History of Vector Analysis

M i c h a e l J. C r o w e

A HISTORY OF

VEC O R ANALYSIS

T h e Evolution of t h e I d e a of a Vectorial System

A

HISTORY

OF

VECTOR ANALYSIS T h e E v o l u t i o n of the Idea of a V e c t o r i a l System

M I C H A E L J. CROWE University

of Notre

Dame

D o v e r Publications, I n c . New York

To

Copyright ©

MARY

ELLEN

1967 b y U n i v e r s i t y o f N o t r e D a m e Press

N e w m a t e r i a l C o p y r i g h t © 1985 by M i c h a e l J. C r o w e A l l rights reserved u n d e r Pan A m e r i c a n a n d International C o p y r i g h t Conventions. Published Lesmill Road,

in

Canada by

General

Publishing Company,

Ltd.,

30

D o n Mills, Toronto, Ontario.

Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, This

London W C 2 H 7EG.

D o v e r edition, first published in

1985,

is an unabridged and

corrected republication of the work first published by the University of N o t r e D a m e Press i n 1967. A n e w Preface has b e e n a d d e d t o this e d i t i o n . M a n u f a c t u r e d i n t h e U n i t e d States o f A m e r i c a D o v e r Publications, Inc., 31 East 2 n d Street, Mineola, N.Y.

L i b r a r y of Congress Cataloging in Publication D a t a C r o w e , M i c h a e l J. A h i s t o r y of v e c t o r analysis. Originally published: Notre D a m e : University of N o t r e D a m e Press,

1967. W i t h c o r r e c t i o n s a n d n e w pref.

Includes bibliographies and index. 1.

Vector analysis—History.

QA433.C76 ISBN

1985

0-486-64955-5

515'.63

I.

Title. 85-13081

11501

Preface

Shortly before becoming President of Harvard in ematician the

Thomas

best k n o w n

Newton have dynasties

Hill

made

vectorial

done

the

system

following of his

1862, the math-

statement concerning

day:

"The

discoveries

of

m o r e for E n g l a n d a n d for the race than w h o l e

of British

monarchs;

and we

d o u b t not that in the great

mathematical birth of 1853, the Quaternions of H a m i l t o n , there is as much

real

promise

of benefit to

mankind as

in any event of Vic-

toria's r e i g n . " L o r d K e l v i n , w r i t i n g w h e n V i c t o r i a was v e r y o l d a n d the

modern

vector

"Quaternions been

done;

system

came

and,

from

very

new,

Hamilton

though

took

a very

different view.

really

good work had

after his

beautifully

ingenious, have been an

m i x e d evil to those w h o have t o u c h e d t h e m in any w a y .

.

un-

. vectors

. . . have n e v e r b e e n of the slightest use to any creature." T h o u g h Kelvin's barbs quaternion

in this attack of the 1890's w e r e d i r e c t e d against the

system,

methods since the

he

had

been

waging

w a r against all

vectorial

1860's.

If a scientist of the present day w e r e forced to take sides in this dispute

on the value of vectorial

methods, he might view Hill

as

overly enthusiastic, but he w o u l d not side w i t h Kelvin. T h e v i e w of vectorial m e t h o d s c h a m p i o n e d by this great physicist has b e e n refuted by the thousands

of uses

that have b e e n f o u n d for vectorial

m e t h o d s . N e a r l y all b r a n c h e s o f classical p h y s i c s a n d m a n y areas o f m o d e r n physics are n o w p r e s e n t e d i n the language o f vectors, a n d the benefits d e r i v e d t h e r e b y are m a n y . V e c t o r analysis has l i k e w i s e proved omy,

a valuable

Despite little

aid

for

many problems

in engineering, astron-

and geometry. the

studied.

importance Not a single

of vector book

analysis,

and

its

not more

history has than

been

a handful of

s c h o l a r l y p a p e r s h a v e u p t o n o w b e e n w r i t t e n o n its h i s t o r y . C o n s e quently m a n y historical errors m a y be f o u n d in the relevant literaiii

Preface

ture. T h e present study was not w r i t t e n in the expectation that all or even

most historical

swered;

questions a b o u t vector analysis w o u l d b e an-

rather it was

written

in the h o p e of presenting an essen-

t i a l l y c o r r e c t o u t l i n e o f t h e h i s t o r y o f t h i s i m p o r t a n t area. I n u n d e r t a k i n g this study I have f r e q u e n t l y b e e n h i n d e r e d by the scarcity of scholarly

studies

numbers,

linear algebra, tensors, theoretical

of the

history

teenth-century mechanics.

This

of such

r e l a t e d areas

as

complex

electricity, and nine-

is of course the c o m m o n plight of

historians of science, a n d I have b e e n consoled by the hope that the present

study

m a y s h e d l i g h t o n t h e h i s t o r y o f o t h e r areas

o f sci-

ence, such as those m e n t i o n e d above. In

this

pects

study I

h a v e c o n c e n t r a t e d o n t h e m o r e f u n d a m e n t a l as-

of vectorial

treated in detail:

analysis;

the

history of the

following topics

is

vector a d d i t i o n a n d subtraction, the forms of vec-

tor multiplication,

vector division

(in

those

systems

where

i t oc-

curs), a n d t h e s p e c i f i c a t i o n o f v e c t o r types. Less a t t e n t i o n has b e e n given to the history of vector differentiation and integration, and the operator V a n d the associated transformation theorems, since these were

for the

work.

No

most part developed originally in a Cartesian frame-

detailed

presentation

of the

complicated

history of the

l i n e a r vector f u n c t i o n has b e e n a t t e m p t e d . T h o u g h the above statement indicates the materials included, it does not sufficiently specify the approach taken in this study. For a n u m b e r o f r e a s o n s I h a v e c h o s e n t o f o c u s (as t h e s u b t i t l e i n d i c a t e s ) on t h e h i s t o r y of t h e i d e a of a v e c t o r i a l system. It s h o u l d not be forgotten that the m o d e r n

system

many

vectorial

created

these

systems e m b o d i e d an

systems

of vector analysis is but one of the in

the

course

of history.

Each of

idea or conception of the form that a

vectorial system can have and should have. A n d it is the history of these ideas that I have tried to describe. To do this I have discussed each

of the

tempted to led

to

these

vectorial

systems

created

before

1 9 0 0 a n d at-

creation,

development,

and acceptance

or rejection of

systems.

The the

the

major

d e t e r m i n e w h a t ideas (mathematical a n d motivational)

history of vectorial analysis m a y in one sense be v i e w e d as

history of systems of abbreviation, since any p r o b l e m that can

b e s o l v e d b y vectorial m e t h o d s can also b e s o l v e d ( t h o u g h usually less

conveniently) by the

vectorial

analysis

older Cartesian methods. T h e history of

may equally w e l l be v i e w e d as the history of a

w a y of looking at physical a n d geometrical entities. Consideration of these

two

aspects

chosen to focus

of the

history

will

help explain w h y I

have

on the evolution of the idea of a vectorial system,

rather than on the history of the major theorems in vectorial analy-

iv

Preface

sis, m a n y o f w h i c h w e r e i n a n y case d i s c o v e r e d b e f o r e a n d o u t s i d e of the vectorial traditions. Concerning

the

references.

The

reader

will

find

that a simple

and

not u n c o m m o n system o f reference has b e e n e m p l o y e d i n t h e text. T h e notes for each chapter are located at the e n d of that chapter; w i t h i n each chapter ordinary note numbers w i l l be f o u n d as w e l l as r e f e r e n c e s t o t h e s e n o t e s o f t h e f o r m (3,11,1; 2 7 ) . T h e l a t t e r a r e r e a d as follows: the first n u m b e r always refers to a note at the e n d of the chapter;

the

numbers to the right of the semicolon always refer to

the page n u m b e r s in the publication indicated in that note. In some c a s e s (as a b o v e ) o n e o r t w o o t h e r n u m b e r s a r e i n c l u d e d t o t h e l e f t of the semicolon; these numbers ( w h e n they occur) refer to the volume

number and

part n u m b e r of the

the reference above is read:

publication

indicated.

Thus

see v o l u m e I I , part I, of i t e m 3 in t h e

notes; consult page 27. T h r o u g h this m e t h o d it has b e e n p o s s i b l e to provide the reader w i t h

many references

that otherwise

could be

i n c l u d e d only t h r o u g h a substantial increase in the size of the book. Concerning sources

quotations

for this

and

translations.

study were books

Since

many

and journals

of

the

of limited circula-

tion I have used quotations rather liberally. All quotations from documents written in foreign languages (French, German, Italian, Russian,

and

cases

Danish)

where

have

previously

been

translated

published

into

English.

translations

In

were

the

few

available,

I

have used these after c h e c k i n g t h e m against the original a n d n o t i n g deviations. T h e sole e x c e p t i o n to this statement occurs in t h e case of Wessel's D a n i s h ; here I have c h e c k e d Nordgaard's E n g l i s h translation

against

translations Concerning

the

French

translation

of Zeuthen.

The

remaining

(the majority) are my o w n . bibliography.

No

formal

bibliographical

been i n c l u d e d in this book. T h e reader w i l l

find

section

has

h o w e v e r that the

sections of notes at the e n d of each chapter w i l l serve rather w e l l as a bibliography raphy

is

for that

greatly

chapter.

Moreover the

d i m i n i s h e d by the

existence

n e e d for a bibliogof a book

that

lists

nearly all relevant p r i m a r y d o c u m e n t s p u b l i s h e d to about 1912; this is

Alexander

Systems

of

Macfarlane's

Mathematics

(Dublin,

Bibliography 1904).

of

Quaternions

Supplements

to

and

Allied

this

un-

c o m m o n l y accurate bibliography w e r e p u b l i s h e d up to 1913 in the Bulletin

of

Quaternions The

the

International

and

Allied

author

wishes

Society Systems

to

express

for of

his

Promoting

the

Study

of

Mathematics.

gratitude

to those

who

have

aided h i m in preparing this study. P u b l i s h e d and u n p u b l i s h e d materials have b e e n o b t a i n e d f r o m libraries too n u m e r o u s to m e n t i o n , a n d this through the kindness of the librarians of the universities of v

Preface

Notre D a m e , Wisconsin, C a m b r i d g e , a n d Yale. Assistance at important

points

has

come

from

Professor Stephen J.

Rogers

of Notre

D a m e University and from Professor D e r e k J. Price of Yale University. Sincere thanks are e x t e n d e d to Professors C. H. B l a n c h a r d a n d William

D.

sor James

Stahlman of the W.

U n i v e r s i t y of W i s c o n s i n a n d to Profes-

B o n d o f P e n n s y l v a n i a State U n i v e r s i t y . T h e s e three

scholars (a physicist, an historian of science, a n d a mathematician) gave generously of their time (in reading the entire manuscript) and of t h e i r w i s d o m (in s a v i n g t h e m a n u s c r i p t f r o m a n u m b e r of errors). To

Professor E r w i n N.

most

sincere

thanks

Hiebert, of the University of Wisconsin, my

for

his

numerous,

detailed,

and

perceptive

c o m m e n t s on the entire manuscript. Portions of the research for this book

w e r e carried out w i t h f i n a n c i a l assistance p r o v i d e d b y funds

administered by C o m m i t t e e on Grants for the Arts and Humanities of the

University of Notre

Notre

Dame,

Dame. Michael J. Crowe

March,

vi

1967

Indiana

Acknowledgments

Grateful lishers

acknowledgment is hereby made to the following pub-

and

libraries

published B.

G.

for permission

to

quote

from

books

and un-

materials:

Teubner Verlag,

Friedrich

Engel,

Hermann

Grassmanns

kalische

Stuttgart,

Grassmanns

for

Leben,

permission

contained

Gesammelte

in

to

quote

Vol.

mathematische

from

Ill

und

of physi-

Werke.

Cambridge

University

published material

in

Library

for permission to quote from

un-

the correspondence of James Clerk M a x w e l l

and Peter Guthrie Tait. C a m b r i d g e U n i v e r s i t y Press for p e r m i s s i o n to quote f r o m Cargill Gilston

Knott,

Life

and

Scientific

Work

of

Peter

Guthrie

Tait.

Ernst B e n n L i m i t e d , L o n d o n , for permission to quote f r o m O l i v e r Heaviside,

Electromagnetic

Macmillan Oliver

&

Co.,

Theory,

Ltd.,

Heaviside,

Vols.

London,

Electrical

I

for

and

III.

permission

to

quote

from

Papers.

T h o m a s N e l s o n a n d Sons, Ltd., L o n d o n , for p e r m i s s i o n to quote from and

Sir

Edmund

Electricity,

Vol.

Yale

Whittaker,

A

History

of

the

Theories

o f Aether

I.

University Library for permission to quote from the u n p u b -

lished material

in the correspondence of Josiah W i l l a r d Gibbs.

Yale U n i v e r s i t y Press for p e r m i s s i o n to q u o t e f r o m L y n d e P h e l p s Wheeler,

Josiah

Willard

Gibbs:

The

History

of

a

Great

Mind.

vii

List

Graph I

Graphs

and

Tables

Q u a t e r n i o n P u b l i c a t i o n s f r o m 1841 to 1900

Graph I I Graph

of

III

Q u a t e r n i o n Books from 1841 to 1900 Annual

N u m b e r of Titles

of Mathematical

Articles and Books, 1868-1909 Graph I V

Grassmannian

Analysis

Publications

113 from

1841 to 1900 Graph V Graph V I Graph VII Graph

VIII

113

G r a s s m a n n i a n A n a l y s i s B o o k s f r o m 1841 to 1900 Q u a t e r n i o n P u b l i c a t i o n s by C o u n t r y

Analysis

Publications

115 by

Country Graph I X Chronology

viii

114 114

Q u a t e r n i o n B o o k s by C o u n t r y Grassmannian

111 112

G r a s s m a n n i a n Analysis Books by C o u n t r y

115 116 256

Contents

Chapter

One

THE

EARLIEST

TRADITIONS

I. I n t r o d u c t i o n II.

1

The Concept of the

Parallelogram

of Velocities

and

Forces

2

III.

L e i b n i z ' C o n c e p t of a G e o m e t r y of Situation

IV.

The

Concept of the

Geometrical

3

Representation

of

Complex Numbers V.

Chapter

5

Summary and Conclusion

11

Notes

13

Two

SIR

WILLIAM

ROWAN

HAMILTON

AND

QUATERNIONS I.

Introduction: Hamiltonian Historiography

17

Hamilton's Life and Fame

19

III.

Hamilton and Complex Numbers

23

IV.

Hamilton's Discovery of Quaternions

27

Quaternions until H a m i l t o n ' s D e a t h (1865)

33

II.

V. VI.

Chapter

Summary and Conclusion

41

Notes

43

Three

OTHER

EARLY

ESPECIALLY

VECTORIAL

GRASSMANN'S

SYSTEMS, THEORY OF

EXTENSION I. Introduction II.

III. IV.

47

August Ferdinand Mobius and His

Barycentric

Calculus

48

Giusto Bellavitis and His Calculus of Equipollences

52

Hermann

Grassmann

and His

Calculus

Introduction

of Extension: 54

V.

G r a s s m a n n ' s Theorie der Ebbe und Flut

60

VI.

G r a s s m a n n ' s Ausdehnungslehre o f 1 8 4 4

63 ix

Contents

VII. VIII.

IX.

T h e Period from 1844 to 1862 Grassmann's

77

Ausdehnungslehre

of

1862

and

the

Gradual, Limited Acceptance of His Work

89

Matthew O'Brien

96

Notes

Chapter

102

Four

TRADITIONS THE

I. II.

IN

MIDDLE

VECTORIAL PERIOD OF

ANALYSIS ITS

FROM

HISTORY

Introduction

109

Interest in Vectorial Analysis

in Various

Countries

from 1841 to 1900 III.

110

Peter Guthrie Tait:

Advocate

and

Developer of

Quaternions IV.

V. VI.

Chapter

Benjamin

117

Peirce:

Advocate

of Quaternions

in

America

125

James Clerk Maxwell: Critic of Quaternions

127

William Kingdon Clifford: Transition Figure

139

Notes

144

Five

GIBBS

AND

HEAVISIDE

DEVELOPMENT

OF

AND

THE

THE

MODERN

SYSTEM OF VECTOR ANALYSIS I.

Introduction

150

II. Josiah W i l l a r d Gibbs III.

Gibbs' Early W o r k in Vector Analysis

152

IV.

G i b b s ' Elements o f Vector Analysis

155

Gibbs' Other W o r k Pertaining to Vector Analysis

158

V. VI. VII. VIII. IX. X.

Chapter I. II. III.

x

150

Oliver Heaviside

162

Heaviside's Electrical Papers

163

H e a v i s i d e ' s Electromagnetic Theory

169

T h e Reception Given to Heaviside's Writings

174

Conclusion

177

Notes

178

Six

A

STRUGGLE

FOR

EXISTENCE

Introduction

IN

THE

1890'S 182

T h e "Struggle for Existence"

183

Conclusions

215

Notes

221

Contents

Chapter

Seven

THE

EMERGENCE

OF

THE

OF VECTOR ANALYSIS: I. II.

III.

SYSTEM

Introduction Twelve

Major Publications

225 in Vector Analysis

from

1894 to 1910

226

Summary and Conclusion

239

Notes

243

Chapter Eight Notes

Index

MODERN

1894-1910

SUMMARY AND CONCLUSIONS

247 255

260

xi

Preface

to

the

Dover

Edition

It is v e r y gratifying that interest in the materials presented in this v o l u m e i s s u f f i c i e n t t o j u s t i f y a s e c o n d e d i t i o n . T h i s has p e r m i t t e d t h e c o r r e c t i o n o f a n u m b e r of small errors and, more importantly, provides an opportunity to b r i n g t o r e a d e r s ' a t t e n t i o n s o m e o f t h e r e l e v a n t s t u d i e s o f s p e c i f i c areas w h i c h h a v e a p p e a r e d s i n c e t h e b o o k ' s first p u b l i c a t i o n i n 1967. Recent researches have shed light particularly on the history of algebra d u r i n g the nineteenth century. T h e most broadly conceived of such works is

Lubos

Novy's

Origins

o f Modern

Algebra.l

British

developments

in

algebra have received most attention, i m p o r t a n t studies having b e e n p u b lished by Harvey W.

Becher,

J.

M.

Dubbey,

Philip C.

Enros,

Elaine

K o p p e l m a n , L u i s M . Laita, a n d Joan L . Richards.2 Interest i n Sir W i l l i a m R o w a n H a m i l t o n ' s a c h i e v e m e n t s i n a l g e b r a has b e e n especially intense. R e s e a r c h i n t h i s a r e a has b e e n a i d e d b y t h e a p p e a r a n c e i n 1967 u n d e r t h e editorship of H. Hamilton's

Halberstam and R.

Mathematical Papers,

tions in algebra.3 T h o m a s L.

E.

Ingram of the third volume of

that volume

being devoted

H a n k i n s has e n r i c h e d

to his publica-

H a m i l t o n i a n schol-

arship by various publications, most notably his e n g a g i n g b i o g r a p h y of the great Irish m a t h e m a t i c i a n a n d scientist.4 T h e scholar most actively engaged in assessing H a m i l t o n ' s place in t h e history of B r i t i s h algebra is H e l e n a M. Pycior,

w h o s e d o c t o r a l d i s s e r t a t i o n i n t h i s a r e a has b e e n f o l l o w e d b y a

number

of studies

of the

contemporaries.5 Jerold algebraic/analytic

researches

W a e r d e n has p r o v i d e d quaternions.7

algebraic

David

ideas

of Hamilton

and

M a t h e w s has p u b l i s h e d a p a p e r o n during

the

a n e w analysis

1830's,6 w h i l e

of Hamilton's

B l o o r has b r o a d l y c o n s i d e r e d

B.

his

British

Hamilton's L.

van

der

1843 d i s c o v e r y o f

Hamilton's algebraic

a p p r o a c h in r e l a t i o n to t h e social, political, a n d p h i l o s o p h i c a l context o f his times,8 whereas T. the genesis and

L.

H a n k i n s a n d John H e n d r y have focused studies on

importance of Hamilton's conception of algebra as the

"Science of Pure Time."9 Arnold R.

N a i m a n i n his d o c t o r a l dissertation

surveyed the role of quaternions

the overall development of mathe-

matics.10 xii

in

Preface to the D o v e r Edition

The fascination

I felt for

Hamilton while

researching this b o o k was

rivaled, if not surpassed, as I l e a r n e d m o r e of his r e m a r k a b l e c o n t e m p o r a r y Hermann Grassmann.

M a n y issues I e n c o u n t e r e d i n s t u d y i n g his m a t h e -

matical creations have b e e n treated in d e p t h by A l b e r t C. doctoral dissertation

is

Lewis whose

a c a r e f u l a n a l y s i s o f G r a s s m a n n ' s Ausdehnungslehre

o f 1844 a n d its sources. D r . L e w i s has n o w p u b l i s h e d s o m e o f his r e s u l t s i n papers on the influence of Grassmann's father and of Schleiermacher on his mathematical system.11 M o r e o v e r , Jean D i e u d o n n e a n d D e s m o n d Fearnl e y - S a n d e r h a v e e a c h p u b l i s h e d essays o n G r a s s m a n n ' s p l a c e i n t h e h i s t o r y of linear algebra.12 R e c e n t r e s e a r c h e s h a v e also d e v e l o p e d n e w p e r s p e c t i v e s o n f i g u r e s less c e n t r a l t h a n H a m i l t o n a n d G r a s s m a n n i n t h i s h i s t o r y . H e l e n a M . P y c i o r has presented ciative

a fresh

Algebra,13

analysis

and

of B e n j a m i n

Hubert

Kennedy

Peirce's has

p i o n e e r i n g Linear Asso-

investigated

James

Mills

Peirce's place in t h e " c u l t of q u a t e r n i o n s " that arose in late n i n e t e e t h century America.14 G.

C . S m i t h i n a r e c e n t p a p e r has u r g e d t h a t M a t t h e w

O ' B r i e n deserves significantly m o r e credit as a pioneer of the

modern

v e c t o r i a l s y s t e m t h a n has t r a d i t i o n a l l y b e e n a c c o r d e d h i m , 1 5 a n d t h e late B. R.

Gossick has

provided

n e w insights on

the contrasting views

of

vectorial methods espoused by O l i v e r Heaviside and L o r d Kelvin.16 T h e first systematic study of the history of Stokes' T h e o r e m a n d of the associated theorems n a m e d after Gauss and G r e e n is d u e to V i c t o r J.

Katz.17 T h e

i n v o l v e m e n t o f R u s s i a n m a t h e m a t i c i a n s w i t h v e c t o r i a l m e t h o d s has b e e n treated b y W . D o b r o v o l s k i j , 1 8 a n d studies o f t h e history o f v e c t o r analysis i n general have b e e n u n d e r t a k e n by A d a l b e r t A p o l i n a n d James W. Joiner, b o t h of w h o m seem to have w r i t t e n w i t h o u t k n o w l e d g e of my book.19 F o r t h e sake o f c o m p l e t e n e s s , m e n t i o n s h o u l d b e m a d e o f t h r e e p u b l i c a tions w h i c h a p p e a r e d before m y book, b u t w h i c h escaped m y b i b l i o g r a p h i c searches. T w o o f these, b o t h f r o m t h e 1930 s , t r e a t t h e h i s t o r y o f c o m p l e x n u m b e r s ; the earlier was w r i t t e n by J. B u d o n whereas the second is E r n e s t Nagel's " I m p o s s i b l e N u m b e r s . " 2 0 T h e latter omission is especially regrettab l e b e c a u s e t h a t essay p r o v i d e s a h i s t o r i o g r a p h i c p e r s p e c t i v e w h i c h w o u l d h a v e e n r i c h e d m y p r e s e n t a t i o n . T h i s i s e v e n m o r e t r u e o f t h e t h i r d essay, that p u b l i s h e d in 1 9 6 3 - 6 4 by the late I m r e Lakatos.21 A l t h o u g h c o n t a i n i n g essentially n o t h i n g r e l e v a n t to the h i s t o r y of vector analysis, Lakatos' n o w f a m o u s essay i s r i c h i n p h i l o s o p h i c a n d h i s t o r i o g r a p h i c i n s i g h t s w h i c h , w e r e I to rewrite this book, w o u l d certainly be included. Some hints as to the d i r e c t i o n I w o u l d t a k e a r e p r o v i d e d i n a h i s t o r i o g r a p h i c essay I p u b l i s h e d i n 1975.22 Persons w h o m a y acquire f r o m this book an interest in f u r t h e r readings in the history of mathematics

may wish

to consult

the

excellent

general

histories of mathematics w r i t t e n by Carl B. Boyer and M o r r i s Kline,23 or for articles on i n d i v i d u a l mathematicians, o f Scientific

Biography,24

Bibliographic

t h e m a n y v o l u m e s o f t h e Dictionary

searches

which

required

days

in xiii

Preface to the D o v e r Edition

t h e e a r l y 1960's c a n n o w b e a c c o m p l i s h e d b y s p e n d i n g a f e w h o u r s w i t h t h e late

Kenneth

Mathematics

O.

Mays

and

with

Bibliography

the

ISIS

and

Research

Manual

of the

History

of

Bibliography.25

Cumulative

In c o n c l u d i n g this u p d a t e d preface, I extend w a r m e s t thanks to the two persons

w h o have

made

this

new edition

possible:

John

W.

Grafton,

Assistant to t h e P r e s i d e n t of D o v e r Publications, a n d James R. L a n g f o r d , D i r e c t o r o f t h e U n i v e r s i t y o f N o t r e D a m e Press. T h e e n c o u r a g e m e n t o f t h e f o r m e r a n d t h e cooperation of the latter are chiefly responsible for this book b e i n g once again available to readers. M i c h a e l J. C r o w e University of Notre

Dame

January, 1985

Notes 1

LuboS

2

Harvey

Novy,

Mathematica, Historia

W.

Origins

7 (1980),

Mathematica,

(1812-1813): (1983),

"Woodhouse,

3 8 9 - 4 0 0 ; J. 4

(1977),

Elaine

Archive for

M.

by Jaroslav T a u e r

Dubbey,

Renewal

Koppelman,

History

trans, Babbage,

295-302;

Precursor of t h e

24-47;

Algebra,"

o f Modern Algebra,

Becher,

Peacock,

and

"Babbage,

Philip

C.

1973).

A l g e b r a , " Historia

Peacock a n d M o d e r n A l g e b r a , "

Enros,

of C a m b r i d g e

(Leyden,

Modern

"The

Analytical

M a t h e m a t i c s , " Historia

Society

Mathematica,

10

" T h e C a l c u l u s o f O p e r a t i o n s a n d t h e Rise o f Abstract

o f Exact

Sciences,

8

(1971),

155-242;

Luis

M.

Laita,

"The

Influence of Boole's Search for a Universal M e t h o d in Analysis on the Creation of His Logic," Annals

o f Science,

Controversy

34

(1977),

between

163-176;

William

Luis

Hamilton

M.

Laita,

and

"Influences

Augustus

De

on

Boole's

Logic:

The

M o r g a n , " Annals o f Science,

36

(1979), 4 5 - 6 5 ; L u i s M . L a i t a , " B o o l e a n A l g e b r a a n d Its E x t r a - l o g i c a l Sources: T h e T e s t i m o n y of M a r y

Everest

Boole,"

History and Philosophy o f Logic,

" T h e A r t and the Science of British Algebra: Truth," 3

H.

Hamilton, 4

Historia

Mathematica,

Halberstam vol.

Ill:

Thomas

L.

7

and

Algebra

(1980),

R.

E.

Hankins,

Sir

William

(1980),

3 7 - 6 0 ; Joan

L.

Richards,

343-365.

Ingram,

(Cambridge,

1

A Study in the Perception of Mathematical

The

Mathematical

England, Rowan

Papers

of

Sir

William

Rowan

1967).

Hamilton

(Baltimore,

1980).

For

a

shorter

b i o g r a p h y w h i c h discusses H a m i l t o n ' s m a t h e m a t i c s o n a m o r e e l e m e n t a r y l e v e l , see: Sean O'Donnell, 5

William

Helena

Algebra ( A

M.

Rowan

Hamilton:

Pycior,

1976 C o r n e l l

The

Portrait

Role

of Sir

of a

Prodigy

William

Hamilton

University doctoral dissertation);

(Dublin,

1983).

in

the

Development

H.

M.

Pycior,

and the

B r i t i s h O r i g i n s o f S y m b o l i c a l A l g e b r a , " Historia Mathematica, 8 ( 1 9 8 1 ) ,

Pycior,

"Early

Criticisms

(1982), 3 9 2 - 4 1 2 ; H . Isis, 6

74 ( 1 9 8 3 ) ,

7

L.

available

in

Approach

to

Algebra,"

23-45;

Historia

H.

M.

Mathematica,

9

211-226. "William Rowan

Archive for

B.

Symbolical

Modern

M . Pycior, " A u g u s t u s D e Morgan's Algebraic W o r k : T h e T h r e e Stages,"

Jerold Mathews,

Analysis, "

of t h e

of British

" G e o r g e Peacock

van

History der

English

of Exact

Waerden,

as

H a m i l t o n ' s P a p e r o f 1837 o n t h e A r i t h m e t i z a t i o n o f

Sciences,

19

Hamiltons

"Hamilton's

(1978),

Entdeckungder

Discovery

177-200. Quaternionen(Gottingen,

of Q u a t e r n i o n s , "

1974).

Mathematics

Now

Magazine,

49

(1976), 2 2 7 - 2 3 4 . 8

David

Nineteenth

Bloor,

Century

"Hamilton

Mathematics,

and

ed.

by

Peacock

on

Herbert

the

Essence of A l g e b r a "

Mehrtens,

Henk

Bos

in

and

Social History o f Ivo

Schneider

( B o s t o n , 1981), p p . 2 0 2 - 2 3 2 . 9

Thomas L. Hankins, "Algebra as Pure Time: William Rowan Hamilton and the Founda-

tions

xiv

of

Algebra"

in

Motion

and

Time,

Space

and

Matter:

Interrelations

in

the

History

of

Preface to the D o v e r Edition

Philosophy and Science,

ed.

by P.

J.

Machamer and

R.

G.

Turnbull (Columbus,

1976),

pp.

327-359. 10

Arnold

R.

Naiman,

The

Role

of Quaternions

in

the

History

of

Grassmanns

of Mathematics

(A

1974

New

York University doctoral dissertation). 11

Albert

C.

Lewis,

An

Historical

Analysis

1975 U n i v e r s i t y o f T e x a s a t A u s t i n d o c t o r a l dissertation); A . Ausdehnungslehre A.

and

Schleiermacher's

Dialektik ,"

Annals

Ausdehnungslehre

of 1844

(A

C . L e w i s , " H . G r a s s m a n n ' s 1844

of Science,

34

(1977),

103-162;

C. Lewis, "Justus Grassmann's School Programs as Mathematical Antecedents of H e r m a n n

Grassmann's in

the

Early

1844

Ausdehnungslehre"

Nineteenth

Century,

ed.

in

by

Epistemological

Hans

Niels

and

Jahnke

Social and

Problems

Michael

of the

Otte

Sciences

(Dordrecht,

1981), p p . 2 5 5 - 2 6 7 . 12

Jean

1-14;

Dieudonne,

Desmond

gebra,"

American

"Hermann Monthly, 13

"The

Mathematical

Grassmann

89

(1982),

Helena

Tragedy

Fearnley-Sander, Monthly,

and

of G r a s s m a n n , ' '

"Hermann 86

the

Linear and

Grassmann

(1979),

809-817;

Prehistory

of

Multilinear Algebra,

and

the Creation

see

also

Universal

D.

8

(1979),

of Linear Al-

Fearnley-Sander,

Algebra,"

American

Mathematical

161-166.

M.

Pycior,

"Benjamin

Peirce's

Linear

Associative

Algebra,"

I sis,

70

(1979),

537-551. 14

Hubert

Kennedy,

"James

Mills

Peirce and

t h e C u l t o f Q u a t e r n i o n s , " Historia

Mathemat-

ica, 6 ( 1 9 7 9 ) , 4 2 3 - 4 2 9 . 15

G.

C.

Smith,

Mathematica, 16

B.

9

R.

"Matthew

(1982),

O'Brien's

Anticipation

of Vectorial

Mathematics,"

Historia

172-190.

Gossick,

"Heaviside

and

Kelvin:

A

Study

in

C o n t r a s t s , " Annals

o f Science,

33

(1976), 2 7 5 - 2 8 7 . 17

Victor

J.

Katz,

"The

History

of Stokes'

Theorem,"

Mathematics

Magazine,

52

(1979),

146-156. 18

W . D o b r o v o l s k i j , " D e v e l o p p e m e n t d e l a t h e o r i e des v e c t e u r s e t des q u a t e r n i o n s dans les

travaux des

mathematiciens

russes

du X I X e

siecle,"

Revue d'histoire des sciences,

21

(1968),

345-349. 19

Adalbert

ralis,

12

Apolin,

(1970),

"Die

357-365;

geschichtliche James

Entwicklung der Vektorrechnung,"

W a l t e r Joiner,

A

History of Vector Analysis (A

Historia Natu-

1971

doctoral

dissertation at G e o r g e Peabody C o l l e g e for Teachers). 20

J.

Budon,

quelques (1933), Ideas, the

175-200; 3

(1935),

Philosophy 21

" S u r la r e p r e s e n t a t i o n g e o m e t r i q u e s des n o m b r e s imaginaires (Analyse de

memoires

Imre

parus

de

220-232; 427-474;

and

History

Lakatos,

1795

Ernest reprinted

of Science

"Proofs

a

1820),"

Nagel, in

(New

and

Bulletin

des sciences

"Impossible

Ernest York,

Nagel,

1979),

Refutations,"

mathematiques,

Numbers,"

ser. the

2,

57

History

of

Teleology Revisited and Other Essays

in

pp.

Studies i n

166-194.

British Journal for

the

Philosophy

of Science,

14

( 1 9 6 3 - 1 9 6 4 ) , 1 - 2 5 ; 1 2 0 - 1 3 9 ; 2 2 1 - 2 4 5 ; 2 9 6 - 3 4 2 . T h e s e e s s a y s h a v e n o w b e e n r e p u b l i s h e d as: Imre

Lakatos,

Proof and

Refutations:

The

Logic

of

Mathematical

Discovery,

ed.

by

John

W o r r a l l a n d E l i e Z a h a r ( C a m b r i d g e , E n g l a n d , 1976). 22

M.

Historia 23

ical 24

J. Crowe, " T e n ' L a w s ' C o n c e r n i n g Patterns ofChange in the History of Mathematics," Mathematica,

Carl

B.

2

Boyer,

Thought from

(1975), A

Ancient

161-166.

History o f Mathematics ( N e w Y o r k , to

Modern

Dictionary o f Scientific Biography,

14

Times

(New

vols.,

ed.

1968) a n d

York, by

Morris

Kline,

Mathemat-

1972).

Charles

Coulston

Gillispie

( N e w York,

Science

Formed from

1970-1980). 25

ISIS

ISIS

Cumulative

Critical

1 9 7 1 - 1 9 8 2 ) ; ISIS 1980);

and

Bibliography.

Bibliographies

1-90:

Cumulative

Kenneth

O.

A

Bibliography

1913-1965,

Bibliography May,

5

of

vols.,

1966-1975, Bibliography

vol. and

the

ed.

History by

I,

Research

of

Magda

ed.

by

Manual

Whitrow

John of the

Neu History

(London, (London, of Mathematics

( T o r o n t o , 1973). XV

CHAPTER

The

I.

Earliest

ONE

Traditions

Introduction The

early

history

of

vectorial

analysis

is

most

viewed within the context of two broad traditions

appropriately

in the history of

science. O n e of these traditions relates to mathematics, the other to physical

science.

T h e first tradition, that w i t h i n the history of mathematics, extends from the time of the Egyptians and Babylonians to the present and consists

in

the

progressive

Throughout time to include

the

broadening of the

concept of number.

c o n c e p t of n u m b e r has b e e n e x p a n d e d so as

not only positive

integers, but negative

numbers,

frac-

tions, and algebraic and transcendental irrationals. E v e n t u a l l y complex

and

higher complex

numbers

(including

vectors)

were intro-

duced. T h e activities of some of the figures in the history of vector analysis m a y be v i e w e d as b e l o n g i n g to this tradition. The also

second tradition, that w i t h i n the history of physical science,

extends

back

mathematical

to

ancient

entities

and

times

a n d consists

operations

that

in the

represent

search for aspects

of

physical reality. This tradition p l a y e d a part in the creation of G r e e k geometry, and the inherited problems.

from

natural philosophers of the seventeenth century

the

Greeks

However

in the

the

geometrical

course

of the

approach

seventeenth

to

physical

century the

physical entities to be represented passed t h r o u g h a transformation. This

transformation

consisted

in

the

shift

in

emphasis

from

such

scalar q u a n t i t i e s as p o s i t i o n a n d w e i g h t to s u c h v e c t o r i a l q u a n t i t i e s as velocity, force, m o m e n t u m , and acceleration. T h e transition was neither abrupt nor was it confined to the seventeenth century. Later developments to

transform

with

in

the

electricity, space

magnetism,

of mathematical

and

optics

physics

into

acted

further

a space filled

vectors.

These two broad traditions converged at a n u m b e r of periods history;

one

such

period

was

in the

in

nineteenth century, and this

1

A H i s t o r y of V e c t o r Analysis

convergence

is

marked

by

the

creation

and

d e v e l o p m e n t of vec-

torial methods. T h e first major three-dimensional vectorial systems were

created in

portant ideas

1840,s.

the

Before

were put forth w h i c h

this

time, however, three im-

l e d t o t h e m a j o r v e c t o r i a l sys-

tems. T h e s e t h r e e ideas are the subject of the present chapter; t h e y are the c o n c e p t of a p a r a l l e l o g r a m of forces, L e i b n i z ' c o n c e p t of a geometry of situation, and the concept of the geometrical representation

of imaginary

II.

The and

Concept

numbers.

of

the

Parallelogram

of

Velocities

Forces

O n e of the most f u n d a m e n t a l m a t h e m a t i c a l ideas in vector analysis

is

the

idea of the addition of vectors.

T h e sum of t w o vectors

w h i c h have a c o m m o n point of origin is defined as the vector originating at the same point and extending to the opposite corner of the parallelogram d e f i n e d by the t w o original vectors. Certain physical entities,

such

as

velocities

a n d forces,

m a y be c o m p o u n d e d

in

a

similar w a y , a n d f r o m this c o r r e s p o n d e n c e stems m u c h of the usefulness

of vector analysis.

T h e idea of a parallelogram of velocities may be found in various ancient forces

authors,8 *

Greek was

not

uncommon

and in

the the

concept

of a

parallelogram

of

sixteenth and seventeenth cen-

turies.9 By the early n i n e t e e n t h century parallelograms of physical entities

frequently

appeared

in

treatises,

and

this

usage

indirectly

led to vector analysis, for this idea p r o v i d e d a striking example of how

vectorial

entities

could

be

used

for physical

applications.

It

s h o u l d not be inferred, h o w e v e r , that all of those w h o used the concept of a parallelogram of physical entities were aware of the idea of a vector or of vector addition.

T h e essential

idea in the parallelo-

gram of physical entities is the construction of a diagram in terms of w h i c h the operations involved in determining the resultant become evident.

The

i d e a o f adding t h e l i n e s n e e d n o t b e i n t r o d u c e d o r w a s

i t (to m y k n o w l e d g e ) e v e r i n t r o d u c e d b e f o r e t h e c r e a t i o n o f vectors. T h u s this

i d e a a l o n e c o u l d n o t a n d a l m o s t c e r t a i n l y d i d n o t directly

s t i m u l a t e a n y o n e to t h e c r e a t i o n of a vectorial system. Its i n f l u e n c e was

indirect b u t i m p o r t a n t , f o r i t w a s t h e f i r s t a n d m o s t o b v i o u s c a s e

in w h i c h vectorial methods could be brought to the aid of physical science. ° T h e system used for n u m b e r i n g notes is described in the preface.

2

The

III.

Leibniz'

Concept

of

a

Geometry

of

Earliest

Traditions

Situation

Gottfried W i l h e l m Leibniz (1646-1716) made many contributions to

mathematics;

among

geometry of situation.

the

In

less

this

well

regard

known

is

Leibniz

his

concept

discussed

of a

the possi-

bility of creating a system w h i c h w o u l d serve as a direct m e t h o d of space

analysis.

Although

the

details

of this

idea were

never fully

w o r k e d out by L e i b n i z , he a d v a n c e d far e n o u g h to be r a n k e d as a conceptual

forerunner of the

essay, w h i c h w a s

history of vectorial Leibniz' main 8,

first

vectorial

first p u b l i s h e d in

analysts.

M o r e o v e r his

1833, p l a y e d a part in the later

analysis.

ideas w e r e contained in a letter dated S e p t e m b e r Huygens.1

1679, a n d w r i t t e n to Christian

In this

letter L e i b n i z

wrote: I am still not satisfied w i t h algebra, because it does not give the shortest methods or the most beautiful constructions in geometry. This is w h y I believe that,

so far as

geometry is

concerned, we

need still

another

analysis w h i c h is distinctly geometrical or linear a n d w h i c h w i l l express situation

[situs]

directly

as

algebra

expresses

magnitude

directly.

And

I

believe that I have f o u n d the w a y a n d that we can represent figures a n d even

machines

and

movements

by

characters,

as

algebra represents

n u m b e r s or m a g n i t u d e s . I am s e n d i n g y o u an essay w h i c h seems to me t o b e i m p o r t a n t . (1; 3 8 2 ) I n his essay, w h i c h was c o n t a i n e d i n the letter, L e i b n i z d e s c r i b e d his

system further: I

have discovered certain e l e m e n t s of a n e w characteristic w h i c h is

entirely different f r o m algebra a n d w h i c h w i l l have great advantages i n r e p r e s e n t i n g t o t h e m i n d , e x a c t l y a n d i n a w a y f a i t h f u l t o its n a t u r e , e v e n w i t h o u t f i g u r e s , e v e r y t h i n g w h i c h d e p e n d s o n sense p e r c e p t i o n . A l g e b r a is the characteristic for u n d e t e r m i n e d n u m b e r s or m a g n i t u d e s o n l y , b u t it does not express situation, angles, a n d m o t i o n directly. H e n c e it is often difficult to analyze the properties of a figure by calculation, and still more difficult to find very convenient geometrical demonstrations and constructions,

even

w h e n the algebraic calculation

is completed.

But

this n e w characteristic, w h i c h follows the visual figures, cannot fail to give the solution, the construction, and the geometric demonstration all at the

same time,

and

in

a natural

w a y and in

one

analysis,

t h a t is,

through determined procedure.

B u t its c h i e f v a l u e l i e s i n t h e r e a s o n i n g w h i c h c a n b e d o n e a n d t h e c o n clusions could

which

not

be

c a n b e d r a w n b y o p e r a t i o n s w i t h its c h a r a c t e r s , w h i c h expressed

in

figures,

and

still

less

in

models,

without

m u l t i p l y i n g these too greatly o r w i t h o u t c o n f u s i n g t h e m w i t h too m a n y points and lines in the course of the m a n y futile attempts one is forced to

make.

This

method,

by contrast, w i l l g u i d e us

surely and without

3

A

History

effort.

of V e c t o r Analysis

I b e l i e v e that by this m e t h o d one c o u l d treat mechanics almost

l i k e g e o m e t r y , a n d o n e c o u l d e v e n test the qualities of materials, because this o r d i n a r i l y d e p e n d s o n c e r t a i n f i g u r e s i n t h e i r s e n s i b l e parts. F i n a l l y , I have no h o p e that we can get v e r y far in physics u n t i l we have f o u n d s o m e s u c h m e t h o d o f a b r i d g m e n t t o l i g h t e n its b u r d e n o f i m a g i n a t i o n . (1; 3 8 4 - 3 8 5 )

His

system

as actually sketched by h i m shows that he by no means

discovered a primitive vector analysis, though the above quotations show

that

he

was

searching

for

s o m e t h i n g a k i n to vector analysis.

L e i b n i z ' s y s t e m c e n t e r e d o n t h e i d e a o f t h e c o n g r u e n c e o f sets o f points.

He used A, B, . . . to represent fixed points and X, Y , . . . to

represent u n k n o w n relation

points.

congruence;

The

thus

he

s y m b o l b was used to express the wrote

A B C b DEF

to

express

that

a

set of t h r e e p o i n t s A, B, C, e a c h of w h i c h w a s a fixed distance f r o m t h e o t h e r t w o p o i n t s , c o u l d b e m a d e t o c o i n c i d e w i t h a n o t h e r set o f similarly fixed points D, E, F. He then discussed locus relations and stated that the locus of points congruent to a

fixed

space

g i v e n t h a t AB

infinite

in

all

directions."

(1;

387)

If it is

point " w i l l be a b

AY,

the p o i n t values of Y w i l l be points on a sphere w i t h center at A and radius

of

whose

points

length

A B C b A B Y

(X)

AB.

The

are

equidistant

determines

relation

AX b BX

from

a circle.

locus

that the

of all

Y's

r e l a t i o n AY

will V

BY

be V

a

determines

and

B,

and

a

the

plane

relation

L e i b n i z then discussed the locus

of points Y satisfying the relation AY "the

A

b BY

straight

CY V

DY

line."

b

CY and c o n c l u d e d that (1;

389)

determines

applied his analysis to four simple problems.

After

showing

a point, Leibniz

O n e of these may be

discussed as typical. T h e p r o b l e m is to show that the intersection of two planes plane,

is a straight line.

and

the

combining before,

relation

these

we

determines

AY

have

T h e relation AY b BY determines one b CY

AY

determines

b BY b CY,

a

second

which,

as

plane.

it was

By

shown

a straight line.10

Proceeding from this s u m m a r y of Leibniz' best-known exposition system,11

of his sis.

First,

w e m a y d i s c u s s its r e l a t i o n t o m o d e r n v e c t o r a n a l y -

Leibniz

deserves

much

credit for suggesting that a n e w

algebra, w h e r e i n geometrical entities are symbolically represented and he

the

symbols

operated u p o n directly, was

desirable.

However,

failed to discover a system in w h i c h geometrical entities could

be a d d e d , subtracted, a n d m u l t i p l i e d . L i k e w i s e he failed to see that A B a n d B A (for e x a m p l e ) can b e v i e w e d a s distinct entities a n d that —AB

could have a significant meaning.

representing runner the

4

of

a

fixed

Mobius

concept

point

and

of a vector.

by

a

T h o u g h his idea of directly

symbol

Grassmann,

he

makes

certainly

him did

a partial forenot

introduce

D e s p i t e the fact that angle considerations

The

Earliest Traditions

d i d not enter into his system, he still m u s t be v i e w e d as h a v i n g constructed a system w h i c h a l l o w e d for the use of co-ordinai^s. L e i b n i z saw that a n e w algebra of the applications failed

to

in

mathematics

develop

Leibniz'

system

practical taken

form

and

sought

in

methods

by

w o u l d have

the physical for these

Couturat,

though

numerous

sciences, but he

tasks.

stated

The in

view

of

relation

to

Grassmann's system, is also a p p l i c a b l e in r e l a t i o n to m o d e r n vector analysis; Couturat wrote:

" I n summary, the calculus of Grassmann

seems to bring fully into reality the geometrical characteristic conceived by Leibniz, and shows that Leibniz' idea was not simply a dream. But there is such a disproportion b e t w e e n Leibniz' conception

of a

system

and

the

very

defective

p r o d u c e d that Grassmann felt a sharp

essay

which

he

actually

distinction should be made

between the ideal conceived and the sketch actually written." Shortly after Jablonowski

1833,

12

w h e n L e i b n i z ' essay was first p u b l i s h e d , the

Gesellschaft expressed their interest in

and enthusi-

asm for the essay by offering a prize for the f u r t h e r d e v e l o p m e n t of Leibniz'

system.

One

mathematician entered the competition and

w o n the prize, e v e n t h o u g h he had created his system before hearing of L e i b n i z ' ideas; this mathematician was Grassmann a n d this incident w i l l be more fully discussed in the third chapter.

IV.

The of

Concept

Complex

Though ence

to

the

term

systems

dimensional

of

the

Geometrical

Representation

Numbers vector

of

space,

analysis

is

mathematics it

should

now

that

not

be

used

may

primarily

be

applied

forgotten

that

in

refer-

in

the

three-

complex

n u m b e r s y s t e m m a y l e g i t i m a t e l y be c o n s i d e r e d as a v e c t o r i a l system.

The

metrical useful

as

two-dimensional representation the

of

vectorial complex

three-dimensional

primary subject of this history. cuss

briefly

system

based

numbers

vectorial

is

on

the

certainly

systems

which

geo-

not are

as the

Nevertheless it is i m p o r t a n t to dis-

the early history of the

geometrical representation

of

complex numbers, not only because the complex n u m b e r system is (broadly

speaking)

a vectorial

system

b u t also

because

Hamilton

discovered quaternions in the course of a search for a t h r e e - d i m e n sional analogue to the complex n u m b e r system. A t least six m e n are c o m m o n l y c r e d i t e d w i t h t h e d i s c o v e r y o f t h e geometrical

representation

of complex

numbers;

t h e y are Wessel,

Gauss, Argand, Buee, M o u r e y , a n d Warren.13 Since the systems crea t e d b y these six m e n are v e r y s i m i l a r a n d are o f l i m i t e d r e l e v a n c e to

the

present

study, they need not all be discussed in

detail.

In

5

A H i s t o r y of V e c t o r Analysis

w h a t follows, the system p u b l i s h e d by Wessel, w h i c h was the earliest a n d a m o n g t h e m o s t i m p r e s s i v e , w i l l b e a n a l y z e d i n s o m e d e p t h ; t h e ideas o f t h e o t h e r f i v e m e n w i l l b e t r e a t e d less f u l l y t h o u g h w i t h special will

be

attention

to

certain

shown

that

some

three-dimensional

aspects of their d e v e l o p m e n t . T h u s it of

vectorial

these

mathematicians

systems

and

that

one

searched

of them

for

influ-

enced Hamilton in an important manner. Although

Hero

of Alexandria

and

Diophantus

in ancient times

h a d encountered the question of the m e a n i n g of the square root of a negative

number,

and although

Cardan

had in his

1 5 4 5 Ars Magna

used complex numbers in computation, nevertheless complex numbers were not accepted by most mathematicians as legitimate mathematical

entities

until

well

into

hardly surprising since numbers

the

nineteenth

such as

century.

This is

V ^ T seem to be neither

less t h a n , greater t h a n , n o r e q u a l to zero. In

modern

mathematics

complex

numbers

are

usually justified

either by representing t h e m in terms of couplets of real numbers or by representing them geometrically. T h e origin of the first method w i l l be discussed in the next chapter. T h e first attempt (which was unsuccessful)

to

represent

complex

numbers

geometrically

was

m a d e in the seventeenth century by John Wallis.14 W h e r e Wallis failed, a Norwegian Caspar Wessel geometrical

(1745-1818)

representation

surveyor succeeded;

in

1799

published the first explanation of the of complex

numbers.15

His

ideas w e r e

p r e s e n t e d before the Royal A c a d e m y o f D e n m a r k i n 1797 a n d publ i s h e d t w o years later in the m e m o i r s of that society.2 Unfortunately, however, Wessel's ematicians until

publication went unnoticed by European math-

1897, w h e n it was r e p u b l i s h e d in a F r e n c h transla-

tion.3 In the first paragraph of his m e m o i r W e s s e l stated: attempt

deals

with

the

question,

"This present

h o w may we represent direction

a n a l y t i c a l l y ; t h a t is, h o w s h a l l w e e x p r e s s r i g h t l i n e s s o t h a t i n a single the

equation length

i n v o l v i n g one u n k n o w n line and others k n o w n , both and

the

direction

of the

unknown

line

may

be

ex-

p r e s s e d . " (2; 55) A s t h i s q u o t a t i o n suggests a n d later passages confirm, Wessel's chief interest was the creation of geometrical methods; his representation of c o m p l e x n u m b e r s was subservient to this aim.

Nonetheless

the

latter p l a y e d a f u n d a m e n t a l role as is indi-

cated by the following statement: treatise]

was

my

" T h e o c c a s i o n for its b e i n g [his

seeking a method whereby I could avoid the im-

possible operations. . .

(2; 57)

After stating that previously only oppositely directed lines could be r e p r e s e n t e d analytically, W e s s e l suggested that it s h o u l d be pos-

6

The

sible to a

find

definition

added

Traditions

methods to represent inclined lines. Wessel then gave of the

if we

Earliest

addition

of straight

lines:

" T w o

unite t h e m in such a w a y that the

right

lines

are

second line begins

w h e r e t h e first o n e e n d s , a n d t h e n pass a r i g h t l i n e f r o m t h e first to t h e last p o i n t o f t h e u n i t e d lines. T h i s l i n e i s t h e s u m o f t h e u n i t e d lines."

(2;

58)

In

the

subsequent

discussion

of addition

Wessel

stated that the same definition can be used in a d d i n g m o r e than t w o (not necessarily coplanar) lines a n d that the order of addition is immaterial.

(2;

59)

Hence Wessel

had

introduced three-dimensional

vector addition and realized the importance of the commutative law for

addition.

Though

Wessel

had

up

to

called the positive unit (our 1

this of x

point •

only

discussed

what

he

1 + yV—I) a n d h a d n o t y e t i n d i -

cated h o w lines in general were to be represented in terms of complex numbers, nevertheless he proceeded to introduce the multiplication of lines.

The

product of two lines

(coplanar with each other

and w i t h the positive unit) was to have a length equal to the product of the lengths of the t w o factors. T h e p r o d u c t l i n e was to be coplanar w i t h t h e t w o factor l i n e s a n d w a s t o h a v e its i n c l i n a t i o n o r d i r e c t i o n angle (defined by reference to the inclination of the positive unit as 0°) e q u a l Wessel

to the

then

L e t 4-1

sum

of the

inclinations

of the

factor lines.

(2; 60)

added:

designate the positive rectilinear unit and +e a certain other

unit perpendicular to the positive unit and having the same origin; then t h e d i r e c t i o n a n g l e o f + 1 w i l l b e e q u a l t o 0°, t h a t o f - 1 t o 180°, t h a t o f + e t o 90°, a n d that o f - e t o - 9 0 ° o r 270°. B y t h e r u l e that t h e d i r e c t i o n a n g l e of the p r o d u c t shall e q u a l the s u m of the angles of the factors, we have: (+1)(+1) = + 1 ;

( + 1 ) ( — 1 ) = —1; ( - 1 ) ( - 1 ) = + 1 ; ( + l ) ( + e ) = +

(-l)(+e) = - e ; ( - l ) ( - € ) = +6;

(+e)(+e) = - 1 ;

e ;

(+e)(-e) = + 1 ;

(+l)(-6) = -

€ ;

(-e)(-e) = - 1 .

F r o m this it is seen that e is equal to V ^ I ; a n d the divergence of the product is d e t e r m i n e d such that not any of the c o m m o n rules of operation are c o n t r a v e n e d . (2; 60) Wessel

stated that any straight line in a plane m a y be represented

analytically showed

how

by

the

expressions

such

expressions

a +

eb

and

are to be

r(cos

v + €

multiplied,

v)

and

divided,

sin

and

raised to powers.

After giving t w o examples of the application of his

methods, Wessel

developed an elementary three-dimensional vec-

tor analysis.

(3; 2 3 - 2 8 )

Wessel began by constructing three mutually perpendicular lines w h i c h passed t h r o u g h the center of a sphere of radius r. W e s s e l specified three

that three radii

of the sphere w h i c h were collinear with the

m u t u a l l y p e r p e n d i c u l a r axes

s h o u l d be

designated by r,

171%

and er and that any point in space c o u l d be designated by a vector of

7

A H i s t o r y of V e c t o r Analysis

t h e f o r m x + r)y + ez.

(3;

23-24)

By a n a l o g y w i t h o r d i n a r y c o m p l e x

n u m b e r s W e s s e l d e f i n e d 1717 a n d e e a s e q u a l t o — 1 . T h e m u l t i p l i c a tion

of vectors corresponded to the rotation and extension of one

v e c t o r by a n o t h e r . T h u s (x

f 171/ + ez)

,,

( c o s u + € s i n u) r e p r e s e n t e d

t h e r o t a t i o n o f t h e v e c t o r x + iqy + e z t h r o u g h t h e a n g l e u a r o u n d t h e 17 or y axis. W e s s e l stated that t h e c o m p o n e n t of t h e v e c t o r that lies on

the

axis

product

of

of t h e

rotation above

is

should 17y +

remain x

cos

unchanged,

u — z

sin

and

thus

u + ex s i n

the

u + ez

cos u . ( 3 ; 2 5 - 2 6 ) T h e s y m b o l , , w a s u s e d t o i n d i c a t e m u l t i p l i c a t i o n . A r o t a t i o n of v d e g r e e s a r o u n d t h e € or z axis w a s e x p r e s s e d in t h e f o l l o w i n g w a y : (x + 1 7 y + ez)

,, ( c o s v + 1 7 s i n v) = ez + x c o s v — y s i n v

+ rjx s i n v + 7)y c o s v . ( 3 ; 2 6 ) R o t a t i o n s a r o u n d t h e 1 7 a x i s c o u l d b e compounded with

rotations

a r o u n d t h e € axis a n d v i c e versa, b u t

r o t a t i o n s a r o u n d t h e axis of t h e p o s i t i v e u n i t (the x axis) w e r e n e v e r discussed by Wessel. T h e reason for this is that serious mathematical

difficulties

were

involved

in

determining how

such

rotations

s h o u l d b e r e p r e s e n t e d , f o r e x a m p l e , t h e p r o d u c t s 17c a n d erj w o u l d h a v e h a d to be defined. W e s s e l p r e s u m a b l y e n c o u n t e r e d these difficulties b u t c o u l d not solve them.16 But even w i t h this limitation on his m e t h o d s Wessel was able to use t h e m to derive a n u m b e r of important results in spherical trigonometry. Wessel's

t h r e e - d i m e n s i o n a l v e c t o r i a l s y s t e m e x h i b i t e d a n a d hoc

character that makes

it appear seriously deficient w h e n compared

to m o d e r n systems; nevertheless, if it is v i e w e d as a creation of the late

eighteenth century, it can only be v i e w e d w i t h awe. Wessel's

treatment of ordinary complex numbers is equally impressive, and it was unfortunate for Wessel a n d for mathematics that his m e m o i r lay b u r i e d for nearly a century. In the early history of complex numbers a striking p h e n o m e n o n occurred:

on three separate occasions t w o m e n i n d e p e n d e n t l y and

simultaneously

discovered

the geometrical representation of com-

plex numbers. In 1806 A r g a n d a n d Buee both p u b l i s h e d independent

treatments

of imaginary

numbers,

and the

same

coincidence

o c c u r r e d i n 1828 w i t h M o u r e y a n d W a r r e n . W h a t i s e v e n m o r e surprising

is

that

Gauss

probably

discovered the

geometrical repre-

sentation of complex numbers at the same time as Wessel. Gauss' tion

first

published

treatment

of the geometrical representa-

of complex numbers appeared in

1831;

17

herein Gauss com-

m e n t e d that h e h a d h a d this i d e a for m a n y years a n d that traces o f i t could Lowell

easily be f o u n d in his Coolidge

investigated

1799 this

"Demonstratio Nova." point

and

showed

18

that

Julian Gauss'

c l a i m was a m p l y s u p p o r t e d b y the fact that s o m e m e t h o d s u s e d i n the

8

1799

paper

seem

"blind

and meaningless"

unless the author

The

Earliest Traditions

already possessed this idea.19 It was t h r o u g h Gauss'

1831 publica-

tion that most mathematicians came into contact w i t h the geometrical

representation of complex numbers, although

of Gauss'

paper

only

in

1852

However Hamilton heard in

(4;

312)

and

Hamilton

Grassmann

1845 that Gauss

heard 1844.20

in

had been

searching

for a "triple algebra" corresponding to the d o u b l e algebra of complex numbers. tion

(4; 3 1 1 - 3 1 2 ) F e l i x K l e i n a r g u e d i n a n 1 8 9 8 p u b l i c a -

that Gauss

Knott

had in

vigorously

accept

the

fact d i s c o v e r e d this.21

denied

geometrical

quaternions,

Ironically

representation

Gauss

but Tait and

himself did

of c o m p l e x

numbers

not

as

a

sufficient justification for them.22 In conclusion it m a y be n o t e d that Gauss' p u b l i c a t i o n w a s t h e shortest, t h e m o s t precise, t h e last, a n d the most influential

o f t h e six i n d e p e n d e n t p r e s e n t a t i o n s .

T h e next p u b l i c a t i o n to be c o n s i d e r e d was the longest, the least precise, the earliest (except for Wessel's), a n d the least influential. On J u n e 20, 1805, a l o n g essay e n t i t l e d " M e m o i r e sur les q u a n t i t e s imaginaires"

was

read before

the

Royal

Society

of London.

The

author was A b b e B u e e a n d his paper was p u b l i s h e d ( w i t h o u t translation)

in

the

1806

Transactions

of

the

Society.23

Royal

treatment of complex numbers was not of high quality; fact

has

expressed

surprise

that

it

was

Buee's

Coolidge in

published.24

The

well-

founded consensus a m o n g those w h o have studied Buee's paper is that

some

ingenuity

mixed

with

much

obscurity

is

to

be

found

there, as w e l l as a near approach to the concept of the m u l t i p l i c a t i o n of directed lines.

H a m i l t o n asserted that B u e e attempted to extend

his m e t h o d s t o space (5; [57]), b u t i f B u e e d i d d o t h i s , h e d i d i t i n a very

unorthodox

manner.

A far s u p e r i o r w o r k also a p p e a r e d in Argand's tites

small

book,

imaginaires

Essai

dans

sur

les

une

1806; this was Jean Robert maniere

constructions

de

representer

les

geometriques.6

quan-

Herein

Argand gave the m o d e r n geometrical representation of the addition and multiplication of complex numbers, and s h o w e d h o w this representation

could

trigonometry,

be

applied to deduce a n u m b e r of theorems

elementary

geometry, and algebra.

in

At this t i m e Ar-

gand d i d not attempt to e x p a n d his m e t h o d s for application to threed i m e n s i o n a l space. F o r seven years A r g a n d shared the fate of W e s sel;

however

unexpected In

in

1813 J.-F.

Gergonne's

1813

attention

was

called to his

book in

a very

way. Frangais published a short m e m o i r in v o l u m e IV of

Annates

de

mathematiques

(6;

63-74),

in

which

Fran-

gais p r e s e n t e d t h e g e o m e t r i c a l r e p r e s e n t a t i o n o f c o m p l e x n u m b e r s . At the conclusion of his paper Frangais stated that the f u n d a m e n t a l ideas in his paper w e r e not his o w n ; he h a d f o u n d t h e m in a letter

9

A

History

of V e c t o r Analysis

w r i t t e n by L e g e n d r e to his (Frangais') brother w h o had died. In this letter L e g e n d r e discussed the ideas of an u n n a m e d mathematician. Frangais

a d d e d that he h o p e d that this mathematician w o u l d make

himself k n o w n and publish his The

unnamed

ideas,

for

Legendre's

Frangais'

paper,

Gergonne

in

results.

mathematician friend

Argand

which

he

had

(6;

in

was Jean

already

p u b l i s h e d his

Robert Argand.

immediately

identified

74)

fact

sent

a

Hearing of

communication

to

himself as the mathematician of

L e g e n d r e ' s l e t t e r , c a l l e d a t t e n t i o n t o h i s b o o k , s u m m a r i z e d its c o n tents, a n d

finally

presented an (unsuccessful) attempt to extend his

system to three-dimensions. lications,

(6; 7 6 - 9 6 ) B e f o r e s e e i n g A r g a n d ' s p u b -

Frangais h a d w r i t t e n a letter to G e r g o n n e containing his

admittedly unsatisfactory attempts to extend the geometrical representation of c o m p l e x n u m b e r s to space.

(6;

96-101) A n d soon after

Argand's publication, Servois published a paper criticizing Argand's attempt and (6;

o u t l i n i n g his o w n ideas on a m e t h o d of space analysis.

101-109)

to

an

Hamilton

anticipation

triplets.

.

. ."

o f the

(5;

[57])

f o l l o w i n g passage

attributed

quaternions,

or

to at

In m a k i n g this

from

Servois least

to

"the an

nearest

approach

anticipation

statement Hamilton

of

had the

Servois' paper in mind.

Analogy w o u l d seem to

indicate that the tri-nominal should be of the

f o r m p cos a + q cos (3 + r cos y, a,

a n d y b e i n g t h e a n g l e s m a d e by a

r i g h t l i n e w i t h t h r e e r e c t a n g u l a r axes, a n d t h a t we s h o u l d h a v e (p cos a + q c o s (3 + r c o s y)(p' c o s a + q' c o s (3 + r' c o s y) = c o s 2 a + c o s 2 /3 + c o s 2 y — 1 . T h e v a l u e s o f p , q , r , p ' , q', r ' s a t i s f y i n g t h i s c o n d i t i o n w o u l d b e absurd;

but

A+

would

B V ^ I ? (7;

they

be

imaginaries,

reducible

to

the

general

form

114-115)

C o n c e r n i n g this

passage

Hamilton

wrote:

T h e s i x N O N - R E A L S w h i c h S e r v o i s t h u s w i t h r e m a r k a b l e s a g a c i t y foresaw,

without

b e i n g a b l e t o determine t h e m ,

the ther; u n k n o w n symbols +i, at

least,

these

latter

him, and furnish

may n o w be

identified with

+ / c , — i , — j , —/c, o f t h e q u a t e r n i o n t h e o r y :

symbols

fulfil

p r e c i s e l y t h e condition p r o p o s e d b y

a n answer t o h i s " s i n g u l a r q u e s t i o n . " I t m a y b e p r o p e r

t o state that m y o w n t h e o r y h a d b e e n c o n s t r u c t e d a n d p u b l i s h e d for a l o n g t i m e , b e f o r e t h e l a t e l y c i t e d passage h a p p e n e d t o m e e t m y eye. (5; [57]) The

series

of articles

in

Gergonne's

letter written by A r g a n d in w h i c h by

Lacroix

(6;

111)

Annales

was

concluded

calling attention to

Buee's

(1806)

this

very

point were

nions

10

is

in

1843.25

strong evidence unknown to The

ideas

that all

Hamilton of the

a

publication.

A r g a n d w r o t e t h a t h e h a d h a d n o k n o w l e d g e o f B u e e ' s w o r k . (6; There

by

he responded to a notice sent in

the

men

123)

discussed up to

w h e n he discovered quater-

next man

to be

discussed were

The

k n o w n to H a m i l t o n as early as thinking,

as

published cal

he

in

repeatedly

1828

Representation

a of

Earliest Traditions

1829 a n d m o r e o v e r i n f l u e n c e d his

acknowledged.

short

book

the

Square

(4;

entitled A Roots

190)

Treatise of

John Warren on

the

Negative

GeometriQuantities.

Warren's

presentation of the geometrical representation of complex

numbers

e x h i b i t e d great care a n d u n d e r s t a n d i n g ; he, u n l i k e B u e e

a n d A r g a n d , was a w a r e o f t h e i m p o r t a n c e o f t h e c o m m u t a t i v e , associative, and distributive laws, t h o u g h he d i d not use these terms.26 W a r r e n discovered his ideas in c o m p l e t e i n d e p e n d e n c e of the other mathematicians

who

wrote

on

the

geometrical

representation

of

complex numbers, b u t he, unlike the majority of t h e m , d i d not discuss t h e e x t e n s i o n o f h i s s y s t e m t o space.27 T h e final independent discoverer of the

geometrical representa-

tion of complex n u m b e r s was the F r e n c h m a n C. V. M o u r e y , w h o in 1828

published

quantites

an

negatives

excellent et

des

the conclusion of his book bra surpassing

treatise

entitled

quantites

La

pretendues

vrai

Theorie

imaginaires.28

des At

M o u r e y stated that there exists an alge-

not o n l y o r d i n a r y algebra b u t also the t w o - d i m e n -

sional algebra created by him.

This

algebra, he stated, extends to

three-dimensions.29 Presumably M o u r e y searched for such an algebra; if he f o u n d it, he d i d not p u b l i s h his discovery.

V.

Summary

and

Conclusion

T h u s w e can say that a t least f i v e m e n , w o r k i n g i n d e p e n d e n t l y o f each other, had by 1831 discovered and p u b l i s h e d the geometrical representation

of

complex

numbers.

These

men

were

Wessel,

Gauss, A r g a n d , W a r r e n , a n d M o u r e y . A t least t w o others, W a l l i s a n d Buee, had c o m e close to the same idea. Wessel, Gauss, Argand, a n d M o u r e y , a s w e l l a s S e r v o i s a n d F r a n g a i s , a n d p e r h a p s B u e e , h a d attempted to

find

higher c o m p l e x n u m b e r s for the analysis of space,

and all had failed. A number of conclusions cussed.

The

first is

m a y be d r a w n f r o m w h a t has b e e n dis-

that the

idea of a graphical representation of

complex numbers was certainly " i n the air" at that time. H o w e v e r , the acceptance of this idea was very slow, and little attention was p a i d to these ideas u n t i l Gauss p u b l i s h e d his p a p e r of 1831.

The

fact that the i d e a was n e g l e c t e d u n t i l Gauss e n t e r e d t h e f i e l d s h o u l d not, I think, be taken as surprising. peatedly

shown

that

radically

new

H i s t o r i a n s of science h a v e reideas

presented only on their

o w n merits are usually neglected. T h e m e n b e f o r e Gauss w e r e all little k n o w n ;

i n d e e d t h e y are n o w k n o w n o n l y because o f t h e i r o n e

great discovery. B u t w h e n Gauss wrote, he w r o t e w i t h the authority

11

A H i s t o r y of V e c t o r Analysis

of one w h o had already acquired fame through impressive work in traditional fields and through his w i d e l y k n o w n prediction of the position of the lost p l a n e t o i d Ceres.

It m a y be n o t e d n o w a n d dis-

cussed later that the pattern exhibited in this instance w i l l recur in the later history of vectorial analysis. S e c o n d , i t has b e e n n o t e d that m o s t o f those w h o w o r k e d o n the geometrical

representation

struct analogous

methods

of c o m p l e x n u m b e r s a t t e m p t e d to confor t h r e e - d i m e n s i o n a l space. T h a t m a n y

e m b a r k e d on this quest illustrates w h a t is probably mathematically o b v i o u s : the search for a system of space analysis was a natural concomitant to the numbers. fore

idea of the

geometrical representation of complex

Up to this point only those w h o m a d e their attempts be-

1831 h a v e b e e n discussed; m a n y others also p u z z l e d over this

p r o b l e m after 1831. A m o n g t h e m was H a m i l t o n , w h o , w o r k i n g precisely in this tradition, discovered quaternions.

12

Notes 1

Gottfried W i l h e l m

Christian Leroy E. first

Leibniz, "Studies

Huygens"

in

Loemker, vol.

published

mathematische

in

und

"Christi.

Leibniz,

I

1833;

Papers

(Chicago, 1956), 3 8 1 - 3 9 6 . the

physikalische

Huygenii

in a G e o m e t r y of Situation w i t h a Letter to

Philosophical

citation

Werke,

as

vol.

aliorumque

given

I,

seculi

pt.

based

Schriften, e d . (which

C.

is

sophic, e d .

I.

translation

Gerhardt, vol.

superior)

as

on II

the

given

are

above)

and

Hermann (Leipzig,

in

and

from

Loemker and

Uylenbroek's

have

text as

trans.

Grassmann, 1894),

Gesammelte

415-416,

celebrium

is

exercitationes

H a g a e c o m i t u m 1833. fasc. I I , p . 6 . " as

1850),

given

in

Leibniz,

Mathematische

17-27, a n d on U y l e n b r o e k ' s text

Leibniz,

Hauptschriften

zur

Griindung

been checked with

the

der

Philo-

1924). Q u o -

Gerhardt's text

g i v e n i n G r a s s m a n n , Werke, v o l .

I, pt.

(cited

I, 417-420.

All quotations have been taken from Martin A. Norgaard's English translation of

the first sixteen sections of Wessel's book; A

in

virorum

text

(Berlin,

ed.

E r n s t Cassirer, trans. A. B u c h e n a u , 2nd. ed., 2 vols. ( L e i p z i g ,

tations

2

his

Letters,

I

XVIII.

mathematicae et philosophiae. Ed. Uylenbroek. Loemker

and

L e i b n i z ' essay a n d letter w e r e

Source

55-66.

Book

I

in

Mathematics,

have also

vol.

used the

I,

ed.

French

see W e s s e l , " O n C o m p l e x N u m b e r s " i n

David

Eugene

Smith

translation of Wessel's

(New

book

York,

1959),

w h i c h is cited in

n o t e (1) a b o v e . T h e t i t l e f o r W e s s e l ' s o r i g i n a l p u b l i c a t i o n i s " O m D i r e c t i o n e n s a n a l v tiske

Betegning,"

Danske

Videnskabernes

by S. Lie

and

it

appeared

Selskabs

Skrifter.

in

vol.

V

Wessel's

(1799)

essay

o f Nye

was

Samling

a f det

rediscovered

Kongelige

in

1895

D. Christensen and C. Juel; it was republished without translation by Sophus in

the

1896

Archiv for

Mathematik

og

Naturvidenskab.

In

this

connection

see

Viggo Brun, "Caspar Wessel et l'introduction geometrique des nombres c o m p l e x e s " in

Revue 3

d'histoire

Caspar

des

Wessel,

sciences, Essai

12

sur

(1959), la

V a l e n t i n e r a n d T . N . T h i e l e , trans. 4

Robert

Perceval

Graves,

20-21.

representation

analytique

de

la

direction,

ed.

H. G. Z e u t h e n and others (Copenhagen,

Life

of

Sir

William

Rowan

Hamilton,

vol.

Ill

H.

1897).

(Dublin,

1889). 5

Sir

William

Rowan

Hamilton,

Lectures

on

Quaternions

(Dublin,

1853).

All

refer-

ences are to H a m i l t o n ' s Preface, w h e r e A r a b i c n u m e r a l s set in p a r e n t h e s e s are u s e d to indicate page numbers. 6

Jean

naires

Robert

dans

contains papers

les

Essai

sur

geometriques,

a reprint of the

on

papers

Argand,

constructions

first

une

2nd

edition

maniere

ed.,

(Paris,

ed.

de J.

representer Hoiiel

les

quantites

imagi-

1874).

This

(Paris,

1806) a l o n g w i t h

selections from the

complex numbers by Frangais, Argand, Gergonne, Lacroix, and Servois,

which

were

originally

published

in

Gergonne's

Annales

des

Mathematiques,

4 ( 1 8 1 3 - 1 8 1 4 ) a n d 5 ( 1 8 1 4 - 1 8 1 5 ) . S e e t h e w o r k c i t e d i n n o t e (7) b e l o w f o r a n E n g l i s h translation series in 7

Hoiiel's

edition.

Jean

Robert

trans. A. S. 8

of Argand's

book;

Hardy

included

less

material

than

Hoiiel

from

the

o f p a p e r s i n G e r g o n n e ' s Annales b u t s u p p l i e d v a l u a b l e c o m m e n t a r y n o t f o u n d

Argand,

Imaginary

Quantities:

Their

Geometrical

Representation,

H a r d y ( N e w Y o r k , 1881).

T h e t h r e e G r e e k a u t h o r s w h o u s e d t h i s c o n c e p t are (1) t h e a u t h o r o f t h e s o - c a l l e d

"pseudo-Aristotelian

Mechanica,"

(2)

Archimedes,

and

(3)

Hero

of Alexandria.

For

13

A H i s t o r y of V e c t o r Analysis the

first

Ages

(Madison,

and

the

nique

Analytique

"On

Spirals"

third

1959),

in

see

4-5,

Marshall

41.

Lagrange,

in

The

On

CEuvres,

Works

Clagett,

The

Archimedes vol.

XI

o f Archimedes,

Science

o f Mechanics

see J o s e p h

(Paris,

trans.

Louis

1888),

Thomas

12,

in

and

Heath

the

Middle

L a g r a n g e , MecaArchimedes,

(New

York,

n.d.),

165. 9

T h e history of this concept is discussed by n u m e r o u s authors; the f o l l o w i n g are

among

the

Maddox J.

most

(New

McCormack

1962);

(4)

tischen 10

(La

A.

For

this

Salle,

Rene

(2)

111.,

Dugas,

Ernst

Mach,

1960);

(3)

"Grundlegung

vol.

example

. .

(1)

1955);

Voss,

Wissenschaften,

plane . has

important:

York,

IV, see

pt.

(1;

I

A

The

History

o f Mechanics,

Science

Max Jammer,

der

Concepts o f Force

Mechanik"

(Leipzig,

390) but note

trans.

o f Mechanics, t r a n s .

in

1901-1908),

J.

R.

Thomas

( N e w York,

Encyklopadie

der

mathema-

43-46.

that L o e m k e r wrote

"AB

b BY for one

w h e r e a s t h e U y l e n b r o e k t e x t ( s e e G r a s s m a n n , Werke, v o l . I , p t . I , 4 2 0 )

(correctly) " A Y « B Y . "

11

There

is

a

fuller

but

similar

vol. V, ed. C. I. Gerhardt (Halle,

exposition

in

Leibniz,

Mathematische

Schriften,

1858), 141-171. M a n y m i n o r statements of L e i b n i z

(for e x a m p l e , statements in letters) are referred to a n d discussed by L o u i s Couturat, La

Logique

de

Leibniz

(Paris,

ideas, particularly as discussed by A. nection

with

Grassmann

E.

Heath,

Leibniz's

in

1901),

they relate

his

which

"The

Geometrical

Characteristic"

Geometrische

Analyse

in

in

12

Louis

Couturat,

There

have b e e n a n u m b e r of studies

that have a i d e d me ruff

Beman,

American

"A

Cajori,

for

"Historical of

1912),

1924);

in

the

Leibniz

(3)

(4)

on

the

in Julian

Analysis

vol.

discussion

27

I,

of

Leibniz'

L e i b n i z ' s y s t e m was also

of Grassmann

Monist,

(Paris,

History of

(1917),

pt.

1901),

I,

a n d Its Con-

36-56,

and

by

321-399.

538.

on the early history of complex numbers

of

Mathematics"

Science,

Graphic

American

46

Monthly,

Coolidge,

Hankel,

der

The

Proceedings

33-50;

(2)

of Imaginaries 19

Geometry

Theorie

in

(1897),

Representation

Mathematical

Lowell

Hermann

full

a m o n g t h e m o s t i m p o r t a n t are (1) W o o s t e r W o o d -

the

Advancement

Note

Wessel"

167-171;

ford,

de

in this study;

Chapter

Association

Time

Logique

a

system.

The

Werke,

13

La

includes

to Grassmann's

o f the

Florian

Before

the

(September-October,

o f the

complexen

Complex

Domain

Zahlensysteme

(Ox-

(Leipzig,

1 8 6 7 ) ; (5) P . S . J o n e s , " C o m p l e x N u m b e r s : A n E x a m p l e o f R e c u r r i n g T h e m e s i n t h e Development

of

263, 340-345; of

Certain

vancement narii

Branches

o f Science

nella

and 36

Mathematics"

of

(1834),

Mathematics

317-345;

Quantities"

Analysis" 185-352;

geometria"

(1898),

Complex

in

Teacher,

47

(1954),

106-114,

257-

(6) G e o r g e P e a c o c k , " R e p o r t o n t h e R e c e n t P r o g r e s s a n d P r e s e n t State in

(7)

in

Battaglini's

(8)

G.

in

Mathematical

of

the

British

Romorino,

Giornale

Windred,

Unfortunately a recent excellent

Report

Angelo

di

Association

"Gli

matematica,

for

the

Elementi

35

(1897),

Ad-

imagi-

242-258;

" H i s t o r y of the T h e o r y of Imaginary and

Gazette,

14

study came

(1929),

533-541.

to my attention too late to take full

a d v a n t a g e o f it. T h i s i s F . D . K r a m a r ' s " V e k t o r n o e i s c h i s l e n i e k o n t s a X V I I I i n a c h a l a XIX

vv"

14

The

matics, e d . 15

be

(in

Russian)

important David

in

passage

Eugene

lstoriko-Matematicheskie

from

Smith,

Wallis

vol.

I

may

Issledovaniia,

be

found

( N e w York,

in

15 A

(1963), Source

Book

225-290. in

Mathe-

1959), 4 6 - 5 4 .

T h e w o r d s " t o p u b l i s h " qualify this statement sufficiently that no m e n t i o n n e e d

made

in

the

text

of Leonard Euler, Charles Walmesley, and Dominique Truel.

T h e basis for attributing the geometrical representation to the first t w o of these m e n is

that it seems f r o m reading their writings on relevant subjects that they probably

had

this

representation.

The

sole

basis

for

mentioning

C a u c h y that T r u e l had this representation as early as Florian Cajori, fore

14

Wessel"

"Historical in

American

Notes on the Mathematical

Truel

is

a

statement

by

1786. F o r f u l l e r discussion see

G r a p h i c Representation of Imaginaries be-

Monthly,

19

(1912),

167-171.

T h e Earliest Traditions 16

Some

of these

difficulties

H a m i l t o n ' s efforts to 17

ische

gelehrte

Friedrich l

will

be

discussed

more

fully

in

Chapter

II, where

a t h r e e - d i m e n s i o n a l vectorial system are treated.

Gauss' untitled publication, w h i c h was a discussion of his " T h e o r i a r e s i d u o r u m

biquadraticum,

Commentatio

Anzeigen

of A p r i l

G a u s s , Werke,

»Ibid.,

19

find

secunda," 23,

vol.

II

1831.

was I

originally

have

(Gottingen,

used

1863),

published

the

text

in

as

the

given

Gotting-

in

Carl

169-178.

175.

Julian

Lowell

Coolidge,

The

Geometry

o f the

Complex

Domain

(Oxford,

1924),

28-29. 20

Hermann

Werke, v o l . 21

Felix

matische

Klein,

o f Edinburgh,

this in

Abbe

Royal

of

24

Coolidge,

25

This

is

VIII

96

Geometry

X,

sur

les

(1806), of

the

implied

Philosophical that

2

see

is

especially

(5;

in

on

the

Gilston

Proceedings

which

Mathe-

Recently

i n Proceedings

Knott,

o f the

Klein

in

Claim

"Pro-

Royal

Society

his

claim,

based

357-362. "Uber

Gauss

(Gottingen,

quantites

Arbeiten

zur

Function-

1922-1933), 55-57.

imaginaires"

the

Domain,

fact

in

Transactions

of

the

24.

that

3rd

in

an

Ser.,

had influenced him;

in

" O n

23-88.

Magazine,

implied

physikalische

Gauss' W e r k e n "

Cargill

criticism"

Complex by

von Tait,

17-23;

1900),

pt.

und

Discovery) of Quaternions"

Schlesinger,

representation of complex

same conclusion Quaternions;

the

(Leipzig,

Ludwig

discussed the authors geometrical

For

"Memoire

in

the

(1900),

document

London,

Quaternions"

Guthrie

24-34.

see

strongly

(not

23

G a u s s , Werke, v o l .

Buee,

Society

Invention

Herausgabe

Peter

a

vol.

point

der

mathematische

397-398.

of Q u a t e r n i o n s :

(1900),

G a u s s , Werke, On

the

Stand

128-133;

o f Edinburgh,

View

23

entheorie" 23

to

Gesammelte

1896), 8 - 9 ,

den

(1898),

Society

Klein's

Grassmann,

(Leipzig,

"Uber

51

Royal

fessor

2 2

II

for Gauss

the

see

pt.

Annalen,

Made of

Giinther

I,

1844

25

paper

(1844),

[31]—[57])

richly historical

as

"On

Hamilton

o f t h e six m e n w h o d i s c o v e r e d t h e

numbers, only Warren was

Hamilton's

(Hamilton,

489-495)

well

as

the

mentioned.

The

p r e f a c e t o h i s Lectures o n work

listed

in

note

(4)

above, w h e r e i n m a n y letters f r o m H a m i l t o n t o D e M o r g a n w e r e p u b l i s h e d i n w h i c h Hamilton

discussed these men.

p a p e r (4; 3 1 2 ) , (2) paper Rowan

(5;

[57]),

Hamilton,

and

vol.

H a m i l t o n e x p l i c i t l y d e n i e d h a v i n g s e e n (1) G a u s s '

M o u r e y ' s b o o k (4; 4 8 9 ) , (3) A r g a n d ' s b o o k (4; 4 3 5 ) , (4) S e r v o i s '

II

(5)

Frangais'

[Dublin,

papers

1885],

(Robert

606).

Perceval

From

the

Graves,

fact

that

Life o f Sir William

Hamilton

had not

read Servois' and Frangais' papers or Argand's book, it seems reasonable to conclude that he had Annales

not read any of the

before

1844.

Hamilton

relevant papers did

not

in

explicitly

volumes IV and V of Gergonne's

deny knowledge

of Wessel,

since

h e never, e v e n after 1843, h e a r d o f W e s s e l , a n d h e d i d not e x p l i c i t l y d e n y k n o w l e d g e of Buee's hand,

paper,

Hamilton

since

he

had already d e n i e d that

attended

the

vancement of Science, and on

the

Recent

(.B.A.A.S.

Report,

Argand's 26

See

Roots

Progress

and

John

o f Negative

the

in

Present

which

papers

Warren, Quantities

it had any merit.

meeting of the

A

from

Treatise

State

Peacock

of

on

the

1828),

Certain

briefly

Gergonne's

(Cambridge,

addition, page 9 for c o m m u t a t i v e

Branches

extent

on

"associative,"

probably

the

the

recognition

first

the

other

discussed

of

(ibid.y

"Report

Analysis" page

228)

Annales.

Geometrical page

Representation

3

for

of

the

commutative

law for multiplication, page

was aware of the associative l a w for m u l t i p l i c a t i o n , a n d page law. T h e importance of this

names

On

British Association for the Ad-

at this m e e t i n g George Peacock presented his and

185-352),

book

1833

Square law

of

18 for a hint that he

13 for the distributive

is that the discovery of quaternions d e p e n d e d to some of the

importance

"commutative,"

historical

and

statement

of these

laws.

"distributive"

was

made

by

in

On a

origin

of the

mathematical

sense

Hermann

the

Hankel,

Theorie

der

15

A H i s t o r y of V e c t o r Analysis complexen

Zahlensysteme

"These have

names

(Leipzig,

have

been

1867),

adopted

footnote

on

universally in

page

3,

not hesitated to transplant t h e m to G e r m a n soil;

was

it

seems

first

i n t r o d u c e d by Sir. W.

R.

he

said,

1840 and hence I

'distributive' and 'commuta-

tive' w e r e introduced by Servois ( G E R G O N N E ' S Ann. vol. V. tive'

where

E n g l a n d since

1814, p . 93); 'associa-

H a m i l t o n . " T h i s statement is re-

peated by both D a v i d E u g e n e S m i t h and Florian Cajori. T h e earliest recognition of the

necessity of proving the commutative

VII, Proposition the term

"associative" is

Connected with lished

in

the

"However, in

l a w for m u l t i p l i c a t i o n is in

Euclid, Book

16. T h e first p u b l i c a t i o n , t o m y k n o w l e d g e , i n w h i c h H a m i l t o n u s e d in the paper " O n a N e w Species of Imaginary Quantities,

a T h e o r y o f Q u a t e r n i o n , " c o m m u n i c a t e d N o v e m b e r 13, 1843, p u b Proceedings

of

the

Royal

Irish

Academy,

2

(1844)

424-434.

He

wrote:

virtue of the same definitions, it w i l l be f o u n d that another important

property of the o l d m u l t i p l i c a t i o n is preserved, or e x t e n d e d to the n e w , namely, that which 27

may

be

called

the

associative

character of t h e

operation.

.

.

."

Ibid., 4 2 9 - 4 3 0 .

At least no extension is suggested in his b o o k or in the t w o s u b s e q u e n t papers

w h i c h he p u b l i s h e d on this subject. Philosophical entitled

Transactions

of

"Considerations

sentation of the Square Geometrical Square

the

H i s t w o later papers w e r e b o t h p u b l i s h e d in the

Royal

of the Roots

Society

Objections

of

London,

119

(1829);

they

were

Raised Against the Geometrical Repre-

of Negative Quantities," pages 2 4 1 - 2 5 4 , and " O n the

Representation of the Power of Quantities Whose Indicies Involve the

Roots

of

251-254) Warren

Negative

Quantities,"

stated that he

pages

had written

339-359.

In

the

first

paper

(ibid.,

his book before he heard of Buee's or

M o u r e y ' s p u b l i c a t i o n ; A r g a n d was not m e n t i o n e d , p r e s u m a b l y because W a r r e n still had not heard of Argand's 28

C.

dues

V.

Mourey,

imaginaires

reprint Buee; have

of his

(Paris, 1828

book.

La 1828).

work.

vrai

Theorie

The

second

Nowhere

des

in

quantites

edition the

work

negatives

et

book

since

quantites

preten-

1861) was used; this was a

does

Mourey mention Argand or

i n fact n o m a t h e m a t i c i a n s are ever m e n t i o n e d i n the book. k n o w n Warren's

des

(Paris,

it was p u b l i s h e d after his o w n .

Mourey could not M o u r e y [ibid., I X )

m a d e the interesting c o m m e n t that his b o o k was an a b r i d g e m e n t of a longer treatise he had written but had not published. 29

16

Ibid.,

95.

CHAPTER

Sir

William

Rowan and

I.

Introduction: The

task

of the

Hamiltonian historian

w o r k of Sir W i l l i a m R o w a n estimates

of his

extreme

dinger

of Hamilton:

While

these

Hamilton Quaternions

Historiography

who

wishes

to

treat any

aspect of the

H a m i l t o n is c o m p l i c a t e d by the fact that

significance for the history of science have varied

between two wrote

TWO

positions.

discoveries

Thus,

(Quaternions,

for example,

etc.)

would

Erwin

suffice

to

Schro-

secure

H a m i l t o n in the annals of both mathematics a n d physics a h i g h l y hono u r a b l e place, s u c h p i o u s m e m o r i a l s can i n his case e a s i l y b e d i s p e n s e d with. For H a m i l t o n is virtually not dead, he h i m s e l f is alive, so to speak, not his m e m o r y . I daresay n o t a d a y passes — a n d s e l d o m an h o u r — w i t h out somebody,

somewhere

writing or printing

on

Hamilton's

this

globe,

name.

pronouncing or reading

or

T h a t is due to his f u n d a m e n t a l

d i s c o v e r i e s i n g e n e r a l d y n a m i c s . T h e H a m i l t o n i a n p r i n c i p l e has b e c o m e the cornerstone of m o d e r n physics, t h e t h i n g w i t h w h i c h a p h y s i c i s t exp e c t s every p h y s i c a l p h e n o m e n o n t o b e i n c o n f o r m i t y . . . . T h e modern development of physics is continually enhancing Hamilton's name. H i s famous analogy b e t w e e n mechanics a n d optics v i r t u a l l y anticipated wave-mechanics,

which

did

not

have to add m u c h to his

ideas, o n l y h a d to take t h e m seriously —a little m o r e seriously t h a n he was

able to take t h e m , w i t h the experimental k n o w l e d g e of a century

ago.

The

central conception of all

modern theory in physics

is

"the

Hamiltonian." If you wish to apply modern theory to any particular probl e m , y o u m u s t start w i t h p u t t i n g t h e p r o b l e m " i n H a m i l t o n i a n f o r m . " T h u s H a m i l t o n i s o n e o f t h e greatest m e n o f s c i e n c e t h e w o r l d has p r o duced.6 In

*

1945 J.

into eclipse.7 above

all

variations.

L.

Synge lamented that Hamilton's fame was passing

Synge cited m a n y aspects of this eclipse b u t stressed

the neglect of Hamilton's contribution to the calculus of He wrote:

" H a m i l t o n was, in fact, a great c o n t r i b u t o r —

probably the greatest single contributor of all t i m e —to the calculus of variations."

(7;

15)

17

A H i s t o r y of V e c t o r Analysis

In

1940 E. T. Whittaker p u b l i s h e d a paper entitled " T h e H a m i l -

tonian ton's

Revival,"

8

reputation

century: verse

in

which

was

he maintained:

touched

about

the

since w h e n , t h e r e has b e e n a steady m o v e m e n t in t h e re-

direction:

one

after

another,

the

i n n o v a t i o n s has b e e n a p p r e c i a t e d . . . ton in

" T h e nadir of Hamil-

beginning of the present

1954:

significance 9

."

of his

" A f t e r Isaac N e w t o n , t h e greatest m a t h e m a t i c i a n of the

E n g l i s h - s p e a k i n g p e o p l e s is W i l l i a m R o w a n H a m i l t o n . . . ." In

1937

titled

great

W h i t t a k e r w r o t e of H a m i l -

E.

T.

Bell

in

his

widely

the

chapter on

Hamilton

presented

Hamilton's

life

as

"An

read

Men

Irish

a tragedy,

o f Mathematics

Tragedy."

10 11

Herein

enBell

in a sense a m a g n i f i c e n t

failure. This

disparity o f v i e w s c o n c e r n i n g H a m i l t o n , w h i c h i n fact dates

back to the nineteenth century, is central to Hamiltonian historiography. T h e m a i n source of this disparity of v i e w s relates to H a m i l ton's

work

on

quaternions.

represented

the

voted

more

than

twenty

quaternions

held

by

Hamilton

mathematics

of the

years

nearly

of his

all

believed

future

and

life

to

that

them.

mathematicians

quaternions

consequently The

of the

de-

view

of

present is

h o w e v e r quite different; the consensus n o w is that the quaternion system

is b u t one of m a n y comparable mathematical systems, and

though value

it for

is

i n t e r e s t i n g as a rather special

application.

system, it offers

little

T h e historian of today must take the above

e v a l u a t i o n of q u a t e r n i o n s as m o s t p r o b a b l y v a l i d , t h o u g h t h e r e remain

sources

of doubt.

Statements qualifying or contradicting this

evaluation — m a d e by such important scientists as E. T. Whittaker,12 George D. Birkhoff,13 and P. A. M. Dirac

14

caution

large

in the

historian,

as

do

the

two

— instill some degree of volumes by Otto F.

Fischer,15 in w h i c h the author attempted to rewrite m u c h of m o d e r n physics

in terms of Hamilton's quaternions.

E. T. Bell's v i e w of H a m i l t o n as a tragic failure certainly s t e m m e d from

the

modern victim

fact that he

felt that

mathematics. of

a

Bell

monomaniacal

deepest tragedy was

quaternions

was

are of little interest to

convinced that

delusion;

he

Hamilton was the

stated

"that

Hamilton's

neither alcohol nor marriage but his obstinate

belief that quaternions h e l d the key to the mathematics of the physical

universe."

much taker

16

of what E. however

The

passed

contributions to

problem

of quaternions

T. Whittaker wrote over

the

mathematical

also stands b e h i n d

concerning Hamilton; Whit-

problem

by

stressing

Hamilton's

physics and by arguing that quater-

nions " m a y even yet prove to be the most natural expression of the n e w physics."

17

T h e present study m u s t stand in the shadow of this dispute con-

18

Sir W i l l i a m R o w a n H a m i l t o n

and Quaternions

cerning Hamilton's greatness; nevertheless it is h o p e d that substantial

progress

toward

following analysis, not possible

to

a

solution

which

argue

may

be

achieved

in

terms

of the

w i l l be d e v e l o p e d m o r e f u l l y later.

that the

quaternion

system

is

the

It is

vectorial

system of the present day; the so-called Gibbs-Heaviside system is the

only system that merits

this distinction.

N o r is it legitimate to

a r g u e (as W h i t t a k e r h a s d o n e ) t h a t t h e q u a t e r n i o n s y s t e m w i l l b e t h e system of a future day.

B o t h of these alternatives are unacceptable;

nonetheless

argued

it

can

be

(though

previously been done) that Hamilton's historically

determinable

path to the

hence to the m o d e r n system.

in

my opinion

this

has

not

quaternion system led by an Gibbs-Heaviside

system

and

In what follows it w i l l be s h o w n that

this was in fact t h e case, a n d thus it w i l l b e c o m e clear that H a m i l ton deserves i m m e n s e credit for his w o r k in quaternions, since this work

led

reasons

to

the

now

w h y this

is

widely

so

used

system

little k n o w n

will

of vector analysis.

also be

discussed.

The

If this

analysis is f o u n d acceptable, it s h o u l d clear up the major p r o b l e m in Hamiltonian

historiography.

II.

s

Hamilton Though

in

Life

and

Fame

general a detailed discussion of a scientist's life n e e d

not be i n c l u d e d in a study such in

regard to

the

fame

Hamilton

attained

and

by

Hamilton

fluenced subsequent events. Hamilton's the

title

The the

during

Some

his

lifetime

indication

of the

strongly

in-

importance of

fame in this history may be attained by a comparison of

pages

title

as this, it is of necessity otherwise

quaternions. T h e reason for this is that

of Hamilton's

page

of

and

Grassmann's

Grassmann's Ausdehnungslehre

first of

major

1844

works.

contained

following: Hermann

Grassmann

Lehrer an der Friedrich-Wilhelms-Schule zu Stettin By

contrast,

the

title

page

of

Hamilton's

Lectures

on

Quaternions

contained: SIR W I L L I A M R O W A N H A M I L T O N , L L . D . , M . R . I . A., F E L L O W OF THE AMERICAN

SOCIETY O F ARTS A N D SCIENCES; O F T H E

SOCIETY

FOR

OF

ARTS

SCOTLAND;

NOMICAL SOCIETY OF LONDON; ERN

SOCIETY

SPONDING

OF

ANTIQUARIES

MEMBER

OF

T H E

HONORARY OR CORRESPONDING OR

ROYAL ACADEMIES

TURIN; LIN;

OF

OF THE THE

OF

ST.

OF

T H E

AND OF T H E AT

ROYAL

ASTRO-

ROYAL NORTH-

COPENHAGEN;

INSTITUTE

OF

CORREFRANCE;

M E M B E R OF T H E IMPERIAL

PETERSBURGH,

BERLIN, AND

ROYAL SOCIETIES OF EDINBURGH A N D DUBCAMBRIDGE

PHILOSOPHICAL

SOCIETY;

T H E

19

A

History

of V e c t o r Analysis

N E W YORK HISTORICAL SOCIETY; T H E SCIENCES CIETIES

AT

IN

LAUSANNE;

BRITISH

A N D

AND

OF

FOREIGN

PROFESSOR OF ASTRONOMY IN T H E AND

ROYAL

William

ASTRONOMER

Rowan

Hamilton

OF

was

was

o r p h a n e d at age

SCIENTIFIC

COUNTRIES;

SO-

ANDREWS'

UNIVERSITY OF DUBLIN;

IRELAND.

born of undistinguished ancestry

on the midnight between August 3 and 4, He

SOCIETY OF NATURAL

OTHER

fourteen,

1805, in D u b l i n , Ireland.

b u t h a d ceased to live w i t h his

parents f r o m the age of three, at w h i c h t i m e he h a d b e e n sent to live with

his

Trim,

uncle,

Ireland.

James

Hamilton,

Hamilton's

uncle,

an a

Anglican

man

clergyman

of education

serving

and

intelli-

gence, d i r e c t e d his n e p h e w ' s p r e u n i v e r s i t y education. T h e success of the uncle as tutor and the brilliance of Hamilton as student were manifested in thirteen, teen

many ways,

Hamilton

"was

languages. . . ."

brew,

Syriac,

French,

Italian,

however

only

18

of w h i c h

in

These

Persian,

the

best k n o w n

is that at age

different degrees acquainted w i t h thirlanguages

Arabic,

were

Sanskrit,

Greek,

Latin,

Hindoostanee,

He-

Malay,

Spanish, and German. T h e study of languages was one

of

Hamilton's

interests,

for

he

also

read

in

geography, religion, mathematics, astronomy, and the best of English

and

foreign

Mecanique but

not

was

in

forces. In

Celeste

literature.

and

significant Laplace's (2,1;

1823

At

detected for this

age

an

sixteen

error

study that the

demonstration

he

began

therein.

It

is

Laplace's interesting

error found by Hamilton

of the law of the parallelogram of

661-662)

Hamilton

entered Trinity College of D u b l i n University.

He h a d placed first in the entrance exam a n d h a d decided that his calling was to science. incredible.

In

knowledge

of Greek,

knowledge optime

was

"became vitations, .

His record at the University bordered on the

second

rare.

celebrity

in

Upon in

the

embarrassing

him.

.

year

the

This

wish

year

Hamilton

was

third

physics.

an

optime

for

another

optime

for his

his

T h e w i n n i n g of even a single

winning

the

intellectual their

awarded year

second

circle

optime,

Hamilton

of Dublin;

number,

poured

and in

in-

upon

. " (2,1; 2 0 9 ) H a m i l t o n r e s o l v e d t o a t t e m p t t o w i n i n h i s f i n a l University was

mer of Ireland. creative

honors

he

his

from

Gold

for

Medals

not fulfilled, was

offered

fessor of A s t r o n o m y at the

in

year

and

of mathematical very

a

his

for the

in

both

classics

during the

and in

science.

s u m m e r after his third

honor of becoming Andrews'

Pro-

University of D u b l i n and Royal Astrono-

H i s s t u d e n t days w e r e also d i s t i n g u i s h e d b y success

endeavors. some

of

He

them.

wrote

numerous

Researches

in

poems science

and

received

begun

in

his

seventeenth year on certain questions in mathematical optics led to

20

Sir W i l l i a m R o w a n H a m i l t o n

his

now famous

1824

and

"Theory

published

of Systems

with

Other important papers

of Rays,"

which

further developments

in

the

same

line

and Quaternions

was

read

four years

in

later.19

of development came in

1830,

1831, a n d 1837. H i s a i m in these papers ( w h i c h e x t e n d to over

three

hundred

science his

in

pages)

terms

mathematical

methods

for

was

of his

methods

use

in

to

reduce

optics

"Characteristic in

optics

dynamics.

to

a

Function."

led

The

Hamilton

mathematical

The

success

of

to extend these

distinguished

historian

of

m e c h a n i c s R e n e D u g a s has s u m m a r i z e d the nature a n d i m p o r t a n c e of Hamilton's

work in

I n short, jealous

optics

and

dynamics:

of the formal perfection w h i c h Lagrange had been

able to give to dynamics, and w h i c h optics lacked, H a m i l t o n undertook the rationalisation of geometrical optics.

He

d i d this by d e v e l o p i n g a

formal theory w h i c h was free of all metaphysics a n d w h i c h , moreover, s u c c e e d e d in a c c o u n t i n g for a l l t h e e x p e r i m e n t a l facts. . . . Then,

returning to

dynamics,

Hamilton presented the

l a w o f varying

action i n a f o r m v e r y l i k e t h a t w h i c h h e h a d d i s c o v e r e d i n o p t i c s . T h u s h e r e d u c e d t h e g e n e r a l p r o b l e m o f d y n a m i c s (for c o n s e r v a t i v e systems) to the solution of t w o simultaneous equations in partial derivatives, or to the determination of a single function satisfying these t w o equations.

Hamilton's

g u i d i n g idea is continuous f r o m his optical w o r k to his

w o r k in d y n a m i c s — i n this fact lies his greatness a n d his p o w e r .

Here

was a synthesis that L o u i s de Broglie was to rediscover a n d t u r n to his o w n account; a synthesis that was, it appears, to be Schrodinger's direct inspiration.20 These

works

certainly

contributed

to

was probably thinking of them w h e n in

Hamilton's

fame;

Jacobi

1842 he referred to H a m i l -

t o n a s " l e L a g r a n g e d e v o t r e p a y s . " (2,111; 5 0 9 ) O f t e n h o w e v e r s u c h highly

mathematical

otherwise

for

basis

new

two

refraction. colleague In

this

At

phenomena

of Hamilton,

he

was and

wrote

siderable

vigorous

and

"perhaps made.

.

the .

most

popular fame.

predicted

on

It was

a theoretical

Humphrey

men

Whewell's

and

Airy

remarkable

friend

in

prediction

predicted (2,1;

635)

a very con-

England and on the

praise

referred

both

time

and

prediction.

predicted.

"excited at the

scientific

636)

finding

had not been

discovery

Lloyd,

to verify Hamilton's

successful,

which

immediate,

he

in optics, internal and external conical

among

(2,1;

21

request,

attempted

third

sensation

not produce

1832

completely a

that this

Continent. . . ."

do

in

Hamilton's

phenomena Graves

papers

Hamilton;

for

to

Hamilton

the

that

was

discovery

has

ever

as

been

. " (2,1; 6 3 7 ) D e M o r g a n w r i t i n g i n 1 8 6 6 s t a t e d : " O p t i c i a n s

had no more i m a g i n e d the possibility of such a thing, than astronomers

had

imagined

the

planet

Neptune,

which

Leverrier

and

21

A

History

of V e c t o r Analysis

A d a m s calculated into existence. t o g e t h e r as,

perhaps,

predictions." No

2 2

the

Pliicker of Bonn

experiment

These two things deserve to rank

t w o most remarkable of verified scientific

i n p h y s i c s has

wrote:

made

such a strong impression on my

m i n d as that of conical refraction. A single ray of light e n t e r i n g a crystal and leaving as a l u m i n o u s cone: this is something unheard of and without

analogy.

wave

Mr.

H a m i l t o n p r e d i c t e d it, starting f r o m the f o r m of the

w h i c h h a d b e e n d e d u c e d b y a l o n g calculation f r o m a n abstract

theory. I confess I w o u l d have h a d little hope of seeing an experimental confirmation theory

of such

which

an

extraordinary

Fresnel's

genius

had

result,

recently

predicted by the created.

But

mere

since

Mr.

L l o y d h a d d e m o n s t r a t e d that the e x p e r i m e n t a l results w e r e i n c o m p l e t e accordance w i t h the predictions of M r . H a m i l t o n , all prejudice against a t h e o r y s o m a r v e l o u s l y l o f t y has b e e n f o r c e d t o d i s a p p e a r . (2,1; 6 3 7 ) T h e fame that came to

Hamilton because

of this

discovery was in-

creased by the fact that it w a s m a d e , l i k e nearly all the discoveries d i s c u s s e d t h u s far, b e f o r e Also was

his

literary

close figures

friendship such as

ridge.

Numerous

worth,

and

had up

H a m i l t o n h a d r e a c h e d his t h i r t i e t h year.

illustrative of, a n d c o n t r i b u t o r y to, H a m i l t o n ' s p o p u l a r fame with

William

Wordsworth

and

other

Maria Edgeworth and Samuel Taylor Cole-

letters

passed

between

Hamilton

and

Words-

each visited the other on m a n y occasions. W o r d s w o r t h

said that H a m i l t o n was one of t w o m e n to w h o m he could look (the

other was Coleridge). To this H a m i l t o n replied:

" I f I am to

look d o w n on you, it is only as L o r d Rosse looks d o w n in his teles c o p e to see t h e stars of h e a v e n r e f l e c t e d . " By

1835

Hamilton's

fame

was

(2,111; 2 3 7 )

established.

In

that year he

was

knighted and received a medal from the Royal Society; in addition he

finished

ton's In

a paper on algebraic couples, w h i c h is the first of Hamil-

publications to be of direct importance for the present study.

1837

h e l d this covery

he

was

elected president of the Royal

position

(1843)

until

1843 to

ment

of quaternions. of

resignation

of quaternions.

from

Honors

his

The

received

Sciences along

Irish Academy and

1845, soon after his dis-

last t w e n t y - t w o years

of his life,

1865, w e r e for the most part devoted to the develop-

all

sorts

continued

these deserves final mention. In ton

in

notice

that the

to

be

bestowed

on

him.

One

of

1865, the year of his death, H a m i l -

newly

founded

National

Academy of

of t h e U n i t e d States h a d e l e c t e d h i m a F o r e i g n Associate,

with

Hamilton's

fourteen

other men.

The

name at the head of the

members

had voted to

place

list of F o r e i g n Associates, pre-

s u m a b l y signifying that in their o p i n i o n he was the greatest living scientist.

In

this

j u d g m e n t does

22

they were

attest to the

probably overly enthusiastic, but their fact,

which

is

very significant for this

Sir W i l l i a m

study,

that

great.

Hamilton's

fame

Rowan

among

Hamilton

his

and Quaternions

contemporaries

had b e e n completed, b u t he was still nearly u n k n o w n . in

was

very

At this same time the majority of Grassmann's scientific w o r k

his

1862

Ausdehnungslehre

now "Professor am

III.

Hamilton

It was

had

Gymnasium

and

Complex

changed zu

only

His subtitle

slightly;

it

was

Stettin."

Numbers

stated previously that H a m i l t o n ' s

was in the tradition of the work done on

discovery of quaternions complex numbers, and in

this regard the history of the geometrical representation of c o m p l e x numbers

and associated ideas

was

given.

But there

was

a second

line of d e v e l o p m e n t in studies on c o m p l e x n u m b e r s that also l e d to quaternions. This line of development was established by H a m i l t o n himself in his titled:

l o n g a n d i m p o r t a n t essay p u b l i s h e d in

1837 and en-

"Theory of Conjugate Functions, or Algebraic Couples; with

a Preliminary Pure Time."

and Elementary Essay on Algebra as the This

paper is

important

in

itself;

Science of

indeed one mathe-

matician referred to it as a greater c o n t r i b u t i o n to algebra than his of quaternions.23

discovery sections:

Hamilton's

paper is divided into three

the first section, w h i c h consists of " G e n e r a l Introductory

Remarks,"

was

written

last;

the

second

section,

an

essay

" O n

Algebra as the Science of Pure T i m e " was w r i t t e n in 1835; a n d the third or

section,

Algebraic

(2,11;

containing Couples,"

his

was

"Theory for

the

of

most

Conjugate part

Functions,

written

in

1833.

144) N e g l e c t of t h e historical s e q u e n c e of t h e c o m p o s i t i o n has

led to

a n u m b e r of historical

Hamilton

began

the

misconceptions.

paper by

writing:

T h e study of Algebra may be p u r s u e d in three very different schools, the Practical, the Philological, or the Theoretical, according as A l g e b r a itself is a c c o u n t e d an I n s t r u m e n t , or a L a n g u a g e , or a C o n t e m p l a t i o n ; according

as

ease

thought,

(the

of operation, agere,

or

the f a r i ,

s y m m e t r y of expression, or

the

sapere,)

is

or clearness

eminently

prized

of

and

sought for. T h e Practical p e r s o n seeks a R u l e w h i c h he m a y a p p l y , t h e P h i l o l o g i c a l person seeks a F o r m u l a w h i c h he m a y w r i t e , the T h e o r e t i c a l p e r s o n seeks a T h e o r e m o n w h i c h h e m a y m e d i t a t e . (3; 2 9 3 ) He then proceeded to

state that t h e a i m o f this p a p e r w a s theoreti-

cal. The

thing

aimed

at,

is

to

improve

the

Science,

not

the

Art

nor

the

L a n g u a g e o f A l g e b r a . T h e i m p e r f e c t i o n s s o u g h t t o b e r e m o v e d , are confusions of t h o u g h t , a n d obscurities or errors of reasoning; riot difficulties of application of an instrument nor failures of symmetry in expression.... F o r i t has n o t f a r e d w i t h t h e p r i n c i p l e s o f A l g e b r a a s w i t h t h e p r i n ciples

of Geometry.

No

candid

and

intelligent

person can

doubt the

23

A H i s t o r y of V e c t o r Analysis truth

of the

chief properties

o f Parallel

Lines,

as

set f o r t h

by E U C L I D

i n his E l e m e n t s , t w o t h o u s a n d years ago; t h o u g h h e m a y w e l l desire t o see t h e m t r e a t e d i n a c l e a r e r a n d b e t t e r m e t h o d . T h e d o c t r i n e i n v o l v e s no

obscurity

nor

confusion

of

thought,

and

leaves

in

the

mind

no

reasonable g r o u n d for d o u b t , a l t h o u g h i n g e n u i t y m a y usefully be exercised in improving the plan of the argument. But it requires no peculiar scepticism to doubt, or even to disbelieve, the doctrine of Negatives and I m a g i n a r i e s , w h e n s e t f o r t h (as i t h a s c o m m o n l y b e e n ) w i t h p r i n c i p l e s like

these:

that

the

numbers plied t h e

that

a

remainder denoting one

greater is

magnitude

less

than

magnitudes

by the

other,

may

be

nothing;

each

less

subtracted from that

two

than

a

negative

nothing,

less,

and

numbers, o r

may be

multi-

a n d t h a t t h e p r o d u c t w i l l b e a positive n u m -

ber, or a n u m b e r d e n o t i n g a m a g n i t u d e greater than nothing; and that a l t h o u g h t h e square o f a n u m b e r , o r t h e p r o d u c t o b t a i n e d b y m u l t i p l y i n g that be

number

by

itself,

is

therefore

always positive,

whether the

number

p o s i t i v e o r n e g a t i v e , y e t t h a t n u m b e r s , c a l l e d imaginary, c a n b e f o u n d

or conceived or d e t e r m i n e d , a n d operated on by all the rules of positive and

negative

they

have

selves

numbers,

negative

squares,

as

i f t h e y w e r e s u b j e c t t o t h o s e r u l e s , although

and

must

therefore

be

s u p p o s e d to

be t h e m -

neither positive or negative, nor yet n u l l numbers, so that the

m a g n i t u d e s w h i c h t h e y are s u p p o s e d t o d e n o t e can n e i t h e r b e greater t h a n n o t h i n g , n o r less t h a n n o t h i n g , n o r e v e n e q u a l t o n o t h i n g . I t m u s t b e h a r d t o f o u n d a S C I E N C E o n s u c h g r o u n d s a s t h e s e . . . . (3; 294)

Hamilton

then

asked

w h e t h e r e x i s t i n g A l g e b r a , i n t h e state t o w h i c h i t has b e e n a l r e a d y u n f o l d e d b y t h e m a s t e r s o f its r u l e s a n d o f its l a n g u a g e , offers i n d e e d n o rudiment which Algebra:

a

may encourage

Science

a hope

properly so called;

of developing a S C I E N C E of strict, pure, a n d i n d e p e n d e n t ;

d e d u c e d b y v a l i d r e a s o n i n g s f r o m its o w n i n t u i t i v e p r i n c i p l e s ; a n d t h u s n o t less a n o b j e c t o f p r i o r i c o n t e m p l a t i o n t h a n G e o m e t r y , n o r less d i s t i n c t , i n its o w n e s s e n c e , f r o m t h e R u l e s w h i c h i t m a y t e a c h o r use, a n d f r o m t h e S i g n s b y w h i c h i t m a y express its m e a n i n g . (3; 2 9 5 ) Hamilton

concluded

ment"

295) a n d elaborated on this idea by writing:

(3;

The folded is

argument into

an

possible,

for

"that

the

chiefly

Intuition

conclusion

independent

rests

the

Pure on

that

the

or

that

Science,

the

of T I M E

existence

notion a

is

such

o f time

Science

of certain

a rudi-

may

be

of Pure

priori

unTime

intuitions,

c o n n e c t e d w i t h that n o t i o n of t i m e , a n d fitted to b e c o m e the sources of a pure Science; and on the actual deduction of such a Science from those p r i n c i p l e s , w h i c h t h e a u t h o r c o n c e i v e s t h a t h e has b e g u n . (3; 2 9 6 - 2 9 7 ) In velop

the

second

the

real

section

of this

number system

concept of time.

paper

on

the

Hamilton attempted to basis

of the

intuition

de-

of the

In this w a y he b e l i e v e d he c o u l d justify the use of

negative n u m b e r s as c o r r e s p o n d i n g to steps in time. It

is

derived

24

generally from

Kant.

believed Such

that

Hamilton's

stress

on

m a y n o t b e t h e case, for Kant's

time

was

name is

Sir W i l l i a m R o w a n

never

mentioned

these

ideas

he

did

Pure

in

mention

Reason

the

paper.

Preface

Kant

(4;

In

to

and

"encouraged

v i e w . . . ." after

in

the

Hamilton's

his

Lectures

wrote

[him]

that

to

[2]) As early as

mentioning geometry:

nected made

became

with by

other."

Hamilton

reading

K a n t four

in

years

Hamilton wrote:

later

exposition

Quaternions

reading

(4;

Kant's

and

of

[2]-[3]) Critique

publish

of this

1827 H a m i l t o n wrote, i m m e d i a t e l y

"The

intimately

each

on

entertain

sciences

adopt here a v i e w of Algebra w h i c h propose)

Hamilton and Quaternions

I

of Space

have

intertwined

(2,1;

229)

letters

it

From

seems

after m a k i n g t h e

and Time

(to

elsewhere v e n t u r e d to and

indissolubly

con-

a n u m b e r of statements

quite

clear that

he

statement.24

above

began

In

1835

"and my o w n convictions, mathematical and meta-

physical, have b e e n so long and so strongly converging to this point (confirmed I

cannot

no

doubt

easily

of late by the

yield

to

the

stare at my strange t h e o r y . "

study of Kant's

authority (2,11;

142)

of those It thus

Pure Reason),

other

friends

seems

that who

that at most

K a n t served as a catalyst for the d e v e l o p m e n t of his ideas a n d as a confirmation

of them.

In the third part of the Conjugate

of his essay is part

is

essay H a m i l t o n p r e s e n t e d his " T h e o r y o f

Functions, or Algebraic Couples." W h i l e the second part generally considered of minor importance, the third

universally

admitted

to

be

of great importance,

for herein

H a m i l t o n d e v e l o p e d c o m p l e x n u m b e r s in terms of ordered pairs of real numbers in almost exactly the same w a y as it is done in m o d e r n mathematics.25 T h e stress

in this section was not on time, although

the interpretation of the couples in terms of time was given. Hamilton

at no point in the paper mentioned Warren or the geometrical

interpretation of complex n u m b e r s ; f r o m this it seems probable that H a m i l t o n believed (like Gauss) that the geometrical representation was

an aid to

intuition,

b u t not a satisfactory justification for com-

plex numbers. Essentially w h a t H a m i l t o n d i d in this section was to set u p o r d e r e d pairs o f r e a l n u m b e r s (a, b ) a n d d e f i n e o p e r a t i o n s o n them.

T h e s e operations w e r e all

numbers.

He

equivalent to In the and

then

complex

THEORY

denotes

showed

an

OF

done

that the

numbers SINGLE

in terms of the rules for real

couples thus considered were

o f t h e f o r m a - h bi.

N U M B E R S , the

IMPOSSIBLE

He

symbol

EXTRACTION,

or a

wrote:

V ^ T is

merely

absurd,

IMAGI-

N A R Y N U M B E R ; but i n the T H E O R Y O F C O U P L E S , the same symbol V - l REAL

is

significant, COUPLE,

square-root

o f the

and

denotes

namely

couple

(—1,

(as 0).

a we In

POSSIBLE have the

just

EXTRACTION, now

latter theory,

seen)

the

or

a

principal

therefore, though

not in the former, this sign V — l may properly be e m p l o y e d ; a n d we m a y write,

if

a, + a2V-l

we

choose,

for

any

couple

(au

a2)

whatever,

(a,,

a2)

=

(3; 4 1 7 - 4 1 8 )

25

A

History

Hamilton

of V e c t o r Analysis

concluded

the

essay b y

writing:

t h e p r e s e n t Theory o f Couples i s p u b l i s h e d . . .

to s h o w .

.

.

that expres-

sions w h i c h seem a c c o r d i n g to c o m m o n v i e w s to be m e r e l y symbolical, and quite

incapable

of being interpreted,

m a y pass

into the w o r l d of

thoughts, and acquire reality and significance, if Algebra be v i e w e d as not

a

mere

Art or

Language,

but as

the

Science

of Pure T i m e .

The

author hopes to p u b l i s h hereafter m a n y other applications of this v i e w ; especially to Equations and Integrals, and to a T h e o r y of Triplets and Sets

of Moments,

Steps, a n d N u m b e r s , w h i c h includes this T h e o r y of

C o u p l e s . (3; 4 2 2 ) The

" T h e o r y of Triplets" that he sought was of course the extension

of the

complex

It is and

number system

clear from

importance

this

to three

paper that

of the

dimensions.

Hamilton

associative,

understood the

commutative,

nature

and distributive

laws.26 T h e majority of mathematicians appreciated the significance of these

laws

had been

only

after

number

developed which

Hamilton's

"Theory"

systems

(especially

quaternions)

did not obey them.

was

poorly received.

Most mathematicians

d i d n o t agree w i t h H a m i l t o n ' s stress o n t i m e , a n d a f e w felt t h e n e e d for

the

d e v e l o p m e n t of c o m p l e x

geometrical

one.

That

Gauss

numbers

and

justification of complex numbers

Bolyai

on

a basis

other than

rejected the

a

geometrical

is almost certainly d u e to the fact

that they both had previously discovered non-Euclidean geometry. W h e n n o n - E u c l i d e a n g e o m e t r y b e c a m e k n o w n (after 1860), mathematicians

then

numbers

became

interested

in

the

in terms of ordered pairs of real

Hamilton's

"Essay"

was

quaternions

numbers.

an

important event in the history of the

for

a

discovery

of

Hamilton

on

direction,

in addition to the

the

development of complex

number

of

quest for higher complex quest in terms

reasons. numbers

First, from

it

set

another

of a m e t h o d of analysis

for t h r e e - d i m e n s i o n a l space. Second, t h r o u g h his m e t h o d of couples at

least

Hamilton

himself became

convinced of the

legitimacy

of

c o m p l e x n u m b e r s , a n d m o r e i m p o r t a n t l y he also o b t a i n e d a m e t h o d that c o u l d be e x t e n d e d in such a w a y as to assure the legitimacy of higher

complex

numbers,

formed

for example by triplets or quad-

r u p l e t s (as i n t h e c a s e o f q u a t e r n i o n s ) . T o p u t i t a n o t h e r w a y , b y t h i s method cantly

H a m i l t o n was prepared, perhaps to discover, m o r e signifi-

to

accept

as

legitimate,

"four-dimensional"

complex

num-

b e r s (as q u a t e r n i o n s ) , e v e n i f n o g e o m e t r i c a l j u s t i f i c a t i o n w e r e t o b e available. from

Support for the above analysis is f o u n d in a letter of 1841

Hamilton to

As

to T r i p l e t s ,

De I

Morgan:

m u s t a c k n o w l e d g e , that t h o u g h I fancied m y s e l f at

one time to be in possession of something w o r t h publishing about them,

26

Sir W i l l i a m R o w a n

Hamilton

and Quaternions

I never could resolve the p r o b l e m w h i c h you have justly signalised as t h e m o s t i m p o r t a n t i n t h i s b r a n c h o f ( f u t u r e ) A l g e b r a : t o assign t w o s y m b o l s O a n d &), s u c h t h a t t h e o n e s y m b o l i c a l e q u a t i o n a + M l + co) = a , +

+ c,&>

shall give the three equations b = bu c = c,

a = au

B u t , i f m y v i e w o f A l g e b r a b e j u s t , i t must b e p o s s i b l e , i n some w a y o r o t h e r , t o i n t r o d u c e n o t o n l y t r i p l e t s b u t polyplets, s o a s i n s o m e s e n s e t o satisfy the s y m b o l i c a l e q u a t i o n a= (a,, a2, . .

.

an);

a being here one symbol, as indicative of one (complex) thought; and f l j , a2, . . . a n d e n o t i n g n r e a l n u m b e r s , p o s i t i v e o r n e g a t i v e ; t h a t i s , i n o t h e r w o r d s , n dates, in t h e c h r o n o l o g i c a l sense of t h e w o r d , o n l y excluding

outward

marks

and

measures,

and

the

notion

of cause

and

e f f e c t . (2,11; 3 4 3 ) Moreover,

after

1843

point of v i e w for his he made this perusal

Hamilton

discovery one

of my

old

stressed

the

importance

discovery of quaternions.

essay,

day w h e n

I

He

of

this

said in fact that

" b e i n g t h e n fresh f r o m a re-

renewed

my

attempts

to

combine

my

g e n e r a l n o t i o n o f sets o f n u m b e r s , c o n s i d e r e d a s s u g g e s t e d b y sets of moments

of time,

with

geometrical

considerations

of points and

lines in t r i d i m e n s i o n a l space. . . . " 2 7

IV. At

Hamilton

s

end

of his

the

Discovery "Essay"

seeking a triplet system. find triplets

as

of

early as

of 1837

not alone

that John system

Hamilton from

at

which

T.

for

in his

1830.

(4;

and

least was

Graves 1836,

similar

(4;

[36]-[37])

De

Morgan,

had

"as

In in

tried

to or

been

1841 which

one

form

in

a

higher

he

was

Morgan

had

received asked

Hamilton

o w n statement

than

correspondence sent

of that year

That

complex

earlier

Graves

Hamilton

Hamilton De

conjecture

property.

perhaps

at w h i c h time to

stated that

shown by Hamilton's

early, had

[39]) T h e

distributive

quest is

Graves space

Hamilton

He h a d , in fact, m a d e d e f i n i t e a t t e m p t s to

entailed abandonment of the was

Quaternions

letter

Hamilton

2 8

on the subject

Hamilton

constructed a

number

myself."

from

a system in

1835.

Augustus

about his

trip-

lets. W i t h this letter w a s a c o p y o f D e M o r g a n ' s 1 8 4 1 p a p e r " O n t h e Foundation

of Algebra,"

2 9

brief discussion of triplets. When lets, t h e

by

1843

Hamilton

framework

within

in (4;

which

De

Morgan

had

included

a

[41]—[42])

began

another intense

which

the

search

search

for trip-

had to be conducted

27

A H i s t o r y of V e c t o r Analysis

was clear to him. T h e following m a y be taken as an outline of the properties that he consciously h o p e d the n e w n u m b e r s w o u l d have. 1. T h e associative property for addition and multiplication. T h u s if

N,

N\

and

(N

+

N')

+

2. T h e N + 3.

N'

N" N"

are

three

and

commutative

=

N'

The

+

N

and

distributive

4. T h e

such

N(N'N")

=

property NN'

=

numbers,

then

N + ( N ' + N") =

and

multiplication.

(NN')N". for

addition

N(N'

+

N'N.

property.

N")

=

NN'

+

NN".

p r o p e r t y that division is u n a m b i g u o u s . T h u s if N a n d N'

are any g i v e n c o m p l e x n u m b e r s , it is always possible to a n d only one n u m b e r X N

and

N')

such

that

(in

general,

NX =

a n u m b e r of t h e

find

one

same f o r m as

N'.

5. T h e property that the n e w n u m b e r s obey the law of the moduli. T h u s if any three triplets c o m b i n e so that {ax

bxi +

+

cj)(a2 +

b2i +

c2j) =

a3 +

b3i +

b22

c 22) =

(a32 +

fo32

c3j,

then (a,2 6. T h e

+

b2

+

c2){a2

+

+

+

c32):

property that the n e w numbers w o u l d have a significant

i n t e r p r e t a t i o n i n t e r m s o f t h r e e d i m e n s i o n a l space. It is well

k n o w n that ordinary c o m p l e x n u m b e r s have all these

properties, w i t h the exception that their geometrical interpretation is for t w o - d i m e n s i o n a l space. In one sense, then, the above is simply

a

detailed

which

statement that

would be

Hamilton

sought for n e w

numbers

directly analogous to ordinary complex numbers.

Of the above properties o n l y the c o m m u t a t i v e property for multiplication h a d to be a b a n d o n e d for quaternions. W i t h limits as restrict i v e a s these H a m i l t o n c o u l d o n l y b e satisfied w i t h quaternions, for, as C. S. Peirce p r o v e d in 1881, " o r d i n a r y real algebra, o r d i n a r y algeb r a w i t h i m a g i n a r i e s , a n d real q u a t e r n i o n s are t h e o n l y associative algebras in w h i c h division by finites always yields an unambiguous quotient." erties

are

30

It is s i g n i f i c a n t to ask at this p o i n t w h i c h of t h e s e p r o p -

retained

for the

scalar (dot)

a n d v e c t o r (cross)

m u l t i p l i c a t i o n s in m o d e r n vector analysis.

product

For the dot product the

associative l a w for multiplication is not relevant, and both the law of the m o d u l i and the unambiguity of division must be abandoned.31 For the must be

cross

product the

associative

and commutative properties

abandoned.32 Moreover division is not unambiguous, and

the l a w of the m o d u l i fails as well.33 F r o m the above comparison of the properties of quaternions and vectors it is e v i d e n t that at least in some ways quaternions not o n l y are s i m p l e r t h a n m o d e r n vectors b u t also entail f e w e r innovations.

28

Sir W i l l i a m R o w a n

In the

period

immediately

after

their

Hamilton

discovery

and Quaternions

quaternions

were

criticized because of the abandonment of the commutative property for m u l t i p l i c a t i o n .

It is an interesting historical speculation in this

regard as to what w o u l d have commutative

and

associative

sion was in general

been

said of a system in w h i c h the

properties

failed,

ent types of multiplication were defined. context of the above attempts to

and

in w h i c h divi-

impossible, and in w h i c h m o r e o v e r t w o differIn any case it was

properties that H a m i l t o n in

in the

1843 r e n e w e d his

triplets.34

find

O n O c t o b e r 16,

1843, H a m i l t o n discovered quaternions.

the best description

Perhaps

of the circumstances surrounding this event is

contained in a letter H a m i l t o n wrote in 1865 to his son A r c h i b a l d H. Hamilton: I f I m a y b e a l l o w e d t o s p e a k o f myself i n c o n n e x i o n w i t h t h e s u b j e c t , I m i g h t d o s o i n a w a y w h i c h w o u l d b r i n g you i n , b y r e f e r r i n g t o a n antequaternionic t i m e ,

when

the

of a V e c t o r ,

conception

you

were as

a

m e r e child, b u t h a d c a u g h t f r o m m e

r e p r e s e n t e d b y a Triplet:

and indeed I

happen to be able to p u t the finger of m e m o r y u p o n the year a n d m o n t h — October, 1843 — w h e n h a v i n g r e c e n t l y r e t u r n e d f r o m visits to C o r k a n d Parsonstown, connected w i t h a M e e t i n g of the British Association, the desire to discover the

laws

of the

multiplication referred to regained

w i t h me a certain strength a n d earnestness, w h i c h h a d for years b e e n dormant, b u t was t h e n on the p o i n t of b e i n g gratified, a n d was occasionally talked of w i t h you. E v e r y m o r n i n g in the early part of the abovecited month, on my c o m i n g d o w n to breakfast, your (then) little brother W i l l i a m E d w i n , a n d y o u r s e l f , u s e d t o a s k m e , " W e l l , P a p a , c a n y o u multiply t r i p l e t s " ? W h e r e t o I of the head:

was always o b l i g e d to r e p l y , w i t h a sad shake

" N o , I c a n o n l y add a n d s u b t r a c t t h e m . "

B u t on the 16th day of the same m o n t h — w h i c h h a p p e n e d to be a M o n day, and a C o u n c i l day of the Royal Irish A c a d e m y — I was w a l k i n g in to attend and preside, a n d your m o t h e r was w a l k i n g w i t h me, along the Royal Canal, to w h i c h she h a d perhaps d r i v e n ; a n d a l t h o u g h she t a l k e d with

me

now

and then,

my m i n d , w h i c h that

I

and a

felt

at

spark

l o n g years

to

gave

once

the

flashed come

yet

a n under-current o f t h o u g h t w a s

going on in

a t l a s t a result, w h e r e o f i t i s n o t t o o m u c h t o s a y importance. forth,

the

An

herald

electric (as

circuit

seemed

to

close;

I foresaw, immediately) o f m a n y

o f d e f i n i t e l y d i r e c t e d t h o u g h t a n d w o r k , b y myself i f

s p a r e d , a n d a t a l l e v e n t s o n t h e p a r t o f others, i f I s h o u l d e v e n b e a l l o w e d to live long enough distinctly to communicate the discovery. Nor could I resist t h e i m p u l s e — u n p h i l o s o p h i c a l as it m a y h a v e b e e n —to cut w i t h a knife on a stone of B r o u g h a m B r i d g e , as we passed it, the f u n d a m e n t a l formula w i t h the symbols, i, j, k; namely i2 = j2 = which

contains

t i o n , has

the

k1 =

ijk = —1,

Solution o f t h e Problem, b u t o f c o u r s e , a s

an

inscrip-

long since m o u l d e r e d away. A more durable notice remains,

however, on the

Council

Books of the A c a d e m y for that day (October

16th, 1843), w h i c h records t h e fact, that I t h e n a s k e d for a n d o b t a i n e d

29

A

History

leave

of V e c t o r Analysis

to

Session:

read

a

Paper

on

Quaternions,

at t h e

First

General

Meeting o f t h e

w h i c h reading took place accordingly, on M o n d a y the 13th of

t h e N o v e m b e r f o l l o w i n g . (2,11; 4 3 4 - 4 3 5 ) Thus

in

a

very

nounced

the

numbers

of t h e

dramatic

discovery form

of

manner

Hamilton

quaternions.

w + ix + jy +

kz,

discovered

These

are

and

an-

hypercomplex

w h e r e w, x, y, a n d z a r e r e a l

n u m b e r s , a n d i, j, a n d /c are u n i t vectors, d i r e c t e d a l o n g the x, y, a n d z axes

respectively. ij = ji

=

The

i, j, a n d /c units obey the f o l l o w i n g laws:

k

jk

- k

kj ii

It is

to be

general

= j j

noted that for t w o

equal

q'q.

=

T h e loss

i

ki

- i

ik

= =

kk

=

= j =

- j

—1

quaternions

q

and

of commutativity in

q',

qq'

does

not in

quaternions, while

it is v e r y i m p o r t a n t h i s t o r i c a l l y , is also significant m a t h e m a t i c a l l y , because this complicates calculations in w h i c h quaternions are used. A l l the other properties discussed above are satisfied by quaternions. T h u s it may be v e r i f i e d that q u a t e r n i o n m u l t i p l i c a t i o n is associative a n d q u a t e r n i o n division

is

unambiguous.

special m e n t i o n ,

These

are

two

important properties

which

bear

since t h e y are not preserved in the algebra of m o d e r n

vectors. There have been a n u m b e r of discussions published on the mathematical present

details

of Hamilton's

purposes

within the Almost

all

that

procedure

need

be

after his

is

his

discovery;

for the

that

Hamilton

worked

discussed above.35

context that has b e e n immediately

in

said

discovery

H a m i l t o n stated that he

"felt that it m i g h t be w o r t h my w h i l e to expend [on quaternions] the labour of at least ten 436)

Hamilton

working

(or it m i g h t be

actually

almost

spent

exclusively on

fifteen)

the

last

y e a r s t o c o m e . " (2,11;

twenty-two

quaternions.

The

years letters

of

his

life

of the first

f e w days after the discovery s h o w that H a m i l t o n felt that his system had

importance

nometry.

for

heat

electricity,36

theory,

sense

I

nions,

from

hope

spherical

trigo-

that

I

am

sions

of their principles, to

me

actually

g r o w i n g modest a b o u t t h e

quater-

m y s e e i n g s o m a n y p e e p s a n d vistas into f u t u r e expan-

pears

to

be

as

I

still

important

must for

century as the discovery of fluxions teenth."

and

(2,11; 4 4 2 ) I n 1 8 5 1 h e w r o t e : " I n g e n e r a l , a l t h o u g h i n o n e

(2,11;

assert that this d i s c o v e r y apthe

middle

of the nineteenth

was for the close of the seven-

445)

In o n e sense at least H a m i l t o n ' s discovery was e p o c h m a k i n g , for quaternions

were

the

number system which His

30

first

well-known

did not obey the

consistent

and

significant

laws of ordinary arithmetic.

" c u r i o u s , a l m o s t w i l d " (as h e c a l l e d i t [2,11; 4 4 1 ] ) d i s c o v e r y m a y

Sir W i l l i a m R o w a n

Hamilton

and Quaternions

be c o m p a r e d to the discovery of n o n - E u c l i d e a n geometry. B o t h discoveries broke bonds

set b y c e n t u r i e s o f m a t h e m a t i c a l t h o u g h t . I m -

mediately

other

after

by Augustus

1843

De

Morgan

new

(who

number

systems

were

(1846).38

T . G r a v e s ( 1 8 4 4 ) (2,11; 4 5 4 - 4 5 5 ) , a n d C h a r l e s G r a v e s This

section

publications 13,

1843,

Irish

will

on

be

concluded

by

a

quaternions through the

Hamilton

Academy,

of

read a paper on which

at

discovered

p u b l i s h e d f i v e n e w systems),37 J o h n

least

discussion

of Hamilton's

year

On

1847.

quaternions

part

was

November

before the Royal

published

in

1844.39

Either this paper or the very similar paper in the July, 1844, issue of the

Philosophical

nions.

(5,25;

Magazine

was

Quaternions,"

delivered

the

In

these

July,

1846,

papers

not analogous (the

w

first

November

Academy and published in in

his

publication

on

quater-

10-13) A m o n g the m o s t i m p o r t a n t papers are his " O n

issue

Hamilton

to

11,

1844,

of the

Philosophical

dealt w i t h

the

the

Royal

Irish

Magazine.

does

(5,29;

26-31)

fact that q u a t e r n i o n s are

ordinary complex numbers

o f w + i x + j y + kz)

to

1847,40 a n d t h e s i m i l a r p a p e r p u b l i s h e d

in that the

scalar part

n o t i n d i c a t e d i s t a n c e on an axis

un-

less, as he h a d s u g g e s t e d earlier, q u a t e r n i o n s be c o n s i d e r e d as f o u r dimensional. A n d on

Thus

account

Hamilton

of the

(writing

facility w i t h

in

the

which

third person)

this

so

stated:

c a l l e d imaginary e x -

p r e s s i o n , o r s q u a r e r o o t o f a n e g a t i v e q u a n t i t y , i s c o n s t r u c t e d b y a right line

having direction

in

space,

and

h a v i n g x,

y,

z f o r its t h r e e r e c t a n g u l a r

c o m p o n e n t s , o r p r o j e c t i o n s o n t h r e e r e c t a n g u l a r axes, h e has b e e n i n d u c e d to call the t r i n o m i a l expression itself, as w e l l as the line w h i c h it

represents,

a

VECTOR.

A

quaternion

g e n e r a l l y o f a real p a r t a n d a vector.

may

The

thus

fixing

be

said

to

consist

a s p e c i a l a t t e n t i o n on

this last part, or e l e m e n t , of a q u a t e r n i o n , by g i v i n g it a s p e c i a l n a m e , a n d d e n o t i n g it in m a n y calculations by a single a n d special sign, appears

to

the

dealing with

author to the

have

subject:

been

although

an

improvement in

the general

his

method

of

notion of treating the

constituents of the imaginary part as coordinates had occurred to h i m in his first researches. Regarded from a geometrical point of v i e w , this algebraically imagin a r y p a r t of a q u a t e r n i o n has t h u s so n a t u r a l a n d s i m p l e a s i g n i f i c a t i o n or representation in space, that the difficulty is transferred to the algebraic a l l y r e a l p a r t ; a n d w e are t e m p t e d t o ask w h a t t h i s last c a n d e n o t e i n g e o m e t r y , o r w h a t i n s p a c e m i g h t h a v e s u g g e s t e d it.41 The

origin

following

of the

word

quotation

in

vector ( a n d t h e w o r d scalar) the

similar

paper

in

is

the

clear from the Philosophical

Maga-

zine. The

a l g e b r a i c a l l y real p a r t m a y r e c e i v e . . . a l l v a l u e s c o n t a i n e d o n t h e

o n e scale o f p r o g r e s s i o n o f n u m b e r f r o m n e g a t i v e t o p o s i t i v e i n f i n i t y ; w e shall

call

it t h e r e f o r e

the

scalar part,

or

simply the

scalar o f t h e q u a t e r -

n i o n , a n d s h a l l f o r m its s y m b o l b y p r e f i x i n g , t o t h e s y m b o l o f t h e q u a t e r -

31

A

History

of V e c t o r Analysis

n i o n , t h e c h a r a c t e r i s t i c S e a l . , o r s i m p l y S., w h e r e n o c o n f u s i o n s e e m s l i k e l y t o arise f r o m u s i n g this last a b b r e v i a t i o n . O n t h e o t h e r h a n d , t h e algebraically

imaginary

part,

being

geometrically

constructed

by

a

straight l i n e or radius vector, w h i c h has, in general, for each d e t e r m i n e d q u a t e r n i o n , a d e t e r m i n e d l e n g t h a n d d e t e r m i n e d d i r e c t i o n in space, m a y be

c a l l e d t h e vector part, o r s i m p l y t h e vector o f t h e q u a t e r n i o n ;

and may

be denoted by prefixing the characteristic Vect., or V. We may therefore say

that

a

quaternion

is

in

general

parts, a n d m a y w r i t e Q = S e a l . SQ + V Q . F r o m the

(5,29;

above

quotations

introduced

cise

mathematical

The than

the

this.

scalar

sense,

analysis.

quotations

In

of its

own

scalar

and

vector

Q = S . Q + V . Q or s i m p l y Q =

it m a y be inferred that it was H a m i l t o n

term

a

sense

Hamilton

also

the

the

term

vector i n

similar

term

its

pre-

radius

vector

before.

however

they

had

and

although

used for m a n y years

above

sum

26-27)

who

had been

the

Q + Vect.

have

mark

the

introduced

a

far

greater

beginning

his

significance

of modern

symbols

S

and

vector

V because

"separation of the real a n d i m a g i n a r y parts of a q u a t e r n i o n is an operation of such frequent occurrence, and may be regarded as so fund a m e n t a l in this theory. . . of his

quaternions xi + yy'

yj + +

(5,29; 26) H a m i l t o n illustrated the use

symbols as applied to the product of the multiplication of two a a n d a',

zk a n d

zz');

V.

in

which

the

a' = x'i + y ' j + z'k, aa'

=

i(yz'

-

z y ' )

scalar parts w e r e 0.

Hamilton wrote: + j(zx'

-

x z ' )

+

Letting a =

" S . aa' = — (xx' +

k(xy'

-

yx').

.

.

(5,29; 30) It is o b v i o u s that these are e q u i v a l e n t to the m o d e r n vector

(cross)

product.42 bols,

product

using them

n o w be used. S.aa'

=

0

negative

of the

modern

scalar (dot)

in

cases

where

the

dot and

cross p r o d u c t w o u l d

Hamilton then proceeded to prove such equations as

when

Another 1847.43

and to the

H a m i l t o n and Tait m a d e very frequent use of these sym-

pair

In these

a

and

of

a'

very

papers

...1881

»

1891

1841 to 1900.

d e r i v e d f r o m this study is that d u r i n g the

period from 1841 to 1900 there w e r e 594 quaternion publications as c o m p a r e d to 217 Grassmannian analysis publications.10 H e n c e 73.2 percent there

of the

were

publications

2.73

quaternion

were

in

the

quaternion

publications

for

each

tradition,

or

Grassmannian

publication. The

results

obtained

strikingly similar. published from

when

only

books

were

considered

was

By actual count there were 38 quaternion books

1841

to

1900, whereas there w e r e

16 books pub-

lished d u r i n g this p e r i o d in the G r a s s m a n n i a n tradition. T h u s 70.4 percent

of the

books

dealt w i t h

quaternions,

or there

were

2.37

quaternion books for each book in the Grassmannian tradition. T h e quaternion

books

Grassmannian

averaged 281

tradition,

249

pages in length;

pages.11

the books of the

F r o m these numbers it may

111

A H i s t o r y of V e c t o r Analysis 8

6

6 5 4

4 3

1841 GRAPH

II.

1871

1861

1851

Quaternion Books from

1841 to

1891

1881

1900.

be inferred that interest in the tradition b e g u n w i t h H a m i l t o n was far greater t h a n that b e g u n w i t h G r a s s m a n n . T h e s e n u m b e r s have been broken d o w n into five-year intervals in Graphs I, II, IV, and V. Graph I shows the n u m b e r of quaternion publications in terms of five-year intervals Hamilton the

from

h i m s e l f are

1841 t o 1900. T h e p u b l i c a t i o n s w r i t t e n b y

indicated by

s o l i d areas.

n u m b e r of quaternion books for the

books by

Hamilton

Graph II

presents

same time intervals.

The

(including a translation and a second edition)

are i n d i c a t e d b y s o l i d areas. From

Graphs

I

and II the following conclusions may be drawn.

I n t e r e s t i n q u a t e r n i o n analysis w a s a t its h i g h e s t l e v e l d u r i n g t h e 1876-1900

period.

The

decrease

in

interest for the period

1881-

1885 indicated by G r a p h I is balanced by the peaking of G r a p h II for the same interval. It is important to note that H a m i l t o n wrote 73 percent of the pre-1866 quaternion publications and 19 percent of all

quaternion

publications.

It w o u l d of course be significant to compare the form of Graph I w i t h a graph s h o w i n g the rate of increase of m a t h e m a t i c a l publications

d u r i n g this

quaternion

time.

publications

Some after

idea of h o w the 1870

compares

rate with

of increase the

rate

of

of in-

crease in mathematical publications in general m a y be obtained by means of the study m a d e by H. S. W h i t e in 1915 based on an analysis

of

Graph

works III

listed

in

the

journal

Fortschritte

der

Mathematik.12

shows the n u m b e r of titles of mathematical articles a n d

b o o k s p u b l i s h e d i n t h e p e r i o d 1868 t o 1909.13 W h e n G r a p h s I a n d I I I are c o m p a r e d , it seems at first sight that interest in evident

quaternions

that

the

was

percentage

declining from

1876 to

of mathematical

1900, for it is

literature

that

was

d e v o t e d to quaternions decreases slightly. B u t this seems to be an

112

Traditions in Vectorial Analysis

1870

GRAPH

III.

Annual

'SO

N u m b e r of Titles

'90

1900

1910

of Mathematical Articles

and Books,

1868-

1909.

erroneous conclusion, for even m o r e

striking than the increase in

the n u m b e r of mathematical publications d u r i n g this period is the increase in the n u m b e r of fields of mathematical research. N u m e r ous fields —such as

non-Euclidean

geometry, mathematical logic,

group theory, as w e l l as m a n y branches of a p p l i e d mathematics — came into p r o m i n e n c e in this period. 68

48

28

16

1841

• GRAPH

IV.

1851

»

1861

>

1871

16

» 188T

Grassmannian Analysis Publications from 1841 to

»

1891

1900.

113

A H i s t o r y of V e c t o r Analysis 4

1841

M 851 GRAPH V.

1861

>1871

Grassmannian Analysis Books from

Graphs mannian

^ 1 8 8 1 — — •

1841

to

1 8 91

1900.

IV a n d V present the results of a s i m i l a r study of Grassanalysis

publications

and

books.

The

G r a s s m a n n are i n d i c a t e d b y t h e s o l i d areas.

works

written by

F r o m these graphs it

becomes clear that the b e g i n n i n g of the m a i n period of interest in Grassmannian

analysis

similar

for

period

other hand, the

came

roughly

quaternions

(1891

fifteen

years

compared

to

later than the 1876).

On the

n u m b e r of Grassmannian analysis publications for

the p e r i o d 1896 to 1900 was approaching the n u m b e r of quaternion publications

for the

same interval.

In regard to Grassmann's per-

358

88

52

52 44

American

British

GRAPH

114

VI.

Quaternion

Publications

French

by Country.

German

Other

Traditions in Vectorial Analysis

10

10 Spanish Russian

8

8

Polish Czech

Portuguese

Japanese

2

British

GRAPH

VII.

Quaternion

German

French

American

Dutch

Books by Country.

sonal contribution it is

noteworthy that he published 25 of the 33

(or 7 6 p e r c e n t ) o f t h e p u b l i c a t i o n s u p t o 1875, a n d 3 3 o f t h e total o f 2 1 7 (or 15 p e r c e n t ) of G r a s s m a n n i a n analysis p u b l i c a t i o n s . A

study

of the

two

fields

in

terms

of interest by country

is

of

significance. T h u s Graphs V I and V I I represent quaternion publications and books respectively as classified into five groups: British, American,

French,

German,

and those

of other countries. Quater-

nion books appeared in ten languages as follows (the n u m b e r after 125

32 28

16

16

GRAPH

French

American

British

VIII.

Grassmannian given

height

Analysis on

this

height on Graph VI

Publications

scale

indicates

indicates

2x

Other

German

by x

Country.

(Note

publications,

that

then

an

if any equal

publications.)

115

A H i s t o r y of V e c t o r Analysis

12

1 American GRAPH

each

IX.

Grassmannian Analysis

language

which

French

indicates

the

German

Italian

Books by Country.

number

of books):

English

(12,

of

1 0 w e r e p u b l i s h e d i n B r i t a i n a n d 2 i n A m e r i c a ) , F r e n c h (8),

German

(8),

Dutch

(2),

Japanese

(2),

P o r t u g u e s e (2), C z e c h o s l o -

v a k i a n (1), P o l i s h (1), R u s s i a n (1), a n d S p a n i s h (1). T h e r e w e r e i n a d d i t i o n a n u m b e r of papers in Italian, at least one in D a n i s h , a n d at least one tive

study

paper was p u b l i s h e d in Australia. F r o m this quantita-

it

may

be

inferred that 60

(books and papers) w e r e British,

percent

of the publications

15 percent w e r e American, 9 per-

cent were French, 8 percent German, with the remaining 8 percent c o m i n g from other countries. This should be considered in relation to

the

fact

that

26

percent

of the

books

on

quaternions

were of

British origin, 5 percent of A m e r i c a n origin (the British books w e r e of course

often

used by Americans), 21

percent

of German

guages.

These

origin,

statistics

and

the

percent of French and 21

remaining were

in

other

lan-

point out that interest in quaternions was

strongest in Britain but was substantial in America, Germany, and France, a n d that it e x t e n d e d to most of the then intellectually productive

countries of the

Graphs

VIII

and

IX

world. represent

Grassmannian

analysis

publica-

tions a n d books r e s p e c t i v e l y that are classified on the same basis of country

of publication.

four countries:

Grassmannian analysis books

each in A m e r i c a and Italy. the

appeared in

12 w e r e p u b l i s h e d in G e r m a n y , 2 in France, and 1 T h e study revealed that 57 percent of

Grassmannian analysis publications appeared in Germany, 18

p e r c e n t i n A m e r i c a , 10.5 p e r c e n t i n b o t h B r i t a i n a n d F r a n c e , w i t h a

116

Traditions in Vectorial Analysis

few

works

appearing

in

Polish,

Italian,

Spanish,

Russian,

and

analysis

was

Czechoslovakian. Thus

it

appears

centralized

in

that

Germany,

interest with

in

less

Grassmannian proportionate

interest outside

Germany than the interest in quaternions outside Britain. It is notew o r t h y that for b o t h systems the country in w h i c h the most interest d e v e l o p e d after the m o t h e r country of the analysis was America.14 This study m a y be summarized by the statement that the interest in quaternion analysis was roughly t w o and one-half times as great as interest in Grassmannian analysis and extended to m o r e countries, w i t h greater interest p r o p o r t i o n a t e l y d e v e l o p i n g i n c o u n t r i e s outside the country in w h i c h the system originated. To this m a y be added the observation that there was substantial interest in quaternions from 1876 to 1900 and that although interest in Grassmannian analysis came s o m e w h a t later, it d i d by the p e r i o d 1 8 9 1 - 1 9 0 0 attain substantial It is

magnitude.

the author's belief that this

quantified study tells no m o r e

than part of the story. It does h o w e v e r s u p p l y a v a l u a b l e perspective i n t o w h i c h d e v e l o p m e n t s d i s c u s s e d i n l a t e r s e c t i o n s m a y b e set.

III.

Peter

Guthrie

Tait:

Advocate

and

Developer

of

Quaternions

T h e i m p o r t a n c e o f T a i t f o r t h i s h i s t o r y i s f o u r f o l d . (1) H e w a s t h e a c k n o w l e d g e d leader of the quaternion analysts f r o m 1865 until his death in

1901. I n d e e d e i g h t b o o k s o n q u a t e r n i o n s ( i n c l u d i n g later

editions, translations, and coauthorships) carried his title page.

name on the

(2) T a i t d e v e l o p e d q u a t e r n i o n a n a l y s i s a s a t o o l f o r re-

s e a r c h i n p h y s i c a l s c i e n c e (as H a m i l t o n h a d n o t ) a n d c r e a t e d m a n y n e w theorems in quaternion analysis w h i c h w e r e capable of b e i n g t r a n s l a t e d i n t o m o d e r n v e c t o r a n a l y s i s . (3) I t w a s p r o b a b l y t h r o u g h Tait that M a x w e l l b e c a m e interested in quaternions.

(4) T a i t w a s

the most important o p p o n e n t of m o d e r n vector analysis. Peter Guthrie Tait was born in

1831 near E d i n b u r g h , Scotland.

In 1841 he entered E d i n b u r g h A c a d e m y w h e r e one year earlier the young

James

Clerk

Maxwell

had

enrolled.

Playmates

in

their

y o u t h , t h e t w o b e c a m e fast f r i e n d s a n d f r e q u e n t c o r r e s p o n d e n t s i n their maturity. was

M a x w e l l ' s 1846 entrance into E d i n b u r g h University

followed by Tait's in

1847, w i t h the o r d e r o f e n t r y b e i n g re-

versed w h e n Tait left for C a m b r i d g e

after one

year at E d i n b u r g h

University, w h i l e M a x w e l l stayed for three. After Tait's graduation in

1852

elected

as a

Senior

Fellow

Wranger

and

of Peterhouse

First

Smith's

College,

writing the first of his m a n y books.

Prizeman

Cambridge,

and

he

was

began

This was coauthored by W. J.

117

A H i s t o r y of V e c t o r Analysis

Steele of a

(who

died

Particle.

Lectures

In

on

ordered

before

1853

Quaternions. your

Athenaeum, caught my

its

Tait As

book,

c o m p l e t i o n ) a n d w a s e n t i t l e d Dynamics

ordered

Tait

on

later

a

copy

wrote

account

of

an

of t h e j u s t - p u b l i s h e d

to

Hamilton:

"when

advertisement

in

I

the

I had NO I D E A w h a t it was about. T h e startling title

eye

in

August

'53,

and as

I

was

going off to shooting

quarters I took it and some scribbling paper w i t h me to beguile the t i m e . . . . H o w e v e r as I t o l d y o u in my first letter I got easily e n o u g h t h r o u g h t h e first six L e c t u r e s . . . ." (1; 126) On

his

return

quaternions, he

to

Cambridge

primarily

was writing.

In

at Queen's

volved

in teaching as Thomas

of the

study of

labor involved in the book

1854 he accepted the Professorship in Mathe-

matics

league

Tait d i d not continue his

because

College, well

Andrews.

Belfast,

Ireland.

Here

he became in-

as in e x p e r i m e n t a l w o r k w i t h his colHe

also

pursued

the

study

of

the

' T h e o r i e s o f H e a t , E l e c t r i c i t y a n d L i g h t . " F i n a l l y i n A u g u s t , 1857, his

interest

in

quaternions

returned

as

a result of reading H e l m -

holtz' famous paper on vortex motion. T h e n as Tait wrote to Hamilton:

"I

suddenly bethought me of certain formulae I had admired

years ago at p. 610 of y o u r Lectures — a n d w h i c h I t h o u g h t (and still think) likely to serve my purpose exactly." to

which

Tait

referred

"