I

THE

THEORY OF

DETERMINANTS

in the

HISTORICAL ORDER

OF DEVELOPMENT

THE THEORY OF DETERMINANTS

IN THE

HISTORICAL ORDER

OF DEVELOPMENT

BY

THOMAS MUIR

FOUR VOLUMES BOUND AS TWO

Volume One: GENERAL AND SPECIAL DETERMINANTS

UP TO 184L

Volume Two: THE PERIOD 1841 TO 1860.

DOVER PUBLICATIONS, INC. NEW YORK NEW YORK

All rights reserved.

This new Dover edition, first published in 1960, is an unabridged and unaltered republi- cation of the following:

Volume One, second edition, originally pub- lished in 1906.

Volume Two, first edition, originally pub- lished in 1911.

Volume Three, first edition, originally pub- lished in 1920.

Volume Four, first edition, originally pub- lished in 1923.

The original work appeared in four volumes. Volume One of the Dover edition contains Vol- umes One and Two of the original edition, and Volume Two of the Dover edition contains Volumes Three and Four of the original edition.

This edition is published by special arrange- ; ment jyith^t. Martin's Press Incorporated.

JAN

*^^gyrry of

l0lT289

Manufactured in the United States of America

Dover Publications, Inc.

180 Varick Street

New York 14, N. Y.

THE THEORY OF DETERMINANTS

in the

HISTORICAL ORDER

OF DEVELOPMENT

VOLUME ONE Part I. GENERAL DETERMINANTS UP TO 1841.

Part II. SPECIAL DETERMINANTS UP TO 1841.

All rights reserved.

PREFACE.

The main object of this work and the contents of it will be found specified in the Introductory Chapter. It is intended for the student who aims at acquiring such a knowledge as can only be got by a study of the subject in the historical order of its development, for the investigator who is* specially interested in this branch of mathematics and wishes to become acquainted with the various lines of attack opened up by previous workers, and for the general working mathematician who requires guide- books and books of reference concerning special domains.

T. M.

Capetown, South Afkica, IQth July, 1905.

CONTENTS	•
PART I.
CHAPTER I niODUCTION,	.	PAOKB 1-5
CHAPTER H
NERAL Determinants, 1693-1779,	.	6-62
/Leibnitz (1693),	. pp. 6-10
Fontaine (1748),			10-11
^Cramer (1750),			11-14
Bezout (1764),			14-17
' Vandermonde (1771),			17-24
Laplace (1772),			24-33
Lagrange (1773),			33-37
(1773),			37-40
(1773),			40-41
Bezout (1779),			4l-o2

CHAPTER in.

General Determinants, 1784-1812,

Hindenburg (1784), (1800),

ROTHE

Gauss

(1801),

pp. 53-55 55-63 63-66

53-79

viii	CONTENTS
MONGE HiRSCH BiNET	(1809), . . . (1809), . . . (1811), . . .	pp. 67-68 69 69-71	PAOKS
Prasse	(1811), . (1811), .	71-72 72-78
Wronski	(1812), . . . CHAPTER IV.	78-79
General Determinants, 1812,	.	80-131
^ BiNET ■^ Cauchy	(1812), . . . (1812), . CHAPTER V.	pp. 80-92 92-131
General Determinants: Retrospect, .		132-133 with Table.

CHAPTER VI.

Gergonne	(1813), .	•'j	pp. 134-135
Garnier	(1814), .		. 135-136
Wronski	(1815), .			136
Desnanot	(1819), .			. 136-148
Cauchy	(1821 \			. 148-150
Scherk	(1825),. .			150-159
SCHWEINS	(1825), .			159-175
	CHAPTER VII. .
ERAL Determinants, 1827-1835, . . . .
Jacobi	(1827), .	pp. 176-178
Reiss	(1829), .		. 178-187
Cauchy	, (1829), .			187-188
Jacobi	(1829), .			188-193
Minding	(1829), .			194-197
Drinkwater	(1831), .			198-199
Mainarui	(18:i2), .			200-206
Jacobi	(1831-33 ,			206-212
»	(1834), .			212-213
>»	(1835),			214

134-175

176-214

CONTENTS

IX

CHAPTER VIII. General Determinants, 1836-1840,

Grunert	(1836),
Lebesgue	(1837),
Reiss	(1838),
Catalan	(1839),
Sylvester	(1839),
MOLINS	(1839),
Sylvester	(1840),
RiCHELOT	(1840),
Cauchy	(1840),
Sylvester	(1841),
Craufurd	(1841),	• .

CHAPTER. IX General Determinants, 1841,

^Cauchy (1841),

^ Jacobi (1841),

/Cauchy (1841),

CHAPTER X

General Determinants : Retrospect,

PART IL CHAPTER XL

AXISYMMETRIC DETERMINANTS, 1773-1841,

pp.

pp.

ROTHE	(1800),
BiNET	(1811).
Jacobi	(1827),
Cauchy	(1829),
Jacobi	(1831),
>j	(1832),
j>	(1833),
>>	(1834),
Lebesgue	(1837),
Jacobi	(1841),
Cauchy	(1841),

pp.

215-219 219-220 220-224 224-226 227-235 235-236 236-238 238-240 240-243 243-245 245-246

247-253 253-272 273-285

290-292 292-293 293-294 295-296 296-297 297-298 298-300 300-301 301-303 304 304-305

215-246

247-285

286-288

with Table.

289-305

X	CONTENTS
	CHAPTER XII.
Alternants, 1771-1841,	.
Prony	(1795), . .		pp.	306-308
CAyCHY	(1812), . .			308-310
SCHWEINS	(1825), . .		-.	311-322
Sylvester	(1839), .			322-325
Jacobi	(1841), . .			325-342
Cauchy	(1841), . .			342-345
	CHAPTER Xni.
Jacobians, 1815-	1841, ....	,	^ ^
Cauchy	(1815), . .		pp.	346-349
>j	(1822), . .			349
Jacobi	(1829), . .			349-352
>i	(1830),			352-354
>i	(1832-33),			354-356
Catalan	(1839), . .			356-358
Jacobi	(1841), .			358-392
Cauchy	(1841), . .			393-394
	CHAPTER XIV.
Skew Determinants, 1827-1845, .	,	, ,
Pfaff	(1815), . . .	pp.	396-401
Jacobi	(1827), . . .		401-405
»	(1845), . .		.	405-406

306-345

346-394

395-406

CHAPTER XV Orthogonants, 1827-1841,

Jacobi

» Cauchy Jacobi

Lebesgue

Catalan

Cauchy

(1827), (1827), (1829), (1831), (1832), (1833), (1837), (1839), (1826), (Postscript),

pp.

410-415 415-424 425-435 435-451 452 453-463 463-467 467-470 470-471

407-471

CONTENTS

XI

CHAPTER XVI.

Miscellaneous,	1811-1841,
Wronski	(1812),
>i	(1815).
»	(1816-17),
»	(1819),
SCHERK	(1825),
SCHWEINS	(1825),
Jacobi	(1835),
Sylvester	(1840),

pp. 472-474 474-476 476-478 478 478-479 479-485 485-487 487

472-487

CHAPTER XVII. Retrospect on Special Forms,

488

Index to the Numbered Results in Part L, .

489

List of Authors whose Writings are Reported on, 490-491

I

CHAPTER I.

INTEODUCTION.

The way in which the material for a history of the theory of Determinants has been accumulated is quite similar to that which has been observed in the case of other branches of science. In the middle of the eighteenth century one of the indepen- dent discoverers of the fundamental idea, viz., Cramer, was fortunate enough to attract attention to it, and in time it became the common property of mathematicians in France and else- where. As it slowly spread it naturally also received accretions and developments, and of the dozen or so of writers who thus handled it in the sixty years that followed Cramer's publication there were of course a few who by a more or less casual refer- ence kept alive the memory of some of their predecessors. It was then taken up by Cauchy, and, thanks to the prestige of his name and to the inherent excellence of his extensive mono- graph, its position as a theory of importance became more firmly assured. The thirty years that followed Cauchy's memoir resembled the sixty that preceded it, save that the number of contributors was considerably larger. Then another great analyst, Jacobij, the most noteworthy of those contributors, produced in Germany a monograph similar in extent and value to Cauchy's, and the importance of the subject in the eyes of mathematicians became still more enhanced. As a consequence, the single decade following gave rise to quite as many new contributions as the preceding three decades had done, and closed with the appearance of the first separately published elementary treatise on the subject, viz., Spottiswoode's. The

2 HISTORY OF THE THEORY OF DETERMINANTS

preface to this contains the first notable historical sketch of the theory, and includes references to the writings of twelve outstanding mathematicians, beginning with Cramer (1750) and ending with the author's own contemporaries, Cayley, Sylvester and Hermite. In the same year (1850) there also occurred something out of the ordinary, for the correspondence between Leibnitz and the Marquis de I'Hopital having been published from manuscripts in the Royal Library at Hanover, the striking discovery was made that more than half-a-century before Cramer's time the fundamental idea of determinants had been clear to Leibnitz, and had been expounded with consider- able fulness by him in a letter to his friend. So strongly attractive had the subject now become to mathematicians that in the single year succeeding the publication of Spottiswoode's short treatise a greater number of separate contributions to the theory made their appearance than in the whole sixty-year period from Cramer to Cauchy. The wants of students every- where had to be attended to : a second edition of Spottiswoode was consequently prepared for Crelles Journal in 1853 ; a text- book by Brioschi was published at Pa via in 1854 ; French and German translations of Brioschi in 1856 ; and an elementary exposition by Bellavitis in 1857. So far as historical material is concerned, the last-mentioned work was of little account ; that of Brioschi resembled Spottiswoode's, the number of references, however, being greater. Of quite a different char- acter was the text-book by Baltzer, which was published at Leipzig the year after the German translation of Brioschi had appeared at Berlin, an important part of the new author's plan being to deal methodically with the history of the subject by means of footnotes. On the enunciation of almost every theorem a note with historical references was added at the foot of the page, the result being that in the portion (thirty- four pages) devoted expressly to the pure theory of determinants about as many separate writings are referred to as there are pages. This was a marked advance, and although during the next twenty years the publication of text-books became more frequent — in fact, if we include those of every language and of every scope, we shall find an average of about one per

INTRODUCTION 3

year — Baltzer's dominated the field; enlarged editions of it appeared in 1864, 1870, and 1875, and the historical notes grew correspondingly in number. Of the other text-books only one, Giinther's, which was published in 1875, sought to follow the historical line taken by Baltzer and to add to the supply of material. Then in 1876 another new departure took place, this being the year in which the first writings were published which dealt with the history alone, the one being an academic thesis by E. J. Mellberg printed at Helsingfors, and the other a memoir presented by F. J. StudniSka to the Bohemian Society of Sciences.

About this time, while engaged in writing my own so-called " Treatise on the Theory of Determinants," I had occasion to look into the question of the authorship and history of the various theorems, and I was reluctantly forced to the conclusion that much inaccurate statement prevailed in regard to such matters and that the whole subject was worthy of serious investigation. A resolution was accordingly taken to set about collecting the titles of all the writings which had appeared on the theory up to the end of 1880. The task was not an easy one, as will readily be understood by those who know how scanty and defective are the bibliographical aids at the disposal of mathematicians, and how often the titles given by investi- gators to their memoirs are imperfect and even misleading in regard to the nature of the contents. The outcome of the search was published in 1881 in the October number of the Quarterly Journal of Mathematics (vol. xviii. pp. 110-149) under the title of "A List of Writings on Determinants." It contained 589 entries arranged in chronological order. Some three or four years afterwards, when there had been time to test the completeness of the earlier portion of the list, the writings included in it were taken up in historical succession and suitable abstracts or reviews of them made for publication in the Proceedings of the Royal Society of Edinburgh ; the first contribution of this kind was presented to the Society in the beginning of the year 1886. At the same time there was being prepared an additional list of writings containing omitted titles, 84 in number, belonging to the period of the

4 HISTOEY OF THE THEORY OF DETERMINANTS

first list, and 176 titles belonging to the further period 1881- 1885. This second list appeared in 1886 in the June number of the Quarteiiy Journal of Mathematics (vol. xxi. pp. 299- 320). In 1890 a collection was made of the contributions, just mentioned, which had up to that date been printed in the Edinburgh Proceedings, and with the consent of the Society was published separately. Unfortunately in that year all this train of work had to be laid aside on account of the pressure of official duties, and ten years elapsed before it could be resumed. It was thus not imtil March 1900 that a second series of analytic abstracts began to appear in the Edinburgh Proceedings, and that the preparation of a third list of writings was methodically undertaken. The period to be covered by this list was the fifteen years 1886-1900; and as the number of writers interested in the subject had in these years continued to increase, and as closer examination of the literature of the previous periods had led to new finds, the resulting compilation was more extensive than the first two put together. It was presented to the South African Association for the Advancement of Science at its inaugural meeting in April 1903 and was pub- lished in the Report; it is also to be found in the Quarterly Journal of Mathematics for December 1904 and February 1905 (vol. xxxvi. pp. l7i-267). The number of titles in the three lists is about 1740 ; they furnish, it is hoped, an almost complete guide to the literature of the theory of determinants from the earliest times to the close of the nineteenth century.

From these later labours it became manifest that it was undesirable in the way of separate publication to issue merely another volume as a continuation of, and similar to, that of the year 1900. The better course clearly was to reproduce the material of that volume along with the intercalations necessi- tated in it by the existence of subsequently discovered papers, and to follow this up in such a way as to give finally within the compass of a reasonably sized volume a full history of the subject in all its branches up to about the middle of the nineteenth century. This is what is here attempted.

The plan followed is not to give one connected history of determinants as a whole, but to give separately the history of

INTRODUCTION 5

each of the sections into which the subject has been divided, viz., to deal with determinants in general, and thereafter in order with the various special forms. This will not only tend to smoothness in the narrative by doing away with the necessity of frequent barkings back, but it will also be of material im- portance to investigators who may wish to find out what has already been done in advancing any particular department of the subject. To this end, also, each new result as it appears will be numbered in Roman figures ; and if the same result be obtained in a different way, or be generalised, by a subsequent worker, it will be marked among the contributions of the latter with the same Roman figures, followed by an Arabic numeral. Thus the theorem regarding the effect of the transposition of two rows of a determinant will be found under Vandermonde. marked with the number xi., and the information intended thus to be conveyed is that in the order of discovery the said theorem was the eleventh noteworthy result obtained: while the mark XI. 2, which occurs under Laplace, is meant to show that the theorem was not then heard of for the first time, but that Laplace contributed something additional to our knowledge of it. Li this way any reader who will take the trouble to look up the sequence xi., xi. 2, xi. 3, &c., may be certain, it is hoped, of obtaining the full history of the theorem in question.

The early foreshadowings of a new domain of science, and tentative gropings at a theory of it, are so difficult for the historian to represent without either conveying too much or too little, that the only satisfactory way of dealing with a subject in its earliest stages seems to be to reproduce the exact words of the authors where essential parts of the theory are concerned. This I have resolved to do, although to some it may have the effect of rendering the account at the commence- ment somewhat dry and forbidding.

CHAPTER 11.

DETEEMINANTS IN GENERAL, FROM THE YEAR 1693 TO 1779.

The writers here to be dealt with are seven in number, viss., Leibnitz, Fontaine, Cramer, Bdzout, Vandermonde, Laplace, Lagrange. Of these the first two exercised no influence on the development of the theory; the real moving spirit was Cramer; Lagrange alone of the others may have been un- affected by this particular part of Cramer's work.

LEIBNITZ (1693).

[Leibnizens mathematische Schriften, herausg. v. C. I. Gerhardt. 1 Abth. ii. pp. 229, 238-240, 245. Berlin, 1850.]

In the fourth letter of the published correspondence between Leibnitz and De L'Hospital, the former incidentally mentions that in his algebraical investigations he occasionally uses numbers instead of letters, treating the numbers however as if they were letters. De L'Hospital, in his reply, refers to this, stating that he has some diflSculty in believing that numbers can be as convenient or give as general results as letters. Thereupon Leibnitz, in his next letter (28th April 1693), pro- ceeds with an explanation : —

"Puisque vous dites que vous av^s de la peine k croire qu'il soit aussi general et aussi commode de se servir des nombres que des lettres, il faut que je ne me sois pas bien expliqu^. On ne s^auroit douter de la generality en considerant qu'il est permis de se servir de 2, 3, etc., comme d' a ou de b, pour veu qu'on considere que ce ne sont pas de nombres veri tables. Ainsi 2.3 ne signifie point 6 mais autant qu' ab. Pour ce qui est de la commodity, il y en a des tres

DETERMINANTS IN GENERAL (LEIBNITZ, 1693) 7

grandes, ce qui fait que je m'en sers souvent, sur tout dans les calculs longs et diflBciles ou il est aise de se tromper. Car outre la commodity de T'epreuve par des nombres, et m^me par I'abjection du novenaire, j'y trouve un tres grand avantage meme pour I'avancement de I'Analyse. Comme c'est une ouverture assez extraordinaire, je n'en ay pas encor parl^ k d'autres, mais voicy ce que c'est. Lorsqu'on a besoin de beaucoup de lettres, n'est il pas vray que ces lettres n'expriment point les rapports qu'il y a entre les grandeurs qu'elles signifient, au lieu qu'en me servant des nombres je puis exprimer ce rapport. Par exemple soyent proposees trois equations simples pour deux inconnues k dessein d'oster ces deux inconnues, et cela par un canon general. Je suppose

10+lla;+12y = 0 (1)

et 20 + 21a; + 22y = 0 (2)

et 30 + 31a; + 32y = 0 (3)

ou le nombre feint estant de deux characteres, le premier me marque de quelle equation il est, le second me marque k quelle lettre il appartient. Ainsi en calculant on trouve par tout des harmonies qui non seulement nous servent de garans, mais encor nous font entrevoir d'abord des regies ou theoremes. Par exemple ostant premierement y par la premiere et la seconde equation, nous aurons :

+ 10.22 + 11.22X

= 0 (4)* -12.20-12.21.. ^ '

et par la premiere et troisieme nous aurons :

+ 10. 32 + 11. 32a;

= 0 (5) -12.30-12.31.. ^

ou il est aise de connoistre que ces deux equations ne different qu'en ce que le charactere antecedent 2 est change au charactere antecedent 3. Du reste, dans un m^me terme d'une meme equation les characteres antecedens sont les memes, et les characteres posterieurs font une m^me somme. II reste maintenant d'oster la lettre z par la quatrieme et cinquieme equation, et pour cet effect nous aurons t

lo-2i	h	lo	22	h
I1.22	\ -	= li	2o	32
l2-2o	3i	h	2i	3o

qui est la derniere equation delivr^e des deux inconnues qu'on vouloit oster, et qui porte sa preuve avec soy par les harmonies qui se remar- quent par tout, et qu'on auroit bien de la peine k decouvrir en

* This is written shortly for +10.22 + 11.22a;=0

+ 10.22 + 11.22a;=01 -12.20-12.21a;=oJ'"

+ The author here slightly changes his notation. What is meant to be indi- cated is

10.21.32 + 11.22.30 + 12.20.31 = 10.22.31 + 11.20.32 + 12.21.30.

8 HISTOEY OF THE THEORY OF DETERMINANTS

employant des lettres a, b, c, sur tout lors que le nombre des lettres et des equations est grand, line partie du secret ~de I'analyse consiste dans la caracteristique, c'est k dire dans I'art de bien employer les notes dont on se sert, et vous voyes, Monsieur, par ce petit echantillon, que Viete et des Cartes n'en ont pas encor connu tons les mysteres. En poursuivant tant soit peu ce calcul on viendra k un theweme general pour quelque nombre de lettres et d'equations simples qu'on puisse prendre. Le voicy comme je I'ay trouve autres fois :

" Datis aequationibus quotcunque sufficientibus ad toUendas quantitates, quae simplicem gradum non egrediuntur, p-o aequatione prodeunte, pimo sumeiidae sunt omnes combinationes possibiles, quas ingreditur una tanium coefficiens uniuscujusque aequationis : secundo, eae combinationes opposita habent signa, si in eodem aequationis prodeuntis latere ponantur, quae habent tot coefficientes communes, qvx)t sunt unitates in numero quantitatum tollendarum unitate minuto : caeterae habent eadem signa.

"J'avoue que dans ce cas des degr^s simples on auroit peut estre decouvert le meme theoreme en ne se servant que de lettres k I'ordinaire, mais non pas si aisement, et ces adresses sont encor bien plus necessaires pour decouvrir des theoremes qui servent k oster les inconnues mont^es k des degres plus haiits. Par exemple, . . . . "

It will be seen that what this amounts to is the formation of a rule for writing out the resultant of a set of linear equations. When the problem is presented of eliminating x and y from the equations

a+bx+cy = 0, d-\-ex+fy = 0, g+hx+ky = 0, Leibnitz in effect says that first of all he prefers to write 10 for a, 11 for b, and so on; that, having done this, he can all the more readily take the next step, viz., forming every possible product whose factors are one coefficient from each equation,* the result being

10.21.32, 10.22.31, 11.20.32,

11.22.30, 12.20.31, 12.21.30;

and thatj then, one being the number which is less by one than

the number of unknowns, he makes those terms different in

sign which have only one factor in common.

The contributions, therefore, which Leibnitz here makes to algebra may be looked upon as three in number : —

(1) A new notation, numerical in character and appearance, for individual members of an arranged group of magnitudes; the two members which constitute the notation being like the

* Of course, this is not exactly what Leibnitz meant to say.

DETERMINANTS IN GENERAL (LEIBNITZ, 1693) 9

Cartesian co-ordinates of a point in that they denote any one of the said magnitudes by indicating its position in the group. (l.)

(2) A rule for forming the terms of the expression which equated to zero is the result of eliminating the unknowns from a set of simple equations. (ll.)

(3) A rule for determining the signs of the terms in the said result. (ill.)

The last of these is manifestly the least satisfactory. In the first place, part of it is awkwardly stated. Making those terms different in sign ivhich have only as many factors alike a» is indicated by the number which is less by one than the number of unknown quantities is exactly the same as making those terms different in sign which have only two f(jbctor» different. Secondly, in form it is very unpractical. The only methodical way of putting it in use is to select a term and make it positive ; then seek out a second term, having all its factors except two the same as those of the first term, and make this second term negative ; then seek out a third term, having all its factors except two the same as those of the second term, and make this third term positive ; and so on.

Although there is evidence that Leibnitz continued, in his analytical work, to use his new notation for the coefficients of an equation (see Letters xi., xii., xiii. of the said correspondence), and that he thought highly of it (see Letter viii. " chez moi c'est une des meilleures ouvertures en Analyse "), it does not appear that by using it in connection with sets of linear equations, or by any other means, he went further on the way towards the subject with which we are concerned. Moreover, it must be remembered that the little he did effect had no influence on succeeding workers. So far as is known, the passage above quoted from his correspondence with De L'Hospital was not published until 1850. Even for some little time after the date of Gerhardt's publication it escaped observation, Lejeune Dirich- let being the first to note its historical importance. It is true that during his own lifetime, Leibnitz's use of numbers in 'place of letters was made known to the world in the Acta Erudi- torum of Leipzig for the year 1700 (Responsio ad JDn. Nic. Fata DuilleHi imputationes, pp. 189-208); but the particular

10 HISTORY OF THE THEORY OF DETERMINANTS

application of the new symbols which brings them into con- nection with determinants was not there given.

In a subsequent volume of Leibnizens mathematische Schrif- ten, — the third volume of the second Abtheilung, — published at Halle in 1863, the following equivalent of the above ' theor^me gdndral ' appears (pp. 5-6) : —

"Inveni Canonem pro tollendis incognitis quotcunque aequationes non nisi simplici gradu ingredientibus, ponendo aequationum numerum excedere unitate numerum incognitarum. Id ita habet.

Fiant omnes combinationes possibiles literarum coeflScientium ita ut nunquam concurrant plures coefficientes ejusdem incognitae et ejusdem aequationis. Has combinationes affectae signis, ut mox sequetur, oomponuntur simul, compositumque aequatum nihilo dabit aequationem omnibus incognitis carentem.

Lex signorum haec ist. Uni ex combinationibus assignetur signum pro arbitrio, et caeterae combinationes quae ab hac diflPerunt coefficien- tibus duabus, quatuor, sex etc. habebunt signum oppositum ipsius signo : quae vero ab hac differunt coefficientibus tribus, quinque, septem etc. habebunt signum idem cum ipsius signo. Ex. gr. sit

10 + lla;+12y = 0, 20 + 21z+22y = 0, 30 + 31a; + 32y = 0; fiet +10.21.32-10.22.31-11.20.32

+ 11 .22.30 + 12.20.31-12.21.30 = 0.

CoeflBcientibus eas literas computo, quae sunt nuUius incognitorum, ut 10, 20, 30."

Although Gerhardt, the editor, states that the original manu- script of Leibnitz, from which this is taken, bears no date, it is very probable to date farther back than 1 693, and not impassible to belong to 1678.*

FONTAINE (1748).

[M^moires donnes a 1' Academic Royale des Sciences, non im- primis dans leurs temps. Par M. Fontaine t de cette Academic. 588 pp. Paris, 1764.]

These memoirs of Fontaine's, sixteen in number, cover a con- siderable variety of mathematical subjects : it is the seventh of

* See also Gerhakdt, K. I., Leibniz fiber die Determinanten, SUzungab

Akad. d. Wiss. (Berlin), 1891, pp. 407-423.

tThe full name is Alexis Fontaine des Berlins. The very same collection was issued in 1770 under the less appropriate title Trait6 de calctd diffirentiel et integral. Vaudermonde is said to have been a pupil of Fontaine's (v. Nouv. Annalea de Math., v. p. 155).

DETERMINANTS IN GENERAL (FONTAINE, 1748) 11

the series which indirectly concerns determinants. There is not, however, even the most distant connection between it and the work of Leibnitz. The heading is "Le calcul integral. Seconde methode," the sixth memoir having given the first method. The date is indicated in the margin.

The matter which concerns us appears as a lemma near the beginning of the memoir (p. 94). The passage is as follows : —

" Soient quatre norabres quelconques

al, a2, aS, aA, at quatre autres nombres aussi quelconques

al, a2, a3, a4 ; faites al a'2 - al a2 = a^l,

a2 a3 - a2 aS = a}'2,

a3 a4 - a3 a4 = a^3,

al a3 - al a3 = a^l,

a2 a4 - a2 a4 = 0^2,

al a4 - al a4 = a^l, vous aurez aH a}2 - an aH + aU d>3 = 0."

Manifestly this is the identity which in later times came to be written

K62I • ks&J - kAI • M4I + Kh\ • 1^2^31 = 0,

and which, so far as we know, appeared first in its proper con- nection in the writings of B^zout. (xxill.) It is curious to note that Fontaine was not satisfied with the lemma in this form, but proceeded to take " autant de nombres quelconques que Ton voudra, al, a2, . . . . , alO, . . . ," and wrote the identity one hundred and twenty -six times before he appended " et cetera," the 126th being

a^6 an - a^e a?n + a>^ a^S = 0.

CRAMER (1750).

[Introduction a I'Analyse des Lignes Courbes alg^briquea (Pp. 59, 60, 656-659.) Geneve, 1750.]

The third chapter of Cramer's famous treatise deals with the different orders (degrees) of curves, and one of the earliest theorems of the chapter is the well-known one that the equation

12 HISTORY OF THE THEORY OF DETERMINANTS

of a curve of the nth. degree is determinable when ^71(71+8) points of the curve are known. In illustration of this theorem he deals (p. 59) with the case of finding the equation of the curve of the second degree which passes through Jive given points. The equation is taken in the form

A + By + Cx + T>yy + Exy+xx = 0;

the five equations for the determination of A, B, C, D, E are written down ; and it is pointed out that all that is neceasary is the solution of the set of five equations, and the substitution of the values of A, B, C, D, E thus found, " Le calcul vdritable- ment en seroit assez long," he says; but in a footnote there is the remark that it is to algebra we must look for the means of shortening the process, and we are directed to the appendix for a convenient general rule which he had discovered for obtaining the solution of a set of equations of this kind. The following is the essential part of the passage in which the rule occurs : —

" Soient plusieurs inconnues z, y, z, v, &c., et autant d'^quations

Ai = Zh + Y^y + X^x + Yh + &c. A2 = Zh + Y^y + X^ + Yh} + &c. A^ = Z^z + Y^y + X^x + YH + &c. A* = Z^z + YV + X^ + V*» + &c. &c.

ou les lettres A^, A^, A^, A*, &c., ne marquent pas, comma k I'Drdinaire, les puissances d'A, mais le premier membre, suppos6 connu, de la premiere, seconde, troisi^me, quatrieme, &c. Equation."

[Here the solutions of the cases of 1, 2, and 3 unknowns are given, and he then proceeds.]

"L'examen de ces Formules fournit cette R^gle g^n^rale. Le nombre des Equations et des inconnues ^tant n, on trouvera la valeur de chaque inconnue en formant n fractions dont le d^nominateur com- mun a autant de termes qu'il y a de divers arrangements de n choses diff6rentes. Chaque terme est compos6 des lettres ZYXV, &c., toujours Rentes dans le mSme ordre, mais auxquelles on distribue, comme exposants, les n premiers chiffres ranges en toutes les mani^res possibles. Ainsi, lorsqu'on a trois inconnues, le d^nominateur a [1x2x3 = ] 6 termes, composes des trois lettres ZYX, qui recoivent successivement les exposants 123, 132, 213, 231, 312, 321. On donne k ces termes les signes + ou - , selon la Regie suivante. Quand un exposant est suivi dans le meme terme, m^diatement ou imm^diate- ment, d'un exposant plus petit que lui, j'appellerai cela un cUrangement.

DETERMINANTS IN GENERAL (CRAMER, 1750) 13

Qu'on compte, pour chaque terme, le nombre des derangements : s'il est pair ou nul, le terme aura le signe + ; s'il est impair, le terme aura le signe - . Par ex. dans le terme Z^Y^X^ il n'y a aucun derangement; ce terme aura done le signe +. Le terme Z^Y^X^ a aussi le signe +, parce qu'il a deux derangements, 3 avant 1 et 3 avant 2. Mais le terme Z^Y^X^ qui a trois derangements, 3 avant 2, 3 avant 1, et 2 avant 1, aura le signe - .

"Le d^nominateur commun ^tant ainsi forme, on aura la valeur de z en donnant k ce d^nominateur le numerateur qui se forme en changeant, dans tous ces termes, Z en A. Et la valeur d'y est la fraction qui a le m^me denominate ur et pour numerateur la quantite qui r^sulte quand on change Y en A, dans tous les termes du d^nominateur. Et on trouve d'une maniere semblable la valeur des autres inconnues."

It is evident at once that the new results here given are —

(1) A rule for fovming the terms of the common denominator of the fractions which express the values of the unknowns in a set of linear equations. (iv.)

(2) A rule for determining the sign of any individual term in the said common denominator (and, included in the rule, the notion of a "derangement"). (iii. 2)

(3) A rule for obtaining the numerators from the expression for the common denominator. (v.)

The problem which Cramer set himself at this point in his book was exactly that which Leibnitz had solved, viz., the elimination of n quantities from a set oi n + 1 linear equations. The solution which Cramer obtained, and which, be it remarked, was the solution best adapted for his purpose, was quite distinct in character from that of Leibnitz. Leibnitz gave a rule for writing out the final result of the elimination; what Cramer gives is a rule for writing out the values of the n unknowns as determined from n of the n -f 1 equations, after which we have got to substitute these values in the remaining (n + l)th. equation. The notable point in regard to the two solutions is, that Cramer's rule for writing the common denominator of the values of the n unknowns (an expression of the 7ith degree in the coefficients) is exactly Leibnitz's rule for writing the final result, which is an expression of the (■?i + l)th degree. Had either discoverer been aware that the same rule sufficed for obtaining both of these expressions, he could not have failed, one would think, to

14 HISTORY OF THE THEORY OF DETERMINANTS

note the recurrent law of formation of them. The result of eliminating w, x, y, z from the equations,

arW + hrX + Cry + drZ = Cr (r = l, 2, 3, 4, 5)

is, according to Leibnitz, if we embody his rale in a later symbolism, \aAcsd,e,\ = 0;

whereas, according to Cramer, it is —

M«2VAI M0^2VAI M«2VAI M«2VAI ^'

and from the collocation of these the one natural step is to the identity

-|«iV3^4«5l = ail«2VAI+M«2«3^Ai+ • • • -eil^zVAI- The fate of Cramer's rule was very diiSerent from that of Leibnitz'. It was soon taken up, and after a time found its way into the schools, where it continued for many years to be taught as the nutshell form of the theory of the solution of simultaneous linear equations. Indeed Gergonne is reported* to have said, " Cette methode ^tait tellement en faveur, que les examens aux ^coles des services publics ne roulaient, pour ainsi dire, que sur elle ; on etait admis ou rejet^ suivant qu'on la possedait bien ou mal."

Finally, the exact difference between Cramer's notation for the coeflBcients of the unknowns and the notation of Leibnitz should be noted, and in connection therewith the fact that when dealing with the subject of elimination between two equations of the mth and -nth degrees in x Cramer uses a notation closely resembling that which Leibnitz employed, viz., [1^] [l^j, &c.

BfiZOUT (1764).

[Recherches sur le degre des t^quations r^sultantes de I'^vanouis- sement des inconnues, et sur les moyens qu'il convient d'employer pour trouver ces Equations. — Hist, de VAcad. Roy. des Sciences, Ann. 1764 (pp. 288-338), pp. 291-295.]

The object of B<^zout's memoir is sufficiently apparent from the title; we may therefore at once give those portions of it

♦By Studniiika. But see Kliigel's WGrterhtich d. rtinen Math., Suppl. II.. p. 67.

DETERMINANTS IN GENERAL (BEZOUT, 1764) 15

which directly concern our subject. On p. 291 is the commence- ment of the following passage : —

"M. Cramer a donne una regie g^n^rale pour les exprimer toutes d^barrassees de ce facteur : j'aurois pu m'en tenir k cette r^gle ; mais I'usage m'a fait connoitre que quoiqu'elle soit assez simple, quant aux lettres, elle ne Test pas de m§me k regard des signes lorsqu'on a au-del4 d'un certain nombre d'inconnues k calculer ; . . . .

Lemme I.

" Si Ton a un nombre n d'equations du premier degr^ qui renferment

chacune un pareil nombre d'inconnues, sans aucun terme absolument

connu, on trouvera par la r^gle suivante la relation que doivent avoir

les coefficiens de ces inconnues pour que toutes ces Equations aient lieu.

"Soient a, b, c, d, &c., les coeflSciens de ces inconnues dans la

premiere equation. a', V, c', d', &c., les coeflBciens des m^mes inconnues dans

la seconde equation, a", h", c", d", &c., ceux de la troisieme & ainsi de suite. " Formez les deux permutations ah & ha & ecrivez ah -ha-, avec ces deux permutations & la lettre c formez toutes les permutations possibles, en observant de changez de signe toutes les fois que c changera de place dans ah & la meme chose k I'egard de ha ; vous aurez

ahc - ach + cah- hoc + bca- cha.

Avec ces six permutations & la lettre d, formez toutes les permutations possibles, en observant de changer de signe a chaque fois que d changera de place dans un meme terme ; vous aurez

ahcd - ahdc + adhc - dabc - acbd + acdh - adch + dach + cahd - cadh + cdab - dcab - bacd + hade - hdac + dhac + bead - bcda + bdca - dhca - chad + cbda - cdba + dcba

& ainsi de suite jusqu'i ce que vous ayez 6puise tous les coefficiens de la premiere equation.

*' Alors conservez les lettres qui occupent la premiere place ; donnez k celles qui occupent la seconde, la meme marque qu'elles ont dans la seconde Equation ; a celles qui occupent la troisieme, la meme marque qu'elles ont dans la troisieme Equation, & ainsi de suite ; egalez en fin le tout k z^ro et vous aurez I'equation de condition cherchee.

" Ainsi si vous avez deux equations et deux inconnues comme

ax +by =0 a'x + b'y = 0 r^quation de condition sera ab' -ha' = 0 ou ah' -a'b = 0 . . . ." In the same way the next two cases are given ; then —

". . . . mais comme ces Equations de condition doivent servir de fonnules pour l'6limination dans les Equations de diff6rens degr6s, il

16 HISTORY OF THE THEORY OF DETERMINANTS

convient de leur donner une forme qui rende les substitutions le moins penibles qu'il se pourra ; pour cet effet, je les mets sous cette forme :

ab' -a'b = 0

(ab' -a'b)c" +{a"b -ab")c' +{a'b" -a"b')c = 0 [(ab' -a'b)c" +(a"b -ab")c' +(a'b" -a"b')c ]d"' + [(a'b -ab')c"' +{ah"' -a"'b)c' +{a"'b' -a'b"')c\d" + [{a"'h - ab'") c" + (a6" - a"b) c'" + {a"b"' - a"'b") c ] d' + [{a'b'" - a"'b') c" + {a"'b" - a"b"') c' + {a"b' - a'b") c'"] d = 0.

Cette nouvelle forme a deux avantages : le premier, de rendre les substitutions k venir, plus commodes; le deuxieme, c'est d'offrir une regie encore plus simple pour la formation de ces formules.

"En effet, il est facile de remarquer 1°, que le premier terme de I'une quelconque de ces equations, est forme du premier membre de I'equation precedente, multiplie par la premiere' des lettres qu'elle ne renferme point, cette lettre etant affectee de la marque qui suit imme- diatement la plus haute de celles qui entrent dans ce meme membre.

" 2°. Le deuxieme terme se forme du premier, en changeant dans celui-ci la plus haute marque en celle qui est immediatement au-dessous & r6ciproquement, & de plus en changeant les signes.

"3°. Le troisieme, se forme du premier, en changeant dans celui-ci la plus haute marque en celle de deux numeros au-dessous & reciproque- ment, & de plus en changeant les signes.

" 4°. Le quatri^me, se forme du premier, en changeant dans celui-ci la plus haute marque en celle de trois numeros au-dessous & reciproque- ment, & changeant les signes, & toujours de meme pour les suivans.

" Par exemple, ....

"D'apres ces observations, il sera facile de voir que I'equation de condition pour cinq inconnues et cinq equations, sera

The latter part of this we are drawn to at once, as it enunciates quite clearly the Recurrent Law of Formation to which attention has above been directed. It has to be observed, however, that the three 'equations of condition' are not in the form got by merely following the ' rule,' and that by deriving each 'terme,' not from the first but from the preceding ' terme ' we should obtain, viz. :

ah' —ah = 0,

(ah' -a'h)c" -{ah" -a"h)c' +{ah" -a"h')c = 0, [{ah' -a'h)c" -{ab" -a"h)c' +{a'h" -a"h')c]d'" -[{ah' -a'b)c'" -{ab'" -a"'h)c' +{a'b'" -a'"h')c]d" + [{ab" -a"b)c"' -{ah'" -a"'b)c" +{a"h'" -a"'h")c-\d' -[{a'b" -a"h')o'" -{a'b'" -a'"b')c" +{a"h'" -a"'h")c']d = 0.

DETEEMINANTS IN GENERAL (VANDERMONDE, 1771) 17

The notable point in regard to the earlier portion is, that Bezout throws his rule of term-formation and his rule of signs into one. In the case of finding the resultant of arX + bry + CrZ = 0 (r=i, 2, 3) his process consists of four steps, viz. : — (l)a,

(2) ab \ —h a,

(3) ah c —a ch +cab —h a c -\-b c a —ch a,

(4) a^b^c^ - a^c^b^ + c^a^b^ - b^a^c^ + b^^c^a^ - c^b^a^ .

The first term of (2) is got from (1) by affixing b, and the second is got from the first by advancing the b one place and changing the sign. The first term of (3) is got from the first term of (2) by affixing c, the second term is got from the first by advancing c a place and changing the sign, and the third is got from the second by advancing c a place and changing the sign ; the last three are got from the second term of (2) in the same way as the first three are got from the first term of (2).

It will thus be seen that while Leibnitz and Cramer direct us to find the permutations in any way whatever, and thereafter to fix the sign of each in accordance with a rule, Bezout requires the permutations to be found by a particular process, and attention given to the question of sign throughout all this process, so that when the terms have been found their signs have likewise been determined.

Bezout's contributions to the subject thus are —

(1) A combined rule of term-formation and! , «. . . ^^ ^ . n . \ (n- 2;-f(iii. 3)

rule or signs. j

(2) The recurrent law of formation of the new functions, (vi.)

VANDERMONDE (1771).

[Memoire sur I'elimination. Hist, de VAcad. Roy. des Sciences (Paris), Ann. 1772, 2« partie (pp. 516-532).]

This important memoir of Vandermonde and that of Laplace, which is dealt with immediately afterwards, both appear in the History of the French Academy of Sciences for 1772, Laplace's

18 HISTORY OF THE THEORY OF DETERMINANTS

memoir occupying pp. 267-376, and Vandermonde's pp. 516-532. There is, however, a footnote to the latter, which states that it was read for the first time to the Academy on 12th January 1771. The part of it which concerns us is the first article, which treats of elimination in the case of equations of the fijrat degree. Vandermonde here writes : —

"Je suppose que I'on represent* par , t , &c., 990 ^^»

I 9 3

„ o q &C., &c,, autant de differentes quantites generales, dont Tune

quelconque soit " una autre quelconque soit r &&, & que le produit

des deux soit design^ a I'Drdinaire par '^ 7

"Des deux nombres ordinaux a & a, le premier, par exemple,

designera de quelle equation est pris le coefficient — & le second

**^ designera le rang que tient ce coefficient dans I'equation, comme on le verra ci-apres.

"Je suppose encore le syst^me suivant d'abreviations, & que Ton fasse

a\P a P a P a\ h~ a.h h .a

a\h\c a.h\c h .e\a c.a\V

a\P\y\^_a P\y\B a f3\y\S a ft\y\S a fi]y\S a\b\c\d a.SJcJd b .c\d\a'*' e,d \a\b d.a\b\c

.|/3|y|8|c| aP\y\8\. a\b\e\d\e\ a.b\c\d[e ' ' ' '

"Le symbole J — L sert ici de caracteristique. Les seules chosos I I a observer sont I'Drdre des signes, et la loi des permutations entre les lettres a, b, c, d, &c., qui me paroissent suffisamment indiquees ci-dessus.

"Au lieu de transposer les lettres a, b, c, d, &c., on pouvoit les laisser dans I'ordre alphabetique, & transposer au contraire les lettres a, P, y, S, &c., les resultats auroient ete parfaitement les memes ; ce qui a Ueu aussi par rapport aux conclusions suivantes.

" Premi^rement, il est clair que ^4^ represente deux tennes diflFerena,

a\ b

I'un positi^ & I'autre negatif, resnltans d'autant de permutations

DETERMINANTS IN GENERAL (VANDERMONDE, 1771) 19

possibles de a & b; que ° * 7 d represente six, trois positifs & trois

a\ b\ c

negatife, r^sultans d'autant de permutations possibles de a, b, & e;

que^JAMi

"^""^ a\b\c\d

" Mais de plus, la formation de ces quantit^s est telle que I'unique cbangement que puisse resulter d'une permutation, quelle qu'elle soit, feite entre les lettres du meme alphabet, dans I'une de ces abreviations, sera un cbangement dans le signe de la premiere valeur.

" La demonstration de cette verite & la rechercbe du signe resultant d'une permutation determinee, dependent generalement de deux pro- positions qui peuvent etre enonc^es ainsi qu'il suit, en se servant de nombres pour indiquer le rang des lettres.

" La premiere est que

1 |2|3f...|mlw + l|...»

1 j2|3i...|TO|TO+l |...»

112 13 1... ln-TO + 1 |n-TO + 2i»-m + 3|...| n

= ±

m|m+liwi + 2|...| n I 1 i 2 |...|to-1

le signe - n'ayant lieu que dans le cas oil n & m sont I'un & I'autre des nombres pairs. " La seconde est que

l|2l3|...lm|TO+l|...|n l|2j3|...tm|m+l|...|«

l!2i3i...!m-l| m I to + 1 Im-|-2 I. . . !»

I|2j3|:..|w-ljm-i-lj m |m+2|..,j»

"II sera facile de voir que, la premiere equation supposee, celle-ci n'a besoin d'etre prouvee que pour un seul cas, comme, par exemple, celui de m = n - 1, c'est-a dire, celui ou les deux lettres transposees sont les deux demieres.

" Au lieu de demontrer generalement ces deux equations, ce qui exigeroit un calcul embarrassant plut6t que difficile, je me contenterai de developper les exemples les plus simples : cela suffira pour saisir I'esprit de la demonstration.

(2^ pages are occupied with verifications for the case of

—\-r, of 7 , and of — i'^rij-) a\b a 1 6 1 c a\b\c \df

" On verra qu'en general la demonstration de notre seconde equation pour le cas n = a, depend de cette meme equation pour le cas n = a — 1,

quel que soit a: d'oii il suit que puisque y-^= ~9iT' ®^® *** generalement vraie. ^ 1 ^ "^ I ^

20

HISTORY OF THE THEORY OF DETERMINANTS

" De ce que nous avons dit jusqu'ici il suit que

H/^|y|g| • • • • 0

a\b\c\d\ . . . ." '

si deux lettres quelconques du mime alphabet sont 4gales entr'elles; car quelque part que soient les deux lettres ^gales, on pent les transposer aux deux derni^res places de leur rang, ce qui ne fera au plus que changer le signe de la valeur ; alors, de leur permutation particuliere, il ne pent, d'une part, resulter aucun changement, puisqu'elles sont ^gales; d'autre part, selon notre seconde equation ci-dessus, il doit en resulter un changement de signe ; cette contradiction ne pent ^tre lev^e qu'en supposant la valeur zero ....

" Tout cela pos^ ; puisque Ton a identiquement,

1 |1 |2^1 1|2 1 j_L2 1 J_|2_ l|2|3""l.2|3"*'2.3i V'^S. 1 |2~ '

2|1 [2_2 1 |2 2 1^2 2 1 |2 1 |2|3~1.2|3'^2.3|l"*'3.1 |2~ '

si Ton propose de trouver les valeurs de ^1 et de ^2 qui satisfont aux deux Equations

1.^1 + 2.^2 + 3 = 0

1.^1 + 2.^2 + 3 = 0, on pourra comparer, & Ton aura

1|2 1|2

^1 =

213

^2

_3|1

112

(Three equations with three unknowns are similarly dealt with.)

" II est clair que ces valeurs n'ont point de facteurs inutiles : mais pour les rendre aussi commodes qu'il est possible dans les applications, & particuli^rement dans celles oil Ton veut faire usage des loga- rithmes, il sera bon d'y employer le plus qu'il se pourra, la multiplication des facteurs complexes. J'observe done 1° que si Ton substitue dans

'^'^'-z, les valeurs des - ^^ P' en ^4-^> d a\ b\ c a\ b

le d^veloppement de

a \ b \ c\d' '~~ ~~ a\ b\ c

aura, en r^duisant & ordonnant, d'apr^s les observations ci-dessus,

on

a\l\y\S^!^a\l a\ b \ c \d

+

b

I,

0 I c

a|j8

c\d

\8 \d'

W

\8

\b

a I c

b\d y\8

d

7 I ^ bTc

DETERMINANTS IN GENERAL (VANDERMONDE, 1771) 21

si de m^me on substitue dans le developpement des ^-V-p-r^J — rl> l^s

a\3\ ISIe al;8l IS a|o|c|a|e|/

valeurs des — 7 P' , ; en "'\,\'\ ,, on aura, en reauisant & ordon-

a\ 0 \ c \a\e a\ 0 \ c \d

nant, d'apres les observations ci-dessus,

a\b c a|/3 7

+

h\ c a

c\d a

a|^|y|g|HC / , I I

a\ b\c\d\e\f \ die a

e\f a

C a|/^ y\8\^\C I

d\e\f a\c b\d\e\f a\d b\c\e\f

siHC I I I I . I III

d\e\f b\d a\c\e\f^ b\e a\c\d\f

+

b\e\f c\e a\ b\d\f c\f a\c\d\e

b\c\f d\f a\b\c\e

b\c\d

+

a\e b\c\d\f a\f b\c\d\e

b\f a\c\d\e

" La loi des permutations & des signes est assez manifesto dans ces exemples, pour qu'on en puisse conclure des developpemens pareils pour les cas de huit & dix lettres, &c., du meme alphabet ; alors, en eniployant les premiers developpemens pour les cas d'un nombre impair de ces lettres, on aura les formules d'elimination du premier degre, sous la forme la plus concise qu'il soit possible.

" Si Ton veut exprimer ces formules, generalement pour un nombre n d'^quations

1 . ^1 + 2 . ^2 + 3 . ^3 + . . . +m.$7n +

1.^1 + 2.^2 + 3.^3+ . . . +m.^m + &c.

+ 71.^/1 + (W+1) = 0 2 2

+ n.$n + {n + l} = 0

la valeur de I'inconnue quelconque ^m, sera renferm^e dans I'^quation suivante, k une seule inconnue

11213

^m

1

1|2|3|.. 2 I 3 I \n-m\n-m + l\n-m + 2\n-m + 3\...\ n

m+l\m + 2\'m + :i\

n+1

1

Iw-l

= 0

le signe + ayant lieu seulement dans le cas oil m & w sont impairs I'un & I'autre."

22 HISTORY OF THE THEORY OF DETERMINANTS

Taking this up in order, we observe first that Vandermonde proposes for coefficients a positional notation essentially the same

as that of Leibnitz, writing „ where Leibnitz wrote 12 or Ij.

Then he defines a certain class of functions by means of their recurrent law of formation — a law and class of functions at once seen to be identical with those of Bezout. A special symbolism is used for the first time to denote the functions; thus, the expression

lo.2j.32 + li.22.3o + l2.2o.3i — lo.22.3i — li.2o.32 — l2.2i.3o, which occurs in Leibnitz's letter, Vandermonde would have denoted by

1|2|3 1 I 2 I 3'

and the result of eliminating x, y, z, w from the set of equations

l^+2^y+3^+4^w=0 (r=l, 2, 3, 4) by

1 |2|3|4 1 I 2 I 3 I 4-

It is next pointed out that permutation of the under row of indices produces the same result as permutation of the upper row, that the number of terms is the same as the number of permutations of either row of indices, and that half of the terms are positive and half negative.

The part which follows this is a little curious. The proposition is brought forward that if in the symbolism for one of the functions a transposition of indices takes place in either row, the same function is still denoted, the only change thereby possible being a change of sign. The demonstration is affirmed to be dependent on two theorems, neither of which is proved, as the proofs are said to be troublesome to set forth. Now it will be seen that the second of these theorems is to the eifect that the transposition of any two consecutive indices causes a change of sign, and that consequently this alone is sufficient for the required demonstration. The first of the auxiliary theorems, in fact, is an immediate deduction from the second, the particular permutation which it concerns being produced by (ti— m+l)(m— 1) transpositions of pairs of consecutive indicea

DETERMINANTS IN GENERAL (VANDERMONDE, 1771) 23

Passing over the illustrations of these propositions, we come next to the theorem that if any two indices of either row be equal the function vanishes identically, and we note particularly that the basis of the proof is that the interchange of the two indices in question changes the sign of the function, and yet leaves the function unaltered.

Upon this theorem the solution of a set of simultaneous linear equations is then with much neatness made t6 depend. In more modem notation Vandermonde's process is as follows: — It is known that

and Ctg I ^1^2 I + ^2 I Cia2 1 + ^^2 I «1^2 I = I «2^lC2 I = ^>

and a 1-^ 4. h \^^ 4- c - 0

hence, if the equations

be given us, we know that

. JaAT l«AI

IS a solution.

This result, moreover, is generalised ; the solution of

r^x^-\-r^^ + . , . + r„a;„ + r„+i = 0 (r=l, 2, , , . w) being fully and accurately expressed in symbols, although the numerators of the values of x^, x^, . . . ,Xn are not in so simple a form as Cramer's rule for obtaining the numerator from the denominator might have suggested.

Lastly, and almost incidentally, Vandermonde makes known a case of the widely general theorem nowadays described as the theorem for expressing a determinant as an aggregate of products of complementary minors. His case is that in which the given determinant is of the order 2m, and one factor of each of the products is of order 2.

Summing up, therefore, we must put the statement of our indebtedness to Vandermonde as follows : —

24 HISTOEY OF THE THEORY OF DETERMINANTS

(1) A simple and appropriate notation for the new functions, e....^|. ^ (VII.)

(2) A new mode of defining the functions, viz., using sub- stantially Bdzout's recurring law of formation. (vm.)

(3) The remark that the ordinary algebraical expression of any of the functions is obtainable by permutation of either series of indices. (ix.)

(4) The remark that the positive and negative terms are equal in number. (x.)

(5) The theorem regarding the effect of interchanging two consecutive indices. i^^-)

(6) The theorem (with proof) regarding the effect of equality of two indices belonging to the same series. (^iif)

(7) A reasoned-out solution of a set of n simultaneous linear equations, by means of the new functions as above defined, (xiii.)

(8) Expression of any of the new functions of order 2m as an aggregate of products of like functions of orders 2 and 2m — 2. (XIV.)

In addition to this, we must view Vandermonde's work as a whole, and note that he is the first to give a connected exposition of the theory, defining the functions apart from their connections with other matter, assigning them a notation, and thereafter logically developing their properties. After Vandermonde there could be no absolute necessity for a renovation or reconstruction on a new basis : his successors had only to extend what he had done, and, it might be, to perfect certain points of detail. Of the mathematicians whose work has thus far been passed in review, the only one fit to be viewed as the founder of the theory of determinants is Vandermonde.

LAPLACE (1772).

[Recherches sur le calcul integral et sur le systeme du monde. Hist, de I' Acad. Roy. des Sciences (Paris), Ann. 1772, 2® partie (pp. 267-376) pp. 294-304. CEtivres, viii. pp. 365-406.]

In the course of his work Laplace arrives at a set of linear equations from which n quantities have to be eliminated.

DETEEMINANTS IN GENERAL (LAPLACE, 1772) 25

This he says can be accomplished by means of rules which mathematicians have given : —

" Mais comme elles ne me paroissent avoir ^t^ jusqu'ici d^montrees que par induction, et que d'ailleurs elles sont impracticables, pour peu que le nombre des Equations soit considerable ; je vais reprendre de nouveau cette matiere, et donner quelques proced^s plus simples que ceux qui sont d6jk connus, pour ^liminer entre un nombre queiconque d'equations du premier degre."

Taking n homogeneous linear equations with the coefficients

la, 16, ic, . . . . %, ^b, ^c, . . . .

he first gives Cramer's rule for writing out what he, Laplace, calls the Resultant, using in the course of the rule the term variation instead of Cramer's term " derangement." Then he gives the "perhaps simpler" rule of Bdzout, and shows that of necessity it will lead to the same result as Cramer's.

The theorem in regard to the effect of transposing two letters is next enunciated, and the blank left by Vandermonde is filled, for a proof of the theorem is given. The exact words of the enunciation and proof are —

" Si au lieu de combiner d'abord la lettre a avec la lettre b, ensuite ces deux-ci avec la lettre c, et ainsi de suite ; c'est-^-dire, si au lieu de combiner les lettres a, b, c, d, e, &c., dans I'ordre a, b, c, d, e, &c., on les efit combin^es dans I'ordre a, c, b, d, e, &c., ou a, d, b, c, e, &c., ou a, e, b, c, d, &c., ou &c., je dis qu'on auroit toujours eu la m^me quantite k la difference des signes prfes.

"Pour demontrer ce Theor^me nommons en general, r^sultante, la quantite qui resulte de I'une queiconque de ces combinaisons, en sorte que la premiere rhultante soit celle qui vient de la combinaison suivant I'ordre a, b, c, d, e, &c., que la seconde r^sultante soit celle qui vient de la combinaison suivant I'ordre a, c, b, d, e, &c., que la troisihme rdsultante soit celle qui vient de la combinaison suivant I'ordre a, d, b, c, e, &c., et ainsi de suite; cela pose, il est clair que toutes ces resultantes renferment le meme nombre de termes, et precisement les m^mes, puisqu'elles renferment tous les termes qui peuvent resulter de la combinaison des n lettres a, b, c, d, e, &c., dispos^es entre elles de toutes les manieres possibles ; il ne peut done y avoir de difference entre deux resultantes, que dans les signes de chacun de leurs termes ; or, il est visible que la premiere r^sultante donne la seconde, si I'on change dans la premiere b en c, et reciproqucment c en b; mais ce changement augmente ou diminue d'une unite le nombre des variations

26 HISTOEY OF THE THEORY OF DETERMINANTS

de chaque terme ; d'ou il suit que dans la seconde r^sultante, tous les termes dont le n ombre des variations est impair, auront le signe +, et les autres le signe - ; partant, cette seconde resultante n'est que la premiere, prise n^gativement.

" II est visible pareillement que . . . ." &c.

The proof is thus seen to consist in establishing (1) that the terms of the one " resultant " must, apart from sign, be the same as those of the other ; and (2) that the terms of the one resultant are either all affected with the same sign as the like terms of the other, or are all affected with the opposite sign, the comparison of sign being made by comparing the number of variations.

After this, the theorem that when two letters are alike the resultant vanishes is established in a way different from Vandermonde's, but not more satisfactory, viz., by considering what B^zout's rule would lead to in that case.

Application is then made to the problem of elimination, and to the solution of a set of linear simultaneous equations, the mode of treatment being again different from Vandermonde's, but this time with better cause. He says —

" Je suppose maintenant que Ton ait les trois Equations

0 = ^a.fi. + ^.fjL + ^c./i",

0 = ^a.fi + %.fj! + '^c./i",

0 = ^a./x + %.^ + h.fi',

je forme d'abord la resultante des trois lettres a, b, c, suivant I'ordre a, b, c, ce qui donne,

W^b.h - M.^c.% + hM^b - ^bMM + ^b.hM - hJbM

ou \i.[^b.h - 2c.36j + 2rt.[ic.36 _ 13.3c] + Sa.\%^c - h.%];

je multiplie ensuite la premiere des equations prt^c^dentes par %.h - H.%, la seconde par ^c.% - ^b.h, la troisi^me par ^.'^c - ^c.'^b, et je les ajoute ensemble, ce qui donne,

0 = fi.[ia.(26.3c - H.%) + 2rt.(ic.36 - ^b.H) + ^a{^b.^c - k.%)]

+ /i'.[16.(26.3c - 2c. 36) + %,(ic.^b - ^b.H) + 36(16. 2c - h.'^b)]

+ /*".[! C.(26.3c - 2c.36) + 2c.(lc.36 - ^.h) + H{^b.h - ^C^)];

or, il suit de ce que nous venons do voir, que les coefficiens de fj.' et fi", sont identiquement nuls, puisqu'ils ne sont quo la resultante des trois

DETERMINANTS IN GENERAL (LAPLACE, 1772) 27

lettres a, b, c, dans laquelle on ^crit b, ou c, par-tout ou est a ; done, on aura pour I'equation de condition demand^e,

0 = ^a.{%.h - ^c.%) + ^a.{h.% - ^.h) + ^a.{^b.h - ^crb);

c'est-^-dire, la r^sultante de la combinaison des trois lettres a, b, c 4galee k zero. On d^montreroit la meme chose, quel que soit le nombre des Equations.

"Pour montrer I'analogie de cette mati^re, avec I'elimination des Equations du premier degr^, je suppose que Ton ait les trois Equations,

^p = ift./x + '^b.fi + h.fx",

2p = %./* + %.fj.' + h.fi",

^p = ^a.fj. + %.ij! + ^c.fi".

Je multiplie, comme ci-devant, la premiere par (^fi.^c - h.%), la seconde par {h.% - ^.h), et la troisi^me par (^b.h - ^cM), je les ajoute en- semble, et j'observe que les coefl&ciens de /x' et de /x", sont indentique- ment nuls dans I'equation qui en resulte ; d'oii je conclus,

_ ^p.C^b.h - ^cM) + Yi^c.% - ^.h) + ^p.(}b}c - hM) ^ ~ ^a{%.h - H.%) + ^a{k.% - ^.h) + ^a(^b:k - ^c.%) '

on voit done que le numerateur de I'expression de fi, se forme du denominateur, en y changeant a en p; on aura ensuite /x' ou /x", en changeant dans I'expression de jj." &c.

This mode of treatment leaves nothing to be desired. It is that which is most commonly employed in the text-books of the present day.

The next point taken up is the most important in the memoir, and requires special attention. It is introduced as "a very simple process for considerably abridging the calculation of the equation of condition between a, b, c" &c. — that is to say, the calculation of a resultant. It is, however, something of much more value than this, involving as it does a widely general expansion-theorem to which Laplace's name has been attached, but of which we have already seen special cases stated by Vandermonde. The theorem may be described as giving an expansion of a resultant in the form of an aggregate of terms each of which is a product of resultants of lower degree. Laplace's exposition is as follows : —

" Je suppose que vous ayez deux Equations,

0 = ifl./x -I- ^.fi ; 0 = ^a.fi + %.fx' ;

ecrivez +ab, et donnez I'indice 1 k la premiere lettre, et I'indice 2 k la seconder I'equation de condition demandee sera 4- ^a.^& - ^5,% = 0.

28 HISTOEY OF THE THEORY OF DETERMINANTS

" Je suppose que vous ayez trois equations ; ecrivez + ab, combinez ce terme avec la lettre c de toutes les mani^res possibles, en changeant le signe de chaque terme chaque fois que c change de place, vous aurez ainsi + abc - axb + cab ; donnez dans chaque terme I'indice 1 k la premiere lettre, I'indice 2 i la seconde, I'indice 3 k la troisieme, et vous aurez +'^a.^.h - ^a-^c.^ft + ^c.^a.^b; cela pos^, au lieu de +^a.^6.^c Ecrivez +(^a.% - '6. 2tt).3c; au lieu de - ' a. ^c. ^6 ecrivez -(^a.% - '^b.^a).h; etaulieude + ^c.^a. ^6 Ecrivez +{^aM - ^^a)M; I'equation de condition demandee sera

0 = (ia.26 - i6.2a).3c - (^aJb - ^Myc + {^a.% - %Hyc.

" Je suppose que vous ayez quatre Equations, ecrivez + ahc - acb + cab, et combinez ces trois termes avec la lettre d, en observant 1° de n'admettre que les termes dans lesquels c precede d; 2° de changer de signe dans chaque terme toutes les fois que d change de place, et vous aurez

+ abed - acbd + acdb + cabd - cadb + cdab ;

donnez ensuite I'indice 1 k la premiere lettre, I'indice 2 k la seconde, &c., et vous aurez

+ ^aM.h.'^d - ^aM^bM +MMMAb

+ ^cM%.*d - ^cMM.^b + hMM.^b ;

cela pos^, au lieu de + ^a.^.h.H Ecrivez

+ (ia.26 - ^.^a).{h.H - HM\

et ainsi des autres termes, et I'equation de condition sera

0 = {hi.% - ^b.H).{H.H - H,*c) - QaM - ^.^a).{h.^d - ^dM)

+ (ia.^6 - i6.*ft).(2c.8rf - H.H) + (2a.36 - %M).Qc.H - ^d.*c)

- {^a^b - %*a).QcM - ^d.^c) + (^a.^b - %*a).QcM - M.h).

"Je suppose que vous ayez cinq equations, ecrivez les six termes + abed- acbd + . . . relatifs a quatre Equations, et combinez-les avec la lettre e de toutes les mani^res possibles, en observant de changer de signe chaque fois que e change de place; donnez ensuite I'indice 1, &c., &c., . . . .; au lieu du terme +^a.^e.^b^eM ecrivez

(ia.36 - ^.Sa).{h.H - H.hye, &c

"Lorsqu'on aura six Equations, on combinera les termes

+ abcde - abeed + &c.,

relatifs k cinq Equations avec la lettre /, en observant 1° de n'admettre que les termes dans lesquels e precede /; 2° de changer de signe lorsque / change de place : on transformera ensuite, par la regie precedente, . . . . "

DETERMINANTS IN GENEEAL (LAPLACE, 1772) 29

Notwithstanding the multiplicity of instances, the rule here illustrated is not made altogether clear. This is due to two causes, — first, the linking of one case to the case before it ; and, second, the want of explicit notification that the letters b,d,f. . . are combined in one way, and the intervening letters c, e, . . . in anotiher. For the sake of additional clearness, let us see all the steps necessary in the case of the resultant of the five equations

arXi + brX2 + 0^X3+ drXi+ 6^x^ = 0 (r=l, 2, 3, 4, 5), and supposing, as we ought to do, that the case of four equations has not been already dealt with. These steps are —

1°. Combining b with a subject to the condition that a

precede 6 : result —

ab.

2°. Combining c with this in every possible way, the sign

being &c. : result —

abc — acb+cab.

3°. Combining d with each of these terms subject to the

condition that c precede d : result —

abed — acbd + acdb + cabd — cadb + cdab.

4°. Combining e with each of these terms in every possible

way: result —

abcde — abced + abecd — aebcd + eabcd

— acbde-{-acbed—

5°. Appending indices : result —

ajb^Csd^e^-a^b^c^e^d^+

6°. Changing a^6„ into (aj)n-bman), c^d, into (c^,-rf^,), &c.: result — (ajb2-b^a2){c^d^-d^c^)e^ - {(i^b^-b^a^){c^d^-d^c^)e^ + . . . . This is the required resultant in the required form.

It is of the utmost importance to notice what is accomplished in 1°, 2°, 3", 4° is simply (a) the finding of the arrangements of a, b, c, d, e subject to the conditions that a precede b, and c precede d, and obtaining each arrangement with the sign which it ought to have in accordance with Cramers rule. The number

30 HISTORY OF THE THEORY OF DETERMINANTS

of necessary directions might thus be reduced to three, viz., (a), (5), (6), in which case (1), (2), (3), (4) would take their proper places as successive steps of a methodic and expeditious way of accomplishing (a).

Laplace appends a demonstration of the accuracy of this development of the resultant of the nth degree, the line taken being that if the multiplications were performed the terms found would be exactly the 1.2,3.... n terms of the resultant, and would bear the signs proper to them as such.

He then goes on to deal with a rule for obtaining a like development in which as many as possible of the factors of the terms are resultants of the third degree.

To do so succinctly he is obliged to introduce a notation for resultants. On this point his words are —

" Je d^signe par {abc) la quantity

dbc - acb + cab - bac + bca- cba,

et par (ab) la quantity ab - ba, et ainsi de suite ; par (^a.^.h) j'indiquerai la quantite (abc), dans les termes de laquelle on donne 1 pour indice k la premiere lettre, 2 k la seconds, et 3 a la troisi^me; par Qa.%), je designerai la quantity (ab) dans les termes de laquelle on donne 1 pour indice k la premiere lettre, et 2 ^ la seconde ; et ainsi de suite."

We can but remark that here again he leaves little room for improvement : his symbolism is essentially that which is still in common use.

The exposition of the rule is as follows : —

"Je suppose maintenant que vous ayez trois Equations, I'equation de condition sera

0 = (la, 26.3c).

" Je suppose que vous ayez quatre Equations ; 6crivez + abc, et combinez ce terme de toutes les mani^rts possibles avec la lettre d, en observant de changer de signe lorsque d change de place, ce qui donne + abed - oMc + adbc - dabc ; donnez I'indice 1 a la premiere lettre, I'indice 2 ^ la seconde, &c., et vous aurez

+ ia.26.3c,4i - ^a.%.H.*c + ^a.H.%.^c - H.^a.^b.*c;

au lieu du terme +'^a.%.h.*d, ^crivez -\-(}a.%.h).*d; au lieu de - H.%.H.*c, ecrivez - Qa.%^c).H, et ainsi de suite, et vous formerez r^quation de condition

0 = Qa.%.h).*d - {^a.%.*cyd + Qa.%.*cyd - (^a.^b.*cyd.

DETERMINANTS IN GENERAL (LAPLACE, 1772) 31

"Je suppose que vous ayez cinq Equations, combinez les termes + abed - abdc + &c., relatifs k quatre Equations avec la lettre e en observant 1° de n'admettre que les termes dans lesquels d precMe e; 2° de changer de signe lorsque e change de place, et vous aurez

+ abode - abdce + abdec + &c.

donnez I'indice 1 i la premiere lettre, I'indiee 2 4 la seconde, &c., et vous aurez

+ ^a.%.HM.^e - ^a.^MM.h + ^a.^^dM^c + &c. ;

ensuite, au lieu de + ^aM.h.*d.^e, ecrivez + (^aM.h).{*d.^e) ; au lieu de - ^^a.^MM.h, ecrivez - {^ayb^c).{^d.^e), et ainsi de suite; et en ^galant k zero la somme de tous ces termes, vous formerez I'^quation de condition demandee.

"Je suppose que vous ayez six equations, combinez les termes + abode - &c., relatifs k cinq Equations avec la lettre /, en observant 1° de n'admettre que les termes ou e precede /; 2° de changer de signe lorsque / change de place : donnez ensuite 1 pour indice k la premiere lettre, ....

"Si vous avez sept equations, combinez les termes +abc.def-&c. relatifs a six equations avec la lettre g de toutes les mani^res possibles ; pour huit equations, combinez les termes relatifs k sept avec la lettre h, en n'admettant que les termes dans lesquels g pr^cSde h, et ainsi du reste."

The really important point in all this is in regard to the manner in vrhich the letters are brought into combination. It will be seen that the set begun with is abc, consequently a precedes b, and b precedes c throughout: then d is combined in every possible way with this : e is combined subject to the condition that d precede e ; / is combined subject to the condition that e precede /: g is combined in every way possible: h is combined subject to the condition that g precede h : and so on. It would appear therefore that the lettres are to be combined in every possible way are d and every third one afterwards, and that each of the other letters is conditioned to be preceded by the letter which immediately precedes it in the original arrangement abcdefghi .... Condensing these directions after the manner of the former case, we should draft the rule as follows : —

(a) Find every possible arrangement of abcdefghi . . . subject to the conditions that in each arrangement we must have a, b, c in their natural order ; d, e, f in their natural order ; g, h, i in their natural order ; and so on.

32 HISTORY OF THE THEORY OF DETERMINANTS

(6) Prefix to each arrangement its proper sign in accordance with Cramer's rule.

(c) Append in order the indices 1, 2, 3, ... to the letters of each arrangement.

(d) Change anbnCr into (a„.6„.Cr), d^exfy into (d^ex-fy), &c.

Without saying anything as to the verification of the developments thus obtained, Laplace concludes as follows : —

"On deconiposeroit de la m^me maniere I'equation R en termes composes de facteurs de 4, de 5, &c., dimensions."

To show how this could be effected would have been a tedious matter, if the method of exposition used in the previous cases had been followed, viz., multiplying instances with wearisome iteration of language until the laws for the combination of the letters could with tolerable certainty be guessed. On the other hand, had Laplace condensed his directions in the way we have indicated, the rule for the case in which as many as possible of the factors are of the 4th degree could have been stated as simply as that for either of the two cases he has dealt with. The only changes necessary, in fact, are in parts (1) and (4), and merely amount to writing the letters in consecutive sets of four instead of two or three.

Further, when the rule is condensed in this way, the problem of finding the number of terms in any one of the new developments — a problem which Laplace solves in one case by considering how many terms of the final development each such term gives rise to — is transformed into finding the number of possible arrangements referred to in part (1) of the rule. Where the highest degree of the factors of each term is 2 and the resultant which we wish to develop is of the nth. degree (which is the case Laplace takes), the number of such arrangements is evidently (L2.3....7i)/(1 . 2/, 8 being the highest integer in n/2 ; if the highest degree of the factors is 3, the number of arrangements is

1.2.3....n (1.2.3)' (1.2)''

where s is the highest integer in 7i/3 and t the highest integer in (n — 3«)/2 ; and so on.

DETERMINANTS IN GENERAL (LAGRANGE, 1773) 33

The facts in reduction of the claim which Laplace has to the expansion-theorem now bearing his name are thus seen to be (1) that the case in which as many as possible of the factors of the terms of the expansion are of the 2nd degree had already been given by Vandermonde; (2) that Laplace did not give a statement of his rule in a form suitable for application to aU possible cases, and, indeed, was not sufficiently explicit in the statement of it for the first two cases to enable one readily to see what change would be necessary in applying it to the next case. Notwithstanding these drawbacks, however, there can be no doubt that if any one name is to be attached to the theorem it should be that of Laplace.

The sum of his contributions may be put as follows : —

(1) A proof of the theorem regarding the efFect of the transposition of two adjacent letters in any of the new functions.

(XI. 2)

(2) A proof of the theorem regarding the efiect of equalizing two of the letters. (xiL 2)

(3) A mode of arriving at the known solution of a set of simultaneous linear equations. (xui. 2)

(4) The name resultant for the new functions. (x^v.)

(5) A notation for a resultant, e.g. {^a.^c.^b). (vn. 2)

(6) A rule for expressing a resultant as an aggregate of terms composed of factors which are themselves resultants. (xiv. 2)

(7) A mode of finding the number of terms in this aggregate.

(XVL)

LAGRANGE (1773).

[Nouvelle solution du probleme du mouvement de rotation d'un corps de figure quelconque qui n'est anime par aucune force acc^^ratrice. Nouv. Meni. de I'Acad. Roy. . . . {de Berlin). Ann. 1773 (pp. 85-128). (Euvres, iii. pp. 577-616].

The position of Lagrange in regard to the advancement of the subject is quite different from that of any of the preceding mathematicians. All of those were explicitly dealing with the problem of elimination, and therefore directly with the functions afterwards known as determinants. Lagrange's work, on the

34 HISTORY OF THE THEORY OF DETERMINANTS

other hand, consists of a number of incidentally obtained algebraical identities which we nowadays with more or less readiness recognise as relations between functions of the kind referred to, but which unfortunately Lagrange himself did not view in this light, and consequently left behind him as isolated instances. With him x, y, z and x, y', z' and £c", y" , ^' occur primarily as co-ordinates of points in space, and not as coefficients in a triad of linear equations ; so that

{xy'z!' + yz'od' + zoiiy" - xz'y" - yx'sf' - zy'x"),

when it does make its appearance, comes as representing six times the bulk of a triangular pyramid and not as the result of an elimination. In days when space of four dimensions was less attempted to be thought about than at present, this circumstance might possibly account for no advance being made to like identities involving four sets of four letters x, y, z, w; x\ y\ z\ w' ; &c.

In this first memoir the algebraical identities are brought together and stated at the outset as follows : —

" Lemme. " 1. Soient neuf quantit^s quelconques

a;, y, z, z', y', z', x'\ y", z" je dis qu'on aura cette equation identique

{xy'z" + yz'x" + zt'y" - xz'y" — yx'z" - zy'x")^

= (a;2 + y2 + z^) (x'^ + y'^ + /2) (x"2 + y"2 + z'"^)

+ 2{xx' + yy' + z£) {xx" + yy" + z£') {x'x" + y'y" + z'z")

- (a;2 + 2/2 + z^) {x'x" + y'y" + z'z"y

- (a;'2 + y'2 + a'2) {xx" + yij" + zz"Y -{x"'^-\-y"'^ + z"^){xz' + yy'-\-zzf.

"CoroUaire 1.

" 2, Done si Ton a antra las neuf quantites precedentes cas six Equations

aj2 j^y2 ^^2 _ fl^ x'x" + y'y" + z'z" = b,

a;'2 +y'^ +z'^ = a', xx" +yy" +zz" = b', x"^ + y"^ + z"^= a", xx +yy' +zz' = b",

DETERMINANTS IN GENERAL (LAGRANGE, 1773) 35

et qu'oii fasse pour abr^ger

^ = y'^'-zy", Tj^z'x" -x'^', (=x'y" -i/x\ fS = J{aa'a" + '2Wb" - ab^ - a'b'^ - a"b"^) ;

On aura de plus les equations identiques suivantes

x'^ + y''n + ^i=0, x"^ + y"r] + ^'C=0, ^^ + rj^ + i^^a'a"-b\

y'i ~ ^'v = ^x' ~ ^'^ "> y"i - ^"v = ^'^ ~ ^ ")

^k-xi = by'-dy'\ z"^-x"i = a"y'-by", x'r) - y'^ = bz' - a'z", x"r] - y"^ = a"z' - bz", qui sont tr^s faciles k verifier par le calcul.

"Corollaire 2. " 3. Si on prend les trois Equations

x$ +yv +4 = P,

xx' +yy' +zz' = b",

xx" + yy" + zz"= b',

et qu'ou en tire les valeurs des quantit^s x, y, z, on aura par les formules connues

^^P {y'z" - z'y") + Vj-nz' - jy')-^ b"(Cy" - ¥') ^ ^ {y'z" - ^y") + V {z'x" - x'z") + c{x'f - y'x'y

^l3{z'x"-x'^') + b'(Cx' -^z) +b"{^^'-Cx") ^~ i (y'^" - ^y") + ^ (^'a^" - ^^') + i{x!y"- y'x"i ^ ^ Pi^y" - y'x") + b'{^y' - r]x') + b"{r}x" - ^y") . ^(y'z" - z'y") + V i^x" - xV) + ({x'y" - y'x") '

done faisant les substitutions de I'Art. pr^c. et supposant pour abr^gcr

on aura

_ p$ + (a"b" - bb')x' + (a'b' - bb")x" X — >

a

I3r] + {a"b" - bb')y' + {a'b' - bb")y" y ■-,

_ PC+ (a"b" - bb')^ + (a'b' - bb")z" ,

In regard to the first identity here (the so-called lemma), the important and notable point is that the right-hand member is the same kind of function of the nine quantities x^+y^-{-z^,

36

HISTORY OF THE THEORY OF DETERMINANTS

xx'-^-yy+zz, xx"+yy"+zz", xx'+yy'+zz, x'^+y"^+z'^, x'x'+yY+zz", xx" + yy" + zz", x'x"+y'y"+z'z", x'"^ + y'"^ + z'"^ as the left-hand member is of the nine x, y, z, x , y', z', x", y" , z". Indeed, without this distinguishing characteristic, the identity would have been to us of comparatively little moment. Possibly Lagrange was aware of it ; but, if so, it is remarkable that he did not draw attention to the fact. It is quite true that Lagrange's identity and the modem-looking identity

X y z	2
X y' z	=
X" y" z"

XX +yy' +zz'

xx" +yy' +ZZ

x'x"+y'y"-\'z'z"

x'

xx' +yy' +zz' x''^ +y'^ +2;'^

xx"-\-yy"+zz" xx"+yy-\-zz" x'"^ +y"^ +z"^

are essentially the same ; but no one can deny that the latter contains on the face of it an all-important fact which is hid in the former, and which in Lagrange's time could be made known only by an additional statement in words. The second identity

is a simple case of one of Vandermonde's, viz., that regarding the vanishing of his functions when two of the letters involved were the same. The third identity

is in modem notation

a.'2 ^y'2 +^'2 x'x"+yY+zV x'x"-\-y'y" + z'z" a;"^ +y"^ +^"2 and is thus seen to be a simple special instance of a very im- portant theorem afterwards discovered. The fourth identity

y'^—^'n = bx' — ax", may be expressed in modern notation as follows : —

\y'y"U	z' z"	2 +	X X	2
z z !	X X		y' y'

y

Wx"

z' I _ d^x" ■\-y'y" -\^z'z" x"

I^Vll *'^ +2/'" +^'^ ^'

and, quite probably, has also ere this been generalised in the like notation.

The fifth identity

X =

/3^+ (a%" - hb')x' + (al/ - hh")x"

DETERMINANTS IN GENERAL (LAGRANGE, 1773) 37

is not so readily transformable, the determinantal theorem which it involves being indeed completely buried. Multiplying both sides by a ; then doing away with a, which seems perversely introduced " pour abreger " when no like symbol of abridgment takes the place of a"b" — hb' or of a'b' — bb" ; and transposing, we have ^^ ^ ^(^^/^/. _ ^2) _ a;'(a"6" -bb') + x"{bb" - afb'),

that is, finally,

\xyV\.\y'z"\ =

		X b" x' a' x" b	b' b a"	J
X	XX -{-yy'	+ zz'
x'	x'^ +y'^	+ z''
x"	x'x	"+y'y	'-\-z'z'

x'x!' ■\-y'y" -\-z'z" x"'^ +y"^ -\-z"'^

which we recognise as an instance of the multiplication-theorem on putting

X y z		1 X x"
x' y z'	X	0 y' y"
x" y" z"		0 z z"

for the left-hand member.

LAGRANGE (1773).

[Solutions analytiques de quelques problfemes sur les pyramides triangulaires. Nowv. Mem. de I'Acad. Roy. . . . (de Berlin). Ann. 1773 (pp. 149-176). (Euvres, iii. pp. 659-692.]

In this memoir also there is a preparatory algebraical portion, the subject being the same as before, and the author's standpoint unchanged. Indeed the two introductions differ only in that the second is a rounding off and slight natural development of the first.

In addition to ^, rj, ^, we have now ^', r{, ^', ^", rj", ^" used as abbreviations for zy" — yz", xz" - zx", . . . . ; in addition to a we have a', a", /3, /3', /3", standing for aa"-b'^ aa'-b"^ b'b"-ab bb"-a'b', bb'-a"b"; and X, Y, Z, X', Y', . . . ., A, A', . . . are introduced, having the same relation to ^, tj, ^, ^' >;',.. . a, a' .... as these latter have to x, y, z, x', y', . . . . a, a, . . . . Lagrange then proceeds : —

38 HISTORY OF THE THEORY OF DETERMINANTS

"3. Or en substituant les valeurs de ^, ^', &c., en x, x', &c., et faisant pour abr^ger

A = xyV + yz'x" + ziiy" - x^y" - yx'z" - zy'xf',

on trouve X = Ax, Y = A//, Z = Aar,

X' = /kc', Y' = Ay', Z' = A2',

X"= Ax", Y"= A/', Z"= Ae",

done mettant ces valeurs dans les dernieres Equations ci-dessus, on aura en vertu des six Equations suppos^es dans I'Art. 1 .

A = A2a, B = A26,

A' = AV, B' = AW,

A"=A2a", B"= A26",

et de Ik il est facile de tirer la valeur de A2 en a, a', a", b, &c. ; car on aura d'abord

^, A a'a"- 13^ a a '

et substituant les valeurs de a, a" et /3 en a, a', &c. (Art. 1)

A2 = aa'a" + 2bb'b" -ab^- a'b"^ - a'T"^ ;

on trouvera la m^me valeur de A2 par les autres Equations. Si on remet dans cette Equation les quantit^s x, y, z, x, &c., on aura la meme equation identique que nous avons donn6e dans le Lemme ci-dessus (p. 86).

"4. II est bon de remarquer que la valeur de A2 pent aussi se mettre sous cette forme

.2 _ aa + a' a + a" a" + 2{ftb + /S'b' + /3"fe" ) ^ 3 '

or si on multiplie cette equation par A2 et qu'on y substitue eiisuite A ^ la place de A\ A' a la place de A^a' et ainsi de suite (Art. pi 6c.) on aura

.,_Aa + AV + A"a"-i-2(B/3fB'j8' + B";8") 3 '

ou bien en mettant pour A, A', &c., leurs valeurs en a, a, &c (Art. 2)

A4 = aa'a" + 2/3/3' /S" - a^^ - a'/3'2 - a"^'2 ;

d'oii Ton voit que la quantity A2 et son carr6 A* sont des fonctions semblables, I'une de a, a', a", b, b', b", I'autre de a, a', a", /3, /3', /3". " 5. De plus, comme Ton a (Art. 3)

xy'z" + yz'x" + zx'y" - xz'y" - yx'z" - zy'x"

= J (aa'a" + 2bb'b" - ab^ - a'b'^ - a"b"^) = A,

DETERMINANTS IN GENERAL (LAGRANGE, 1773) 39

et qu'il y a entre les quantites a;, y, z, x', &c., et a, a', a", b, &c., les memes relations qu'entre les quantites ^, rj, ^, ^', &c., et a, a a", fS, &c. (Art. 1), on aura done aussi

$vT + va" + c^w - ^r v - ^rr' - cvr

= 7 (aa'a" + 2^^'^" - ayS^ - a'^'2 - a"/3"2) = A2. Done on aura cette equation identique et tr6s remarquable

= (x2/V + y^'ji' + 2;a;'y" - a;^;'?/" - y7^^' - zy'x"YJ'

The remaining portion is of little importance ; its main contents are four sets of nine identities each, viz. : —

1. xi+x'i'-\-x"i" = ^, 2/^+2/'f +2/"f' = 0, &c.

2. xi+yr, +z^ =A, x'i+y'f]+z'^ =0, &c.

A Besides the fact that Art. 3 contains a proof of the Lemma of the previous memoir, we have to note the new identity

X = Ax, which in modern determinantal notation is

xz zx'

yx' xy'

= X I xy'2^' I ,

— a simple special instance of the theorem regarding what is nowadays known as "a minor of the determinant adjugate to another determinant."

The last two lines of Art. 4 by implication make it almost certain that Lagrange did not look upon

xy'z" + yz'x" ■\-zx^y" —xz'y" — yx'^ — zy'x^' and aaV+266'6"-a62 -a'6'2 _^'y"i.

as functions of the same kind.

The new theorem in Art. 5, which Lagrange justly characterises as " very remarkable," is in modern determinantal notation

yz

zy" yz

zx

\xz"

2^03'

yx" xy'

X	y	z
X	y	z'
X	y	z

40

HISTOEY OF THE THEORY OF DETERMINANTS

— a simple instance of the theorem which gives the relation, as we now say, " between a determinant and its adjugate."

In regard to the remaining identities which we have numbered (1), (2), (3), (4), we note that (1) and (3) are not new, although

(3) is here given almost in the form desiderated above (pp. 36-37) ; (2) involves the fact that A is the same function of x, x', x", y, y', y", z, z', z", as it is of x, y, z, x' , y', z', x", y" , z" \ and

(4) may be transformed as follows : —

xL = a^ + 6Y + ^'r. a y z h" y z y y" z"

xx' + yy' + zz' y' xx" + yy" + zz" y

so that it may be considered as another disguised instance of the multiplication-theorem, the determinant just reached being equal to

X y	z		X	0	0
x' y'	z'	X	y	1	0
X' y"	z"		z	0	1

LAGRANGE (1773).

[Recherches d'Arithmetique. Nouv. Mem. de I'Acad. Roy. . . . {de Berlin). Ann. 1773 (pp. 265-312).]

This is an extensive memoir on the numbers " qui peuvent etre representees par la formule Bt^ + Ctu + DuV At p. 285 the expression

py^-^-^qyz+rz^

is transformed into Ps^ + 2Qsaj + Ra;-

by putting 2/ = Ms + Nil-,

and z=7n8+'nx,

and Lagrange says —

" . . . . je substitue dans la quantite PR - Q^ les valeurs de P, Q et R, et je trouve en effa9ant ce qui se d^truit

PR-Q2 = (^r-22)(M7i-Nm)2; . . . ."

DETEEMINANTS IN GENERAL (BEZOUT, 1779) 41

which we at once recognise as tlie simplest case of the theorem connecting (as we now say) the discriminant of any quantic with the discriminant of the result of transforming the quantic by a linear substitution.

Putting now in compact form all the identities obtained from the three preceding memoirs of Lagrange, we have —

( 1 ) {xy'z" + yzx" + zxy" — xz'y" — yx'z" — zy'x!'f

= aa'a' + 2bh'b" — ab^ — (ib''^ — a"b"'^, (xvii.) where a = x^-{-y^ + z^, a'= ....

(2) £^ + r]~ + ^^ = aa"-b^, where ^=y'z"-zY, r}=^. . . . (xviii.)

(3) y'^—zrj — hx' — a'x". (xix.)

(4) i^ = ax + ^"x'-\-^'x", wherea = aV-62, fi" = . . . . , and A = xy'z" + yzx" + zx'y" — xz'y" — yx'z" — zy'x". (xvii. 2)

(5) X = A^, where X = r,'^"-^'r,". (xx.)

(6) {xy'z" + yz'x" + zx'y" — xz'y" — yx'z" — zy'x"y-

= &r + €i" + ^i'r," - aV - riiT - tn'i". (XXI.)

(7) PR-Q2 = (pr-^2)(Mn-Nm)2, (xxii.) if _p(Ms + Na;)2 + 2g(Ms-fNa;) {ms+nx)-\-T{ms-\-7ixf

= Ps^ + 2Qs« + Rcc^ identically.

BEZOUT (1779).

[Theorie G^n^rale des Equations Algebriques, §§ 195-223, pp. 171-187 ; §§ 252-270, pp. 208-223. Paris.]

In his extensive treatise on algebraical equations Bezout was bound, as a matter of course, to take up the question of elimination ; and, as he had dealt with the subject in a separate memoir in 1704, one might not unreasonably expect to find the treatise giving merely a reproduction of the contents of the memoir in a form suited to a didactic work. Such, however, is far from being the case. He merely mentions the necessary references to the work of Cramer, himself, Vandermonde, and Laplace ; and then adds —

" Mais lorsqu'il a ete question d'appliquer ces diff^rentes m^thodes au probleme de l'6limination, envisage dans touts son etendue, je me

42 HISTOEY OF THE THEORY OF DETERMINANTS

suis bientdt apper9u qu'ils laissoient tous encore beaucoup k desirer du c6t6 de la pratique."

His main objection to the said methods is that when one has to deal with a set of equations of no great generality, with coefficients, it may be, expressed in figures —

"II faut construire ces formules dans toute la gendralit^ dent les Equations soot susceptibles, et faire par consequent le m^me travail que si les Equations avoient toute cette generality.

(197). Au lieu done de nous proposer pour but seulement, de donner des formules gen6rales d'6limination dans les Equations du premier degre, nous nous proposons de donner une r^gle qui soit indiff(6rem- ment et egalement applicable aux equations prises dans toute leur g6n6ralite, et aux Equations considerees avec les simplifications qu'elles pourront offrir : une regie dont la marche soit la mSme pour les unes que pour les autres, mais qui ne fasse calculer que ce qui est absolument indispensable pour avoir la valeur des inconnues que Ton cherche : une regie qui s'applique indifF^remment aux equations num^riques et aux Equations litt6rales, sans obliger de recourir a aucune formula. Telle est, si je ne me trompe, la r^gle suivante.

" Ehgle g4n6rale pour calculer, tcnUes d, la fois, ou s^parSment, les valeur s des inconnues dans les Equations du premier degr6, soit litUrales soit numiriques.

"(198). Soient u, x, y, z. &c., des inconnues dont le nombre soit n, ainsi qui celui des Equations.

"Soient a, 6, c, d, &c., les coeflBciens respectifs de ces inconnues dans la premiere Equation.

"a', h\ c', d', &c., les coefficiens des m§mes inconnues dans la seconde Equation.

"a", b", c", d", &c., les coefficiens des m^mes inconnues dana la troisieme Equation : et ainsi de suite.

"Supposez tacitement que le terme tout connu de chaque Equation soit affect^ aussi dune inconnue que je represente par /.

"Formez le produit uzyzt de toutes ces inconnues 6crites dans tel ordre que vous voudrez d'abord ; mais cet ordre une fois admis, conservez-le jusqu'^ la fin de I'operation.

" Echangez successivemeiit, chaque inconnue contre son coefficient dans la premiere Equation, en observant de changer le signe a chaque echange pair : ce resultat sera, ce que j'appelle, xine premiere ligne.

"Echangez dans cette premiere ligne, chaque inconnue, contre son coefficient dans la seconde equation, en observant, comme ci-devant, de changer le signe a chaque echange pair: et vous aurez une seconde ligne.

"Echangez dans cette seconde ligne, chaque inconnue, contre son coefficient dans la troisieme Equation, en observant de changer le signe k chaque echange pair : et vous aurez une troisikme ligne.

DETERMINANTS IN GENERAL (BEZOUT, 1779) 43

"Coiitinuez de la meme maniere jusqu'a la dernifere Equation inclusivement ; et la derniere ligne que vous obtiendrez, vous donnera les valeurs des inconnues de la maniere suivante.

" Chaque inconnue aura pour valeur une fraction dont le numerateur sera le coefficient de cette meme inconnue dans la derniere ou n' ligne, et qui aura constamment pour denominateur le coefficient que I'inconnue introduite t se trouvera avoir dans cette m^me w* ligne."

The application of this very curious rule is illustrated by a considerable number of varied examples, of which we select the second —

" (200). Soient les trois Equations suivantes

ax +by +CZ +d =0,

a'x + h'y + c'z +d' =0,

a"x + b"y + c"z + d" = 0. " Je les ecris ainsi

ax +by +CZ +dt = 0,

a'x +b'y +c'z +d't = 0,

a"x + b"y + c"z + d"t= 0.

Je forme le produit xyzt.

Je change successivement x en a, y en b, z en c, t en d, et observant

la regie des signes, j'ai pour premiere ligne

ayzt - bxzt + cxyt - dxyz.

Je change successivement x en a, y en b', z en c', t en d', et observant la regie des signes, j'ai pour seconde ligne

{ab' - a'b)zt -{ad - a'c)yt + {ad' - a'd)yz

+ {be' - b'c)xt - {bd' - b'd)xz + {cd' - c'd)xy.

Je change successivement x en a", y en b", z en c", t en d", et observant la regie des signes j'ai pour troisieme ligne

[{ab' - a'b}c" - {ad - a!d)b" + {bd - b'd)a"^t

- \{ah' - a'b) d" - {ad' - a'd) b" + {bd' - b'd) a"] z + \{ad - a'c)d" - {ad' - a'd)d' + {cd' - dd)a"]y

— [{bd - b'd)d" - {bd' - b'd)d' + {cd' - dd)b"]x. D'ou (198) je tire

_ - [{bd - b'c)d" - {bd' - b'd)c" + {cd' - dd)b"] {ab' - a'b)d' - {ad - a'c)b" + {bd - b'd) a" '

_ + [{ad - a'c) d" - {ad' - a'd) d' + {cd' - dd) a"] {ab' - a'b)d' - (ad - a'c)b" + {bd - b'c)a"]

_ - [{aV - a'b) d" - {ad' - a'd) b" + {bd' - b'd) a"] „ {ab' - a'b)c" - {ad - a'c)b" + {bd - b'c)a" '

44

HISTOEY OF THE THEORY OF DETERMINANTS

Among the other examples are included (1) one in which the coefficients in the set of equations are given in figures ; (2) one in which some of the coefficients are zero ; (3) one showing the simplification possible when the value of only one unknown is wanted; (4) one showing the signification of the vanishing of one of the " lignea " ; (5) one showing the signification of the absence of one of the unknowns from the last " ligne " ; and (6) one or two concerned with the allied problem of elimination.

B^zout nowhere gives any reason for his rule; it is used throughout as a pure rule-of -thumb : its effectiveness being manifest, he leaves on the reader the full burden of its arbi- trariness. The unreal product xyzt at the very outset must have been a sore puzzle to students, and none the less so because of the certainty which many of them must have felt that a real entity underlay it.

To throw light upon the process, let us compare the above solution of a set of three linear equations with the following solution, which from one point of view may be looked upon as an improvement on the ordinary determinantal modes of solution as presented to modern readers.

The set of equations being

ax -\-hy +CZ -^d =■ 0 ax -\-h'y +CZ -\-d' = 0 a"x+b"y + c"z+d"= 0

we know that the numerators of the values of x, y, z, and the common denominator are

bed		a c d		a b d		a b c
b' c' d'	+	a' c' d'	—	a' b' d'	+	a' b' c'
b" c" d"	>	a" c" d"	>	a" b" d"	>	a" b" c"

They are therefore the coefficients of «, y, z, t in the determinant

abed a' b' c d' a" b" c" d" xyzt

Thus the problem of solving the set of equations is transformed into finding the development of this determinant. In doing so

, or A say.

DETEEMINANTS IN GENEEAL (BEZOUT, 1779) 45

let us use [xyz] to stand for the determinant of which x, y, z is the last row, and whose other rows are the two rows immediately above x, y, z in A: similarly let [zt] stand for the determinant of which z, t is the last row, and its other row the row c", d" immediately above z, ^ in A ; and so on in all possible cases, including even [xyzt], which of course is A itself. Then clearly we have

[xyzt] = a [yzt] — b [xzt] + c [xyt] — d [xyz] ( 1 )

Developing in the same w^ay the four determinants here on the right side, we have as our next step

[xyzt]= a{h'[zt] -c'[yt] + d'[yz]) - h{a[zt] - c[xt] + d'[xz]) + c(a'[yt]-b'[xt]+dlxy])

- d(a'[yz] - b'[xz] + c[xy]),

= (a¥ — a'b)[zt] — {ac' — a'c^yt] + {ad' — ad)[yz] + (be' - b'c)[xt] - (bd' - b'd)[xz] + {cd' - c'd)[xy\

Again, developing the six determinants [zt], [yt], .... in the same way, and rearranging the terms, we have finally

[xyzt]= {{ab'-a'b)c" -{ac' -a'c)b"+{bc' -b'c)a"]t -{{ab'-a'b)d"-{ad'-a'd)b"+{bd'-b'd)a")z + {{acf - a'c)d" - {ad' - a'd)c" + {cd' - c'd)a"]y

- {{bc'-b'c)d"-{bd'- b'd)c" + {cd' -c'd)b"}x.

But the coeflBcients of x, y, z,t in [xyz t] were seen on starting to be the numerators and the common denominator of the values oi X, y, z in the given set of equations : hence

-{{bc'-b'c)d"-{bd'-b'd)c"-\-{cd'-c'd)b"} ^ ~ {{ab' - a'b)c" - {ac' - a'c )b" + {bc'-b'c)a"y

2/= • •

z=

Now it is at once manifest that the successive developments here obtained of the determinant [xyzt] are letter by letter identical with the successive " lignes " obtained by B^zout from the unreal product xyzt ; but that instead of having one arbitrary step succeeding another, as in the application of Bdzout's rule, there is here a fluent reasonableness characterising the whole

46

HISTORY OF THE THEORY OF DETERMINANTS

process.* As for the peculiarities requiring elucidation in the series of special examples above referred to, they are seen, when looked at in this light, to be but matters of course.

Not only so, but it will be found that the translation of xy into \xy\, &c., is an unfailing key to much that follows in Bezoufc in connection with the subject. For example, let us take the wide extension of the rule which is expounded later on in the treatise, in a section headed

Considerations utiles pour ahreger consid^rablement le calcul des coejfficients qui servent d Velimination.

There are in all fifteen pages (pp. 208-223, §§ 252-270) devoted to the subject. The contents of three paragraphs will give a sufficiently clear idea of the nature of the whole. The notation used is identical with that of Laplace, e.g.,

{ah') = ah'-a%

{ah'c") = {ab'-a'h)c" - {ah"-a"h)c + {a'h" -a"h')c,

Two of the three selected paragraphs stand as follows : —

"(264.) Cette mani^re de proc6der au calcul des inconnues, en les grouppant, n'est pas applicable seulement k notre objet; elle peut en g^n^ral ^tre appliqu^e dans toutes les equations du premier degre.

♦ If the fact at the basis of the process were made use of nowadays, it would be advantageous, of course, in the first instance to simplify the determinant as much as possible. For example, the equations being {B^zout, p. 178)

			2a; + 4y + 52 = 3x + 5y + 22 = 5ar + 6y + 42 =	221 30 k 43 J
we might proceed as	follows	:—
	2 3 5	4 5 6	5 -22 2 -30 4-43	=	0 2 1 1 0 -3	11 -3 -3	-6 -8 9
	X	y	2 t		X y	2	t
=3	0 1 0	0 0 - -1 -	9 0 -4 -5 -1 3	= 27	0 0 1 0 0 -1	1 0 0	0 -5 3
	X	y	2 t		X y	2	t

=27{-< + 02-3y-5a;}; whence x = 5, y = 3, 2=0.

DETERMINANTS IN GENERAL (BEZOUT, 1779) 47

" Si Ton avoit, par exemple, les quatre equations suivantes

ax +hy +CZ +dt +e =0, a'x +b'y +c'z +d't +e' =0, a"x + h"y + c"z + d"t + e" = 0, a"'x + h"'y + c"'z + d"'t-\-e"'= 0.

En se rappellant que chaque inconnue a pour valeur le coefficient qu'elle se trouve avoir dans la derniere ligne, divise constamment par celui que I'inconnue introduite aura dans cette meme ligne, on verra bientot qu'on peut reduire le calcul a chercher le coefficient de I'une quelconque des inconnues dans la derniere ligne ; parce que de la meme maniere qu'on en aura calcule un, on calculera de meme tous les autres : ou m^me, lorsqu'on en aura calcule un, on pourra en d^duire tous les autres, lorsque les equations auront toute la generality possible. Or pour avoir la valeur du coefficient d'une des inconnues dans la derniere ligne, la question se reduit a calculer la valeur du produit des autres inconnues. Mais pour ne pas se tromper sur les signes, il faudra toujours ne pas perdre de vue, la place que cette inconnue est censee occuper dans le produit de toutes les inconnues. Ainsi, dans le cas present, au lieu de calculer generalement la derniere ligne pour avoir xyztu, je calcule seulement cette derniere ligne pour yztu : et pour I'avoir de la maniere la plus commode, je grouppe en cette maniere yz . tu, et je procede comme il suit, au calcul des lignes, observant que y est cens6 k la seconde place.

Premiere ligne. -bz.tu-yz. du,

Seconde ligne. + (6c') .tu-hz. d'u + h'z .du + yz. (de'),

Troisi^me ligne. -(he') .d"u + {hc") .d'u-hz .(d'e")-(b'c") .du + b'z. (de")-b"z.(de'),

Quatri^me ligne. +(5c') . (d"e"') - (be") . (d't"') + (be"') . (d'e") + (b'c") . (de'")

- (b'c'") . (de") + (h"e"') . (de') ; c'est le coefficient de x dans la derniere ligne.

"Pour avoir celui de w, je calculerois de mSme la valeur de xyzt, en le grouppant ainsi, xy . zt, et je trouverois pour valeur du coefficient de u dans la derniere ligne, la quantite

{ah').{c"d"') - {ab").{c'd"') + {ab"').{c'd") + {a'h").{cd"') - {a'b"').(cd") + {a"b"').(cd') ; D'ou je conclus

^ -f (be') . (d"e"') - (be") . (d'e'")+(bc"') . (d'e")+(b'e") . (de'") - (b'c'") . (de")+(b"e"') . (de') ^ ~ (ab') . (c"d"') -(ab"). (e'd'") + (ab'") . (e'd") + (a'b") . (cd'") - (a'b'") . (cd")+ (a"b"') . (cd')

et ainsi de suite.

(265.") Si j'avois les cinq Equations suivantes —

ox +by +CZ +dr +et +f = 0, a'x +b'y +c'z +d'r +e't +/' = 0, a"x +b"y +c"z +d"r +e"t +f" = 0, a"'x + b"'y + c"'z + d"'r + e"'t +/'" = 0, a"x + b^^y + <^^z + d"r + &H +/''' = 0.

48

JIISTORY OF THE THEORY OF DETERMINANTS

Je calculerois, par exeniple, le coefficient de x dans la derniere ligne, en calculant yzr . tu, ou yz . rtu, ou yz .rt . u.

Si j'avois six (Equations dont les inconnues fussent x, y, z, r, s et f, je calculerois, par exemple, le coefficient de x, en calculant ou yz . rs . tu, ou yzrs . tu, ou yzr . stu, et ainsi de suite.

The next paragraph deals with an illustrative example. The twelve equations —

Aa + A'a' + A'V = 0

Ab + A'b'+A"b" =0

Ac + A'c' + A"c" +Ba + B'a + B"a" = 0

+ Bb +B'b' +B"b" =0

+ BC + BV+BV' =0

+ Bd + B'd' + B"d" + Ca + C'a' + C"a" = 0

+ Cb + C'b' + C"b" =0

+ Cc+CV+C"c" =0

+ Cd + C'd' + C"d" + Da + B'a' + D"a" = 0

+ m+Ub' + -D"b"^.0

+ Dc+DV+D"c" = 0

Ad + A'd' + A"d" + Da + D'a' + D"a" = 0)

are given, and what is required is the result oi; the elimination (equation de condition) of the twelve quantities — A, A', A", B, B', B", C, C, C", D, D', D". This is found (the as in the last equation being misprints for d's) to be —

(aUc") . [{hc'd'J - {ah'c'Jiah'd")] = 0.

The two paragraphs quoted (§^ 264, 265) show that B^zout could obtain with considerably increased ease and certitude any one of Laplace's expansions of numerator and denominator. What it accomplished in the illustrative example is virtually, in modern symbolism, the reduction of

a a a .

h h' h" . . .

c cf c" a a' a

h h' h"

G C C .

d d' d" a a' a'

d d' d"

h h' h" . . ,

c c c" .

d d' d" a a' a''

. . . b b' h"

. . . c c' c'

. . . d d' d"

DETEEMINANTS IN GENERAL (BEZOUT, 1779)

49

to the form | ab'c" \ . | bc'd" ^ - | ab'c" ^ . | ab'd" \. Although this can be done nowadays with ease by means of Laplace's expansion-theorem in its modern garh, it may be safely affirmed that Laplace himself, using his own process, would not have succeeded in making the reduction. Considerable importance thus attaches from more than one point of view to B^zout's curious " rule."

The only other section with which we are concerned bears the heading Methode pour trouver des fonctions d'un nombre quelconque

de qvxintiUs, qui soient zero par elles-memes. In the second paragraph of the section the principle is explained as follows : —

" (216) Concevons un nombre n d'equations du premier degr^ renfer- mant un nombre n+l d'inconnues, et sans aucun terme absolument connu.

" Imaginons que I'on augmente le nombre de ces Equations de I'une d'entr'elles ; alors 11 est clair que ce que nous appellons la derni^re ligne sera non seulement Tequation de condition necessaire pour que ce nombre n + 1 d'equations ait lieu ; mais encore que cette Equation de condition aura lieu ; en sorte qu'elle sera une fonction des coefficiens de ces Equation's, laquelle sera zero par elle-m^me.

"Voila done un moyen tres-simple pour trouver un nombre ti + I* de fonctions d'un nombre n + 1 de quantites, lesquelles fonctions soient zEro par elles-m^mes."

For example, the pair of equations

ax +by +CZ =0\

a'x + b'y + c'z = Oj

is taken, the first equation is repeated, and for this set of three

equations the equation de condition is found to be

{ab' — a'b)c — {ac'—a'c)b + (bc' — b'c)a = 0.

"Or il est clair que la troisieme equation n'exprimant rien de different de la premiere, cette derni^re quantity doit ^tre zero par elle-m^me : done si on a ces deux suites de quantites

a, b, c a', b', c' on pent ^tre assure qu'on aura toujours

{ab'-a'b)c - (ac'-a'c)b + {bc'-b'c)a = 0.

* Should be n.

50 HISTORY OF THE THEORY OF DETERMINANTS

" Et si au lieu de joindre la premiere Equation, c'eftt ^t^ la seconde, nous aurions trouv^ de m6me

(a6'-o'6)c' - {ac'-a'c)V + {he' -h'c)a' = 0."

Similarly in regard to the quantities

a, b, c, d

a', h', c', d'

a", h", c", d"

the identity

[{ah'-a'b)c" -{ac' -a'c)h" + {hc' -h'c)a"]d '

^[{ah'-a'b)d"-{ad'-a'd)h"-\-{hd'-'h'd)a"^c + [{ac'-a'c)d"-{ad'-a'd)c" +{cd' -c'd)a"]fe -[{be' -b'c)d" -{W -b'd)c" +{cd' -c'd)b"^a = 0 and two others are established, the general theorem of course being merely referred to as easily obtainable.

Thus far there is in substance nothing new. What we have obtained is simply a different aspect of Vandermonde's theorem, that when two indices of either set are alike the function vanishes, or, as we should now say, a determinant with two rows identical is equal to zero. Indeed the identities are used by Vandermonde in B^zout's form when solving a set of simul- taneous equations. But what follows is important. By taking two of these identities

{ab'-ab)c -{ac'-a'c)b +{bc'-b'c)a =0

{ab' - a'b) c' - (ac' - a'c)6' + (6c' - b'c)a: = 0,

multiplying both sides of the first by d', both sides of the second

by d, and subtracting, there is obtained in regard to the

quantities

a, b, c, d

a', 6', c', d' the identity

{ab'-ab){cd'-c'd) - {ac'-a'c){bd'-b'd) + {bc'-b'c){ad'-a'd) = 0

Similarly by taking the three next identities before obtained, which for shortness we may write in modern notation,

\ab'c"\d - \ab'd"\c + \ac'd"\b - \bc'd"\a = 0, I ab'c" \d' -\ ab'd" | c' + | ac'd" \b' - \ bc'd" | a' = 0, I ab'c" \d"-\ ab'd" I c"+ I ac'd" \b" - \ bc'd" I a"= 0,

DETERMINANTS IN GENERAL (BEZOUT, 1779) 51

tiiere is deduced in regard to the quantities

a, 6, c, dy e

a', 6', d*, cT, ^

a'\ h'\ €\ <i", ^ the identities

\ah'd'\\dtl I - |o6'«r|.!«^ I + \cuidr\.\M \ - \lMfdr\.]cui' I = 0,

\ab'(r\.\€Ur I - |a6'<r'|.|oS" I + \ac'd"\.\he" j - \bc'd:'\.\ae" \ = 0, \ab'<r\.\d^/'\ - lab'dTllee"] + \ae'd"\.\h'e"\ - \bc'd"\.\a'e"\ = 0.

Finally these last three identities are taken, both sides of the first multiplied by f", both sides of the second by — /', both sides of the third by /, and then by addition there is obtained in regard to the quantities

a, 6, c, d, e, f

a\ h\ c', <f , «', f

o", 6", c", cT, /', f

the identity

\ah'c"\.\deT\ - yib'd"\.\(^r\ -\- \aui'dr\.\heT\ - \hc'dr\.\a€'f"\ = 0. The subject of what may appropriately be called vanishing aggregaies of detemiinant-products is not pursued farther, the ooikdadiiig paragraph being

'* (223) En Yoili assez poor faire connottre la route qa'on doit tenir, pour trouver ces sortes de theoremes. On voit qu'il y a una infinite d'autres combinaisons a faire, et qui dormeront chacane de nouvelles fonctions, qui seront z6ro par elles-mSmes : mais cela est fEicile k trouver actuellement."*

* It is very carious to observe, in passing, that altlioagh Bezoat does not obtain all his vanishing aggregates directly by means of the principle which he so carefully states at the commencement, nevertheless every one of them can be so obtained. He does not extend the principle beyond the case where only one of the original equations is repeated. If, however, we take the equations

ax +by +CZ +«fac =0,

a'z + 6'y + e't + d'uc = 0,

repeat both, of them so as to hare a set of four, and then proceed by the mAhodt pomr abriger to find the Sqmatiam de condition, we obtain

|«6'|.|ar| - |oc'|.|6d'| + \adr\.\bc'\ + \bc'\.\ad'\ - |M'i.|oc'| + lafj.joi'j = 0, i.e. 2{|a6'|.i«i'l - loc'l.IM'l + \atr\.\bc'\} = 0. This is the identity near the foot of p. 51, and others are readily aeea to be obtainable in the same way.

52 HISTORY OF THE THEORY OF DETERMINANTS

Our second list of Bdzout's contributions thus is : —

(1) An unexplained artificial process for finding the numerators and denominators of fractions which express the values of the unknowns in a set of linear equations, or for finding the resultant of the elimination of n quantities from n-\-l linear equations, — a process especially useful when the coeflicients have particular values. (ii. 3 + iii. 4 + iv. 2)

(2) An improved mode of finding Laplace's expansions, especi- ally (but not exclusively) useful when the coeflBcients have particular values, (xiv. 3)

(3) A proof of Vandermonde's theorem regarding the effect of the equality of two indices belonging to the same set. (xii. 3)

(4) A series of identities regarding vanishing aggregates of products. (XXIII. 2)

CHAPTER -III.

DETERMINANTS IN GENERAL, FROM THE YEAR 1784 TO 1812.

The writers of this period are eight in number, viz., Hindenburg, Rothe, Gauss, Monge, Hirsch, Binet, Prasse, Wronski. Of these the first two and Prasse, belonging as they did to the so-called Combinatorial School, were not independent of one another; Hirsch was a mere expositor ; and the others were authors who had not specially studied the subject, but who had attained results in it in the course of other investigations.

HINDENBURG, C. F. (1784).

[Specimen analyticuvi de lineis curvis secundi ordinis, in delvycidationem Analyseos Finitorwm Kaestneriance. Auctore Christiano Friderico Rudigero. Gum, "praefatione Garoli Friderici Hinndenhurgii, professoris Lipsiensis. (pp. xiv-xlviii.) xlviii + 74 pp. Lipsice.']

One of the problems dealt with by Riidiger being the finding of the equation of the conic passing through five given points (" coejiicientium determinatio Traiectoriae secundi ordinis per data qiiinque puncta "), Hindenburg, in his preface, takes occa- sion to show how the generalised problem for ^n{n-\-2>) points has been treated, pointing out that it is, of course, immediately dependent on the solution of a set of simultaneous linear equations. He directs attention to the labours of Cramer and Bezout, specially lauding the method of the latter given in the treatise of 1779. Then he says — " Haec de Opere Bezoldino in universam,, quod plurimis adhuc Lectoribus nostris ignotum,

54 HISTORY OF THE THEORY OF DETERMINANTS

erit, dicta sujfflciant Nunc Regulam ipaam proponam." .... The seventeen pages which follow, contain a tolerably close Latin translation of the Regie gin^rale pour calcvZer . . . . , and the Mdhode pour trouver . . . . , pp. 172-187, §§ 198-223, which have been expounded above. Cramer's rule is next given, the second mode of putting it being in words, and the iirst as follows : —

"Sint plures Incognitse z, y, x, w, &c. totidemque Aequationes simplices indetenninatae

Ai = Zh + yiy + X^x + W% + &c. A2 = Z% + Y2j/ + X^x + W% + &c.

A^ = Z*z + Y*y + X'^x + Whv + &c. &c. &c. &c. &c. &c. &c. Erit, . . . ., positis terminorum signis, ut praecipitur in fine Tabulae, pag. seq. A YXWVUT ....

Permut (1, 2, 3, 4, 5, 6, 7, . . . .)

z =

Permut (1, 2, 3, 4, 5, 6, 7, . . . .) Z YXWVUT . . . ."

(VII. 3)

The similar expressions for y, x, w, v, u, t are given, and then

the " regula signoi^ra." After an illustrative example, the

question of the sequence of the signs is taken up.

"Quod si itaque +sg{l, 2, 3, . . . , n) denotet signorum vicissitu- dines, quibus hie aflBciuntur Permutationum a nunieris 1, 2, 3, ... n singulae species, et -s^(l, 2, 3, . . . n) signa contraria vel opposita: appatet fore

s^(l, 2) =+sg{l) -sg{l)

sg(l,2,S) =+sg{l,2) -sgil,2) +6-^(1,2)

sg{l, 2, 3, 4)= +sg{h 2. S)-sg{l, 2, 3) + sg{l, 2, 3) -5^(1, 2, 3)

unde, quia

sg(l) est	+ .fi	icile eruitur
sg{l, 2) esse	+ -
sg{h 2, 3) ...	. + -	- + + -
sg{\, 2, 3, 4) . . .	. + -	-++--++-	- +
	+ -	- + + + + -	- +

and it is pointed out that the first sign is always +, and the last + or — according as the number 1 + 2 + 3+ . . . + (m — 1) is even or odd.

DETERMINANTS IN GENERAL (HINDENBURG, 1784) 55

Bearing in mind that Hindenburg wrote his permutations in a definite order, this remark regarding the sequence of signs entitles us to view him as the author of a combined rule of term-formation and rule of signs, which may be formulated as follows : —

Write the permutations of 1, 2, 3, . . . , n in ascending order of magnitude as if they were numbers ; make the first sign + , the second — , the next fair contrary in sign to the first paix, the third pair contrary in sign to the second pair, the next six (1.2.3) contrary in sign to the first six, the third six contrary in sign to the second six, the fourth six contrary in sign to the third six, the next twenty-four (1.2.3.4) contrary in sign to the first twenty-four, and so on. (ii. 4 + ill. 5)

EOTHE, H. A. (1800).

[Ueber Permutationen, in Beziehung auf die Stellen ihrer Elemente. Anwendung der daraus abgeleiteten Satze auf das Eliminationsproblera. Sammlung comhinatorisch- analytischer Abhandlungen, herausg. v. G. F. Hindenburg, ii. pp. 263-305.]

Rothe was a follower of Hindenburg, knew Hindenburg's preface to Rlidiger's Specimen Analyticum, and was familiar with what had been done by Cramer and Bezout (see his words at p. 305). His memoir is very explicit and formal, proposition following definition, and corollary following proposition, in the most methodical manner.

The idea which is made the basis of it, that of place-index {" Stellenexponent "), is an ill-advised and purposeless modifica- tion of Cramer's idea of a "derangement." The definition is as follows: — In any permutation of the first n integers, the place-index of any integer is got by counting the integer itself and all the elements after it ivhich are less than it. For example, in the permutation

6, 4, 3, 9, 8, 10, 1, 7, 2, 5 of the first ten integers, the place-index of 9 is 6, and that of 7 is 3, The counting of the integer itself makes the place- index always one more than the number of " derangements "

56 HISTORY OF THE THEORY OF DETERMINANTS

connected with the integer. This necessitates the introduction of a corresponding modification of Cramer's " rule of signs," viz,

"3. Willkiihrlicher Satz. Jede Permutation der Elemente 1, 2. 3, . . . , r, werde mit dem Zeichen + versehen, wenn entweder gar keine, oder eine gerade Menge gerader Zahlen, unter ihren Stellenexponenten vorkommt ; mit dem Zeichen - hingegen, wenn die Menge der geraden Zahlen, unter den Stellenexponenten ungerade ist." (ill. 6)

It is diflScult to suggest any justification for the changes here introduced. The author himself refers to none. Indeed, in the very next paragraph he points out that to ascertain whether there be an even number of even integers among the place- indices is the same as to diminish each of the place-indices by 1, and ascertain whether there be an even number of odd integers, that is, whether the sum of the odd integers be even. He then concludes —

" Man kann also auch die Regel so ausdriicken : Jede Permutation bekommt das Zeichen + wenn die Summe der um 1 verminderten Stellenexponenten gerade, - hingegen, wenn sie imgerade ist."

This is simply Cramer's rule, and it is the only rule of signs employed henceforward in the memoir, the expression " die Summe der um 1 verminderten Stellenexponenten," occurring over and over again as a periphrasis for "the number of derangements."

The next four pages are occupied with a very lengthy but thorough investigation of the theorem that two pei^iutations differ in sign if they be so related that either is got from the other by the interchange of two of the elements of the latter. Strictly speaking, however, the proposition proved is something more definite than this, viz. —

// in a permutation of the integers 1 . 2, . . . r thei^e be d integers intermediate in place and value between any two, A and B, of the integers, the interchanging of the said two would increase or diminish the number of inversions of order by 2d + l. (III. 7)

The proof consists in finding the sum of the place-indices for the given permutation in terms of d as just defined, c the number of elements less than both A and B and situated between them, / the number of such elements situated to the right of B, and '

DETERMINANTS IN GENERAL (ROTHE, 1800) 57

e the number of elements between A and B in value and situated to the right of B; then finding in like manner the sum of the place-indices for the new permutation ; and finally comparing the two sums. The concluding sentence is as follows : —

" Denn da . . . . , so ist die Summe der Stellenexponenten der zweyten Permutation um d + e+1 -e + d oder urn 2d+l grosser als bey der ersten Permutation ; folglich gilt das auch bey der Summe der um 1 verminderten Stellenexponenten, da bey beyden Permu- tationen r einerley ist. Also ist die eine Summe gerade, die and ere ungerade, folglich haben nach (4) beyde Permutationen verschiedene Zeichen."

As immediate deductions from this, it is pointed out that

The sign of any one permutation may be determined when the sign of any other is known, by counting the number of interchanges necessary to transform the one 'permutation into the other ; (ill. 8)

and that

If one element of a permutation be made to take up a new place, by being, as it were, passed over m other elements, the sign of the new permutation is the same as, or different from, that of the original according as m. is even or odd. (iii. 9)

A third corollary is given, but it is, strictly speaking, a self-evident corollary to the second corollary, and is quite unimportant.

Rothe's next theorem is —

The permutations o/ 1, 2, 3, . . . . , n beivg arranged after the manner in which numbers are arranged in ascending order of magnitude, any two consecutive permutations will have the same sign, if the first place in which they differ be the (4n-t-3)*'* or (4n -f- 4)*^ from the end, and will be of opposite sign if the said place be the (4n-|-l)*'^ or (4n -|- 2)*^ /rom the end. (ill. 10)

Thus if the permutations of 1, 2, 3, . . . ., 10 be taken, and arranged as specified, two which will occur consecutively are 8, 4, 9, 3, 10, 7, 6, 5, 2, 1 8,4,9,5, 1,2,3,6,7,10; and as the first place in which these differ is the 7*^^ from the end, it is affirmed that the signs preceding them must be alike. The mode of proving the theorem will be readily understood by

58 HISTORY OF THE THEORY OF DETERMINANTS

seeing it applied to this illustrative example. Taking the permutation

8, 4, 9, 3, 10, 7, 6, 5, 2, 1,

and interchanging 3 and 5 we have the permutation

8, 4, 9, 5, 10, 7, 6, 3, 2, 1, and thence by cyclical changes the permutation

8, 4, 9, 5, 1, 2, 3, 6, 7, 10, the number of alterations of sign thus being l+(5 + 4+3 + 2 + l) i.e. 1 + 1(5x6), — an even number.

Annexed to the theorem is the following corollary, which is not essentially different from Hindenburg's proposition regarding the sequence of signs, —

Jf the permutations of 1, 2, 3, ..., n — 1 he arranged after the manner in which numbers are arranged in ascending order of magnitude, and also in like manner the permutations of 1, 2, 3, . . . ., n—1, n, then those permutations of the latter arranged set which begin with r, say, have in order the same signs as the permutations of the former arranged set, or different signs, according as t is odd or even. (iti. 11)

For example, arranging the permutations of 1, 2, 3, each with its proper sign in front, we have

+ 1.2,3 -1.3,2

-2,1,3 .^.

+ 2,3,1 ^ '

+ 3, 1, 2 -3,2,1; then arranging those permutations of 1, 2, 3, 4 which begin with 3 say, each with its proper sign, we have

+ 3,1.2,4 -3,1,4,2

o, ^, 1, 4 /gv

+ 3,2,4,1 ^ ^

+ 3,4,1,2 -3,4,2,1;

DETERMINANTS IN GENERAL (ROTHE, 1800) 59

and the two series of signs are seen to be identical, 3 being an odd number. Viewing this quite independently of the theorem to which it is annexed, it is evident that a change of sign at any point in the series (A) implies a change at the corresponding point in the other series, and consequently attention need only be paid to the first sign of (B) as compared with the first sign of (A). Now the first sign of (A) must necessarily be always plus, there being no inversions ; and the first sign of (B) depends on the changes necessary for the transformation of the natural order 1, 2, 3, 4, into 3, 1, 2, 4. The truth of the corollary is thus apparent.

A second corollary is given, but it is of still less consequence, the difierence between it and the first being that in the arranged set (B) the place whose occupant remains unchanged may be any one of the n places. (ill. 12)

The next few paragraphs concern the subject of " conjugate permutations " {verwandte Permutationen), — apparently a fresh conception. The definition is —

Two permutations of the numhers 1, 2, 3, . . . , n are called CONJUGATE when each number and tlie number of the pkuce which it occupies in the one permutation are interchanged in the case of the other permutation. (xxiv.)

For example, the permutations

3, 8,5,10,9,4,6,1,7,2 (A)

8,10,1, 6,3,7,9,2,5,4 (B)

are conjugate, because 3 is in the 1*'' place of (A) and 1 is in the 3"^ place of (B), 8 is in the 2"'^ place of (A), and 2 is in the S^^ place of B, and so on in every case. The first theorem obtained is —

Conjugate permutations have the saTne sign. (ill. 13)

This is proved in a curious and interesting way, a special conjugate pair being considered, viz., the pair just given as an example. To commence with, a square divided into 10 x 10 equal squares is drawn, the vertical rows of small squares being numbered 1, 2, 3, &c. from left to right, and the horizontal rows 1, 2, 3, &c. from the top downwards. The permutation

3, 8, 5, 10, 9, 4, 6, 1, 7, 2

60

HISTORY OF THE THEORY OF DETERMINANTS

12345678 9 10

is then represented by putting a dot in each of the horizontal rows, in the first under 3, in the second under 8, and so on ; so that if the rows be taken in order, and the number above each dot read, the given permutation is obtained. For the represen- tation of the conjugate permutation nothing further is necessary : we obtain it at once if we only turn the paper round clockwise until the vertical rows are horizontal, and read off in order the numbers above the dots. In the next place the number of " derangements " belonging to the permutation 3, 8, 5, .... is indicated by inserting a cross in every small square which is to the left of one dot and above another ; thus the two crosses in the first hoi-zontal row corre- spond to the two " derangements " 32, 31 ; the six crosses in the second horizontal row to the six " derange- ments " 85, 84, 86, 81, 87, 82 ; and so on. Then it is observed that if we turn the paper and try to indi- cate the "derangements" of the conjugate permutation by inserting a cross in every small square which is to the right of one dot and above another, we obtain exactly the same

crosses as before. The signs of the two permutations must thus be alike.

Immediately following this, the 24 permutations of 1, 2, 3, 4 are given in a column, each one having opposite it, in a parallel column, its conjugate permutation. The existence of aelf- conjugate permutations, e.g., the permutation 3, 4, 1 2, is thus brought to notice, and the substance of the following theorem in regard to them is given : —

If U„ be the number of self-conjugate permutations of the first n integers, then

U„ = Vn-i + {n-l)Un-2 (XXV.)

where Uj = 1 and Ug = 2.

This, however, is the only one of his results which Rothe does not attempt to prove.

1

2 3 4 5 6 7 8 9 10

X X .

XX X X X X .

XX X .

XX X XX X .

XX X XX

X X

X X

X

DETERMINANTS IN GENERAL (ROTHE, 1800) 61

In the second part of the memoir, which contains the applica- tion of the theorems of the first part to the solution of a set of linear equations, there is not so much that is noteworthy. Methods previously known are followed, the new features being formality and rigour of demonstration. The coeflScients of the equations being

11, 12, 13, .... , Ir 21, 22, 23, . . . . , 2r

rl, r2, rS, . . . . , rr it is noted, as Vandermonde had remarked, that the common denominator of the values of the unknown may be got in two ways, viz., by permuting either all the second integers of the couples, 11, 22, 33, . . . . , rr, or all the first integers: but this is supplemented by a proof, that if any term be taken, e.g.,

16.24.33.47.51.68.79.82.95 vrith the couples so arranged that the first integers are in ascending order, and the sign be determined from the number of inversions in the series of second integers, then the sign obtained xvill be the same as would be got by arranging the couples so as to have tlie second integers in ascending order, and determining the sign from the inversions in the series of first integers. The proof rests entirely on the previous theorem, that conjugate permutations have the same sign; indeed the new proposition is little else than another form of this theorem. (ill. 14)

The desirability of an appropriate notation for the cofactor, which any one of the coefficients has in the common denominator, is recognised,* and the want supplied by prefixing f to the ■coefficient in question ; for example, the cofactor of 32 is denoted by f32.

It is thus at once seen that the denominator itself is equal to

In. fin + '2.n.i'ln + .... + rn.irn, or n\.in\ + n^.inl + .... + nr.inr. (vi. 2)

Also by this means one of Bezout's (or Vandermonde's) general theorems becomes easily expressible in symbols, viz.,

In.ilm + 2n.f2m + •••■+ rn.irm = 0, (xii. 4)

* Lagrange's use of a corresponding letter from a different alphabet must not be forgotten.

62 HISTORY OF THE THEORY OF DETERMINANTS

the proof of which is given as follows. In all the terms of f Im, every one of the integers except one occurs as the first integer of a couple, and every one of the integers except m occurs as the second integer of a couple : consequently, in every term of In.ilm the first places of the couples are occupied by the integers from 1 to r inclusive, while in the second places m is still the only integer awanting and n occurs twice. Suppose then all the terms of

In.flm + 271. f 2m + .... + m.irm

so written that the first integers of the couples are in ascending order of magnitude, and let us attend to a single term

'pw -qn-

in which the two couples, having n for second integer, are the I?"' and q^^. If we inquire from which of the expressions In.flm, 2n,f2w, .... this term comes, we see that it is a term of both pn . ipm and qn . iqm, and must, therefore, occur twice. Further, we see that in ^n . f qm it has the sign of the term

• pm • -qn-

of the common denominator, and that in qn . ipm, it has the sign of the term

-pn- -qm-

of the common denominator. But these two terms of the common denominator have dififerent signs : consequently

In.flm + 2w.f2m + .... + rnArm

consists of pairs of equal terms with unlike signs, and thus vanishes identically. (xii. 4)

These preparations having been attended to, the set of r equations with r unknowns is solved by Laplace's method ; and a verification made after the manner of Vandermonde. It is also pointed out, that if the solution of a set of equations, say the four

fiXy+ox^+px^-^ qx^ = 8^

DETERMINANTS IN GENERAL (GAUSS, 1801)

63

be Xj^—Asj^+ Bs2+ CS3+ DS4

x^=Es^+ Fs2+ GS3+HS4

OJg = ISj + KS.2 + LSg + MS4

ic^ = Nsi + 0^2 + Psg + Qs^^ then the solution of the set

^2/1+ /2/2+%3+ 02/4 = '^2 C2/i +5^2/2+ ^2/3 +^2/4 = '^3

which has the same coefficients differently disposed, will be

2/1 = Avi 4- E-Wg + IV3 + Nv^^ 2/2 = B Vj + Fv^ + Kvg + 0^4

3/3 = CVl + 0^2 + LVg + PV4

2/4 = Dvi+Hv2+Mv3+QvJ ; (xxvi.)

and hence, that the solution of a set having the special form acCi + bx^ + cx^ + dx^ = s^' bxi +6X2 + /% + gx^ = §2

CXj^ + /iC2 + ^3 + '^^4 = *3

(iiKi + 9x2 + ix^ + jx^ = s^^ will itself take the form, viz.

As^ + Bsg + Csg + DS4 = aJi Bsj + Es2 + Fsg + GS4 = x^ Csj + Fsg + Hsg + IS4 = x^ Dsj+Gsg+Isg + J84 = a;J . (xxvi. 2)

GAUSS (1801).

[Disquisitiones Arithmeticce. Auctore D. Carolo Friderico Oauss. 167 pp. Lips. Werke, I. (1863) Gottingen.]

The connection of Gauss with our theory was very similar to that of Lagrange, and doubtless was due to the fact that Lagrange had preceded him. The fifth chapter of his famous work, which is the only chapter we are concerned with, bears the title " De formis cequationibusque indeterminatis secundi gradus," and its subject may be described in exactly the same words as Lagrange used in regard to. his memoir Recherches

64 HISTORY OF THE THEORY OF DETERMINAISPTS

d'Arithm^tique (1773 : see above), viz. "les nombres qui peuvent ^tre represent^s par la formule Bt^+Ctu-\-I>u\" Gauss writes his form of the second degree thus — axx+2bxy+cyy; and for shortness speaks of it as the form (a, b, c). The function of the coefficients a, b, c, which was found by Lagrange to be of notable importance in the discussion of the form, Gauss calls the " determinant of the form," the exact words of his definition being

"Numerum bb-ac, a cuius indole proprietates formae (a, b, c) imprimis pendere in sequentibus docebimus, determinantem huius formae uocabimus." (xv. 2)

Here then we have the first use of the term which with an extended signification has in our day come to be so familiar. It must be carefully noted that the more general functions, to which the name came afterwards to be given, also repeatedly occur in the course of Gauss' work, e.g., the function aS - /Sy in his statement of Lagrange's theorem (xxii.)

6'6' - a'c' = (66 - ac){aS - jByf. But such functions are not spoken of as belonging to the same category as bb — ae. In fact the new term introduced by Gauss was not " determinant " but " determinant of a form," being thus perfectly identical in meaning and usage with the modem term " discriminant."

Notwithstanding the title of the chapter Gauss did not confine himself to forms of two variables. A digression is made for the purpose of considering the ternary quadratic form (" formam ternariam secundi gradus"),

axx + a'x'x' + a"x"x" + 26a; V + 2b' xx" + 2b" xx', or as he shortly denotes it

/a a, a"\ V6, 6', 6'7.

In the matter of nomenclature the following paragraph of this digression is interesting, —

" Ponendo bb - a'a" = A, b'b' - aa" = A', b"b" -aa' = A", ab - b'b" = B, a'b' - bb" = B', a"b" - bb' = B",

DETERMINANTS IN GENERAL (GAUSS, 1801) 65

oritur alia forma

/A A' A"\ p

\B B' B'7 • • • • -^ quam formae

/a a' a"\ ,

\b b' h") ■ ■ ' • J

ddjunctam dicemus. Hinc rursus inuenitur, (xxvii.)

denotando breuitatis caussa numerum

abb + a'b'b' + a"b"b" — aa'a" - 2bb'b" per D, BB-A'A" = aD, B'B' - AA" = «'D, B"B" - AA' = a"D, AB - B'B" = bB, A'B' - BB" = b'D, A"B" - BB' = b"T>, unde patet, formae F adjunctam esse formam

^oD, a'D, a"D>

/aD, aD, a D\ \bJ), b'J), b"D).

Numerum D, a cuius indole proprietates formae ternariae / imprimis pendent, determinantem huius formae uocabimus ; (XV. 2)

hoc modo detenninans formae F sit = DD, sive aequalis quadrato determinantis formae /, cui adjuncta est."

In this there is no advance so far as the theory of modern determinants is concerned, the identities given being those numbered (xx) and (xxi) under Lagrange. On the same page, however, an extension is given of Lagrange's theorem (xxii), regarding the determinant of the new form obtained by effecting a linear substitution on a given form. Gauss' words in regard to this are —

" Si forma aliqua ternaria / determinantis D, cuius indeterminatae sunt X, x', x" (puta prima = a;, &c.) in formam ternariam g determinantis E, cuius indeterminatae sunt y, y', y", transmutatur per substitutionem

*^^"^^ X =ay +(Sy' +yy",

X = ay + /3'y' + y'y",

x" = a"y 4- (By + y"y",

ubi nouem coefficientes a, /?, &c. omnes supponuntur esse numeri integri, breuitatis caussa neglectis indeterminatis simpliciter dicemus, / transire in g per substitutionem (S)

«> A y

"', ^', 7

«", ^", y" atque / implicare ipsam g, siue sub f contentam esse. Ex tali itaque suppositione sponte sequuntur sex equationes pro sex coefficientibus

66 HISTORY OF THE THEORY OF DETERMINANTS

in g, quas apponere non erit necessarium : hinc autem per calculum facilem sequentes coriclusiones euoluuntur :

" I. Designate breuitatis caussa numero

o./3'y" + Py'o-" + yo-'P" — yP'o-" — °-y'P" — P°-'y" per k inuenitur post debitas reductiones

E = kkl), (XXII. 2)

When freed from its connection with ternary quadratic forms the theorem in determinants here involved is

If Afl = ttQ ttf)'^ + ftjai^ + 0502*+ ib^iO^ + 26, Oq 02 + ^b^Qa^ , Ai = aA^ + Oift^ + 02/3/ + 2fco/3,/32 + 2b^PoP^ + 2b^^^ , Aa = ttoTo'^ + ai7i^ + QaVa'^ + 26o7i72 + 26i7o7!i + S^aToTi .

Bo = O(^o7o + «iJ3i7i + 02/3272 + K (^iJ^ + /SsVi) + &i iPoJa + AYo) + ^i (i3o7i + /3i7o) . Bi = Oo7o*o + ai7iai + 0^72*2 + ^'0 (71*2 + 7a«i) + W (7002 + 72*o) + ^2(7oai + 7i«o). B2 = Co ao/3o + <h aift + cufi^2 + K («i/3« + «2/3i ) + ^1 (sAs + «2/8o) + ^2 («o/3i + ai/3o),

then

X (oo/^iy-a + ^oyia2 + roai^2 " 7o^i«2 " aoyi/32 " ^o^iJif- As thus viewed it is an instance of the multiplication-theorem, the product of three determinants (in the modem sense) being expressed as a single determinant.

The multiplication-theorem is also not very distantly connected with the following other statement of Gauss : —

"Si forma ternaria / formam ternariam /' implicat atque haec formam/": implicabit etiam / ipsam /". Facillime enirn perspicietur, si tr&nseat

/ in /' per substitutionem j /' in /" jier substitutionem

8", e", C

a', i8', y a", P", y"

f transmutatum iri per substitutionem

aS +/88' +y8", at +^c' + ye", a( + ftC +yC' a 6 + 13' 8' + y'h", a'e + /3'e' + y'c", a' C+ft'C + YC"

a"6 + ft"8' + y"S", a"c + /3'V + 7'V', a"C+l3"C + y"C"" (xvir. 3)

DETERMINANTS IN GENERAL (MONGE, 1809) 67

MONGE (1809).

[Essai d'application de I'analyse a quelques parties de la g^om^trie ^l^mentaire. Joum. de I'J^c. Polyt, viii. pp. 107-109.]

Lagrange, as we have already seen, was led to certain identities regarding the expression

xy'z" + yz'x" + zx'y" — xz'y" — yx'z" — zy'x"

in the course of investigations on the subject of triangular pyramids. The position of Monge is that of Lagrange reversed. From the theory of equations he derives identities connecting such expressions, and translates them into geometrical theorems. The simpler of these identities, as being already chronicled, we pass over. At p. 107 he takes the three equations

a-^u + l\x + C{y + d^z + e^ = 0

a^u+h^-\-c^y+d^z + e.^ = 0

a^u + 63a; 4- c^y + d^z + e^ = 0,

and eliminating every pair of the letters 11, x, y, z, obtains the six equations

)8M,+ aic+P = 0 (1)

ya;+/82/ + Q = 0 (2)

6y+yz-{-^ = 0 (3)

az+ <5u + N = 0 (4)

yu- ay+ S = 0 (5)

^z- ^a;+R = 0 (6); the ten letters

«, /3, y, S, M, N, P, Q, R, S

being used to stand for the lengthy expressions which we nowadays denote by

|V2^3l» kiCzO'al' kiMsl' kiVsl'

I ^iVs I ' I ^1^2«3 I . ! CW2«3 I . - I «1<^2«3 I . I 0^lC2«3 I ' I \'^2^Z I •

Then, taking triads of these six equations, e.g., the triads (1), (2), (5) he derives the identities

68

HISTOEY OF THE THEORY OF DETERMINANTS

aQ+/3S-yP = 0^ 5P+aR-^N = 0 -yN+ <SS+aM = 0 -^M+yR+6Q = 0j

or

-lV2<^3!-M2«3l +

-|«lC2<^3l-l«lV3l + KMsl-klVsl - |aiV3l-M2«3l = ^^

(xxiii. 3)

which in their turn, he says, by processes of elimination, may be the source of many others. For example, each of the four being linear and homogeneous in a, /8, y, S, these letters may all be eliminated with the result

or

RS + QN-PM = 0,

I «lC2e3 I • I ^1^2«3 I - I M2«8 I • I ^1 Vs I " I ^A^3 I • I ^ih^Z I = ^■

Also, eliminating P from the first and second, S from the first and third, Q from the first and fourth, and so on, we have

-i8yN + <5aQ+ )8^S + ayR = 0, a/3M. + y ,?P - ^yN - SaQ = 0, a^M-y^P+ /3^S-ayR = 0, &c. &c.

I.e.

-\a^c^d^\

a^d^ I • 1 &iC2«3 I - I "^A^S I • I ^lC2<^3 i I ai&2C3 I • I ^l<^2«3 I + I ^^lC2^3 I • I 01^2^3!

&c. &c.

ci^d^e^

^1^2*3

} = o,

(XXVIII.)

Monge does not pursue the subject further. His method, however, is seen to be quite general ; and we can readily believe that he possessed numerous other identities of the same kind. This is borne out by a statement in Binet's important memoir of 1812. Binet, who was familiar with what had been done by Vandermonde, Laplace, and Gauss, says (p. 28C): — "M. Monge m'a communique, depuis la lecture de ce mdmoire, d'autres the'orfemes trfes-remarquables sur ces r^sultantes; mais ila ne sont pas du genre de ceux que nous nous proposons de donner ici."

DETERMINANTS IN GENERAL (HIRSCH, 1809) 69

HIRSCH (1809).

[Sammlung von Aufgaben aus der algebraischen Gleichungen, von Meier Hirsch (pp. 103-107). xvi + 360 pp. Berlin.]

The 4th Chapter Von der Elimination u. 8. vj., contains five pages on the subject of the solution of simultaneous linear equations. These embrace nothing more noteworthy than a statement, without proof, of Cramer's rule, separated into three parts (iv., iii. 2, v.), and carefully worded.

BINET (May 1811).

[M^moire sur la theorie des axes conjugu^s et des momens d'inertie des corps. Journ. de VJ^cole Poly technique, ix. (pp. 41-67), pp. 45, 46.]*

In this well-known memoir, in which the conception of the moment of inertia of a body with respect to a plane was first made known, there repeatedly occur expressions, which at the present day would appear in the notation of determinants. There is only one paragraph, however, containing anything new in regard to these functions. It stands as follows : —

" Le moment d'inertie minimum pris par rapport au plan (C) a pour

ABC-AF2-BE^-CDH2DEF

g%BG-TP) + hHAC-W) + t^(AB-D^) + 2gh(EF-CD) + 2gi(DF-BE) + 2hi{DE-AF )

Si, dans le num^rateur,

ABC - AF2 - BE2 - CD2 + 2DEF

on remplace A, B, C, &c. par 'Smx^, Imy^, &c. que ces lettres representent, on a

'Lm3?'2my^^mz^ - 1mx^(2myzy - '2my^ (2mxzy

- '2mz^ (^mxyY + 22mxy'2mxz2myz,

et Ton peut s'assurer que cette expression est identique k

'2mm'm"{xy'z" + yz'x" + zx'y" - xz'y" - yx'z" — zy'x"y;

*An abstract of this is given in tlie Nouv. Bull, des Sciences par la Sociiti PhilomatiqvA, ii. pp. 312-316.

70

HISTORY OF THE THEORY OF DETERMINANTS

par une transformation analogue, on peut ramener la quantity

^2(BC-F2) +/i2 (AC-E2) +i2 (AB-D2)

+ 2gh(EF - CD) + 2^i(DF - BE) + 2hi(DE - AF), k celle-ci

'2,mm'[g{yz' - zy') + ^(^o;' - x^) + i{xy' - yx!)Y"

Now the numerator referred to would at the present day be written

A D E

DBF

E F C .

and since 2ma;2, &c. stand for mx^+7n^x^-\-m^^+ • • ., &c-> the first identity may be put in the form

mxz + m^x^Zi + vrufcf^ + . . myz + m^y^z^ + m^^ + 2

= mmjTn^

«2 2/2

^2

+ mm^m^

mxz + TWjXjZj + TTtjX^ + .

myz + m^y^Zi + m^^ + . mz^ +mjZi^ +mK^ +. 2

*^1

2/1

Z,

2/3

Zo

+

(XVIIL 2)

where x^, y^, - • • are for convenience written instead of x\ y", ... It will been seen that this is an important extension of a theorem of Lagrange, the latter theorem being the very special case of the present obtained by putting 'in = m^ = 7rh^=\, and m^ = 7n^= . . . =0, — a fact which is brought still more clearly into evidence if, instead of the left-hand member of the identity, we write the modem contraction for it, viz.

mx my

rrijX^

^22/2

mz m.z, m^z,

in

"2 -"2

^32/3 m^z^

X	X,	X^	X^ . . .
y	2/1	2/2	2/8 •• •
z	«1	2^2	«8 ...

Again the denominator

/(BC-F2) +h^ AC-E2) +i2 (AB-D2)

+ 2gr/i(EF- CD) + 2(/i(DF - BE)+ 2/ii(DE - AF)

being in modem notation

. (J h i

y A D E

h D B F

i E F C

DETERMINANTS IN GENERAL (BINET, 1811)

71

the second identity may be written

9

h 7nxy+in^x{y^ + . i rnxz -\-m^Xj^z^-{-.

h mxy + mjXjy^+. my^ -{-'niiyx +• myz-\-m^y^z^+.

= rri7n.

g X x^	2	g X x^	2	g X^ 332
h y yi	+ mw2	'^ y 2/2	+ m{m2	h' Vi 2/2
i z z^		i z z^		i z^ z^

mxz+m^Xj^z^ + . myz+m^y^z^ + . TTIZ^ +m^z^ +.

+ .. (xxix.)

This is also an important theorem, and is not so much an extension of previous work as a breaking of fresh ground.

BINET (November 1811).

[Sur quelques formules d'algebre, et sur leur application a des expressions qui ont rgipport aux axes conjugues des corps. Nouv. Bull, des Sciences 'par la Society Philomatique, ii. pp. 389-392.]

In this paper Binet returns to the consideration of the first of the two identities which have just been referred to, writing it now in the form

2 {xy'z" - xz'y" + yz'o^' - yx'z" + zxy" - zy'x'J = 2x222/^222 - 2a;2 {^yzf - Xy^ ( Hxzf - llz^CZxyf + 21xylxz^yz. He puts it in the same category as the identity 2(y'z-zyy = ^y^lz^-(^yz)\ which he speaks of as being then known. Further, he says

" Ces deux formules sont du meme genre que la suivante

2 f 'ux'y"z"'~ux'z"y'"+uy'z"x"'-ui/'x"z"'+iiz'x"y"'-vz'y"x"'+xy'u"z"'-xy'z"u'"^ * < + xz'y"u"'-xz'u"y"'+xti'z"y"'-xti'y"z"'+yz'u"x"'-yz'x"u"'+yu'x"z"'-yu'z"x"' V t+ yx'z"u"'-yz'u"z"'+zu'y"x"'-zu'x"y"'+zx'y"u"'-zx'u"y"'+zy'x"u"'-zy'u"x"'J = 2^22x22^22^2 _ 2ti22a;2 (2^2)2 - 2^22^2(2x2)2 - l.v?l.z^ (2xyf

- ^x^ly^i^iuzf - 2x22^2 (2Mi/)2 - l.y^'Zz^ (luzf

+ 2'2u'^^xy^xz2yz + 1'2x'^'2uy2uz1yz + 2'2y^^ux^uz2ixz

+ 22222t/x2My2x3/ + (2mx)2 (^yzf + {l.uyf (2xzf + (2uz)\^yy ■

- 2^ux^y'2yzl:zu - 2'2,uy^yz^zx1au - 2^yl<yx1xzl>zv,"

72

HISTORY OF THE THEORY OF DETERMINANTS

-a result which in modem notation would take the form

2/1

2^1

U2

2/2

U3

2/3

2,

+

2 "3 UX + -M.^^^^ + • .

2/1 2^1

2/2

2^0

2/4

Z,

+

a'2/ + a;i2/i + .. fl;2;+a;j2^j +••

(XVIII. 3)

^2/+'"'i2/i+-- uz+u^z^ + .

x^ +x^- +.. xy+x{y^-\-.. xz+x^z^+.

y^+ 2/1^+.. 2/^ + 2/12^1+.

2/2; +2/1% + .. 2^ +^1^ +.

It is thus clear that, in November 1811, Binet was well on

the way towards a great generalisation. He even says that the

three identities may be looked upon

"comma les trois premieres d'une suite de formules construites d'apr^s une meme loi facile k saisir."

He merely indicates, however, the mode of proof he would adopt for the results obtained, and refers to possible applications of them in investigations regarding the Method of Least Squares (Laplace, Connaissance des Terns, 1813) and the Centre of Gravity (Lagrange, M^m. de Berlin, 1783). The mode of proof need not be given here, as it turns up again in the far more important memoir in which the theorem in all its generality falls to be considered.

PRASSE (1811).

[Commentationes Mathematicse. Auctore Mauricio de Prasse. 120 pp. Lips., 1804, 1812. (Pp. 89-102 ; Commentatio vii.* : Demonstratio eliminationis Cramerianae.)]

Of previous writings the one which Prasse's most resembles is Rothe's. There is less of it, and it shows less freshness ; but there is the same stiff formality of arrangement, and the same effort at rigour of demonstration.

* Separate copies of the Demonstratio diminationis Crameriance are also to be found, bearing the invitation title-page :

Ad memoriam Kregelio-Stembachiajiam in avditorio philosophorum die xviii JiUii MDCCCXi. h. ix cdebrandam invitant ordinum AcademicB Lips. Decani seniores conterique ad-^essores .... Demonstratio eliminationis Crameriance.

It is these copies which fix the date. See Nature, xxxvii. pp. 246, 247.

DETERMINANTS IN GENERAL (PRASSE, 1811)

73

The definition of a permutation (variatio) being given, the first problem (which, however, is called a theorem) is propounded, viz., to tabulate the permutations of a, ^,y,8, . . . (" Variationuin ex elementis a, j8, y, . . . constructarum et in Glasses com- binatoi'ias digestarum Tabulam parare "). The result is

s}

a/3 07	a5"
)3a /37	^5
ya. 7|3	75-
Sa 5/3	Sy)
a/87	a/35>
a7/3	075
a5/3	a57
pay	pad
^7»	PyS
pSa	pdy ^
7«J3	yad
7/Sa	ypS
7*0	ydp
«o/3	8a7
5/3a	spy
57a	dyp)
	a^75
	a,357
	aypd
	aydp
	aSPy
	aSyP
	PayS
	Pa5y
	PyaS
	Pyda
	pSay
	Pdya yaps
	yadp
	yPad
	7/35o
	ydap
	ydpa
	dapy
	dayp
	dpay
	dpya
	Syap
	dypa^

74

HISTORY OF THE THEORY OF DETERMINANTS

The first row of the permutations involving two letters is got by taking the first letter of the previous row and annexing each of the others to it in succession and in the order of their occurrence ; the second row is got in like manner from the second letter ; and so on. Similarly the first row of permutations involving three letters is got from a^ the first obtained permutation of two letters, the second row from ay the next obtained permutation of two letters, and so on.*

The second problem (and on this occasion actually so designated) is somewhat quaint in its indefiniteness, viz., to prefix to each permutation the sign + or the sign — , so that the sum of all the permutations involving the same number of letters (>1) may vanish {"Singulis Variationibus, omissis repetitionibus, sigva -\- et — ita praejigere, ut summa secundoe et cujuslibet classis insequentis evanescat"). There is no indefiniteness or multiplicity about the solution, which in substance is : — Make the permutations in every row of the preceding table alternately + and — , the first sign of all being +, and the first permutation of every other row having the same sign as the permutation from which it was derived. In this way the table becomes

+ a, -	P. +y	-S
+ 0/3,	-ay.	+ ad
-pa,	+Py,	-/35
+ 7a,	-7/3,	+ yS
-da.	+ 5/3,	-h.

+ a/37.	-a/3«
-«7/3,	+ ayS
+ a«/3,	-aiy
-po-y,	+ Pad
+/»y«.	-Pys
-pSa,	+PSy
+yap,	-yad
-7/3a,	+ 7/33
+ ySa,	-75/3
-5a/3,	+ day
+ 5/3a,	-Spy
— 57a,	+ hP

* It will be seen that the order in which the permutations come to hand in this process of tabulation is the order in which they would be arranged according

DETERMINANTS IN GENERAL (PRASSE, 1811)

75

+ a^yS ^

- a/357

- 07/35 + 075/3 + a5/37 -aS7/3

- ^ayd + /3a57 + /37a5 -/375a — /35o7 + /357a

+ 70/35

- 7a5/3

- 7/3a5 + 7/35a + 75a/3

- 75/3a

- 5a/37 + 5a7/3 + 5/3o7

- 5/370

- 570/3 + 57/3a /

A proof by the method of mathematical induction (so-called) is given that with these signs the sum of all the permutations of any group vanishes.

Up to this point the essence of what has been furnished is a combined rule of term-formation and rule of signs. (11. 5 -|- iii. 15) In connection with it Bezout's rule of the year 1764 may be recalled.

The third problem is to determine the sign of any single permutation from consideration of the permutation itself. The solution is: — Under each letter of the given permutation put all the letters which precede it in the natural arrangement and which are not found to precede it in the given permutation; and make the sum -f or — according as the total number of such letters is even or odd.

to magnitude if each permutation were viewed as a number of which o, /3, 7, 5 were the digits, a being -<:/3->c7<:5 ("ordo lexicographicus," " lexicographische AnordnuDg" of Hindenburg).

76 HISTORY OF THE THEORY OF DETERMINANTS

" ExEMP. Datse complexiones sint hae :

eyS/?, Saey, «Sya, 8/8cy.

Literae secundum I subjiciantur

a a a a a . /3y8 a a a . a a a a

/?^/? /3y ^^^ P.y

7 7 77 7

S S

quarum numeri sunt

9 6 9 7

qui complexionibus datis praefigi jubent signa

+

The proof that this rule of signs, which is manifestly nothing else than Cramer's, leads to the same results as the previous rule, is quite easily understood if a particular permutation be first considered. For example, let the sign of the particular permutation S^ay be wanted. Following the first rule, we should require to note four different members, viz.,

(1) the no. of the column in which S^ay occurs in the 4th group,

(2) „ „ S^a „ 3rd „

(3) „ „ 5/8 „ 2nd „

(4) „ „ 8 „ 1st „

The first of these numbers being 1, we should infer that in fixing the sign of S^ay in the fourth group there had been no change from the sign of S^a in the third group; the second number being also 1, we should make a like inference ; the third number being 2, we should infer that in fixing the sign of 5/3 in the second group there had been 1 change from the sign of 6 in the first group; and finally, the fourth number being 4, we should infer that in fixing the sign of S in the first group there had. been 3 changes from the sign of o in that group. The total number of changes from the sign of a in the first group being thus 34-1 + 0 + 0, i.e., 4, the sign would be made +. Now the 3 in this aggregate is simply the number of letters in the first group which precede S, the 1 is simply the number of letters taken along with S before /3 comes to be taken along with it to form 5/8 in the second group, and the two zeros correspond

DETERMINANTS IN GENERAL (PRASSE, 1811) 77

to the fact that S^a on the third group and S^ay on the fourth group have no permutation standing to the left of them. Conse- quently to count the number of changes (3 + 1 + 0 + 0) from the sign of a in accordance with the first rule is the same as to count the number of letters placed under the given permutation, thus,

S^ay a a. .

y

in accordance with the second rule.

Another point of resemblance between Rothe and Prasse is thus made manifest, viz., that they both refused to accept Cramer's rule of signs as fundamental, preferring to base their work on a rule equally arbitrary, and then to deduce Cramer's from it.

In case it may have escaped the readet, attention may likewise be drawn to the fact that Prasse prefixes a sign not only to permutations involving all the letters dealt with, but also to any permutation whatever involving a less number ; so that in reckoning the sign of aS^, say, the full number of letters from which a, 8, ^ are chosen must be known.

A theorem like Hindenburg's is next given, viz., If the permuta- tions of any group be separated into sub-groups (I) those which begin with a, (2) those which begin with /3, and so on, then the series of signs of the Srd, 5th, and other odd sub-groups is identical with the series of signs of the Ist sub-group, avd the signs of any one of the even sub-groups is got by changing each sign of the first sub-group into the opposite sign. (in. 16)

It is more extensive than Hindenburg's in that it is true of permutations which involve less than all the letters, provided such permutations have had their signs fixed in accordance with Prasse's rule. The proof depends, of course, on the first rule of signs, and consists in showing that if the theorem be true for any group it must, by the said rule, be true for the next group. It will be remembered that Hindenburg gave no proof.

Following this is Rothe's theorem regarding the interchange of two elements of a permutation, or rather an extension of the

78 HISTORY OF THE THEORY OF DETERMINANTS

theorem to signed permutations involving less than the whole number of letters. The proof is as lengthy as Uothe's, even more unnecessary letters than Rothe's c, f, e being introduced, (in. 17)

The last theorem is Vandermonde's (xii.) ; and this is followed by two pages of application to the solution of simultaneous linear equations.

No reference is made by Prasse to Hindenburg, Rothe, or Vandermonde.

WRONSKI (1812).

[Refutation de la Th^orie des Fonctions Analytiques de Lagrange. Par Hoen^ Wronski. (pp. 14, 15,..., 132, 133.) 136 pp. Paris.]

In 1810 Wronski presented to the Institute of France a memoir on the so-called Technie de I'Algorithmie, which with his usual sanguine enthusiasm he viewed as the essential part of a new branch of Mathematics. . It contained a very general theorem, now known as " Wronski's theorem," for the expansion of functions, — a theorem requiring for its expression the use of a notation for what Wronski styled combinatory sums. The memoir consisted merely of a statement of results, and probably on this account, although favourably reported on by Lagrange and Lacroix, was not printed. The subject of it, however, turns up repeatedly in the Refutation printed two years later ; and from the indications there given we can so far form an idea of the grasp which Wronski had of the theory of the said sums.

At page 14 the following passage occurs: —

" Soient Xj, Xg, Xg, &c. plusieurs fonctions d'uno quantity variable. Nommons somme combinatoire, et d^signons par la lettre hebraique sin, de la maui^re que voici

to[A«Xi. A'Xg. A'Xg . . . A'-X^], (XV. 3) (vii. 4)

la somme des produits des differences de ces fonctions, composes de la maniere suivante : Formez, avee les exposans u, b, c, . . . , p des diff6rences dont il est question, toutes les permutations possibles ; donnez ces exposans, dans chaque ordie de leurs permutations, aux differences cons^cutives qui composent le produit

AX1.AX2.AX3 . . . AX,;

donnez de plus, aux produits s^par^s, formes de cette maniere, le signe positif lorsque le nombre de variations des exposans a, b, c, etc.,

DETERMINANTS IN GENERAL (WRONSKI, 1812) ' 19

consid^res dans leur ordre alphab6tique, est nul ou pair, et le signe n^gatif lorsque ce nombre de variations est impair ; enfin, prenez la somme de tous ces produits s6par6s. — Vous aurez ainsi, par exemple,

M3[A-XJ = A-X„ tt3[A-Xi . A'Xg] = A«Xi . A^Xg - A*Xi . A'^X^,

The new name, combinatory sum, and the new notation, did not originate in ignorance of the work of previous investigators, for memoirs of Vandermonde and Laplace are referred to. The only fresh and real point of interest lies in the fact that the first index of every pair of indices is not attached to the same letter as the second index, but belongs to an operational symbol preceding this letter, and is used for the purpose of denoting repetition of the operation. This and the allied fact that the elements are not all independent of each other, A^X^ and A^X^^ for example, being connected by the equation

A2Xi = A(AiX,),

indicate that Wronski's combinatory sums form a special class with properties peculiar to themselves.

CHAPTER IV.

DETERMINANTS IN GENERAL IN THE YEAR 1812.

Here we have the record of only one year and of only two authors to deal with ; but the authors, Binet and Cauchy, are of supreme importance, and the product of the year probably exceeded that of all the years that had gone before.

BINET (November 1812).

[M^moire sur un systfeme de formules analytiques, et leur application a des consid(^rations g^om^triques. Journ. de I'^c. Polyt, ix. cah. 16, pp. 280-802, . . .]

It would seem as if the above-noted frequent recurrence of functions of the same kind had led Binet to a special study of them. In the memoir we have now come to, his standpoint towards them is changed. They are viewed as functions having a, history: for information regarding them, the writings of Vandermonde, Laplace, Lagrange, and Gauss are referred to : they are spoken of by Laplace's name for them, resultantes d deux lettres, a trois lettres, d quatre lettres, &c. ; and the first twenty-three pages of the memoir are devoted expressly to establishing new theorems regarding them.

Of these the fundamental, and by far the most notable, is the

afterwards well-known Tnultiplication-theorem. It is enunciated

at the outset as follows : —

" Lorsqu'on a deux systemes de n lettres chacun, et nous supposerons chaque systems ecrit avec une seule lettre portant divers accens, qui scrviront k ranger dans le meme ordre les deux systemes ; on peut

former avec ces lettres un nombre n—- — de resultantes k deux lettres,

DETERMINANTS IN GENERAL (BINET, 1812) 81

en ne prenant dans le second terme de chacune que des lettres portant les memes accens que celles du premier. Si, avec deux autres systemes de lettres, on forme encore des resultantes k deux lettres, et qu'on les multiplie chacune par sa correspondante obtenue des deux premiers systemes, c'est-^ dire, par celle dont les lettres portent les memes accens ; la somme des produits de toutes ces resultantes correspon- dantes sera elle-meme une r^sultante a deux lettres, dont les termes ou lettres seront des sommes de produits des el^mens des deux systemes portant les memes accens. Avec deux groupes de trois systemes de n lettres chacun, on peut former semblablement deux series de resultantes k trois lettres ; faisant ensuite la somme des produits de celles qui se correspondent par les accens de leurs lettres, on aura encore une r^sultante k trois lettres. Pareille chose ayant lieu pour des resultantes k quatre lettres, &c., on peut conclure ce theor^me : Le produit d'un nombre quelconque de sommes de produits * de deux resultantes corre- spondantes de mSme ordre, est encore une r^sultante de cet ordre."

(xvii. 4 + XVIII. 4)

The mode of proof adopted is lengthy, laborious, and not very- satisfactory, except as affording a verification of the theorem for the cases of " resultantes " of low orders. It rests too on certain identities, the demonstration of which is open to similar criticism. All that Binet says regarding these absolutely essential identities is (p. 284)—

" Je repr^senterai par 2a la somme a' + a" + a" + &c., des quantit^s a', a", a'", &c. ; par 2a6 la somme des produits ab + a'b' + a"b" + &c., dans chacun desquels les lettres a et 6 ont le mSme accent ; par 2a6' la somme a'b" + h'a" + a'h'" + &c., 1^ tous les produits d'un des a par un des h, portent un accent different de celui de a ; par '2ab'c" la somme a'h"c"' + h'd'a"' + c'a"h"' + &C., et ainsi de suite. Cela pose, on verifie aisement les formules suivantes ;

2a&' = lalb - 2a6,

'2ab'e" = 2a262c + 22a&c - 2a2Z»c - 262ca - 2c2a&,

'2ab'c"d"' = lalh^cld - 62abcd

- 2a262a? - 2a2c25(i - 2a2(/2Jc

- "Zc^dlfab - 'Zb^dlac - 262c2a<i + "^ablfcd + ^2uc^bd + 2a6t2&c

+ 22a2icrf + 2'2b'Ecda + 11c2dab + 2^2d^bc,

'2ab'c"d"'e'' = 2a2A2c2c?2e + &c., &c."

♦ There is an extension here which one is scarcely prepared for, viz., *He produit d'un nombre quelconque de sommes de produits," instead of la somme d'un nombre de produits.

t Meant for 2ad.

82 HISTORY OF THE THEORY OF DETERMINANTS

It is thus seen that not only is no general proof of the identities given, but that even the law of formation of the right-hand members of the identities themselves is left undivulged. The exact words employed in the demonstration of the first case of the multiplication-theorem are (p. 286) —

" Avec un nombre n de lettres y', y", y'", &c, at un m^me nombre de /, z", z"\ &c, on pent former n ——— r^sultantes k deux lettres {y', z"),

(y'> ^")> &c- (y"> ^") &c. ; ayant form6 pareillement avec les lettres, v', u", v"', &c., {■', t"> C"> &c., les resultantes (v', ^"), (v', ^"'), &c., iy'i C"')> <^c-> consid^rons la somme 2(y, ^')(v, ^') des produits des resultantes qui se correspondent par les accens dans les deux systemes. On voit, en developpant, par la multiplication, chacun des termes de cette somme, qu'elle revient a

2yv . z'(' - Ssu . y'C.

A ces deux demi^res int^grales, on pent appliquer la transformation indiqu^e par la premiere des formules de I'art. 1 : on parvient ainsi k

2(y, z'){v, C) = 2yvlzC-^zv^yC

Ce dernier membre pouvant ^tre assimil6 a Ik forme (y, z'), il en r^sulte que le produit d'un nombre quelconque de fonctions, telles que 2(y, z'){v, ('), est lui-mdme de la forme (y, z')."

The application here of the identity

2a6'= 2a26 — Safe requires a little attention. The result of multiplication and classification of the terms is

or, as it might preferably be written, ^

and this we know from the said identity

= [2^. 2if- 2(^.27)] - [20";7.2^-2(^.^)],

which, because of the equality of 1>{yv . z^) and 2(3i; . y^), becomes

2yi).2iJ- Iz'v.ly^.

The inherent weak points, however, of the mode of demon- stration stand out more clearly when the next case comes to be considered, viz., the case for resultants of the third order. From the three sets of n letters

DETEEMINANTS IN GENERAL (BINET, 1812) 83

y> y\ y", ....

z, z, z , ....

all possible "resultantes k trois lettres" are formed, and each resultant is multiplied by the corresponding resultant formed from other three sets of n letters,

i, i\ i", ....

V, V, V , ....

^. r. r', ....

Each of these \n{n — l'){n — 2) products consists of 36 terms, there being thus 671(71 — 1) (72,-2) terms in all. But these 67i(n — l)(7i — 2) terms are found to be separable into six groups,

^i^- ^-^ixi.y'v'.zf'n^ +^{yi.z'v'.x"n,

so that the result which we are able to register at this point is

^{^. y\ 0(e ^'> n = ^^i ■ y'v . zT + ^yi- z'v . x"^"

+ 2;s^ . x'v' . y'T - ^^i ■ z'v . y'T V -lyi.x'v'.z'^r-llzi.yV.xT-

To the right-hand member of this the substitution

2a6'c" = 2a262c + 22a6c - 2a26c - 26Sca - ^c^ab is now applied six times in succession ; that is to say, for

Xxi.yV.zT

and the five other term-aggregates which follow, we substitute

'Lx$'Zyvlz^+2'Z{xi.yv.z^)

- Ixilliyv . z^)-lyv^(z^. xi)-Xz^l(xi. yv)

and five other like expressions. By this means we arrive, " toute reduction faite," at

2(a;, y', z"){^, v', ^") = ^x^lyu^z^+^yilzv^x^+Xz^^xvl^y^ - llxilzv^y^- lyilxvlz^-'ZziXyvXx^,

which is the result desired.

It is easy to imagine the troubles in store for any one who might have the hardihood to attempt to establish the next case in the same manner.

84 HISTORY OF THE THEORY OF DETERMINANTS

If Binet's multiplication-theorem be described as expressing a sum of products of resultants as a single resultant, his next theorem may be said to give a sum of products of sums of resultants as a sum of resultants. The paragraph in regard to it is a little too much condensed to be perfectly clear, and must therefore be given verbatim. It is (p. 288) —

" D^signons par S(y', z") una somme de r^sultantes, telle que

(y;,0+(y;.0+(y.;,0+&c.;

c'est-^-dire,

yX' - ^!y'/ + y,X" - ^.'y." + y.X," - ^JyJ + &c. ;

et continuons d'employer la caracteristique 2 pour les int^grales relatives aux accens sup^rieurs des lettres. L'expression

2[S(y,.').S(.,0] devient par le d^veloppement de chacun de ses termes, et en vertu de la premiere formule de I'art, 1 ou de celle du no. 4,

2y^v^ S^r^^^ - ^zv^ ^y,i, + %,/*', ^^„^, ~ ^'^//■"/^^z/C + &C.

+ 2y,v 2^,C. - '^^.-",^yL + '^y.y.i^^.L - 2^.", AX„ + &c.

+ &C.

En indiquant done par S^ des int6grales qui supposent, dans chaque terme, les m^mes accens inf^rieurs aux lettres du m^me alphabet, ces accens pouvant 6tre ou non les memes pour celles des alphabets differens, on pourra ^crire la pr^cedente suite, en faisant usage de ce signe, ce qui donne

2[S(y, z')S(v, O] = Si[2yv2^C-2^v2yt]. Cette nouvelle quantity est encore de la forme S {y\ z"), en sorte qu'on peut dire que le produit de fonctions, telles que

2{S(y,/)S(v,0}, sera lui-m^me de la forme S(y', z")."

This, if I understand it correctly, may be paraphrased and expanded as follows : —

Take the product of two sums of s resultants, viz.

{| 2/iV I + 1 2/2V I + I VzW I + ...■ + i VsW \)

or 2|2/,V|.f|u,^^/|,

«=i »=i

where, it will be observed, all the resultants in the first facto:

are obtained from the first resultant | y-^z-^ \ by merely changing

the lower indices into 2, 3, . . . , s in succession, and that th

DETEEMINAliTS IN GENEEAL (BINET, 1812) 85

second factor is got from the first by writing v for y and ^ for z. Then form all the like products whose first factors are

l2/iVl I2/1VI. • • • , l2/i"-VI;

these being along with | yi^Zi^\ the ^n{n — l) resultants derivable from the two sets of n quantities

2/i'. Vi^ yi> . . . . , 2/1"

The sum of these ^n{n—l) products may be represented, if we choose, by n=n r .=# .=. -1

m< n

Now if the multiplications be performed, there will be 8^ terms in each product, and the theorem we are concerned with has its origin in the fact that the sum of all the first terms of the products is expressible as a resultant by applying the multiplication- theorem, likewise the sum of all the second terms, and so on, the result being an aggregate of 8^ resultants. For if we fix upon a particular term of the first product, say

^^'^''"^ \ynW\-\-.V\

which arises from the multiplication of the h*'^ term of the first factor by the k^^ term of the second factor, then take the corre- sponding term of the other products, and write down their sum

I ynW\ • I v,%'\ + I y,W \-\v,'^,'\+ + 1 y,-'^n^ I . I u,»-^^,» I,

it is manifest that this sum is by the multiplication-theorem

yHW+yHW+- . . .+2/.%'^ z^W+zuW+' ■ • .+2^^"^*"

yn'^.'+yu'^.'+ +yH%^ Zh'^,'+z,V+ +^n%'' •

Consequently since h may be any integer from 1 to s, and k likewise any integer from 1 to s, the theorem arrived at is accurately expressed in modern notation as follows : —

U3 L »=i .=1 J

*=» *=» -.1 1 I

= 22

ykW+yKW+- • .+2/aX" z,w+zhW+- ■ •+2^'*^*" yHV+yHV+- ■ '+yHH>r znV+^K%'+- • •+^."^*"

o^ tt

Vh yh • ■ ■ ' yn

^h ^h ■ • • • ^h

Vk V*-'-' Vu*

^k ^k • • • ' il

86 HISTORY OF THE THEOEY OF DETERMINANTS

It is easily seen to be true of resultants of any order, as Binet himself points out. (xxx.)

When s is put equal to 1, it degenerates into the extended multiplication-theorem.

The theorem which follows upon this, but which is quite unconnected with it, may be at once stated in modern notation. Itis—

If 2| aJi2/2^3 1 denote the sum of the resultants obtainable from the three sets of n quantities

tCj X2 X^ .... Xn

2/1 2/2 2/3 • • • • 2/*

^1 ^2 ^3 • • • • ^m

and 2 1 x^y^ \ denote the like sum obtainable from the first two sets, then

2 1 x^y^^ I = 2a; . 2 I yi2r2 1 + Sy . 2 1 ^lajj I + Sar. 2 I x^y^ |. (xxxi.)

This is arrived at by writing out the terms of 2 1 y-^z^ \,oi'Z\ z^x^ |, and of 2 1 x^y^ \ in parallel columns, thus

I2/1 2^2! \^i «2l la^i 2/2I I2/1 ^sl 12^1 x^\ la^i 2/3!

\yn-iZn\ |«n-i«ni | a;n-i2/n | ;

then deriving n results from the members of the first row by multiplying by x^, y^, z^ respectively and adding, multiplying by ajg, 2/2? 2^2' ^^^ adding, and so on ; then treating the second and remaining rows in the same way; and then finally adding all the 71.^71(71 — 1) results together. Ekch of these results is a vanishing or non-vanishing resultant of the 3"* order, and it will be found that each non-vanishing resultant occurs twice with the sign + and once with the sign — .

This process is readily seen to be simply the same as per- forming the multiplications indicated in the right-hand member of (xxxi.), i.e.y -: , „

{x^+x^+.	' + ^n){\yyZ^\ + l^i^sl +.	.+ |y«-i«»|)
+(2/i+2/2+-	• '■^yn){\z^X2\ + \z^X^\+.	.+ \Zn-xXn\)
+ (2^1+ ^8 + . .	' + 2n){\xiy^\ ■\-\x{y^\ +.	.+ \Xn.xyn\),

DETERMINANTS IN GENERAL (BINET, 1812) 87

summing every three corresponding terms in the products, and writing the sum as a vanishing or non-vanishing resultant. There would be n.^n(n — l) resultants in all ; but as each suffix occurs n — 1 times in the second factors and once in the first factors, there must be in each product n — 1 terms having the said suffix occurring twice: consequently there must be n—1 resultants vanishing on account of this recurrence, and therefore altogether n{n — l) vanishing resultants. Of the non- vanishing resultants, — in number equal to n . ^n{n — l)—n(n—l), or ^n(n—l){n — 2), — each one of the form

I ^kVi^i 1 where h<k<l must be accompanied by two others,

\Xj,y^Zi\ and [xiy^z^l, and the sum of these is

I oc^yk^i I - 1 x„y^i I + I Xj,y^, I , i.e., I x^y^i I .

The final result is thus the sum of the resultants of the form

I ^x^hVi^i i where h<k<l, and i = 3, 4, . . . ,n, the number of them, as we may see from two different stand- points, being in(n-l){n-2).

Returning to the series of identities,

^3 I yi^2 I + 2/3 I 2^ia'2 I + 2^3 I ^l3/2 I = I ^l2/2^.<J I '

«4 i 2/l^2 I + 2/4 I ^1^2 I + 2=4 I a'iy2 I = I ^iVi^i I > &LC. &C.

which by addition give the result

Ix^ly^z^l + ^yl\z^x^\ + I,zl.\x^y^\ = ^Ix^y^Zs] ,

Binet next raises both sides of all of them to the second power, and obtains

32|a:,2/2^3|2 = ^^^\y,z^f + ^y^I.\z,x^f + ^z^^\x,y,f ^

+ 21yzY.{\z^x^\.\x^y^\)+2^zx1{\x^y^\.\y^z^\) Uxxxii.) -\-21acy^{\y^z^\.\z^x^\).]

88 HISTORY OF THE THEORY OF DETERMINANTS

:,}

Substituting for 2|2/i^2p» 2|^l'^2!^ • • • • > their equivalents as given by the multiplication-theorem, he then deduces

21^12/2^3!^ = Sflj^S^/^S^^ + TLyzLzxLxy — l.x^{^yzf

-ly^iXzxf - i:z\^yf,

not failing to note that this is not a fresh result, but merely a case of the multiplication-theorem in which the factors are equal. By putting the right-hand member here into the form

XyHlz^lx^-{Xyz)^} + XzHXx^Xy^-ilxyf}

- ljc^{2y^lz^-(lyz)^} + 2i:yz{^zxlxy-2yz^x^},

there is next arrived at the first identity of the set

= ly^i:\z,x,\^+I,z'X\x,y,\'-^x^l\y,z,\^ + 22yzI,\z,x,\\x,y,\A

= 2^22|a;^y^|2+2a;22|y^^^ |2_ 22^22|^^^J2+ 22^x2|a;,2/2|| 2/1^2!, [(xxxiiiO

= lx^^y,z,\^+2y'l\z,x,\^-lz^i:\x,y,\'+2^yX\y,z,\\z,x,\,]

and immediately from these the set

2|a;i2/22;3p = 2a;22|2/i2^2p+2«a;2|cBi2/2M2/i2^2l+2a;2/2|2/i2'2M2^ia'2lA

= 'Ey^l\z^x^\^ + Xxyl\y^Z2\.\z^X2\ + ^yzX\zjX^\.\Xiy^\UxxxiY.]

= 2^22 1 aj^y/ + 2yz^\ z^x^ \ . \ x^y^ \ + J.zx1. \ x^y^ | . | ^i^a I -I

We may note in passing that either of these sets leads at once to the initial theorem

Znx,y,z^\^ = ^x^nyv^.i? + ^y'nz^x^\^ + ^z^nx,y^f + 21yz1.\z^x^\.\x{y^\ + 21zx1\x^y^\.\y^z^\ + ^1xy^y^z^\.\z^x^\,

and that with the multiplication-theorem already established this reverse order would be the more natural.

The next step taken is the formation of resultants of the 2"* order from elements which are themselves resultants of the 2°* order ; viz., just as from the three rows of n quantities

x^ x^ x^ .... Xf^

2/12/22/3 • • • • 2/«

\

DETERMINANTS IN GENERAL (BINET, 1812) 89

there were formed the three other rows of ^n{n — l) quantities

yi^2

z,x,

1**'2 I '

2/l2^3l. ..••, IVl^nl, 12/223!' .-.., \yn-iZn\,

ZyC^ I J . • • • > I Z-^n I , I Z^^ I , . . . . , \ Zn^ ^Xn | ,

laJi^al' l^'iysl. ...., !a;i2/n|, 1 2^22/3 1. ••••, kn-i2/n|, so from the latter three other rows of quantities

1 ^1^2 1	^i^z\
|a'i2/2l	^{yz\
^12/2 1	a'i2/3l
1 yi^2 1	2/1^3 1
12/12^2 1	2/1^3 1
1 %«'2 1	2l«'3 1

1 ^fn-a^^n 1	Zn-	■v^n
|a;n-22/«l	Xn-	■l2/n
|a5»-22/n|	Xn-	l2/«
1 ^n-is^^w 1	Vn	-l^w
|y»-2^n|	Vn	-l^«
1 2^n-2a;n 1	Zn-	-l^n

are formed, the number in each new row being clearly

i.e., i(n+l>i(n-l)(7i-2).

The new quantities are, of course, not written by Binet in the form

but the fact that they are resultants of the 2*"* order is carefully noted. Each of them is shown to be transformable, by a theorem which may be viewed as an extension of a result given by Lagrange, so as to have two of the elements resultants of the S""* order, and the other resultants of the 1^* order. This is done by taking, for example, the identities

^n I ViZj I + 2/ J Zi^i I + Zj,\Xiyj\ = \ x^y,z^ \ , a^fc I Vi^i I + y* I z^x^ I + 2; J aji^^ I = I x^y^z^ \ ,

multiplying both sides of the first by %, and both sides of the second by x^, subtracting, and writing the result in the form

1 ^^kVh \\zf<^j\+ l^k^hW Xiyj \ = x^\ x^y^Zj \ - x^\ x^y^Zj \ ,

~ I ^* x^

1 1 x^yiZj I I x^yiZj

90 HISTORY OF THE THEORY OF DETERMINANTS

where of course it has to be noted that in many cases one of the resultants of the S^^ order will vanish. The quantities, therefore, to be dealt with, are

^1 I ^l2/2«3 I , . • . , flJfc I X^yiZj \- X^\ XtViZj I , . . . ,Xn\ Xn-iVn-iZn 1 1 Vl I «^l2/2^3 I > • • • > 2/* I Vh'^t^J \- Vhl Vl^i^S I , • . • . yn I Xn-iVn-i^n \ ', ^1 I ^l2/2^3 I . • • • . 2;^ I Zj^Xiy^ \ - Zj,\ Z^tyj | , . . . , 0„ | Xn-iVn-i^n |.

By raising each of the elements of the first row to the second power, taking the sum and simplifying, we could, we are told, show that the result would be

lXj^X\x^y^^\^.

Very prudently, however) another process is chosen. It is re- called that the quantities in the third triad of rows are related to those in the second as those in the second are related to those in the first, and that consequently the required sum of squares of resultants is, by the multiplication-theorem itself, expressible as a resultant, viz.,

where the elements of the resultant on the right are sums of products of quantities in the second triad of rows. Then the same theorem is used to make a further step backwards, viz., to express each of these three sums of products of resultants as a resultant whose elements are sums of products of the quantities in the first triad of rows, the effect of the substitution being

2|| 2^13^2 1 , 1 x,y,\\' = {lz,'i:x,^-{^z,x,f}{Ix,'ly,^-(Ix,y,n

- {Xz^x^Zx^y^ - Xy^z^'Ea^^}l

Simple multiplication transforms this into

^ 2 f I^^Zy.'lz,^ - ^y^'(^z,x,f - 2z,^(I^y,f | ^ ^ I +2Xy^ZjI.ZjX^'Lx^yj^ - Iac^\'Ly^z^^ r

which, by still another use of the multiplication-theorem, we

know is equal to

^M^^x^y^z^f,

(xxxv.)

DETEEMINANTS IN GENERAL (BINET, 1812) 91

The set of six results of which this is one, is

2Z,2 =20,2 ^x,y,z,\\ J.Y^Z^='Ly^z^I.\x^y^z^f, SZ^Xi = S^i^i 2|a;i2/203p,

if, for shortness, we denote the quantities of the third triad of rows by

Xj, Xg, ....

Y Y

Zp Zg, . . . .

Following these, and deduced by means of them, is an equally noteworthy theorem regarding the sums of squares of all the resultants of the third order, which can be formed from the quantities of the second triad of rows. Denoting these quantities I temporarily by £ i

6l> 62' • • ' • *lv *]2> • ' ' • Sl> S2' • • • •

we know (xxxii.) that

+ 22Y,Z,.2^,^, + 22Z,X,.2^,^,

-whence, by using the set of six results just obtained, we have

32 I ^,,2^3 1^

' '^' «l 1+ 2^rj,^,.ly,z, + 21^,i,.Zz,x, + 22^,,;,.2a:i2/J and therefore, again by (xxxiii.)

2|#i'72^3l'={2|a;i2/2^3in'. (XXXVI.)

It is finally pointed out that from the third triad of rows there might, in like manner, be formed a fourth triad, and

92 HISTORY OF THE THEORY OF DETERMINANTS

analogous identities obtained ; also that, instead of starting with three rows, we might start with four,

^l> ^2' ^3' • • • • J ^n ^Jj, X^, X^, . . . . , Xfi

Vv 2/2. 2/3, ..... 2/«

^l> ^2» ^8» • • • • » ^ri)

form from them other f our

I ^iV^z \>

I 2/l2'2<3 I ,

I ^li^s \>

I ^v^iVz I > ♦

thence in the same way a third four, and in connection therewith establish the identity (xxxi. 2)

S<i2 I x^y^z^ I - Sa;i2 | y^zj^^ \ + S^iS | z^t^^ \ - llz^l \ t^x^y^ \ = 0 and other analogues. (xxxii. 2 + xxxv. 2)

The rest of the memoir, 52 pages, consists of geometrical applications of the series of theorems thus obtained.

CAUCHY (1812).

[Memoire sur les fonctions qui ne peuvent obtenir que deux valeurs ^gales et de signes contraires par suite des trans- positions op^r^es entre les variables qu'elles renferment. Journ. de I'Ec. Polyt, x. Cah. 17, pp. 29-112. (Euvres (2) i.]

This masterly memoir of 84 pages was read to the Institute on the same day (30th November) as Binet's memoir, of which we have just given an account. The coincidence of date has to be carefully noted, because the memoirs have in part a common ground, and because there is a presumption that the authors, knowing this beforehand, had, in a friendly way, arranged for simultaneous publicity. Binet's words on the matter are (ix. p. 281) —

DETERMINANTS IN GENERAL (CAUCHY, 1812) 93

"Ayant eu derni^rement occasion de parler k M, Cauchy, ing^nieur des ponts et chauss^es, du th^or^me g6n6ral que j'ai 4nonc^ ci-dessus, il me dit ^tre parvenu, dans des recherches analogues k celles de M. Gauss, k des th^oremes d'analyse qui devaient avoir rapport aux miens. Je m'en suis assur^ en jetant les yeux sur ces formules : mais j'ignore si elles ont la m^me g^n^ralit^ que les miennes : nous y sommes arrives, je crois, par des voies tr^s-diff^rentes."

And Cauchy 's corroboration is (p. Ill) —

" J'avais rencontr6 I'^te dernier, k Cherbourg, ou j'6tais fix6 par les travaux de mon ^tat, ce theor^me et quelques autres du m^me genre, en cherchant k g^neraliser les formules de M. Gauss. M. Binet, dont je me f6licite d'etre I'ami, avait 6t6 conduit aux m^mes r^sultats par des recherches diff6rentes. De retour k Paris, j'etais occup6 de poursuivre mon travail, lorsque j'allai le voir. II me montra son th^or^me qui 6tait semblable au mien. Seulement il d^signait sous le nom de r^sultante ce que j 'avals appel6 determinant."

Cauchy prefaces his memoir by another, entitled

Sur le nombre des valeurs qu'une fonction peut acqu^rir lorsqu'on y permute de toutes les manieres possibles Us quantites qu'elle renferme.

This latter must to a certain extent be taken into account, because it serves to show the point of view which he considered most natural for examining the subject, and also the exact position held by the functions now called determinants, when functions in general come to be classified according to the number of values they are able to assume in certain circumstances.

At the outset of it the writings of Lagrange, Vandermonde, and Ruffini are referred to; the fact is recalled that the maxi- mum number of values which a function can acquire by inter- changes among its n variables is 1 . 2 . 3 . . . . w ; also that when the maximum is not obtained, the actual number must be a factor of the maximum ; and then proof is given of the very notable theorem that the number of values cannot be less than the greatest prime contained in n without being equal to 2. It is pointed out likewise that functions capable of having only two values are known from Vandermonde to be constructible for any number of variables. For example, the number of

94 HISTORY OF THE THEORY OF DETERMINANTS

variables being three, a^, a^, a^, all that is needed is to form their difference-product

(ag-tta) (as-^i) («2-«i) or aj^ttg + ob^cbi + o^dz — (as^^i + a2^<^3 + *i^*2)»

when it is found that either of the parts

or a^a^-\-a^a^+a^a^,

is an instance of a function capable of only two values by per- mutation of the variables ; the result indeed of any permutation being merely that the one function passes into the other. Further, the whole expression

a^a^ + «2^<*l + *1^^3 ~" (^8^*1 + ^2^<^3 + <*l^<^2)

is another example, the difference between the two values which it can assume being however a difference of sign merely. As a reference to the title of the memoir of November 1812 will show, it is functions of this latter class which Cauchy there considers.

At the commencement he contrasts them with functions which suffer no change whatever by permutation of variables, that is to say, symmetric functions : and, noting the fact, afterwards ascertained, that the new functions consist of terms alternately + and - , and that were it not for this alternation of sign they would be symmetric functions, he decides to extend the term " symmetric " to them, and having done so, seeks to distinguish them from ordinary symmetric functions by calling them "fonc- tions sym^triques altern^es," and calling the other " fonctions sym^triques permanentes." Cauchy's view of determinants may therefore now be described by saying that he considered them as a special class of alternating symTuetric functions.

To include them, however, either the adoption of a convention is necessary, or an extension of the definition must be made. For example, a^b^ — a^b^ is not an alternating function, unless the elements be so related that the interchange of a^ and a^ necessitates the interchange of 6^ and b^ at the same time; or unless the definition be so worded that interchange shall refer

DETEEMINANTS IN GENERAL (C^CHY, 1812) 95

to suffixes, not to letters. Cauchy selects the former course his words being (p. 30)

" . . . . concevons les diverses suites de quantites

dv	«2. • • •	• . . a.
K	h^ ...	. ., ft.
<^v	H, . . .	. , c.

tellement li^es entre elles, que la transposition de deux indices pris dans Tune des suites, necessite la m^me transposition dans toutes les autres ; alors, les quantites

Oj, Cj, . . . , OjJ ^2' • • • ' ^3' ^3> • • • •

pourront §tre consid^r^es comme des fonctions semblables de

ttj, a^, Ag, . . . . ; et par suite, les fonctions de

^1' "1' ^1' • • • » ^2' ^2' ''2' ' • • » ^»f ^W ''nJ • • • •

qui ne changeront pas de valour, mais tout au plus de signe, en vertu de transpositions op6r6es entre les indices 1, 2, 3, .... n, devront etre rangees parmi les fonctions symetriques de a^, aj, . . . , a„, ou, ce qui revient au meme, des indices 1, 2, 3, . . . , n. Ainsi

O'lh + 0"p2 + <^zh + 2C1C2C3,

O'xh + ^2^3 + ^zK + "2^1 + «3*2 + «1^3 '

cos(aj - a^Q,os{a^ - a3)cos(a2 - a^,

seront des fonctions symetriques permanentes, la premiere du second ordre et les autres du troisieme ; et au contraire,

<hh + Hh + hK - "2^1 - ^\h - <*3*2> sin («! - Og) sin (aj - Og) sin {a^ - a^)

seront des fonctions symetriques alternees du troisieme ordre."

The question of nomenclature being settled there next arises the question of notation. This also is decided on the ground of the resemblance of the functions to symmetric functions. It being known that any symmetric function is representable by a typical term preceded by a symbol indicating permutation of the variables, e.g.

S(«i^2) ^^ ^^(^1^2) standing for a-J)2 + (i2\

and S^ (O'lh^) standing for a-J)^ + a^b^ + 0.3^ + ^2^1 + %^2 + ^1^3 >

96 HISTORY or THE THEORY OF DETERMINANTS

also, that any non-symmetric function may be taken as the typical term of a symmetric function, the question arises whether the like may not be true of alternating functions. A lengthy examination of the latter point leads to the conclusion that any non-symmetric function K cannot be the originating or typical term of an alternating function unless it satisfies a certain condition, viz., that it be such that any value of it obtained by an even number of interchanges of indices will be difierent from any other value obtained by an odd number of interchanges. Should, however, this condition be satisfied, and Ka, K,s, K^, .... be all the values of the former kind, and K^, K^, K^, .... all the values of the latter kind, then

(K<,-}-K^ + K,+ ) - (K,+K^ + K,+ )

is an alternating function and is appropriately representable by

S(±K)

if the indices appearing in K alone are to be permuted, and by

S"(±K)

if the indices to be permuted be 1, 2, 3, . . . , ?i. For example, taking the typical term a^^ we have

and ^^{±a-p^ = a^b^+a^h^+a^bj^ — a^b^ — a^b^ — a^h^, = S'(=Fa,b,) = S'(=Pa,b,)=

S^{±a^b^) is an impossibility, as when there are four indices a^b^ does not satisfy the condition required of a typical term ; indeed, Cauchy notes that the number of indices in any term must either be the total number or 1 less.

The number of permutations being even, it is clear that the number of + terms Ka, K|8, . ... is the same as the number of negative terms K^, K^, (x. 2)

a generalisation of a remark of Vandermonde's.

Further, since K^, K^, .... are all the terms that arise from an even number of transpositions, and K^, K^, .... all those that arise from an odd number of transpositions, it is plain that

DETERMINANTS IN GENERAL (CAUCHY, 1812) 97

any single transposition performed upon each of the terms of the function

(K.+Kp+Ky+ ) - (K,+K^+K,+ )

must change it into

(K,+K^+K,+ ) - (K.+K +K,4- )

— this is, in fact, the proof that it is an alternating function — consequently each of the parts

K, + K^ + K^+ ,

Kx+K +K,+ ,

belongs to the class of functions which have only two different values.

Also it is evident that if throughout the function any par- ticular index be changed into another and no fv/rther alteration made, the resulting expression must he equal to zero, (xii. 5) a theorem regarding alternating functions which is the general- isation of a theorem of Vandermonde's.

We have lastly to note, that the criterion which determines whether a particular K belongs to the class Ka, K^, .... or to the class K^, K^, .... is incidentally shown to be reducible to a more practical form. For example, if the term be K^, and it be derivable from K, say, by the change of the suffixes 1, 2, 3, 4, 5, 6, 7 into 3, 2, 6, 5, 4, 1, 7, that is to say, in Cauchy's language by means of the substitution

1, 2, 3, 4, 5, 6, ^

/I, 2, 3, 4, 5, 6, 7\ \3, 2, 6, 5, 4, 1, 7/,

we transform this substitution into a "product" of "circular' substitutions, viz., into

G;^;D-(n) •©■(?)

and subtracting the number of "factors," 4, from the total number of suffixes 7, make the sign + or — according as this difference is even or odd.

Here the subject of general alternating functions may be left for the present. What remains of the first part of the memoir, refers to special cases, which naturally fall to be considered

98 HISTORY OF THE THEORY OF DETERMINANTS

in another chapter of our history. At the close of the part Cauchy says (p. 51) —

" Je vais maintenant examiner particuli^rement une certaine esp^ce de fonctions sym^triques altem^es qui s'ofFrent d'elles-m^mes dans un grand nombre de recherches analytiques. C'est au moyen de ces fonctidns qu'on exprime les valeurs g^n^rales des inconnues que renfer- ment plusieurs equations du premier degr^. Elles se representent toutes les fois qu'on a des Equations k former, ainsi que dans la th^orie g^nerale de l'6limination."

The writings of Laplace, Vandermonde, Bezout, and Gauss are referred to, and from the latter the name "determinant" is adopted.

The second part bears the title —

Des fonctions symetriques altern^es designees soils le nom de ddterminans. (xv. 4)

and opens with the following explanatory definition (p. 51) —

"Soient o^, Og, . . . , a„ plusieurs quantit^s diflP^rentes en nombre ^gal k n. On a fait voir ci-dessus qu'en multipliant le produit de ces quantiW8,ou a,a^,....a^

par le produit de leurs differences respectives, ou par

{a^ - a^){a^ - a^) . . . (a„ - a^){a^ - flg) • • • («« - O2) • • • K " <»«-i).

on obtenait pour r^sultat la fonction sym^trique altern^e

qui par consequent se trouve toujours ^gale au produit

a^a^a^ a^

X (ajj - a,)(a3 -a^)... (a„ - a^){as -a^)... (a, - ^2) • • • K " ««-i)-

Supposons maintenant que Ton ddveloppe ce dernier produit, et que dans chaque terme du d6veloppement on remplace I'exposant de chaque lettre par un second indice ^gal k I'exposant dont il s'agit, en ecrivant par exemple a^.. au lieu de a/, et a,^ au lieu de a/, on obtiendra pour r6sultat une nouvelle fonction sym^trique altern^e, qui, au lieu detre representee par

sera representee par

le signe S etant relatif aux premiers indices de chaque lettre. TpUe

DETEEMINANTS IN GENERAL (CAUCHY, 1812) 99

est la forme la plus g^nerale des fonctions que je d^signerai dans la suite sous le nom de (Utermirmns. Si I'on suppose successivement *

n=l, 71 = 2, &c

on trouvera

- ay^a^.^^.^ - a^.ia^.jfiy^ - a2-i*i-2'*8-3»

&c

pour les d6terminans du second, du troisi^me ordre, &c "

In regard to this it is important to notice that there are really two definitions given us. The latter, viz., that involved in the symbolism of alternating functions,

S(±ai.^a^.2a^.^ a„.„)

contains nothing more than Leibnitz's rule of formation and an improved rule of signs. The former is new and may be paraphrased as follows : —

If the multiplications indicated in the expression a^a^a^ ... an x(«2-<*i)(«3-ai)- • .{dn-aiKa^-a^)- ■ • {an-a^). . .(a„-a„_,) be performed, and in the result every index of a power be chaTiged into a second suffix, e.g., a/ i'<^io a,.,, the expression so obtained is called a determinant, (in. 18), (viii. 2)

and is denoted by q/,^ „ ^ „ \ /vtt e^\

In this definition the rule of signs and the rule of term- formation are inseparable — a peculiarity already observed in the case of Bezout's rule of 1764.

After the definitions various technical terms are introduced. The n^ difierent quantities involved in

are arranged thus

S(± 01.108.203.3 . . .	. «»•«)
*11» %-2' *1-S' •	. . . Oi.„
^2*1» ^2-2» **2*8' •	• • • fltj-n
%•!» ^'S' %'8» •	• • • «8n
&C
J^n-V *n-2' *n-3' •	• • %-n

')^n=2, n=3, &c. is meant.

100 HISTORY OF THE THEORY OF DETERMINANTS

" sur un nombre 6gal k n de lignes horizontales et sur autant de colonnes verticales," and as thus arranged are said to form a symmetric system, of order n. The individual quantities a^.^, &c., are called the terms of the system, and the letter a when free of suffixes the characteristic. The " terms " in a horizontal line are said to form a suite