A History of Mathematical Notation. Vol II

CARNlCtIB INSTITUTE

OF TECHNOLOGY

A HISTORY OF MATHEMATICAL NOTATIONS VOLUME

II

NOTATIONS MAINLY IN HIGHER MATHEMATICS

A HISTORY OF

MATHEMATICAL NOTATIONS BY

FLORIAN CAJOBJ,

PH.D.

Professor of the History of Mathematics

University of California

VOLUME

II

NOTATIONS MAINLY IN HIGHER MATHEMATICS

THE

OPESN

COURT PUBLISHING COMPANY CHICAGO ILLINOIS *

COPYRIGHT 1929 BY

THE OPEN

Goinrr PUBLISHING

COMPANY

Published March 1929

Second Printing September 1930 Third Printing January 1952

PWNTfJfW

CHICAGO

PREFACE TO THE SECOND VOLUME The larger part of this volume deals with the history of notations in higher mathematics. The manuscript for the parts comprising the two volumes of this History was completed in August, 1925, but since then occasional alterations and additions have been made whenever

new material or new researches came to my notice. Some parts of this History appeared as separate and educational journals, but

tific

later the articles

articles in scien-

were revised and

enlarged. I

am indebted to Professor R.

Karpinski for aid in the

C. Archibald and to Professor L. C. arduous task of reading the proofs of this

volume.

FLORIAN CAJORI UNIVERSITY OF CALIFOBNIA

TABLE OF CONTENTS INTBOOTCTION TO THE SECOND VOLTTME PABAGBAPH9 I.

TOPICAL SURVEY OF SYMBOLS IN ARITHMETIC AND ALGEBRA 388-510

(ADVANCED PART)

388-94 388

Letters Representing Magnitudes

Greek Period Middle Ages

389 390

Renaissance Descartes in 1637

391 392

Different Alphabets Astronomical Signs

393 394

Vietainl591

The

Letters

and

TT

Euler's

Use

of

395-401 395

e

Early Signs for 3.1415 First Occurrence of Sign

396 397 398

ic

v

Spread of Jones's Notation Signs for the Base of Natural Logarithms

The Letter

399

400 401

e

B. Peirce's Signs for 3.141 of the Dollar

The Evolution

and 2.718

Mark

Different Hypotheses

Evidence in Manuscripts and Early Printed Books

.

Modem

.

Dollar

Mark

in Print

Conclusion Signs in the Theory of Numbers Divisors of Numbers, Residues

........

Congruence of Numbers Prime and Relatively Prime Numbers

Numbers Numbers Figurate Numbers Suras of

.

.

.

DIophantine Expressions

Number Fields Perfect Numbers Mersentte's Numbers fermat's Numbers Cotes's

Numbew

406-20 407 408 409 410

4H

Partition of

,

402-5 402 403 404 405

....,**.

.,.,..*.,,. *

,

vll

.

.

.

.

412 413 414 415 416 417 418

TABLE OF CONTENTS PABAGBA.PH8

Bernoulli's Numbers Euler's Numbers

Signs for Infinity and Transfinlte Numbers Series. Signs for Continued Tractions and Infinite

.

.

-

'

Continued Fractions Tiered Fractions Infinite Series

Signs in the Theory of Combinations

Binomial Formula Product of Terms of Arithmetical Progression Tandennonde's Symbols Combinatorial School of Hindenburg Ivramp on Combinatorial Notations ^

....

Signs of Argand and Amp-lre

Thomas

Jarrett

Factorial

n

Subfactorial

N

Continued Products Permutations and Combinations Substitutions

Groups Invariants and Covariants

Dual Arithmetic Chessboard Problem Determinant Notations Seventeenth Century Eighteenth Century Early Nineteenth Century

45968

Modern Notations Compressed Notations Jacobian Hessian

.

Cubic Determinants Infinite Determinants Matrix Notations Signs for Logarithms Abbreviation for "Logarithm" Different Meanings of log x, Ix, and

Power

of a Logarithm Iterated Logarithms

Marking the Characteristic Marking the Last Digit Sporadic Notations

Complex Numbers

419 42 421 422-38 4 434 435 ^ 439-58 4 ^9 440 441 443 44o 446 447 44S 45 451 452 453 454 456 457 458

Lx

.

.

.

459 460 461 462 463 464 465 466 467 468 469-82 469 470 472 473 474 478 479 480

TABLE OF CONTEXTS

ix PABAGiSAPHS

Exponentiation

481

.

Dual Logarithms

482

483-94

Signs of Theoretical Arithmetic 37 Signs for "Greater" or "Less

483

Sporadic Symbols for "Greater" or ''Less

33

484 485 486 487

Improvised Type Modern Modifications Absolute Difference

~

^

Other Meanings of and Sporadic Symbols Signs for Absolute Value

489

A Few Other

491

492

Zeroes of Different Origin General Combinations between Magnitudes or Symbolism for Imaginaries and Vector Analysis

Square Root of Minus One De Morgan's Comments on V 1 Notation for a Vector Length of a Vector

Symbols

for the

493

Numbers

.... ....

495-510 495 501

502 504

.

Equality of Vectors Products of Vectors

II.

494

505

506

Certain Operators Rival Vector Systems

508

Attempts at Unification

509

Tensors

510

507

SYMBOLS IN MODERN ANALYSIS

511-700 511-37

Trigonometric Notations Origin of the

Modern Symbols

and

for Degrees, Minutes,

Seconds Signs for Radians

Marking Triangles Early Abbreviations of Trigonometric Lines Great Britain during 1602-18 European Continent during 1622-32

511

515

....

Great Britain during 1624-57 Seventeenth-Century English and Continental Practices are Independent England during 1657-1700 The Eighteenth Century Trigonometric Symbols of the Eighteenth Century Trigonometric Symbols of the Nineteenth Century

Less

Common

Trigonometric Functions

Quaternion Trigonometry Hyperbolic Functions Parabolic Functions

516 517 518 519 520 521

522 524 .

.

525

.

.

526 527

528 529 531

TABLE OF CONTENTS PABAGBAPHS

Inverse Trigonometric Functions John HerscheFs Notation for Inverse Functions . Martin Obmy s Notation for Inverse Functions Persistence of Rival Notations for Inverse Functions .

Inverse Hyperbolic Functions Powers of Trigonometric Functions

Differentia!

.

.

.

.

....

.

Survey of Mathematical Symbols Used by Leibniz Introduction Tables of Symbols Remarks on Tables

.

.

.

.

...

and Integral Calculus

1*

Introduction

2.

Symbols

. for Fluxions, Differentials, and Derivatives a) Total Differentiation during the Seventeenth and

&)

Eighteenth Centuries. Newton, Leibniz, Landen, Fontaine, Lagrange (1797), Pasquich, Grtison, Arbogast, Kramp Criticisms of Eighteenth-Century Notations.

c)

Total

532 533 534 535 536 537

538-65 538 542 563-65 566-639 566 567

567-78 579

Woodhouse, Lacroix, Lagrange during- the Nineteenth Century. Barlow, Mitchell, Herschel, Peacock, Differentiation

Crelle, Cauchy (1823, 1829), M. Ohm, Cauchy and Moigno (1840), B. Peirce, Carr,

Babbage,

Peacock, Fourier d) Partial

Differentials

582

and

Partial

Derivatives.

Monge, Condorcet, Legendre, Lagrange (1788), Lacroix, Da Cunha, L'HuiHer, Lagrange (1797), Arbogast, Lagrange (1801), CreUe, Barlow, Cauchy, M. Ohm, W. R. Hamilton, W. Bolyai, Cauchy and Moigno, C. G. J. Jacobi, Hesse, B. Peirce, Strauch, Duhamel, Carr, M4ray, Muir, Mansion 3. Symbols for Integrals, Leibniz 4. Early Use of Leibnizian Notation in Great Britain. 5. Symbols for Fluents: Later Notations in Integral CalEuler, Karsten, Fontaine,

.

.

.

.

culus.

593

620 621

Newton, Reyneau, Crelle, Euler, Fourier, H. Moore, Cauchy's Residual

Volterra, Peano, E.

Calculus

622

6.

Calculus Notations in the United States

7.

Symbols for Passing to the limit. L'HuiKer, Weierstrass Oliver, Riemann, Leathern, DiricMet, Pringsheim, Scheffer, Peano, W. H. Young

630

*

.

8.

The Sign --

9.

Concluding Observations

.

631 638 639

TABLE OF CONTENTS

xi

PA2AG3APHS

640, 641

Finite Differences

640

Early Notations Later Notations

641

642-66

of Functions

in

Symbols Theory A. Symbols for Functions in General B. Symbols for Some Special Functions Symmetric Functions Gamma and Beta Functions Elliptic Functions Theta Functions Zeta Functions

..,,..

-

.

Power

.

.

.

.

Series

Laplace, Lame",

and Bessel Functions

Logarithm-Integral, Cosine-Integral, etc

Symbols in Mathematical Logic Some Early Symbols

The Sign for "Therefore" The Sign for "Because" The Program of Leibniz

.

642 647 647 649 651 656 659 661 662 665

667-99 667 668 669 670

Signs of

H.Lambert

671

G. J. von Holland G. F. Castillon J. D. Gergonne

672 673 674 675 676 677 678 679 680 681 682 684 685 686 687 688 692 693 695 696 697 698

Bolyai

Bentham A. de Morgan

.

G.Boole

W. S.

Jevons Macfarlane C. S. Peirce Ladd-Franklin and Mitchell B,. G. Grassmann E. Schroeder

J.H. MacColl G. Frege G. Peano A. N. Whitehead E. H. Moore

Whitehead and Hussell P. Poretsky L.Wittgenstein

Remarks by Eignano and Jourdain

A Question

,

699

TABLE OF CONTENTS

xii

PARAGRAPHS III.

SYMBOLS IN GEOMETRY (ADVANCED PART) 1. Recent Geometry of Triangle and Circle,

2.

700-711

700

etc

Geometrographie

701

Signs for Polytiedra

702

Geometry of Graphics Projective and Analytical Geometry Signs for Projectivity and Perspectivity Signs for Harmonic and Anharnionic Ratios

703 704

.... .

.

.

.

.

.

.

.

.

Descriptive Geometry "

Analytical Geometry Pliicker's

.

Equations

The Twenty-seven Lines on a Cubic Surface The Pascal Hexagram IV.

THE TEACHINGS OF HISTORY

705 706 707 70S 709 710 711

712-50

A. The Teachings of History as Interpreted by Various

712-25 712 713

Writers. Individual Judgments

D. Andre Quotations from A. de Morgan

Review

of

J.

W.

714

L. Glaisher

D. E. Smith A. Saverien

715

...

...

C. Maclaurin

717

Ch. Babbage E. Mach

718

B. Branford

720

A. N. Whitehead

721

H. F. Baker H. Burckhardt

722

P. G. Tait

724

723

O.S.Adams

724 725

Committee

of

Symbols Invention of Symbols Nature of Symbols Potency

719

A British

B. Empirical Generalizations on the Growth of Mathematical Notations

Forms

716

of

Symbols and Spread

727-33 727 727 728

State of Flux

729 730 731

Defects hi Symbolism Individualism a Failure

732 733

Selection

of

Symbols

TABLE OF CONTENTS

xiii

PABAGBAPHS

C. Co-operation in Some Other Fields of Scientific Endeavor Electric Units

in

Mathematics

....

In Vector Analysis In Potential and Elasticity

738

To Be Reached by International Committees

the Only Hope for Uniformity of Notations

ALPHABETICAL INDEX

736 737 739

In Actuarial Science

E. Agreements

734

735

Star Chart and Catalogue

D. Group Action Attempted

734

.

.

.

740

ILLUSTRATIONS PARAGRAPHS

107. B. PEIRCE'S SIGNS

FOE 3.141 .... AND 2.718

400

108.

FROM J. M.

109.

PILLAR DOLLAR OF 1661

402

110.

FORMS THAT ARE NOT DOLLAR SYMBOLS

402

111.

SYMBOLS FOR THE SPANISH DOLLAR OR PESO TRACED FROM MANUSCRIPT LETTERS, CONTRACTS, AND ACCOUNT-BOOKS

403

112.

THE MODERN DOLLAR MARK IN THE MAKING

403

113.

DOLLAR MARKS IN L'HOMMEDEEU'S DIARY, 1776 FROM CHAUNCEY LEE'S AMERICAN ACCOMPTANT,

PEIRCE'S TABLES, 1871

401

.

114.

.

FROM CHATJNCEY LEE'S AMERICAN ACCOMPTANT, PAGE MULTIPLICATION TABLE FOR SEXAGESIMAL FRACTIONS.

117.

MARKING THE GIVEN AND REQUIRED PARTS OF A TRIANGLE, THE GIVEN AND REQUIRED PARTS OF ANOTHER TRIANGLE,

120.

NOTATION IN TRIGONOMETRY

142

.

404

.

115.

119. ILLUSTRATING GIRARD'S

.

403 1797.

116.

118.

.

.

,

.

.

404

.

.

513

1618 1618 .

.

518 519

A PAGE OF ISAAC NEWTON'S NOTEBOOK SHOWING TRIGONOMETRIC SYMBOLS

121. LEIBNIZ' 122. LEIBNIZ'

522

MSS DATED FIGURE IN MSS DATED FIGURE IN

OCTOBER

26, 1675

OCTOBER

29, 1675

123.

FROM AJRBOGAST'S

124.

MANUSCRIPT OF LEIBNIZ, DATED OCTOBER 29, His SIGN OF INTEGRATION FIRST APPEARS

125. G.

Calcul des Derivations (1880),

PAGE

Bow's NOTATION

.... ....

570

xxi

578

1675, IN

FREGE'S NOTATION AS FOUND IN His Grundgesetze I, PAGE 70

UME 126.

518

.

.

570

WHICH 620

(1893),

VOL687

702

INTRODUCTION TO THE SECOND VOLUME It has been the endeavor to present In the two volumes of this History a fairly complete list of the S3nnbols of mathematics down to the beginning of the nineteenth century, and a fairly representative selection of the symbols occurring In recent literature In pure mathematics. That we have not succeeded in gathering all the symbols of

modem

mathematics

is quite evident. Anyone hunting, for even an modern mathematical literature Is quite certain bag symbolisms not mentioned In this History. The task of making

hour, in the jungle of to

a complete collection of signs occurring in mathematical writings down to the present time transcends the endurance of a single Investigator. If such a history were completed on the plan of the present work, it would greatly surpass this in volume. At the present time the designing of new symbols is proceeding with a speed from, antiquity

that

Is

truly alarming. Diversity of notation is bound unnecessarily to retard the spread of a knowledge of the new results that are being reached in mathematics. What is the remedy? It is hoped that the material here presented will afford a strong Induction, facilitating the passage from the realm of conjecture as to what constitutes a wise course of procedure to the realm of greater certainty. If the contemplation of the mistakes in past procedure will afford a more intense conviction of the need of some form of organized effort to secure uniformity, then this History will not have been written in vain.

ADDENDA PAGE

28, line 3, add the following: In the Commonplace Book of Samuel B. Beach, B.A., Yale, 1805, now kept In the Yale University Library, there Is given under the year 1804, "the annual expence about $700," for the upkeep of the lighthouse in New Haven. The dollar mark occurs there in the conventional way now current. Prof. D. E. Smith found the symbol $ very nearly in the present form in DabolPs Schoolmaster's Assistant, 4th edition, 1799, p. 20. Dr. J. M. Armstrong of St. Paul, Minn., writes that in the Medical Repository, New York (a quarterly publication), Vol. Ill, No. 3, November and December, 1805, and

January, 1806, p. 312, the $ is used as it is today. 29, line 6, add the following: Since this volume was printed, important additional and confirmatory material appeared in our article "New Data on the Origin and Spread of the Dollar Mark" in the Scientific Monthly, September, 1929, p. 212-216. PAGE 145, line 1, for Gioseppe read Giuseppe Moleti PAGE 323, lines 8 and 9, for in G. Cramer found earlier read in Claude RabueFs Commentaires sur la Geometric de M. Descartes, found also. Lyons, 1730, and in G. Cramer In the alphabetical index insert Mahnke, D., 542, 543, 563.

PAGE

.

.

.

.

.

.

TOPICAL SURVEY OF SYMBOLS IN ARITHMETIC

AND ALGEBRA (ADVANCED PART) LETTERS REPRESENTING MAGNITUDES 388. Greek period. The representation of general numbers by letters goes back to Greek antiquity. Aristotle uses frequently single capital letters, or two letters, for the designation of magnitude or

number. For example, he says: "If A is what moves, B what is being moved, and P the distance over which it was moved, and A the time during which it was moved, then the same force A, in the same time could

move

the half of

B

A "BT any how much time

twice as far as F, or also in half the time

1 2 exactly as far as T." In other places he speaks of the

force," "the time

EZ"

In another place he explains

and trouble may be saved by a general symbolism. 3 Euclid 4 in his Elements represented general numbers by segments of lines, and these segments are marked by one letter, 5 or by two letters placed at the ends of the segment, 6 much the same way as in Aristotle. Euclid used the language of line and surface instead of numbers and their products. In printed editions of the Elements it became quite customary to render the subject more concrete by writing illustrative numerical values alongside the letters. For ex-

ample, Clavius in 1612 writes (Book VII, Prop. 5, scholium) "A, f f," and again (Book VIII, Prop. 4), "A, 6.5, 5.C, 4.D, 3." In Robert Simson's translation of Euclid and in others, the order of the English Alphabet is substituted for that of the Greek, thus A B T A

D

H

E Z 9, etc., in Euclid others. 7 1

2

Aristotle Physics

areABCDEFGH, etc.,

in

Simson and

vii. 5.

Ibid. viii. 10.

3 Aristotle Anolytica posteriora L 5, p. 74 a 17. Reference taken from Gow, History of Greek Mathematics (Cambridge, 1884), p. 105, n. 3. 4

Euclid's Elements,

5

Euclid's Elements, p. 194-98.

Book 7. Book 7, Prop. 3

6

Euclid's Elements,

7

A. de Morgan in Companion

Book

7,

(ed. J. L.

Heiberg), Vol. II (1884),

Prop. 1 (ed. Heiberg), Vol. to the

II, p.

188-90.

British AlmanaCj for 1849, p. 5. 1

A HISTORY OF MATHEMATICAL XOTATIOXS

2

it was Apollonius of Perga who, like Archiinto numbers divided groups or myriads and spoke of double, medes, and so and finally of the "K fold" myriad. This on, myriads, triple 1 According to Pappus,

wish general expression of a myriad of as high an order as we may marks a decided advance in notation. Whether it wr as really due to Apollonius, or whether it was invented by Pappus, for the more eleT gant explanation of the Apollonian system, cannot now be determined. But Apollonius made use of general letters, in the manner observed in 2 Euclid, as did also Pappus, to an even greater extent. The small Greek letters being used to represent numbers, Pappus employed the

Greek

"The

3 capitals to represent general numbers. Thus, as Cantor says, possibility presents itself to distinguish as many general mag-

4 nitudes as there are capital letters." 5 It is of some interest that Cicero, in his correspondence, used have already seen that letters for the designation of quantities.

We

Diophantus used Greek letters for marking different powers of the unknown and that he had a special mark jj? for given numbers. We have seen also a symbol ru for known quantities, and yd and other symbols for unknown quantities (Vol. I, 106). 389. Middle Ages. The Indian practice of using the initial letters of words as abbreviations for quantities was adopted by the Arabs of the West and again by the translators from the Arabic into Latin. As examples, of Latin words we cite radix, res, census, for the unknown and its square; the word dragma for absolute number. In Leonardo of Pisa's Liber abbad (1202), 6 the general representation of given numbers by small letters is not irncommon. He and other writers of the Middle Ages follow the practice of Euclid. He uses letters in establishing the correctness of the rules for proving operations by casting out the 9's. The proof begins thus: "To show the foundation of this proof, let ,a.b. and .&. h>

W W

when ^ (

)

)

,

but by taking x*= 9z+10, where 27>25.

391. Vieta in 1591. The extremely important step of introducing the systematic use of letters to denote general quantities and general numbers as coefficients in equations is due to the great French alge-

work In artem analyticam isagoge (Tours, uses which are primarily representatives of letters 1591). capital lines and surfaces as they were with the Greek geometricians, rather than pure numbers. Owing to this conception, he stresses the idea of braist F. Vieta, in his

He

of the terms in an equation. However, he does not conhimself to three dimensions; the geometric limitation is aban-

homogeneity fine

Bruno Berlet, Adam Riese, sein Leben, seine Rechenbucher recknen. Die Cos von Adam Riese (Leipzig, 1892), p. 35-62. 1

2

Grammateus, Rechenbuch

mentar-Mathematik, Vol. II (2d 3

J.

4

M.

Tropfke, op. Cantor, op.

tit.,

(1518), Bl.

GUI;

J.

seine Art zu

Tropfke, Geschichte der Ele-

ed., 1921), p. 42.

Vol. II (2d ed., 1921), p. 42.

ciL, Vol. II

und

(?d ed., 1913), p. 489.

LBTTEKS REPEESENTIXG MAGNITUDES doned, solidum.

5

and he proceeds as high as ninth powers soHdo-solidoThe homogeneity is illustrated in expressions like "A

planum

+Z in B,"

the

term

the A is designated planum, a "surface/" so that be of the same dimension as is the second term, Z times B. If a letter B represents geometrically a length, the product of two B's represents geometrically a square, the product of three B s represents a cube. Vieta uses capital vowels for the designation of unknown quantities, and the consonants for the designation of known quantities. His own words are in translation: "As one needs, in order that one may be aided by a particular device, some unvarying, fixed and clear symbol, the given magnitudes shall be distinguished from the unknown magnitudes, perhaps in this way that one designate the required magnitudes with the letter A or with another vowel Ey 7, 0, 1 U, F, the given ones with the letters B, G, D or other consonants." first

may

J

392. Descartes in

1637.

A

geometric interpretation different

from that of Vieta was introduced by Rene Descartes in his La g&ometrie (1637). If 6 and c are lengths, then be is not interpreted as an area, but as a length, satisfying the proportion bc:b = c:l. Similarly,

With

-

is

a

c

line satisfying the proportion -:l c

Descartes,

if

b represents a given

= 6:c.

number it

is

number; a negative number would be marked Hudde2 who first generalized this procedure and let a tive

always a posib.

It

letter

was

B

J.

stand

a number, positive or negative. 393. Different alphabets. While the Greeks, of course, used Greek letters for the representation of magnitudes, the use of Latin letters became common during the Middle Ages. 3 With the development of other scripts, their use in mathematics was sometimes invoked. In 1795 J. G. Prandel expressed himself on the use of Latin type in algefor

braic language as follows:

"Why tion, while 1

2

Latin and Greek letters are chosen for algebraic calculaGerman letters are neglected, seems, in books composed in

Vieta, Isagoge (Tours, 1591), J.

Hudde, De

volume of F. Van

(Amsterdam, 1659),

Tom.

I,

fol. 7.

reductions aequationum (1657), published at the end of the first Schooten's second Latin edition of Rene* Descartes' Geometrie p. 439.

Vol. II (1907), p.

1,

See G. Enestrom in Encyclopedic des scien. math., n. 2; also Bibliotheca mathematica (3d ser.), Vol. IV

(1903), p. 208; The Geometry of Descartes, (Chicago, 1925), p. 301. 3

by Smith

See, for instance, Gerbert in (E-uvres de Gerbert,

p. 429-45.

&

Latham, Open Court

par A. OHeris

(Paris, 1867),

A HISTORY OF MATHEMATICAL NOTATIONS

6

our language, due to the fact that thereby algebraic quantities can be instantaneously distinguished from the Intermixed writing. In Latin, French and English works on algebra the want of such a convenience was met partly by the use of capital letters and partly by the use of italicized letters. After our German language received such development that German literature flourishes in other lands fully as well as the Latin, French and English, the proposal to use German letters in Latin or French books on algebra could not be recounted as

a singular suggestion." "The use of Greek letters in algebraic calculation, which has found wide acceptance among recent mathematicians, cannot in itself encumber the operations in the least. But the uncouthness of the Greek language, which is in part revealed in the shape of its alphabetic characters, gives to algebraic expressions a certain mystic appear1

ance." 2

Charles Babbage 3 at one time suggested the rule that all letters that denote quantity should be printed in italics, but all those which indicate operations should be printed in roman characters.

The

detailed use of letters

letters will try.

and

of subscripts

and superscripts

of

be treated under the separate topics of algebra and geome-

4

That even highly trained mathematicians may be attracted or by the experience on the theory of

repelled by the kind of symbols used is illustrated of Weierstrass who followed Sylvesters papers

algebraic forms until Sylvester began to employ Hebrew characters which caused him to quit reading. 5

394. Astronomical signs. We insert here a brief reference to astronomical signs; they sometimes occur as mathematical symbols. The twelve zodiacal constellations are divisions of the strip of the celestial 6 sphere, called the "zodiac"; they belong to great antiquity. these constellations are as follows: symbols representing

The

1

Johann Georg Prandel's Algebra (Miinchen, 1795),

3

Charles Babbage, art. "Notation/' in Edinburgh Encyclopedia (Philadelphia,

p. 4.

2

Ibid., p. 20.

1832). 4

Consult Vol. I, 141, 148, 176, 188, 191, 198, 342, 343; Vol. II, 395-401, 443, 444, 561, 565, 681, 732. 6

E. Lampe in Naturwissenschaftliche Rundschau, Vol. XII (1897), quoted by R. E. Moritz, Memorabilia mathematica (1914), p. 180. 6 Arthur Berry, Short History of Astronomy (New York, 1910), p.

W. W.

p. 361;

13,

14;

Bryan, History of Astronomy (London, 1907), p. 3, 4; R. Wolf, Geschichte der Astronomie (Mtinchen, 1877), p. 188-91; Gustave Schlegel, Uranographie chinoise, Vol. I (Leyden, 1875),

Book V, "Des zodiaques et

des plantes."

LETTERS EEPRESEXTIXG MAGNITUDES

7

T

Aries, the

==

Libra, the Balance

&

Taurus, the Bull Gemini, the Twins

^l

Scorpio, the Scorpion

^3

S

Cancer, the Crab Leo, the Lion

m

Virgo, the

n 2

The

Ram

Sagitarius 7 the Archer

I

Capricornus, the Goat Aquarius, the Water-Bearer

?

^

Maid

signs for the planets, sun,

moon,

Pisces, the Fishes

etc.,

are as follows:

O

Sun

11

Jupiter

C

Moon

b

Saturn

Earth

&

$

Mercury Yenus

23

Ascending node Descending node Conjunction

B

Mars

5



.

.

=

the equation

by

Jacobins words: "1st nemlich,

um

----

=

L

In

jj diese

VeraUgemeinerung

fiir

die

eine ungerade Zahlff'f" quadratischen Reste anzudeuten, p irgend Primzahlen bedeuverschiedene oder . wo /, /', f" gleiche so dehne ich die schone Legendre'sche Bezeichnung auf zusam.

.

.

.

.

.

.

,

ten,

/x\ -j,

mengesetzte Zahlen p in der Art aus, dass ich mit

(

wenn x zu p

---- bezeichne." Legendre's (j\ (^] (j\ 3 and Jacobi's symbols have been written also (N\c) and (x\p). A sign 4 similar to that of Legendre was introduced by Dirichlet in connection Primzahl

ist,

das Product

c~l

with biquadratic residues; according as

wrote

]

(

=+1

or

1,

A;

Another symbol analogous to that

the

number +1,

or

1,

of

A.

M. Legendre, Essai

or

1

(mod.

c),

he

Legendre was introduced by numbers; he designated by

according as k

residue of m, such that one has fcto-^s 1

=+1

Dickson5 writes

respectively.

Dirichlet 6 in the treatment of complex

rm\

4

ffcl

is,

or

is

not, quadratic

(mod. m), k and

m being

sur la theorie des nombres (Paris, 1798), p. 186.

XXX

(1846), p. 172; Werke, Vol. VI Jacobi, Crelle's Journal, Vol. der Konigl. Akademie d. Monatsberichten, from 262. Reprinted (Berlin, 1891), p. Berlin vom Jahr 18S7. See L. E. Dickson, op. ciL, Vol. Ill, p. 84. Wissenschaften 2

C. G.

3

G. B. Mathews, Theory of Numbers (Cambridge, 1892), p. 33, 42.

4

G. L. Dirichlet, Abhand.

J.

m

d.

K. Preussi&ch. Akad.

d.

Wissensch. (1833), p. 101-

21; Werke, Vol. I (1889), p. 230. 5

L. B. Dickson, op. dt., Vol. II, p. 370.

6

G. L. Dirichlet,

(1889), p. 551.

Crelle's Journal, Vol.

XXIV

(1842), p. 307; Werke, Vol. I

THEORY OF NUMBERS

31

complex integers and p the norm of m. Elsenstein employed the 1

sign

I

in biquadratic residues of

complex numbers to represent the

complex unit to which the power n^p~ l! (mod. m) is congruent, where m is any primar}' prime number, n a primary two-term prime number different from TTZ, p the norm of m. Jacobi 2 lets A {x} be the excess of the number of divisors of the form 4??i+l of x, over the number of divisors of the form 4?n+3, of x. His statement is "Sit B {x} numerus factorum ipsius x, qui formam 3

C^

4m +3 habent, Glaisher3 represents this excess, for a number n, by E(n). Dickson4 lets T (ri) be the excess T (n) or E' of the sum of the rth powers of those divisors of n which (whose

4M+1

habent,

facile patet, fore

numerus factorum, qui formam

A^=B

'

C (x

{x )

V

E

complementary

divisors) are of the

form

4w+l

sum

over the

of the

rth powers of those divisors which (whose complementary divisors) are of the form 4m+3. He also lets A(n) be the sum of the rth powers

n whose complementary divisors are odd. Baehand Landau6 lets T(ri), be the number of divisors of

of those divisors of

mann5

lets t(ri),

the positive integer n; Landau writes 2"(n)=ST(n), C = Euler's constant 0.57721 .... and R(x) = cT(x)x log -(2C-l)x, x>0.

= 7 (l) + T'(2) + -

Dickson

TM

lets

the sign

r(n)

....

+T(n)>

where

Bachmann8 designates by 5(n) the number of distinct prime divisors of n. Landau and Dickson9 designated by Og(T) a function whose quotient by g(T) remains nuis

the number of divisors7 of n.

merically less than a fixed finite value for all sufficiently large values of r. Before them Bachmann 10 used the form 0(n). The constant C, referred to above as "Euler's constant," was in1

G. Eisenstein,

2

C. G.

Crelle's Journal, Vol.

J. Jacobi,

XXX

(1846), p. 192.

Werke, Vol. I (1881), p. 162, 163; Dickson, op.

tit.,

Vol.

I,

p. 281.

W.

3

J.

4

L. E. Dickson, op. ciL, Vol.

5

P.

L. Glaisher, Froc.

London Math. I, p.

Bachmann, Encyklopddie

d.

Soc., Vol.

XV (1883), p.

104-12.

296.

math. Wissensch., Vol. I (Leipzig, 1898-

1904), p. 648. 6

Edmund

Landau, Nachrichten von der K.

Gesellschaft d.

Wissensch. ZM

Gottingen, Math.-Phys. Klasse (1920), p. 13. 7

L. E. Dickson, op.

8

P. Bachmann, op.

9

tit., rit.,

Vol.

L. E. Dickson, op. tit., Vol. Vol. I (1909), p. 31, 59. ,

10

I, p.

279.

p. 648. I, p.

305; Landau, Handbuch der

.... Prim-

P. Bachmann, Analytische Zahlentheorie (Leipzig, 1894), p. 401.

A HISTORY OF MATHEMATICAL NOTATIONS

32

1 troduced by Euler who represented it by the letter (7. We mention of numbers. it here, even though strictly it is not part of the theory Mertens2 designated it by a German capital letter E. It is known also 33 Mascheroni 4 in 1790 designated it by the as "Hascheroni's constant/ 5 6 letter A. This designation has been retained by Ernst Pascal. Gauss 7 "E. or W. Shanks adopted the designation wrote ^o=~

0,5772156. Eul. constant." The letter

E for "Eulerian constant" was adopted by Glaisher in 1871 and by Adams in 1878. Unfortunately, E is in used to designate danger of being confused with E2m sometimes 9

8

,

Eulerian numbers. In formulas for the number of different classes of quadratic forms 10 of negative determinant, Kronecker introduced the following nota-

by some later writers: n is any positive odd integer; r is any positive integer of the integer; m is any positive is form 8fc 1; s any positive integer of the form 8&+1; G(ri) is the number of non-equivalent classes of quadratic forms of determinant n, in which at is the number of classes of determinant TZ,; F(ri) tion which has been adopted

odd

is odd; X(ri) is the sum of all sum of all divisors of n; -&(ri) is the sum of >i/n, minus the sum of those which are

one of the two outer coefficients

least

divisors of n; $(n)

the divisors of i/n and of the divisors of the form 8fc3 which are \/n;

the

number

of

is

the

< y"n and of the divisors of the form 8k number of

of those of the

divisors of

form 4k

1

;

3 which n of the form 4&+1, minus

$(ri) is

the

number

of divisors

3fc+l, minus the number of those of the form half the number of solutions in integers of n =

n of the form

f

(j>(ri)

is

(n) 1

L. Euler,

p. 14-1-6;

M.

3fc

1

;

Commentani academiae Petropolitanae ad annum 1786, Tom. VIII, tit., Vol. Ill (2d ed.), p. 665; Vol. IV (1908), p. 277.

Cantor, op.

LXXVII

2

F. Mertens, Crete's Journal, Vol.

3

Louis Saalschiitz, Bernoullische Zahlen (Berlin, 1893),

(1874), p. 290. p. 193.

4

L. Mascheroni, Adnotationes ad calculum integralem Euleri (Pavia, 1790-92), Vol. I, p. 11, 60. See also Euleri Opera omnia (1st ser.), Vol. XII, p. 431. 6

Ernst Pascal, Repertorium

6

C. F. Gauss, Werke, Vol. Ill (Gottingen, 1866), p. 154.

7 * fl

10

d.

hoheren Mathematik, Vol. I (1900), p. 477.

W. Shanks, Proc. Royl Soc. of London, Vol. XV (1867), p. 431. J. W. L. Glaisher, Proc. Roy. Soc. of London, Vol. XIX (1871), p. J.

C. Adams, Proc. Roy. Soc. of London, Vol.

L. Kronecker in Crete's Journal, Vol.

LVII

XXVII

515.

(1878), p. 89.

(1860), p. 248.

THEORY OF NUMBERS

33

%=

2 2 of solutions in integers of +3.64?/ in which positive, negative, and zero values of x and y are counted for

^'(n) is half the

number

,

both equations.

Some

of the various

new

notations employed are indicated

by the

following quotation from Dickson: "Let x*fc) be the sum of the fcth powers of odd divisors of x; x'k(%) that for the odd divisors >j/ir; Xi'Oc) the excess of the latter sum over the sum of the kth powers of /; the odd divisors 1, but to be ( 1)* if m is a product of k distinct primes >1, while &i = L Mertens5 writes pn, Dickson, 6 ju(n), for the b m of Mobius. This function is sometimes named after Mertens.

-

Dirichlet 7 used the sign

,

when n and

5 are integers

and s^n>

YL

to designate the largest integer contained in -. Mertens 8

and

later

s

authors wrote write U-

for the largest integer

[x]

or [a: 6]. Dirichlet 10 denoted

the complex

number

a+fo",

rgrc.

Stolz

and Gmeiner

2 2 by N(a+bi) the norm a +6 of

a symbolism used by EL

J. S.

Smith 11 and

others. 1

L. E. Dickson, op.

2

M.

3

L. E. Dickson, op.

4

p.'

ctt.,

Vol.

I,

p. 305.

Mathematische Annalen, Vol.

A. F. Mobius in

Vol. IV,

ait.,

Vol.

I,

LX

(1905), p. 471.

p. 105, 109; see also Vol. II, p. 768.

Crette's Journal, Vol.

IX

(1832), p. Ill; Mobius, Werke,

598.

5

F. Mertens, Crelle's Journal, Vol.

6

L. E. Dickson, op. tiL, Vol.

7

p.

Lercli,

G. L. Dirichlet, AWiand.

d.

I,

LXXVII

(1874), p. 289.

p. 441.

K.

Preussisch. Akad. d. Wissensch. von 1849,

69-83; Werke, Vol. II (1897), p. 52. 8

F. Mertens, op.

cti.,

Vol.

LXXVII

(1874), p. 290.

Stolz und J. A. Gmeiner, Theoretische Arithmetik, Vol. I (2d ed.; Leipzig, 1911), p. 29. 9

10

G. L. Dirichlet,

11

H.

J. S.

Crette's Journal, Vol.

XXIV

(1842), p. 295.

Smith, "Report on the Theory of Numbers," Report British Associa-

tion (London, 1860), p. 254.

A HISTORY OF MATHEMATICAL NOTATIONS

34

The designation by ft of a multiple of the integer n is indicated in the following quotation from a recent edition of an old text: "Pour exprimer un multiple d'un nombre nous mettrons un point audessus de ce nombre:

un

ainsi

...

327

signifie

un multiple de 327

multiple coinmun aux deux nombres

...

,

a,c

...

signifie

1 a, et c."

408. Congruence of numbers. The sign 5= to express congruence numbers is due to C. F. Gauss (1801). His own words are:

of integral

"Numerorum congruentiam hoc signo, =, in posterum denotabimus, 16=9 (mod. 5), modulum ubi opus erit in clausulis adiungentes, 7= 15 (modo II)." 2 Gauss adds in a footnote: "Hoc signtim propter

magnam

analogiam quae inter aequalitatein atque congruentiam in-

venitur adoptavimus.

ment,

Ob eandem caussam

ill.

Le Gendre

in

com-

laudanda ipsum aequalitatis signum pro conquod nos ne ambiguitas oriatur imitari dubitavi-

infra saepius

gruentia retinuit,

mus."

The objection which Gauss expressed to Legendre's double use of the sign = is found also in Babbage 3 who holds that Legendre violated the doctrine of one notation for one thing by letting = mean: = (1) ordinary equality, (2) that the two numbers between which the placed will leave the same remainder when each is divided by the same given number. Babbage refers to Peter Barlow as using in his Theory of Numbers (London, 1811) the symbol // placed in a horiis

407). Thus, says Babbage, Legendre used the 1 (mod. p), and the symbolism a = Gauss 1, Barlow the symbolism a fep. Babbage argues that "we ought not to multiply the number of mathematical symbols without necessity."

zontal position (see

symbolism a*=

tt

tt

A

more recent writer expresses appreciation of Gauss's syrabol: "The invention of the symbol ss by Gauss affords a striking example

of the advantage which may be derived from appropriate notation, '4 in the development of the science of arithmetic. 7

and marks an epoch

Among the earliest writers to adopt Gauss's symbol was

Rramp

C.

of Strassbourg; he says: "J'ai adopte de meme la signe de congruence, propose par cet auteur, et compos6 de trois traits parall&les, au lieu de 1 Claude-Gaspar Bachet, Problemes plaisants Labosne; Paris, 1874), p. 13.

et

d&ectdbles (3d ed.,

2 C. F. Gauss, Disquisitiones arithmeticae (Leipzig, 1801), art. 2; Vol. I (Gottiagen, 1863), p. 10.

3

par A. Werke,

Charles Babbage, art. "Notation" in the Edinburgh Cyclopaedia (Phila-

delphia, 1832). 4

G. B. Mathews, Theory of Numbers (Cambridge, 1892), Part

I,

sec. 29.

THEORY OF

35

Ce signe nf a paru essentiel pour toufe cette partie de Panalyse, admet les settles solutions en nombres enters, taut positifs qne ~ Is sometimes used for "incongraent" 2 negatifs." The sign 409. Prime and relatively prime numbers. Peanos designates a T prime by A P Euler* lets 3-D stand for the number of positive integers 4eux. qui

1

.

D

not exceeding 'D which are relatively prime to ("denotet character Ti-D multitudinem istam numeromm ipso minorum, et qui cum eo

D

nullum habeant divisorem communem"). Writing n for D, Eider's xD was designated (n) by Gauss, 5 and T(ri) (totient of ri) 6 by Sylvester. Gauss's notation has been widely used; it is found in DedekincFs edition of Dirichlet's Vorlesungen t^&er Zahlentheorie7 and in Wertheim's Zahlentheorie.* Jordan9 generalized Euler's ^rD function, and represented by [n, k] the number of different sets of k (equal or distinct) positive integers gn, whose greatest common divisor is prime to n. In place of Jordan's [n, k] Story 10 employed the k 11 symbol ( n\ some other writers #*(n), and Dickson Jk(ri). Meissel designates by $(n) the nth prime number, 12 so that, for instance, ^(4) =5, and by rev (reversio) the function which is the 13 opposite of ^, so that "rev ^(X)==T^ rev (x)~x" and by E rev (m) = 14 t?(m) the number of primes in the natural series from 1 to m inclusive, function

T

X

CL Kramp,

"Notations," Siemens d'AnthmMique Unwerselle

(Cologne,

1808). 2

See, for instance, L. 1923), p. 38.

E. Dickson, Algebras and Their Arithmetics (CMcago,

G. Peano, Formidaire matMmatique, Tom. IV (1903), p. 68. Acta Acad. Petrop., 4 II (or 8), for the year 1755 (Petrograd, 1780), p. 18; Commentationes arithmeticae, Vol. II (Petrograd, 1849), p. 127; L, E. Dickson, op. 3 4

ctt.,

Vol.

I, p.

61, 113.

5

C. F. Gauss, Disguisitiones ariihmeticae (Leipzig, 1801), No. 38. See also article by P. Bachmann and E. Maillet in Encyclopedic des sciences mathematiqiLes,

Tome 6

I,

VoL

J. J.

7

III (1906), p. 3.

Sylvester, Philosophical

Magazine (5th

ser.),

VoL

XV

(1883), p. 254.

P. G. L. DiricHet, Vorlesungen uber Zahlentheorie, herausgegeben

Dedekind (3d

ed.;

von R.

Braunschweig, 1879), p. 19.

8

G. Wertheim, Anfangsgrunde der Zahlentheorie (Braunschweig, 1902), p. 42.

9

C. Jordan, Traite des substitutions (Paris, 1870), p. 95-97.

10

W.

11

L. E. Dickson, op.

12

E. Meissel,

13

E. Story, Johns

E. Meissel, 14 E. Meissel,

HopMns University Circulars cti., VoL I, p. 147, 148.

}

Crelle s Journal, ibid., p. 307. ibid,, p.

313.

VoL XLVHI,

9

p. 310.

Vol. I (1881), p. 132.

A HISTORY OF MATHEMATICAL NOTATIONS

36

2 which Dickson 1 represents by 0(m). Landau writes v(x) for the number of primes ^x in the series 1, 2, ---- [x], where [x] is the largest that x. He also lets f(x) be a simpler function of x, such ,

integer

3.^00

.

8 R. D. Carmichael uses H{y] to represent

the index of the highest power of the prime p dividing y, while Strids4 m to denote the index of the highest power of the prime p berg uses

H

which divides ml

Sums

410. as

a sign

of numbers.

Leibniz 5 used the long letter

of integration in his calculus,

of integers.

J

not only

but also as the sign for the

For example, he marked the sum

of triangular

sum

numbers

thus:

"1+3+ 6+10+etc. = l+4+10+20+etc.= fjx 1+5+15+25+etc. = This practice has been followed by some elementary writers; but certain modifications were introduced, as, for example, by De la Caille

6

who

lets

f stand for the sum

Jo C(K) stand for the

lets

for the

sum

of certain

sum

numbers,

of their cubes, etc.

of the rath

J

for the

Bachmann 7

powers of the divisors of

fc,

m

mJm

8 of Cambridge used the being any given odd number. A. Thacker n W W = z where is an integer and n a .... 1 +z notation 4>0) +2 9 of the ftth powers of the sum be lets Dickson k (ri) integer.

+

,

positive



Vol.

1

L. E, Dickson, op.

2

Edmund Landau, Handbuch und

(Leipzig 3

tit.,

I, p.

429.

der Lehre von der Verteilung der Primzahlen

Berlin), Vol. I (1909), p. 4.

B. D. Canniehael, Bull Amer. Math.

<E, Stridsberg, Arkiv for Matematik, See L. E. Dickson, op. ciL, Vol. I, p. 264.

Soc., Vol.

XV

(1909), p. 217.

Astr., Fysik., Vol.

VI

(1911),

No.

34.

5 G. W. Leibniz, "Historia et origo calculi differeritialis" in Leibnizens Mathematische Schriften (ed. C. I. Gerhardt), Vol. V (1858), p. 397. 6 D. de la Caille, Lectiones elementares mathematicae .... in Latinum tra-

a C(arolo) S.(cherffer)e S. J. (Vienna, 1762J, p. 107. Encyklopadie d. Math. Wiss., Vol. 1 (1900-1904), p. 638.

ductae 7

.

.

.

.

8

A. Thacker,

9

L. E. Dickson, op.

Crelle's Journal, Vol. tit.,

Vol.

I,

XL

p. 140.

(1850), p. 89.

THEORY OF

37

the integers ^n and prime to n. Sylvester 1 in 1866 wrote Syfl to express the sum of all the products of j distinct numbers chosen from 2 writes Sn m for Sylvester's 1, 2, ....,{ numbers. L. E. Dickson -

,

also

Sn

sign

F(a K) stands

+ (p-l)*.

l*+2+

The' sign Sn has been variously used for the sum of the nth powers of all the roots of an 3 algebraic equation. The use of S for "sum" is given in 407,

and

Sj ti ,

The

nomial, a and

for

}

for a

N being integers.

4

homogeneous symmetric polyDickson5 introduces the symbol JL

''The separation of two sets of numbers by the symbol JL shall denote that they have the same sum of fcth powers n. Chr. Goldbach noted that a+ft+d, a+y+8, j3+y+ for fe = 1, , in the following quotations

:

B^a+d, 0+, y+3 a+ft+y+3."

8,

}

411. Partition of numbers. Euler's "De partitione numeronim" 6 contains no symbolism other than that of algebra. He starts out with

and

indi-

indicates in

what

the product (l+a^r)(l+^)(l+a^a)(l+^)(l+a^j8r), cates its product

l+Pz+Qz*+Rz*+Sz*+,

etc.

and considers the term Nxnzm "whose coefficient number n may be the sum of m

various ways the the series a, ft y,

etc.,

by

5, e,

different terms of

f etc." ,

7

Chrystal uses a quadripartite symbol to denote the number of partitions. "Thus, P(n\p\q) means the number of partitions of n

*

into p parts, the greatest of which is g" "P(n\ >q) means the number of partitions of n into any number of parts, no one of which is to exceed q; Pu(n\>p\^)j the number of partitions of n into p or any less number of unequal parts unrestricted in magnitude." "P(n|$l, 2, 22 , ;

|

2s ... .) the number of partitions of n into any number of parts, each part being a number in the series 1, 2, 22 2s ... ." C. G. J. Jacob! represents the number of partitions, without repe8 tition, of a number n, by N(n=l*Xi+2*X2+3*x^, x^Q. Sylvester, ,

,

,

1 J. J. 2

Sylvester, Giornale di matematiche, Vol.

L. E. Dickson, op.

3

See, for instance, 1834), p. 133, 134.

tit.,

Vol.

M. W.

IV (NapoK,

Drobiseh, Roh&re numerische Gleichungen (Leipzig,

4

L. E. Dickson, op. c&, Vol.

6

L. E. Dickson, op. at, Vol. II, p. 705.

I,

p. 84.

6 L. Euler, Introductw in analysin infinUorum, Vol. 1797), chap, xvi, p. 253.

7

1866), p. 344.

I, p. 95, 96.

I,

Editio

Nova (Lugduni

G. Chrystal, Algebra, Part II (Edinburgh, 1889), p. 527, 528.

8 J. J.

"On the

Sylvester, Collected Mathematical Papers, Vol. II (Cambridge, 1908), Partitions of Numbers/' p. 128.

A HISTORY OF MATHEMATICAL NOTATIONS

38

using the terms "denumerant" and "denumeration," writes CUT

"QU in its explicit form