is to be allowed in Exchequer as well as other courts. There is the Precedent of Poolye's case for Cambridge. Which is expressely allowed. Boxters case: Which by a writ of Privy seal (which is of record in the Exchequer, ex parte rememoratoris Reginae) is superseded.

10

To the objection, that they will not allow it in our case, where our jurisdiction is concerned. Wee may plead. The charter of 12 Apr. 10 Ed. 3. marked N 3. at the marks in the margin +. +. That the Prohibitio regis non curret &c. That the chancellor is not to be molested for false imprisonment by any writs from above.

15

4 other (1) cl breaks off (2) courts 6 case [allowed del] for 7 seal add. 11 Wee (1) pr breaks off (2) may 760

Vicechancellors: i.e. John Fell, q.v. 380

169. WALLIS to JENKINS 21/[31] January 1667/8, enclosure A writ thereupon directed to the Chancellor &c. 27 Apr. 48. Edw. 3. (of which you have a Copy out of the Proctors Old Statute book writ about 250 years ago) Authorizing the Chancellor to proceed notwithstanding such writs.

5

761

And a clause in K. Hen. 8 Charter to that purpose. Provided the Vicechancellor submitted to that way of Appeal provided in the University. Which end in chancery. The French composition 22 Edw. 3. (marked P. fase. XIII. 1.) That the Chancellor is to proceed in cases that concern the University as a University. Farendons case; indicted for an assault on the Marshals-man of the Kings bench executing the Kings writt. Farendon762 was Comissary or (as we now call it Vicechancellor) & is claimed by the Chancellor & allowed, in Hen. 4. time. You have the Record at large.

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Dr Chaces case, 9 Hen. 6.763 in the printed year-Book. (Where the jurisdiction was questioned to amerce townsmen for not paving the streets.) Hee claims cognizance of his own cause in his own court, (where that particular is of large debated by the judges) & it is allowed. Painters764 case: 7 King James. Where the question was of Jurisdiction; whether the Vicechancellor might amerce the Bailifs for Night-walking

4 the | Vice del] Chancellor 6-8 And a clause . . . in chancery, add. 11 as as University corr. 14 in (1) Edw. 4 (2) Hen. 4. 18 of (1) the cause (2) his 21 might (1) amit (2) amerce 761

K. Hen. 8 Charter: i.e. the charter of Henry VIII of 1 April 1523 ('Wolsey's charter'); cf. Wallis's Extracts from the University Archives enclosed in WALLIS-JENKINS 28.I/[7.II].1667/8. 762 Farendon: i.e. William Farington (d. 1420), chancellor's commissary at the University of Cambridge 1401. 763 Dr Chaces case, 9 Hen. 6.: cf. the correction in WALLIS-JENKINS 21/[31].I.1667/8. 764 Painters: i.e. Thomas Painter, bailiff; TYACKE (ed.), History of the University of Oxford IV, 113-14.

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170. WALLIS to JENKINS, 28 January/[7 February] 1667/8 who pretended the Kings business, as hues & cryes &c. And it was allowed in the Kings bench to be of the same nature with that of Dr Chace765. And Painter remanded till the Vicechancellor release him. 5

10

The market in particular, which is here the case, The whole power the Town hath in the market (at lest the cheif) depends on that clause in their charter of Edw. 3. Clericus Merkati nostri se non intromittet &c. Which is taken from them, (after the great conflict upon their submission) and granted to the University in the same words. Et quod Clericus Mercati se in presentia nostra vel hereditorum nostrorum de cetero non intromittat infra burgum ilium aut suburbia ejusdem de officio suo exercendo de aliqua se eundem burgum tangente vel suburbia. 10 Jan. 32 Edw. 3. You have the copy of the town charters by themselves. And of this in the University, in that of K. Charles766.

170. WALLIS to LLEWELYN JENKINS Oxford, 28 January/[7 February] 1667/8 Transmission: W1 Draft of letter sent: OXFORD University Archives WP7/16/1, f. 86r-89v (f. 86V-88V enclosure, f. 89r-89v blank). W2 Letter sent: KEW The National Archives PRO SP 29/233, No. Ill, 12 pp. (pp. 2 and 11 blank, pp. 3-10 enclosure) (our source). On p. 12 at 90° to address in Jenkins's hand: '16 Reasons against the Leasing of Felon's Goods to the Citty of oxford by Dr Wallis 28. Jan. 67.' Postmark on p. 12: 'JA/29'. Answered by: JENKINS-WALLIS 30.I/[9.II].1667/8. Enclosure: Extracts from the University Archives. The background to the present letter is evidently the attempt on the part of the city of Oxford to bring the conflict with the University to an end, fearing that custom and trade might be taken away from townsmen. After both sides failed to reach an agreement, it was decided that the vice-chancellor should proceed in the courts of justice against the city. See WOOD, Life and Times II, 128-9.

2 of the (1) Cha breaks off (2) same 3 remanded (1) to (2) till 765

that of Dr Chace: cf. the correction in WALLIS-JENKINS 21/[31].I.1667/8. that of K. Charles: i.e. the 'great charter' of Charles I of 3 March 1636; cf. Wallis's Extracts from the University Archives 766

382

170. WALLIS to JENKINS, 28 January/[7 February] 1667/8 Oxford Jan. 28 1667./8.

Sir, I suppose Mr Vicechancellor767 hath given you an account of Yesterday's Convocation; in which, hee gave a brief Narrative of what hath lately passed in order to a Treaty with the Town. Whereupon Delegates were appointed to consider of it, but not to conclude: (not cum nuda relatione ad domum, which the masters would not passe; but cum relatione ad domum et approbations ejusdem.) After which, he sent for mee at night, & desired mee to draw up somewhat and send to you, concerning those dangerous consequences which are apprehended in making a Lease to the Town, of Felons goods &c, with the Powers appurtenant thereunto. Which last night & this morning I have been doing, as the shortness of Time would permit. Which that it might be the more cleare to you; I have caused the Words of our charters which grant us those Rights & Powers, to be transcribed768; and then, what did, for the present, occurr to mee of danger in putting over those Powers to the Town. (Though I will not undertake to be so well skilled in the Law, as in so short a time to foresee all.) What use you are to make of it; I presume you have Instructions from Mr Vice-chancellour. I understand allso from him, that the Habeas Corpus for Fish Lme769 is come down; which I suppose will be sent up to you. To which it will be necessary (out of the Instructions I sent) to get a Return drawn up by the best Clarks you can find, to be ready to annex to the Writ: And this, the sooner the better. For we are not so far to trust to the Treaty on Foot, as to be surprised by being too secure. And I think it would be very Necessary that Justice Brown770 were, before hand, acquainted with the Particulars of the Return which I drew up, that he may have timely leisure to consider of them, (being yet strange to him,) which in so short a time as it will be pleading at the Bar, cannot be done.

13 permit. (1) & because it was blamed and not so likely to be ready, I may which that serve what use you are to make of it; I presume you have W1 17 as |in so short a time add] to foresee jthem del. all.) 767

Vicechancellor: i.e. John Fell, q.v. transcribed: i.e. the enclosed Extracts from the University Archives. r&9 Fish Line: i.e. Fish Lyne. See WALLIS-JENKINS 26.X/[5.XI].1667. 770 Brown: i.e. Sir Samuel Browne (d. 1668). 768

383

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171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure

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In the Exchequer; I hope care hath been taken that wee bee not surprised there, as having deserted our Plea for not pleading to it on the first Saturday in this Term, as was the order of last Term. If the suit there, be wholly withdrawn by consent; I think there is no danger of it. But if onely suspended; wee must be vigilant that wee be not surprised. I could wish allso that wee had a Copy (if not an Exemplification under Seal) of Dr Chace's case at the Kings Bench, in Hillary term 8° Hen. 6. out of the Records. For though, in the Printed Year-book, the pleadings favour us; yet it is not there expressely sayd, that the cause was dismissed. And the case is extremely pertinent to that now in the Exchequer. Sir, I hope you will excuse mee for the troubles I give you in these affairs, from Sir

Your humble servant, John Wallis.

15

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These For Dr LLuellin Jenkins Judge of the Court of Admiralty at Exeter house London.

[12]

171. WALLIS to LLEWELYN JENKINS Oxford, 28 January/[7 February] 1667/8, enclosure: Extracts from the University Archives Transmission:

W Draft of paper sent: OXFORD University Archives WPj/16/1, f. 86V-88V. w Paper sent (transcript in scribal hand with corrections and additions by Wallis): I hope (1) so much care . . . taken as (2) care ... that 6 Copy (1) of th breaks off (2) (if 7 at the Kings Bench, add. II mee |(I hope) del] for 384

171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure KEW The National Archives PRO SP 29/233, No. 1111, 2 pp., and No. Ill II, 5 pp. (our source). Enclosure to: WALLIS-JENKINS 28.I/[7.II].1667/8.

The Words of King Henry the 8th his Charter771 concerning Felons Goods &c. 1° April. 14° Hen. 8.

[I, p. 11

At insuper de uberiore gratia nostra damus et concedimus praedictis cancellario et scholaribus dictae universitatis quaecunque annum, Diem, strepum, Vastum, Deodandum et Thesaurum Inventum, catalla felonum, utlagatorum, fugitivorum damnandorum felonum de se et felonum in exigendis positorum sive aliter delinquentium pro quo bona sive catalla sua amittere sive forisfacere debeant infra villain Oxoniae praedictam et suburbia ejusdem cujuslibet et quorumcunque ligeorum sive subditorum nostrorum. Nee non omnium bonorum et catallorum quorumcunque manuopere captorum, et quod bene liceat eisdem cancellario, commissario, deputato sive vicem gerenti et scholaribus et successoribus suis omnia praedicta [fines, amerciamenta, redemptiones exitus, forisfacturas,] Deodanda, Thesaurum inventum, annum, diem, vastum, catalla felonum, utlagatorum, fugitivorum, damnandorum, felonum de se, felonum in exigendis positorum et caetera praemissa quaecunque et qualitercunque forisfacta, sive forisfacienda et omnia quae ad nos, haeredes et successores nostros de dictis finibus, amerciamentis, exitibus, die, et Anno vasto, streppo, Deodandis, catallis felonum, utlagatorum, fugitivorum, damnandorum, felonum de se, et in exigendis positorum, sive aliter (ut praedicitur) forisfactis quoquo modo pertinere possit per ipsos et Ministros suos levare, colligere percipere et habere tarn per manus suas proprias, sive assignatorum suorum, sive Deputatorum suorum quam per manus vicecomitis, Escaetoris et aliorum Ofnciariorum nostrorum quorumcunque et seipsos in seisinam inde ponere absque calumpnia, impetitione vel impedimento nostro vel haeredum nostrorum Officiariorum vel Ministrorum nostrorum, vel Haeredum nostrorum praedictorum aut aliorum Justiciariorum,

1-3 The Words ... Hen. 8. add. Wallis 14 [fines . . . forisfacturas,] add. Wallis 19 dictis finibus, amerciamentis, exitibus, add. Wallis 771

King Henry the 8th his Charter: i.e. the charter of 1 April 1523 ('Wolsey's charter'). 385

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171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure Escaetorum, Vicecomitum, Coronatorum, Majorum Ballivorum, Constabulariorum, seu Ministrorum nostrorum vel Haeredum nostrorum quorumcunque. The words of King Charls's Charter in 3° Marti 11° Car. I.772

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Insuper cum Dominus Henricus illius nominis Octavus, Antecessor noster per literas suas patentas praerecitatas praefatis cancellario et scholaribus dictae universitatis et successoribus suis (inter alia) dederit et concesserit quod ipsi quaecunque annum, diem, strepum, vastum, Deodandum, The10 saurum Inventum, catalla felonum utlagatorum, fugitivorum damnandorum, felonum de se, et felonum in exigendis positorum sive aliter delinquentium pro quo Bona sive catallaj sua amittere sive forisfacere deberent [I, P. 2] Infra villain Oxoniae et suburbia ejusdem cujuslibet et quorumcunque ligeorum sive subditorum suorum. Nee non omnium Bonorum et catallo15 rum quorumcunque manuopere captorum in proprios Universitatis usus per se vel Officiarios et Ministros suos colligerent et levarent ac omnia alia circa eadem bona et catalla felonum ac aliorum ibidem mentionatorum exequerentur et facerent, prout in praedicta Antecessoris nostri charta plenius continetur. Sciatis ulterius quoddam nos ex Ampliori et uberiori 20 gratia nostra special! ac ex certa scientia ac mero motu nostris pro nobis Haeredibus et Successoribus nostris iisdem cancellario, Magistris, et Scholaribus et successoribus suis in perpetuum damus et concedimus per praesentes, quoddam ipsi eadem omnia et singula Bona et catalla felonum aliorumque praementionatorum ac debita etiam Jura et credita sive per 25 scripta vel alio quovis modo ad ipsos pertinentia tarn infra praedictam Universitatem Oxoniensem et ipsius praecinctum, quam etiam Infra Civitatem Oxoniae et suburbia ejusdem accidentia sive emergentia habeant et teneant sibi et successoribus suis libere et quiete et pacifice absque compute seu aliquo alio proinde nobis Haeredibus et successoribus nostris 30 reddendo seu faciendo. Ita quod ubi et quandocunque iisdem Cancellario 2-3 quorumcunque. |Et Insuper volumus et per praesentes concedimus quod praedictus cancellarius et scholares ac omnes eorum servientes et cujuslibet eorum serviens aut eorum seu universitatis dictae Ministri firmarii et Tenentes sui, et eorum quilibet ubicunque fuerint exonerati et quieti sint de quibuscunque prisis chiminagiis captionibus carriagiorum Equorum Carractarum plaustrorum. del. Wallis] rr2

King Charls's Charter in 3° Marti 11° Car. 1.: i.e. the 'great charter' of Charles I of 3 March 1636. 386

171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure Magistris et scholaribus, vel Cancellarii locum tenente et successoribus suis certo vel probabiliter innotuerit aliquem vel aliquos crimen feloniae commisisse aut in suspitionem feloniae incidisse infra universitatem praedictam ejusve praecinctum vel infra Civitatem Oxoniae ejusve suburbia. Tune Cancellarius ejusdem universitatis vel ejus locum tenens per seipsum, vel ejusdem universitatis procuratores eorumve Deputatos officiarios vel Ministros suos hac in parte appunctuatos vel appunctuandos, in quamcunque domum aut alium locum ubi hujusmodi bona et catalla aut alia praementionata, Nee non scripta et chirographa eorundem felonum aut Delinquentium Debita jura et credita quovismodo tangentia sive concernentia esse, abscondi aut in quorumcunque manibus detineri contigerit libere et impune ingredi liceat et scrutinio sive examine super iisdem facto et habito eadem omnia sic inventa vel in posterum invenienda ut Cancellario Magistris et scholaribus Universitatis praedictae forisfacturae virtute et vigore praesentis istius nostrae concessionis in quorumcunque custodia fuerint vendicare, arrestare, capere, seisire et impune asportare et post eorum debitam convictionem in usum proprium ejusdem universitatis convertere. Resistentes autem et contradicentes tanquam eos qui Cancellario ejusve Commissario et procuratoribus aliisve officiariis et Ministris Universitatis praedictae in executione officiorum suorum Resistunt et contradicunt secundum leges et statuta Regni nostri, vel secundum Jura et consuetudines ipsius Universitatis debite castigare et punire. These are the Rights & Powers which are by the intended Lease for 200 years to be passed over to the Town, for a Noble a year. For which Concession, the Town offer by their Indenture (to which we are to be parties,) to acknowledge a Power in the Vice-chancellor & Proctors 1. to punish according to the Laws of the Kingdom such as they find Wandering after Ten of the clock at night in summer, or Nine of the clock in Winter, & before four in the morning. 2. To search Inns Taverns and notoriously suspected houses, and such as they find there Debauched, to punish according to the Lawful usages of the University. 3. Such as they shal find notoriously debauched, or resisting the Vice-chancellor or Proctors, they promise not to abbet or defend, but suffer them to be punished according [II, p. 1] to the present statutes of the University.

5 vel ejus . . . ejusdem universitatis add. Wallis 23-34 These axe ... the University, add. Wallis 26 1. add. 387

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171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure Reasons against the passing away the Universitys Right in Felons Goods &c, with the Powers concerning the same by lease to the Citty of Oxford for 200 yeares at the Rent of a Noble a yeare. 5

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1. There is no priviledg or Right which the University hath which is more certaine, then what they have to these, (no not the power of holding convocations and graunting degrees) they being thinges which cannot passe from the crowne by prescription, (which is all the citty have to pretend) but by words of Expresse graunt which the University have. But the citty have not; Which is a thing so certaine and knowne, that no lawyer in England will doubt of it. See the Abbot of Strata Marcella's case. And former decisions for the University. 2. It is the whole Royalty for the university that is intended thus to passe. Which is a thing of that Nature, that No man who hath but the spirit of a Gentleman would part with it for a much greater value then what he makes of it. 3. It is such a Royalty as scarce any Corporation in England doth Enjoy, beside this University, (No not the Citty of Londont selfe.) For though divers Corporations may be found which have some of these thinges, yet very few, (if any) that have all. 4. It would be very unreasonable; when as K. Henry the Eight hath graunted us such a Right which to the Citty was Never graunted: & K. Charles by his Charter (obtained by the great care of the late Archbishop Laud, our then Chancellor) hath so much Enlarged it, perhaps beyond all president, that the University should now make no better use of this, which for theire sake was taken from the Kinges Almoner, then to passe it away to the Citty of Oxford for a Noble a yeare. 5. Whereas it is pretended, that it hath beene a matter of contention, and that wee Never made any thing of it. I say as to the first, that there is no power or priviledg, which wee have but is and hath beene soe. And if makeing it a matter of contention will wrest of it out of our hands. Wee must part with all; even that of holding convocations and makeing statutes for our owne Body. For the towne have not only made complaints in parliament and otherwise against us oftentimes, but even preferred

8 praesumption corr. Wallis 9 pretend) and corr. Wallis 25 precedent W 388

171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure Indictments against the Chancellor, Procter and Convocation for so doing; as holding unlawfull Conventicles, contrary to the laws of the land, & incurring the penalties of Praemunire or the like. Which occasioned the Charter of K. Henry the Fourth for the steward of the University who might hold pleas of such Indictments. At which Charter the Towne was so much offended. 6. The Matters of contention are not hereby avoided; But Rather increased. For, (beside that the Excercise of this power by the towne will sure create new causes of contention;) the other perticulars now actualy in contest as the power of the Market, and of priviledged men's trading, & theire arresting our Bedles for Executing the vicechancellors warrant, [II, p. 2] are not at all provided for; But in Effect deserted, even then when wee have spent above an hundred pound in defence of our Rights therein, and are fair ennough to have them decided for us. Which thinges breakeing off in the posture they now are, will seeme Rather to passe against us. So that all our Rights therein must bee deserted too, or else wee must contend for them at a greater disadvantage. And the power of the night watch, which is the only thing thought hereby to bee secured, will by this graunt if it passe, bee made more disputable then ever. Of which by and

by-

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7. As to that of our makeing nothing of it, wee must upon the same Reason part with all the Rest, For what hath the University (as to mony) made of the Maior and Citizens takeing theire yearly Oath of fealty to us: which yeilds us not a penny and that of theire offerring on Saint Scholastica's Day, which is But 42 pence to the University, (and 21 pence to the vicar of Saint Marys.) Why should wee not therefore for another Noble passe away this Right also? And the like for the Assises of Bread and Beere. And of weights and measures, and the Inspection of the Streets and pavements. Which have beene (and are) matters of contention and do Realy cost more then wee make of them, And the like of the clarkship of the Market (if at least it bee not to passe by this graunt) why should it

1 Proctors W 5 Indictments. (1) \\ The (2) At 12 when wee scarce spent an hundred corr. Wallis 13 have (1) fair inough to have them (2) spent W 18 thing (1) the pretended (2) thought W 19 passe, jwill del. bee 26 Why should . . . Right also? add. w, add. in margin W 389

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171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure not bee graunted away? Since it doth not yeild the University any thing? And doth not so much as pay the Universities charges to the dark of the Market. 8. The thing is not so inconsiderable as is pretended. For the University have Really Received very considerable summes on this account, 5 sometimes Threescore and Tenn or four score poundes at a time. And what ever at any time the University have actualy seised, the towne Never had the Confidence to sue for: But suffered us to injoy it as knowing they had no Right to set forth as evidence in the case; and have only by vio10 lence laid handes on what they can get. And though the University have not converted all these summes to theire owne proper use yet some part thereof they have so imployed, and part of it by way of charity, for the payment of Just debts, (which must otherwise have beene lost) and for Releife of widdowes and Orphants of such as have made away themselves, and for other charitable uses; According to the true contentment of such 15 forfeitures, which were otherwise to come to the Kinges Almoner for such uses. From whom they were taken to be given to us, and when heretofore there hath beene differences concerning such thinges It was not Betweene the University and Citty But Betweene the Kinges Almoner and the University, the Citty never interposing, for the Goods of Felons de se &c. 20 till since the Kinges Almoner did Relinquish them to the University. To whom they do belong if not to us. 9. If it were no more But a power Reserved in us for an equitable Releife of the widdowes and children of persons whose Estates are soe forfeited, it were at least a Christian work to doe so much of charity. For 25 though the law do make forfeit the Estates of such persons, yet there is equity that the Innocent widdowes and children should be considered.) [II, p. 3] And even the disposing of thinges thus forfeited belonging ordinerly to the Kinges Almoner it may be well presumed to be so Intended. Only that it should be left to his or the Kinges discretion how far and in what 30 way to do it. But if wee put it out of our handes into the Cittys they are not to expect the least Releife. But to be totaly Ruined (for it is cleare ennough by the Experience of theire practice what mercy is to be found at theire handes.) And though sometimes Rather then the persons should seek for shelter from the Universities Right, the Citty Baileifs are faine to 35

11 owne (1) use (2) proper use 15 contendment corr.

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171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure accept of some composition, (knowing that they have no title) wee shall soone see what use they will make of it; and how little soever it may be thought worth, while it is ours it will soone bee found a very considerable purchase to them. 10. If it were indeed nothing worth: yet were it better to keep it for nothing, then to sell it for Nothing, that so wee might at least transmit the Right to our successoures as it hath beene Derived to us, and if wee do not think it worth while to effect our Right therein yet let us not by our act preclude those who may come after us. Who certainly will deservedly blame this generation when they shall see so faire a Royalty (of which they shall then find the Town to make so much advantage) passed away by us for a Noble a yeare. Which will not by them be looked upon as a Rent, But Rather as a perpetual Reproach, for haveing passed it away at so cheap a Rate. Nor need wee feare our Not doing it will be a matter of charg or contention. For if wee do but tacitely permit the towne to usurp upon us without our passing it away by an Act of our owne, that contention will soe long cease. Nor will they Revive the contention so long as wee are willing to be silent. 11. If it be thought a thing soe little worth, wee might do well first to try whether the towne will be content to passe away theire No-Right at so cheap a Rate as wee are Ready to passe away our Right. And if they will I should advise for peace sake, that wee may be purchasers. If not, it is a signe they look upon it as better worth. 12. When as it hath beene hitherto thought a great oversight in our predecessours that when the Royalty of North-gate hundred was to be sold, the University suffered the towne to buy it, when they might themselves have purchased it: (Because thereby the towne Inlarged theire Jurisdiction to our prejudice:) Wee do hereby extend it not only to North-gate hundred and Holy-well, But to Saint Clements and the whole precincte

I no Title) yet when wee shall have given them a legall-title; wee shall soon see what they will make of it. And how W 9 Act (1) prejudice (2) preclude W II Town add. Wallis 14 it add. Wallis 15 but so silely permit corr. Wallis 19 (1) 11. Whereas it hath been hitherto thought a great oversight in our predecessors; yet when the Royalty of North-gate Hundred was to be sold. (2) 11. If it W

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171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure

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of the University. As far as Saint Bartholomews Bagly-wood, Botly, and Gofford Bridge, which Jurisdiction though, while in the Universities hand It hath beene moderatly used, yet when the towne shall have it, both wee and our Neighboures will soone find the Inconveniency. For the Rights and powers Appurtenant thereunto, which are now about to be passed away extend to the whole praecincts of the University, which extend some miles further then those of the Towne. 13. The clarkship of the market is hereby (in effect) graunted away. For that depending cheifly on forfeitures, (which the University officers do commonly give to the poor) All these fall into the Townes hands under those general words in our charters sen aliter delinquentium pro quo bona sua sive catalla amittere sen foris facere debent. Which will extend to these and to all other forfeitures whatsoever: suppose by praemunire or many other wayes beside Felonyes and Treasons. 14. The Right of the Night watch, which is the only thing thought hereby to be secured to us, is so far from being secured that is indeed given away, so far as it was wont to bee controverted. For the controversy betweene the University and the towne about the Night watch heretofore, was not, whether the Vicechancellore & proctors may walk and punish Noctivagatores. For that was alwayes unquestionable till just Now, But, whether they only and not the Major and Baileifs also. And even this hath airways hitherto been judged for us, as in the parliament 18° Edw: 3. and other parliaments; and at Lawe in Paynters case at the Kinges Bench 7° Jacobi; and at the Councill table, by the consent of lot bodyes 10° Jacobi: beside our charters Ed: 3. and many more. But this graunt doth evidently passe that away; alloweing to the towne (as our officers, though not thencforth under our correction) at lest an equal power therein (and

1 Bartlilomews corr. ed. 8 away ( 1 ) the profit of (2) . For that 9 For | the profit of add] that W 9 University (1) doth commonly (2) officers 10 commonly {—} to corr. Wallis 16 secured by us corr. Wallis 16 indeed taken or seeken away corr. Wallis 21 hath also corr. Wallis 26 alloweing (1) power therein (and in much more) (2) to 27 thenceforth add. W

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[II, p. 4]

171. WALLIS to JENKINS, 28 January/[7 February] 1667/8, enclosure in much more) under the pretence of searching for Felons, or for goods writings or evidences belonging to such; even in Colledges and scollers chambers and studies. And what is like to be the effect of such concurrent powers (when the proctors with theire company and the baileifs with theirs, shall walk at once) may be easily foreseene by what happened 29° Edw: 3. when upon a much lesse pretence of Authority the townes-men in warlike manner slew so many of the schollers and committed so great outrages for which all their Rights were seised into the Kinge's hands, and so much of theire former powers taken from them. And if the powers mentioned in King Charles Charter as hereunto appertaining, be but Real; it will soone be evident whether the towne be fit to be intruded with such powers; and whether Archbishop Laud would have taken care to have them so fully expressed had he thought that wee would have passed them to the towne and then be confronted with our owne charters. Who will not spare to stretch it to the utmost and turne our owne charters upon [II, p. 5] us. 15. So much of the Night watch as is pretended to be secured to us (that it may be lawfull for the Vicechancellor and procters to search and punish Noctivagatores) is not at all advanced by it. For if wee have the power without theire acknowledgment, wee need it not. If not: theire concession signifies Nothing: For it is not a thing in theire power to graunt. Nor will theire acknowledgment preclude any single person from bringing his Action to try our power so oft as they please: Nor will it be any plea in law against such an Action. No nor hinder them from countenancing, under hand, any who will soe molest us. Beside that it is so lamely expressed as that (if wee be parties to the Indenture) it will do us more hurt then good: Wee seeming thereby to accept of that as our whole Right which they there acknowledg. 16. No Act of Parliament of doubtfull consequence is usualy made more then a Probationer at first till Experience may discover what is not 1 in add. 2 even in Colleges & Scholars chambers fe Studies add. in margin W 4 (when the Proctors with their company, & the Bailifs with theirs shall walk at once,) add. W 4 company (1) when (2) and 8 all the rights corr. Wallis all their Rights were seized into the Kings hands, and add. in margin W 14 Who will not spare to stretch it to the utmost; & turn our own Charters upon us. add. W

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172. BROUNCKER: Solution, [December 1667/January 1667/8]

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at first foreseene. It would therefore be very imprudent for us, at the first dash, to passe away such a Right accompanied with such powers for 200 yeares: When it is not possible to foresee what mischeifes may follow upon it. These which I have mentioned being but some few of what an expert Lawyer might shew us. And all the consequences of Removing the ancient boundes of such powers as these, nothing but time can discover: when it will be part Remedy or Redresse.

172. WILLIAM BROUNCKER: Solution to Dulaurens's Problem [December 1667/January 1667/8] Transmission:

C Copy of note in Oldenburg's hand: LONDON Royal Society Classified Papers 24, No. 11, p. 1. This copy of Brouncker's solution to Dulaurens's problem, which had been read at the meeting of the Royal Society on 12 December 1667 (BIRCH, History of the Royal Society II, 226), is possibly identical with (a copy of) the note which was enclosed in OLDENBURG-BOYLE 28.1/[7.II].1667/8 (OLDENBURG, Correspondence IV, 120-3) and intended for Wallis; see BOYLE-OLDENBURG 1/[11].II.1667/8 (OLDENBURG, Correspondence IV, 139-40, 139).

Problema transmissum Parisiis, et lectum coram Societate R. die 10

Numero datis Circuli cbd radio eg, et inscripta bd extra circulum porrecta ad libitum, ita tamen ut ad ductam cf acta parallela by circuli circumferentiam secet in puncto y; Verum, non autem vero proximum (quod quidem per sinuum canones facile fieri potest) valorem exhibere lineae by.

1 for us add. W

5 might ( 1 ) discover (2) shew us 5 And | all add. \ the consequences W

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173. JENKINS to WALLIS, 30 January/[9 February] 1667/[1668]

Solutum a Nobilissimo Vice-comite Brouncker, ut sequitur.

CG = R BD = D sit x quaelibet quantitas rationalis, minor differentia Radii et Lineae datae. BY

DF =

173. LLEWELYN JENKINS to WALLIS [London], 30 January/[9 February] 1667/[1668] Transmission:

C Letter sent: OXFORD University Archives WP7/16/1, f. 91r-91v. On f. 91V endorsement in unknown hand: '30 Jan. 1667', and postmark: 'JA/30'. Reply to: WALLIS-JENKINS 28.I/[7.II].1667/8.

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174. WALLIS to OLDENBURG, 1/[11] February 1667/[1668] Thursd. 30. Jan. 67 Sir

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I have your Line773, and your Reasons774 for which I humbly and heartily thank you. Wee'l provide as well as we can for the H. Corpus. & if Mr Just. Brown775 be accostable he shall not be unacquainted with what your line intimates: we shall not, I hope, be surprisd in the Exchequer & I will look after Dr Chace his Case. I beg Mr V: Chancellor776 that the Delegacie will please to think of some Persons more capable then I am to serve them. I shall not fayle to give them and their subdelegates the very best & uttmost of my service, in this & all other occasions. I shall make the best use I can of your Reasons, among our friends & Adversaries (for such also we have) here and shall remain Sir

Your most humble & thankfull servant L. J.

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[91V]

For the Reverend Dr Wallis D. D. publique Professor & Gustos Archivorum at his House neer the Scholes in Oxford.

174.

WALLIS to HENRY OLDENBURG Oxford, I/[11] February 1667/[1668] Transmission:

W1 Paxt draft of letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 30r (written on OLDENBURG-WALLIS 24.XII.1667/[3.I.1668]) (our source).—printed: OLDENBURG, Correspondence IV, 83. 773

Line: i.e. WALLIS-JENKINS 28!/[7!I].1667/8. Reasons: i.e. the 'Reasons against the passing away the Universitys Right in Felons Goods', which were enclosed in Wallis's letter of 28.1/[7.II]. 1667/8 (Extracts from the University Archives). 775 Brown: i.e. Sir Samuel Browne (d. 1668). 776 Mr V: Chancellor: i.e. John Fell, q.v. 774

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174. WALLIS to OLDENBURG, 1/[11] February 1667/[1668] W2 Letter sent: LONDON Royal Society Early Letters Wl, No. 36, 4 pp. (p. 4 blank) (our source). On p. 1 beneath address in Oldenburg's hand: 'Rec. Febr. 3.' Postmark on p. 1: 'FE/3'. — printed: OLDENBURG, Correspondence IV, 141-2. Reply to: OLDENBURG-WALLis 24.XII.1667/[3.I.1668]. Answered by: OLDENBURG-WALLIS 4/[14].II.1667/8.

(W1) Febr. 1. 1667. 777

The French Probleme if understood of rational! numbers, is to imperfectly expressed. For the demand being onely verum valorem exhibere lineae BY; If meant of rational numbers, it should have been numero ratio(na)le exhibere or at lest numero exhibere. B(ut) I find, my Lord Brouncker hath satisfied778 it in that sense: so (that) I need not spend further thoughts about it. J. Wallis.

(W*)

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1C

Oxford Febr. 1. 1667.

[3]

Sir

Your last779 to niee I have observed, & accordingly have put your Transactions Numb. 16.780 into Latine. Which hath been finished some weekes, & when you please shall be sent. The French Probleme, which you say is to be understood of rational Numbers: was (if so) too imperfectly expressed. For the demand being 4 onely (1) of (2) verum 6 exhibere \exhibere del. or 16 of (1) whole (2) rational 777

French Probleme: i.e. the problem of Dulaurens contained in OLDENBURG-WALLIS 10/[20].XII.1667. 778 my Lord Brouncker hath satisfied: cf. BROUNCKER Solution to Dulaurens's Problem, XII.1667-I.1667/8. 779 Your last: i.e. OLDENBURG-WALLIS 24.XII.1667/[3.I.1668]. 780 Transactions Numb. 16.: i.e. Philosophical Transactions No. 16 (6 August 1666), which contained Wallis's hypothesis of the tides (WALLis-BoYLE 25.IV/[5.V].1666), the appendix to this hypothesis (WALLis-OLDENBURG 18/[28].VII.1666), and Wallis's review of Hobbes's De principiis et ratiocinatione Geometrarum (WALLIS-OLDENBURG 24.VII/[3.VIII].1666).

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174. WALLIS to OLDENBURG, 1/[11] February 1667/[1668] onely verum valorem exhibere lineae BY: if ment of rational numbers; it should have been nume.ro rationali exhibere, or at lest nume.ro exhibere. But since that exposition, I have spent no more thoughts on it; (choosing rather to imploy my first leisure hours on your other task.) But I find, by 5 yours781 lately to Mr Boyle, that My Lord Brounker hath satisfied it in that sense. The Correspondences, which you say are desired, I like very well of. As to the books lately printed, I presume Mr Boyle hath given you an account782 what is to be expected hence. For those skilled in Arabick; we 783 10 have here, (beside Dr Pocok,) Mr Clark one of our Bedles & Mr Hyde784 our Library keeper, (both heretofore imployd in the Biblia Polyglotta785,) Mr Huntington786, a fellow of Merton College; Mr Bernard787 a fellow of St Johns College, & now Procter; Mr Marsh788 a fellow of Exeter College; Mr Smith789 of Magdalene College; & some others. There is elsewhere, 5 that (1) Mr (2) My Lord 9 what is to be expected hence add. 10 Clark |one of our Bedles add] & Mr Hyde |our Library keeper add] , (both (1) sometim breaks off (2) heretofore 781

yours: i.e. OLDENBURG-BOYLE 28.1/[7.II]. 1667/8 (OLDENBURG, Correspondence IV, 120-3). This letter apparently contained Brouncker's solution of Dulaurens's problem, which was destined for and passed on to Wallis by Boyle; see BOYLE-OLDENBURG l/[ll].II.1667/8 (OLDENBURG, Correspondence IV, 139-40, 139), and BROUNCKER Solution to Dulaurens's Problem XII.1667-1.1667/8. 782 account: see BOYLE-OLDENBURG 29.XII.1667/[8.I.1668] (OLDENBURG, Correspondence IV, 93-5, 94-5). Boyle's attempts in Oxford to obtain a catalogue of books printed in England during the previous twelve years were not successful. Oldenburg had requested the catalogue for Carcavi. 783 Mr Clark: i.e. Samuel Clarke. 784 Mr Hyde: i.e. Thomas Hyde. 785 Biblia Polyglotta: i.e. S.S. Biblia Polyglotta. Complectentia textus originales, Hebraicos, cum Pentateucho Samaritano, Chaldaicos, Graecos, versionumque antiquarum Samaritanae, Graecae sept., Chaldaicae, Syriacae, Vulg. Lai., Arabicae, Aethiopicae, Persicae quicquid comparari poterat ... Ex mss. antiquiss. undique conquisitis optimisque ex impressis summa fide collatis, ed. Brian WALTON, 6 vols., London 1655-7. Walton was assisted, among others, by Pocock, Clarke, and Hyde. 786 Mr Huntington: i.e. Robert Huntington (1637-1701), orientalist, B.A. 1658, M.A. 1663 at Merton College; later bishop of Raphoe, DNB. 787 Mr Bernard: i.e. Edward Bernard (1638-96), student of oriental languages and mathematics, B.A. 1659, M.A. 1662 at St John's College, later appointed Savilian professor of astronomy; made proctor of the University of Oxford in 1667. DNB. 788 Mr Marsh: i.e. Narcissus Marsh (1638-1713), M.A. 1660, B.D. 1667, and D.D. 1671 at Exeter College, later archbishop of Armagh, DNB. 789 Mr Smith: i.e. Thomas Smith (1638-1710), B.A. 1661 and M.A. 1663 at the Queen's

398

174. WALLIS to OLDENBURG, 1/[11] February 1667/[1668] Mr Nicholas Graves790 (or Edward) brother to John Graves who put out the Persian Grammar & some other bookes in Arabick; & some others whom I know not. The businesse of this, is particularly to desire that some notice be taken of the Spring-tydes which happen at this New-Moon; and the next Full moon; & (if it be not out of memory) what was the last Full-moon for at one of these three times are to bee (according to my hypothesis) the highest Tydes for this time of the year. (Higher than those in March next.) at London & Chatham, as well as on all the shore of Kent. I am Yours &c. J. Wallis. [2]

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To Count Ubaldini's last letter791, I doubt whether I shall give very good account from hence; but I shall not be unmindfull of it.

[1] These

For Mr Henry Oldenburgh in the Palmal near St James's London.

15

2 & some add.\ others 5 at add. 5 next (1) New (2) Full 7-13 for at one . . . unmindfull of it. written at 9(f to the left of first part of letter 1 to bee add. College, fellow of Magdalen College from 1666 onwards. He went to Constantinople as chaplain in 1668. DNB. 790 Mr Nicholas Graves: probably Thomas Greaves D.D. (1612-76), oriental scholar, fellow of Corpus Christi College, Oxford, rector of Dunsby in Lincolnshire, and prebend of Peterborough Cathedral in 1660, DNB. He had two elder brothers: Sir Edward Greaves M.D. (1608-80), physician to Charles II, and John Greaves (1602-52), Savilian Professor of Astronomy from 1643 to 1648. DNB. 791 Count Ubaldini's last letter: now missing. On Ubaldini see WALLis-OLDENBURG 12/[22].II.1666/7.

399

176. WALLIS to OLDENBURG, 8/[18] February 1667/8

175. HENRY OLDENBURG to WALLIS Oxford, 4/[14] February 1667/8 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 8/[18].II. 1667/8. Reply to: WALLIS-OLDENBURG l/[ll].II.1667/8. Among other things, Oldenburg requested of Wallis that he forward him his translation of No. 16 of the Philosophical Transactions.

176. WALLIS to HENRY OLDENBURG Oxford, 8/[18] February 1667/8 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 38, 2 pp. (our source). Postmark on p. 2: 'FE/10'.—printed: OLDENBURG, Correspondence IV, 159-60. Reply to: OLDENBURG-WALLIS 4/[14].II.1667/8. Answered by: OLDENBURG-WALLIS 11/[21].II. 1667/8. Enclosure: WALLIS Solution to Dulaurens's Problem.

Oxford Febr. 8. 1667./8.

Sir,

5

I have sent you herewith, my solution792 of the French Problem in Rationails. Which doth not disagree from that of my Lord Brouncker793, though it's like his processe (which his paper doth not expresse) and mine, were not the same. I have allso sent you (according to yours794 of Febr. 4,) the Latin 5 processe (1) fe (2) (which 7 Latin translation of Numb. 16. originally deleted, but not replaced 792

solution: i.e. the enclosed Solution to Dulaurens's Problem. that of my Lord Brouncker: cf. BROUNCKER Solution to Dulaurens's Problem XII.1667-I.1667/8. 794 yours: i.e. OLDENBURG-WALLIS 4/[14]. 11.1667 793

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176. WALLIS to OLDENBURG, 8/[18] February 1667/8 translation of Numb. 16. which went yesterday from hence (with some other things) to Dr Jenkins (Judge of the Admiralty) at Exeter house: whom I have desired to deliver it to you when you shall call for it. It is a smal parcell covered with brown paper & directed with an endorsement, for yourself. As to the Books of Steno, & Borellus795, which you mention: I like the designs well; & hope they may be well performed. As to the other796, I confesse I am suspicious whether it will answere the title (the rather, because the Probleme is so lamely expressed797, that it makes me doubt whether the Author have the dexterity to expresse his notions clearly.) I wish your Building798 good successe; but cannot promise that I shal here be able to get you subscriptions toward it; nor do I presume to deliver my opinion o(f) the design, (knowing so little of it,) but take for granted so many wise men see very good reasons to undertake it, & ways how to get it perfected. I am sorry I sent you not the advertisement of marking spring tides, a fortnight since; (because its possible that the last full moon might be the most considerable;) but as to that of this New moon, I hope it came time inough (because springtydes at London, usually happen 2 or 3 days after the change & full,) and for the next full-Moon, you have sufficient warning. No more at present but that I am Your most affectionate friend & humble servant, Jo: Wallis.

2 house: (1) it is (2) whom 3 it add. 10 successe; (1) though I (2) but 11 here add. 13 good add. 17 moon, (1) & the (2) I 795

Books of Steno, fe Borellus: i.e. STENSEN, Elementorum myologiae specimen, Florence 1667, and BORELLI, De vi percussionis liber, Bologna 1667. 796 the other: almost certainly DULAURENS, Specimina mathematica duobus libris comprehensa ..., Paris 1667. Oldenburg was still awaiting the two copies which Justel had dispatched no later than in December 1667. See OLDENBURG-WALLIS 10/[20].XII.1667. 797 the Probleme is so lamely expressed: cf. WALLIS-OLDENBURG 1/[11].II.1667/8. 798 Building: i.e. a college for the Royal Society, which the Society planned to build with the aid of subscriptions. The form for subscribing contributions was approved at the meeting of the council on 30 January 1667/8; see BIRCH, History of the Royal Society II, 243-4.

401

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177. WALLIS: Solution to Dulaurens's Problem, 8/[18] February 1667/8 In my Lord Brounkers solution799, x is the same with my r — b — c; at d with him, is 2b in mine: and his BY, is my 2y. These For Mr Henry Oldenburgh, 5 at his house in the Palmal near St James's London.

[2]

177. WALLIS: Solution to Dulaurens's Problem Oxford, 8/[18] February 1667/8 Transmission:

W Paper sent: LONDON Royal Society Early Letters Wl, No. 39, 1 p. (our source).— printed: OLDENBURG, Correspondence IV, 160-2 (Latin original), 162-4 (English translation) . Enclosure to: WALLis-OLDENBURG 8/[18].II. 1667/8.

Oxon. Febr. 8. 1667./8.

10

Quod ad Problema ex Gallia propositum: Numero datis Circuli CBD, radio CG, et inscripta BD, extra circulum porrecta ad libitum; ita tamen ut ad ductam CF acta parallela BY, circuli circumferentiam secet in Y: Verum valorem exhibere lineae BY.

1-2 In my .. . my 2y. written at 90° on left margin 799

my Lord Brounkers solution: cf. Brouncker's Solution to Dulaurens's Problem. 402

177. WALLIS: Solution to Dulaurens's Problem, 8/[18] February 1667/8

Solutio mea nupera800 haec fuit; Datis numeris r = CG, 2b = BD, et sumpto / = DF: Erit Quadratum semissis rectae BY.

Hoc est (extracta ubique radice quadratica) Petitur jam, ut BY sit rationalis. Dico 1°. Si id vellet Problema; dicendum fuisset, verum valorem numero rationali exhibere- saltern numero exhibere- non simpliciter, exhibere. 2°. Ut tamen hoc fiat; Expositis r, 6, (ex hypothesi) rationalibus; ita sumendus erit / rationalis (quo sit 262 — r2 + bf rationalis,) ut sit V/ : r2 + f2 + 2bf : rationalis. Hoc est, ut sit r2 + f2 + 2bf quadratus. Hujus autem quadrati radix (propter 6 < r) erit minor quam r + /. Esto r + / — c (sumpto c rational!:) Cujus Quadratus r 2 + / 2 + c2 + 2rf —

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2rc - 2cf = r2 + f2 + 2bf. Adeoque c2 + 2rf - 2rc - 2cf = 2bf. Hoc est,

2rf - 2bf - 2cf = 2rc - c2. Adeoque / =

15

Sumpto itaque c quovis rationali (saltern qui minor sit quam r —6, vel

major quam 2r;) positoque

Erit

CORR. ED.

3 800

Solutio mea nupera: see WALLIS-OLDENBURG 13/[23].XIL1667. 403

by,

177. WALLIS: Solution to Dulaurens's Problem, 8/[18] February 1667/8

5

rationalis. Dico autem, qui minor sit quam r — b, vel major quam 2r: Quoniam, si major sit c quam r — b, erit 2r — 2b — 2c negativus; adeoque, nisi tarn magnus fuerit ut sit etiam Ire — c2 negativus, erit / negativus: Hoc est, Punctum F quod supponitur ultra D, continget citra D; ut in fig. 3 et 4.

Casus Figurae 2ae, (quern non animadvertisse videtur proponens,) tune continget quando BD exponitur minor quam subtensa Quadrantis, sumiturque BF non tanta ut sit arcus GB major quadrante. Atque sic habes plenariam Problematis solutionem. 10

Sin tibi videbitur expedire solutionem meam, (sine vel analyseos processu vel demonstratione,) nudam transmittere: Sic fiat. Datis numeris r = CG, 26 = BD, sumptoque ad libitum / = DF: Erit

BY Sin porro petatur, ut BY sit rationalis;

3 sit c add.\ quam 3 2r 2 - 26 - 2c corr. ed. 9 habes (1) Prob breaks off (2) plenariam

404

178. WALLIS to JENKINS, 11/[21] February 1667/8 Expositis r, b, rationalibus; sumatur c rationalis quilibet qui sit vel ajor qua minor quam r — 6, vel major quam 2r: fiatque / = Eritque rationalis. Jo. Wallis.

178. WALLIS TO LLEWELYN JENKINS Oxford, 11/[21] February 1667/8 Transmission:

W Letter sent: KEW The National Archives PRO SP 29/234, No. 145, 1 f. (originally folded; address on lower half at 180°; verso blank). Endorsement in Jenkins's hand: 'Dr Wallis about our suits with the Town. 10. Febr. 67.' Postmark: 'FE/12'. Answered by: JENKINS-WALLIS 13/[23].II.1667/8.

Oxford Febr. 11. 1667./8.

5

Sir,

I understand from Mr Vicechancellor801 that (at lest not before eleven of the clock) Fish Line's counsell had not moved for the Return of the Habeas Corpus. If they did not afterwards move, it will be yet our concernment (I conceive) that the return be put into the Court, & to move to have the Cause dismissed upon the Return. Which if it may be obtained, will be a good confirmation of the Precedent in Painters case802. I fear their artifice will be to put off the business by delay's; & if this return be not put into Court this Term, all this processe is lost. And if the cause should onely expire without a decision, wee shall have nothing remaining there on record to be a precedent for future cases. You will, I trust, excuse this trouble, from 7 understand (1) tha breaks off (2) from 8 clock) (1) the (2) Fish 12 good (1) fu breaks off (2) confirmation 801

Mr Vicechancellor: i.e. John Fell, q.v. Painters case: on the case of the bailiff Thomas Painter cf. Wallis's Notes relating to the allowance of the claim in the Exchequer enclosed in WALLIS-JENKINS 21/[31].I.1667/8. 802

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179. OLDENBURG to WALLIS, 11/[21] February 1667/8 Sir Yours to serve you Joh: Wallis.

5

These For Dr Lluellin Jenkins, Judge of the Court of Admiralty, at Exeter-house London.

179. HENRY OLDENBURG to WALLIS London, 11/[21] February 1667/8 Transmission:

C Letter sent: OXFORD Bodleian Library MS. Add. D. 105, f. 31r-32v (f. 32r blank) (our source). Paper torn at bottom of f. 31, causing loss of (at least) one line of text (f. 31r) and of signature (f. 31V). Postmark on f. 32V: 'FE/11'.—printed: OLDENBURG, Correspondence IV, 172-3. Reply to: WALLis-OLDENBURG 8/[18].II.1667/8.

London Febr. 11. 1667/8. 10 Sir,

15

Mr Collins Joyns with MyLord Brounker and yourself in finding fault with the manner of expressing the French probleme; which perhaps the Author did to embarasse the facility of it. I hope, to see MyLord Brounker on Thursday next, and then He will see your solution and processe. Mean time, I had a letter803 from the Author, telling me, that he proposed it not, as if he Judged the solution of it very difficult, but that from it some fine proprieties of the Cercle might be discoverd, that no man yet had spoken off. And when he shall have seen our solutions, he saith, he will returne his owne and a second, found by a friend804 of his at 17 some (1) g breaks off (2) fine 803letter: without doubt DlILAURENS-OLDENBURG [2J/12.II.1667/8

(OLDENBURG,

Correspondence III, 335-6), which Dulaurens had mistakenly dated 1667, but was endorsed by Oldenburg as received on 10 February 1667/8. 804 friend: not identified. 406

179. OLDENBURG to WALLIS, 11/[21] February 1667/8 Paris, together with the consequences, he deduceth from thence. His new Book805, I find, is already on the way: When it corns, such persons as you will soon discover his abilities. There are now arrived two Exemplars of the New Petrus Blesensis806, to which you were pleased to contribute so much; one, I suppose, is for the publick library of Oxford, the other, for 5 yourself. They are not yet in my hands, though I have seen the packet at [31V] Mr Scots807, with whose {—}| then conveigh it to Oxford, God willing. Our friends here are careless enough in observing the Tydes. I went on Tuesday, Wednesday and Thursday last to Whitehall- and Westminsterstairs, to look the High-water mark of the Spring-tides of those dayes, and 10 understood by the Water-men, that on Tuesday-night and Wednesdaymorning the Spring-tide were between 2. and 3. foot higher than the ordinary Tides, though one of the Water-men pretended, the cause of it to be the then fierce blowing North-westerly wind, and the Landfluds: But on Wednesday night and Thursday-morning the Wind was westerly and 15 calme, and yet the Tydes continued as high as before. I shall endeavor to engage others to observe with me in other places the Tydes at the approaching full Moon, as you direct. I have not yet had leisure to call at Exeter house for your Latin Numb. 16.808 but I heartily thank you for it. 20 809 You will find in my letter to Mr Boyle a relation of a strange Earthquake in Persia, hapned not long since, which produced a Lake

10 High-water mark of (1) those (2) the 12 Spring-tide (1) was (2) were 13 the cause |of add. it to be the then (fierce add] blowing 21 find (.7) by (2) in 805 new Book: i.e. DULAURENS, Specimina mathematica duobus libris comprehensa, Paris 1667. 806 New Petrus Blesensis: i.e. PETRUS BLESENSIS, Opera omnia, ed. Pierre de Goussainville, Paris 1667, two copies of which Justel had sent from Paris; cf. JUSTELOLDENBURG [19J/29.II.1667/8 (OLDENBURG, Correspondence IV, 188-90, 188). For Wallis's contribution in collecting information on transcripts of works of Petrus Blesensis in English libraries see WALLIS-OLDENBURG 3/[13] JV.1666 and subsequent letters. 807 Mr Scots: i.e. the intermediary between Oldenburg and Justel. 808 Latin Numb. 16.: i.e. Wallis's Latin translation of Philosophical Transactions No. 16. Wallis had sent his translation to Jenkins, who delivered it to Oldenburg not later than on 13 February; see JENKINS-WALLIS 13/[23].II.1667/8. 809 letter to Mr Boyle: i.e. OLDENBURG-BOYLE ll/[21].II.1667/8 (OLDENBURG, Correspondence IV, 169-71).

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180. JENKINS to WALLIS, 13/[23] February 1667/[1668] where was none, and dryed up another, that was before, and was full of fish. I am in hast

Sir Your very humble and faithf.

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[32V]

For his much honor'd friend Dr John Wallis Savilian Professor of Geometry in Oxford.

180. LLEWELYN JENKINS to WALLIS [London], 13/[23] February 1667/[1668] Transmission:

C Letter sent: OXFORD University Archives WP7/16/1, f. 93r-94v (f. 93V and 94r blank). On f. 94V endorsement in unknown hand: '13 Feb. 1667', and postmark: 'FE/13'. Reply to: WALLIS-JENKINS ll/[21].II.1667/8.

13 Febr. 67.

Sir 15

Your Papers810 have been delivered to Mr Oldenbergh. Mr Serjeant Holloway811 is positive that we ought cannott goe on or doe any act this Terme: he having inform'd himself (as he sayes) by Mr Mandy the Protonotary that the first act or Motion regularly necessarily must be Line's812. What his opinion will be next Terme I know not: sure he must change or else we shall in a mischief otherwise inextricable. But he undertakes

810

Your Papers: i.e. Wallis's Latin translation of Philosophical Transactions No. 16; see OLDENBURG-WALLIS ll/[21].II.1667/8. 811 Holloway: i.e. Charles Holloway. 812 Line's: i.e. Fish Lyne. 408

181. WALLIS to COLLINS, early 1668? to satisfie Mr V. Chancellor813 & all Persons Concern'd when he comes down. I am

Sir Your most humble servant L. Jenkins

[94V] For the Reverend Dr Wallis D.D. at his House neer the Scholes in Oxford.

181.

WALLIS to JOHN COLLINS early 1668? Transmission:

Manuscript missing. Existence and date: Referred to in WALLIS-COLLINS 15/[25].II.1667/8. This letter was clearly written after WALLIS-COLLINS 5/[15].II. 1666/7 and before WALLIS-COLLINS 15/[25].II. 1667/8. The manner of reference suggests it had been sent recently.

182. WALLIS to JOHN COLLINS Oxford, 15/[25] February 1667/8 Transmission:

W Letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, 196r-196av (f. 196ar blank) (our source). On right margin off. 196av in unknown hand: 'Dr Wallis de Angulo Contactus'. Postmark on f. 196av: 'FE/17'.—printed: RIGAUD, Correspondence of Scientific Men II, 485-8. Answered by: CoLLiNS-WALLis 25!I/[6!II]. 1667/8. 813

Mr V. Chancellor: i.e. John Fell, q.v.

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182. WALLIS to COLLINS, 15/[25] February 1667/8 Oxford. Febr. 15. 1667./8.

Sir, I have this evening received Leotaud's814 Cyclomathia815 which you sent mee; & have read over that part of his second Book816 which is particularly directed against mee817; and some of the former propositions which by 5 their titles seemed most to concern the business, as the 4 th , 5 th , 6 th , 7th (& some others) with their demonstrations. And I find (as my last818 did presage,) that, (beside the cavilling at some phrases & expressions, which hee would strein to such a sense as to make appear absurd,) there is 10 nothing material in it that seems to need any answere. The whole stress of what hee sayth depends on what he repeats out of Clavius819, that the Angle of Contact, & the Right-lined Angle, are Heterogeneal, & not capable of Proportion; which they would found on 5 Def. 5 Euclid820. Which they would have to be, not a definition of Homogeneal Quantities, (which is the whole drift of that Definition) but of such Homogeneal 15 Quantities as have proportion one to another: Supposing (contrary to the 4th definition,) that, of Homogeneous quantities, some have & some have not proportion to one another. To which I had spoken so fully in my Treatise de Angulo Contactus821, cap. 5, 6, 7, 8, as that the reading of those chapters is Answer inough to all of this kind. Of all which he takes very little or no notice. And if wee should grant that to be the meaning of the definition, as he would have it, (which indeed it is not;) yet this could serve him at the most, but onely as to the Angle of Contact; not as to that of a Semicircle, or other segment of a Circle. For it is manifest, that both a 4 particularly (1) written (2) directed 17 & some |have add. not 22 definition, (1) which hee would have, (2) as 814

Leotaud's: i.e. Vincent Leotaud, q.v. Cyclomathia: i.e. LEOTAUD, Cyclomathia sen multiplex circuli contemplatio tribus libris comprehensa, Lyons 1663. 816 second Book: i.e. 'Liber secundus. Explicatur Anguli contactus natura ex Euclidis, caeterorumque Geometrarum rnente: alienaque de eo aliquot Recentiorum sententia clare ac evidenter refellitur' (157-234). 8lr which . . . mee: cf. WALLIS-HUYGENS 31.VIII/[10.IX].1668. 818 my last: i.e. WALLis-CoLLiNS early 1668 ?. 819 Clavius: i.e. Euclidis elementorum libri XV . . . , ed. CLAVIUS, 2 vols., Rome 1574, vol. I, 154r. 820 5 Def. 5 Euclid: i.e. EUCLID, Elements V, def. 5. 821 De Angulo Contactus: i.e. WALLIS, De angulo contactus, et semicirculi, disquisitio geometrica, Oxford 1656. 815

410

182. WALLIS to COLLINS, 15/[25] February 1667/8 Right Angle may be so multiplied as to exceed any Angle of a segment; & that the Angle of any segment may be so multiplied as to exceed a Right Angle: And therefore these at lest must be such Homogeneal quantities as have proportion one to another, by that definition. Which granted, doth directly overthrow his whole hypothesis; as himself is aware. And though I had urged this, clearly & strongly inough in my Treatise, especially cap. 7, yet of this (because it did pinch too close) hee takes no notice. For all those arguments drawn from the 1 prop. 10 Euclid822. & 2a prop. I1 Archimedis de Sphaera et Cylindro823, (which are the foundation of all those demonstrations whether of the Ancients or of the moderns,) which proceed (as they speak) by way of Exhaustion. (Which do all suppose; that, it is a sufficient proof of Equality, to prove that quantities differ less than by any assignable part.) Hee thinks it a sufficient evasion, to say, that a Right Angle may exceed that of a Semicircle, by such an excesse as is lesse than the Infinitesima pars of either. Which evasion, if it be allowed, will as well elude all those demonstrations by way of Exhaustion, [196V] or| by Inscription & Circumscription of Right-lined & Curve-lined figure, (so frequent amongst both the Ancients & the moderns.) For how easy is it to answer (if this be allowed) to that of Archimedes (for instance) de dimensione circuli; That, a Circle is bigger (or lesse, if you will,) than the Triangle which he proposeth824 as equal to it, but not by any assignable quantity, but by somewhat that is lesse than the Infinitesima pars thereof. And the like to all demonstrations of that kind. To my last Argument, from Opticks; he hath no other evasion, but to

1 any (1) seg breaks off (2) Angle 4 have |(even in his sense) del. proportion 6 especially add. 8 & (.7) 1 prop. 2 (2) 2 a prop. I1 10 all add. 10 ) add. ed.

16 allowed, (1) doth (2) will 16 those (1) Arg breaks off (2) demonstrations 20 (or ... will,) add. 21 not | bigger del] by 24 To (1) that (2) my 822

1 prop. 10 Euclid: i.e. EUCLID, Elements X, prop. 1. de Sphaera et Cylindro: i.e. ARCHIMEDES, De sphaera et cylindro I, prop. 2. 824 proposeth: i.e. ARCHIMEDES, De dimensione circuli, prop. 1. 823

411

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668]

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deny825, that, in speculis curvis, Angulus incidentiae est aequalis angulo reflectionis. Who, I presume, is the first that ever did deny it. To an Argument of Gregories, or Ainscombs, from that of Euclide826, si auferatur plusquam dimidium, atque ex reliquo plusquam dimidium, &c. (1 El. 10.) Hee tells827 (ad prop. 7.) that it is not to be left to the Demonstrator, to make such ablations, by what way he thinks fit; but that his Adversary is to direct, in what way such ablations shall bee made. Which discovers so great a weakness; as if either he did not know what it is to demonstrate; or else meant, of design, to prevaricate. But I suppose, hee might have been ingaged, unawares, in his former Examen828, against Ainscomb, or Gregory; which rather than to retract, hee resolved to defend as well as he might; & was obliged to take mee in by the way. But inough at present, from Sir

your friend & servant John Wallis

15

20

[196av]

For Mr John Collins, Accountant for his Majestic at the Excise Office in Bloomsbury London.

183. WALLIS to VINCENT LEOTAUD Oxford, 17/[27] February 1667/[1668] Transmission:

E1 First edition of (missing) letter sent: WALLIS, A Defense of the Treatise of the Angle of Contact, 79-88 (our source). E2 Second edition: WALLIS, Opera mathematica II, 638-45. 4 auferatur jplusquam add. \ dimidium, 825

deny: i.e. LEOTAUD, Cydomathia, 232-3. that of Euclide: i.e. EUCLID, Elements X, prop. 1. 82r tells: i.e. LEOTAUD, Cydomathia, 177. 828 former Examen: i.e. LEOTAUD, Examen circuli quadraturae hactenus editarum celeberrimae quam Apollonius alter, magno illo Pergaeo non minor Geometra R. P. Gregorius a Sancto Vincentio Societatis lesu, exposuit, 2 parts, Lyons 1653-5. 826

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] This letter, a reply to Leotaud's Cyclomathia, was first sent to Oldenburg in London who then transmitted it to his correspondent in Paris, Henri Justel. Cf. WALLIS, A Defense of the Treatise of the Angle of Contact, 79: 'It was some years after the Book was published, before I saw it: When I had seen it, I wrote a Letter to himself, in Answer to it. Which Mr. Henry Oldenburg (at my request) sent to his Correspondent at Paris, to transmit to Leotaud. This Letter, Mr. Oldenburg told me soon after, that his Correspondent had received at Paris, and would take care of it. But whether or no it came to Leotaud's hands, I am not certain; having since heard nothing of it: And, not long after, I heard that Leotaud was dead.' It would appear that this letter was conveyed by Henri Justel to Leotaud. See JUSTEL-OLDENBURG [25!V]/5.V.1668; OLDENBURG, Correspondence IV, 333-5. Cf. WALLIS-HUYGENS 31.VIII/[10.IX].1668.

Clarissimo Viro D. Vincentio Leotaudo, Delphinati, Joannes Wallis, Oxoniensis, S. Oxoniae, Feb. 17. 1667. Stilo Angliae.

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Incidi, Vir Clarissime, nudius tertius, in Cyclomathiam tuam; ante quinque annos, ut videtur, impressam; sed nuperrime (quantum audio) hue allatam. Quam inspiciendam obtulit Amicus quidam meus830, harum rarum peritus; eo praesertim nomine, quod me ibidem animadverterit a te notatum. Quod fecit ut ad ejusdem Librum Secundum, qui me spectare dicebatur, me statim converterim; omissis Primo & Tertio, quibus de aliis rebus agitur. Quid feci, ut me ibidem in arenam vocaveris, nescio; neque solicitus inquire. Moleste forsan tuleris (utut vos inter vos dissentire non iniquum judicetis) quod contra Clavium vestrum (Jesuitam) ego (non Jesuita) nonnulla scripserim831. Adeoque vestra interesse putaveris, ut ex vestra Societate non-nemo causam ejus (sive justam sive injustam) utcunque defendendam susciperet. Quod tamen non erat necesse ut contra me faceres, qui non soleo Clavii vestri iniquus esse aestimator: Cujus etiam causam (ut & Gregorii San-Vincentiani vestri) contra Meibomium susceperim832; eumque alias, prout res tulerit, passim defendo. Quanquam enim in Re13 fecit E2 829

Cyclomathiam: i.e. LEOTAUD, Cyclomathia sen multiplex circuli contemplatio tribus libris comprehensa, Lyons 1663. 830 Amicus quidam meus: i.e. John Collins. Cf. WALLis-CoLLiNS 15/[25].II.1667/8. 831 scripserim: i.e. WALLIS, De angulo contactus, especially chapter 5-8. 832 susceperim: i.e. WALLIS, Adversus Marci Meibomii, De Proportionibus dialogum, tractatus elenchticus.

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ligionis negotio nos a vobis diversa sentiamus, non proinde necesse erit, ut dissentiamus in Mathematicis: Ubi non Authoritatibus res agenda est; sed, Demonstrationibus. Vel fieri potest, ut controversiis quae Tibi cum Aynscomio tuo intercesserant implicitus, non commode te expedire posse putaveris, nisi & me simul in partes vocaveris; cum ea, quae de Angulo Contactus & Semicirculi scripseram, Gregorii de Sancto Vincentio, Aynscomii, & Tacqueti, placitis quibusdam a te oppugnatis, favere videantur. Quaeque si prius vidisses, fieri fortasse posset, ut inde abstinuisses plane, & ne Aynscomium tuum ea de re solicitasses: (utut ubi temere forsan manum conseruisses, defensionem malueris utcunque moliri, quam videri palinodiam canere.) Quippe quam suscepisti contra Gregorii de S. Vincentio quadraturas controversiam833, non minus feliciter expedire potuisses, licet hanc intactam praeteriisses; quae cum ilia connexa non est: Praesertim, ubi sequiorem partem tuendam susceperis; cum, in ea de quadraturis, potiorem suscepisse videaris. Quanquam enim ego Gregorium San-Vicentianum, pro Mathematico minime imperito habeam; ut qui multa & acute & solide scripserit; (cujusque causam, ut dictum est, contra Meibomium, ne rogatus quidem, susceperim:) Quadraturas tamen absolvisse non existimo. Vel denique (quod potius speraverim) sine partium studio, mero veritatis intuitu, potueris hoc fecisse: Eadem libertate, qua & ego soleo ab aliis nonnunquam in paucis discedere, quos alias defendendos existimo: Praesertim cum te a probris, ut plurimum, abstinuisse videam. Quicquid sit quae contra me multis scripseris, paucis diluenda visum est: Neque enim prolixa Refutatione opus erit; sed potius brevibus stricturis. Quippe in angustum res redacta est: Num scilicet Angulus Contingentiae sive Contactus, ad Angulum Rectilineum (aut etiam ad alios Curvilineos & Mixtos) comparatus, pro Homogeneo habendus sit, an pro Heterogeneo (saltern quoad Rationem) & nullius Rationis capaci. Quippe in hoc unico, & Tu & Clavius Praesidium collocatis, & speratis Asylum; contra ea in contrarium prolata Argumenta, quae alias (ne vobis quidem diffitentibus) Demonstrativa habenda erunt. Si enim & Quantus fuerit (quod vos vultis) & Angulo Rectilineo Homogeneus; non erit quod Absurda ilia possitis declinare, quae non modo Pefetarras834 833

controversiam: i.e. LEOTAUD, Examen circuli quadraturae hactenus editarum celeberrimae quam Apollonius alter, magno illo Pergaeo non minor Geometra R. P. Gregorius a Sancto Vinventio Societatis lesu, exposuit, Lyons 1653. S3 *Peletarius: i.e. Jacques Peletier (1517-82). The arguments he employed in his con-

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] & Ego, (ne Savilium porro & Vietam memorem) sed & Vestrates SanVincentianus, Aynscomius, & Tacquetus, cumulate vobis objiciunt. Quam quidem rem, quanquam in meo De Angulo Contactus & Semicirculi tractatu, cap. 5, 6, 7, &c. me satis confecisse puto, ni praejudicium tibi oculos perstrinxisset: Quoniam tamen tu illud nondum assequi videri velis, te sequar, ut quae superest caligo, si fieri possit, etiam a tuis oculis discutiatur. Propositionem tuam primam835 ego hactenus concede: Nempe, Quo jure quis Quantitatem infinite extensam Imaginari velit, eodem & infinite diminutam imaginandam permittere debeat. Idque utroque sensu, quo Infiniti vox solet occurrere. Si enim, per Infinitum, intelligatur Indefinitum, seu quantumlibet magnum; quo sensu, apud Geometras, Recta Infinita hoc est, quantumlibet longa, vel quantum opus est longa, vel ducta supponitur, vel ducenda praescribitur: Quo jure Rectam quantumlibet Longam possibilem esse ponimus, eodem & quantumlibet Brevem possibilem esse, concedendum erit. Quippe prout supponitur Continuum posse in Infinitum continuari, ita & in Infinitum dividi; hoc est, nullos vel Continuationis vel Divisionis states esse terminos, ultra quos procedi sit impossibile. Si vero, per Infinitum, intelligatur id quod sit Absolute Infinitum Actu; (puta quod totam possibilitatem habeat in actum reductam:) Etiam hie concede, quo jure quis, hoc sensu, imaginari velit Infinite-Magnum, etiam Infinite-Parvum Imaginandum esse. Sed, Imaginandum potius dico, quam Datum iri. Ad Secundam Propositionem836 quod spectat; Concede, Infiniti ad Finitum, nullam esse [Finitam] Rationem: (Neque etiam Indefiniti ad Definitum rationem Definitam.) Dico tamen: Quo jure quis Quantum Infinitum imaginari velit, eodem & Infinitam Rationem imaginandam esse. Adeoque, Infiniti ad Finitum, aut etiam Finiti ad Infinite-exiguum, rationem esse dico Infinite-magnam: Finiti ad Infinitum, vel etiam Infiniteexigui ad Finitum, rationem Infinite- exiguam. Neque tibi in contrarium suppetias feret quinta Definitio quinti Euclidis837: Dicam utique, Infiniteexigui Infinite-Multiplum, expositum quodvis Finitum aequare posse, nedum superare. Atque ego pari jure admittendum postulabo Infinite-Mulflict with Clavius over the horn angle are quoted in CLAVIUS (ed.), Euclidis elementorum libri XV, 3rd ed., Cologne 1591, 133-45. 835 Propositionem tuam primam: i.e. LEOTAUD, Cyclomathia, 157-61. 836 Secundam Propositionem: i.e. LEOTAUD, Cyclomathia, 161-2. 83r quinta . . . Euclidis: i.e. EUCLID, Elements V, def. 5.

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tiplum, quo tu vel Infinite-Magnum vel Infinite-exiguum. Dum autem quaeris838, num Rationalis futura sit haec Ratio, an Irrationalis: Utrumvis dicas, perinde est: (quippe, quodcunque fuerit hoc Infinitum, quod ad Finitum unum habiturum est Rationem Rationalem; idem ad Finitum aliud, Rationem habebit Irrationalem; modo ilia Finita duo, Rationem habeant ad invicem Irrationalem:) Hujusmodi siquidem minutias omnes, absorbet ipsa Infinitude. Atque perinde est acsi peteres, si quis imaginari velit Numerum Infinitum, num futurus sit ille Par an Impar? Vel Tripartibilis, an secus? &c. At interim, quo jure tu vel Infinite-Magnum, vel Infinite-|Parvum imaginaberis; eodem ille: Vel [81] Infinite-Multum, vel etiam Infinite-Paucum (ipsa Unitate in Fractiones divisa) imaginabitur. Ad Prop. 3.839 Concede, Quantitates etiam Finitas esse, quarum nulla potest esse ad invicem Ratio. Tales utique sunt quantitates quae sunt ad invicem Heterogeneae, puta, Linea & Planum, Planum & Solidum, Linea & Solidum: item Angulus & Linea, Angulus & Superficies, Linea & Tempus, Tempus & Pondus, atque hujusmodi alia cum Heterogeneis comparata. Est utique Ratio (per def. 3. El. 5.840) Homogeneorum ea relatio quae est Kara Tr^AtKorr/ro;. Homogenea vero, seu (quod per def. 3. tantundem valet) Rationem invicem habentia, sunt (per def. 5.) ea quae Multiplicata possunt se mutuo superare. Quoniam itaque Hora Temporis, utcunque Multiplicata, nunquam aequabit vel superabit Libram Ponderis; Hora & Libra, seu Tempus & Pondus, Heterogenea censenda erunt, adeoque non Rationem ad invicem habentia. Atque de reliquis similiter. Estque haec unica Homogeneorum definitio, quae apud Euclidem uspiam extat. Hinc discas, Curvam quamvis & Rectam, utut Dissimiles, Homogeneas tamen esse; quoniam exposita Curva quaevis ita Multiplicari potest, ut expositam quamvis Rectam superet; & vice versa. Sic Curvilineum & Rectilineum; puta Circulum & Quadratum: Expositus utique Circulus si exposito Quadrato nondum major sit, erit saltern ipsius Duplum, Triplum, vel aliud aliquod Multiplum, quadrato illo majus; & vice versa. Lineam vero & Superficiem Heterogeneas esse; quoniam Linea, cum nihil habeat Latitudinis, quantumvis Multiplicata, necdum habebit; (quippe nihili Duplum, seu alias Multiplum, est adhuc nihil;) adeoque nee fiet Superficies. 838

quaeris: see LEOTAUD, Cyclomathia, 161. Prop. 3.: i.e. LEOTAUD, Cyclomathia, 162-5. 840 def. 3. El. 5.: i.e. EUCLID, Elements V, def. 3. 839

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] Atque hinc speciatim discas, Angulos Pianos omnes, sive sint Rectilinei, sive Curvilinei, sive Misti, (qui ullius sint Magnitudinis,) invicem Homogeneos esse. Sunt utique vel Aequales, vel Majores, vel Minores exposito Rectilineo; & quidem si Minores, possunt saltern Multiplicati Majores fieri; & vice versa. (Quod ne tu quidem de quovis negaveris, excepto solo Angulo Contactus.) Cumque tu hoc in Angulo Contactus desideratum animadvertis; id non eo fit quod Heterogeneus sit, sed quod non sit Quantus. Miror autem ego te, hominem Mathematicum, existimare posse, turn totum Angulum Rectum, turn (quern hujus partem esse vis) Angulum Semicirculi, cuivis Recto Homogeneum (per 5 def. 5.841) reliquum vero quern facis Angulum Contactus Heterogeneum esse. Quasi quidem fieri possit, ut & Totum, & Ablatum, sed non & Reliquum, eidem alicui sit Homogeneum. Cum vero tu existimas, Quantitatum invicem Homogenearum, alias habere, alias non habere, rationem ad invicem; atque has ab illis Euclidem definitione 5. disterminasse: Hoc ipsum est quod & Clavio primum, atque post ilium Tibi, aliisque multis fraudi fuit. Euclides utique jam ante (def. 3.) turn Rationes omnes, Homogeneorum esse; turn & Homogeneorum omnium, esse ad invicem Rationem; non minus definiverat. Quoniam vero Homogenei vocem, non prius ab illo usurpatam, nedum in superioribus definitam, sed quae definitione omnino indigeret, hie usurpaverit; hanc alteram Homogeneorum, seu (quod ipsi est Iao5vva^tov} Rationem ad invicem habentium, definitionem subjungit. (Quae & in Graecis Codici[82] bus immediate subjungitur Tertiae; cur autem Clavius hanc| Quintam fecerit, interposita Quarta quae est in Graecis Octava, ego nescio.) Euclides itaque, neque ob eum finem quern tu insinuas, neque frustra tamen, sed justis de causis hanc quintam interposuit definitionem. Nempe, ut quid per Homogenea seu Rationem invicem habentia, significatum velit, definiret. Sed & per hanc ipsam Definitionem, & per 1 Prop. 10.842 determinat; Homogenei cujusvis nullam esse posse tarn exiguam partem, quae Multiplicata non possit totum superare. Quae itaque tu hie ex Aynscomio843 habes844, Suppono, inter duas 23 ipse corr. ed. 841

5 def. 5.: i.e. EUCLID, Elements V, def. 5. 1 Prop. 10.: i.e. EUCLID, Elements X, prop. 1. 843 ex Ayncomio: i.e. AYNSCOM, Expositio ac deductio geometrica, 5. 844 habes: see LEOTAUD, Cyclomathia, 163. 842

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] Magnitudines esse aliquam Rationem, idem esse ac, duas Magnitudines secundum quantitatem (Kara rKr\\i^OTr\Ta) posse comparari: Rursum, illas Magnitudines sic posse comparari, de quibus did potest, haec major aut minor est ilia: Unde consequens est, Magnitudines illas, juxta de/. 5 5. lib. 5. Eucl.8^5 ejusmodi esse, ut una aliquoties sumpta possit alteram aequare vel superare: Adeo sunt & veritati & Euclidis menti consona, ut quo te vertas cunque nunquam sis evasurus. Propositionem quartam846, (quae est Euclidis Definitio Anguli Plani) ego admitto. Quod, Angulus Planus, est duarum linearum in piano se 10 mutuo tangentium, & non in directum jacentium, alterius ad alteram inclinatio. Adeoque haec saltern tria requiri judico, Quod in Piano lineae se mutuo tangant, Quod ad invicem inclinentur, Quodque non in directum jaceant: Et propterea, si vel in directum jaceant, (ut cum una sit alterius 15 continuatio,) vel nulla sit ad invicem inclinatio, (ut in Parallelismo, & cum altera alteri immergitur;) Angulum vel nullum fieri, vel nullius Magnitudinis: Sed &; horum alterum contingere, quoties ita concurrunt lineae, ut, licet continuentur, se mutuo non secabunt: Et propterea Contactus Angulum nullius esse Magnitudinis.

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Verbi gratia: Parallelae rectae A0, CD, Angulum non constituunt, turn quia nondum se mutuo tangunt, turn quia nulla est alterius ad alteram inclinatio, sed Parallelismus. Si vero AB, retento Parallelismo, deorsum ferri intelligatur donee ipsi CD occurrat: Tactus quidem net, sed non Angulus; propter nullam alterius ad alteram inclinationem: Nee, utcunque continuatae, se mutuo secabunt, sed altera alteri immergetur. 845

def. 5. lib. 5. End.: i.e. EUCLID, Elements V, def. 5. Propositionem quaxtam: i.e. LEOTAUD, Cyclomathia, 165-7.

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] Sin, eodem retento Parallelismo, transferri intelligatur AB in situm DE- Tactus etiam sic net, non autem Angulus, cum altera sit alterius continuatio. Eadem vero AB, & (non Parallela) FG, invicem inclinantur; sed Angulum non constituunt, quia nondum se mutuo tangunt. Sin, eadem retenta inclinatione, sursum moveatur AB vel FG deorsum, donee invicem occurrant; Angulus net: (erit utique linearum occurrentium, nee in directum positarum, Inclinatio.) Sed &, propter illam ad invicem inclinationem, si continuentur se mutuo secabunt. In lineis Curvis, cum Curva Rectae non possit congruere, eadem [83] tamen Analogia accommodanda erit.j

Rectae HIK, subsit LMN Semicirculus; cujus supremum punctum, rectaeque HIK proximum, sit M: Manifestum est, turn varias Peripheriae partes varium respectu ipsius HIK rectae situm habere, turn quae propius sunt ad M propius ad Parallelismum accedere; ej usque propterea in ipso M situm, Parallelism! instar habendum; rectamque HIK, utpote huic situi Parallelam, si retento hoc Parallelismo deorsum ferri intelligamus, donee in M peripheriae occurrat; non tarn secabit ilia peripheriam, aut ad illam inclinabitur, quam super ipsum M punctum aKXivuq jacebit, Angulum vel nullum vel nullius Magnitudinis efficiens, (pariter atque AB recta ad rectam CD demissa,) propter nullam utrobique inclinationem. (Quern nullius Magnitudinis Angulum, Angulum Contactus dicunt,) Si vero ulterius adhuc demittatur eadem H IK recta; in binis semper punctis, 23 binis subinde punctis E 419

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(sed ubi alius est Peripheriae situs, ad illam rectam, quam in M fuerat) secabit; Angulos faciens Rectilineis vel aequales vel proportionales. Miraris autem tu, (pag. 209.) Tantae apud Me Authoritatis esse Peletarium, ut cum eo ausim affirmare, [eas solas Lineas indinari in puncto concursus, quae, si producantur, se mutuo secabunt,] quam cum Euclide sentire [duas quaslibet lineas quomodocunque concurrentes, mutuo indinari; sive, quod idem est, Angulum constituere.] Ego vero Peletarii Authoritate non moveor, (ut neque Clavii,) sed Argumentis, & rei necessitate. Miror autem ego te existimare posse847, Eudidem sentire, duas quaslibet lineas quomodocunque concurrentes mutuo indinari, seu (ut ais) quod idem est Angulum constituere. Ego (cum Euclide) duos casus excipio: Immersionem & Continuationem; (propter nullam utrobique Inclinationem, sed potius Parallelismum.) Si feratur AB in situm CD, non Angulum cum hac faciet (saltern nullius Magnitudinis,) sed Immersionem: Si ad situm DE; non Angulum, sed Continuationem: Si ad situm his intermedium; partim immergetur, partim continuabit ipsam CD: Angulum certe non constituet (saltern non ullius Magnitudinis) in sensu Euclidis, (qui non per Tangentium tactum, sed per Tangentium inclinationem, definit Angulum) cum nulla sit concurrentium Inclinatio. Tu si secus sentias, fruere tuo sensu. Sed & te male habet, (pag. 166.) quod dixerim ego, Recentiorum aliquot magnos viros, & ex veteribus fortasse nonnullos, de Angulo Contactus ita locutos esse, ac si haberet Anguli quantitatem; non dixerim Omnes. Atque exclamas, Notum esto ac definitum, Omnes Geometras turn Antiquos turn Recentiores, veritatis & observantiae gratia, Euclidis sententiae subscripsisse. Festina lente, Vir Clarissime; (quippe haec aXa^ovsiav potius sonant atque jactantiam, quam Mathematical!! Demonstrationem.) Tune Omnia Omnium, turn Antiquorum turn Recentiorum, scripta legisti? Omniaque ibidem lecta, turn animadvertisti probe, turn probe meministi? Ego certe, qui nee Omnia Legi, nee Lectorum omnia Memini, cautius loqui soleo. Sed nee observantiae gratia, sed Veritatis & Demonstrationum, soleo Geometris subscribere. Tu forte Clavio, observantiae gratia; ego Peletario, Veritatis tantum gratia subscribe. [84] 15 ad sit am DE corr. ed. 29 Tune corr. ed. 847

Miror . . . posse: see LEOTAUD, Cydomathia, 209. 420

183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] Quaenam autem fuerit Euclidis sententia, nondum inter te & me convenit. Ego Euclidem saltern, & Apollonium, ex meis partibus stare existimo. Demonstrant utique Angulum Contactus (ille ad Circulos, hie ad Sectiones Conicas) saltern Minorem esse quam infinite exiguum, (neque uspiam dicunt, aliquid habere Magnitudinis.) Quod autem tale est, ego quidem (Euclidis authoritate fretus, Def. 5. El. 5. & Prop. 1. El. 10.) Non- quantum esse existimo. Verum quidem est, Euclidem non totidem verbis pro me pronunciasse: Nequid tamen in contrarium dicat (nedum demonstret) caute abstinet, turn ad Prop. 16. turn ad Prop. 31. Lib. 3.848 (quod Capite 2. ostenderam.) Atque Argumentis aliunde ex eo petitis, ad meas partes trahetur. Caeterique Graeci (quantum scio) omnes, uno excepto, vel de hoc negotio plane tacent, vel ita caute pronunciant, ut meis potius partibus favere videantur. Tu si uno illo plures ex Graecis noveris, (quos ego vel non legi, vel non animadvert!, vel non memini) qui Angulum Contactus positive-quantum esse, aperte dixerint: Opitulare, quaeso, nescientiae meae; mihique benignus indica. Fieri quidem potest ut plures sint, (ideoque dixi, fortasse nonnullos:) Ego, praeter unicum, neminem novi. Ex Recentioribus Latinis, plures agnosco tecum sentire: Non Omnes tamen. Quippe praeter Peletarium (quern mihi, credo, concedes) Tres saltern ex tuis, San- Vincentianus84^9, Aynscomius850, atque Tacquetus851, (quos ut magnos viros praedicas) secus atque tu sentiunt: Atque ex aliis, SAVILIUS852 saltern & VIETA853 (Viri certe tuis non minores) quibus & Flussatam854: addas (nobilem Euclidis Interpretem.) Qui quamvis non eodem modo se omnes expediant, a Te tamen Omnes diversa sentiunt (atque a Clavio tuo.) Atque in hoc saltern omnes consentiunt mecum; Quod impossibile est, ut & Angulus contactus positivam habeat Anguli 848

Prop. 16. . . . Lib. 3.: i.e. EUCLID, Elements III, prop. 16 and 31. San-Vincentianus: i.e. ST-VINCENT, Opus geometricum, lib. 8, 870-1. 850 Aynscomius: i.e. AYNSCOM, Expositio ac deductio geometrica, cap. 2, 5-13. 851 Tacquetus: i.e. TACQUET, Elementa geometriae planae ac solidae. Quibus accedunt selecta ex Archimede theoremata, Antwerp 1665, 86-93. 852 SAVILIUS: i.e. SAVILE, Praelectiones ire.sde.cim in principium Elementorum Eudidis, Oxford 1621, 54-5. 853 VIETA: i.e. VIETE, Variorum de rebus mathematicis responsorum, liber VIII, Tours 1593, cap. 13; Opera, mathematica, 386-7. 864: Flussatam: i.e. FRANCiscus FLUSSATES CANDALLA (Francois de Foix, Comte de Candale), Elementa geometrica: Libris XV; ad germanam geometriae intelligentium e diversis lapsibus temporis iniuria contractis restituta . . ., Paris 1566. 849

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] Magnitudinem, quae tamen utcunque Multiplicata nunquam vel minimum aequet rectilineum, nedum excedat; & simul constent Def. 5. Lib. 5. & Prop. 1. Lib. 10. cum Def. 3. Lib. 5. Euclidis. 5

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Sed & Catoptricos, ad unum omnes, existimo ex meis partibus esse. Quippe qui & uno ore consentire videntur, Angulum Incidentiae & Angulum Reflectionis aequales esse, turn qui ad Speculum fiunt, turn qui ad Planum Tangens. Quod fieri non potest, nisi vel Anguli Contactus nullius sint Magnitudinis, vel semper aequales: Quorum utrumvis, in speculis Parabolicis, Ellipticis, Hyperbolicis, & (praeter Sphaerica) curvis omnibus, tibi pariter adversantur, atque pro me concludunt. Dum vero tu (Pag. 232.) negas, in hujusmodi Curvis speculis Angulum Reflexionis Angulo Incidentiae aequalem esse: Tu certe primus es qui hoc dixeris, nee eris audiendus. Ubi vero tu Euclidem existimas (Pag. 164.) ne syllabam quidem perperam tradidisse: Sed nee Libraries quicquam vel addidisse vel immutasse, (quod Tacquetum subdubitasse dicis, Pag. 200.) sed erga hunc Geometriae parentem observantiores semper fuisse, quam ut ejus opus tarn absolutum quoquo modo temerare non vererentur, (Pag. 204.) Nae tu homo credulus es atque EvrrsiOr/s, qui haec sentis! (quod itaque in Theologicis minus mirabor.) Videris tu certe, Euclidis Codices Manuscriptos nunquam vidisse; (quorum vix duos reperias, qui non ab invicem multum different) sed neque Graecum editum; Cujus editor saepius innuit, turn Codices suos variasse (ut ne de ordine vel numero propositionum semper consentiant) turn se nonnunquam, praeter omnium quos habuit Codicum fidem, nonnulla immutasse. Ego certe Euclidem, siquis alius, maxime veneror, (nee apud eum quicquam scio cui non assentior; tantum abest ut me neglecti Euclidis insimulasse debeas.)| Agnosco utique Celebrem Geometram; sed & Homi- [85] nem, nee •deb'Kvevaov. Quod vero ne syllabam posuerit ipse, quae possit in melius mutari; neque librariorum vel incuria vel audacia mutatum quicquam: Rhetorice forsan dici poterit; certe non Geometrice. Quippe ego non pauca, (in libris quos habemus) & omissa, & addita, & loco mota, nullus dubito. Et quanquam mihi non necesse sit, ad rem praesentem, ut haec dicam; cum nihil apud Euclidem occurrat quod mihi adversatur, (nisi tu Clavii paraphrasin & additamenta pro Euclide habeas:) Tua tamen vel maxime interest haec dicere. Nisi enim & 5. Def. 5. & 1. Prop. 10. obliterentur, tua constare non poterunt. 422

183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] Ad Prop. 5.855 Concedo tibi, Angulo competere Quantitatum Affectiones: Sed & ego tibi permitto, ut Quantitatem rotunde appelles. Quo verbo nunc vulgo dicimus, quod Euclides p,£j£'&os dixerit. Cum enim Anguli sint ad invicem Rationum capaces; etiam nejefir] dicendi erunt, per 3. Def. 5. El. Ad Prop. 6.856 Ego tibi non concedo, eum Angulum majorem statim esse, cujus crura, post aliquam a puncto concursus distantiam, magis divaricantur: (manifestum utique est, Acutum rectilineum, minorem sic futurum Angulo Contactus:) Nisi, retenta ea quae in concursu fuerat directione utriusque, idem fiet. Rationes ego in meis Cap. 3. & 4. attuli: Nee opus est ut hie repetam; cum tu nihil hie affers quod eorum vires imminuat. Adeoque & Prop. 7.857 Ut falsam rejicio: Tuisque Gregorioh Aynscomiohactenus saltern assentior, ut impossibile dicam Angulo Contactus positivam Magnitudinem concessum iri, quin Geometrica Principia destruantur. (ut autem aliis Angulis vera Magnitude concedatur, nihil impedit.) Eorumque Argumenti aciem (quo probant, Semicirculorum omnium Angulos aequales esse, adeoque & Contactus Angulos vel saltern aequales esse vel potius nullius Magnitudinis) tu nullis viribus obtundes. Quod utique tu opponis (Pag. 177.) In exhaustionibus, (ubi plus quam dimidium aufertur, atque ex residue plus quam dimidium, atque sic deinceps) subtractiones illas, non pro suo Demonstratoris arbitrio, sed arbitrio Adversarii iniri debere: Ridiculum est sophisma. Quotusquisque (quaeso) est, ex Demonstratoribus per exhaustiones, qui Adversarium consulit, quo pacto velit ille ablationes fieri? Num Archimedes, in Dimensione Circuit vel in Quadratura Parabolael vel de Sphaera & Cylindrol vel uspiam alibi, hoc facit? Num Euclidesl Num quispiam alius, seu veterum seu recentiorum? Apage has ineptias! Consule tu primam decimi Euclidis, & discas inde exhaustiones ineundi methodum. Prop. 8.858 Duae Magnitudines inaequales, quarum discrimen tale est, ut quantumlibet Multiplicatum neutram possit superare vel adaequare; nullam inter se rationem habere possunt: si pro nullam inter se rationem habere possunt; dixisses, sunt impossibiles; vera fuisset propositio; quam demonstrasses ex Prop. 1. El. 10. Sed prout tu illam enuncias, absurda 10 fiat E2 855

Prop. Prop. 857 Prop. 858 Prop. 856

5.: 6.: 7.: 8.:

i.e. LEOTAUD, i.e. LEOTAUD, i.e. LEOTAUD, i.e. LEOTAUD,

Cyclomathia, Cyclomathia, Cyclomathia, Cyclomathia,

167-8. 168-73. 174-7. 178-84.

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est, & sui destructiva. Quippe quae Magnitudines inaequales sunt, Rationem habent. Ipsa enim inaequalitas est ratio. Ipsaque illarum Differentia (quod tu Discrimen vocas859) qua altera alteram superat; Homogeneas esse indicat. Heterogenea quippe inter se non comparantur: Vel die tu, si possis, quo excessu, Hora Temporis, superat Libram Ponderis?| [86] Neque aliud supponit Euclides, (Prop. 1. El. 10.) quam ut Magnitudines sint inaequales, quo affirmet, turn ipsas, turn earum per continuam subductionem ortas Differentias, ita Multiplicari posse ut utramvis superent. Quippe si inaequales sint, Rationem habent; adeoque per 3. Def. 5. sunt Homogeneae, (turn ipsae quidem, turn partes suae) adeoque poterit utriusvis quaelibet particula sic Multiplicari, ut reliquam superet, per 5. Def. 5. Dico, per 5. Def. 5. Non per postulatum lib. 10. Quippe quod tu memoras libri decimi postulatum, non Euclidis est, sed Clavii, postulatum: Et quidem plane superfluum. Continetur utique in 5. Def. 5. Et non nisi ob hanc definitionem perperam intellectam, a Clavio insertum. Dum vero tu Propositionem, ut a te propositam, in lineis demonstrare satagis, operam ludis. Impossibile utique est, ut sit lineae pars aliqua (nisi tu Punctum vis esse Partem lineae) quae vel ad totam vel ad reliquam (Homogenea ad Homogeneam) non habeat Rationem, vel etiam tantilla sit, ut non possit multiplicata totam superare: per Prop. 1. El. 10. vel 5. Def. 5. Quodque tu San- Vincentianum& Aynscomiumsibi persuasum habere dicis, Duas quaslibet Magnitudines, quibus competit inter se comparari secundum majus & minus, eo ipso rationem aliquam inter se habere, adeoque debere, per Def. 5. se mutuo superare si saepius repetantur: Omnino verum est. Quodque tu in contrarium profers, nullius est momenti. Illud speciatim quod habes, de Homogeneis quoad quantitatem, sed non quoad rationem; haberi forsan possit inter Sophistarum A070'paxi&S acuta distinctio, (ubi verbis tantum agitur) sed non in Geometrarum schola; ubi non nuda vocabula, sed rerum pondera & demonstrationes spectantur. Nam eo ipso quod sint, quoad quantitatem Homogenea, ra9 turn |ut del. ed.\ earum 10 utrumvis corr. ed. 23 El add. ed. 859

vocas: see LEOTAUD, Cydomathia, 179.

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] tionem habent, per 3. Def. 5. Item inaequalia esse, nee tamen Rationem habere, est contradictio in terminis. (Nisi quo sensu nihil & aliquid sunt inaequalia.) Item Datis Magnitudinibus, datur eorum Ratio; Dataque Ratione Totius ad Partem suam, datur ejusdem & ad Reliquam Ratio; per 1. & 5. Datorum Euclidis Item Angulus Rectus ad Angulum Semicirculi, etiam te judice, rationem habere debet, (per 5. Def. 5.) quoniam utervis ita multiplicari potest ut reliquum superet: Sed & per tuam hanc, Prop. 8. Rationem non haberet, utpote quorum differentia (quam tu facis Angulum Contactus) non potest ita multiplicari ut utrumvis superet. Habebit igitur, & non habebit: Quod est Absurdum. Prop. 9.860 Vera est, Si A ad B rationem habeat, atque B ad (7, etiam A ad (7, rationem habebit, sed & ad B + C. (sunt utique omnes Homogeneae) Sed mihi non officit. Prop. 10.861 Anguli Segmentorum similium nullam inter se rationem habere possunt: Falsa est. Sunt utique Aequales. Quod quidem, in Circulis Aequalibus; ipse fateberis. Ego etiam in Circulis inaequalibus affirmo: Nee tu potis eris refutare. (Propositiones utique praecedentes aliquot unde hoc infers, nihili sunt.) Sed & possunt multiplicati se mutuo superare: Ergo [87] rationem habent; per 5. Def. 5. Prop. II.862 Anguli duorum segmentorum inaequalium ejusdem Circuli; & segmentorum dissimilium in Circulis diversis; rationem inter se habere non possunt: Falsa est. Possunt utique multiplicati se mutuo superare: Ergo Rationem habent; per 5. Def. 5. Demonstratio tua nihili est; quia futilibus superstruitur. Prop. 12. & 13.863 Verae sunt: Sed mihi non officiunt. Prop. 14.864 Quae Definitio est: & Prop. 15.865 Quae illi accommodatur: satis inter se conveniunt: Sed non, cum aliorum loquendi formulis. Sed mihi non officiunt. Prop. 16.866 Nullus Angulus diversae speciei lineis comprehensus, ad

860

Prop. Prop. 862 Prop. 863 Prop. 864 Prop. 865 Prop. 866 Prop. 861

9.: i.e. LEOTAUD, Cyclomathia, 184-5. 10.: i.e. LEOTAUD, Cyclomathia, 185-6. 11.: i.e. LEOTAUD, Cyclomathia, 187-8. 12. fe 13.: i.e. LEOTAUD, Cyclomathia, 188-9 and 189-91. 14.: i.e. LEOTAUD, Cyclomathia, 191-3. 15.: i.e. LEOTAUD, Cyclomathia, 193-7. 16.: i.e. LEOTAUD, Cyclomathia, 197-234.

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668]

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alium quemvis Angulum, Rationem ullam habere potest: Omnino falsa est: Turn quia futilibus superstruitur; turn propter 5. Def. 5. Certum utique est, ita multiplicari posse utrumvis comparatorum, ut reliquum superet. Vides itaque, quam ampla seges propositionum falsarum, (etiam contra 5. Def. 5. tuo sensu intellectam) ex infelici tuo lolio pullulaverit. Tuo, inquam: Quamquam enim Clavius tibi in aliquibus praeiverit; non tamen sustinuit ille tot monstra proferre. Die igitur in posterum, quod omnino dicendum est; Anguli Contactus Magnitudinem nullam esse: atque videbis haec omnia monstra protinus disparere, omniaque in Geometria belle convenire. Vel si tu id malis, die esse Minorem quam infinite-exiguum; utpote minimo possibili Rectilineo minorem; (quod ab Euclide demonstratum esse, ne tu negaveris) quod mini perinde satisfaciet. Quippe, quod demonstratum est, minus esse quam infinite-exiguum, haberi solet pro nonquanto, unde tota Exhaustionum doctrina pendet. Vel etiam, (quo tibi maxime faveam) si circulum haberi vis pro Polygono Rectilineo laterum numero-infinitorum; & Tangentem, pro recta per Polygoni Angulum transeunte, rectae ab ejus Centro Perpendiculari: Die Angulum Contactus esse, Infinitesimam partem duorum Rectorum: (seu -^R-} Quippe tantus erit uterque Angulus externus contactu illo factus; per Calculum a me, Cap. 12. institutum. (Quo tamen minor esse debet, certe non major, Angulus Contactus Circuli.) Verum si tu hoc dixeris; dicendum etiam erit, Peripheriam Circuli non habendam pro una linea in directum continuata, (prout tu, Pag. 221.) sed, totidem Angulorum esse quot est Laterum: Hoc est, in quovis Peripheriae puncto Angulum constitui, aequalem duobus rectis dempta infinitesima parte quatuor rectorum, (vel 2R — ^R.) Quippe tantus erit quilibet Angulus istius Polygoni. Sin tu velis (ut Pag. 221.) ut haec Tra/j.TroXvjui'ia, in Peripheria, evanescere censeatur in Non-Angulum, sed continuam ejusdem lineae directionem, (pariter atque cum duo crura Anguli Rectilinei explicata, cessante Angulo, fiant continua recta:) pariter censendus erit externus ille Contactus Angulus, quasi complicatis cruribus, etiam in non-angulum, seu Angulum nullius magnitudinis, transire. Dumque Peripheria pro una continuata linea censeatur; censendus erit Angulus Contactus pro non1 ullum corr. ed. 2 superstuitur corr. ed. 3 utramvis corr. ed. 32 fiunt E'2

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183. WALLIS to LEOTAUD, 17/[27] February 1667/[1668] angulo. Argumenta mea non repeto; (ex TVactatu meo de Angulo Contactus petenda) ut quae adhuc inconcussa manent, nee opus habeant ut denuo statuminentur. Nam (praeter quasdam verborum captiunculas, quae nullius momenti sunt, neque responsionem merentur) totum illud, quod tu contra eorum aliqua (omissis reliquis) movere satagis, hoc unico nititur fundamento, Quod (ex 5. Def. 5. perperam intellecta) existimes, Magnitudinum invicem Homogenearum (etiam finitarum) alias habere, alias [88] non habere, rationem ad\ invicem. Quod quidem fundamentum, cum in praecedentibus subversum sit, plurimisque absurdis gravatum, quae tu ut justas inde consequentias deducis; quae huic superstruuntur, simul ruunt. Sed nee Argumenta nova superaddo (quae tamen in promptu sunt) utpote supervacanea; cum res ipsa jam abunde sit confecta. Hanc unam tamen, de novo, adjungam demonstrationem.

Curvam quamvis AE, recta contingens AT, Angulum Contactus faciet EAT; qui immotus maneat. Atque huic congruus, motu continue ferri

13 supervacanda corr. ed. 14 adjunjam corr. ed. 15-16 faciat E2

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184. OLDENBURG to WALLIS, 25 February/[6 March] 1667/8

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intelligatur, a situ CAR, per DAS, ad EAT, porroque ad FAV. Manifestum est, (propter Angulos Curvilineos CAD, DAE, EAF, aequales rectilineis RAS, SAT, TAV,) quantum hoc motu rectae AR, demitur angulo RAT, tantum motu curvae AC continue demi angulo CAT: totisque tandem demptis, transitum iri ad angulos ex contraria parte positos, ut TAV, TAP. Fiet autem hie transitus (ab angulo a sinistra ad angulum a Dextra) totis demptis, vel eodem utrobique momento; adeoque (propter aequales utrobique ablationes) aequales ab initio fuerint CAT, RAT, (utpote aequalibus ablationibus absumpti) & propterea CAR nullius magnitudinis; (quod nos dicimus) vel non eodem momento. Quo autem momento AR ad AT pervenit, (Adeoque AC ad AE) exhauritur angulus RAT: Si autem non eodem momento exhauritur totus CAT; esto hoc paulo serius, (quippe citius fieri, ne tu dixeris) recta AR existente in AX, (quippe rectam AT transiisse, necesse erit, cum serius sit quam dum AR fuerit in AT) & AC in AC. Erit igitur angulus CAR seu EAT, (quo CAT superat rectilineum RAT) aequalis ipsi EAG, seu TAX: Angulus Contactus, rectilineo: Quod est absurdum. Eodem igitur momento fit utrobique transitus: Adeoque angulus Contactus est nullius magnitudinis. Quod erat demonstrandum. Tu interim vir Clarissime, aequo animo feras velim, quod non iniquo raptim exaravi. Vale.

184. HENRY OLDENBURG to WALLIS 25 February/[6 March] 1667/8 Transmission:

Manuscript missing. Existence and date: Mentioned in the postscript of OLDENBURG-BOYLE 25!I/[6!II]. 1667/8; OLDENBURG, Correspondence IV, 205-8, 207. Answered by: WALLis-OLDENBURG 29.II/[10.III]?. 1667/8. This letter accompanied two copies of PETRUS BLESENSIS, Opera omnia, ed. Pierre de Goussainville, Paris 1667, which Justel had sent from Paris for Wallis and for the Bodleian Library. In return, as Oldenburg pointed out, Justel desired a copy of Wallis's 4 tantundem E2 12 memento corr. ed. 17 memento corr. ed.

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185. COLLINS to WALLIS, 25 February/[6 March] 1667/8 Grammatica linguae Anglicanae. See OLDENBURG-BOYLE 25.II/[6.III].1667/8; OLDENBURG, Correspondence IV, 205-8, 207.

185. JOHN COLLINS to WALLIS 25 February/[6 March] 1667/8 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-COLLINS 27.II/[8.III]. 1667/8. This letter apparently contained an account of DULAURENS, Specimina mathematica, duobus libris comprehensa, Paris 1667.

186. WALLIS to JOHN COLLINS Oxford, 27 February/[8 March] 1667/8 Transmission:

W Letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 197r-197v (our source). On left margin of f. 197V at 90° to address in Collins's hand: 'About Dulaurens and Leotaud', and at top at 180°, again in Collins's hand: 'Vera Circuli et Hyperbolae quadratura in propria sua Proportionis Specie inventa et demonstrata a Jacobo Gregorio Ab S Patavii'.—printed: RlGAUD Correspondence of Scientific Men II, 488-90. Reply to: COLLiNS-WALLis 25.II/[6.III].1667/8.

Oxford Febr. 27. 1667./8. Sir,

Yours867 Febr. 25. I received this morning. The account which you give mee of Du Laurens his Algebra868, answers my expectation. For by the 3 account (1) of (2) which 867

Yours: i.e. COLLINS-WALLIS 25.II/[6.III].1667/8. Algebra: i.e. DULAURENS, Specimina Mathematica, duobus libris comprehensa, Paris 1667. 868

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glorious Title which was represented; & the great weight layd upon a slight Probleme869, & that so lamely proposed that it was hard to pick out what he meant: made mee think there was no great matter to be expected from him. The problem you mention about the length of a line cutting an Ellipse given in specie, & cutting one of the Axes with an angle given: You well observe, to be none of mine870; but one proposed (it seems) by such an other Algebrist, as Du-Laurens. And it may be solved by the same method in an Hyperbola, as I have done in the Ellipse, with little more then changing the signes + and —. And in a Parabola, yet more easyly. The streightening a Curve871 was done by Mr Neil, (& after him, by Dr Wren & my Lord Brounker,) a good while before Heurat872: (& I suppose both proceeded upon the grounds I mentioned in my Arithm: Infinit:) And it was commonly known to divers of our English mathematicians before Heurat's came abroad. With my answer873 to Leotauds Cavills, I perceive you are satisfyed. I confesse I thought Leotaud (though I had not read much of him) had been a better Geometer, till I saw that piece. But since I find that hee assents to opinions in Geometry venerationis gratia', I shall take him to be as Orthodox in Geometry, as in Divinity, confining his opinions in the one to father Clavius, as in the other to the Popes authority. Of Slusius874 I have a much better opinion. And did you not tell mee that hee is reprinting875 this summer, I should have desired you to have bought his works for mee, (because I have them not,) & sent down. But I think, I shall see you at London about the middle of Easter term. Fabry's Opticks876 I have not yet 17 hee (1) takes up (2) assents to 21 not add. 22 mee, (1) fe (2) (because 23 (because |I add. ed.\ have 869

Probleme: on Dulaurens's problem and Wallis's solution cf. OLDENBURG-WALLIS 10/[20].XIL1667 and subsequent letters. 8ro none of mine: cf. WALLIS-OLDENBURG 30.III/[9.IV].1668. 871 streightening a Curve: see WALLIS, Tractatus duo, 91-6; Opera mathematica I, 5514. This controversy originated in 1659 and was rekindled in 1673 by the publication of Huygens's Horologium oscillatorium. 872 Heurat: i.e. Heuraet's Epistola de transmutations curvarum linearum in rectas of 13 January 1659 (new style), printed in DESCARTES, Geometria, ed. F. v. Schooten, vol. I, Leiden 1659, 517-20. 873 answer: i.e. WALLIS-LEOTAUD 17/[27].II.1667/8. 874 Slusius: i.e. Rene Francois de Sluse (1622-85). 875 reprinting: i.e. the second edition of Sluse's Mesolabum, printed in Liege in 1668. 876 Opticks: i.e. FABRI, Synopsis optica, Lyons 1667. 430

186. WALLIS to COLLINS, 27 February/[8 March] 1667/8 had leisure to examine. I doubt allso hee admits much in Mathematicks venerationis gratia, as well as Leotaud; which makes him insist on the Earths stability; But it is in him more pardonable; because here the Popes authority is interposed, (having condemned the Copernican Hypothesis of Heresy;) But whether an Angle of Contact and of a circular segment, be homogeneous or heterogeneous to right-lined Angles, the Pope hath not yet determined. Mr Gregory's peece877 about the Quadrature of the Circle, Ellipse, & Hyperbola; I have looked over: And, for ought I discern upon a slight view of it, (not having strictly examined every proposition,) it seemes to be truly inough performed according to the method proposed. And his Prop. 32. shews his way of calculating an Hyperbolick Space. I am of your opinion, that it would advance the sale of the book, to have a clear & easy methode layd down for the operation, for any space proposed. But I think it best to bee done by himself, who is most Master of his own Notion therein; & hath allready (I presume) considered of abbreviating Methods; For, to calculate a Table after the method of this 32d proposition, would be a long work. What you mention, of solving equations of the 3d 4th or superior degrees, Trigonometrically; I do well inough approve of. For the process of Angular Sections, resolving themselves into such Equations; The one will be but the inverse of the other. This at present from, yours &c. John Wallis. [197V] These For Mr John Collins, his Majesties Accountant, at the Excise Office in Bloomsbury, London.

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1 allso add. 3 stability; (1) & (2) But 6 to (1) that of (2) right-lined 9 ought add. 12 an (1) Elliptick (2) Hyperbolick 13 sale |of add. ed.\ the 14 methode (1) pro breaks off (2) layd 877

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peece: i.e. GREGORY, Ve.ro, circuli et hyperbolae quadratura, Padua 1667.

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189. WALLIS to MORAY?, ? February 1667/8

187.

WALLIS to HENRY OLDENBURG 29 February/[10 March] ? 1667/8 Transmission:

Manuscript missing. Existence and date: Mentioned in OLDENBURG-BOYLE 3/[13].III. 1667/8; OLDENBURG, Correspondence IV, 223-5, 225. Reply to: OLDENBURG-WALLIS 25!I/[6!II]. 1667/8.

188. ROBERT MORAY ? to WALLIS ? February 1667/8 Transmission:

Manuscript missing. Existence and date: Referred to in WALLIS-BROUNCKER 4/[14].XL1668. Answered by: WALLIS-MORAY? 7.II.1667/8. In this letter, almost certainly from Sir Robert Moray, Wallis was asked to supply his opinion on Gregory's Vera circuli et hyperbolae quadrature, which accompanied it.

189. WALLIS to ROBERT MORAY ? [Oxford], ? February 1667/8 Transmission:

W Part quotation (from memory) of missing letter sent in WALLIS-BROUNCKER 4/[14]. XI.1668: LONDON Royal Society Early Letters Wl, No. 65, p. 1 of 6 pp. Reply to: MORAY 7-WALLis 7.II.1667/8. The member of the Royal Society who sent Wallis a copy of Gregory's Vera circuli et hyperbolae quadratura in February 1667/8 is almost certainly Sir Robert Moray. Like Gregory, with whom he shared episcopalian principles, Moray had studied at St. Andrews and it was probably through his favour that the young mathematician was appointed to the newly-created professorship there in 1668. Evidence suggests that they had been in contact during Gregory's stay in Italy and possibly also beforehand. See

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189. WALLIS to MORAY?, ? February 1667/8 COLLINS-GREGORY ?.II.1667/8 (RIGAUD, Correspondence of Scientific Men II, 174-9) and COLLINS-GREGORY 30.XII.1668/[10!.1669] (TURNBULL, James Gregory, 54-8). Since Moray had for a long time been a friend of Huygens, and was also acquainted with Wallis, it is conceivable that he suggested to Gregory that he submit copies of his work on the quadrature of the circle and the hyperbola to these two mathematicians for their opinion. It appears that Gregory sent two copies of the book to England and that these arrived in early February 1668. Probably they were addressed to Moray, although Collins suggests that they were sent to Gregory's friend, the bookseller Samuel Thompson. See COLLINS-GREGORY ?.II.1667/8; RIGAUD, Correspondence of Scientific Men II, 174-9, 174. One of these, intended as a copy for reprinting, was given to Thompson, who lent it to Collins. Collins in turn showed it to Brouncker, who then gave a short account of the work at the meeting of the Royal Society on 27 February 1668. See BIRCH, History of the Royal Society II, 253. The other copy would clearly have been sent to Wallis at the same time. Assuming Moray was the sender, he would as patron understandably have asked the Savilian professor to give his opinion privately, thus explaining the reference to a 'familiar letter' in WALLIS-BROUNCKER 4/[14].XL1668. Wallis also gave a short account to Collins at the same time. See WALLIS-COLLINS 27.II/[8.III].1668. Huygens published his opinion under the title 'Examen de Vera Circuli & Hyperboles Quadratura' in the Journal des Scavans of 2 July 1668 (new style), 52-6. See HUYGENS-WALLIS [3]/13.XL1668.

/... ] a Book lately published at Padua by Mr James Gregory now a Member of this Society, entituled, Vera Circuli et Hyperbolae Quadratura in propria sua proportionis specie. This Book soon after it came over into England, was by another Member of this Society sent to mee, desiring (in general terms) my opinion of it. And after a slight perusal of the whole, to see what matter it conteined, and examining the Demonstration of the more leading propositions, the account I gave him in a familiar letter (of which I kept no Coppy) was, as I remember, to this purpose; that it seemed to mee to contein divers things (so far as I could judge upon a slight perusall) ingeniously demonstrated though obscurely; amongst which was a new methode of approximation for the squaring of the Circle, which was allso equally applicable to the Ellipsis and Hyperbola. And more than to this purpose I do not remember that I did write. /".../

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190. WALLIS to OLDENBURG, 7/[17] March 1667/8 (i)

190.

WALLIS to HENRY OLDENBURG Oxford, 7/[17] March 1667/8 (i) Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 40, 2 pp. (our source). Endorsed by Oldenburg on page 2 beneath address: 'Rec. March 9. 67.' Postmark on p. 2: 'MR/9'.—printed: OLDENBURG, Correspondence IV, 229-30, 230-1 (English translation). Enclosure: WALLIS-OLDENBURG 7/[17].III.1667/8 (ii).

Oxford. March. 7. 1667./8. Sir,

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I was desired878 by Mr Collins (a member of our Society) to look over Monsr Fabry's Synopsis Optica879, & to give you an account of the Contents thereof. Which is as follows. Honorati Fabri Synopsis Optica, Eorum plurima quae in Opticis, Catoptricis, et Dioptricis tractari solent, breviter attingit. viz. De modo visionis, Angulo visorio, atque Magnitudinum et Distantiarum aestimatione; a prop. 1, ad 10. De Perspectivae et Scenographiae fundamento, 11 — 15. De Objecto in motu, vel per foramen conspecto, aliisve accidentibus affecto 16. — 19. De Speciebus et Lumine per foramen trajectis; umbrarum projectione, et circulorum caelestium; Saturnique phaenomenis per principia optica explicandis. 20 — 24.

6 plurima add. 7 breviter (1) exhibet, viz. (2) attingit. 8 visionis, (1) et (2) Angulo 11 motu, (1) vel (2) vel 13 foramen (1) c breaks off (2) trajectis (a) 20 — 21. (6) ; (oa) et (66) umbrarum projectione, et circulorum caelestium; (oao) 20 — 22. (bbb) Saturnique 878

desired: possibly in CoLLiNS-WALLis 25!I/[6!II]. 1667/8. See WALLIS-COLLINS 27.II/[8.III].1667/8. 879 Synopsis Optica: i.e. FABRI, Synopsis optica, in qua ilia omnia quae ad opticam, dioptricam, catoptricam pertinent, . . . demonstrantur, Lyons 1667. 434

191. WALLIS to OLDENBURG, 7/[17] March 1667/8 (ii) De Luminis Reflexione, speculisque planis, concavis et convexis, Sphaerico, Cylindrico, Conico, Parabolico, Elliptico, Hyperbolico; figuraeque in illis conspectae loco, situ et magnitudine. 25 — 40. De Luminis diffusione et Refractione, in aqua, in vitris, planis, sphaericis, Ellipticis, Hyperbolicis; ubi de Telescopic, Perspicillis ocularibus, Microscopic, Polyscopio, et de coloribus. 41 — 53. De Refractionibus, et Parallaxibus, aliisque phaenomenis caelestibus, et speciatim cometis. 54 — 58. In appendice, agitur, de Hypothesi Motuum caelestium quam ipse commentus est, ut Systemati Ptolemaicae consentaneam, sed demptis orbibus solidis. Accedunt tres Epistolae; in quibus agitur, de nuperis nonnullis Saturni, Jovis, et Martis observationibus; de Maculis in Jove et Marte observatis, unde colligitur eorum circa suos axes conversio; umbrisque Satellitum in Jovis disco observatis.

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[2] These

For Mr Henry Oldenburg, at the Pellmell, near St James's London.

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191.

WALLIS to HENRY OLDENBURG Oxford, 7/[17] March 1667/8 (ii) Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 43, 2pp. (our source). Endorsements at top of p. 1: 'read April. 2. 1668. enterd L.B. 2. 197.' at top right, and 'a translation of the foregoing.' at top left. On right margin of p. 2 Oldenburg has noted beneath signature at 90°: '(./) Dr Wallis's letter concerning tydes, read before the Society April 2. 1668. and orderd to be registred (2) An Extract (additional del] 1 convexis, (1) sive Sphaericis (2) Sphaerico 5 Hyperbolicis; (1) lentibusque in (2) in (3) ubi de 6 de (1) d breaks off (2) coloribus. 7 Refractionibus, (1) et (2) et Parallaxibus (a) caelestibus, (b) , aliisque 10 ut (1) tamquam { — } (2) Systemati 14 Satellitum (1) Jovialium (2) in

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191. WALLIS to OLDENBURG, 7/[17] March 1667/8 (ii) of a letter of Dr Wallis to M. Oldenburg Concerning Tydes.'—printed: OLDENBURG, Correspondence IV, 231-3. wl Copy of letter sent: LONDON Royal Society Letter Book Original 2, pp. 197-200. w2 Copy of wl: LONDON Royal Society Letter Book Copy 2, pp. 236-9. E First edition of letter sent (with alterations according to the instructions given in WALLIS-OLDENBURG 17/[27].III.1667/8 (i)): Philosophical Transactions No. 34 (13 April 1668), 652-3 ('A Letter. Written by Dr. John Wallis to the Publisher, concerning the Variety of the Annual High-Tydes, as to several places; with respect to his own Hypothesis, deliver'd No. 16, touching the Flux and Reflux of the Sea.'). Answered by: OLDENBURG-WALLis 10/[20].III.1667/8. Enclosure to: WALLIS-OLDENBURG 7/[17].III.1667/8 (i). As was requested by Oldenburg in his letter of 10/[20].III.1667/8, Wallis, who had not kept a copy of the present letter (see WALLIS-OLDENBURG 17/[27].III.1667/8 (i)), prepared a Latin version on 17/[27].III. 1667/8. Correspondingly, that version differs from this in a number of respects. See WALLIS-OLDENBURG 17/[27].III. 1667/8 (ii). Contrary to the endorsement, it was the Latin and not the English version, which was read before the Royal Society on 2 April 1668; see BIRCH, History of the Royal Society II, 262.

March. 7. 1667./8. Oxford. Sir

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In my Hypothesis for Tydes880, you may remember, that I cast the Annual High-Tydes not on the Two Aequinoxes, about the 11th of March & September; nor yet on the Apogaeum & Perigaeum of the Sun, about the middle of December & June; but (as proceeding from a complication of those two causes) on a Midle time between the Perigaeum and the two Aequinoxes; (like as is the greatest inaequality of the Natural days, preceding from a complication of the same causes:) And, particularly, for the coast of Kent (& consequently the rivers of Thames & Medway) about the beginning of November, & February: Which agrees with obser-

1-3 Sir, In my Hypothesis E 4 High-Tydes |for the Coast of Kent, (fe consequently in the Rivers of Thames, & Medway,) del. \ not 6 middle of (1) Jul breaks off (2) December 7 between |Dec. 11, fe Mart. 11; & between del. the Perigaeum and the two Aequinoxes; (1) and particularly (2) (like 9 from (1) the lik breaks off (2) a complication of (a) both (b) the same 880

Hypothesis for Tydes: i.e. WALLIS-BOYLE 25.IV/[5.V].1666, printed in Philoso ical Transactions No. 16 (6 August 1666), 263-81. The relevant sections follow the paragraph beginning with 'But it is now . . . ' . 436

191. WALLIS to OLDENBURG, 7/[17] March 1667/8 (ii) vations on those coasts: and particularly with that of yours881 of Febr. 5. this year. The last year, when I was present in the Royall Society, I remember, an account882 was brought us, of the Annual High-Tydes on the Severn, & at Chepstow-bridge, to bee about the beginning of March & end of September. Which though they do not agree with the particular times on the coast of Kent; yet in the general they agree thus far; that the one is about as much before the one Aequinox, as the other is after the other Aequinox. You now acquaint mee883 with High-tydes about Febr. 22. about the coast of Plimouth (if I mistake not:) which is later than that of the Coast of Kent, but sooner than that on the Severn. And I doubt not but in other parts of the world, will be found other varietyes. The reasons of these varieties are (as I have formerly signified,) to bee attributed, to the particular position of those parts, rather then to the Generall Hypothesis. Of which this, in brief, may serve for some account at present. The General Hypothesis of the Earths diurnal motion from West to East, would cast that of the Waters (not following so fast) from East to West; which causes the constant current within the Tropicks (where the Circles are greatest) westward from the Coast of Africk to the Coast of America, (which is allso the cause of the constant Eastern Brize blowing in those

1-2 and particularly . . . this year. add. 2 The last (1) I (2) year, 6 they (1) th breaks off (2) have this (3) agree 8 with (1) a very (2) High-tydes about Febr. 22. about the (a) coasts (6) coast 9 (if I mistake not:) missing in E 15 diurnar corr. ed. 881

yours: see OLDENBURG-WALLIS 11/[11[.II1667/8 account: i.e. the account given by Henry Powle at the meeting of the Royal Society on 12 December 1666, upon which a paper was read at the meeting on 19 December 1666; BIRCH, History of the Royal Society II, 133-4. Wallis, who commented on these observations in WALLIS-OLDENBURG 19/[29].1.1666/7, had probably been present at one of these meetings. 883 You now acquaint mee: i.e. by OLDENBURG-BOYLE 3/[13].III. 1667/8 (OLDENBURG, Correspondence IV, 223-5, 225). Oldenburg explicitly told Boyle to pass on to Wallis the information on Colepresse's latest observation on high-tides at Plymouth, contained in COLEPRESSE-OLDENBURG 22.II/[3.III]. 1667/8 (OLDENBURG, Correspondence IV, 197-9, 199). Already in an earlier letter (COLEPRESSE-OLDENBURG 7/[17].1.1667/8; OLDENBURG, Correspondence IV, 105-7) Colepresse had sent the Royal Society observations on the tides, which were subsequently printed in Philosophical Transactions No. 33 (16 March 1667/8), 632-3. 882

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191. WALLIS to OLDENBURG, 7/[17] March 1667/8 (ii)

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parts:) But the sea thus beating on the Coast of America is cast back, as with an Eddy, on either hand; & consequently returnes from the American shoar Eastward towards the coast of Europ; where, the Parallel circles to the Equator being lesse, & consequently the diurnal motion slower, doth not cast the Waters so strongly westward, as between the Tropicks, & so not strong inough to overcome the Eddy, which it meets with from the other motion. Which gives the Sea a North-Easterly motion, (on these coasts,) as to its usuall course: (which allso I take to be the cause, why North-Easterly winds are more frequent then any others on our coasts.) The Current therefore of our Seas being North-Easterly; we are next to consider at what times it runs more to the North, & at what more to the East. When it runs most Northerly, it runs up the Irish sea, & so up the Severn: when most Easterly, it runs straight up the Chanel, & so to the coast of Kent: when between these; it beats against Devonshire & Cornwall, & those parts. Wee are therefore to consider (as to the annuall periods) that the Annual motion of the Earth in the Zodiack, & the Diurnal in the Aequator, are not precisely in the same direction, but make an Angle of 23^ grad. at the Aequinoxes, but run as it were parallel at the Solstices; and as they be near or farther from these points, so is the inclination varied. Which several directions of motion, do cause the compound motion of both, to vary from the East & West, more or lesse according as the suns position is further or nearer the Solstices: And therefore nearer to the Aequinoxes, this inclination doth cast the| constant current of our Seas more to the North & South; & further from it, more to the East & West. Which is the reason why the current up the Irish sea, is near to the Aequinoxes, (at the beginning of March & end of September;) and up the Chanel or narrow seas, farther from it,

1 as add. 3 shoar add. 5 as (1) under the (2) between the Tropicks, & (a) therefore (6) so 7 a (1) Norther breaks off (2) North-Easterly 8-9 (which allso . . . on our coasts.) del. Oldenburg (on Wallis's instructions) The Current therefore E 12 When it (1) lies (2) runs 13 straight add. 17 direction, add. 25 current (1) up t breaks off (2) up

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course:

[2]

191. WALLIS to OLDENBURG, 7/[17] March 1667/8 (ii) (at the beginning of Febr. & of November;) and against the Coasts of Devonshire & thereabouts, at some intermediate time. And thus much I thought convenient to signify upon this occasion. Desiring allso that the Spring tides now next coming may be observed; whether they be not less than those at work about the 5th of February; & that you will please to keep notes of such observations as you meet with of this kind. I remember that the great wind884 which happened about Febr. 18. about 5 or 6 years agoe; was a North Easterly wind; which came off from Sea about Devonshire & Cornwall, & so crosse the land by Oxford & so to London. Whether it were at the time of the menstrual Spring-tydes, I do not remember. And I do not much question, but that the same things which cause extraordinary Tydes; may likewise cause extraordinary winds: though not allways at the same times at the same place: the one being caused by an Eddy of the water beating against Land; the other by a like reverberation of the Air against Mountains or high hills. But in the water it is more observable then in the Ayr. This in hast from Yours &c Joh: Wallis.

I and (1) on the (2) towards (3) against the Coasts of Devonshire (a) , at s breaks off (&)& 2-19 I thought fit to signifie upon this occasion. Dat. Oxford the 7. of March An. 1667/8. E 3 convenient (1) so (2) to 8 about (1) 5 (2) 5 or 6 II much add. 15 hills. (1) An breaks off (2) But in the water it is more observable then in the (a) water. (6) Ayr. 884

great wind: i.e. the storm of 17/18 February 1661/2; see The Diary of John Evelyn, ed. DE BEER, III, 316.

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193. WALLIS to OLDENBURG, IT/[27] March 1667/8 (i)

192.

HENRY OLDENBURG to WALLIS 10/[20] March 1667/8 Transmission:

Manuscript missing. Existence and date: Referred to in OLDENBURG-BOYLE 17/[27].III. 1667/8 (OLDENBURG, Correspondence IV, 248-50, 250) and WALLIS-OLDENBURG 17/[27].III. 1667/8 (i). Answered by: WALLIS-OLDENBURG 17/[27].III.1667/8 (i). In this letter, Oldenburg apparently requested that Wallis send him a Latin version of his latest letter concerning his hypothesis of the tides (WALLIS-OLDENBURG 7/[17].m.l667/8(ii)).

193.

WALLIS to HENRY OLDENBURG Oxford, 17/[27] March 1667/8 (i) Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 42, 2pp. (our source). On p. 2 beneath address in Oldenburg's hand: 'Rec. March. 18. 1667.' Postmark on p. 2: 'MR/1{8}'.—printed: OLDENBURG, Correspondence IV, 252. Reply to: OLDENBURG-WALLIS 10/[20].III. 1667/8. Enclosure: WALLIS-OLDENBURG 17/[27].iII.1667/8 (ii).

Oxford March. 17. 1667./8.

Sir,

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I thank you for yours885 of Mar. 10. with the information in it. I have according to what you did therein intimate, sent you886 the summe of mine887 of March 7. (for the words I have not) in Latine. In my English, which before I sent you, you may please to blot out that Parenthesis [which may bee the occasion why the North-East Winds blow so frequently on our coast] and likewise that concerning the High winds in February 4 or 885

yours: i.e. OLDENBURG-WALLis 10/[20].III.1667/8. sent you: i.e. the enclosed WALLIS-OLDENBURG 17/[27].III.1667/8 (ii). 88r mine: i.e. WALLIS-OLDENBURG 7/[l7].III. 1667/8 (i). 886

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194. WALLIS to OLDENBURG, IT/[27] March 1667/8 (ii) 5 years ago: For it was (I think) in Febr. 18. 1662./3. and, if so, it was near the time of Neap-tides, not of the Spring-tydes; & so nothing to the purpose. The Latine though I now send it; yet I am not willing to venture it to the publike, till my Lord Brounker have first well weighed it, & do approve it. I expect to hear, how the Spring-Tydes happened at London, at the Aequinox last past. Which I expect should be higher then ordinary, but not so high as at the beginning of February. I think it would now be time to think of Printing Mr Horrocks's papers888. By Eddy, I mean, Aqua (repercussa) per circuitum retrocurrens. At Billingsgate, (because of the Graves-end Barges,) they do constantly attend the High-water, both by night & by day. Were it not to get some porters, or water-men, or some appertaining to Innes near it, who use there to attend, to keep a Diary of the Time & Hight of each Tyde, for a Year or more together? This in hast from Yours John Wallis. [2] These For Mr Henry Oldenburgh, in the Old Palmal, near St James's London.

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194.

WALLIS to HENRY OLDENBURG Oxford, 17/[27] March 1667/8 (ii) Transmission:

W Letter sent: LONDON Royal Society Eaxly Letters Wl, No. 41, 2pp. (our source). At top left of p. 1: 'read April 2: 68. enter'd LB. 2. 163.'. At top middle of p. 1 Oldenburg has noted: 'Prom Dr Wallis to M. Oldenburg (1) Concerning (2) About the hapning of I Febr. |18. add] (1) 166{3./4.) (2) 1662./3. 4-6 & (1) ap breaks off (2) do approve it. I expect to hear, how the (a) Tydes (6) Spring-Tydes happened at London, (oa) upon (66) at II or some appertaining to Innes near it, add. 888

papers: i.e. astronomical papers of Jeremiah Horrox. See Wallis's letters to Oldenburg of 6/[16].IV.1664, 30.IV/[10.V].1664, and 21.IX/[1.X].1664.

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194. WALLIS to OLDENBURG, IT/[27] March 1667/8 (ii) the (a) Tydes (6) highest Annual Tydes in the Interniediat times between the Perigee of the Sun and the Aequinoxes.' wl Copy of letter sent: LONDON Royal Society Letter Book Original 2, pp. 163-5. w2 Copy of wl: LONDON Royal Society Letter Book Copy 2, pp. 196-9. E First edition (with several major alterations): WALLIS, Opera mathematica II, 755-6 ('De aestu maris, epistola tertia. Ad D. Oldenburgium scripta, 7° Martii, 1667. Stilo Angliae.'). Enclosure to: WALLis-OLDENBURG 17/[27].III. 1667/8 (i). Already on 7/[17] March, Wallis had sent Oldenburg a letter in English in which he gave an account of the latest observations on the high tides reported to him and showed how they were accommodated by his hypothesis. Oldenburg subsequently requested this Latin version, which Wallis produced without a copy of the former and which therefore differs from that. It was probably written on or shortly before 17/[27] March; nevertheless Wallis gave it the date of the original English letter. The present letter was read at the meeting of the Royal Society on 2 April 1668. See BIRCH, History of the Royal Society II, 262.

Oxoniae Martii 7. 1667./8.

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In mea (Vir Clarissime) de Marinis Aestibus Hypothesi889, conjicio ego (uti meministi) maximos Tumores Annuos, non in bina Aequinoctiorum tempora (circa Martii et Septembris diem undecimum, stil. vet. contingentia;) neque in Apogaei et Perigaei Solis tempora (circa medium Junii et Decembris:) Sed, in tempora Perigaeo et Aequinoctiis intermedia; (Oriuntur utique ex earundem causarum complicatione; turn hi Tumores Annui, turn Dierum Naturalium Inaequalitates maximae, sub eadem tempora contingentes;) Et speciatim, ad Kantiana littora, (et propterea in Fluviis Thamesino et Medwayano,) sub initia Februarii et Novembris. Quod quidem Observatis satis respondet; eisque nominatim quae tu modo ad dies Febr. 3, 4, 5, te Londini habuisse refers890. Dum autem, praeterita Aestate, in Regia Societate nostra aderam;

4 (circa |undecimum del] Martii W (circa 11 Martii, & 13 Septembris, St. vet. E 5 Apogaei aut Perigaei E 7 Annuui W corr. ed. 9 propterea, credo, in Fluviis etiam Thamesino E 10 Medwayano, sub W corr. ed. 889

Hypothesi: i.e. WALLIS-BOYLE 25.IV/[5.V].1666, printed in Philosophical Transactions No. 16 (6 August 1666), 263-81. 890 refers: see OLDENBURG-WALLIS ll/[21].II.1667/8.

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194. WALLIS to OLDENBURG, IT/[27] March 1667/8 (ii) ibidem nunciatum891 memini, Annuos hos Tuniores ad Sabrinae fluvium, et adjacentem Ponteni Chepstowensem, sub Martii initium, et Septembris finem, contigisse solitos, (quos St Davids stream et Michaelmas stream appellant:) Non quidem eisdem plane temporibus quo ad littora Kantiana contingunt (quos vocant Candlemas stream et Allholland stream) sub initia Februarii et Novembris: sed saltern (quod utrobique convenit;) alterum quasi tantundem ante Aequinoctium Vernum, quantum alterum post Aequinoctium Autumnale. Tandem mini nuncias892, in proximis literis, circa diem 22 Februarii jam elapsi, praetumidos Aestus in littore Devoniensi (prope Plimutham) contigisse: Hoc est, serius quam pro littore Kantiano, citius autem quam pro Sabrinae Ostiis. Sed et alias alibi terrarum varietates contingere solitas, nullus dubito. Hujusmodi autem varietates, ubique terrarum contingentes, cum non tarn ex generali Hypothesi, quam particular! locorum et littorum situ, aliisque circumstantiis mille modis variatis dependeant; earum omnium causas reddere, ob rationes jam aliquoties dictas, non in me suscipio.

1 ad Sabrinae (1) fl breaks off (2) fluvium, W 3 contingere solitos, E 3-4 et (1) appe breaks off (2) Michaelmas stream appellant:) (a) Qui quamquam (b) Non quidem eisdem jplane add.\ temporibus W 4-9 appellant:) non eo tempore quo ad littus Kantianum, (quos vocant Candlemas Stream & Allholland Stream,) sub initia Februarii & Novembris contingunt. || Hactenus tamen convenit; tantundem quasi Ante Aequinoctium Vernum alteros, atque alteros Post Aequinoctium Autumnale, utrobique contingere. 11 Tandem milii narras, in E 6 sed |saltern add] (quod (1) utrob breaks off (2) in utrisque (3) utrobique W 13-444, 1 dubito. || Cur ego hujusmodi varietatum, ubique terrarum contingentium, causas reddere, nolim in me suscipere; jam anteliac dictum est aliquoties: Utpote quae non tarn ex generali Hypothesi, quam ex particular! locorum & littorum situ, aliisque circumstantiis, mille modis variatis, dependent. || Quod autem E 15-16 locorum (1) et littorum (2) et littorum situ, (a) dependeant (6) aliisque circumstantiis mille modis variatis dependeant; earum (ao) omnes (66) omnium W 891

ibidem nunciatum: a reference either to the account of observations concerning tides given by Henry Powle at the meeting of the Royal Society on 12 December 1666 or to Powle's paper read at the meeting on 19 December. See the comment on the corresponding passage in WALLis-OLDENBURG 7/[17].III. 1667/8 (ii). 892 nuncias: i.e. in OLDENBURG-BOYLE 3/[13].III.1667/8 (OLDENBURG, Correspondence IV, 223-5, 225), which contained information on Colepresse's observations on high tides at Plymouth, made on 22 February 1667/8. See the comment on the corresponding passage in WALLIS-OLDENBURG 7/[17].III.1667/8 (ii). 443

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194. WALLIS to OLDENBURG, IT/[27] March 1667/8 (ii)

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Quod autem ad modo memoratas attinet: haec forte non incommode dici poterunt. Cum Tellus diurno motu ab Occidente Orientem versus volvatur; Aquaque propterea, (quae tardius sequitur,) quasi retro relicta, ab Oriente versus Occidentem rejici videatur; maxime intra Tropicos, ubi, propter Parallelos majores, diurnus Telluris motus est concitatior; (quod in causa esse creditur, cur molliores aurae ab Oriente quasi quotidie flantes, ibi terrarum sentiri soleant:) Ordinarius aquarum cursus esse debet ab Africano littore Americanum versus. Sed adverse littore Americano repercussae, (utrinque versus Polos declinantes,) circuitu facto, inde ad Europaeum littus, contrario quasi motu retrocurrunt: utpote ubi, propter parallelos minores, adeoque tardiorem Telluris motum diurnum, non tarn fortiter Occidentem versus rejiciuntur aquae, ut ab Americano littore repercussis obsistere valeant. Unde fit, ut Maris cursus, tamquam inde ad nos delati, quasi Euro-Boream spectare debeat: (ita tamen ut, ubi hue prope advenerit, circuire possit juxta nostra littora, Aequatorem versus, ut hoc circuitu restituatur id quod ab Africano littore continue cursu Occidentem versus decedit: Hoc utique postulare videtur elementi fluidi Libramentum.)! [2] Id autem porro inquirendum est, Quibus anni temporibus cursus hie, Orientem et Septentrionem interjectus, magis ad Septentrionem declinat;

3 versus (pro Copernicana hypothesi) censetur converti: Aquaque E 5 ferri videatur E 9 versus, sed aquae adverse E 10 facto (Eddy nostri vocant,) inde E 11 quasi |motu add.\ (1) feruntur: (2) retrocurrunt: utpote W quasi motu feruntur. Utpote E 13 repercussis resistere E 15-20 debeat. Indeque factum judico quod retrocurrens hie motus, postquam Norwegiae Promontorium praeteriverit (in Septentrionem rejectus) in Littore Suecico (& quod inde ad Orientem porrigitur) vix ullum efficit notabilem Aestum. Prout etiam in mari Caspio nullus observatur Aestus Marinus (ut quod cum Oceano non conjungitur:) nee quidem notabilis in Mari Baltico (quo per Fretum Sundicum haud facile penetrat retrocurrens Aqua quae Aestum faceret:) nee (ob similem causam) in Mediterranei Maris partes interiores, & Euxino Ponto. || Id autem E 20 hie (ab Americano littore repercussus) Orientem E 444

194. WALLIS to OLDENBURG, IT/[27] March 1667/8 (ii) et, quibus magis ad Orientem. Quippe cum magis ad Septentrionem declinat; in Mare Hibernicum directe fertur, adeoque ad Sabrinae Ostia: Ubi autem magis ad Orientem vergit; directe fertur in Fretum Britannicum, adeoque ad littus Kantianum: Interea vero, dum cursu his intermedio fertur; Devoniae littus, tractumque adjacentem, fortius impetit. Animadvertendum insuper est, Telluris motum Annuum (per Zodiacum,) et Diurnum (secundum Aequatorem) non eodem vel parallelis planis fieri; sed ad invicem inclinari, angulum facientes grad. 23^ circiter. Sed et, Tellure circa puncta Solstitialia versante, motus hi sunt quasi parallel!; alibi vero se mutuo plus minusve decussant, prout propius aut remotius ab Aequinoctiis Tellus abest. Et consequenter, qui ex ambobus componitur motus, (sive accelerando, sive retardando, quorum illud noctu fit, hoc interdiu,) cum utroque participans; prope Solstitia, in Orientem directe tendit; alibi, ad Septentrionem vel Meridiem vergit, (alterum de die, alterum de nocte,) et maxime circa Aequinoctia. Atque hinc fieri videtur, quod Tumores Annui ad Sabrinae Ostia, (propter Mare Hibernicum, inter Euro-boream et Austro-zephyrum porrectum,) propius ad Aequinoctia contingunt, sub initium Martii et finem Septembris: sed in Kantiano littore, (propter Fretum Britannicum, inter Orientem et Occidentem porrectum,) longius utrinque ab Aequinoctiis (adeoque ad Solstitium Brumale prope,) sub Februarii et Novembris initia:

1 Quippe cum magis add. ad W Quippe (nos quod spectat) quum magis ad E 2 Hibernicum (1) fertur (2) directe W 4 Kantianum, fe (quod vocatur) Rumney Marsh: Interea E 5-6 impetit: Indeque factum, ut citius in Hibernicum mare & Sabrinae ostia, serius in Mare Britannicum fe littus Kantianum, (post Aequinoctium Autumnale) pertingant Annui aestus maximi, fe intermedio tempore ad littus Dcvonicnse. \ \ Animadvertendum E 6 motum (1) Diurnum (per Zodiacum,) et Annuum (2) Annuum (per Zodiacum,) et Diurnum W 9 Sed, Tellure E 10 mutuo (1) decussant (2) plus W decussantes E 11-16 qui ab ambobus componitur motus, (cum utroque participans,) prope Solstitia, in Orientem directe tendit: alibi, ad Septentrionem vel Meridiem vergit, fe maxime circa Aequinoctia. || Atque hinc fit, quod E 12 motus add. W 14 (alterum de die, alterum de nocte,) add. W 20 Aequinoctiis, sub Februarii E

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196. WALLIS to OLDENBURG, 30 March/[9 April] 1668 Atque ad Littus Devoniae cum tractu adjacente, temporibus intermediis.

195.

HENRY OLDENBURG to WALLIS 26 March/[5 April] 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in WALLIS-OLDENBURG 2/[12].VII. 1668 (ii). In this letter, Oldenburg requested that Wallis provide an account of Dulaurens' Specimina mathematica. To this end he sent a copy of that work with the letter. See WALLIS-OLDENBURG 30.III/[9.IV].1668 and WALLIS-OLDENBURG 2/[12].VII. 1668 (ii).

196.

WALLIS to HENRY OLDENBURG Oxford, 30 March/[9 April] 1668 Transmission:

Wl Letter sent: LONDON Royal Society Early Letters Wl, No. 44, 4 pp. (our source).— printed: OLDENBURG, Correspondence IV, 285-91. W2 Latin translation of W1: LONDON Royal Society Early Letters Wl, No. 45, 4 pp. E First edition of Latin translation (with some passages omitted): Philosophical Transactions No. 34 (13 April 1668), 654-5 ('Another Letter Written by the same Hand, concerning some Mistakes, to be found in a Book lately publish'd under the Title of Specimina Mathematica Francisci Du Laurens, especially touching a certain Probleme, afnrm'd to have been proposed by Dr. Wallis, to the Mathematicians of all Europe, to solve it.') (first part), and Philosophical Transactions No. 38 (17 August 1668), 748-50 (second part). In the present letter, Wallis provides an account of Dulaurens's Specimina mathematica as requested by Oldenburg. Oldenburg chose at first to publish only part of the Latin translation (up to the paragraph ending '... in the face of all the World, a thing so false.' in the English original), in which Wallis rejects the idea that he was the author of the probleme on the ellipse. Only after Dulaurens printed his reply entitled Responsio . . . ad epistolam D. Wallisii ad darissimum virum Oldenburgium scriptam (Paris [1668]), rejecting Wallis's account of the history of the problem, did Oldenburg publish 'what 1 intermediis. Quippe in eas partes, illis temporibus, ab Americano littore ad Europaeum Recurrens maris motus, composite Telluris motu, detorquetur. || Atque haec est, quae impraesentiarum occurrit, hujus Phaenomeni non improbabilis ratio. Vale. E 446

196. WALLIS to OLDENBURG, 30 March/[9 April] 1668 (out of respect of the same [i.e. Dulaurens]) was supprest ever since that Vindication was printed, with which it then came joyned' (Philosophical Transactions No. 38 (17 August 1668), 748). See also DULAURENS-OLDENBURG [13J/23.V.1668 (OLDENBURG, Correspondence IV, 398-400) and JUSTEL-OLDENBURG [13J/23.V.1668 (ibid., 402-4, 402) for preliminary comments by Dulaurens on the first part of Wallis's letter. Dulaurens's printed reply was sent to Wallis in a packet accompanied by OLDENBURG-WALLIS 30.VI/[10.VII].1668. See also JUSTEL-OLDENBURG [27.VJ/6.VI.1668 (OLDENBURG, Correspondence IV, 427-9, 428).

Oxford March. 30. 1668.

Sir, I received from you893, a few days since, Dulaurens his Specimina Mathemafe'ca894; which I did presently peruse, that I might (as you desire) give you some account of it. The Title, mythinks, promises much more than the book doth perform. A great part of his first booke, seems to be taken out of Mr Oughtred895 & my self896, (though he doth not there so much as name either of us,) & that so evidently, that he doth many times not onely retain the peculiar phrases & forms of speaking, but the very same Notes &; Symbols. And much of the second book, out of Vieta, Schooten, & the Tracts by him published897, (but these he thinks fit to mention898.) There are, in it, divers things unsound; &; many more, unaccurate inough. But what are those genuine principles, & new Elements 9 not onely add. 893

received from you: i.e. with OLDENBURG-WALLis 26.III/[5.IV].1668. Dulaurens his Specimina Mathematical i.e. DULAURENS, Specimina mathematica duobus libris comprehensa. Quorum primus syntheticus agit de genuinis matheseos principiis in genere, in specie autem de veris geometriae elementis hucusque nondum traditis. Secundus vero de methodo compositionis, atque resolutionis fuse disserit, & multa nova complectitur, quae subtilissimam analyseos artem mirum in modum promovent, Paris 1667. 895 Oughtred: presumably OuGHTRED, Clavis mathematicae, London 1631 (and subsequent editions). 896 my self: presumably WALLIS, Arithmetica infinitorum, Oxford 1656. 897 Vieta, Schooten, fe the Tracts by him published: i.e. VlETE, In artem analyticam isagoge, Tours 1592, reprinted in VIETE, Opera mathematica, ed. Fr. v. Schooten, Leiden 1646; SCHOOTEN, Principia matheseos universalis, ed. E. Bartholin, Leiden 1651, reprinted in DESCARTES, Geometria, ed. Schooten, II, Leiden 1661, 1-47; SCHOOTEN, Appendix de cubicarum aequationum resolutione, in DESCARTES, Geometria, ed. Schooten, I, Leiden 1659, 345-68; HUDDE, Epistola de reductione aequationum, in DESCARTES, Geometria, ed. Schooten, I, Leiden 1659, 401-506. 898 mention: see e.g. DULAURENS, Specimina mathematica, 165, 172, 174, 191, 211, 213-17. 894

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of Geometry, hitherto undiscovered, (which his Title promises) I do not find. Much less can I be of his opinion, when he tells us, pag. 141, that he thinks there is no man who would not prefer his Few, before All Euclides, Elements. In the close, he doth mee manifest injury; in affirming of mee what is grossely false. Hee hath, there, a particular Appendix899, with this specious Title, Solutio Problematis a D. Wallisio totius Europae Mathematicis propositi, sed prius ad generale revocati, Anno MDCLVIII, eodem tempore quo propositum est. After this Title, it thus follows. Problema D. Wallisii. Datis Ellypseos maximis Diametris, turn puncto in transversa ejus Diametro assignato, reperire in numeris segmenta lineae intra Ellypsim terminatae, et per datum punctum transeuntis, atque datum angulum cum dicta Diametro facientis. Verum quia praepositae quaestionis solutio aeque facilis est in numeris, ac in lineis, (ut postea apparebit,) melius facturum me judicavi, si prius demonstrationem Analyticam hie afferrem, ex qua tarn Numerica, quam Geometrica sequeretur ad Problematis solutionem pertinens effectio. Atque ut haec solutio cum faenore detur, speciale D. Wallisii Problema, ad generale sic revoco.

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(After which Praeface, he proceeds to give as a Probleme in his own words, with a solution of it, for 7 pages together.) To this I say, 1. The proposing a chalenge of this nature, to all the Mathematicians in Europe; is a Vanity, that I was never yet guilty of: & 'tis like inough, that I never may be. 2. If I had thought fit (for Ostentation) to make such a chalenge; certainly I should have made choise of something, which should have been either of more difficulty, or of more concernment, than I take this to be: which a very indifferent Algebrist, at first sight, may solve in half an hour. 3. I never yet did, nor perhaps ever shall, propose either that Problem, or any to that purpose, (to the best of my remembrance,) to any one person: much less to all the Mathematicians in Europe. Nor is there any thing of truth in what he doth here affirm of mee. 4. There was indeed once a question somewhat like it, proposed to mee, (& I gave a present solution of it,) but never by mee proposed to any. 899

Appendix: i.e. DULAURENS, Specimina mathematica, 249-55.

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196. WALLIS to OLDENBURG, 30 March/[9 April] 1668 And how far I am concerned in it, you have in a Letter900 of mine to the Lord Vicount Brounker, dated May 11, 1658, (the same year that Dulaurens mentions) and the same year Printed in my Commercium Epi901 [2] stolicum , pag. 171. in these words,| Sub initium Februarii jam proxime elapsi, amicorum non nemo cui forte occurrebam sero vesperi, quaestionem sequentem mihi porrexit in scriptis: quam jam nuperrime, ab eodeni intelligo, typis vulgatam esse, cum hac Epigraphe, Spectatissimos viros Matheseos Professores, et alios praeclaros in Anglia Mathematicos, ut Problema solvere dignentur, Jean de Montfert maxime desiderat. Extremis Ellipseos Diametris, distantia centri ab aliquo puncto in axi transverso, ubi linea eundem secet sub angulo dato, in numeris datis: segmenta ejusdem lineae (si opus est) productae, et intra transversum axem et ellipsin terminata, in numeris exhibere. Hanc ego quaestionem, suam ratus (neque enim vel innuebat ille, vel ego turn sciscitabar, cujus erat,) paulo adhuc universalius expositam, sub hac fere quae subest forma (neque enim ipsissima verba memini) postero mane solvebam; neque eram de ilia ultra solicitus, (ut quae res nee magnae difficultatis videbatur, nee momenti:) quam et, quod jam audio, varii variis modis solvebant, utut eorum solutiones nondum viderim. (After which follows, a short & clear solution of mine, with the Demonstration of it, of that Probleme more universally proposed: allowing onely for the Errors of the Presse, which the Reader who understands it, will easyly rectify.) And this is all that I ever had to do in it. But how Mons. Dulaurens can pretend this to be, a proposing of that Probleme, by mee, to all the Mathematicians in Europe; I leave to the judgement of any who can understand Latine, though he should understand nothing of mathematicks. The person who shewed it mee, was Dr Rawlinson902, (first, in writ900

Letter: i.e. WALLis-BaouNCKER 11/[21].V.1658. Printed in my Commercium Epistolicum: i.e. WALLIS, Commercium epistolicum, Oxford 1658, 168-74. The passage quoted by Walk's is to be found on 171-2. 902 Dr Rawlinson: i.e. Richard Rawlinson (d. 1668), fellow of the Queen's College, Oxford and associate of Christopher Wren and Samuel Hartlib. 901

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ing; afterwards, in Print.) What that Jean de Montfert903 is, who did propose it, I do not know; But his printed Paper, at that time, was in London common inough, and proposed to divers other Mathematicians beside my self, (as a Probleme sent out of France;) and was solved by divers of ours in London. And one of them (Dr Christopher Wren, then Astronomy Professor at Gresham-College; now, at Oxford,) printed & published his Solution; and, with it, (in the same Paper,) by way of Return, a Probleme by him proposed, to the Mathematicians in France, (as this had been from thence to us in England;) to which there is none of them have yet (that I hear of) returned any Solution. His own Solution of it, is since published904 in my Treatise De Cycloide, pag. 72, 73. Now when all this is publike & notorious; I cannot but wonder, with what confidence (or gross negligence,) he should publish, in the face of all the World, a thing so false. But the truth is, Mons. Dulaurens, is much more negligent of what hee writes, than doth become a Mathematician. Of which we need go no further for instance then this Probleme of Mons. De Monfert, put into Dulaurens's Words. Where why he should (constantly) write Ellypsis with y, I will not examine, (permitting him to spell as he please;) Nor, why he gives us one turn without another to answere it, (because this concerns the Grammar of it, not the Geometry.) But why he should, instead of Extremis Ellipseos Diametris, (that is, the Greatest & the Lest,) substitute Diametris Maximis, (as if an Ellipse had more Greatest diameters then One;) he ought to give us some good account, or else it will be thought a Negligent expression. With the like negligence he puts in transversa ejus Diametro, instead of in Axe transverso; as if either an Ellipse had no other Transverse Diameters but the Axes; or else, that a point assigned in any other Transverse Diameter would indifferently serve; where no other Diameters are given but the two Axes. He might as well have sayd, ubivis intra Ellipsin assignato; for there is no Point within the Ellipse, which is not in some Transverse Diameter. Again, the Segments of the Line sought being required in Numbers; the things given were to be designed

5 them (1) (Dr Wren (2) (Dr 7 by way of Return, add. 18-21 Where why . . . not the Geometry.) omitted in E 903 904

Jean de Monfert: possibly a pseudonym of Pascal. See PASCAL, (Euvres VIII, 136. published: i.e. WALLIS, Tractatus duo, Oxford 1659, 72-3.

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196. WALLIS to OLDENBURG, 30 March/[9 April] 1668 [3]

in Numbers allso:| And so, instead of datis Ellypseos diametris Maximis, he should have sayd, Ellipseos Diametris Extremis (not Maximis] per Numeros designatis; or, in Numeris datis. And instead of turn assignato puncto in transversa ejus diametro; (where puncto in Numeris assignato would not be proper:) he should have sayd, Punctoque in utrovis Axe transverso (not transversa diametro) per suam vel a Centra, vel a Vertice, distantiam, Numero designatam assignato, (or words to that purpose.) And instead of Segmenta lineae intra Ellipsim terminatae, (which doth not design either end, either of the Line, or of its Segments;) it should have been Segmenta rectae Ellipsi (not intra Ellipsim) terminatae, in puncto illo sectae; or Segmenta rectae per punctum illud transeuntis, huic Axi (sen Puncto) et Ellipsi interjecta; or, Rectae Segmenta Ellipsi et puncto illo terminata; or something to that purpose, which might design as well the End of the streight Line, as the Point in which it is divided: neither of which are by his words determined, but left to conjecture. And so many negligent expressions, in one short proposition, are so much the more unpardonable, because De Montferts probleme, which Dulaurens doth thus imperfectly represent, was much better expressed. Which therefore he should not have varied, unless it had been for the better. Nor do I see what excuse he can pretend: For where ever he mett with this Probleme, (whether in De Montfert's printed Probleme, or in Dr Wren's printed Solution, or in my printed Letter,) he could not but see the Problem better expressed; and, withall, that it was Jean de Monfert, (not Johannes Wallisius,) that did propose it. But there is again as great a negligence, in that which next follows. For when he tells us presently, that it is equally easy to solve the question in Numbers and in Lines, (& therefore would advance his skill, by shewing how it may be done in Lines allso;) It is much otherwise. For to do it in lines, is of no more difficulty, then, to draw a streight line which shall cut another at angles given; (for if a streight line be so drawn, which in the point assigned shal so cut the Axis; the Ellipse, without further construction, doth terminate its Segments:) Which De Montfert, it seemes, was not so weake a Geometer, as to think it a Problem worth proposing;

4 ejus add. 20 For (1) whethe breaks off (2) where 32-452,1 Which De Monfert . . . to solve it. omitted in E 32 De Monfert, (1) was (2) it

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though Dulaurens think it a credit to be able to solve it. I might adde also; that, where as it is proposed, to find the Segments; his whole process doth onely give us the Greater Segment; not the Lesser, nor the Whole Line. 'Tis true, the Lesser Segment may be easyly found, (& so might the Greater allso;) but he should have told us How, since he undertook so to do. The applying this Problem to other Conick Sections; is no new difficulty. For the same process which I shew in the Ellipse, will (mutatis mutandis) be equally applicable to any curve line, whose ordinates are known: (and that whether the Diameter be cut within, or without, the Curve:) and a very indifferent Algebrist will know how to accommodate it to the particular case. I do not trouble you with so diligent a Collection of the like Negligences throughout the Book; Because I am not now writing a Confutation, but Negligentiae Specimina; & these (for so short a Probleme) may suffice. I might shew the like Negligence in that Probleme of his own; which you were pleased to send mee, a while since, to solve. Which was so ambigiously, & imperfectly expressed, (and so unlike a Mathematician,) that it was more difficult to determine the sense of it, than to solve the Probleme. But of this I have given you some account heretofore. | [4] I shal onely adde a few Specimina of the same Negligence which appeares frequently throughout his book. As pag. 67. where he tells us, that as Two streight Lines, do not comprehend a Space, so neither Two Plains. Hee should have sayd, Nor Three Plains. For, as Fewer Streight Lines than Three cannot conclude a Superficial; So neither can Fewer Plains than Power, conclude a Solide Figure. Nor is it very accurate, when he tells us (in the next words) that Two Plains cannot meet in Two Places: For in All Places of the same Line which is their Common Section, they do meet; but no where else. But his Description of an Angle, (in the same page,) is yet more negligent: which he defines to be Duarum pluriumve, ejusdem Speciei, magnitudinum, (ad unum punctum collectarum, &c,) brevissima distantia. For, (to say nothing of Time, Weight, Strength, &c, which he doth elsewhere905 allow to be Magnitudes; &; are, in Euclide's phrase, p,ejedri;)

14 Book; (1) (which would be endless (2) Because 16-20 I might shew .. . some account heretofore, omitted in E 905

elsewhere: see DULAURENS, Specimina mathematica, I.

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196. WALLIS to OLDENBURG, 30 March/[9 April] 1668 taking Magnitude in the strictest sense, for Extensive Corporeal Magnitude; Hee doth himself acknowledge, that it is not to be understood of Bolides, but onely of Lines & Surfaces] & therefore gives us a Retractation, pag. 89. Again, instead of ejusdem Speciei, he should rather have sayd ejusdem Generis, (for so he means,) that is Homogeneous quantities: For a streight line & a Curve, (being Homogeneous, or ejusdem Generis,) though not ejusdem Speciei, do make an Angle: & the like of superficies specie differentes: And even a Line to a Plain, (or other Surface,) may have its Angle of Inclination; (as well as a Plain to a Plain;) though not so much as Homogeneous. Again, having thus defined an Angle, he tells us in the next page, (pag. 68.) that, if those magnitudes be two Lines, the Angle is a Plain Angle; as if he had never heard of Spherical Triangles; or as if no Superficial Angle could be made but in a Plain. Again, Duarum pluriumve, is not safely sayd of Lines: For such meeting of Three Lines, are not an Angle, but Two Angles at lest; more then Two, not concurring to make One Angle. Again, having defined his Angle in generall, (pag. 67,) by Duarum pluriumve; he tells us particularly of a Solide Angle, pag. 68, that it may be comprehended Una vel pluribus, (giving instance for that of One, in the Vertex of a Cone;) as if One were duae vel plures. Again, what doth he mean by brevissima distantial In the Point of Concurse, there is no Distantia; out of it, there is no Minima: (For there can be none assigned, than which there may not be a Lesse.) But indeed his whole Notion of an Angle, is unsound: which is not to be defined by Remotio, or Distantia, but by Inclinatio; as he might have learned from Euclide's Definition. So, pag. 171. He tells us, that in these two forms of Quadratick Equations, aa — ca + dd — 0, and aa + ca + dd — 0, the Roots are both Affirmative: which is far otherwise. In the Former of them the Rootes are indeed both Affirmative; but in the Latter, they are both Negative. If these few, for a tast, do not suffice; it will be easy to furnish a more ample service. But whether they proceed from Incuria or Inscitia, (as he

1 strictest (1) strict breaks off (2) sense, 5 (for so he means,) add. 5 quantities: (1) For streight fe crooked (2) For 6 Generis,) (1) do make an Angle, though, not ejusdem Speciei: (2) though 7 of (1) Surfaces (2) superficies 18 be (1) compl breaks off (2) comprehended 21 there I is add. ed.\ no Distantia

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197. BROUNCKERto OLDENBURG, March? 1667/8 distinguisheth of Errors, pag. ult.90&) I will not determine: (I doubt, there is somewhat of Both:) nor give you, at present, any further trouble. Yours &c. John Wallis.

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For my honoured friend, Henry Oldenburg, Esquire.

197. WILLIAM BROUNCKER to HENRY OLDENBURG [London], March? 1667/8 Transmission:

E1 First edition of (missing) letter sent: Philosophical Transactions No. 34 (13 April 1668), 646-9 ('The Squaring of the Hyperbola by an infinite series of Rational Numbers, together with its demonstration') (our source). E2 Latin translation of E1: WALLIS, Opera mathematica III, 656-9. The occasion for publishing this quadrature was without doubt the publication of Gregory's Vera circuli et hyperbolae quadratura, on which Brouncker himself spoke at the meeting of the Royal Society on 27 February 1668/9; BIRCH, History of the Royal Society II, 253. The introduction to E1 suggests that Brouncker had long been in possession of the result: 'What the Acute Dr. John Wallis had intimated, some years since, in the Dedication of his Answer to M. Meibomius de proportionibus, vid. That the World one day would learn from the Noble Lord Brounker, the Quadrature of the Hyperbole; the Ingenious Reader may see performed in the subjoyned operation, which its Excellent Author was pleased to communicate, as followeth in his own words.' Cf. OLDENBURGBOYLE 17/[27].III.1667/8; OLDENBURG, Correspondence IV, 248-51, 248.

My Method for Squaring the Hyperbola is this:

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Let AB be one Asymptote of the Hyperbola EdC; and let AE and BC be parallel to th'other: Let also AE be to BC as 2 to 1; and let the Parallelogram ABDE equal 1. See Fig. 1. And note, that the Letter x every where stands for Multiplication.

1-2 (I doubt, . . . further trouble, omitted in E 906

pag. ult.: i.e. DULAURENS, Specimina mathematica, 256 (not paginated).

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Supposing the Reader knows, that EA. a£. KH. CB. &c. are in an Harmonic series, or a series reciproca, primanorum sen arithmetice proportionalium (otherwise he is referr'd for satisfaction to the 87, 88, 89, 90, 91, 92, 93, 94, 95, prop. Arithm. Infinitor. Wallisii907: I say

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in infinitum For (in Fig. 2, & 3) the Parallelog.

And (in Fig. 4.) the Triangl.

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907 Arithm. Infinitor. Wallisii: i.e. WALLIS, Arithmetic/1 infinitorum, Oxford 1656, prop. 87-95.

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| And that therefore in the first series half the first term is greater than the sum of the two next, and half this sum of the second and third greater than the sum of the four next, and half the sum of those four greater than the sum of the next eight, &c. in infinitum. For \dD = br + bn; but bn > fG, therefore \dD > br + fG, &c. And in the second series half the first term is less then the sum of the two next, and half this sum less then the sum of the four next, &c. in infinitum. That the first series are the even terms, viz. the 2 d , 4 th , 6 th , 8 th , 10th, &c. and the second, the odd, viz. the 1st, 3d, 5 th , 7th, 9 th , &c. of the following series, viz. ^ ^3, ^, ^ ^, g^, &c. in infinitum = 1. Whereof a being put for the number of terms taken at pleasure, -^TT^ is the last, ^n- is the sum of all those terms from the beginning, and ^kthe sum of the rest to the end. 456

[647]

197. BROUNCKERto OLDENBURG, March? 1667/8 That | of thefirstterme in the third series is less than the sum of the two next, and a quarter of this sum, less than the sum of the four next, and one fourth of this last sum less than the next eight, I thus demonstrate. Let a = the 3d or last number of any term of the first Column, viz. of Divisors,

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And 48o4 — 192a3+240a2—96a = Excess of the Numerator above Denomin. But The affirm. That is, 48o4 + 240a2 Because a4 + 5a2 a3 + 5a

> > > >

the Negat. 192a3 + 96a 4a3 + 2a 4a2 + 2

if a > 2. 15

Therefore Therefore | of any number of A: or Terms, is less than their so many [648] respective B, that is, than twice so many of the next Terms. Quod, &c.

By any one of which three Series, it is not hard to calculate, as near as you please, these and the like Hyperbolic spaces, whatever be the Rational Proportion of AE to BC. As for Example, when AE is to BC, as 5 to 4. (whereof the Calculation follows after that where the Proportion is, as 2 to 1. and both by the third Series.) First then when (in Fig. 1.) AE.BC :: 2.1.

10

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197. BROUNCKER to OLDENBURG, March? 1667/8

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2 x 3 x 4) 1. (0.04166666664 x 5 x 6) 1. (0.00833333336 x 7 x 8) 1. (0.00297619048 x 9 x 10) 1. (0.001388888810 x 11 x 12) 1. (0.000757575712 x 13 x 14) 1. (0.000457875414 x 15 x 16) 1. (0.000297619016 x 17 x 18) 1. (0.000204248418 x 19 x 20) 1. (0.000146198820 x 21 x 22) 1. (0.000108225122 x 23 x 24) 1. (0.000082345224 x 25 x 26) 1. (0.000064102626 x 27 x 28) 1. (0.000050875128 x 29 x 30) 1. (0.000041050930 x 31 x 32) 1. (0.000033602132 x 33 x 34) 1. (0.000027852034 x 35 x 36) 1. (0.000023342636 x 37 x 38) 1. (0.000019756638 x 39 x 40) 1. (0.000016869140 x 41 x 42) 1. (0.000014518042 x 43 x 44) 1. (0.000012584344 x 45 x 46) 1. (0.000010979346 x 47 x 48) 1. (0.000009636148 x 49 x 50) 1. (0.000008503450 x 51 x 52) 1. (0.000007541552 x 53 x 54) 1. (0.000006719354 x 55 x 56) 1. (0.000006012556 x 57 x 58) 1. (0.000005401458 x 59 x 60) 1. (0.000004870460 x 61 x 62) 1. (0.000004406862 x 63 x 64) 1. (0.0000040002-

0.0416666666 0.0113095237

0.0029019589

0.0007306482

0.0001829939

21 42 x 43 x 44) 1. (0.0000025843- corr. ed.

458

197. BROUNCKERto OLDENBURG, March? 1667/8 0.0416666666 0.0113095237 0.0029019589 0.0007306482 3) 0.0001829939 (0.0000609980 0.05679179 + 0.00006100 0.05685279 < EdCy But 0.0007306482 0.0001829939 0.0000458315

10

Therefore 0.05679179 + 0.00004583 + 0.00001528 0.05685290 > EdCy.

15

For, it has been demonstrated that | of any terme in the last Column is less than the terme next after it; and therefore that, of the last terme, at [6491 which you| stop, is less than the remaining terms, and that the total of these is less than | of a third proportional to the two last. And therefore ABCyE being =0.75 0.75 and EdCy > 0.05685279- and < 0.05685290 And ABCdE is < 0.69314720- and > 0.69314709 But when AE.BC :: 5.4. or as EA. to KH. then will the space ABCE. or now, the space AHKE (AH = \AB.) be found as follows. 8 x 9 x 10) 1 (0.0013888888 16 x 17 x 18) 1 (0.0002042484 18 x 19 x 20) 1 (0.0001461988 32 x 33 x 34) 1 (0.0000278520 34 x 35 x 36) 1 (0.0000233426 36 x 37 x 38) 1 (0.0000197566 38 x 39 x 40 1 0.0000168691

5

20

25

0.0003504472

0.0000878204 30

459

197. BROUNCKERto OLDENBURG, March? 1667/8

5

0.0013888888 0.0003504472 3) 0.0000878204 (0.0000292735 0.0018271564 + 0.0000292735 0.0018564299 < Eab But 0.0003504472 0.0000878204 0.00002200737

10

15

20

Therefore 0.0018271564 + 0.0000220074 + 0.0000073358 0.0018564996 > #06 Therefore EMb. (Fig. 4.) being =0.025 Eab > 0.0018564299EMba (Fig.4.) or EKM (Fig.l.) > 0.02685643AHKM< 0.22314356-

0.025 & < 0.0018564996 < 0.02685650 > 0.22314349

Therefore 3 ABCdE= 2.07944154 Therefore the Logar. of 10 and AHKE = 0.2231435is to the Log. of 2 ABCdE (when AE.BC:: 10.1.) = 2.3025850as 2.302585 to 0.693147

1 0.003888888 corr. ed. 21 ABCdE (when AE.BC :: 10.1.) = 2.025850 corr. ed.

460

198. [WALLIS?] to OLDENBURG, March/April ? 1668

198. [WALLIS ?] to HENRY OLDENBURG [Oxford], March/April? 1668 Transmission:

w Copy (in unknown hand) of (missing) note sent: LONDON Royal Society Early Letters Wl, No. 37, 1 p. At top of page, in the same hand: 'I have communicated an Extract of yours of the 10th of march last, to one of our Analysts, whose opinion concerning it is this.' On reverse in Oldenburg's hand: 'Answer to Du Laurens about the Analyt. way of Demonstrating.' This short paper constitutes a reply to a question raised by Dulaurens in DULAURENSOLDENBURG [29.IIj/10.III.1667/8, OLDENBURG, Correspondence IV, 214-15. Oldenburg translated the mathematical section of Dulaurens's letter into English {Royal Society Early Letters L5, No. 15), possibly to disguise the author. The style employed in the answer suggests that it might have originated from Wallis.

The Solution of a geometricall Probl. requires Construction & demonstration, & that synthetically by arguing with knowne quantities onely, not analytically, which argueth by knowne & unknowne mixt together. The proper office of the Analysis in solving a Probl. is to find out a Canon to direct how the Construction may be made, but the Canon doth not alwaies discover all the Determinations requisite to make the Probl. & its Construction possible, for experience will shew that oftentimes in forming the Construction, even of plane Problemes, such objections will start & rise up against the possibility of the Construction, which cannot be solved by any thing apparent either in the Canon or the Analysis. So if a Triangle be to be made of three right lines, although one of them be rightly found out by Construction according as the Canon directs & the other two were first given in the Probl. yet before the Triangle can be described it must be proved synthetically that every two of those 3 lines taken together as one right line are longer than the third, which many times is a worke of greater difficulty than the finding out of the Canon. The Analyticall Resolution indeed if geometrically argued may be evidence enough to convince a Geometrician whether he be an Algebrician or not that the Canon or Theor. deduced from it is true. But a Geometrician will hardly allow a

14 as one right line add. 19 allow a (1) pro breaks off (2) geometricall

461

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199. PHILIPS to WALLIS, 6/[16] April 1668

5

geometricall Probl. to be sufficiently solved unlesse it be performed by Composition, towards the finding out whereof Analysis is an admirable guide, but not sufficient of it selfe to make a compleat solution of a geometricall Probl. And as to a Syntheticall regresse by the steps of the Analysis of a geometricall Probl. it is not alwaies so easy to goe backwards as forwards, nor is it possible, when any terme in any Equation or Analogy of the Analysis exceeds geometricall dimensions.

199.

HENRY PHILIPS to WALLIS [London], 6/[16] April 1668 Transmission:

C First part of letter sent (figure and remaining part containing tables and last two paragraphs missing): LONDON Royal Society Early Letters PI, No. 52, 2 pp. At top of p. 1 in Oldenburg's hand: 'A letter written to Dr John Wallis by Mr Henry Philips, containing his Observations about the True Time of the Tydes.', and: 'Read before the Society Apr. 9. 1668.' Endorsement on left margin: 'Entered L.B. 3.367.' c1 Copy of letter sent (first part in scribal hand, tables and last two paragraphs in Oldenburg's hand): LONDON Royal Society Letter Book Original 3, pp. 367-71. c2 Copy of c1: LONDON Royal Society Letter Book Copy 3, pp. 454-9. E First edition of letter sent: Philosophical Transactions No. 34 (13 April 1668), 656-9 ('A Letter written to Dr. John Wallis by Mr. Henry Philips, containing his Observations about the True Time of the Tides') (our source). The present letter, in which Philips gives an account of his observations concerning the true time of the tides, was read at the meeting of the Royal Society on 9 April 1668; BIRCH, History of the Royal Society II, 264.

Worthy Sir, Being desired by Mr. O.908 to give in, what informations I could, concerning the Tides, I have made bold to present this Paper to 10 your Consideration; which though it have little or no relation to your more curious Philosophical Experiments, yet, I hope, will be of very good

2 an (1) eas breaks off (2) admirable 6 it add. 908

Mr. O.: i.e. Henry Oldenburg.

462

199. PHILIPS to WALLIS, 6/[16] April 1668 use for the finding out the True time of the Tides at all times of the Moone, which is (I conceive) of as great concernment, as any thing in the Motion of the Tides. For, this time of the Tides, though it be a very necessary thing to be known, yet is very rudely and slightly reckoned up by most Seamen and Astronomers; most of them reckoning, as if the Moone being upon such a set point of the Compasse (as the Seaman calls it) or so many houres past the Meridian (as the Almanack-Makers reckon) it were High-Tide in such and such a Port at all times of the Moone. And thus they reckon the Tides every day to differ constantly 48 m. As for instance; A South- West Moone makes a full Tide at London, that must be understood, that it is High-Tide at London when the Moon is three hours past the Meridian. Now this is true indeed at the New and Full Moon, but not at other times of the Moone, which few take any notice of: only Mr. Booker909 had wont to give this Caveat, that about the first and last quarters of the Moone, the Neap-tides did not flow so long as the Spring-tides by one point of the Compasse; but he gives no rule to proportion the difference. But observing this more narrowly, I find, that at London the Tides fall out at the least two points, that is, one hour and an half sooner, in the Quarters then in the New and Full Moone. Now this being a very considerable difference of time, which might very well make many Seamen and Passengers to lose their Tides, I set my self to watch this difference of the time of the Tides, and to find out some Rule, how to proportion the time of the Tides between the Spring-tides and the Neap-tides, and I [657] found by many trialls, that the true time of the Tides might be found| out to be somewhat shorter and shorter, from the New and Full Moone unto the Quarters; yet not in an equall manner, neither gradually decreasing from the New and Full Moone untill the Quarters; but rather, that there was some little difference of alteration both at the New and Full Moones, and also at the Quarters; and that the greatest difference fell out in the midst between them, agreeing very well to a Circular proportion, after this manner: (See Fig. 5.) First, Divide a Circle into 12 Equal parts, or hours, according to the Moones motion or distance from the Sun, from the New Moone to the Full. Secondly, Let the Diameter of the Circle be divided into 90 parts or 909 Booker: i.e. John Booker, an almanack-maker. See TAYLOR, Mathematical Practitioners, 362.

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199. PHILIPS to WALLIS, 6/[16] April 1668

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min. that is, according to the time of the difference of Tides between the New or Full Moone, and the Quarters; which is one hour and an halfe. Thirdly, Make perpendicular lines cross the Diameter of the Circle, from hour to hour. Fourthly, Reckon the time of the Moones coming to the South in the circumference of the Circle, and observe the Perpendicular-Line, that falls from that point upon the Diameter; and the proportionall Minutes, cut thereby, will shew, how many Houres, or Minutes are to be substracted from the time of Hightides at the New and Full Moone, that so you may have the true time of the Tides that present day. For Example; At London, on the day of New and Full Moone, it is High-Tide at London at 3 of the Clock, that is, when the Moone is three hours past the Meridian: and so by the Common Rule, the Moone being about four dayes old, it will be South about three of the Clock, and it will be High-tide three houres afterwards, that is, at 6 of the Clock. But now by this Rule, if you count this time of the Moones coming to the South in 464

199. PHILIPS to WALLIS, 6/[16] April 1668 the Circumference, the perpendicular-line, which comes from 3 to 9, cuts the Diameter in the halfe, at 45 min. which shews, that so much is to be abated from the time of High-tide in the New and Full Moones; So that it is High-tide 45 min. before 6 of the Clocke, that is, at 5. hours 15 min. and not at 6 of the clock, according to the common-Rule. The like you may do for any other Port or place, knowing the| time [658] of High-water at the New and Full Moon in that place: And you may do it the more readily, if you set down the time of Highwater at the New and Full Moon under the Diameter, as I have done for London, where it is high-tide at III. of the clock. So that when the Moon is South at III. of the clock, the perpendicular cuts the diameter at II. hours 15. m. which added to the time of the Southing makes it V. hours 15. m. and so when the Moon is South at IX. of the clock, by adding 2 h. 15. m. you have the time of highwater, which is XL of the clock and 15m. And thus you may easily make a Table, which by the Southing of the Moon, shall readily tell you the time of High-tide at any time of the Moone, as I have done here for London: To which all other places may be reduced to correspond. Moon South

Tide

Moon

Tide

Moon

Tide

London

London

H. 3 3 3 3 3 3 3 4 4 4 4 4 4 4 4 4 5 5

South H. M. VI 0 10 20 30 40 50 VII 0 10 20 30 40 50 VIII 0 10 20 30 40 50

London

H. M. XII 0 10 20 30 40 50 I 0 10 20 30 40 50 II 0 10 20 30 40 50

South H. M. Ill 0 10 20 30 40 50 IV 0 10 20 30 40 50 V 0 10 20 30 40 50

M. 0 9 18 27 36 45 54 2 9 16 23 30 37 44 50 57 3 9

H. 5 5 5 5 5 5 5 5 6 6 6 6 6 6 6 7 7 7

M. 15 21 27 33 40 46 52 59 6 13 20 28 36 44 53 2 11 20

465

H. 7 7 7 8 8 8 8 8 9 9 9 9 9 10

M. 30 41 52 4 14 25 36 48 0 13 26 39 53 6

10 10 10 11

20 33 47 1

Moon South

H. M. IX 0 10 20 30 40 50 X 0 10 20 30 40 50 XI 0 10 20 30 40 50

5

10

15

Tide London

20

H. M. 11 15 11 11 11 12 12 12 12 1 1 1 1 1 2 2 2 2 2

29 43 57 10 24 37 50 3 16 29 42 54 3 16 27 38 49

25

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199. PHILIPS to WALLIS, 6/[16] April 1668 [659]

20 23 30 31 33 34 35

In In In In In In In

column column column column column column column

'Octo.': 3 34 c1 E corr. ed. 'Sept.': 8 38 c1 'Jan.': 6 5 corr. ed. 'Sept.': 4 44 c1 'Mar.': 12 12 c1 'Jan.': 11 42 c1 In column 'July': 4 06 c1 'Aug.': 5 2 c1 In column 'Dec.': 8 36 c1

466

200. CONSTANTIJN HUYGENS to WALLIS, [21 June]/I July 1668

These things I have found to fall out right at London for many years, and so I suppose they may in other places. If the difference be not so much between the Neap-tides and Spring-tides in other places, the Diameter must be divided into fewer parts. As for the highest Tides to happen two or three dayes after the full Moon, I have not made much observation of it, and see little reason for it, but the time thereof agrees here with. And high Spring-tides are not alwayes alike; this year I have not observed any. I should be glad to hear, how these rules hold in other places, that so this true time of the Tides may be more punctually known.

5

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April 6. 1668.

200. CONSTANTIJN HUYGENS to WALLIS

The Hague, [21 June]/l July 1668 Transmission:

C Draft of (missing) letter sent: THE HAGUE Koninklijke Bibliotheek KA XLIV, pp. 618-19, No. 507 (our source). At top left of p. 618 in Huygens's hand: 'Doctor! Joanni Wallisio. Oxon. 1. Jul. 1668.'—printed: CONSTANTIJN HUYGENS, Bnefwisseling III, 233. c Copy of C: THE HAGUE Koninklijke Bibliotheek KA XLV, f. 177v-178r. The present letter is a letter of recommendation written by Constantijn Huygens for his nephew, Maurits le Leu de Wilhem, at the time on a journey through Europe. Cf. WALLIS-HUYGENS 31.VIII/[10.IX].1668.

Amplissime doctissimeque vir, de tot humanitatis vestrae officiis, quibus me indignum hospitem excepistis, cum ante triennium ad vos excurrerem910, praecipuum illud existimavi, quod rogare me, et quidem serio, ut videbatur, dignati estis ut

5 higest corr. ed. 13 indignum (1) (—) (2) hospitem 15 ut videbatur, add. 910 excurrerem: i.e. Constantijn Huygens's visit to Oxford in 1664. Cf. WALLISOLDENBURG 3/[13].XI.1668; WALLIS-HUYGENS 31.VIII/[10.IX].1668.

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200. CONSTANTIJN HUYGENS to WALLIS, [21 June]/I July 1668

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opusculorum, quae in lucem subinde aliud agens dedi, exemplaria illustrissimae vestrae Bibliothecae sinerem inferri. Memini equidem me respondere, magnam eorum partem mihi vernaculam esse, vobis barbaram, non destitistis tamen ab incepto: ut qui complurium lucubrationum exocticarum copiam in illo omnis disciplinae et eruditionis horreo dicebatis asservari. Parui itaque, et recepi facturum me quod hodie praesto. Non expecto ut quaeratis, quamobrem toto triennio malum nomen fuerim. Dicant hoc dissidia nostra, et acerrima et pudendissima fratrum odia, quibus invicem nos laceratum ivimus. Nimirum, viri amicissimi ne in amicorum numero essetis mihi, neque Regnorum vestrorum, neque hujus Reip. ratio passa est. Quod felix, faustum ac perpetuum sit, deferbuit ille aestus, et, ut voveo speroque, amantium ira amoris redintegratio erit. Redeo itaque ad vos eodem vultu atque animo, quo ante non solos tres, sed et paene quinquaginta annos Oxonios colles salutavi911. Accipite, rogo, constantis amicitiae leves tesseras, quando hoc voluistis; par est quid intrivistis etiam vos exedere. Mihi non exiguum operae pretium erit in turba tot insignium virorum vestra autoritate censeri, nisi quidem vos oblati loculi in illo Theatro tandem coeperit poenitere. Quod si factum est, quaeso et obtestor per candorem vestrum ne id latere sinatis Juvenem912 hunc, meum e sorore nepotem, qui his munusculis passus est se onerari. Non gravabitur autori reportare si quid vobis agnoverit minus gratum esse. Ipsum interea vestrae omnium comitati ex animo commendo:| Tuae im- [619]

2 equidem add. 6 recepi (1) hoc (2) facturum |me add. quod (a) ecce (6) hodie 8 et ( 1 ) acus (2) acerrima 9 amicissimi (1) in. hostium (2) ne in amicorum 10 neque (1) hui Reip. conditio (2) hujus Reip. ratio 13 atque animo add. 15 est (1) exedere vos quid intrivistis. (2) quid intrivistis etiam vos exedere. 16 pretium (1) est (2) erit 17 virorum (1) (—} (2) |vestra autoritate add.\ censeri, nisi jquidem add.\ vos 18 in (1) Theatro vestro dum poeniteret (2) illo Theatro tandem coeperit poenitere 18 factum |est, add.\ (1) rogo (2) quaeso et obtestor 20 qui (1) (his) (2) his 20 onerari. (1) Ipsum (2) Non gravabitur (a) domum referre (6) autori reportare 911

salutavi: i.e. Constantijn Huygens's visits to Oxford in 1618 and 1622. Juvenem: i.e. Maurits le Leu de Wilhem (b. 1641), nephew of Constantijn Huygens the Elder. Cf. WALLIS-HUYGENS 31.VIII/[10.IX].1668. 912

468

201. OLDENBURG to WALLIS, 30 June/[10 July] 1668 primis, mi Wallisi, et excellent! Collegae Bibliothecario vestro. Gestit in magnorum virorum amicitias et consortia admitti et, si qua fides Avunculo, sua virtute, doctrina, morumque suavitate ac modestia invenietur illis non indignus. Vale vir maxime et me ama, quasi nunc ex postliminio. Scrib. Hagae Com. Calendis nostris quintilibus MDCLXIIX.

201. HENRY OLDENBURG to WALLIS 30 June/[10 July] 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 2/[12].VII. 1668 (i) and WALLIS-OLDENBURG 6/[16].VIL1668. This letter contained a copy of DULAURENS, Responsio ... ad epistolam D. Wallisii, [1668]. It was received by Wallis on 2/[12] July.

202. WALLIS to JOHN COLLINS early July 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in WALLIS-COLLINS 21/[31].VII. 1668 and COLLINSPELL 18/[28].VIL1668. Evidently written before COLLINS-BRERETON 11/[21].VII.1668. Answered by: CoLLiNS-WALLis 14/[24].VII. 1668. As emerges from COLLINS-PELL 18/[28].VII. 1668, this letter contained as an enclosure Wallis's catalogue of 145 errata which he had found in proofs of Brancker's Table of Incomposit Numbers, less than 100,000. Collins made a copy and sent this to Pell as an enclosure to COLLINS-PELL 18/[28].VII. 1668. Pell in turn sent a list of those

1 in add. 2 consortia (1) intromitti (2) admitti 2 Avunculo (1) est, (2) , sua 3 invenietur (1) ( } (2) illis non indignus 4 ama, (1) vel (2) quasi nunc

469

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203. WALLIS to OLDENBURG, 2/[12] July 1668 (i) ten errata which had not been recognized previously to Brancker (PELL-BRANCKER 21/[31].VII. 1668). However, they arrived too late to be incorporated into the table as printed (with separate pagination) at the end of An Introduction to Algebra, London 1668. The errata (with one exception) were first published by Wallis in A Discourse of Combinations, Alternations, and Aliquot Parts, London 1685 (136), together with twenty others, corresponding to the list Collins added to the foot of his copy.

203. WALLIS to HENRY OLDENBURG Oxford, 2/[12] July 1668 (i) Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 46, 2 pp. (our source). On p. 2 Oldenburg has noted to the right of address at 90°: 'For Transactions 1. The writ of Stubs913. 2. Gregory of sight914. 3. Dr. Wallis'.—printed: OLDENBURG, Correspondence IV, 488-9. Reply to: OLDENBURG-WALLis 30.VI/[10.VII].1668. Enclosure: WALLis-OLDENBURG 2/[12].VIL1668 (ii).

Oxford July. 2. 1668.

Sir,

5

I received yours915 this morning (for which I thank you) & have since finished & transcribed the inclosed916, as here you have it; in answere to the paper917 inclosed in yours. Which I purpose to send you to morrow morning by the Coach, together with the Printed paper & the former

5 the (1) inclosed (2) paper 913

writ of Stubs: probably STUBBE, 'The Remainder of the Observations made in the formerly mention'd Voyage to Jamaica, publisht Numb. 36', printed in Philosophical Transactions No. 37 (13 July 1668), 717-22. 914 Gregory of sight: possibly 'An Extract of a Letter concerning an Optical Experiment, conducive to a decayed Sight, communicated by a Worthy person, who found the benefit of it himself, printed in Philosophical Transactions No. 37 (13 July 1668), 727-9, and 'An Extract of another Letter from the same hand, confirming the contents of the former; and adding some other Observations about Sight', printed ibid., 729-31. 915 yours: i.e. OLDENBURG-WALLIS 30.VI/[10.VII].1668. 916 inclosed: i.e. WALLIS-OLDENBURG 2/[12].VII.1668 (ii). 9lr paper: i.e. DULAURENS, Responsio . . . ad epistolam D. Wallisii ad darissimum virum Oldenburgium scriptam, [1668]. 470

204. WALLIS to OLDENBURG, 2/[12] July 1668 (ii) letter918; rather then by Post, both because this comes sooner, & because the whole would be too big a packet for the Post. This I now send, with the former letter, is as much as will be convenient for you to insert into one occurrence. If he919 have not inough of this; he may have more hereafter if there be occasion. If he have too much of this, he may thank himself. I

5

am Sir yours &c John Wallis. [2] These For my worthy friend, Mr Henry Oldenburgh, at his house about the middle of the Old Palmal near

10

St. James's

15

London.

204.

WALLIS to HENRY OLDENBURG Oxford, 2/[12] July 1668 (ii) Transmission:

W Paper sent: LONDON Royal Society Early Letters Wl, No. 47, 3 pp. (our source). — printed: OLDENBURG, Correspondence IV, 489-92 (Latin original), 492-5 (English translation). E Printed version (with corrections and amendments incorporated): Philosophical Transactions No. 38 (17 August 1668), 744-7 ('Some Animadversions, written in a Letter by Dr. John Wallis, on a printed Paper, entitul'd Responsio Francisci du Laurens ad Epistolam D. Wallisii ad Cl. V. Oldenburgium scriptam.'). Enclosure to: WALLIS-OLDENBURG 2/[12].VII.1668 (i). The present letter, although addressed to Oldenburg, is in fact a reply to Dulaurens's Responsio ad epistolam D. Wallisii, which in turn was a response to the pub1 Post, (1) as well (2) both 1 & (1) the (2) because 918 former letter: i.e. WALLIS-OLDENBURG 30.III/[9.IV].1668, the remaining parts of which were now to be printed. 919 he: i.e. Dulaurens.

471

204. WALLIS to OLDENBURG, 2/[12] July 1668 (ii) lication of the first part of WALLIS-()LDENBURG 30.III/[9.IV].1668, containing Wallis's review of Dulaurens's Specimina mathematical, in Philosophical Transactions No. 34 (13 April 1668). The printed version of the present letter, including the corrections and amendments given in WALLIS-()LDENBURG 4/[14].VII. 1668 and in WALLISOLDENBURG 6/[16].VII. 1668, appeared in Philosophical Transactions No. 38 (17 August 1668), where it is followed by the second part of WALLIS-OLDENBURG 30.III/[9.IV].1668.

Oxonii Julii 2°. 1668. Doctissimo Amicissimoque D. Hen. Oldenburgio, S. 5

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Epistolam tuam920 (Vir Clarissime,) quae Dulaurensii impressam chartarn921 (quae me spectabat) habuit inclusam; accepi hodie. Cui et statim respondendum censui: neque enim longa deliberatione opus est. Expetebas Tu a me non ita pridem literis tuis922, (quod meministi probe,) ut, quid ego de Dulaurensii libro923 turn nuper edito (quern cum illis literis in eum finem mittebas) sentirem, tibi paucis exponerem. Quod cum ego privatis ad te literis924 post quatriduum missis fecerim, (quippe hoc amico expetenti negandum non putavi,) tu harum partem aliquam typis vulgandam paulo post curasti925, (earn nempe quae illatam mihi injuriam expostulabat,) reliqua reticendo. Haec homini bilem movent. Quibus ego haec summatim repono. Turn mihi fuisse liberum, amico expetenti, libere quid sentirem exponere: turn et Te arbitrio tuo usum esse. Speciatim vero, quod ad injuriam spectat quam mihi factam quere6 longa missing in E 7 Expectabas E 7 (quod meministi probe,) missing in E 10 post quatriduum missis missing in E 12 typis vulgandam paulo post curasti, add. W 13 reliqua, in sui, credo, gratiam, reticendo. E 920

Epistolam tuam: i.e. OLDENBURG-WALLIS 30.VI/[10.VII].1668. impressam chartam: i.e. DULAURENS, Responsio . . . ad epistolam D. Wallisii, [1668]. 922 literis tuis: i.e. OLDENBURG-WALLIS 26.III/[5.IV].1668. 923 libro: i.e. DULAURENS, Specimina mathematica duobus libris comprehensa, Paris 1667. 924 literis: i.e. WALLIS-OLDENBURG 30.III/[9.IV].1668. 925 harum partem aliquam vulgandam paulo post curasti: i.e. the publication of the first part of WALLIS-OLDENBURG 30.III/[9.lV].1668 in Philosophical Transactions No. 34 (13 April 1668). 921

472

204. WALLIS to OLDENBURG, 2/[12] July 1668 (ii) bar; dum me tanquam Thrasonem926 aliquem inducit, Problema leviculum (et quidem, prout ipsius verbis exponitur, ridiculum,) totius Europae Mathematicis proponentem; quo de me triumphum ageret, monstrando, quam ille potis sit solvere: Non diffitetur, errasse se927. (Et quidem res ipsa clamat: quippe non ego, sed Montfertius nescio quis Gallus, illud quod innuit Problema, Anglis proposuerat: quod varii variis modis solutum dederunt; inter quos et ego.) Hoc tantum causatur, quod Amiens928 nescio quis tale quid (non dicit, id ipsum,) ipsi retulerat. Sed quorsum est, ut Amicum advocet; cum, quid ego ea in re fecerim, jam palam prostet in scriptis meis editis. Et quidem suspicor suum tale quid quod ab Amico acceperat, non aliud fuisse, quam, hujusmodi Problema in scriptis meis a me Solutum extare; pro quo ille (pari atque in reliquis negligentia) a me Propositum substituit; additque de suo, Totius Europae Mathematicis, quo et jactantior Thraso, et Triumphus suus sit illustrior. Utut sit: hoc eum male habet, quod non simpliciter negaverim me istius Problematis authorem esse; sed, (quod garrulitatem vocat,) mea verba cum suis juxta ponendo, ostenderim, quam mihi manifesto fuerit injurius. De caeteris autem; Non placet ei, quod de suis Ego Censuram instituerem: Hoc est, Nollet ut ego tibi petenti dicerem, quid de libro edito [2] sentirem. Sed quidni liceat? Nam et idem alios sentire, tu etiam nosti.| Male habet etiam929, quod eum censuerim plus fronte polliceri, quam opere absolvent; (nempe hoc mollius sonare putaram, quam si dixissem Parturiunt monies, &c.930) Sed et tu alios juxta mecum sentire, Fastuoso Titulo931 Librum haud satis respondere, non ignoras; nedum in exauctorati Euclidis vices successurum. Neque prius illis fidem faciet, rem se-

7 dederint E 1 Amicus quidam tale E 14 suus (1) sit ill breaks off (2) sit W 17 ponendo, (1) ostenderem (2) ostenderim W 21 polliceri, (1) quam (2) quam W 22 putaram, add. W 926

Thrasonem: i.e. the boastful soldier in Terence's Eunuchus. Non diffitetur, errasse se: see DULAURENS, Responsio ad epistolam D. Wallisii, 6-7. 928 Amicus: i.e. Prenicle; see DULAURENS-OLDENBURG [13]/23.V.1668 (OLDENBURG, Correspondence IV, 398-400, 399). 929 Male habet etiam: see DULAURENS, Responsio ad epistolam D. Wallisii, 1. 930 Parturiunt montes, &c.: HORACE, Ars poetica 139. 931 Fastuoso Titulo: cf. the complete title given in WALLIS-OLDENBURG 30.Ill/ [9.IV].1668. 92r

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204. WALLIS to OLDENBURG, 2/[12] July 1668 (ii)

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cus esse; quam viderint, Genuina Matheseos Principia, et Elementa Vera, (quae hucusque nondum tradita insinuat,) ab eo felicius tradi, quam ea tradiderint superiores. Dixeram, partem magnam ex Oughtredi meisque scriptis (utut neutrius meminerit) desumptam VIDERI; (nempe propter multa quae nobiscum habet communia, et peculiares loquendi formulas ipsaque eadem Symbola passim retenta;) item ex Vieta, Schotenio, aliisque ab eo editis, quorum et subinde meminit. Sed quo animo quove consilio haec dixerim, dicit932 se plane non capere. Dicam: (imo res ipsa dicit, quippe hoc inde directe sequitur:) Nempe ut tibi ostenderem, sua non esse Nova omnia, et hucusque nondum tradita. Rem ipsam quoad caeteros agnoscit, (quibus et Hariotum accenset;) Mea tantum scripta, dicit933 se non legisse; quod excusatum petit. (Esto. Et habeo excusatum. At interim non eo magis inter hucusque nondum tradita censenda erunt, quod ipse non legerit.) Et quidem, Oughtredum quod spectat, enumerat aliquammulta quae jam fatetur ex ipso quasi verbatim variis in locis transcripta; atque excusatum it quod Authorem non nominaverit. (Unde me conjecturam non temere fecisse, satis constat.) Sed negat ea partem magnam (respectu totius) dicenda esse. (Patior itaque ut, pro parte magna, modo id dictum malit, partem potissimam rescribat.) Verum ego, non Numero Verborum, sed rerum pondere, partem magnam aestimo; nee ea tantum ex Oughtredodesumpta Videri existimo, quae totidem Verbis apud eum extant; sed totam earn doctrinam, utut aliis verbis expositam, quae ab ipso jam ante multos annos tradita fuerit, quamque ex eo hausisse videri possit hie Author: licet hie pluribus forte paginis, quam ille lineis, rem eandem explicaverit. Id itaque dictum velim; Magnam partem earum Rerum quae hie traduntur, apud Oughtredum(ne et mea scripta interponam) vel totidem verbis extare, vel verbis tantundem significantibus, vel inde posse levi negotio deduci; ut non pro Rebus hucusque nondum traditis censeri debeant: Sed et hinc desumptam (rem ipsam quod spectat, licet variata nonunquam verborum formula,) videri' propter easdem non raro peculiares loquendi

2 ab ipso E 7 ab (1) eod breaks off (2) eo W 10 et (1) hactenus (2) hucusque W 24 licet |hie add.\ pluribus (1) fortasse (2) forte W 932 933

dicit: see DULAURENS, Responsio ad epistolam D. Wallisii, 5. dicit: see DULAURENS, Responsio ad epistolam D. Wallisii, 4. 474

204. WALLIS to OLDENBURG, 2/[12] July 1668 (ii) [3]

formulas retentas, ipsaque eadem non raro symbola. Quae quidem tam| aperta sunt vestigia unde haec desumpta sint; ut jam non possit ipse non fateri, utut nomen prius retinerit. Atque eadem, servata proportione, de caeteris quos dixeram intellige. Non enim ego utrobivis intenderam crimen Plagii, (quod ipse amoliri vellet,) sed ut tibi dicerem, quod res est, Principia sua, quatenus sana sunt, turn et aliis fuisse pridem cognita, turn et ab aliis dudum tradita, rem ipsam quod spectat, utut sub aliis verborum formulis; neque jam primitus detecta, atque hucusque nondum tradita. Sed et tibi digitum intendi, apud quos authores haec eadem negotia reperias ipse; et quidem, prout ego sentio, non minus feliciter exposita. Cui concinit, quern ad me de illo characterem scripto misit Vir quidam Mathematicus934, tibi non ignotus, (Dulaurentio non inferior,) priusquam ego librum videram, nee interogatus quidem: Algebram (inquit) Dulaurentii, ad D. Oldenburg transmissam, vidi: qui autem Tua Chartesiique et hujus Interpretum scripta viderunt, Authorem credo hunc non sunt admiraturi: Quasi quidem ego, non tarn Censere dicendus sim, quam Consentire. Sed conqueritur porro935, quod dixerim inibi reperiri aliqua parum sana, et minime accurata multo plura. Quorum alterum jam fatetur ipse, (ut non tibi fuerim hac ex parte iniquus index;) alterum non-dum. Neque tarn conqueritur quod haec censuerim, quam quod hujus censurae causas in publicum non protulerim; quippe si hoc fecissem, turn de publico, turn de seipso, gratiam (inquit) meruissem. Hoc autem crimine tuum est me levare. Quippe ego neutrum horum in publicum protuli; sed at Te utrumque. Rogatus utique a te sententiam; ego privatis ad te literis936, et quid censerem paucis indicavi, et cur ita. Addebam scilicet (non quidem justam totius libri confutationem; neque enim id agebam: sed) pauca specimina eorum quae cursim legenti occurrebant vel parum sana, vel minus 1 tarn missing in E 3 reticuerit. E 10 Cui sententiae concinit, E 12 (Dulaurensio, credo, non inferior) E 18 alterum add. W 19 iniquus Judex E 21 non (1) protulerit (2) protulerim W 23 levare. Rogatus enim a Te sententiam, Ego datis ad Te literis E 934

Vir quidam Mathematicus: almost certainly John Collins, who had sent an account of Dulaurens's Specimma mathematica in COLLINS-WALLIS 25.II/[6.III].1667/8. 935 conqueritur porro: see DULAURENS, Responsio ad epistolam D. Wallisii, 5. 936 privatis ad te literis: i.e. WALLIS-OLDENBURG 30!II/[9!V].1668.

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accurate dicta. Quo autem consilio tu, cum partem horum in publicum emiseris (quo forte illatam mihi injuriam utcunque elueres,) reliquum reticueris, (quod in illius gratiam factum putaverim,) cum et totum vel emittere vel reticere potuisses; tu melius noveris. Quoniam vero et ille hoc expetit; per me licebit ut tota Epistola, prout scripta fuerat, quae et tuae potestatis (utpote ad te scripta) facta est, simul prodeat. Ut judicet orbis literatus, numnon justas habuerim ita censendi causas, utut stricturis brevibus insinuatas. Neque enim novas ego jam adjungo; turn quod liber ipse mihi nunc prae manibus non sit; turn, si esset, non nova hanc ob causam recensione censerem indigere. Opprobria, reliquamque quam habet maledicendi copiam, non attingo: quoniam haec non aliud demonstrant magis, quam impotentem scribendis animuin: et me minus quam ilium feriunt. Tu interim (Vir Clarissime) Vive, et Vale. Tuus &c Johannes Wallis.

3 cum et . . . potuisses; missing in E 3 vel emittere vel add. W 5 et (1) ju breaks off (2) tuae W 8 insinuatas; atque resciscat ipse, postquam iram decoxerit, quam inibi & libere & candide egerim; libere tecum, & cum illo, satis candide. Novas ego jam non adjungo, turn E 10 recensione add. W 10-11 indigere. Neque enim mihi tune erat in animo ad vivum omnia resecare, necdum est. Id olim forte fiet, si necesse videbitur; quod non fore autumo, quippe non tanti res est. Quod ad Problema spectat, quod a me Freniclio, ut difficile, propositum innuit937, atque ab ipso solutum; rem secus atque est narrat. Patet utique, Scriptis editis, neque Freniclio a me propositum fuisse, neque ut difficile, Problema quod insinuat, aut etiam ut magni momenti; sed apud alium (cum Ego de Freniclio nih.il inaudiveram) obiter insinuatum, tanquam Fermatiano simile; (Vid. Commercium Epstolicum pag. 35. lin. 4. & seqq.) Quod autem Ego Problema meum depreciatum iveram, arripuit Freniclius, sponte sua, ut satis elegans, & solutione sua dignum. Quae quam aliena sint ab iis, quae hie narrat Du Laurens, cum ipse videas, non possum non rogare, ut imposterum velit ille in Historicis enarrandis fidelius agere, atque in tradendis Mathematicis accuratius. || Opprobria E 12 magis, add. W 937

Problema . . . , quod a me Freniclio ... propositum innuit: see DULAURENS, Responsio ad epistolam D. Wallisii, 7. Cf. WALLIS-DIGBY 21.XI/[1.XII].1657, especially the passage, to which Wallis explicitely refers in the following. 476

205. WALLIS to OLDENBURG, 4/[14] July 1668

205. WALLIS to HENRY OLDENBURG Oxford, 4/[14] July 1668 Transmission:

W Letter sent: LONDON Royal Society Eaxly Letters Wl, No. 48, 2 pp. (our source). Postmark on p. 2: 'JY/6'.—printed: OLDENBURG, Correspondence IV, 495-6.

July. 4. 1668. Oxford. Sir,

I sent you, on Friday morning by Moor's Coach, a packet conteining, a Letter938 in answere to Dulaurens's his printed paper, with the other papers returned as you desired. Which I hope was safely received. You may in that Answere, make these few alterations, if you think fit. In the end of the second paragraph; for reliqua reticendo, you may say reliqua (in sui, credo, gratiam) reticendo. In the fourth paragraph, about the middle, after the word Amicus, instead of nescio quis, put quidam. Toward the end of the 2d page; for Dulaurensio non inferior, say rather Dulaurensio, credo, non inferior. And presently after, for hujus interpretum, say ejus interpretum. Toward the end of the next paragraph, after these words, stricturis brevibus insinuatas, adde (in the same period,) atque resciscat ipse (postquam iram decoxerit) quam inibi et Libere, et Candide, egerim. (Libere, tecum, et, cum illo, satis Candide.) And begin the next period thus, Novas ego jam non adjungo, (instead of Neque enim novas jam adjungo,) and in the end of that paragraph, after indigere. adde Neque enim mihi tune erat in animo ad vivum omnia resecare; necdum est. To which you may subjoin this Paragraph. 6 may add. 8 gratiam (1) reti breaks off (2) reticendo 9 middle, (1) for (2) after 19 of (1) it, after in breaks off (2) that paragraph, 938

Letter: i.e. WALLIS-OLDENBURG 2/[12].VIL1668 (ii). Wallis dispatched this letter on 3 July 1668 in a packet together with WALLIS-OLDENBURG 2/[12].VII.1668 (i), Dulaurens's Responsio ad epistolam D. Wallisii, and WALLIS-OLDENBURG 30.III/[9.IV].1668. 477

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Quod ad Problema spectat, quod a me Freniclio, ut difficile, propositum innuit939, atque ab ipso solutum: rem secus atque est narrat. Patet utique (scriptis editis) neque Freniclio a me propositum fuisse, neque ut Difficile, Problema quod insinuat; aut etiam, ut magni momenti: sed apud alium (cum ego de Freniclio nihil inaudiveram) obiter insinuation, tanquam Fermatiano simile, (vide Commercium Epistolicum, pag. 35. lin. 4 et seqq.) Quod autem ego Problema meum depreciatum iveram; arripuit Freniclius, sponte sua, ut satis elegans, et solutione sua dignum. Quae, quam aliena sunt ab eis quae hie narrat Dulaurens, cum ipse videas: non possum non rogare, ut in posterum velit ille, in Historicis enarrandis fidelius agere, atque in tradendis Mathematicis accuratius. And what other alterations you, or My Lord Brouncker (to whom my service) shall think necessary: I permit to your discretions. Resting, Your very humble servant, John Wallis.

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These For Mr Henry Oldenburgh, in the Pelmell, near St. James's London.

[2]

3 utique (1) in scriptis (2) (in scriptis editis) (3) (scriptis editis) (a) nee (6) neque 4-7 apud (1) alium (2) alium |(cum ego de Freniclio nihil inaudiveram) add] obiter insinuatum, (a) ut Perm breaks off (b) tanquam Fermatiano simile. |(vide Commercium Epistolicum, pag. 35. lin. 4. et seqq.) add] Quod 7 depreciatum (1) (—) (2) iveram; arripuit Freniclius, |sponte sua, add. ut 10 velit (1) illius (2) ille, 11 in (1) tra breaks off (2) tradendis 13 think add. ed. 939

Problema . . . , quod a me Freniclio ... propositum innuit: see DULAURENS, Responsio ad epistolam D. Wallisii, 7. Cf. WALLIS-DIGBY 21.XI/[1.XII].1657, especially the passage, to which Wallis explicitely refers in the following. 478

206. OLDENBURG to WALLIS, 4/[14] ? July 1668

206.

HENRY OLDENBURG to WALLIS 4/[14] ? July 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 6/[16].VII. 1668. This letter apparently contained a report on new notions about mean proportionals from Dulaurens, of which Oldenburg had been informed in JuSTEL-OLDENBURG [27.VI]/7.VII.1668 (OLDENBURG, Correspondence IV, 477-9, 478).

207. WALLIS to HENRY OLDENBURG Oxford, 6/[16] July 1668 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 49, 2 pp. (our source). At top of p. 1 beneath date: 'Extr: enter'd LB. 2. 234'. At top right of p. 2 above address in Oldenburg's hand: 'An Extract of Dr Wallis's letter concerning Dr Wilkins's universal Character.', and beneath address, again in Oldenburg's hand: 'Received July 8. 68.' Postmark on p. 2: 'JY/8'.—printed: OLDENBURG, Correspondence IV, 508-9. w1 Extract of letter sent: LONDON Royal Society Letter Book Original 2, p. 234. w 2 Copy of w1: LONDON Royal Society Letter Book Copy 2, p. 274. Reply to: OLDENBURG-WALLIS 4/[14]?.VII.1668.

Oxford July. 6. 1668.

Sir, To your packet940 of Tuesday last (for which I thank you) which came to mee on Thursday before noon (when our Post was ready to go away) I dispatched an answere941 the same day before supper, & sent it you the next morning by Moor's coach; which should have been with you

6 should (1) be (2) have been 940 941

your packet: i.e. OLDENBURG-WALLIS 30.VI/[10.VII].1668. answere: i.e. WALLIS-OLDENBURG 2/[12].VIL1668 (i) and (ii).

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on Saturday: But I perceive by yours942 of (I suppose) Saturday night (though without date) you had not then received it. I gave you since notice of it (that it may be called for, if not timely delivered,) by mine943 of the last post: which also intimated some smal alterations which you may make. Nor do I think of adding more. Yours, which I received to day, doth not seeme to mee to alter the case at all. The Means Proportionall which the Letter944 intimates, do (certainly) not concern this book945. For I do not remember any thing of that nature (if the meaning be, of 2 mean proportionals between two extremes given, to be found Geometrically:) and having since sent for the book (which was out of my hands) I do not find any such thing in it. But it is (I suppose) some other thing that passeth in writing amongst them946. However; if he undertake that task, I am ready to venture one the Mathematicians side; qui gageroint qu'il se tromperoit9^7. I was thinking after those words, in the former Additions, — ad vivum omnia resecare; Necdum est; of adding Id olim forte fiet, si necesse videbitur. But I think they may be as well left out, unless you adde this allso, Quod non fore autumo; quippe non tanti res est. For I think time may be better imployed, than in large answering of such Books; though there bee matter inough to furnish it. If I must needs be put to that work (or to more replies) it would be convenient that wee know what those new notions are about mean proportionals, which he pretends to: which wil, I suppose, furnish us with new matter. Hevelius's Cometography948 I shal be glad to see.

5 more (1) , unlesse (2) . Yours 12 if lie undertake that task, add. 14-15 words, (1) tow breaks off (2) in the former Additions, — \ad vivum omnia resecare; add. Necdum est; of adding (a) Id olim (b) Id 17 think (1) time (2) time 22 which wil, I suppose, furnish us with new matter add. 942

yours: i.e. OLDENBURG-WALLis 4/[14]?.VIL1668. mine: i.e. WALLis-OLDENBURG 4/[14].VII. 1668. 944 Letter: i.e. JuSTEL-OLDENBURG [27.VIJ/7.VII.1668 (OLDENBURG, Correspondence IV, 477-9, 478). 945 book: presumably Dulaurens's Specimina mathematica. 946 them: i.e. Dulaurens and Huygens. See JUSTEL-OLDENBURG [27.VIJ/7.VII.1668 (OLDENBURG, Correspondence IV, 478). 947 qui gageroint qu'il se tromperoit: see JUSTEL-OLDENBURG [27.VIJ/7.VII.1668 (OLDENBURG, Correspondence IV, 478). 948 Hevelius's Cometography: i.e. HEVELIUS, Cometographia, Danzig 1668. 943

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208. WALLIS to OLDENBURG, 8/[18] July 1668 I suppose Dr Millington949 may be willing to have one of his Selenography950 ; of which therefore I shall speake to him. Dr Wilkins951 his Book952 I have perused, & judge well of it. The thing doubtless is fesible, that is, that a Philosophicall Language may be contrived, explicable either by sounds, by letters, or by other marks: which may have many advantages of any language yet extant. But that any one such Language (for there may be infinites of such possible) shal so obtain, as to become Universal!, I must say (as he doth) that I have but very slender expectations. However; what he hath done may be of very good use, though That should never come to pass. I was (before I came out of town) to seek you at your lodgings, to have come out of your debt; but found you not at home. I shall be ready to do it on the first occasion. I am Sir Your affectionate friend & humble servant Joh: Wallis. [2] These For Mr Henry Oldenburgh, at his house in the Palmal, near St James's London.

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208. WALLIS to HENRY OLDENBURG Oxford, 8/[18] July 1668 Transmission:

W Letter sent (with corrections, amendments, and instructions for the printer entered later by Oldenburg): LONDON Royal Society Early Letters Wl, No. 50, 4pp. (our source). At top right of p. 1 in Oldenburg's hand: 'Dr Wallis's Account of Mr Mercators Logarithmotechnia produced and read at the Society July 16. 1668.' Beneath date: 'Entered R.B. 4. 01.' On p. 4 in Oldenburg's hand: 'Produced and read in the Society July 16. 1668. and ordered to be entered.' 949

Dr Millington: i.e. Thomas Millington. Selenography: i.e. HEVELIUS, Selenographia, Danzig 1647. 951 Dr Wilkins: i.e. John Wilkins, q.v. 952 Book: i.e. WILKINS, Essay towards a Real Character and a Philosophical Language, London 1668. 950

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208. WALLIS to OLDENBURG, 8/[18] July 1668 w1 Copy of letter sent: LONDON Royal Society Register Book Original 4, pp. 1-4. w2 Copy of w1: LONDON Royal Society Register Book Copy 3, pp. 82-6. E Printed version, based on WALLIS-BROUNCKER VII/VIII.1668 (missing) containing the amended and slightly expanded text of W and bearing its date: Philosophical Transactions No. 38 (17 August 1668), 753-6 ('Logarithmotechnia Nicholai Mercatoris: discoursed of in a letter written by Dr. J. Wallis to the Lord Vis-count Brouncker') (our source). Three days after Wallis had sent the present letter, in which he gives an account of Mercator's Logarithmotechnia, he wrote to Oldenburg again, requesting that a number of alterations be made in the case of its being accepted for publication in the Philosophical Transactions. He also suggested that the amended version of the letter be communicated to the Society and registered. Oldenburg duly carried out the instructions and the paper was read at the meeting of the Society on 16/[26] July and entered in the Register Book. See BIRCH, History of the Royal Society II, 306. Wallis later sent a copy of the letter, incorporating alterations and additions, to Brouncker (WALLIS-BROUNCKER VII/VIII.1668). He reports on this in WALLISOLDENBURG 3/[13].VIII. 1668, and therefore it is probable that the letter was sent shortly before this time. There, Wallis indicates to Oldenburg that he would like this amended version to be published in the Philosophical Transactions. It appeared in the August issue, but with the date of the original letter to Oldenburg. Also published in that issue was Wallis's letter to Brouncker of 5/[15] August 1668, containing the proof to which he had referred at the end of the earlier letter.

(W)

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Clarissimo Doctissimoque Oxoniae Julii 8°. 1668. D. Henrico Oldenburgh. S. Incidebam heri Vir Clarissime in D. Mercatoris Logarithmotechniam953 nuper editam954. Quae ita mihi perplacuit ut non prius dimiserim quam perlegissem totam. Phraseologiam quam sub initio habet, seu loquendi formulam, unam aut alteram, mutatam mallem; quippe terminorum ali4 heri (Illustrissime Domine) E 5 placuit E 6-483, 4 totam. Et quamquam pauca quaedam, Phraseologiam quod spectat seu loquendi formulas nonnullas, mutata mallem; sunt tamen ipsa sensu suo sana: Eaque E 953

Logarithmotechniam: i.e. MERCATOR, Logarithmotechnia, London 1668. nuper editam: Mercator's Logarithmotechnia was first communicated to the public on 5/[15] August 1667. The printed edition of 1668 included additional propositions on the quadrature of the hyperbola and the discovery of sums of logarithms (prop. 17-19). It was bound with Ricci's Exercitatio geometrica de maximis et minimis [Rome 1666] at the request of Collins. See COLLINS-PELL 5/[15].IX.1668, British Library MS. Add. 4278, f. 342 r ~ v . 954

482

208. WALLIS to OLDENBURG, 8/[18] July 1668 quot levis immutatio, quantum ego judico, turn rem ipsam, turn sensa sua, aptius explicata daret. Verum illud necesse non est; nam et ea quam habet loquendi formula, rite intellecta, (atque ita ut ipsam intellectam velit,) sensum sanum habet: Eaque quae superstruitur doctrina Logarithmos expedite atque subtiliter construendi, perspicue satis atque ingeniose traditur: ut non videam quid illic additum optarem. Et (quae huic subjungitur955) Quadratura Hyperbolae, elegans admodum est atque ingeniosa. Nempe in hunc sensum. Postquam in Hyperbola MBF, (cujus Asymptotae AN, AH, ad angulum rectum coeant) ostenderat (prop. 14.) Rectangula BIA, FHA, spA etc. (ductis BI, FH, sp, etc. parallelis asymptotae AN,) invicem esse aequalia; adeoque latera habere reciproce proportionalia; (quae nota est Hyperbolae proprietas:) Positis AI = BI = 1, et HI = a; ostendit (prop. 15.) FH — ^5 (Nempe propter Analogiam AH. AI :: BI .FH: hoc est, 1 + a .1 :: I . ^q^-) Sed et (quod dividendo 1, per 1 + a, ostenditur,) j^ = 1 — a + a2 — a3 + a4 &c. (continuatis deinceps in infinitum, ipsius a potestatibus, alternatim negatis et affirmatis.)

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Cumque hoc perinde obtineat ubicunque ultra punctum / ponatur

6 ut non videam quid illic additum optarem. missing in E 7 Quae huic subjungitur Quadraturae Hyperbolae E 8 Nempe ad hunc sensum. V. Fig. 1. E 9 AN, AE corr. Oldenburg (on Wallis's instructions) 10 Rectangula BIA, FHA, spA, (reliquaque hujusmodi,) invicem alt. Oldenburg (on Wallis 's instructions) 16 +a4 add. 16 in infinitum missing in E 955

quae . . . subjungitur: i.e. Prop. 17, Quadraxe Hyperbolam. 483

208. WALLIS to OLDENBURG, 8/[18] July 1668

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H: Positis, ut prius, AI — 1; hujusque continuatione qualibet, ut IT — A; quae intelligatur in aequales partes innumeras dividi, quarum quaelibet, ut Ip, pq, &c, dicatur a; adeoque Ip, Jg, &c, sint a, 2a, 3a, &c, usque ad A: Quae his respondent rectae (seu parallelogramma totidem, latitudinem a habentia,) ps, qt, &c, usque ad ru, (spatium BIru complentia,) erunt 1 -a +a2 -a3 +a4 &c. 1 -2o +4o2 -8o3 +16o4 &c. 1 -3a +9a2 -27a3 +81a4 &c. et sic deinceps usque ad 1 -A +A2 -A3 +A4 &c.

Cum itaque sint 1 + 1 + 1 &c (usque ad ultimum) a + 2o + 3o &c (usque ad A) a2 + 4a2 + 9a2 &c (usque ad A 2 ) a3 + 8a3 + 27a3 &c (usque ad A 3 ) a4 + 36o4 + 81a4 &c (usque ad A4) et sic deinceps:

= = = = =

(quod ostendit ille, prop. 16. estque a me alibi demonstratum:) Recte colligit, (prop. 17) expositum spatium Hyperbolicum BIru = A— gA 2 + jjA3 — |A4 + |A5 &c. Adeoque si (assignato ipsi A — Jr, valore suo in numeris, ut res postulaverit,) distribuantur in duas classes, A, |A3, ^A 5 , &c (potestates Amrmatae,) et ^A 2 , |A4, &c. (potestates negatae;) harumque Aggregatum ex Aggregate illarum| subducatur: Residuum erit [2] ipsum BIru, spatium Hyperbolicum. Nequis autem operam lusum iri existimet, propter addendorum seriem in utraque classe infinitam; adeoque non absolvendam: Huic incommodo medelam (tacitus) adhibet: ponendo A = 0,1, vel = 0,21, aliive fraction! Decimali aequalem, quae minor est quam 1: (hoc est, sumpta Ir minore quam AI — 1:) Quo fit, ut posteriores ipsius A potestates, tot gradibus infra integrorum sedem descendant, ut merito negligi possint. Exempli gratia; Positis AI — 1, et Ir — 0, 21: erit

4 rectae add. 4-5 rectae ps, qt, &c. usque ad ru, (spatium BIru complentes) sunt, E 15 a 4 + . . . = |A5 missing in E 25 Hinc E 27 aequalem, adeoque minorem quam E

484

208. WALLIS to OLDENBURG, 8/[18] July 1668

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Quae est brevis Synopsis, Quadraturae suae satis elegantis. Dissimulandum interim non est; siquis totius BIHF spatii, (cujus latus IH longius intelligatur quam AI,) quadraturam postulet; rein non ita feliciter successuram; propter medelam, quam modo diximus, malo minus sufficientem. Cum enim jam ponenda sit A > 1; manifestum est, posteriores ipsius potestates, altius in Integrorum sedes penetraturas, adeoque minime negligendas. Huic autem incommode, levi constructionis immutatione, facile subvenitur. Caeteris utique ut prius constructis; Quadrandum exponatur HFur spatium; (cujuscunque fuerit longitudinis AH assignata, puta major minorve quam AI; sumptoque ubivis inter A et H puncto r; puta ultra citrave punctum I.) Ponantur autem (non, ut prius, AI = 1, et Ir = A; sed) AH = 1, et Hr = A, quae intelligatur in aequales partes innumeras dividi, quarum quaelibet sit a. Erunt itaque, post AH = 1, reliquae deinceps decrescentes, 1 — a, 1 — 2a, 1 — 3a, &c usque ad Ar — I — A. Item, propter aequalia rectangula FHA, urA, BIA, &c, puta = 62; erit 1,2 i,2 k2 k2 k2 HF = y, reliquaeque deinceps j^, jz^;, fir^ &c, usque ad ru = ^ZTA? spatium HFur coniplentes. Factaque divisione,

13 Synopsis Quadraturae suae satis elegans. E 20 subvenitur. Vid. Fig. 1. E 23 assignata missing in E 24 AI; vel huic aequalis: sumptoque E 25 punctum I, vel in ipso I puncto:) Ponantur E 26 partes add. 29 aequalia add. 31 coniplentes. (Quae oninia ostensa sunt, in niea Arithmetica Infinitorum, prop. 88, 94, 95.) E

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208. WALLIS to OLDENBURG, 8/[18] July 1668

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reperietur ^ = b2 + b2a + b2a2 + b2a3 + 62o4 &c, Hoc est b2 in 1 + 10 a + a2 + a3 + a4 &c, (sumptis ipsius a potestatibus continue sequentibus, affirmatis omnibus.) Cumque de reliquis idem sit judicium: Erunt rectae omnes, ipsis HF & ru interjectae,

15

b2 in et sic deinceps, usque ad [3]

Omniumque summa, 20

Exempli gratia: Erunt A= 0, 21 Positis AH = I . Hr = A = 0,2l AI = b = 0,l adeoque D BIA = b2 = 0,01

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18 Omniumque Aggregatum, A + f A2 + f A3 + \A* + \A6 fee, in 62 = FHru. (per Arithm. In fin. prop. 64.) E

486

208. WALLIS to OLDENBURG, 8/[18] July 1668

Horumque summa ducta in Est

5

Qualium

Rhombo AHGN

Atque haec sunt quae Tibi hac de re ideo scribenda duxi, quoniam ipsi D. Mercatori minus familiariter notus sum. Quae tamen si ipsi impertire velis; non displicebit, credo, haec suae Quadraturae facta accessio. Quae autem de Hyperbola cujus Asymptotae angulos rectos faciunt dicta sunt, alii cuivis tarn facili negotio accommodantur, ut non sit quis ea de re quicquam ultra monere. Vale. Tuus ex animo, Johannes Wallis.

6 Quadr: AHGN corr. Oldenburg (on Wallis's instructions) 9 accessio. (1) Vale. (2) Quae autem 487

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208. WALLIS to OLDENBURG, 8/[18] July 1668 (E, ending)

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Exempli gratia: Positis AH = I . Hr = A = 0,21 AI = b = 0,1 Adeoque b2 = 0,01

Erunt

[755]

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Horum summa ducta in b2 Exhibet

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[756]

Rectus. Quadrato, si angulus A sit ^ Rhombo, J ^ Obhquus. Quae quidem tarn absoluta est tamque expedita Hyperbolae quadratura, ut nesciam an melior sperari debeat. Atque haec sunt quae hac de re scribenda duxi. Quae si D. Mercatori impertiveris; non displicebit, credo, haec suae Quadraturae facta accessio. Posse haec ad Logarithmorum inventionem accommodari, non est quod moneam: Sed & ad Summam Logarithmorum inveniendam956: quam inquirit ille prop. 19.) Nempe, Positis AH = 1, AI = IB = 6, (ut prius) planoque BIHF = pi. Erit pi - b2 + b3 = BIps + BIqt + BIru, &c. usque ad BIHF. Si autem non ab ipsa BI incipiatur; sed ultra citrave, puta a ps: Posita pH = a & psFH = pi. erit (universaliter) pstq + psur &c (usque ad psFH) = pi — ab2: qualium 1, aequetur cubo ipsius AH.) Quod

Uuahum 1. = AHGJM

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5 |A4 = 0,00048623- corr. ed. 11 ±A10 = 0,0000000017- corr. ed. 16 Qualium 1. = ANGN corr. ed. 956

Summam ... inveniendam: i.e. Prop. 19, Invenire Summam Logarithmorum.

488

209. WALLIS to OLDENBURG, 11/[21] July 1668 alias, si opus erit, demonstrabitur. Tu interim, Illustrissime Domine, Vale. Oxon. d. 8. Julii, 1668.

209.

WALLIS to HENRY OLDENBURG Oxford, 11/[21] July 1668 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 51, 2 pp. (our source). On p. 2 beneath address in Oldenburg's hand: 'Ace. jul. 13. 68.' At bottom of p. 2 Oldenburg has noted at 180° to address: 'not to be entered'. Postmark on p. 2: 'JY 13'.—printed: OLDENBURG, Correspondence IV, 525-6.

Oxford July. 11. 1668. Sir,

I have given you the trouble of so many letters of late, that I am allmost ashamed to trouble you with this. Which is but to desire, that in case you think fit to print that957 which I sent you by the last Post: You would please to adde (if it be not too late) just before Vale, (or instead of it.) Nempe pro Substituendum erit,

Qualium 1. = AHGN, Quadrato; Qualium 1. = AHGN, Rhombo.

5 you (1) so (2) the 957

that: i.e. WALLIS-OLDENBURG 8/[18].VII. 1668.

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209. WALLIS to OLDENBURG, 11/[21] July 1668

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Which few words, make the quadrature Universal to All Hyperbole's; which in Mercator's is particularly fitted to that onely where A is a, Right Angle. And now I hope the Quadrature of the Hyperbola is so compleat; as that wee can hardly expect a better. And I think it will be no dishonour to our Society, to have this work perfected by members of our own. In conformity to this Addition; it will be convenient, in the line just before the last paragraph, to insert the letters AHGN- & to read it thus, Qualium 1.00000000000 = AHGN, Quadrato rectae AH. And, if the scheme be not drawn, let it be onely such as this inclosed;958 (for the line DE, (& the rest of Mercator's scheme,) I make no use of at all.) But then; (in the beginning of the 3d Paragraph, Postquam &c: in stead of Asymptotae AN, AE, you must say Asymptotae AN, AH, (putting the letter H for E.) And, in the next line; instead of (reliquaque hujusmodi) put, &c. (ductis BI, FH, sp. &c. parallelis Asymptotae AN.) This being done; you have both Mercators Quadrature, & my Additional; full, clear, & short: & the Figure allso cleared of what could well be spared. If these amendments should come too late (which I am apt to think they will not, unless the other did so;) it is not much material; for all was sound before, this onely makes it a little the more neat. Yours &c J. Wallis. I think it will not be amisse to communicate my letter (amended as is here directed) to the Society at their next meeting; whereby it may be Registred, & thereby become the more Theirs.

25

These For Mr Henry Oldenburg, at his house in the Palmal, near St James's London.

[2]

6 convenient, |in the line add] just before the last (paragraph, add] to 12 must add. 13 line; (1) instead (2) instead 15 both add. 958 this inclosed: not found. Cf. the figure contained in the printed version of WALLISOLDENBURG 8/[18].VII. 1668.

490

210. COLLINS to BRERETON, 11/[21] July 1668

210. JOHN COLLINS to WILLIAM BRERETON London, 11/[21] July 1668 Transmission:

C Copy of (missing) letter sent in Pell's hand: LONDON British Library MS. Add. 4398, f. 148r.

From Brooke house f? July XL 1668 Dr Wallis hath examined the table of Incomposits & found many faults therein more than are mentioned in the Errata959. Whereof I intend to send Dr Pell the particulars960. Dr Wallis is sorry that seeing Dr Pell hath done so well, he hath done no more. I have received a letter961 from Dr Pell directed to my house, and hope by the next Poste to return an answere. In the interim may I beseech your Lordship to represent my humble service to the Dr. I should gladly know the Drs opinion962 of Mr Mercators Logarithmotechnia. Hugenius hath wrote something963 in the French transactions against Mr Gregories assertion that the Circle & Hyperbole are quantities non-Analytick that is, whose Dimensions cannot be expressed by any Aequation. To which Mr Gregorie was not long a drawing up a Reply964, 1 f? add. 2 table of (1) L breaks off (2) Incomposits 9 of (i) Dr (2) Mr 959

hath examined . .. Errata: see WALLIS-COLLINS early VII. 1668. I . . . particulars: He did this in COLLINS-PELL 18/[28].VIL1668. 961 letter: i.e. PELL-COLLINS 6/[16].VII. 1668 (LONDON British Library MS. Add. 4278, f. 126v-127r). 962 Drs opinion: In fact, Pell did not approve of Mercator's book. See PELL-COLLINS 29.VIII/[8!X].1668 and PELL-COLLINS 6/[16].IX.1668. Cf. BIRCH, History of the Royal Society II, 306. 963 wrote something: i.e. Huygens's critical review of J. Gregory's Vera circuli et hyperbolae quadratura, entitled 'Examen de Vera Circuli & Hyperboles Quadratura, in propria sua proportionis specie inventa & demonstrata a Jacobo Gregorio Scoto, in 4°. Patavii', Journal des Scavans (2 July 1668), 52-6; HUYGENS, (Euvres completes VI, 228-30. This piece was written as a letter to Gallois, dated [22.VIJ/2.VII.1668. See ScRIBA, 'Gregory's Converging Double Sequence', Historia mathematica 10 (1983), 274-85 and HUYGENS-WALLIS [3]/13.XI.1668; HUYGENS, (Euvres completes VI, 27881. 964 Reply: i.e. GREGORY, 'Mr. Gregories Answer to the Animadversions of Monsieur 960

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212. WILKINS to WALLIS, 13/[23] July 1668

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which may come into the English Transactions. He hath also squared the Conchoide, Cissoeid and hath found an Hyperbole equall to a figure made of all Tangents standing as perpendiculars on a line of equall parts. Hence, as the adding of Secants made a Logarithme tangents: So the adding of tangents makes a Logarithme Secant. Dr Wallis hath duply'd965 to Du Lauren's Vindication.

211.

HENRY OLDENBURG to WALLIS 13/[23] July 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 16/[26].VII. 1668. This letter apparently concerned the question of when best to publish Wallis's animadversions on Dulaurens's Responsio.

212. JOHN WILKINS to WALLIS London, 13/[23] July 1668 Transmission:

C Letter sent (in real character, except address): OXFORD Bodleian Library Savile A.4 (iv)r-(iv)v (our source). Postmark on f. (iv) v : '(JY}/14'. W Wallis's transliteration of (7: OXFORD Bodleian Library Savile A.4 (iv)r (our source). Answered by: WALLIS-WILKINS 16/[26].VII. 1668.

2 made |of add] all |such del] Tangents Hugenius upon his Book, De vera Circuli & Hyperbolae Quadratura; as they were publish'd in the Journal des S§avans of July 2. 1668', Philosophical Transactions No. 37 (13 July 1668), 732-5. This piece was written as a letter to Oldenburg, dated 13/[23].VIL1668. 965 duply'd: i.e. WALLIS-OLDENBURG 2/[12].VIL1668 (ii). This letter was published in Philosophical Transactions No. 38 (17 August 1668), 744-7.

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212. WILKINS to WALLIS, 13/[23] July 1668 Beneath Wilkins's letter, written in his 'real character' is Wallis's transliteration in Latin letters. Underneath this is the draft of his reply to Wilkins in 'philosophical language', acompanied by the transliteration thereof (WALLIS-WILKINS 16/[26]. VII. 1668).

(C)

t(iv) v ] For the Reverend Doctor John Wallis at Oxford

(W)

5

Friend (beloved person) I-am-saluting-you in-the-Character real,-that-by-this-means I-maybe-provoking an-answer-from-you of-thesame-kind which-you-can-be-performing without-difficulty, if-you-not-have-been -wanting leisure for-reading my-book966, in-hope of-an-Epistle 7 may- (1) provoke-you (2) be-provoking 10 of- (1) a-letter (2) an-Epistle 966

book: i.e. WlLKINS, An Essay towards a Real Character, and a Philosophical Language, London 1668. This was licensed by the council of the Royal Society on 13/[23].IV.1668. See BIRCH, History of the Royal Society II, 265.

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213. WALLIS to MORAY, 14/[24] July 1668 from-you, I-am-under-writing-my-self Lyndyn 1668 month 7th day 13th

Your-friend most-loving Dzhon Wilcinz

213.

WALLIS to ROBERT MORAY Oxford, 14/[24] July 1668 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 52, 2pp. (p. 2 blank) (our source). At top of p. 1: 'Read July 16: 68. Entered LB 2. 238.' At top of p. 2 Oldenburg has noted: 'A Letter of Dr Wallis to Sir R. Moray, Concerning some Experiments of Motion.' w1 Copy of letter sent: LONDON Royal Society Letter Book Original 2, pp. 238-9. w2 Copy of w1: LONDON Royal Society Letter Book Copy 2, pp. 277-8. The present letter, concerning Wallis's thoughts on Borelli's De vi percussionis, was read by Sir Robert Moray himself at the meeting of the Royal Society on 16/[26] July 1668. See BIRCH, History of the Royal Society II, 306-7.

Oxford July. 14. 1668.

5

Sir,

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I have tired Mr Oldenburg with so many letters of late, that I am willing to give him a little respite & take this opportunity of congratulating your return967 from Scotland; which, I doubt not, will add some vigour to the proceedings of the Royall Society, who could not but be sensible of the absence of so active a Member. And (that my letter may not look like a bare complement) I take the liberty hereby to suggest the repeating of some experiments, which Borellus (in his late book968 de vi percussionis pag. 269, 270, 271, 272,) tells us that he hath made. The result 12 the (1) tryall (2) repeating 96r

return: Since 1661, Moray had been Lord of the Exchequer for Scotland, but often spent time in London. He presented to the Royal Society 'some curiosities which he had brought with him out of Scotland' on 2/[12] July and therefore had probably arrived in London shortly before then. See BIRCH, History of the Royal Society II, 302-3. 968 book: i.e. BORELLI, De vi percussionis, Bologna 1667.

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214. COLLINS to WALLIS, 14/[24] July 1668 of them, is to this purpose: That, which way so ever a heavy body be violently cast, (upward, downward, horizontally, or at any angle of inclination thereunto,) the naturall motion of Descent by reason of its gravity, (which with the motion of projection makes up its compound motion,) is still the same. It is a thing which I have oft had thoughts of, as very well worthy a severe Examination by triall; & what I meet with in him, doth renew those thoughts. The particular ways of tryall I will not take upon mee to prescribe, (whether that of Borellus himself, or any other;) But if you think it as proper, as I do, that the thing be tryed by a committee of the Society; you will (I doubt not) find proper ways of doing it. To these I may adde another of his experiments, pag. 114. 115. about a Pendulum; whose string being stopped about the middle in its motion, the bullet swinging onely by the lower half of the string shall (he sayth) spend the same time in finishing its excursion, as if the string at its whole length had continued to move freely. I doe not inlarge in describing the experiments, but refer to the places where they are more fully described, & subscribe myself Your very humble servant John Wallis.

214.

JOHN COLLINS to WALLIS 14/[24] July 1668 Transmission: Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-COLLINS 21/[31].VII. 1668. Reply to: WALLIS-COLLINS early VII.1668. Collins evidently sent Wallis news on James Gregory, who was spending the summer in

1 way add. 1 heavy (1) be (2) body 6 in add. 9 tryed (1) I doubt not but (2) by a committee of the Society; 15-19 con|tinued . .. Wallis. at 90° in left margin 16 experiments, (1) because they are sufficient (2) but

16 are add.

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215. WALLIS to WILKINS, 16/[26] July 1668 London, in particular concerning the emerging controversy between him and Huygens following the publication of Vera circuli et hyperbolae quadratura (Padua 1667). He also commented on Wallis's catalogue of corrections to Brancker's table of incomposite numbers.

215.

WALLIS to JOHN WILKINS Oxford, 16/[26] July 1668 Transmission:

Wl First draft (in philosophical language) of (missing) letter sent: OXFORD Bodleian Library Savile A.4 (iii)r. W2 Second draft (in philosophical language) of (missing) letter sent: OXFORD Bodleian Library Savile A.4 (vi) r . W3 Final draft (in philosophical language with transliteration) of (missing) letter sent (written on WILKINS-WALLIS 13/[23].VII. 1668): OXFORD Bodleian Library Savile A.4 (iv)r (our source). Reply to: WiLKiNS-WALLis 13/[23].VIL1668.

Sir,

I have-been receiving your letter, written in the character real, and I-am sending to you an answer, written in the language Philosophical, that you may-be-able-to know I-am understanding this and that, and I am 5

Oxford, year 1668. month 7. day 16.

6 year add.

496

Your friend and servant John Wallis

216. WALLIS to OLDENBURG, 16/[26] July 1668

216.

WALLIS to HENRY OLDENBURG Oxford, 16/[26] July 1668 Transmission:

W Letter sent: LONDON Royal Society Eaxly Letters Wl, No. 53, 2 pp. (our source). On p. 2 beneath address in Oldenburg's hand: 'Ace. 17. julii 1668.', and at bottom at 180°, again in Oldenburg's hand: 'not to be entered'. Postmark on p. 2: 'JY/17'.—printed: OLDENBURG, Correspondence IV, 553-4. Reply to: OLDENBURG-WALLis 13/[23].VII.1668. The letter enclosed a now missing specimen of a reply to Dulaurens's Responsio ad epistolam D. Wallisii, which was more extensive than WALLIS-OLDENBURG 2/[12].VII. 1668 (ii).

July. 16. 1668. Oxford.

Sir, I received yours969 of the 13th instant. Notwithstanding which I am still of opinion that it were more proper to put that brief answere to Dulaurens into this months transactions. Because I am not of opinion to make a solemn business of it; (such things being better answered by neglecting.) and should I do so, it would be too long to insert into a transaction. I send you here a specimen970 of somewhat on the first three or four leaves, which I have done since your letter. And should I on this occasion go to make a new collection, it would be expected it should be full; & in the nature of a just confutation, which I do not think the Book to deserve. You may adde, if you please, somewhere to this purpose; Quae autem tanquam Nova, a se inventa enumerat; patior ut ii qui haec prius non noverunt, atque tanti aestimant, ab illo discant. Qui autem trita vident et trivialia vel etiam falsa, pro novis et magnis venditata; ita censebunt ut eos sentire par est; etiam me non monente. But I am no way of opinion, 5 transactions. (1) Both that it (2) Because 6 (such things . . . neglecting.) add. 9 which I ... letter add. 13 tanquam \Nova, add] a se inventa (1) enumer breaks off (2) enumerat; 15 vel etiam falsa add. 969 970

yours: i.e. OLDENBURG-WALLIS 13/[23].VIL1668. specimen: i.e. the now missing enclosure.

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217. WALLIS to OLDENBURG, 18/[28] July 1668

5

either to defer it; or to make a solemn business of it: But shall rather let it by as it doth. The Paper I send, I have no copy of; & therefore must have it again if there be occasion. I have onely time to adde (lest the post be gone) that I am Yours &c John Wallis.

These For Mr. Henry Oldenburgh 10 in the Pallmall near St. James's London.

[2]

217. WALLIS to HENRY OLDENBURG Oxford, 18/[28] July 1668 Transmission:

W1 First draft (broken off): LONDON Royal Society Early Letters Wl, No. 55, pp. 1-4 of 8 pp. (our source). At top of p. 1 in Oldenburg's hand: ' ( 1 ) Other Animadversions Written (2) A second letter (a) by (b) of Dr John Wallis on (ao) a (66) the same printed Paper (aoa) entitul'd Responsio Francisci Du Laurens ad Epistolam D. Wallisii ad Cl. Oldenburgium scriptam (bb) of Franciscus Du Laurens mention'd in the next foregoing Transactions.' W2 Paper sent: LONDON Royal Society Early Letters Wl, No. 56, 4 pp. and No. 55, pp. 5-8 of 8 pp. (our source). Foot of both leaves of No. 56 cut off (now W3). W3 Two fragments of the first two leaves of W2: LONDON Royal Society Early Letters Wl, No. 57, middle and lower of three fragments (our source). C Oldenburg's introduction to E2 and part transcript of middle fragment of W3 (verso): LONDON Royal Society Early Letters Wl, No. 57, upper fragment (our source). El First edition of first part of paper, based on W1: Philosophical Transactions No. 39 (21 September 1668), 775-9 ('A second Letter of Dr. John Wallis on the same printed Paper of Franciscus Du Laurens'). E2 First edition of second part of paper, based on W2 and W3, middle (verso only) and lower (recto and verso): Philosophical Transactions No. 41 (16 November 1668), 825-32 ('A Continuation of Dr. Wallis his second Letter, publish't in Numb. 39, to the Printed Paper of Mr. Du Laurens'). Enclosure to: WALLis-OLDENBURG 20/[30].VII. 1668. The background to the existence of the two main manuscripts W1 and W2 is explained

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217. WALLIS to OLDENBURG, 18/[28] July 1668 in WALLIS-OLDENBURG 7/[17].XI.1668. On recognizing that Wl was too long, Wallis produced a shorter version W2 and then inadvertently sent both versions to Oldenburg as enclosures to WALLIS-OLDENBURG 20/[30].VII. 1668. Wallis did not become aware of the mistake until Oldenburg returned the sheets concerned in OLDENBURG-WALLIS early XI.1668. In that letter, Oldenburg asked how to proceed with the publication of Wallis's paper, the first part of which had in the meantime been printed in Philosophical Transactions No. 39 (21 September 1668), based on the incomplete draft W1. Wallis sent both versions back to Oldenburg on 10/[20] November 1668 (see WALLISOLDENBURG 7/[17].XI.1668) with the instruction to 'omitt what is in the first leaf of the two sheets [i.e. of W2], till you come to Porro (ut minutiora quaedam praeteream) &x. And print on, the next (in this months transactions) as being a part of the former letter of July. 18. but omitted in the last either by a mistake, or for want of room. [... ] For my meaning was that the whole letter of July 18. should have come together. Only (as it now happens) the first part was intended to be a little more contracted then as it is now printed'; WALLIS-OLDENBURG 7/[17].XI.1668. A corresponding introductory comment ((7) was inserted by Oldenburg before the continuation of Wallis's letter, now based on W2, in Philosophical Transactions No. 41 (16 November 1668), 825-6.

(W1) Dixeram971, Vir Clarissime, in fine cujusdam ad Te Epistolae, nonnisi ex multis pauca, ea esse specimina, eorum quae apud Dulaurensium occurrunt vel parum sana vel minime accurate tradita. Si vis, ut paucis illis annumerem plura, Obsequor, modo ne petas, ut Omnia. Ab initio itaque ut ordiar; Quantum (inquit §1.) vocamus id omne, quod Extensionem vel Distinctionem in se recipit; eadem videlicet Quantitatis denominatione duabus his rerum affectionibus significandis accommodata. Quantitatis itaque vocem jam definivit, qua nempe significatione

3 eorum add. 4 tradita. (1) Petis jam (2) Si vis, ut paucis illis (a) adjungam (&) annumerem plura, (oa) quo (bb) Obsequor (aaa) itaque, (sed tantum (bbb) , |(sed tantum non invitus,) add. and del.\ modo 5 Omnia. jQuanquam etiam illi petition! tantum non invitus obsequor turn quod ego non aliorum sphalmatis recensendis delector; (1) turn (2) (nee, nisi provocatus, cum aliis severus esse soleo;) turn quod (a) Erratis (b) in Mathesi minus est (oa) periculi necess breaks off (bb) ab Erratis periculi; cum nemo, paulo attentior, ab illis ipse non carere possit; saltern si monitus ut cum delectu agat. del] 6 §1. add. 971

Dixeram: i.e. WALLIS-OLDENBURG 4/[14].VII. 1668. Already in the Latin version of WALLIS-OLDENBURG 30.III/]9.IV].1668, published in Philosophical Transactions No. 34, Wallis remarks on Dulaurens's Specimina mathematical 'Occurrunt inibi aliqua parum sana, fe, minime accurata multo plura'.

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intellectam vult. Eandem (§2) dividit (ut alii) in Continuum, et Discretam seu Separatum, quam Multitudinem vocat. Mox autem (§.3.) Quantitas, inquit, Separata, etiam in rebus omni quantitate destitutis locum habet. Die, quaeso, Quomodo possunt res illae Quantitatem habere, quae sunt omni quantitate destitutael vel (quod statim occurrit) Subjectum Quantitatis Discretae, Quantitatis expers essel Sed exempla subjicit, Duo Puncta, Tria Momenta, Decem Spiritus, &c. At vero annon Duo Puncta, (etiam juxta ilium,) Distinctionem habent? adeoque Quantitatem discretaml Quomodo igitur Omni quantitate destituta dicit? Vel, si non habent, Quomodo dici poterunt discretae Quantitatis Subjectum? Dices forsan, haec eum velle, Puncta Singula non esse Quanta, esse autem Bina. Esto. At peto, an in Punctis singulis locum habeat Discreta Quantitas? sintve singula discretae quantitatis Subjectum? Si sic; turn non sunt omni quantitate destituta: Si non; quomodo hinc ostenditur, quod Subjectum quantitatis discretae, sit Quantitatis expersl Sed et, ut de Singulis prospiciamus. Dicit ille §3, Quantitatis ideo esse expertia, eo quod in unoquoque subjectorum nee ulla distinctio sit, nee extensio. Taceo hie unumquodque dici quantitatis subjectum (quod tamen quantitatis expers esse vult:) Quoniam id expressius dicitur §4 Numerorum, inquit, natura, non simplicem magnitudinem repraesentat, sed ex pluribus Quantitatibus aggregatam. Si itaque Duo Puncta sint ex pluribus Quantitatibus aggregata; horum singula sunt Quantitates, non omni quantitate destituta. Vides tu, quam haec inter se belle conveniunt. Porro, §2, Distinctio, inquit, arguit plurium rerum aggregatum, Dis-

1 (§2) |mox del. dividit 2-3 vocat. (1) Hanc autem Quantitatem Separatum, etiam in rebus omni quantitate destitutis locum habere dicit. (2) Mox autem (§.3.) Quantitas, inquit, Separata, etiam . . . locum habet. 9 disc.re.tam1 (1) Cur (2) Quomodo 13 an (1) Puncta singula (2) in Punctis singulis 14 Quantitas'! (1) sintve Discretae si sic (2) sintve 15 ostenditur, (1) subjectum quantitatis discretae esse (2) quod subjectum quantitatis discretae, sit 17 Quantitatis (1) esse expertia, quoniam (2) ideo esse expertia, eo quod 19 dici (1) subjectum quantitatis (2) quantitatis subjectum, 24 Vides tu, quam haec inter se belle conveniunt. add. 24 bene El

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217. WALLIS to OLDENBURG, 18/[28] July 1668 junctis inter se partibus compositum; ut est Populus, Acervus, et unumquodque eorum, quorum Paries propriis extremitatibus terminantur et ab alterius fine discretae sunt: Atque hujusmodi rerum congeries peculiari nomine Multitudo dicitur. Habes hie turn Distinctions, turn Multitudinis definitionem: Et utrobique requiruntur Paries disjunctae, et quae suis extremitatibus terminatae ab aliis sint Discretae, non Continuae: Et §5, Quantitati Discretae, nimirum Multitudini, praecipue convenit Divisio: Dividi autem nihil aliud est quam Separari, sen Distingui: Multitudinis autem naturam in Distinctions positam esse vidimus. At statim §6, Dis[2] tinctio, inquit, a Divisione differt,\ &c. unde si Distinctio pro quodam genere accipiatur, erit Divisio pro ejus specie assumenda. At vero, si Dividi nihil aliud sit quam Distingui; etiam Divisio nihil aliud erit quam Distinction Quomodo igitur Distinctio a Divisione differtl Aut, altera Genus, altera Species ejusdem censenda erit? Eo, inquit, differunt, quod Distinctio quamcunque Pluralitatem exprimit; Divisio vero earn solam, quae a rerum separabilitate exoritur. Imo vero non quamcunque Pluralitatem exprimit Distinctio; sensu suo, sed rerum tantum, quae sunt inter se Disjunctae, extremitatibus propriis terminatae, atque ab aliis discretae; Quippe hoc arguere modo dixerat Distinctionis vocem. Quae vero ita sunt, non modo Separabilia, sed et acta Divisa sunt et separata: Nee erit Distinctio Divisionis genus quoddam, sed Divisionem in sua significatione includet. Subdit §7. Neque Actualis tantummodo Divisio Multitudini concedenda, verum etiam Potentialis tribuenda videtur; non enim unquam ita divisa est, ut pluribus aliis modis secari non possit: Veluti numerus Duo-

1 compositum; (1) et max, cujus, inquit, partes pr breaks off (2) ut 4 hie turn (1) Disju breaks off (2) Disting breaks off (3) Distinctionis 5 disjunctae, (1) quaeque (2) et quae 9 statim add. 12-13 etiam (1) Distinctio nihil erit quam Divisio (2) Divisio nihil jaliud add] erit quam Distinction Quomodo igitur (a) Distinctione (b) Distinctio a Divisione differt? (aa) vel (bb) Aut 17 Pluralitatem (1) exprimit Distin breaks off (2) exprimit Distinctio; sensu suo, 19 atque add. 19 discretae; (1) D breaks off (2) exprimit D breaks off (3) Quippe 20 sed et acta (1) separata (2) separata sunt (a) et (b) seu divisa (3) Divisa sunt et separata: 23 Divisio (1) multitud breaks off (2) Multitudini

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217. WALLIS to OLDENBURG, 18/[28] July 1668 denarius non ita divisus est in paries suas Duodecimas, ut in tertias, quartas, sextas, et adhuc alias quasdam sine nomine dividi nequeat. Verum mihi dicat velim, (ut ab exemplo suo non recedam,) an Duo Puncta (quam esse Multitudinem satis diserte dixerat,) sint hujusmodi Divisionis capacia? Nempe, ut nunquam ita divisa esse possint, quin et pluribus adhuc 5 modis secari possint! Vel etiam, an Duodecim puncta (quoniam et hunc numerum jam insinuat) postquam in multitudinis partes duodecimas, (nedum in tertias, quartas, sextas, aliasque Assis partes, quae suis nominibus censeri solent, quales sunt Quincunx, Septunx, Bes, Dodrans, Decunx, 10 et Deunx,) distributa fuerint; in alias adhuc partes sine nomine dividi queant? Quippe si non possint, retractandum videtur quod hie dicitur: Si possint, retractandum erit quod modo dixerat §3. in horum unoquoque subjectorum nullam esse vel distinctionem vel extensionem; secandum utique erit, punctorum quodvis. Postquam autem haec dixerat, §7, quodque hanc actualem, virtua15 lemque divisionem Multitudo sibi principaliter assumit, (cum Magnitudo potentialem tantum derivatamque participet;) ne sibi non statim contradiceret; subjungit §8. Ex hoc discrimine sequitur, Cunctam vim Multitudinis, (quae certo, inquit, determinatoque partium numero conflatur,) modum in Divisione recipere, sive Divisionis terminos habere, ultra quos 20 sectio amplius procedere nequit; Magnitudinem vero Divisionem in infinitum admittere. Die, quaeso, (modo Oedipus sis,) qui haec simul constare possint, Nunquam ita dividi posse multitudinem, quin et pluribus aliis modis secari possit; Habere tamen certos divisionis terminos, ultra quos sectio nequit amplius procederel Sed et eadem pluribus prosequitur, Nam, 25 inquit, quando Multitudo dividitur, quia in ilia partium numerus determinatur, necesse est divisionis modos in eodem determinari; et consequenter potentiae divisivae vim, quae tune ad certos illos modos contrahitur alligaturque, aliquando exhauriri et tandem omnino sisti, quando videlicet ad

5 esse (1) pos breaks off (2) possint 5 possint, quia El 8 sextas, (1) aliasque (2) aliasque Assis (a) qu breaks off (&) partes 10 Deunx,) (1) in alias (2) distributa 12 horum add. 16 assumit, (1) Magnitudo (2) (cum Magnitudo \vero del.\ potentialem 18 §8. (1) Cunctam vim Mu breaks off (2) Ex 22 quaeso, |tu, del] (modo

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217. WALLIS to OLDENBURG, 18/[28] July 1668 [3] ultimam usque divisionem perventum erit. Atque ad hunc sensum plura.j

Putan' haec sana esse posse omnia, quae ita contradictionibus scatent? Turn §9.10. Multitudinis ortum et incrementum, a Continuae Magnitudinis Divisione, arcessere satagit. Satis frigide. Quippe ex suis omnino principiis oritur Multitude, a continui natura plane diversis. Nisi et decem spirituum (quos modo memorabat) Multitudinem, ex Continui Sectione ortam censeri velit. Rectius Euclides972' Multitudini, non ex divisione Continui, sed ex Unitatum compositione, ortum asscribit; 2 def. 7. Et quidem, si quando accidit, continuum aliquod in membra dividi; non eo magis Multa dicenda erunt, quam si continua non fuissent: est utique ad hoc, omnino accidentarium, quod aliquando fuerint unum. Imo vero: Nunierus Quadrantum in eodem Continue, Quaternarius non minus jam est, quam post sectionem erit. Nee rectius §11.12. a Divisions rerumque Distinctions, Aequalitatem, Inaequalitatem, aliaque similia, primum orta, dicit. Quippe eorum, quae ab invicem satis distincta sunt, fieri potest ut nulla sit vel Aequalitas, vel Inaequalitas, (puta Temporis, et Lineae;) Aequalitas enim atque Inaequa-

1 perventum erit. (1) Itaque (2) At in magnitudinis divisione, idem accidere nequaquam potest &c. Itaque manifestum est, Multitudinis divisibilitatem longe aliam esse ab ea divisibilitate quae in Magnitudine reperitur: haec enim infinitam cum sit virtutis nullam unquam secandis corporibus finem imponit: alteram vero limitatem habens potentiam &c (3) Atque ad hunc sensum plura. 2 (1) Putas tu (2) Putan' tu corr. ed. 4 incrementum, (1) arcessere satagit a continui (2) a Magnitudinis Div breaks off (3) a Continuae Magnitudinis 5 Quippe (1) omnino (2) ex suis omnino principiis oritur (a) a cont breaks off (b) Multitude 6 Nisi (1) valet ut (2) et 8-10 ortam (1) vel breaks off (2) censeri velit. jRectius Euclides; ... asscribit; 2 def. 7. add] Et quidem, (a) quo (6) si 10 in (1) partes (2) membra 11 ad hoc, add. 12 Imo vero: (1) equidem Quadrantum numerus, (2) Numerus Quadrantum in eodem continue, (a) non minus est Quaternarius, quam postquam (&) Quaternarius non minus 17 sunt, (1) fieri potest ( ( 2 ) fieri 972

Euchdes: i.e. Elements VII, def. 2.

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217. WALLIS to OLDENBURG, 18/[28] July 1668 litas, non est nisi inter Homogenea; cum tamen et Heterogenea sint inter se satis Distincta. Sed et ejusdem Continui Duo semisses sunt invicem Aequales, et Triens Quadrante Major, quamquam (suo sensu) neque Distincta sint neque Divisa; Quippe Dividi nihil aliud est, quam separari sive distingui, §5; hoc est, §2. inter se disjungi, propriisque extremitatibus 5 terminari, atque ab invicem discreta esse, non continua. Et quidem Aequale esse, vel Inaequale, omnino abstrahit ab eo quod res continuae sint vel disjunctae. Duae Horae Continuae, non minus sunt inter se aequales, quam si essent totis annis Disjunctae. Sed et Idem est sibi-ipsi Aequale, 10 (quippe tantundem est quantum ipsum est; quae vera est Aequalitatis notio:) sed a seipso Distinctum esse vel Disjunctum, non erit dicendum. Item, si Aequalibus A, B, addatur commune C, tota fiunt aequalia A + C et B + C (propter communem notionem, Si aequalia aequalibus addantur, tota erunt aequalia;) cum tamen C, quod utrobique additur, non sit a se distinctum. Item, aggregata ilia A + C, B + C, aequalia esse posse nemo 15 negabit, quamquam non sint ita propriis extremitatibus terminata, ut ab alterius fine sint discreta, (quod ad Distinctionem requiritur §2.) sed sint in eodem C communicantia. Sed et Totum parts, sua majus esse, nemo non dixerit; utut ab ea Distinctum atque Disjunctum non sit. Perperam itaque concludit, Aequalitatis atque Inaequalitatis principium non aliunde 20 quam ab ipsa Divisions petendum. Quippe Aequale esse vel Inaequale, non minus est immediata Entis affectio, quam Continuum esse vel Discretum. Neque illud ab hoc ortuni, magis quam hoc ab illo. Quod autem ait, §13. Aequalitatem atque Inaequalitatem certas esse Quantitatum inter se habitudines; omnino verum est. Sed novum non est; 25 quippe hoc ipsum norunt omnes; ipseque jam olim definiverat Euclides973, nenipe turn has, turn rationes alias, Homogeneorum invicem habitudines esse, 3 def. 5. Quod vero mox sequitur, Sola quantitatum inter se compara-

l Homogenea; (1) sed et (2) cum 3 quamquam (1) Distincta non sint: suo sensu (2) (suo sensu) neque (3) (suo 4 nihil (1) esse ait (2) aliud est 5 §5; (1) sive (2) inter (3) sive (4) hoc est, §2. 14 tamen jipsum del] C 19 utut (1) Totum a parte sua (2) ab 22 Discretum (1) ; ut et vel Unum esse vel Multa Neque alterum ab (2) . Neque 26 Euclides, |3 def. 5. del] nempe 27 habitudines \certas del] esse 973

Euclides: i.e. Elements V, def. 3. 504

217. WALLIS to OLDENBURG, 18/[28] July 1668 tio, non sufficit ut aequales inaequalesve dicantur; sed inter se secundum corporis sui molem conferri debent: Quapropter Aequalitatem Inaequali[4] tatemque certas esse Quantitatum\ inter se juxta corporis molem amplitudinemve comparatarum habitudines, colligere licet: Novum quidem est; sed Verum non est: nee omnino admittenda est haec sua Aequalitatis et Inaequalitatis definitio. Certum utique est, Aequalia esse posse atque Inaequalia, quae corporis molem nullam habent, adeoque non possunt secundum hanc comparari: Sic duo tempora, aequalia possunt esse vel inaequalia; Sed et duo puncta duobus punctis, tria momenta tribus momentis, decem spiritus decem spiritibus; numero aequales esse possunt; utut non habeant, secundum quam comparentur, corporis molem. Requiritur utique ad Aequalitatem atque Inaequalitatem, non quidem sola quantitatum comparatio; sed et ut quantitates comparatae sint Homogeneae; (quod ex Euclide modo diximus;) ut juxta corporis molem amplitudinemve comparentur, non requiritur: ob causas modo dictas. Neque tarn ego ipsi hac in re contradico, quam ipse sibi. Quippe (tanquam dictorum §13 oblitus,) subjungit §14, inter duos numeros, duasve lineas, (quae molem corpoream non habent,) aequalitatem vel inaequalitatem reperiri; non vero inter numerum atque lineam. At vero, qui fieri potest, ut numerus numero (qui corporis molem non habent) aequalis sit vel inaequalis, si ad hoc requiratur, ut juxta corporis molem comparentur? Imo vero, quae molem habent corpoream, non est necesse ut quoad hanc comparentur, quoties Aequalia dicantur vel Inaequalia. Possunt utique duo corpora, longitudine aequalia, vel aeque alta, vel aeque lata, vel aequaliter inclinata, vel etiam aeque gravia dici; quae quoad corporis molem sunt inaequalia. Sic duo montes, duobus muribus, sunt aequales numero seu multitudine; utut non magnitudine seu corporis mole.

8 compaxaxi add. 8 tempora, (1) duae lineae, duo numeri, &c (2) aequalia possunt esse vel inaequalia; (a) item duae (&) Sed 11 molem. (1) \\ Neque tarn ego sibi (2) Requiritur (a) quidem, (&) utique 13 comparatio; verum etiam, ut alt. Oldenburg comparatio; verum etiam ut E1 14 diximus;) (1) apud (2) ut juxta

18 lineas, ( (1) utut (2) quae 19 fieri (1) potest (2) possit (3) potest, ut numerus numero (a) aequ breaks off (b) (qui corporis 22 habent add.

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Verum quidem est, inter lineam et numerum (quatenus tales) aequalitatem aut inaequalitatem non reperiri; (utpote quantitates heterogeneas:) neque inter alias, quam quae sunt ejusdem generis quantitates. Verum hoc non ille primus docuit, sed Euclides olim; (ut modo dictum est;) atque post ilium alii, nedum qui illo fuerunt superiores. (Quamquam hoc ipsum a Dulaurensio vix tuto dicitur; ut qui Aequalitatem atque Inaequalitatem, non in ipsa Quantitatum Homogenearum natura fundatam esse vult, sed in rerum Distinction^ adeoque Heterogeneis, utpote non minus inter se distinctis, pariter convenire debent, atque Homogeneis:) Id autem requiritur, ut quoad saltern illud mensurae genus, quo comparantur, sint homogeneae; ut autem illud sit moles corporea, non requiritur. Sic Triangulum Pyramidi aeque altum dici posse, (sin altitudine aequale,) nemo non dixerit; utut haec corporis molem habeat, illud non habeat: Quoniam quoad altitudinem (secundum quam comparantur) Homogenea sunt, utut alias heterogenea. Dum vero ille, ad Aequalitatem aut Inaequalitatem requiri docet, ut juxta corporis molem comparentur; tu mecum juxta judicabis, credo, hoc minus Sanum esse. (Sufficit utique ut juxta Longitudinem, Latitudinem, Altitudinem, angulum Inclinationis, Curvedinem, Durationem, Numerum, Vim, Pondus, Celeritatem, aut aliud quodcunque demum fuit quantitatis genus comparentur, quod utrique comparatorum commune sit; non minus quam juxta molem corporis.) Vides itaque quo tendunt ipsius nova Principia, hucusque nondum tradita, quorum hoc unum est.

1-3 aequalitatem (1) et (2) aut inaequalitatem non (a) intercedere; (b) reperiri; (utpote quantitates (ao) heterog breaks off (bb) heterogeneas:) (aoa) cum (666) neque inter alias, quam quae sunt (aaaa) unius (bbbb) ejusdem, 5-9 (Quamquam hoc . . . atque Homogeneis:) add. 9 Id (1) saltern (2) autem requiritur, ut quoad jsaltem add.\ illud 12 Pyramidi (1) Altitudine aequale (2) aeque altum (a) (sin (b) dici 13 corporis add. 17 minus (1) esse (2) Sanum 18 Altitudinem, (1) Inclinationis (2) angulum 19 Numerum, |Vim, add.\ Pondus, jCeleritatem, add.\ aut aliud (1) atque (2) quodcunque 20 demum (1) fuerit (2) fuit 506

217. WALLIS to OLDENBURG, 18/[28] July 1668 [56, p. 1] Doctissimo Amicissimoque Viro D. Henrico Oldenburg S.

Oxoniae Julii 18. 1668.

Dixeram974 (Vir Clarissime) in fine cujusdam ad Te Epistolae; nonnisi ex multis pauca ibidem esse specimina, eorum quae apud Dulaurensium occurrunt, vel parum sana, vel minime accurate tradita. Petis, ut paucis illis plura annumerem: quo illud constet. Obsequor itaque; modo ne petas ut Omnia, sed tantum non invitus, quia bonas horas melius collocatum iri puta quam in his expandis. Ut itaque ab initio exordiar; Quantum definit (§1) id omne quod Extensionem vel Distinctionem in se recipit: extensione innuens Quantitatem Continuum] Distinctione, Discretam, (quam et Quantitatem Separatam, et Multitudinem appellat,) in quae duo genera (sicut alii) Quantitatem distribuit. Mox tamen subjicit §3. Discretam quantitatem, etiam in rebus omni quantitate destitutis locum habere; et Discretae quantitatis Subjectum, quantitatis expers esse. Quae contradictiones in terminis videntur implicare. Et, ne se expediat dicendo, ea sigillatim sumpta non esse quanta, quorum Aggregatum quantum sit: non modo se hinc excludit dicendo, in Uniquoque subjectorum, nee ullam distinctionem esse nee extensionem; ubi unumquodque (sigillatim sumptum) hujus Subjectum esse

5 pauca (1) ea (2) ibidem 6 accurate (1) dicta (2) tradita 7 quo illud constet. add. 9 quam (1) altercando (I?) in 12 et | Quantitatem add] Separatam 18 quanta, |(ut duo puncta, trio, momenta, decem spiritus,) del] (1) non modo sibi hie excludit eadem (2) quorum 19 subjectorum, (1) nee extensionem (2) nee 20-508, 2 sumptum) jliujus add. Subjectum esse innuitur: sed et (1) idem expresse dicitur §4, (a) Numerum (b) Numerorum naturam, (aa) divisam esse atque compositam, et (66) divisam esse atque compositam, (aoa) et (bbb) et ex pluribus Quantitatibus aggregatam, repraesentare: (2) expresse §4, Numerorum, inquit, natura ex pluribus Quantitatibus aggregatam, repraesentat: (aaaa) Si itaque Numeri (bbbb) ubi 974

Dixeram: i.e. WALLIS-OLDENBURG 4/[14].VII. 1668. Already in the Latin version of WALLIS-OLDENBURG 30!II/[9!V].1668, published in Philosophical Transactions No. 34, Wallis remarks on Dulaurens's Specimina mathematical 'Occurrunt inibi aliqua parum sana, fe, minime accurata multo plura'. 507

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innuitur: sed et expresse §4, Numerorum, inquit, natura ex pluribus Quantitatibus aggregatum, repraesentat: ubi eadem, etiam sigillatim sumpta, Quantitates dicuntur, quorum Aggregatum est multitude. Unumquodque igitur et Quantitatis Subjectum est et Quantitas; et tamen omni quantitate destitutum, ipsum Quantitatis subjectum est quantitatis expers. Similiter, cum Distinctionem, (qua Multitudinis, seu discretae quantitatis naturam designat,) sic definiverat §2, Distinctio arguit rerum aggregatum, Disjunctis inter se partibus compositum; ut, inquit, est populus, acervus, et quicquid eorum, quorum paries propriis extremitatibus terminantur, et ab alterius fine discretae sunt: dixeratque §5, Dividi, nihil aliud est quam separari, sive distingui. Quo sibi statum contradicat; subdit §6. Distinctio a Divisione differt: et, si Distinctio pro quodam Genere accipiatur; erit Divisio pro ejus specie. Item, §7, Multitudo, inquit, nunquam ita divisa est, ut pluribus aliis modis secari non possit. Veluti Numerus Duodenarius &c. Cui statim contradicit §8. Ait enim, cunctam vim multitudinis certo determinatoque partium numero conflari; et modum in divisione recipere, sive divisionis terminos habere, ultra quos sectio procedere nequit; Item, Quando multitude dividitur, quia in ilia partium numerus determinatur, necesse est etiam divisionis modos determinari; et consequenter potentiae divisivae vim, (quae ad certos illos modos contrahitur, alligaturque,) aliqando exhauriri, et tandem omnino sisti, quando videlicet ad ultimam divisionem perventum erit: Et mox, Multitudinis, inquit, divisibilitas; limitatam habens potentiam, circa praefinitas quasdam divisionis rationes solummodo versatur, et determinatis certarum partium spatiis inclusa\ coercetur. Quae omnia, [57 (m)1 supra dictis, directe repugnantur.

3 dicuntur, add. 3-5 Unumquodque igitur . . . quantitate destitutum, (a) et quantitatis expers (b) ipsum Quantitatis subjectum est quantitatis expers. add. 6 (1) Item (2) Similiter 7 aggregatum, (1) distinctis (2) Disjunctis 12 Divisione add. 15 possit. (1) veluti Numerus duodenarius, non ita divisus est in (2) cui statim contradicitur §8. (3) Veluti 21 exhauriri, et tandem add. 26 repugnantur. (1) \\ Multitudinis ortum et incrementum, a continua Magnitudinis Divisione arcessere videtur (quasi lianc origi breaks off (2) \\ Et §6,

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217. WALLIS to OLDENBURG, 18/[28] July 1668 Et §6, Distinctio^ inquit, quamcunque pluralitatem exprimit, et non earn solam quae a rerum separabilitate exoritur. Quod supra traditis repugnat. Si enim Distinctio arguit (ut earn ipse definiverat §2) paries inter se disjunctas- et quidem ita propriis extremitatibus terminatam quamlibet, ut ab alterius fine discreta sit: Quam, quaeso, aliam pluralitatem [56, p. 2] potest exprimere quam quae a rerum separabilitate oritur?| Porro, §10, Multitudinis originem et incrementum, a Divisione continui, petitum Numerique infinitudinem, hinc esse vult, quia continui divisibilitas est interminabilis. Satis frigide. Quippe Multitude ex suis principiis oritur, non minus quam Magnitude ex suis. Neque ex sectione continui, sed ex compositions Unitatum, numerum ortum esse, jam olim definiverat Euclides975, 2 def. 7. Numerique infinitude, non tarn ex infinita continui divisibilitate oritur, quam quia tanta non potest esse Unitatum multitude quin adhuc Unitas alia componi possit, nedum plures. Certumque est, ea Multa esse posse, quae Continua nunquam fuerunt, (ut duo Puncta, tria momenta, decem spiritus, &c, ne ab exemplis suis §3 discedam.) Et quidem siquando continget continuum aliquod in membra dividi; hoc, quantum ad Multitudinis naturam, est plane accidentarium; quippe haec non minus fuissent Multa, etiamsi non fuissent aliquando continua. Et quidem in eodem continue, nondum secto, Quadrantum numerus non minus est Quaternarius, quam olim erit cum fuerit divisum. Abstrahit atque simplicior Numerorum natura, ab eo quod, quae Numerantur, vel

I exprimit, (1) et non earn (2) non autem earn. (3) et 3 enim (1) Distinguit (2) Distinctio 5 pluralitatem (1) exprimit, (2) potest exprimere 7 a (1) Dist breaks off (2) Divisione 10 non minus add. 10 suis. (1) Neque ex (2) Neque II olim (1) docuit (2) definiverat 12 2 def. 7. (1) Possuntque ea Multa esse, quae Continua nunquam fuerunt: (ut duo puncta, tria momenta, decem spiritus, fee, ne ab illius exemplis §3. discedam) (2) Numerique 13 quia (1) tanta Unitatum multitude nulla sit cui non possit adhuc alia U breaks off (2) tanta 14 Unitas add. 14 possit (1) . (2) , nedum plures. 22 quod, (1) Numerata (2) quae Numerantur 975

Euclides: i.e. Elements VII, def. 2. 509

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continua sint, aut aliquando fuerint, vel divisa. Nee rectius §11, a Divisione rerumque distinctione, aequalitatem et inaequalitatem primum oria, dicit; et §12, aequalitatis et inaequalitatis principium non aliunde quam ab ipsamet Divisione petendum. Certum utique est, Aequalia multa esse, vel inaequalia, quae neque Divisa sunt neque Distincta; saltern prout ipse Distinctionem definit; hoc est, ita ab invicem disjuncta, et propriis extremitatibus terminata, ut alterum ab alterius fine Discretion sit: Quippe, ejusdem continui, duo semisses, sunt invicem aequales, utut non discreti, sed continui. Et duae Horae continuae non minus inter se aequales sunt, quam si totis annis interpositis Discretae fuissent. Sic Totum quodvis est majus sua parte, quamvis nee disterminentur: Et aequale omnibus simul sumptis; hoc est, idem sibi ipsi aequale (quamvis hie §12, neget hoc esse posse et distinctionem suppositorum requiri asserit.) Item A + B, et B + C, aequalia possunt esse, vel inaequalia, utut in eodem C communicent. Sed et ipsius B est aequale sibi, (hoc est, tantumdem est atque ipsum est; quae vero est aequalitatis notio;) utut a se distinctum atque divisum non fuerit. Quotusquisque enim est qui non hujusmodi demonstrationem apud Euclidem et alios passim viderit; positis A et C invicem se aequalibus, addatur utrique idem B; eruntque A + B et B + C invicem aequalia, propter communem notionem Si aequalia aequalibus adduntur &c. Sed et hie ipse §16 Aequalitatem Identitatis quandam speciem esse dicit; siquid itaque eum ipse B, non modo sit ejusdem magnitudinis (quod ad aequalitatem sufficit) sed et idem ipsum Ens; erit certe, a fortiori, ipse B aequale. Sed et e contra; Numerus et

2 (1) Similiter (2) Nee rectius §11, |12 del] a 5 sunt add. 7 disjuncta, et (1) terminis (2) propriis 7 ut (1) ab (2) alterum 9-14 Et duae ... requiri asserit). add. in margin 11 fuissent. (1) Sic Totum quodvis (2) Sic Totum quodvis 15 B est (1) sibi (2) aequale 17 qui |non add] hujusmodi 22 itaque (1) hie (2) eum 24 contra; (1) quae |sunt add.\ ab invicem satis Distincta, non tamen Aequalia statim sunt vel Inaequalia. (2) Numerus

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217. WALLIS to OLDENBURG, 18/[28] July 1668 Linea, sunt quidem inter se satis Distincta, non tamen sunt vel Aequalia vel Inaequalia; utpote quantitates invicem Heterogeneae, quarum ad invicem rationem non esse, definiverat olim Euclides976, 3 def. 5. sed et hie fatetur §14. Perperam itaque asserit, non aliunde quam ab ipsamet divisions petendum esse, Aequalitatis et Inaequalitatis principium. Et quidem non minus est immediata Entis affectio, Aequale esse vel Inaequale, (utpote quod ex Homogeneae quantitatis essentia fluit,) quam vel Continuum v [57 (m) ] esse vel Divisum; non illud ex hocce derivatum. | Porro (ut minutiora quaedam praeteream, ne nimius sim, quae tamen ipsa reprehensionem merentur) inter alia, quibus Aequalitatem ob Moderationis virtutem laudat, Inaequalitati interim vitio vertens, quod Excessu et Defectu laboret; haec occurrunt §18. Tanta est aequalitatis moderatio; ut eas non solum quas afficit quantitates augeat, minuat, multiplied, atque dividat, nulla facta in ipsis quoad aequalitatem mutatione; sed etiam, ut quantitates ab inaequalitate affectas per similes operationes tractando, in[56, p. 3] tactam in illis inaequalitatis notam relinquat. Id credo vult, (nisi velit rhetoricando fucum facere,) eandem, quae 9 [In left margin Oldenburg's instruction for the printer:] Here you are to begin to print on.

1 non tamen (1) vel Aequalia sunt vel Inaequalia (2) sunt vel Aequalia vel Inaequalia 4 quam ab (1) (—} (2) (—} divisione (3) ipsamet divisions petendum esse, (a) pr breaks off (b) Aequalitatis 5 quidem (1) minus immediata est Entis affectio (2) non 7 essentia (1) fluere (2) fluit 16 relinquat. |Et paulo post, Sic ergo Aequalitas seipsam primo, deinde inaequalitatem per quaelibet augmenta vel decrementa modo aequalia deducere valet, nullo aut aequalitatis aut inaequalitatis detrimento. del] 17 (1) Id credo vult, (nisi velit rhetoricando fucum facere,) (a) si expositae (6) si 976

Euclides: i.e. Elements V, def. 3.

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217. WALLIS to OLDENBURG, 18/[28] July 1668 prius erat vel aequalitatem vel inaequalitatem, manere immutatam; quaecunque facta fuerit utrinque vel aequalium additio aut subductio, vel per aequalia multiplicatio aut divisio. Missis autem Aequalibus, de inaequalibus dispiciamus. Inaequalitatis nota quam vult, (ni fallor,) est ipsa inaequalium Differentia; et, hanc intactam relinqui, est, eandem manere quae 5 prius fuerat. (Quippe hoc turn ipsa verba spectare videntur, turn argumentum ejus.) Quod quidem in Additione et Subductione, verum est; puta, si expositis 10 et 6, addantur utrinque 2, ut fiant 12 et 8; vel subducantur 2, ut fiant 8 et 4; eadem intacta manet inaequalitatis nota, seu Differentia 10 (4.) Non autem in Multiplication et Divisione; Quippe si per 2 vel multiplicentur, ut fiant 20 et 12; vel dividantur, ut fiant 5 et 3; Differentia fit illic 8, hie 2; neutrobique (quae prius erat) 4. Argumentum ejus est merum sophisma, (quod plus habet in conclusione quam in Praemissis.) Hoc (inquit) facile colligitur ex inaequalitatis ad aequalitatem revocandi ratione: ut enim inaequales quantitates ad aequalitatem perveniant, necesse est 15 addi minori, vel a majori detrahi, ipsarum quantitatum Differentiam: sed,

expositae quantitates, sive aequales fuerint, sive inaequales, aequalium admittent vel additionem vel subductionem, aut per aequalia vel multiplicationem vel divisionem; eadem quae prius fuerat vel aequalitas vel inaequalitas immutata manet. (oa) Ego vero, (ut de aequalibus taceam;) in inaequalibus rem secus esse pronuntior (bb) Missis autem Aequalibus, (aaa) de Inaequalibus paulo videamur quo (bbb) de Inaequalibus despiciamus; num. eadem quae prius fuerat semper maneat Inaequalitas. Inaequalitatis nota, quam vult, (ni fallor,) est ipsa Inaequalium Differentia: Hanc autem intactam manere, est eandem quae prius esse. (Quippe hue spectat, quod affert argumentum.) Hoc autem (aaaa) Aequalium Additionem et Subductionem, (bbbb) ubi Aequalium facta est Additio et Subductio, vero est: non autem ubi per aequalia facta est vel Multiplicatio vel Divisio. (aaaaa) vel (bbbbb) Puta si expositis 10, et 6, fiat utrobique binarii vel additio, ut fiant 12, et 8; vel subductio, ut fiant 8 et 4; eadem quae prius manet diferentia 4. Si vero (aaaaaa) fiat utrinque vel m breaks off (bbbbbb) utraque per 2, vel multiplicentur, ut fiant 20 et 12; vel dividantur, ut fiant 5 et 3: differentia (aaaaaaa) saltern sit (bbbbbbb) fit illic 8, hie 2, non (aaaaaaaa) (ut prius) (bbbbbbbb) (quae prius erat) 4. (2) Id credo 1-3 immutatam; |quaecunque facta . . . aut divisio. add. (1) Inaequalitatis (2) Missis 5 intactam (1) manere, (2) relinqui, 10 Divisione; (1) puta (2) Quippe 12 neutrobique (1) (pri breaks off (2) (quae prius 14 ex (1) aequalitatis (2) inaequalitatis 15 enim (1) inaequalitates (2) inaequales

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217. WALLIS to OLDENBURG, 18/[28] July 1668 per communem aequalium Additionem vel Subductionem, (vides de Multiplicatione et Divisione nihil dici,) neque minor quantitas, majoris differentia augetur, neque major eadem differentia contrahitur; cum idem utrique inaequalitatis parti adjiciatur vel dematur; (quod in Multiplicatione et Divisione non fit:) Haec sunt praemissa; videamus conclusionem: [57 an Ergo (inquit) sive aequalium additione, aut Multiplicatione, sive aequalium detractione vel Divisione, inaequales quantitates augeantur minuanturve, (vides quomodo in conclusione se insinuant Multiplicatio et Divisio, quae in praemissis non erant;) nunquam hac rations in aequalitatem incident, hoc est, manebit semper in ipsis inaequalitas; (vides quomodo jam languet illud, manere intactam, in nudum manere; sed mox resumet vires: Verum hoc non est quod erat probandum, inaequalitatem manere aliquam, sed manere intactam; et praemissa, quatenus quicquam probant, hoc probant, propter idem utrique parti adjectum vel demptum: Sed pergit rhetoricando;) Sic ergo, inquit, Aequalitas seipsam primo, deinde inaequalitatem, per quaelibet augmenta vel decrementa, modo aequalia, (quod interim per aequalia multiplicando, vel dividendo non obtinetur, ut ipse putaverat,) deducere valet, nullo vel aequalitatis vel inaequalitatis detrimento: (vides, resumptis viribus, languidum illud manere, in cum nullo detrimento manere, jam erigi: quod, per rhetoricam variationem phraseos, idem significet, quod prius, intactam relinqui, et nulla facta mutatione.} [56, p. 4] Haec autem fusius aliquanto deduxi, ut| videas, quam, in Demonstrando vacillet hie Mathematicus Rhetoricaster. Mox autem §19, quoniam Inaequalium quantitatum una Major, sive Excedens; altera minor, sive Deficiens dicitur; haec autem Excessus atque

I communem (1) Aequalium (2) aequalium 3 cum idem (1) (—} (2) utrobique (3) utrique 6 aequalium additione, (1) vel (2) aut 7 inaequales quantitates augeantur minuanturve, add. in margin II illud, |manere add. intactam (1) per (2) in 14 parti \vel add] adjectum 14 Sed (1) (—} (2) pergit 16 quod (1) aequaliter (2) interim 17 obtinetur, (1) quo (2) ut 19 (vides, |jam, del.\ resumptis 20 rhetoricam (1) phraseos immutationem, (2) variationem phraseos, 24 autem §19, add. 25 dicitur; (1) sint (2) haec

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Defectus nomina, aberrationes a media significant, (quod vitiorum est;) hoc est, ab Aequali, (cujus itaque modo laudata Moderatio, virtus erit:) quo tandem in vituperatae Inaequalitatis favorem se insinuet; missis his (quae imperfectam Inaequalitatis naturam respiciunt) nominibus; alia (inquit) hi termini nomina sortiuntur; nam qui Major est, Totum dicitur; qui Minor, Pars: (quasi quidem Partis nomen, non pariter imperfectam naturam insinuet, atque Minoris:) Adeoque (novis definitionibus) Totum definit esse quantitatem majorem ad minorem et homogeneam collatam; Partem vero, minorem esse quantitatem ad majorem et homogeneam comparatam. Sed omnino fallitur hie novus Definitor; qui Totum et Partem, tantundem significare autumat, atque Majus et Minus: Verum quidem est Totum sua Parts, majus esse, (et Partem Toto minorem:) Sed non vice versa, Omne Minus cujusque Majoris Partem esse, quod hie insinuat. Lunam ego Tellure Minorem existimo; Sed non existimo Telluris Partem esse. Hoc ilium forte decepit, quod videret apud Euclidem977, 1 def. 5, Partis nomen, peculiar! signification, prout Multiplo opponitur, pro eodem atque Submultiplo, seu aliquota parte, (uti nunc loquimur,) usurpari. Sed aliud significat Pars prout, peculiar! sensu, est correlatum Multipli: (I def. 5.) aliud, prout, vulgata signification, opponitur Toti, (9 ax. I.)978 (nempe, illud quod, cum reliquo, componit Totum.) Atque ex his, inquit, manifestum fit, Totum majus esse sua parte; (quod est Euclidis Axioma Nonum:) Omnino quidem; nempe si Totum et Pars, idem significant atque Majus et Minus. Sed et inde pariter manifestum est, Euclidem fuisse Asinum; Nempe si in illo Axiomate, hoc solum dictum velit, Majus, majus

7 definitionibus), (1) definit (2) Totum 8 ad minorem add. 13 versa, (1) omne Maj breaks off (2) Omne 14 non add. 16 pro (1) submulti breaks off (2) eodem 18-19 prout (1) Multiplo opponitur (2) , peculiari sensu, est (a) Multipli (b) correlatum Multipli: (1 def. 5.) aliud, prout (ao) opponitur (66) , vulgata 20 reliquo, jactu del. componit 21 Axioma (1) 9 um (2) Nonum:) 24 si (1) Axiomate illo (2) in illo Axiomate 24 velit, (1) Majus est majus (2) Majus, majus est 977

Euclidem: i.e. Elements V, def. 1. (9 ax. 1.): i.e. Elements I, ax. 9.

978

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217. WALLIS to OLDENBURG, 18/[28] July 1668 est minore. Quod, si forte, pro Definitione, ferri posset; saltern Axioma esset plane ridiculum. Deinde §.21, Commensurabilitatis et Incommensurabilitatis fontem aperire satagit, (eadem felicitate, qua Multitudinis, et Aequalitatis sive Inaequalitatis originem quaesivit:) Nempe, Quando pars aliquoties sumpta totum suum praecise constituit, Aliquota dicitur: Atque haec pars est toti suo commensurabilis: Belle quidem. Annon vero est hie egregius Defini[57 an tor, qui Partem commensurabilem,\ eandem esse Definit, atque Partem aliquotam? Verum, quid si Pars aliqua non possit aliquoties sumpta totum suum praecise constituere? (puta si sit ut 4 ad 6;) An propterea non erit commensurabilis? Quid item, si duae sumantur quantitates quarum altera alterius non sit pars? Num propterea non possunt esse commensurabiles? Vel etiam duae quantitates invicem aequales; (quarum itaque altera alterius pars esse non possit, cum non sit minor;) Annon erunt commensurabiles? Die tu potius; Duas pluresve quaslibet quantitates (sive altera alterius pars aliquota sit, sive non aliquota, sive ne pars quidem,) commensurabiles esse, si ulla quantitas assumi possit (utut ab eis omnibus diversa) quae singulas possit aliquoties repetita adaequare. Noli autem commensurabilitatem coercere ad earn solam, quae est inter Partem aliquam aliquotam, Totumque illud cujus ea pars sit. Quippe hoc non est [55, p. 5] Commensurabilitatis fontem aperire, sed obturare.| Mox autem §24, Partem Aliquantam (quae ab Aliquota distinguitur)

1 Quod (1) fortasse (2) , si forte 7 commensurabilis: \Illa enim, velut mesura, seipsam primo, deinde aliquoties repetita totum suum mensurat; Quae cum ipsummet hac ratione adaequet, necesse est mensuram et commensurabilitatem in Aequalitate fundari. del. \ Belle quidem. 7 est hie (1) Definitor (2) egregius Definitor, qui Partem commensurabilem, (a) tantundem (6) eandem 9 aliquotam? (1) Verum si paxs aliqua aliquoties sumpta (2) Verum, 9 Verum quod E2 12 pars? (1) Annon possunt liae com breaks off (2) Num 12 commensurabiles? (1) Si sumantur (2) Vel 13 quarum jitaque add. \ altera alterius (1) non sit pars: ne suo quidem sensu; hoc est, non sit minor;) (2) pars esse non possit, cum non sit minor;) 15 Duas (1) quaslibet pluresve (2) pluresve quaslibet 17 esse, (1) quas (2) si 17 ab (1) utraque diversa (2) eis omnibus diversa) quae (a) utramque (b) singulas 19 commensurabilitatem (1) ad (2) coercere

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sic Definit. Quando vero Pars, aut quantumlibet exigua hujus partis portio aliquoties sumpta, toti suo aequalis fieri nequit, sed vel ipsum semper excedit, vel ab eo semper deficit, tune Aliquanta vocatur. Atque haec pars, inquit, est Toti suo Incommensurabilis. Si ego cum singulis, quae passim occurrunt, verbis imperite positis, litem movere vellem; infinitus essem. Haec autem Definitio ita multis scatet mendis, ut ea prius amovenda sint, quam id dicat, quod ille dictum vellet. 1°. Perperam dicitur, sed vel ipsum semper excedit, vel ab ipso semper deficit] et satis absurde. Impossible enim est ut pars ea, ejusve portio, sic sumpta, vel semper excedat, vel semper deficiat. Verbi gratia; 1 ad \/5, talis pars est, qualeni ille vellet. Sed non vel semper excedit aliquoties sumpta, (nam 1, semel, vel bis sumpta, minor est quam -\/5;) vel semper deficit, (nam ter vel pluries sumpta, major erit; est enim \/5, major quam 2, et minor quam 3:) sed aliquando excedit, aliquando deficit, semper autem vel excedit vel deficit, nunquam aequalis est; atque hoc ipsum est quod ille dictum vellet. Pro his itaque verbis, vel semper excedit vel semper deficit; reponendum erit, semper vel excedit vel deficit. 2° Perperam etiam, disjunctive, dicitur Quando pars, aut hujus partis portio, nequit, &c. Quippe hoc semper contingit, ut vel ipsa pars, vel saltern hujus aliqua Portio, nequeat aliquoties sumpta toti aequalis fieri: Adeoque per hanc definitionem, pars omnis dicenda esset turn Aliquanta, turn Incommensurabilis cum toto suo. Verbi gratia, si Pars sit ad Totum suum, ut 4 ad 6; non potest ea toties sumi ut toti sit aequalis; nam semel sumpta, minor erit; bis sumpta, major: Si sit ut 4 ad 8; pars quidem ea bis sumpta, toti aequabitur, sed ejus portio, 3, nequit ita sumi ut aequalis fiat; nam bis sumpta, minor erit; ter sumpta, major

5 essem add. 8 semper (1) excedat (2) excedit 10 talis est, quam E2 11 aliquoties sumpta, add. 12 \/5;) (1) neque (2) vel 132, (1) sed (2) et 17 deficit. (1) 2° Quod disjunctive (2) 2° Perperam 19 hujus (1) Portio (2) aliqua 20 per hanc definitionem, add. 22 6; (1) nequit haec aliquoties su breaks off (2) non 23 Si (1) pars (#) sit 24 quidem add.

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217. WALLIS to OLDENBURG, 18/[28] July 1668 quam 8: et quidem semper, vel pars ipsa, vel ejus aliqua portio, (saltern in quantitate continua;) ita se habebit. Itaque pro eo quod disjunctive dicitur, Pars, aut hujus partis portio, nequit; dicendum erat copulative, neque pars ipsa, neque hujus partis portio, potest. 3° Neque hoc sufficit; fieri enim potest, ut turn ipsa pars, turn ipsius aliqua portio, (nedum aliquammultae portiones,) ita se habeant, nee tamen ea pars sit incommensurabilis. Verbi gratia, Si pars sit ad totum, ut 4 ad 5; non potest ipsa pars sic sumi (nam semel sumpta, minor est; bis sumpta, major illo Toto:) sed neque ipsius portio 3 vel 2; (nam portio 3 semel sumpta, minor est quam 5; bis sumpta, major: et 2; bis sumpta minor; ter sumpta, major:) potest tamen ejus alia portio, nempe 1, sic sumi; (nam portio 1 quinquies sumpta, toti 5 aequatur.) Neque hie opem feret, inferta clausula quantumlibet exigua; certum enim est, in parte, quae vel maxime commensurabilis sit, sumi posse portiones quantumlibet exiguas, quae non modo totum non metiantur, sed ne commensurabiles sint. Dicendum igitur, neque pars ipsa, neque ulla hujus partis portio, &c. (Quod ita limitandum erit ut mox dicetur.) 4° Superest adhuc aliud mendum, quod majoris est momenti, et imperitiam arguit. Quippe si haec constet definitio, omnino nulla pars erit cum toto suo incommensurabilis. Nam in ea quae vel maxime sit incommensurabilis, sumi poterit portio aliqua (nedum innumerae) quae Totum mensurant. Verbi gratia, Latus Quadrati ad Diagonium suum, est incommensurabile; vel (ut hie loquitur) est pars ejus incommensurabilis: Sumi tamen potest Lateris aliqua portio, quae Diagonii Dimidio, vel Quadranti aequetur: quae itaque bis aut quater sumpta, Toti aequabitur. Quod videtur hie Definitor non animadvertisse; cui vel maxime prospiciendum erat.

1 (saltern. (1) si Totum sit (2) in quantitate 2 eo quod disjunctive dicitur, add. 3 nequit; (1) repondendum erit, (2) dicendum erat copulative, 7 potest (1) vel (2) ipsa pars 8 sumpta, |neque del. minor est; bis sumpta, |neque del. major 9 (nam | portio add] 3 10 major:) (1) sufficit si ulla ejus (2) potest 11 nam portio add. 13 enim est, add. 16 (Quod . . . dicetur.) add. 17 mendum, (1) majoris adhuc momenti, quod (2) quod majoris est momenti, et 18 haec jconstet add] definitio, (1) nulla (2) omnino 23 potest (1) in Latere (2) Lateris

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Non enim sufficit ad commensurabilitatem, ut partis aliqua Portio mensuret Totum, (quod semper net,) sed ut partis aliquae Pars aliquota totum mensuret: Pro portio itaque reponendum erit pars aliquota. Suntque haec quatuor menda, tanti momenti singula, ut eorum nullum non evertat totam definitionem: et quartum omnium maxime; quod ego non Incuriae, sed Inscitiae (prout ipse distinguit) imputandum existimo. [55, p. 6] Sed esto Definitio, vel maxime ad mentem suam, sic reformata; Quando Pars ad Totum suum ita se habeat, ut neque pars ipsa, neque ulla hujus partis pars aliquota, quantumlibet exigua, possit, aliquoties sumpta; Toti suo aequalis fieri, sed semper vel ipsum excedit, vel ab eo deficit, tune Aliquanta vocatur. Atque haec pars est toti suo Incommensurabilis. Haec, inquam, Definitio sic reformata (quae apud ipsum erat misere deformis) admitti potest pro Partis Incommensurabilis definitione. Si vero sit etiam definitio Partis Aliquantae: Die tu mini, quaeso, (modo Oedipus sis,) Qualem ego partem dicam, numerum 4, numeri 6? Pars Aliquota non est, per §21, quia non aliquoties sumpta totum praecise constituit, (nam semel sumpta, minor est; bis sumpta, major:) Neque est Aliquanta Pars, per jam defmita; quamquam enim non possit ipsa, potest tamen ipsius aliquota pars, ut 2 vel 1, aliquoties sumpta, toti aequalis fieri; (nam 2 ter sumpta, vel 1 sexies, aequantur toti 6.) Cum itaque neque Pars Aliquota sit, nee Aliquanta, (partem autem omnem vel Aliquotam vel Aliquantam dicendam, hactenus censuerint homines,) Die mihi, Quam dicam? Sed neque Pars Commensurabilis est, per §21, (quippe commensurabilem non aliam definit ille, quam Aliquotam;) Nee Incommensurabilis, per jam definita. Ecqua igitur? At interim hie Definitor; qui Partem Commensurabilem, idem esse facit cum Aliquota; et partem Aliquantam, idem cum Incommensurabili; male se habitum conqueritur, quod apud eum nonnulla reperiri parum sana dixerim.

4 ut (1) n breaks off (2) eorum nullum non (a) totam (&) evertat 5 omnium (1) maximum (2) maxime 11 vocatur. (1) Estque (2) Atque 13 vero sit (1) et (2) etiam 18 enim (1) p breaks off (2) non 19 pars, (1) puta (2) ut 23 Pars add. 23 (quippe (1) non aliam (2) commensurabilem 518

217. WALLIS to OLDENBURG, 18/[28] July 1668 Statim vero, §26, (ne sibi non, ut solet, contradiceret,) Numeros omnes, invicem esse commensurabiles, affirmat; quoniam omnes mensural Unitas. Quae quidem vera sunt; sed prius traditis contraria. Quippe ille non alias definiverat Commensurabiles quantitates, quam quarum altera sit altering aliquota pars: multi autem numeri ita se non habent; puta 4 et 6. Neque illas commensurabiles dixerat, quas aliqua tertia commensurat, (quod definivisse oportuit,) sed quarum altera mensural reliquam, sitque ejus aliquota pars. Adeoque utut 1 sit ad 4 et ad 6, commensurabilis, (quoniam utrumque metitur,) non tamen erit (per illius tradita) numerus 4 ad 6 commensurabilis, quorum neuter metitur reliquum, sitve ipsius aliquota pars. Eandem enim ille, et Partis Aliquotae, et partis Commensurabilis, defmitionem fecerat, §21; sicut et (illi contradistinctam) partem Aliquantam, eandem esse definit atque Incommensurabilem, §26. Quae quidem ego inter ipsius Nova Principia, hucusque nondum tradita (necdum recipienda,) annumeranda censeo. Sed et §25, Commensurabilitatis et Incommensurabilitatis fontes, porro investigatum it. Omnis, inquit, numerus juxta possibiles quae sunt in eo sectiones divisus, tandem relinquit Unitatem, sen particulam sui minimam. Docuimus enim, inquit, omnem numerum divisibilitatis suae terminos habere, ultra quos sectio non procedit. Fateor haec dixisse, (docuisse, non dico: Ecquis enim ante nescivit.) Sed et contraria docuit, (nempe, si quis Discere velit,) Ait enim, §7, Multitudo nunquam ita divisa est, ut pluribus aliis modis secari non possit. Veluti numerus Duodenarius non ita divisus est in paries duodecimas, ut in tertias, quartas, sextas, et adhuc alias quasdam sine nomine dividi nequeat. Sed esto; ea jam dicit. Quid postea? Ergo (infert) ex naturali numerorum structura commensurabilitas exurgit. Commensurabilitas, inquam, Numerorum, ex sua numerorum natura exsurgit, (non minus quam ex sua Linearum natura, Commensurabilitas Linearum;) Hoc est, ex numerorum natura fit, quod illis (quae et aliis quantitatibus convenit) conveniat Commensurabilitas;

2 esse add. 5 pars: \(1) cum (2) multi autem numeri ita se non habent; puta 4 et 6. add.\ Neque 9 numerus add. 16-17 Commensurabilitatis ... investigatum it. add. 27 exurgit. (1) Commensurabilitatis Num. breaks off (2) Commensurabilitas, inquam, Numerorum, (a) ex eorum natura exsurgit (b) ex 29 Linearum; (1) sicut ex omnium omnino rerum natura exsurgit, quod eas, quas habent, habeant affectiones.) (2) ) Hoc est,

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(sicut et ex omnium omnino rerum natura oritur, quod eas, quas habent, habeant affectiones;) et quidem omnibus; (quoniam omnes mensurat unitas.) Sed Commensurabilitatis simpliciter (quae et aliis quantitatibus cum numero communis est) non minus ex sua cujusque quantitatis natura, vel ipsa Quantitatis, quae est omnibus communis, petenda est| ratio. Sed, et [55, p. 7] ait, ex naturali Magnitudinis constitutions, Incommensurabilitas exoritur. Recte quidem. Sed et Commensurabilitas. Sed et pariter ea quae in Sonis est, et quae in Ponderibus, vel Durationibus, turn Commensurabilitas turn Incommensurabilitas, ex ipsa Sonorum, Ponderum, Durationum, &c, constitutione exoritur. Quippe omnium horum naturae ita sunt comparatae, ut Soni, Pondera, Tempora, &c, sint Incommensurabilitatis capacia: sed et Commensurabilitatis non minus. Quod vero ille persuasum iret; Incommensurabilitatis quae in Magnitudinibus est, rationem ex magnitudinum natura petendam; illius autem quae in eisdem est Commensurabilitatis non ex ipsa magnitudinis, sed ex Numerorum natura oriri: omnino est ridiculum. Non minus enim est ex Magnitudinis natura, ut possit in partes Commensurabiles dividi, quam ut possit in Incommensurabiles. Quod et eo magis absurdum est, quod ea quae jam est numerorum constitutio, ex humano institute oritur. Sed et, si ipsi adhibenda fides, ipsa Numerorum natura (adeoque et horum Commensurabilitas) ex continui divisione oriri putanda erat. §10. Sed, caeteris missis, videamus quam hie Demonstrator probat, (non quidem Incommensurabilitatem ex magnitudinis natura ortam, sed) omnino ullas esse posse magnitudines Incommensurabiles. (Quamquam enim ego illud non negem, sed aliunde probari posse sciam: Nego tamen eum, etsi hoc probandum suscipiat, omnino probasse.) Omnis, inquit, Magnitudo in infinitum divisa non relinquit particulam, quae propterea quod

4 non minus add. 5 quae (1) omnibus communis est, (2) est omnibus communis, 5 et add. Sed ait E2 7 et (1) Incommensurabilitas (2) Commensurabilitas 9 Durationum, (1) et (2) fee, 14 petendam; (1) Illius a breaks off (2) illius 14 est add. 18 numerorum (1) instit breaks off (2) constitutio 19 Sed et, si ... erat. §10. add. 20 ex (1) Magnit breaks off (2) continui divisione oriri (a) di breaks off (b) putanda 22 probat, (1) magnitudines omnino (2) (non

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217. WALLIS to OLDENBURG, 18/[28] July 1668 parva sit secari non possit, quin ilia in infinitum secta infinitas efficit particulas, quarum singulae in infinitas minores sectiles sunt, ut res finem habitura non sit, si quis minutias omnes consectari velit. (Quippe hoc est, quod aliter dici solet, continuum esse divisibile in semper divisibilia.) Nunquam igitur, inquit, ex infinita magnitudinis divisions, ad aliquam particulam devenietur, quae minima dici debeat: quae pro communi omnium mensura sumi queat. Esto. Hallucinatur autem omnino si hinc oriri sentiat Incommensurabilitatem: Non enim ex sections interminabili, sed ex modo sectionis, probasse oportuit Incommensurabilitatem. Certum enim est sectionem in infinitum continuari posse, sine ulla incommensurabilitate (Crassamque arguit naturae Incommensurabilitatis ignorationem, hoc nescire.) Verbi gratia. Si exposita recta (aliave magnitude) intelligatur continua bisectione dividi quousque libet: certum est, commensurabilem illam esse dimidiis suis, et dimidiorum dimidiis, et sic deinceps in infinitum, utut ad minimum nunquam pervenitur, (quod Tyro quilibet in mathematicis, facile demonstrabit; tantusque Magister non debuit ignorare.) Nam aliquotae partis aliquota pars (quantumlibet continuetur sectio) erit et Totius aliquota pars; et omnes invicem commensurabiles. (Quod-

3 velit. |Esto. del] (Quippe 4 divisibilia.) (1) Sed quid porro. Nunquam ex infinita igitur, inquit, Sed omnino (2) Hallucinatur autem omnino, (3) Nunquam 6 quae pro . . . queat. add. in margin 12-13 Si exposita (1) magni breaks off (2) recta ( (a) aliaque (&) aliave magnitude) intelligatur continua bisectione (aa) in infinitum (bb) dividi 14 dimidiis, (1) & si breaks off (2) et sic 15 infinitum, (1) utcun breaks off (2) utut 17 ignorare.) jSimiliter in Trisectione, Quadrisectione, aliisque in infinitum innumeris sectionum formis; (1) utut ad mini breaks off (2) sectionibus rationalibus quibuscunque. Sed pergit Nee propterea, inquit, certa determinataque dabitur in magnitudine particula quae pro communi omnium magnitudinum mensura sumi queat, nam si talis daritur, earn minimam (a) fore (b) esse necesse foret, quandoquid in minima seetitur atque determinatur. (Estque haec tota ipsius probatio) sed nee est necesse ut ad (aa) minimum (bb) minimum possibile pervenitur. Sufficit ut demonstretur omnia bisegmenta, et bisegmentarum bisegmenta, et quae sequuntur in infinitum, fore semper partes (aaa) rationales, quod satis (constat) (bbb) toti commensurabiles nee unquam bisecando ad partem (aaaa) irrationalem (bbbb) inncommensurabilem perveniri posse (aaaaa) , sed (bbbbb) . del]

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que de Bisectione dicitur; de aliis sectionibus in partes commensurabiles, pariter ostendi potest, etiam in infinitum continuatis.) Nunquam igitur, hac ratione, ad Incommensurabilitatem pervenietur. Adeoque argumentuni ejus, ab interminabili divisibilitate continui, ad partium Incommensurabilitatem; non modo non probat quod susceperat probandum; sed probat eum Commensurabilitatis et Incommensurabilitatis naturam non satis intelligere. Quod ex proxime dicendis confirmabitur. | [55, p. 8] Nam §29. Utrum fortuito oblata Problemata sive Theoremata, in quibus Commensurabilitas vel Incommensurabilitas ex ipsis terminis non statim apparet, Geometrica solum an vero Numerica simul sint, id est utrum solis magnitudinibus, an et numeris etiam accommodari possint, hac (inquit) ratione dignosces. Si ad illorum constructionem arbitraria tantum requiratur quantitatum Divisio, vel Multiplicatio, indubitabile signum est, ipsa de utraque quantitatum specie simul exponi: Si vero per appositam in quaestione conditionem determinatae vel multiplicationes vel divisiones necessariae sint ad quaesitum efficiendum, tune generales Commensurabilitatis vel Incommensurabilitatis regulae docebunt, utrum numerorum essentiae talibus multiplicationibus ferendis idoneae sint. Admodum imperite. Quippe nullae vel Additiones, vel Subductiones, vel

1 de (1) quavis alia sectione pariter obtinet, quae sit in partes divisa (2) aliis sectionibus in partes commensurabiles, pariter (a) obtinet (b) ostendi potest, 2 Nunquam . . . pervenietur. add. 4 continui, (1) non modo non probat (2) ad partium 7 satis add. 15 quaestione add. 16 generales (1) Commensurabilitates vel Incommensurabilitates (2) Commensurabilitatis vel Incommensurabilitatis 18 Admodum imperite. missing in E2 19 imperite. jEstque hoc indubitale signum (ut cum eo loquar) Commensurabilitatis et Incommensurabilitatis naturam, liuic minime perspectam esse. del. \ Quippe 19 Quippe (1) nulla unquam vel Additio, vel Subductio, vel Multiplicatio, vel Divisio (2) nullae vel Additiones, vel Subdictiones, vel etiam Multiplicationes, vel Divisiones, (inter

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217. WALLIS to OLDENBURG, 18/[28] July 1668 etiam Multiplication.es, vel Divisiones, (inter terminos invicem commensurabiles peractae,) ullam unquam Incommensurabilitatem inducent. Oritur utique haec ex Radicum extractionibus' (quoties nempe faciendae requirantur, nee absolvi possunt.) Adeoque si nulla requiritur Radicum extractio, (seu quod huic tantundem est;) sed Additionibus, Subductionibus, Multiplicationibus, et Divisionibus, (inter terminos commensurabiles peragendis,) quaecumque demum illae, vel qualescunque fuerint, peragenda sint omnia: nullus erit Incommensurabilitatis metus. Admodum igitur imperite, et absurde satis, de Multiplicationibus et Divisionibus, hac in re, praecepta tradit. Quod et indubitabile signum est, (ut cum ipso loquar) Commensurabilitatis atque Incommensurabilitatis naturam, huic minime perspectam esse. Porro §31. Rationem definit esse, determinatam quandam aequalitatis, inaequalitatisve, speciem. Cujus contrarium verum est. Sunt enim Aequalitas et Inaequalitas, species rationis. Mox §32. Cum Rationem in Arithmeticam et Geometricam divideret; De Ratione indiscriminatim pronunciat Rationis terminos in infinitum augeri posse, manente semper eadem ratione: quasi idem in Arithmetica ratione (quae Differentiis aestimatur, non Quotientibus,) pariter verum esset atque in Geometrica. Sed taedet plura commemorare. Haec interim eorum aliqua sunt (nee tamen omnia) quae in ipius Libri primi Capite primo, notanda censui. Ex quibus possis de reliquis conjecturam facere. Totum vero librum ita recensere atque ad examen vocare, mini neque vacat, neque animus est: Sed neque operae pretium fore autumo. Haec autem sunt de quibus gloriatur;

2 inducent. (1) ; (ut |adeoque add] absurde admodum ad de Multiplicationibus et Divisionibus hac in re praecepta tradit.) (2) . Oritur 3 ex (1) nuper at breaks off (2) Radicum 3 nempe add. 10 ut add. 13 Rationem (1) seu Propor breaks off (2) definit esse (a) defin breaks off (6) determinatam 18 idem (1) Arithmetica ratione seu proportione (2) in Arithmetica ratione 24-524,1 animus est: (1) Tu vero vides (a) quaenam ea sunt de (6) qualia sunt ea de quibus ipse gloriatur, quae adhuc observavit nemo, quae hucusque nondum tradita. Tu (2) Sed neque .. . gloriatur; quae (oa) adhuc observavit nemo (inquit) (66) adhuc observavit nemo; quae hucusque nondum tradita.

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217. WALLIS to OLDENBURG, 18/[28] July 1668 quae adhuc, observavit nemo; quae hucusque nondum tradita. Tu vero boni consulas; et Vale. Tims, Johannes Wallis.

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(C] A Continuation of Dr Wallis his second Letter, publish't in Numb. 39, to [57 (u) r ] the Printed Paper of M. Du Laurens.

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This other part of Dr Wallis's second Letter to M. Du Laurens, though written and sent to the Publisher at the same time, when the first part was, yet came not then abroad, upon a consideration intimated in Numb. 38, p. 750; and the same could not find room in these Tracts, till this month, when 'tis publish't, rather from a desire, further to comply with the said Du Laurens, demanding the reasons of the Animadvertor's Censure, than from any propension to disputes. The Publisher can bona fide assure the Author of the Paper, here further animadverted upon, of the reality of what is here affirm'd and profess'd by him, and in particular, that the originall of this, what follows, came to his hands but a day or two after its Date, which was July 18. 1668. the same, which was mentioned Numb. 38. p. 750. | [57 (u) r ] The Letter itself is thus continued, Porro (ut minutiora quaedam praeteream, ne nimius sim, quae tamen ipsa reprehensionem merentur) inter alia, quibus Aequalitatem ob Moderationis virtutem laudat, Inaequalitati interim vitio ver-

6 publish't add. 7 of (1) Monsr (2) M. 8 This (1) second (2) other 11 till (1) now, (2) this month, 12 publish't (1) much (2) rather 12 desire, (1) to {—} (2) further to comply with (a) the demand of the said Du Laurens, insisting to (b) the said Du Laurens, demanding 14 any (1) Inclination to (2) propension 17 the originall of add. 17 follows, (1) hath been in his hands ever since the first (2) came 20 itself (1) follows thus, (2) continues thus, (3) is thus continued,

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218. OLDENBURG to WALLIS, 18/[28] July 1668 tens, quod Excessu et Defectu laboret, haec occurrunt §18. Tanta est aequalitatis moderatio; ut eas non solum, quas afficit quantitates, augeat, minuat, multiplicet atque dividat, nulla facta in ipsis quoad aequalitatem mutatione, sed etiam, ut quantitates ab inaequalitate affectas per similes operationes tractando, intactam in illis inaequalitatis notam relinquat. Id, credo, vult

218.

HENRY OLDENBURG to WALLIS 18/[28] July 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 20/[30].VII. 1668. In this letter, Oldenburg apparently informed Wallis that his short reply to Dulaurens (i.e. WALLIS-OLDENBURG 2/[12].VIL1668 (ii)) could not be included in the Philosophical Transactions for July. At the same time, he appears to have offered to announce that Wallis's piece would appear in the subsequent issue. In addition, the letter contained news from France on Huygens's disclosure of mistakes in Dulaurens's Specimina mathematica. Of this, Oldenburg had been informed by JUSTEL-OLDENBURG [4]/14.VII.1668 (OLDENBURG, Correspondence IV, 499-502, 499-500).

219. JOHN COLLINS to JOHN PELL London, 18/[28] July 1668 Transmission:

C Letter sent: LONDON British Library MS. Add. 4278, f. 340r-340v. On f. 340r beneath date in Pell's hand: 'Received Jul. 20 th .' Postmark on f. 340V: 'JY/18'. Enclosure: Wallis's Catalogue of Errata in Brancker's Table of Incomposite Numbers. Answered by: PELL-COLLINS 29.VIII/[8.IX].1668.

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219. COLLINS to PELL, 18/[28] July 1668 Reverend Sir

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I yesterday received a Letter979 from Mr Branker in answer to Mr Merry's980 Scruples981. Mr Branker supposeth I kept a Coppy of his obiections, but I assure you I did not, nor had leisure to read them at all. Mr Branker likewise sent up 19 faults982 more (saith he) to be added to those in page 198, and am now so confident that my table is thoroughly corrected that out of it I dare Collect the enumeration of Incomposits alone mentioned page 193 and thinkes of sending it by some other opportunity. The last 5 lines in the former Page are the Errata as sent by Mr Branker. All the rest above, are the Errata as examined by Dr Wallis who you see hath enlarged them to a greater Number. This I promised sooner to send, but had not leisure. You write to know what you are in my Debt983. I answer my Lord Brereton hath promised to pay it all, and entred into Bond so to doe. Whereto there is but one Witnesse who is his Lordships Servant. And that being paid there will be nothing due. I should be glad to receive your censure of Mercators Logarithmotechnia and of Riccio984. And when his Lordship intends to returne, be pleased to Mind him to bring up all his bookes of Trade. As Baker985 of the Smyrna trade &c, and also the

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Letter: i.e. BRANCKER-COLLINS c.!6/[26].VII. 1668 (not found). Merry's: i.e. Thomas Merry (d. 1683), English mathematician. Following his death, Wallis received from Collins a large manuscript tract, in which Merry expounds Hudde's rules. Wallis's plan to publish the manuscript was not carried out, though he did print excerpts. See WALLIS, Algebra, 142; Additions and Emendations, 157-62; Opera mathematica II, 150-5. See also COLLINS-PELL 9/[19].IV.1667 (RiGAUD, Correspondence I, 127) and COLLINS-WALLIS 21/[31].III.1670/1. A list of Merry's works is given in BIRCH, History of the Royal Society IV, 326-7. 981 Scruples: i.e. Merry's objections to RAHN, An Introduction to Algebra, translated out of the High Dutch into English by T. Brancker ... Much altered and augmented by D[r]. Pfell]., London 1668. 982 19 faults: Collins has copied these at the foot of Wallis's Catalogue of Errata. 983 Debt: Collins appears to have regularly sent money to Pell. 984 Mercators Logarithmotechnia, and of Riccio: i.e. MERCATOR, Logarithmotechnia and RlCCI, Exercitatio geometrica de maximis et minimis, which were printed together at London in 1668; see PELL-COLLINS 29.VIII/[8.IX].1668. 985 Baker: not identified. 980

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220. COLLINS to PELL, 18/[28] July 1668, enclosure Seiur de Taneur986 on the 10 of Euclid gallice which I have a desire to see. I cease enlargement and subscribe my selfe Brooke house July 18th 1668

Your most affectionate servant JC

[340V] To the Reverend Doctor John Pell at the house of the right honourable the Lord Brereton at Brereton in Stone bagg Cheshire

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JOHN COLLINS to JOHN PELL 18/[28] July 1668, enclosure: Wallis's Catalogue of Errata in Brancker's Table of Incomposite Numbers Transmission:

C Copy of (missing) paper sent in Collins's hand: LONDON British Library MS. Add. 4398, f. 175r. Enclosure to: COLLINS-PELL 18/[28].VII. 1668. Some time after receiving Wallis's catalogue of errata, which were enclosed in WALLISCOLLINS early VII.1668, Collins produced a copy which he enclosed in his letter to Pell. Beneath the table he has entered five lines of errata received from Brancker himself, all of which had also been recognized by Wallis. Pell has marked off those ten errata in Wallis's catalogue which are not contained in Brancker's earlier lists. Cf. PELLBRANCKER 21/[31].VIL1668 and PELL-COLLINS 29.VIII/[8.IX].1668, where Pell points out that Brancker had mistakenly written 40597 for 40599.

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Seiur de Taneur: i.e. Jacques-Alexandre Le Tenneur (fl. 1639-48). His Traite des quantites incommensurables, ou sont decidees plusieurs belles questions des nombres rationaus et irrationaus. Les erreurs de Stevin refutees. Et le Dizieme livre d'Eudide, illustre de nouvelles demonstrations plus faciles et plus succinctes que les ordinaires, et reduit a 62 propositions. Avec un Discours de la maniere d'expliquer les sciences en francais was published in Paris in 1640. See PELL-COLLINS 28.X/[7.XI.]1668 and note to CAVENDISH-MERSENNE 22.VIII/[1.IX].1640 (MERSENNE, Correspondance X, 99).

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220. COLLINS to PELL, 18/[28] July 1668, enclosure

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Page Number for set 3 5579 7 P 9211 5 19 61 9287 19 37 9799 40 41 6 10199 7 P 11 10813 13 7 —47 37 47 13201 23 43 8 14873 73 107 9 17563 3 7 41 17981 P 10 18903 7 3 18907 3 7 18909 7 3 11 20983 3 P 12 23203 3 P 23381 193 103 13 24011 3 13 25093 13 23 25873 23 P 14 27233 31 113 27517 7 P 15 28201 3 P 28203 3 P 29599 blank P 16 30167 71 97 31001 -29 29 17 32297 71 P 33259 97 79 33409 47 P 33591 7 3 33593 3 7 18 34089 7 3 34209 23 3 35089 3 P 37583 19 13 7 3 5 8 11 16

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set 7 11 29 P P 3 P 3 P 43 17 7 3 466 3 P P 7 3 7 7 7 3 P 17 P 3 7 79 P 151 13 73 173 P 3 139 P 7 29 3 P

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for 72383 3 72557 73 72601 97 73023 P 73051 7 73481 179 73493 P 73913 P 75653 151 78199 P 80333 67 80561 17 80663 P 83123 103 84311 57 85909 137 86699 281 86993 79 91189 P 91707 P 91793 23 92701 3 92703 7 92773 163 93101 151 52 93161 93719 7 94769 41 95371 281 95797 P 96109 3 97487 3 97903 3 98099 26 98551 39 99443 17

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137 79 7 41

set P 37 79 3 11 197 7 7 P 11 11 13 11 101 59 P 181 P 7 3 13 7 3 113 157 59 P 97 283 13 13 13 13 263 139 277

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221. WALLIS to OLDENBURG, 20/[30] July 1668

221.

WALLIS to HENRY OLDENBURG Oxford, 20/[30] July 1668 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 54, 2 pp. (our source). At top of p. 2 at 90° to address in Oldenburg's hand: 'not to be entered', and beneath address: 'Ace. d. 21. julii 68.'—printed: OLDENBURG, Correspondence IV, 560. Reply to: OLDENBURG-WALLis 18/[28].VII.1668. Enclosure: WALLIS-OLDENBURG 18/[28].VII.1668.

Oxford July. 20. 1668. Sir,

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I had finished the inclosed987 on Saturday, but because the pacquet would have been too big to send by the Post, I send it not till this morning by the carrier. If you have thought fit to insert what I sent before988, into the Transactions of this Month, I think that will be enough for once. And if that be not inough for the whole, you have this in store against another month if there bee occasion.

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Yours &c. J. Wallis.

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Since I wrote that above I received yours989 of July. 18.1 am not much displeased that what I wrote, is left out this time; though I had rather it had been in. Chiefly because I would not have too much of one thing at a time in it. The expedient you offer, will serve the turn; especially if it were mentioned in the transactions. That of Mr Gregory990 might possibly 4 too big (1) for (2) to send by 8 occasion. (1) whi breaks off (2) Yours 15 in the I in the del. ed.\ transactions 98r

inclosed: i.e. WALLIS-OLDENBURG 18/[28].VII.1668. sent before: i.e. WALLIS-OLDENBURG 2/[12].VIL1668 (ii). 989 yours: i.e. OLDENBURG-WALLIS 18/[28].VIL1668. "°That of Mr Gregory: i.e. Gregory's written reply to Huygens's review of his De vera circuli et hyperbolae quadratura, printed in Philosophical Transactions No. 37 (13 July 1668), 732-5 ('Mr. Gregories Answer to the Animadversions of Monsieur Hugenius upon his Book, De vera Circuli & Hyperbolae Quadratura . . . ' ) . Cf. COLLINS-BRERETON 11/[21].VIL1668. 988

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222. WALLIS to COLLINS, 21/[31] July 1668 require more hast then mine. I find by your note from France991; that I was not mistaken in what I conjectured992 of his mean proportionals. Certainly he is no competent undertaker of such designs: & but a very indifferent superficial mathematician. And I think it would be more honour then his book deserves: For mee to confute it. Of the inclosed, I keep no copy. [2] These For my worthy friend, Mr Henry Oldenburgh, at his house, about the middle of the Pelmel, near St James's London.

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222. WALLIS to JOHN COLLINS Oxford, 21/[31] July 1668 Transmission:

W Letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 198r-198av (f. 198V and 198ar blank) (our source). At top right off. 198av in Collins's hand: 'About finding the Area of an Hyperbola by Logarithmes'. Postmark on f. 198av: 'JY/22'.—printed: RIGAUD, Correspondence II, 490-2. Reply to: CoLLiNS-WALLis 14/[24].VIL1668.

Oxford July 21. 1668.

Sir, I received yours993 of July 14. & thank you. What I wrote994 of Mr Mercator, was not intended to his disadvantage at all; but the contrary. For, though I know him not, I have a good esteem for him. What concerns the 1 by yours (1) of (2) note corr. ed. 4 it (1) is (2) would 991

your note from France: i.e. JusTEL-OLDENBURG [4]/14.VII. 1668, of which Oldenburg had informed Wallis in his letter of 18/[28].VII. 1668. 992 conjectured: see WALLIS-OLDENBURG 6/[16].VII. 1668. 993 yours: i.e. COLLINS-WALLIS 14/[24].VIL1668. 994 wrote: presumably in WALLIS-COLLINS early VII. 1668.

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222. WALLIS to COLLINS, 21/[31] July 1668

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Logarithms, I did so far onely consider (as I read it over once, in some hast,) whether what was done were well; & it seemed to mee so to be, & very ingenious; & so much as would be sufficient for a good expedite way of calculating Logarithms; but I did not so far study it as to consider whether or how it might bee further improved. For the whole was but an afternoons work. Onely having found the Quadrature of the Hyperbola, a little lame (that is, not so full as wished it were,) I did that night consider how it might be improved, & the next morning wrote that to Mr Oldenburg995. Because I was unwilling, to leave to foreigners the perfecting of that which was by ours carried on so far. That this expedient in squaring the Hyperbola, must need afford a proportional help in the Logarithms; is evident: because they depend one upon the other. And his last proposition, of finding a summe of Logarithms; may be much expedited from some other principles that I have not room here to mention. In sum if (as I put it) we make AH = 1. AI = IB = b. HI = A. and the plain BIHF = pi. then is pi - b2 + 63 = BIps + BIqt + BIrn &c to BIHF.

What I wrote996 of the Errata in Mr Brankers Table; was not to find fault with the correcting; but to make a supply of his table of Errata, which in such a businesse is material to the Reader: & cost mee allmost as much pains as calculating the whole Table anew (saving the time of writing it over) for I used the same method to examine as I would have done to calculate a new one. I can say nothing to the comparing997 of Mr Gregorie's with that M. Huygens, because I do not remember that I have seen that of Huygens. 16 BIHF (1) ; which is summe of the Logarithms (2) . \\ What 20 pains as add. calculating 995

wrote . . . Oldenburg: i.e. WALLIS-OLDENBURG 11/[21].VII.1668. wrote: i.e. in WALLIS-COLLINS early VII.1668. 997 the comparing: cf. COLLINS-BRERETON 11/[21].VII.1668. 996

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223. PELL to BRANCKER, 21/[31] July 1668 But wonder that Huygens should write against him unprovoked, being himself an ingenious modest man, as I have hitherto apprehended him: & certainly Gregorie's piece is not contemptible. The other particulars you mention of Mr Gregories, are good; &, I suppose, likely inough to be true. In printing my things998: I had rather you make use of Mr Oughtreds note of Multiplication x than that of *; the other being the more simple. And if it be thought apt to be mistaken for X, it may helped by making the upper & lower angles more obtuse x . The Post is going; so that I can adde no more but that I am yours &c. John Wallis. [198av] These For Mr John Collins at his house next the three Crowns in Blomesbury-market. London.

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223. JOHN PELL to THOMAS BRANCKER Brereton, 21/[31] July 1668 Transmission:

C Letter sent: LONDON British Library MS. Add. 4278, f. 81r-82v (f. 81V and f. 82V blank). Answered by: BRANCKER-PELL 25.VII/[4.VIII].1668. These additional errata, taken from Wallis's catalogue, were too late to be incorporated into RAHN, An Introduction to Algebra, translated out of the High Dutch into English by T. Brancker . . . Much altered and augmented by D[r]. P[ell]., London 1668. Apaxt from the first, and with the addition of twenty others, they were published by Wallis in A Discourse of Combinations, alternations, and Aliquot Parts, London 1685, 136.

5 of add. 998

my things: probably the manuscript of Wallis's Mechanica sive de motu tractatus geometricus, which Collins was seeing through the press. See COLLINS-PELL 21!V/[1.V].1668, British Library MS. Add. 4278, f. 334r and WALLIS-COLLINS 8/[18].IX.1668.

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224. BRANCKER to PELL, 25 July/[4 August] 1668 July 21 Mr Collins in his letter999 of July 18th sent Dr Wallis his Catalog1000 of 145 Errata in your Table. That is, TEN more than your first & second Collection containes. They are these Page

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7 16 28 31 33 34 41 49

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For

Set

—47 37 47 31001 -29 29 54449 71 P 55609 3 P 60701 01 101 64499 13 P 1 67993 P 80561 17 13 96109 3 13 97487 3 13

He addes that he hath your answere to Mr Merry's scruples. I assure you, saith he, I did not keep a coppy of his objections nor had leisure to read them at all. Brereton, 7° 21 20

[82r]

to T.B. at Newchurch to be left with Mr Peter Yates mercer in Middle Wiche

224.

THOMAS BRANCKER to JOHN PELL 25 July/[4 August] 1668 Transmission:

C Letter sent: LONDON British Library MS. Add. 4278, f. 87r-87v. At bottom off. 87r Pell lias written: 'I received this on Munday August 3.'

19 Brereton in cipher 999

letter: i.e. COLLINS-PELL 18/[28].VII.1668. Catalog: see enclosure to COLLINS-PELL 18/[28].VII. 1668.

1000

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224. BRANCKER to PELL, 25 July/[4 August] 1668 Reply to: PELL-BRANCKER 21/[31].VII.1668. Honoured Sir Yours1001 of the 21 I had the 22d with Drs Ws. ten Errata not observed by me. I am very glad a Person of such eminent Fame and Industry hath been at the pains to review the Table. And yet perhaps it is not correct. But for these ten, the first of them I saw but sleighted: the 2d and 5* are, in my copyes, not errata: the 3 last are all that I yet know to have escaped my red pen unduely: the remaining 4 were left un-redded, but I saw them not when I went over all to find all that were not marked red. I note these things so particularly that you might the better conjecture at the reason of these oversights, and direct me to any other way more sure, and more likely to maintain attention in the proof. Which if you do I shall once more prove the table: for I yet doubt its exactnesse. I purpose to be at Br: as soon as I well can: But, partly bee: of Mr F. Ch: Mathem:1002 and partly bee: our Lad is gone from us, I think it will not be till the middle of the 2d week to come. I pray be pleased to tell Mrs Linds:1003 that I yesterday found St Johns wort in great plenty, and in flour. If she desire it I can send her as much as in reason she can use this year. But I must find somebody here to employ about it. Possibly it were better for her to send some one early in the morning hither and he need not be above \ an hour gathering of a convenient burden. I am briefe bee: I know not through what hands this is to passe. If you please to send at any time to Mr Peter Yates Mercer in Middle Wiche. He can soon convey it to me. (or, it comes safely by Joh: Faringdon when you have that opportunity.) We are all well and both of us prefer our respective salutations &c. d

There hath been a 2 attempt but lesse violent July 25. 68.

Your ever obliged T.B. V.S.S.

5 for (1) this (2) these 12 prove (1) it (2) the table 12 yet (1) fear (2) doubt 21 burden. (1) But if she will (2) I 1001

Yours: i.e. PELL-BRANCKER 21/[31].VIL1668. 1002 p QJ^ Mathem: j e Frallcis Champante. 1003 Mrs Linds: presumably an apothecary at or near Brereton.

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225. WALLIS to BRAMSTON, 27 July/[6 August] 1668 [87V]

For Dr Pell at Brereton Hall

225.

WALLIS to JOHN BRAMSTON Oxford, 27 July/[6 August] 1668 Transmission:

W Draft letter: OXFORD Bodleian Library MS. Savile a. 11, f. 89V.

Oxford July. 27. 1668. Sir 5

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Being so wholly a stranger to those of the Gentry in Essex; I was incouraged by the Warden1004 of Wadham College to make my applications to yourself (though a Stranger) as a civil & generous person; in a business wherein my self & Dr Christopher Wren (the two Savilian Professors of Mathernaticks in this University) are concerned. Trusting that (the business being just) you will excuse the confidence that a Stranger takes in giving you this trouble. There lyes in Rettenden, a farm called Littlehays, now in the occupation of one Henry Spitty1005; which is part of the Lands given by Sir Henry Savile for the maintainance of his two Professors. The whole Rent whereof being part of our Stipend; wee have taken it to be exempted from the Assessements for the Monthly Contribution, by a Proviso (confirmed, for about twenty years last past, in all Acts of that nature; as well since his Majesties Restauration as before;) whereby it is Provided, That nothing contained in the sayd Acts shall be extended to charge any - Reader - in either of the Universities, for or in regard of 10 the (1) trouble (2) confidence 14 whole add. 18 is (1) Prop breaks off (2) Provided 18 nothing (1) therein contained (2) contained in the sayd Acts 1004

Warden: i.e. Gilbert Ironside, the younger (1632-1701), warden of Wadham College 1667-92; vice-chancellor of the University of Oxford 1687-9. Thereafter bishop of Bristol (1689-91) and of Hereford (1691-1701). DNB. 1005 Spitty: see WALLIS and CHRISTOPHER WREN-ESSEX COMMISSIONERS 27.VII/[6. VIII].1668. Cf. WALLIS-? 3/[14].IX.1700.

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225. WALLIS to BRAMSTON, 27 July/[6 August] 1668 any Stipend, Wages, or Profit whatsoever arising or growing due to them in respect of their places & imployments in the sayd universities. Which hath been allowed a good Plea, & the Professors for the time being have, from the first, constantly injoyed this exemption, now for about twenty years; not only in Essex, but in all other counties wherein wee are con- 5 cerned; viz. in the Counties of Northampton, Kent, & Glocester. But wee are now informed from our Tenant, that now very lately wee have been assessed for our rents issuing out of those Lands, & the mony levied by distresse. (Occasioned, I suppose, by the endeavours of some, who, by charging of us, would ease themselves.) Wee have, upon this occasion, 10 addressed a Letter1006 to the Commissioners for that purpose. Which our Tenant hath direction, if he have opportunity, to show you; & to receive your instructions. I do not know, whether, about six years since, My Lord chancellor1007 did not write you a letter in our behalf, (the same thing being then in question;) Hee did, at lest, promise us so to do if there 15 should bee need. But this I know, that it was then adjudged for us; &, since that time till now, wee have not been troubled. And if to the justice of our cause, you shall please to adde the favour of your assistance; I trust we shal find the same successe again; and you will very much oblige

Sir, For the Right Worshipful Sir John Brampston in Essex

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Your very humble servant John Wallis.

1 them (1) by virtue of (2) in respect of their \sayd del] places (a) & (b) or im breaks °ff (c) & imployments 2 universities. (1) And wee have, upon this plea, [constantly add.\ injoyed that exceniption, (2) Which 3 have (1) injoyed this exemption, now (2) , from 4 for (1) about (2) about 7 that (1) wee have been very lately assessed (2) now very lately wee have been assessed 11 for (1) redresse (2) that 12 opportunity, (1) to let you see (2) to 13 your (1) d breaks off (2) instructions, (a) Not (b) I 13 years (since, add] (1) (the same thing (2) My Lord chancellor did not jthen del] write 15 question (1) & adjudged for us;) (2) ;) Hee 1006 1007

Letter: i.e. WALLIS and WREN-ESSEX COMMISSIONERS 27.VII/[6.VIII].1668. Lord chancellor: i.e. Edward Hyde, first earl of Clarendon (1609-74).

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226. WALLIS et al. to the ESSEX COMMISSIONERS, 27 July/[6 Aug.] 1668

226.

WALLIS and CHRISTOPHER WREN to the COMMISSIONERS OF THE MONTHLY ASSESSMENTS IN THE COUNTY OF ESSEX Oxford, 27 July/[6 August] 1668 Transmission:

W Draft letter: OXFORD Bodleian Library MS. Savile a. 11, f.89r.

Oxford July. 27. 1668. Gentlemen,

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We understand from our Tenant Henry Spitty1008 of Rettendon in the County of Essex, that a distress hath been lately taken for monyes assessed toward the Monthly contribution, for the farm which he there hath, called Little-hays. And though, wee beleeve, it was not by any direct order of yours to that purpose; but rather by the mistake of the Collectors or Assessors: Yet you are the persons to whom wee are to apply for redresse. You may please therefore to understand, that the lands aforesayd were given by Sir Henry Savile to the University of Oxford (not to any particular college) in trust1009, for the maintenance of the two Mathematick Professors, or Publike Readers of Geometry & Astronomy in the sayd University; & that the whole rent thereof, be it more or lesse,

3 in the County of Essex add. 4 assessed (1) to the (2) toward 5 faxm (1) of Little-hays which he hath. (2) which he there hath, called Little-hays. 6 wee |do del.\ beleeve 7 rather add. 10-11 to the University . . . in trust, add. 12 Publike add. 12 of (1) Astron breaks off (2) Geometry 13 sayd add. 1008

Spitty: see WALLIS-BRAMSTON 27.VII/[6.VIII].1668. Cf. WALLIS-? 3/[14].IX.1700. given by ... in trust: see Statutes of the University of Oxford, ed. Griffiths, Oxford 1888, 249-50. 1009

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226. WALLIS et al. to the ESSEX COMMISSIONERS, 27 July/[6 Aug.] 1668 is the stipend or wages of the sayd publike Readers; & that whatsoever is charged on the sayd rents, is charged on the sayd Stipend or wages. You may please allso to consider that both in the Act of Parliament for the Royall Ayd (to which that allso for the Additional supply doth referr,) and all other Acts for monthly contributions not only since his Majesties happy restitution, but even in the late times of Usurpation, it is & hath been airways expressely Provided That nothing contained in the sayd Acts shall be extended to charge any — Reader in either of the Universities, for or in regard of any Stipend, Wages, or profit whatsoever, arising or growing due to them in respect of their places & imployments in the sayd Universities. By virtue of which Proviso (continued from time to time for about twenty years last past) the Rents of the sayd lands (as being the Stipend of the sayd Publike Readers) have airways been exempted from such assessements (& so reputed) as well in the late times of usurpation, as ever since his Majesties happy Restitution, even to this time. And when at any time, by mistake, (as about eighteen years ago,) the sayd lands have been assessed; upon application to the Commissioners for that purpose it hath airways been redressed, & the mony (when any hath been levyed) repayd again. And allthough the Proviso hath not been thought to extend to all College lands, (because the rents thereof are not onely for

1 whatsoever is (1) assessed (a) thereon, is (6) on the sayd rents, (2) charged on the sayd rents, is 2 on the (1) Stipend or wages, of the sayd Professors (2) sayd Stipend or wages. 3 that (1) as well (2) both 4 allso (1) of (2) for 4 referr,) (1) as (2) and 6 it (1) is (2) is fe hath been allways 8 shall (1) ext breaks off (2) be extended 10 sayd (1) University. (2) Universities, (a) By virtue of which Proviso (oa) , the sayd lands (66) (continued from time to time) the Rents of the sayd Lands have been (oao) ther breaks off (bbb) hitherto (aaaa) as breaks off (bbbb) exempted, (& so reputed,) from (b) By 12 the (1) sa breaks off (2) Rents 13 allways |been add] exempted (1) hitherto (& so reputed) from such assessements, (2) from such assessements (& so reputed) 15 as (1) ever since (2) since (3) ever since 15 even add. 16 (as about eighteen years ago,) add.

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226. WALLIS et al. to the ESSEX COMMISSIONERS, 27 July/[6 Aug.] 1668

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the Stipends of the Heads & Fellows of the sayd Colleges, but due to the Colleges at large, as well for those as for divers others uses;) yet to these (which are not College lands, but of another nature) it hath allways been allowed to extend; because here the whole rent, bee it more or lesse, is the Stipend of the sayd Publike Readers, & not applicable to other uses; & therefore clearly within the words of the Act. The same justice & favour hath been in like manner allowed as all along in all other Counties where wee have been concerned, viz. in the Counties of Northampton, Kent, & Glocester. And when, upon the change of the Commissioners, presently after his Majesties return, the Rents of these lands in Essex were once assessed: The commissioners who then were (the same wee suppose, for the most part, with those that now are,) upon a like application with this, did us the favour (without any further attendance) to discharge that Assessement, & give such order as that wee have never since been troubled till this time. Since therefore it is no new case, but what hath been so often, & in so many places adjudged in favour of us, & which is in itself so clear: wee trust, that those worthy persons who are now commissioners, will not be found either less favourers of learning, or less friends to the Universitie, than those who have been Commissioners, in worse times, or in other places, or lesse than yourselves have formerly been. We have therefore directed our Tenant to wait upon you; & desire you will please to give him & us such redresse in this case, that wee may

2 to these . . . another nature) add. 4 extend; (1) to these (which are not (a) College) (6) in the nature of Coll: lands) (2) because |here add.\ the 6 Act. (1) Nor hath it been onely so allowed (2) The same (a) favour fe (b) justice 7 as all along add. 7 Counties (1) wherein (2) where 9 when (1) as presently upon the change of the Commissioners in the County of Essex, (2) upon the change of the Commissioners, presently 10 the Rents of add. 13 this, (1) {—} (2) did 13 attendance) (1) to give us such redresse (2) to 14 order as add. that (1) till now (2) wee 21 therefore (1) giv breaks off (2) directed

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227. WALLIS to BROUNCKER, July/August 1668 not our selves be put to the trouble of so long a journy to look after it. And you shall oblige, Gentlemen, Your very humble servants, John Wallis D.D. Professor of Geometry 1 Christopher Wren D.L. Professor of Astronomy > in the University of Oxford. For the Right Worshipful the Commissioners of the Monthly Assessements in the County of Essex.

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227. WALLIS to WILLIAM BROUNCKER Oxford, July/August 1668 Transmission:

E First edition of missing letter sent: Philosophical Transactions No. 38 (17 August 1668), 753-6 ('Logarithniotechnia Nicholai Mercatoris: discoursed of in a letter written by Dr. J. Wallis to the Lord Vis-count Brouncker') (see WALLIS-OLDENBURG 8/[18].VII. 1668, where alterations and modified ending of E are reproduced). In this letter, Wallis apparently sent Brouncker this amended and expanded version of his letter to Oldenburg of 8/[18].VII.1668 at the end of July or the beginning of August, as he reports in WALLIS-OLDENBURG 3/[13].VIIL1668. There he also expresses the wish that this version be printed in Philosophical Transactions. It was duly published as a letter to Brouncker, though with the date of the earlier letter, together with the demonstration he had promised at the end; see WALLIS-OLDENBURG 8/[18].VII. 1668. See also WALLIS-BROUNCKER 5/[15].VIII. 1668.

1 our selves add.

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229. WALLIS to OLDENBURG, 3/[13] August 1668

228.

HENRY OLDENBURG to WALLIS July/August 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-OLDENBURG 3/[13].VIII. 1668. In this letter, Oldenburg apparently reported on the response to Wallis's letter to Moray of 14/[24] July, which had been read at the meeting of the Society on 16/[26] July. He also requested Wallis's opinion on Huygens's review of Gregory's Vera circuli et hyperbolae quadratura and of Gregory's reply, recently published in Philosophical Transactions. Probably both pieces were enclosed to this purpose. See WALLIS-BROUNCKER 4/[14].XL1668.

229.

WALLIS to HENRY OLDENBURG Oxford, 3/[13] August 1668 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 58, 2 pp. (our source).— printed: OLDENBURG, Correspondence V, 3-4. Reply to: OLDENBURG-WALLS VII/VIII.1668.

Oxford Aug. 3. 1668.

Sir,

5

I received yours1010 by your friend1011; & shall be ready to serve him for your sake, & his own; for hee seems to be an ingenious person. The particulars of your letter, most of them, I have answered in one1012 to my Lo. Brouncker. The experiments1013 of Borellus I am well contented should 6 well add. 1010

yours: i.e. OLDENBURG-WALLIS VII/VIII.1668. friend: not identified. 1012 one: presumably WALLIS-BROUNCKER 3/[13].VIIL1668. 1013 experiments: i.e. pendulum experiments described in BORELLI, De m percussionis, Bologna 1667 (see WALLIS-MORAY 14/[24].VII. 1668), and discussed at the meeetings of the Royal Society on 16 and 23 July 1668 (BIRCH, History of the Royal Society II, 1011

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229. WALLIS to OLDENBURG, 3/[13] August 1668 be true, & I am ready inough to beleeve them, (as others who write de motu projectorum are wont to do:) yet (if it have not been done allready) I thought they might have been repeated for confirmation, that wee may (in point of experiment) trust our own eyes. But, it seems something of that kind hath been done allready. Borellus his book I have read; &, like him well. And, though I have not so far examined him, or am so satisfyed, as to avouch it all: yet there is, at lest, very much of it good: &, I hope so of the rest, though I cannot say that I am satisfyed in all. Mr Gregories reply1014 to M. Hugens, I am well inough satisfied with: having compared it with what was before written, & what M. Hugens objected. I do a little wonder, that M. Hugens should write against him unprovoked, unlesse there had been more cause then what I yet see: For (except a little slip in one example, which was not materiall,) the rest is Hugens mistake, not his. And, that there was a conformity between the Hyperbolick spaces, & the Logarithmes; tis true, is not new in Mr Gregory; but neither was in M. Hugens; But was, before either of them, in Gregorius de Sancto Vincenfo'o1015; from whence I suppose both of them, and others allso, have taken the notion first; though managed it variously. You will see it in my Commercium Epistolicum Epist. 39. 401016. What you have of mine1017 concerning Du Laurens. I did beleeve

4 of that (1) hath bee breaks off (2) kind 5 &, (1) find that (2) like him well. And, though I have |no corr. ed. so fax examined him, or (a) so (b) am 7 much |of it add] good; (1) (into) (2) (in) (3) &, 16 Gregorius (1) the (2) de 18 first; (1) they (2) though managed it (a) differently (6) variously. 20 did (1) fear (2) beleeve 306-9). Oldenburg presumably had sent Wallis much the same information on these experiments as he did in his letter to Boyle of 27. VII/[6. VIII]. 1668 (OLDENBURG, Correspondence IV, 569-71, 571). 1014 Mr Gregories reply: i.e. 'Mr. Gregories Answer to the Animadversions of Monsieur Hugenius upon his Book, De vera Circuli fe Hyperbolae Quadratura . . . ' , Philosophical Transactions No. 37 (13 July 1668), 732-5, in reply to Huygens's review of Gregory's De vera circuli et hyperbolae quadratura, printed in Journal des Scavans (2 July 1668 new style), 52-6. 1015 in Gregorius de Sancto Vincentio: see SAINT-VINCENT, Opus geometricum, 'liber sextus de hyperbola', esp. 586 (prop. 109) and 594-7 (prop. 125-30). 1016 Epist. 39. 40: i.e. WALLIS-DIGBY 5/[15].V.1658 and WALLIS-BROUNCKER 11/[21]. V.1658. 1017 mine: i.e. WALLIS-OLDENBURG 18/[28].VII. 1668.

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229. WALLIS to OLDENBURG, 3/[13] August 1668

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would be more then would bee fit for one time to insert: and therefore could have been contented you had put part of it in this month, & referred the rest to the next. Mr Gregories 10th Proposition1018, as to what it undertakes (if I mistake not his meaning) seems well inough demonstrated; viz. that those converging series, cannot bee so terminated by Analytical operation as he proposeth. But (what Mr Hugens exceptions do oppose) that there can be no other Analytick means of squaring the Circle or Hyperbole; is not (at lest there) affirmed, & therefore was not to bee proved. What you cite out of the Italian1019, concerning the business of Transfusions (unless there be more than what is transcribed) seems onely to speak of the thing as a ridiculous fansy; not as if any such thing were then practised, or attempted; nor proposed in order to any experiment to be made. My Letter1020 about Mr Mercator, I have a little altered in the Copy1021 sent to my Lord; according to which you may please to insert it: And, whether, as sent to my Lord or to yourself, you may do as you please. No more but that I am Yours &c. J. Wallis.

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These For Mr Henry Oldenburg, in the Palmal near St. James's London.

[2]

I insert: (1) which made mee wish rather (2) and therefore could have been contented 3 the rest add. 8 other add. II is (1) d breaks off (2) transcribed) 14 Letter (1) to (2) about 1018

Mr Gregories 10th Proposition: i.e. GREGORY, De vera circuli et hyperbolae quadratura, prop. 10. 1019 cite out of the Italian: i.e. in Philosophical Transactions No. 37 (13 July 1668), 731-2, where Oldenburg had quoted from a tract printed at Rome, in which it was stated that blood transfusion had been known to Andreas Libavius (15407-1616). 1020 Letter: i.e. WALLis-OLDENBURG 8/[18].VII.1668. 1021 Copy: i.e. WALLis-BROUNCKER VII/VIII.1668.

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230. WALLIS to BROUNCKER, 3/[13] August 1668

230. WALLIS to WILLIAM BROUNCKER 3/[13] August 1668 Transmission:

Manuscript missing. Existence and date: Referred to in WALLIS-BROUNCKER 6/[16].VIII. 1668 and WALLISOLDENBURG 3/[13].VIII. 1668. Enclosure: Investigations on Fermat's Negative Theorem. Wallis used the present letter to send Brouncker his investigations on Fermat's negative theorem for n — 3, which holds that a rational cube cannot be divided into two further rational cubes, and on the indeterminate quadratic equation otherwise known as Pell's equation. The form of presentation clearly indicates that they were sent as an enclosure to a now missing letter written at the same time, but only posted the following day. Cf. WALLIS-BROUNCKER 6/[16].VIIL1668.

231. WALLIS to WILLIAM BROUNCKER 3/[13] August 1668, enclosure: Investigations on Fermat's Negative Theorem Transmission:

Wl Final draft of paper sent (Part I): OXFORD Bodleian Library MS. Don. d. 45, 116r-117r (paginated 219-221) (our source). At top right of f. 116r in Wallis's hand: 'anno 1666 or 1667'. W2 Early draft of paper sent (Part II): OXFORD Bodleian Library MS. Don. d. 45, f. 117V-119V (paginated 222-226). W3 Early draft of paper sent (Part II): OXFORD Bodleian Library MS. Don. d. 45, f. 135r-135v (paginated 257-258). At top right off. 135r in Wallis's hand: 'vid. pag. 222.' W* Final draft of paper sent (Part II): OXFORD Bodleian Library MS. Don. d. 45, f. 136r-137v (paginated 259-262) (our source). At top off. 136r in Wallis's hand: 'Aug. 1. 1668. Sent to my Lo. Brouncker (with that of pag. 219) Aug. 3. 1668.' W5 Draft of paper sent (Part II, English): OXFORD Bodleian Library MS. Don. c. 49, f. 14r-14v. W6 Draft of paper sent (Part II, English): OXFORD Bodleian Library MS. Don. c. 49, f. 15r-15v. El Reworked version of paper sent (Part I, English): WALLIS, Algebra, 372-4. E2 Latin translation of E1: WALLIS, Opera mathematica II, 427-9. Enclosure to: WALLIS-BROUNCKER 3/[13].VIII. 1668.

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231. WALLIS to BROUNCKER, 3/[13] August 1668, enclosure As indicated by Wallis's note on W1, the earliest drafts of this paper probably date from 1666 or 1667 and were possibly motivated by plans in France to have all or part of Fermat's works printed; see JuSTEL-OLDENBURG [22.IVJ/2.V.1668 (OLDENBURG, Correspondence IV, 330-2). The content of the paper of which the first part deals with Pell's equation and the second part with Fermat's negative theorem for n = 3, largely corresponds to parts I and II of WALLIS-BROUNCKER VIII?. 1668. The latter was probably intended for publication in the Philosophical Transactions.

(Wl) Theorema Fermatianum, cum Problemate annexe.

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Exposito n numero quovis integro non-quadrato: Dantur quadrat! in integris innumeris, ut a 2 , qui in expositum non-quadratum ducti, assumpta unitate, efficient na2 + I numerum quadratum. Proponitur Theorema demonstrandum; et construendum Problema; Exposito n, numeros a invenire. Investigatio Theorematis.

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Si sit na + 1 numerus quadratus integer; Erit ^/ : na2 + 1, (ejusdem Latus) Numerus integer, qui major sit quam o\/n, et minor quam a\Jn + (Quippe quadratus ab illo, esset na2; quadratus ab hoc, no2 + 1 + Cumque horum differentia sit unitate minor (propter a, n, nu2 meros integros;) oportet ^/ : na +1, ilium esse integrum qui surdum a^/n proxime superat; et quidem excessu minore quam Sit m, numerus integer, surdo ^Jn proxime major: hujusque ad ilium complementum p — m — ^/n, < 1. Adeoque surdi a^/n, complementum ad am, erit ap = am — a^/n.

4 (1) Sit n expositus numerus integer quilibet non-quadratus (2) Exposito n numero quovis integro non-quadrato 4 quadrat! (1) numero infiniti, qu breaks off (2) in integris innumeris, ut 6 Proponitur (1) demonstrandum Theorema; (2) Theorema demonstrandum; 10 quadratus (1) ; Erit hujus latus, ^/ : na2 +1 : (2) integer; Erit \/ : na2 + 1, (ejusdem Latus) Numerus (a) quadr breaks off (b) integer

15 ilium (1) in breaks off (2) esse 16 quam (1)

(2)

546

231. WALLIS to BROUNCKER, 3/[13] August 1668, enclosure

Sit /, numerus integer, surdo a^n proxime major. Cujus quidem surdi, ad am, complementum ap, cum possit esse majus unitate, (utut enim sit p < 1, possit tamen ap unitatem superare;) Sit z, integer proxime minor quam ap; qui itaque hinc demptus, relinquit surdi a^/n ad I (integrum proxime majorem) complementum ap — z = I — a^/n < < 1. 2 Sit r = 2^; erit itaque ^ > ap — z; et a p — az < r. Et propterea a < adeoque p < Erat autem et z < ap, adeoque | Est ergo ^ < p < Qu°d omnino possibile esse constat; quoniam ita sumi possunt a, z, integri, ut sit |, turn minor quam p, turn hinc tamen deficiat differentia quae sit data minor; adeoque tali ut sit Dato autem ejusmodi uno numero o, habentur alii infiniti, per ea quae ostendimus in Commercio Epistolico1022, Epist. 14. 16. 17. 19.1023 et alibi. Ejusque (ut problemati satisfiat) quomodo inveniantur, ibidem V [116 ] ostendimus. | Demonstratio. Numero surdo ^Jn (utpote radice non-quadrati) sit m numerus integer proxime major: Cujus itaque excessus, unitate minor erit; nempe 1 quidem (1) jsurdi add.\ complementum (2) surdi, 5 complementum add. 6 r = 277^, (1) adeoque (2) erit itaque 15 problemati (1) satisfaciat (2) satisfiat 1022

Commercio Epistolico: i.e. WALLIS, Commercium epistolicum, Oxford 1658. Epist. 14. 16. 17. 19.: i.e. BROUNCKER-WALLIS 22.X/[1.XI].1657 (letter 14), WALLIS-DIGBY 21.XI/[1.XII].1657 (letter 16), WALLIS-BROUNCKER 17/[27].XIL1657 (letter 17), and WALLIS-BROUNCKER 20/[30]. 1.1657/8 (letter 19). 1023

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231. WALLIS to BROUNCKER, 3/[13] August 1668, enclosure m — \Jn < 1. Ponatur p — m — ^/n; et r — ^THSumantur porro integri duo z, a, ita comparati, ut sit | < p < 5

' Z' k°c es^' u^ §it P maJor quam |, sed minor quam Quod fieri omnino posse, constat. Certum utique est, sumi posse, in numeris rationalibus, fractum ^, qui tarn prope deficiat ab exposito irrationali p, ut hunc superet y=p;) ut sit Hoc set (posito

Hoc est (demptis aequalibus) 10

majori excessu quam sit y,(qui Poterit supert Hoc est; ut esse quantumvis exiguus, seu dato minor:) Hoc est (sumptis quadratis) superet -jj^ majori excessu quam ^ + y 2 ; hoc est, ut sit ^ > ^+y2; seupr > zay+a2y2: putapr — s = zay+a2y2; Hoc est (resolvendo

15

aequationem) radix aequationis ay = ^/ : ^z2 + pr — s : —\z. Hoc itaque diviso per y ita utcunque sumpto ut a sit numerus rationalis, atque tarn magnus ut | minor sit p- habebitur a qui sumpto z respondeat. Q a sic inventus sit fractio; reducantur z, a, ad eandem denominationem, et communis denominator abjiciatur.

5-549,1 constat. (1) \\ Cum enim ita sumi possint z, a, ut ^-, turn data quantitate p minor sit, turn ab ea deficiat differentia quantumvis exigua, seu quae sit data minor: Esto ilia quantumvis exigua differentia, y: Adeoque ^ + y — p. Id porro requiritur, ut sit f + y < V--z2+*pr-.+z. Hoc egt; 2z + 2ay < ^/ : z2 + 4pr : +z; seu z + lay < ^J : z2 + 4pr : Hoc est (sumptis quadratis) z2 + 4zay + 4a 2 y 2 < z2 + 4pr; Hoc est 4,zay + 4a 2 j/ 2 < 4pr; seu zay + a2y2 < pr. Quod fieri posse, ex Aequationum doctrina constat. Esto enim zay + a2y2(< pr) = pr — s. Manifestum est, (propter pr — s quantitatem positivam, adeoque s < pr,) turn aequationem illam zay + a2y2 — pr — s, non esse impossibilem, (quippe in ilia forma, nulla est impossibilis;) turn et radicem ay, verum habere valorem affirmativum, sumptis z, a, conformem: Nempe (a) a^. possun^ igitur ita sumi z, a, numeri, ut dictum est. \\ Sumptis itaque (2) jCertum utique . . . denominator abjiciatur. in margin\ \\ Sumptis itaque 5 est, |ita del. sumi 7 est, |ut ^ tarn prope deficiat ab |p, ut ^pr > eundem superet del.\ (posito 9 poterit |poterit del. esse 11 sit add. 12 seu pr > zay+a2y2: ( 1 ) Hoc est (per aequationum doctrinam) sumpto z ad placitum, radix ay < *J : |z + pr : —z (2) puta 13 aequationis (1) ay + a2y2 (2) ay = ^/ : |z2 + pr — s : \ — ^z add.\. Hoc itaque (a) per y (b) diviso

548

231. WALLIS to BROUNCKER, 3/[13] August 1668, enclosure

[nr]

Sumptis itaque, ut dictum est, z, a, numeris: Propter | < p; ad z < an: erit an — z, quantitas positiva, seu maior quam 0. Item; propter p < erit lap < ^/ : z2 + 4pr : +z. et 2 2ap — z < -y/ : z + 4pr. Hoc est, (sumptis utrinque quadratis) 4o2p2 ^apz + z2 < z2 + 4pr: Adeoque 4a2p2 — 4apz < 4pr: Hoc est, a2p — az 5^2a> eP3a' ^C' usclue ad j^ = AS infinitae, (nempe, si ad Verticem usque processum continuaveris;) vel, usque ad -£^ — C(3, (posito DC = A,) si continuaveris usque ad C/3, ubivis intra AS et D(3 sumptam. (Adeoque omnium Aggregatum, ^ + -^ + ^^ + ^^ &c, est ipsum DC(3(3 planum.) Manifestum itaque est, (et ibidem prop. 94, ostensum1028,) si intelligantur singulae df3, in suas a vertice distantias Ad, ductae; hoc est, -^in a, ^ in 2a, (et sic de reliquis;) erunt omnia rectangula Ad(3, hoc est, rectarum d(3 momenta respectu AS, (intellige, facta ex magnitudine in distantiam ducta;) seu plana semiquadrantalem Ungulam (cujus acies AS] complentia, (eisdem d/3 rectis perpendiculariter insistentia;) invicem aequalia. Quippe singula — b2. (Quorum cum unum sit AIV8 quadratum, erit AI = b.) Adeoque, Totius AD/3/36 (plani infiniti,) seu omnium d/3 illud complentium, momentum respectu rectae AS (ut axis aequilibrii;) seu Ungula semiquadrantalis eidem ADfifiS insistens (aciem habens AS;) sunt totidem 6 repetam. |V. Fig. 2. add. Oldenburg] 8 sed (1) ex. (2) a parte 1025 literis meis: i.e. WALLIS-OLDENBURG 8/[18].VIL1668. Cf. WALLIS-BROUNCKER VII/VIII.1668. 1026 Ostensum est: i.e. WALLIS, Arithmetica infinitorum, 71 (prop. 95). 1027 appello: i.e. WALLIS, Arithmetica infinitorum, 68-9 (prop. 88). 1028 ostensum: i.e. WALLIS, Arithmetica infinitorum, 71 (prop. 94).

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b2' hoc est, db2. (Ungula magnitudinis finitae piano infinitae magnitudinis insistens.) Ejusque pars piano AC(3(3'8 insistens (propter AC = d — A.) similiter ostendetur aequalis ipsi d — A in b2 ductae; hoc est, db2 — Ab2. Adeoque pars reliqua, ipsi DC(3'[3 insistens, aequalis ipsi Ab2. Quod itaque est ejusdem DC/3/3 momentum respectu Ad. Atque hoc momentum per plani DC/3/3 magnitudinem, puta per pi, divisum; exhibet plani distantiam Centri gravitatis ab A8, ^-: adeoque distantiam ejusdem a D/3, d — Haec itaque a Df3 distantia, in pi (plani magnitudinem) ducta; exhibet dpi — Ab2 ejusdem DC/3/3 momentum respectu D/3; seu Ungulam eidem DC'/?/? insistentem cujus acies sit D(3. Haec denique Ungula (cujus altitude, in D/?, nulla sit, sed, in C/3, ipsi DC aequalis;) si ex planis ipsi DC[3[3 parallelis conflari intelligatur; erunt ea, CD(3(3 , Cd[3[3, et sic deinceps; hoc est, aggregatum omnium Cd/3/3, Cd/3/3, usque ad CD/3/3. Sunt autem ea plura (ut ex Gregorii de Sancto Vincentio, aliorumque post ilium, doctrina1029 constat,) tanquam Logarithm! Arithmetice proportionalium Cd, Cd, &c. usque ad CD] (seu a, 2a, 3o, &c usque ad A.) Ergo ungula ipsa, est eorundem Aggregatum. Hoc est (posito D = 1,) [2] dpi — Ab2 = pi — Ab2. Quod ostendendum erat.l Porro; cum sit &c. (Quod dividendo 2 b per d — a, patebit:) vel, posito d = 1, (quo ipsius d potestates omnes deleantur, seu 1 &c in b2. et similiter in 1 + 2o + 4a + 8a &c; et similiter in reliquis: Erunt omnes d(3, (spatium DC[3[3 complentes,) Adeoque (per Arithm. Infin. prop. 641030) omnium Aggregatum, seu ipsum DC[3[3 spatium, erit

infr 2 et sic deinceps usque ad

4 aequalis add. 16 ut ex (i) doctrina (2) Gregorii 24 4a262 + 8aV con: ed. 1029

ex . . . doctrina: see SAINT-VINCENT, Opus geometricum, 'liber sextus de hyperbola', esp. 586 (prop. 109) and 594-7 (prop. 125-30). 1030 per . . . prop. 64: i.e. WALLIS, Arithmetica infinitorum, 52-3 (prop. 64).

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233. WALLIS to BROUNCKER, 5/[15] August 1668 Ideoque, Plani DC(3(3 momentum respectu D/3; seu semiquadrantalis Ungula eidem insistens cujus acies sit D(3~ seu planorum aggregatum ipsam constituentium; seu Loe;arithmorum summa quos ea repraesentant, Qualium Cubus (seu Rhombus solidus) ACEo sit 1. Si vero non ponatur d = 1, sed cujuscunque magnitudinis: erit saltern

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in ft2. = pi. Qualium 2

d = ADE8 Quadrato vel Rhombo. Ungulaque (ut prius) dpi — Ab2. Qualium e?3 — AD Ed Cubo, vel (si angulus A sit obliquus) Rhombo solido. Cumque A (posito d = 1) vel e (quicunque ponatur valor ipsius d) sit minor quam 1, (propter A < d:) illius potestates posteriores ita continue decrescunt, ut tandem negligi possint; planique valor pi exhibeatur quantumlibet vero propinquus. Atque haec est (Illustrissime Domine) Methodi, quam innuebam, ex meis principiis deductio, et demonstratio brevis. Vale. Tui observantissimus Johannes Wallis.

1 respectu add. 8 Qualium d2 . . . Rhombo. add. 10-11 Qualium d3 . . . Rhombo solido. add. 12-15 Cumque A ... propinquus. add. in margin 17 Vale. |Oxon. d. 5. Aug. 1668. add. Oldenburg]

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234. WALLIS to BROUNCKER, 6/[16] August 1668

234.

WALLIS to WILLIAM BROUNCKER Oxford, 6/[16] August 1668 Transmission:

W Letter sent: LONDON Royal Society Early Letters Wl, No. 60, 2pp. Enclosure: WALLIS-BROUNCKER 5/[15].VIII.1668. Clearly, Wallis originally intended to enclose a letter for Oldenburg in the present letter. There is no other evidence to suggest that such a letter was sent, as an enclosure or otherwise.

Oxford Aug. 6. 1668 My Lord,

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This1031 should have come by Thursday's Post1032 (according to the first date) but company coming in, deteined mee so long that before I had transcribed it, the Post was gone. I hope your Lordship will find the Demonstration without mistake, & such as will satisfy your Lordship (& then, I know it ought to satisfy others:) If it prove otherwise; Your Lordship will oblige mee by letting mee know so much. That which your Lordship suspected, (when it came without the Demonstration,) I think was not an error: for in pi — ab2, since that pi doth as well increase as a, the whole quantity is not diminished by the increase of a, as it would have been if pi had been a standing quantity. I sent your Honour a former Pacquet1033 by Tuesday's post, which I hope your Lordship received: Your Honour will excuse mee that I trouble you with another so soon. I am My Lord,

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Your Lordships very humble & obliged servant, John Wallis. 12 quantity. |I have left Mr Oldenburg's open, that I need not transcribe what is sayd to him. del.\ I sent 14 excuse (1) you (2) mee 1031

This: i.e. WALLIS-BROUNCKER 5/[15].VIII. 1668. should . . . Post: i.e. should have arrived on 6/[16] August, which was a Thursday. 1033 former Pacquet: i.e. WALLIS-BROUNCKER 3/[13].VIII.1668, enclosing Wallis's investigations on Fermat's negative theorem. 1032

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235. WALLIS to BROUNCKER, 8/[18] August 1668 [2] These For the Right Honourable William Lord Vicount Brouncker, President of the Royall Society London.

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235. WALLIS to WILLIAM BROUNCKER 8/[18] August 1668 Transmission:

Manuscript missing. Existence and date: Covering letter to Wallis's demonstration of Fermat's other negative theorem, which holds that there is no right-angled triangle in numbers whose area is a square.

236. WALLIS to WILLIAM BROUNCKER 8/[18] August 1668, enclosure: Demonstration of Fermat's Other Negative Theorem Transmission:

W Draft of paper sent: OXFORD Bodleian Library MS. Don. d. 45, f. 138r. At top right Wallis has written: 'Sent to my Lo: Brounker Aug. 8. 1668.' Enclosure to: WALLIS-BROUNCKER 8/[18].VIII. 1668. The content of the paper largely corresponds to part IV of WALLIS-BROUNCKER VIII?.1668, which was probably intended for publication in the Philosophical Transactions.

Aliud Theorema Fermatianum Negativum, Quod ibidem demonstrandum proponebatur. Nullum est in numeris Triangulum Rectangulum cujus Area

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236. WALLIS to BROUNCKER, 8/[18] August 1668, enclosure sit numerus Quadratus.1034 Demonstratio. Demonstrationem hujus, in Commercio Epistolico Epist. 44.1035 dederam his verbis.

In exposito schemate (cujus constructio patet) Trianguli Rectanguli BCD latera non possunt esse numeris effabilia, nisi AD DE sint inter se ut numeri plani similes, (secus enim, qui ab ipsis fit, non erit numerus quadratus, ejusque radix BD efFabilis,) hoc est, ut numeri quadrati inter se. Esto ut 2a 2 , 2e2. Erunt igitur CB, CD, BD, ut a2 + e 2 , a2 - e 2 , 2ae. Adeoque CD, ^BD, ut a2 — e 2 , ae. Et proinde (cum duorum quadratorum differentia, atque eorundem medius proportionalis, non possint esse plani similes,) qui ab ipsis fit, (hoc est, area trianguli,) non potest esse numerus quadratus. Quod erat demonstrandum.

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Haec demonstratio quo minus pro legitima habeatur, hoc tantum objicitur; quod non demonstraveram assumptum Lemma, Duorum quadratorum differentiam, eorumque medium proportionalem, (nempe a2 — e 2 et ae,) non posse esse numeros pianos similes.

3 Epist. 44. (1) hunc (2) dederam 7se (1) aut (2) ut

8 ejusque (1) latus (2) radix 18 differentiam, 1034

(1) et eo breaks off (2) eorumque

Aliud . . . Quadratus: see FERMAT-DIGBY [28.III]/7.IV.1658. Epist. 44.: i.e. WALLIS-DIGBY 20/[30].VI.1658.

1035

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237. COLLINS to WALLIS, 15/[25] August 1668 Putaram ego, Lemma illud, apud tantos viros quibuscum res ag batur admissum iri, neque expressa demonstratione indigere. Id autem, quoniam hoc expetunt, sic demonstro. Intelligantur a 2 , e 2 , quadrat! inter se primi, seu in eadem ratione minimi; (quippe vel tales sunt, vel, facta ad communem mensuram applicatione, ad tales reduci possunt, non mutata ratione rectarum AD, DE.) Ergo et a, e, inter se primi, (quippe siquis hos duos metitur numerus, idem et a 2 , e2, metietur.) Ergo et a, e, a — e, item a, e, a + e, singuli singulis inter se primi; (nani siquis vel horuni vel illorum duos metitur numerus, idem et tertium metietur; quippe, qui totuni et ablatum metitur, metitur reliquum; quique ablatum et reliquum metitur, metitur totum.) Cumque a — e sit ad utrumque duorum o, e, primus; primus item erit ad ae. factum ex illis. Similia turn ostendetur, a + e primum esse ad eandem ae quoniam ita est turn ad e turn ad a. Cum itaque ae primus sit turn ad a + e, turn ad a — e; primus item erit ad o2 — e2 ex illis factum. Non sunt igitur a2 — e 2 , et ae, plani similes; quippe similes plani non possunt esse inter se primi; (intellige, nisi sint quadrati; qui casus hie non obtinet.) Quod erat demonstrandum.

237.

JOHN COLLINS to WALLIS 15/[25] August 1668 Transmission:

Manuscript missing. Existence and date: Mentioned in and answered by WALLIS-COLLINS 25.VIII/[4.IX]. 1668. In this letter, Collins apparently sent Wallis news of Oldenburg's marriage to Dora

1-2 ego, (1) apud viros tantos, Lemma illud, etiam absque diserta demonstratione (2) Lemma illud, apud tantos viros | quibuscum res agebatur add] admissum 3 quoniam (1) non admittunt, (2) hoc expetunt, 8 siquis (1) eorum (2) vel horum vel illorum 12 primus; (1) erit (2) primus 12 ad jeandem add. and del\ ae. factum 13-14 primum esse (1) ae. (2) ad eandem ae. (3) ad eandem ae, quoniam ita est turn ad e turn ad a. (a) Cumque sit ae primus (b) Cum itaque ae primus sit 16 primi; (1) sed vel alter alterum metietur, vel saltern his similium minimus metietur utrumque. (2) (intellige, nisi sint quadrati; qui casus hie non obtinet.)

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238. WALLIS to COLLINS, 25 August/[4 September] 1668 Katherina Dury, Oldenburg's former ward. He also informed Wallis of the state of printing of his Mechanica sive de motu, which he was seeing through the press. Finally, he reported on Huygens's rule for squaring the hyperbola by means of logarithms, which the Dutch mathematician had recently published at the end of his review of Gregory in the Journal des Scavans (2 July 1668), 56.

238. WALLIS to JOHN COLLINS Oxford, 25 August/[4 September] 1668 Transmission:

W Letter sent: CAMBRIDGE Cambridge University Library MS. Add. 9597/13/6, f. 199r-199av (f. 199V and 199ar blank) (our source). At top off. 199av at 90° to address in Collins's hand: 'About an Hyperbolick Space'. Postmark on f. 199av: 'AU/26'.— printed: RIGAUD, Correspondence of Scientific Men II, 492-3. Reply to: COLLINS-WALLS 15/[25].Vin.l668. Oxford Aug. 25. 1668.

Sir,

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I received yours1036 of the 15th instant. I shall be very glad of any happyness for Mr O.1037