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MATHEMATICAL PAPERS

CAMBRIDGE UNIVERSITY PRESS

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THE COLLECTED MATHEMATICAL PAPERS

OF

JAMES JOSEPH SYLVESTER

F.R.S., D.C.L., LL.D., Sc.D.,

Honorary Fellow of Sf John's College, Cambridge j

Sometime Professor at University College, London ; at the University of Viijiiii* i

at the Royal Military Academy, Woolwich ; at the John* Hopkins Univenity, Baliimorc

and Savilian Professor in the University of Oxford

VOLUME IV

(1882— 1897)

Cambridge

At the University Press 1912

Cambttligc :

PRINTED BY JOHN CLAY, M.A. AT THE UNIVERSITY PRESS

PREFATOEY NOTE

rriHE present volume contains Sylvester's Constructive Theory of Partitions, papers on Binary Matrices, and the Lectures on the Theory of Reciprocants. There is added an Index to the four volumes, and a Biographical Notice of Sylvester. The Mathematical Questions in the Educational Times are as yet unedited, but an Index to them is appended here. I have to acknowledge the kindness of Dr J. E. McTaggart, F.B.A., who secured for me the loan of the Essay on Canonical Forms, from the Library of Trinity College, Cambridge, for Vol. i, and that of Mr R. F. Scott. M.A., Master of St John's College, Cambridge, for the use of the volume called The Laws of Verse, from which the matter contained in the Appendix to Vol. ii was reprinted, who supplied also the Autograph on the Frontispiece of this Volume. To the latter gentleman, as well as to Major P. A. MacMahon, Professor E. B. Elliott and Sir Joseph Larmor, I owe my best thanks for reading through the Biographical Notice. In carrying through the task of editing the Papers, I have, in general, thought it most fitting not to oflFer any remarks of my own in regard to Sylvester's text, though many times at a loss to know how best to act. In the Appendix to Vol. I I have departed from this rule, giving there an account of Sylvester's chief theorems in regard to determinants. For two other cases the reader may find notes, Proceedings of the London Mathematical Society, Vol. iv, Ser. II (1907), pp. 131—135, and Vol. VI (1908), pp. 122—140; these refer respectively to the paper No. 36, p. 229, and to the paper No. 74, p. 452, both m Vol. II of the Reprint. Many corrections of errors in the printing of algebraical formulae have been introduced, though many, it is to be feared, still remain ; but no alterations of Sylvester's statements have been made without definite indication, by square brackets or otherwise. To the Readers and Staff of the University Press the very greatest credit and gratitude for their watchful carefulness are assuredly due, many of the corrections in the volumes being due to them.

H. F. BAKER. June 1912.

TABLE OF CONTENTS

PAGES

Portrait of J. J. Sylvester .... Frontispiece

Medallion Head of Biographical Notice

Biographical Notice xv xxxvii

1. A constructive theory of partitions, arranged

in three acts, an interact and an exodion 1 83

(American Joaroal of Mathematics 1882, 1884)

2. Sur les nombres de fractions ordinaires in-

egales qu'on pent exprinier en se servant de chiffres qui n'excedent jias un nombre donne 84 87

(Comptes Bendus de I'Acadimie des Sciences 1883)

3. Note sur le theoreme de Legendre citS dans

une note instrde dans les Comptes

Rendus 88—90

(CompteB Rendas de I'Acad&uie des Sciences 1883)

4. Sur le prodtiit ind^fini \—x.\—af.\—af... 91

(Comptes Bendns de I'Academie des Sciences 1883)

5. Sur un theoreme de partitions ... 92

(Comptes Rendas de I'Acadimie des Sciences 1883)

6. Preuve graphique du theoreme d' Eider sur

la jHirtition des nombres pentagonaux . 93, 94

(Comptes Rendus de I'Acad^mie des Sciences 1U83)

7. Demonstration graphique d'un theoreme

d'Euler concernant les partitions des

nombres 95, 96

(Comptes Bendas de I'Acad^mie des Sciences 1883)

8. Sur un theoreme de imrtitions de nombres

complexes contenu dans un theoreme de

Jacobi 97 100

(Comptes Bendas de I'Acadimie des Sciences 1883) 8. IV. 5

Vlll

Contents

PAGES

9. On the number effractions cotdained in any "Farey seHes" of which the limiting number is given 101 109

(Philosophical Magazine 1883)

10. On the equation to the secular inequalities in

the planetary theory . . . . 110, 111

(Philosophical Magazine 1883)

11. On the involution and evolution of quater-

nions 112 114

(Philosophical Magazine 1883)

12. On the involution of two matrices of the

second order 115 117

(Soathport British Association Beport 1883)

13. Sur les quantites formant un groupe de

nonions analogues aux quaternions de

Hamilton ...... 118—121

(Comptes Bendas de I'Acad^mie des Sciences 1883)

14. On quaternions, nonions, sedenions, etc. . 122 132

(Johns Hopkins University Circulars 1884)

15. On involutants and other allied species of

invariants to matrix systems . . 133 145

(Johns Hopkins University Circulars 1884)

16. On the three laws of motion in the world of

universal algebra 146 151

(Johns Hopkins University Circulars 1884)

17. Equations in matrices 152, 153

(Johns Hopkins University Circulars 1884)

18. Sur les quantity formant un groupe de

nonions analogues aux quaternions de

Hamilton 154 159

(Comptes Bendus de 1' Academic des Sciences 1884)

19. Sur line note recente de M. D. Andre . 160, 161

(Comptes Bendus de 1' Academic des Sciences 1884)

20. Sur la solution d'une classe tres Stendue

d'iquMtions en quaternions ... 162

(Comptes Bendas de I'Acad^mie des Sciences 1884)

Contents ir

PAGES

21. Stir la coi^respondance entre deux esphces

differentes de fonctions de deux systemes de quantites, correlatifs et egalement nombreux 163 165

(Comptes Bendas de I'Acad^mie des Sciences 1884)

22. Sur le theoreme de M. Brioschi, relatif aux

fonctions symetriques .... 166 168

(Comptes Beudus de I'Acad^mie des Sciences 1884)

23. Sfiir line extension de la loi de Harriot

relative aux equations algebriques . 169 172

(Comptes Bendas de I'Acad^mie des Sciences 1884)

24. Sur les equations monothetiques . . . 173 175

(Comptes Bendus de I'Acadimie des Sciences 1884)

25. Sur Vequation en matrices px = xq . . 176 180

(Comptes Bendas de 1' Academic des Sciences 1884)

^26. Sur la solution du cas le plus general des Equations lin^aires en quantites binaires, c'est-d-dire en quaternions ou en matrices du second ordre 181, 182

(Comptes Bendas de I'Acad^mie des Sciences 1884)

Stir les deux methodes, cdle de Hamilton et celle de Vatiteur, pour r^soudre Vequation lindaire en quaternions .... 183 187

(Comptes Bendas de I'Acad^mie des Sciences 1884)

Sur la solution explicite de Vequation quad- ratique de Hamilton en quaternions ou en matrices du second ordre . . . 188 198

(Comptes Bendas de I'Acad^mie des Sciences 1884)

29. Sur la resolution gendrale de Vequation

lindaire en matrices d'un ordre quelcon-

qiie 199—205

(Comptes Bendus de I'Acad^mie des Sciences 1884)

30. Sur Vequation lineaire trinome en matrices

dun ordre quelconque .... 206, 207

(Comptes Bendas de I'Acad^mie des Sciences 1884)

62

Contents

31. Lectures on the principles of universal

algebra

(American Journal of Mathematics 1884)

32. On the solution of a class of equations in

quaternions

(Philosophical Magazine 1884)

33. On Hamilton's quadratic equation and the

general unilateral equation in matrices

(Philosophical Magazine 1884)

34. Note on Captain MacMahon's transforma-

tion of the theory of invariants .

(Messenger of Mathematics 1884)

35. On the D'Alembert-Camot geometrical para-

dox and its resolution ....

(Messenger of Mathematics 1885)

36. Sur line nouvelle theorie de formes algehriques

(Comptes Bendus de I'Acadfimie des Sciences 1885)

37. Note on Schwarzian derivatives .

(Messenger of Mathematics 1886)

38. On reciprocants

(Messenger of Mathematics 1886)

39. Note on certain elementary geometrical no-

tions and determinations

(Proceedings of the London Mathematical Society 1885)

40. On the trinomial unilateral quadratic equa-

tion in matrices of the second order .

(Quarterly Journal of Mathematics 1885)

41. Inaugural lecture at Oxford, on the method

of reciprocants . . .

(Nature 1886)

42. Lectures on the theory of reciprocants

(American Journal of Mathematics 1886 )

43. Sur les reciprocants purs irrMuctiUes du

quatrieme ordre

(Comptes Eendus de I'Aoad^mie des Sciences 1886)

PAGES

208—224

225—230

231—235

236, 237

238—241 242—251 252—254 255—258

259—271

272—277

278—302 303—513

514

Contents xi

PAGES

44. Sur une extenmon du theorenie relatif au

nomhre d^invariants asyzygetiques dim type donne a une classe de formes ana- logues 515 519

(Comptes Bendas de rAcad^mie des Sciences 1886)

45. Note sur les invariants differentiels . . 520 523

(Comptes Bendas de I'Acad^mie des Sciences 1886)

46. Stir Tequation differentielle d'une courhe

d'ordre quelconque 524 526

(Comptes Bendas de I'Acad^mie des Sciences 1886)

47. Stir une extension d'un theorenie de Clebsch

relatif atix courbes du quatrieme degre 527, 528

(Comptes Bendas de 1' Academic des Sciences 1886)

48. On the differential equation to a curve of

any order 529, 530

(Natare 1886)

49. On the so-called Tschirnhausen tratuforma-

tion 531—549

(Crelle's Joamal fiir die reine nnd angewandte Mathematik 1887)

50. Stir une decouverte de M. James Hammond

relative d, une certaine s4rie de nomhres qui figurent dans la tMorie de la trans- formation Tschirnhausen . . . 550 552

(Comptes Bendas de I'Acad^mie des Sciences 1887)

;51. On Hamilton's numbers .... 553 584

(Philosophical Transactions of the Boyal Society of London 1887, 1888)

62. Sur les nombres dits de Hamilton . . 585 587

(Compte Benda de I'Assoc. Franijaise (Toalouse) 1887)

63. Note on a proposed addition to the voca- bulary of ordinary arithmetic . . 588 591

(Nature 1888)

54. On certain inequalities relating to prime numbers 592 603

(Nature 1888)

55. Sur les nombres parfaits .... 604 606

(Comptes Bendas de I'Acad^mie des Sciences 1888)

xii Contents

PAGES

56. Siir une classe sp^dale des diviseurs de la

somme d\me serie giomMrique . . 607 610

(Comptes BenduB de TAcad^mie des Sciences 1888)

57. Sur T impossibility de I'existence d'un nonibre

parfait impair qui ne contient pas au

moins 5 diviseurs premiers distincts . 611 614

(Comptes Bendus de TAcad^mie des Sciences 1888)

58. Sur les nomhres parfaits .... 615 619

(Comptes Eendus de 1' Academic des Sciences 1888) (Mathesis 1888)

59. Preuve Mementaire du thdoreme de Dirichlet

sur les progressions arithmStiques dans

les cas oil la raison est 8 on 12 . . 620 624

(Comptes Bendus de I'Acad^mie des Sciences 1888)

60. On the divisors of the sum of a geometrical

series whose first term, is unity and common ratio any positive or negative integer 625 629

(Nature 1888)

61. Note on certain difference equations which

possess an unique integraV . . . 630 637

(Messenger of Mathematics 1888 9)

62. Sur la reduction biorthogonale d'une forme

lineo-lineaire a sa forme canonique . 638 640

(Comptes Eendus de I'Acadfimie des Sciences 1889)

63. Stir la correspondance complete entre les

fractions continues qui expriment les deux racines d'une equation quadratique dont les coefficients sont des nombres rationnels 641 644

(Comptes Eendus de I'Acad^mie des Sciences 1889)

64. Sur la representation des fractions continues

qui expriment les deux racines d'une

Equation quadratique .... 645, 646

(Comptes Bendus de I'Acad^mie des Sciences 1889)

Contents xiii

PAGES

65. Sur la valeur d'une fraction continue finie

et purement periodique .... 647 649

(CJomptes Bendus de TAcad^mie des Sciences 1889)

66. A new proof that a general quadric may he

reduced to its canonical form {that is, a linear function of squares) hy means of a real orthogonal stibstittition . . 650 653

(Messenger of Mathematics 1890)

67. On the reduction of a bilinear quantic of the

nth order to the form of a sum of n products by a double orthogonal substi- tution 654—658

(Messenger of Mathematics 1890)

68. On an arithmetical theorem in periodic

C07itinued fractions 659 662

(Messenger of Mathematics 1890)

69. On a funicular solution of Buff on' s "problem

of the needle" in its most general form 663 679

(Acta Matbematica 1890—1)

70. Sur le rapport de la circonference an dia-

mktre 680, 681

(Ck>mptes Bendns de I'Acad^mie des Sciences 1890)

71. Preuve que ■n ne pent pas etre racine

d'une equation algebrique h coefficients

entiers . 682—686

(Comptes Bendns de I'Aoad^mie des Soienoes 1890)

72. On arithmetical series 687 731

(Messenger of Mathematics 1892)

73. Note on a nine schoolgirls problem . . 732, 733

(Messenger of Mathematics 1893)

74. On the Goklbach-Euler theorem, regarding

prime numbers 734 737

(Nature 1896—7)

xiv Contents

FAQES

75. On the number of jyroper vulgar fractions in their lowest terms that can he formed with integers not greater than a given number 738 742

(Messenger of Mathematics 1898)

Index to Professor Sylvester's contributions

TO "Mathematical Questions from the

Educational Times" 743 747

Index to the four volumes of the "Collected Mathematical Papers " of James Joseph Sylvester 748 756

BIOGEAPHICAL NOTICE*.

Lord of himself and blest shall prove

He who can boast "I've lived to-day, To-morrow let dispensing Jove

Cast o'er the skies what tint he may.

" Sunshine or cloud ! the work begun

And ended may his power defy, He cannot change nor make undone

What once swift Time has hurried by."

Law of Verte, p. 73 (from Horace).

James Joseph Sylvester was boru in London on 3 September 1814, 1814 of a family said to have been originally resident in Liverpool. He was among the youngest of several brothers and sisters, and the last to survive. His father, whose name was Abraham Joseph, died while he was young. His eldest brother early in life established himself in America and assumed the name of Sylvester, an example followed by all the brothers.

If we attempt to realise the scientific circumstances of the time of Sylvester's birth by recalling the dates of some of those whose work might

* The chief aathority for the oatward facts of Sylvester's life used in this record is the Obituary Notice by Major P. A. MacMahon, B.A. , F.B.8., Royal Society Proceedings, Lxni, 1898, p. ix. There is also an article in the Dictionary of National Biography, by Professor E. B. Elliott, F.B.S. and Mr P. E. Matheson, M.A., which gives a list of authorities, and an earlier article by Major MacMahon, Nature, 25 March 1897. Other sources of information are referred to in the course of the following.

rvi Biographical Notice

naturally come before him, either in connexion with his subsequent career at Cambridge, or with his own later investigations, we find it difficult to make a choice. Of Englishmen Henry Cavendish (1731 1810) was dead, Thomas Young (1773—1829) was forty-one, Faraday (1791—1867) was twenty-three, and had just exchanged (in 1813) a bookbinder's workshop for the laboratory of the Royal Institution, Sir John Herschel (1792 1871) was twenty-two, and George Green (1793 1841), who was afterwards to be examined with Sylvester at Cambridge, was twenty-one. Cayley, with whom he was to be so much associated, was born in 1821, and was Senior Wrangler in 1842. The year 1814 was " the year of peace," and was the year in which Poncelet (1788 1867) returned to Paris from the Russian prison in which he had recon- structed the theory of conic sections; Lagrange (1736 1813) had just died, but there were living Laplace (1749 1827), Legendre (1752—1833), Fourier (1768—1830), Ampere (1775—1836), Poisson (1781—1840), Fresnel (1788— 1827), Cauchy (1789—1857). J. C. F. Sturm (1803—1855), whose theorem was to have such an importance for Sylvester, was eleven years his senior ; Hermite's life extended from 1822 to 1901. In Germany there were Gauss (1777 1855), whose Disquisitiones Arithmeticae is dated 1801, Steiner (1796—1863), von Staudt (1798—1867), Jacobi (1804—1851), W. Weber (1804—1891), Dirichlet (1805—1859), Kummer (1810—1893), while Weier- strass was born in 1815 ; and then there were Helmholtz (1821 1894), Kirchhoff (1824—1886), Riemann (1826—1866), and Clebsch (1833—1872). In Italy Brioschi, who took part in the development of the theory of in- variants, was born in 1824 and died in 1897 ; and the name of Abel (1802 1829) cannot be omitted. All these, and many others, went to form the atmosphere in which Sylvester's life was spent.

Until Sylvester was fifteen years of age he was educated in London from the age of six to the age of twelve with Mr Neumegen, at Highgate, subsequently, for a year and a half, with Mr Daniell at Islington, then, for five months, at the University of London (afterwards University College), where apparently he met Professor De Morgan, who (except from 1831 to 1835) taught at this institution fi"om 1828 to 1867 ; for Sylvester speaks in 1840 (l 53) of having been a pupil of De Morgan's. His gift for Mathe- matics seems undoubtedly to have been apparent at this time ; for Mr Neumegen sent him at the age of eleven to be examined in Algebra by Dr Olinthus Gregory, at the Royal Military Academy, Woolwich, and it is recorded that this gentleman was writing to Sylvester's father two years later to enquire for him, with a view to testing his progress in the interval. 1829 In 1829, at the age of fifteen, Sylvester went to Liverpool ; here he attended the school of the Royal Institution, residing with aunts. The Institution, it appears, was founded in 1814, largely by the exertions of William Roscoe (1753 1831), and its school in 1819 ; it must not be confounded with the Liverpool Institute, which grew out of the Mechanics Institute, founded in

Biographical Notice xvii

1825, by Mr Huskisson. The Head-master at this time was the Rev. T. W. Peile, afterwards Head-master of Repton, and the mathematical master was Mr Marratt. A contemporary at the school was Sir William Leece Drinkwater, afterwards First Deemster, Isle of Man. At this school Sylvester remained less than two years. In February 1830 he was awarded the first prize in the Mathematical School, and was so far beyond the other scholars that he could not be included in any class. While here, also, he was awarded a prize of 500 dollars for solving a question in arrangements, to the great satisfaction of the Contractors of Lotteries in the United States, the question being referred to him by the intervention of his elder brother in New York. At this early period of his life, too, he seems to have suffered for his Jewish faith at the hands of his young contemporaries ; possibly this may account for the episode recorded, of his running away from school and sailing to Dublin. Here, with only a few shillings in his pocket, he was accidentally accosted by the Right Hon. R. Keatinge, Judge of the Prerogative Court of Ireland, who, having discovered him to be a first cousin of his wife, entertained him, and sent him back to Liverpool.

The indications were by now sufficient to encourage him to a mathe- 1831 matical career. After reading for a short time with the Rev. Dr Richard Wilson, sometime Fellow of St John's College, Cambridge, afterwards Head- master of St Peter's Collegiate School, Eaton Square, London, Sylvester was entered* at St John's College on 7 July, as a Sizar, commencing residence on 6 October 1831, when just over seventeen, his tutor being Mr Gwatkin. He resided continuously till the end of the Michaelmas Term, 1833, though he seems to have been seriously ill in June of this year. For two years from the beginning of 1834 his name does not appear as a member of the College, and apparently he was at home on account of illness. In January 1836 he was readmitted, this time as a Pensioner, and resided during the Lent and Michaelmas Terms, being also incapacitated in the intervening term. In January 1837 he underwent his final University examination, the Mathe- matical Tripos, and was placed second on the list. The first six names of that year were Griffin, St John's ; Sylvester, St John's; Brumell, St John's; Green, Gonville and Caius ; Gregory, Trinity, and Ellis, Trinity. Of these, George Green, bom at Sneinton, near Nottingham, in 1793, was already the author of the famous paper, " An essay on the application of Mathematical Analysis to the theories of Electricity and Magnetism," which was published at Nottingham, by subscription, in 1828. He died in 1841, more than fifty years before Sylvester.

Of the general impression which Sylvester produced upon his con- temporaries at Cambridge, it is difficult to judge. It is recorded that he attended the lectures of J. Gumming, Professor of Chemistry in the

* The Eagle, the College Magazine, «x (1897), p. 603. A list of Sylvegter's scientific dis- tinctions is given in this place (p. 600).

xviii Biographical Notice

University from 1815 to 1861, and, as required by College regulations, the Classical lectures of Bushby. We know how keen was his interest in Chemistry many years later in Baltimore (cf. his paper on The New Atomic Theory, lii 148) : and his writings furnish evidence of the pleasure he took in introducing a Classical allusion. When he became Editor of the Quarterly Journal of Mathematics ia 1855 he secured the printing of a Greek motto on its title-page :

o Ti ovaia rrphs yfve(Tiv, fTrumffif) irpos niariv Ka\ hiavoia npos fiKa<riav eort ;

later on, the American Journal under his care also had (iv 298) a Greek motto :

npayfiaTav (\ty)(Os oi ^XfTro/ifvav ;

in his older age the reading and translation of Classical authors was one of his resources.

He was, in later life at least, well acquainted with French, German and Italian, and rejoices (ii 563) because these with Latin and English "may happily at the present day be regarded as the common property and inherit- ance of mathematical Europe." He was also much interested in Music. We are told that at one time he took lessons in singing from Gounod, and was known to sing at entertainments given to working men. " May not Music," he asks (ll 419), "be described as the Mathematic of sense, Mathematic as Music of the reason ?..." Or again (ill 128), " It seems to me that the whole of aesthetic (...) may be regarded as a scheme having four centres, ..., namely Epic, Music, Plastic and Mathematic " ; and he advocated " a new method of learning to read on the pianoforte " (ill 8).

Of his interest in general literature, and his keen relish for a striking phrase, no reader of his papers needs to be reminded. To his first long paper on Syzygetic Relations, published in the Philosophical Transactions of the Royal Society (i 429), he prefixes the words

How charming is divine philosophy !

Not harsh and crabbed as dull fools suppose,

But musical as is Apollo's lute

And a perpetual feast of uectar'd sweets.

Where no crude surfeit reigus I

In his paper on Newton's rule, also in the publications of the Royal Society (II 380), he quotes

Turns them to shapes and gives to airy nothing A local habitation and a name.

In his Constructive Theory of Partitions (iv 1) he leads off with

seeming parted, But yet a union in partition ;

Biographical Notice Tax.

the Second Act, in which the Partitions are transformed by cunning opera- tions performed on the diagrams which represent them, is introduced by

Naturelly, by composiciouns Of anglis, and slie reflexiouns ;

as the plot thickens he begins to feel more need of apology, and Act III

begins with

mazes intricate, Eccentric, intervolved, yet regular Then most, when most irregular they seem ;

while, when he comes to the Exodion, and feels that, after fifty-eight pages, direct appeal may have lost its power, he takes refuge in Spenser's fairyland with the lines

At which he wondred much and gan enquere What stately building durst so high extend Her lofty towres, unto the starry sphere.

Of his clever sayings we all remember many : " Symmetry, like the grace of an Eastern robe, has not unfrequently to be purchased at the expense of some sacrifice of freedom and rapidity of action " (l 309) ; or again, in support of the contention, that to say that a proposition is little to the point is not to be taken as demurring to its truth (ll 725), " I should not hesitate to say, if some amiable youth wished to entertain his partner in a quadrille with agree- able conversation, that it would be little to the point, according to the German proverb, to regale her with such information as how

Long are the days of summer-tide And tall the towers of Strasburg's fane,

but should be surprised to have it imputed to me on that account that I demurred to the proposition of the length of the days in summer, or the height of Strasburg's towers." More direct still (ill 9), disclaiming the idea that the simplicity of Peaucellier's linkwork should discredit the difiiculty of its discovery, " The idea of the facility of the result, by a natural mental illusion, gets transferred to the process of conception, as if a healthy babe were to be accepted as proof of an easy act of parturition." Some others will be found referred to in the index.

It is also recorded that among the friends of his earlier life was H. T. Buckle, author of the History of Civilisation, with whom, in addition to more serious reasons for sympathy, chess playing was a link of friendship.

Whether the many sides of Sylvester's character, indicated by these gleanings from his later life, were much in evidence at Cambridge, we do not know. The intellectual atmosphere of the place at the time was extremely vigorous in some ways. The Philosophical Society was founded in 1819, largely on the initiative of Adam Sedgwick and J. S. Henslow, and obtained a Charter in 1832; its early volumes are evidence of the great

XX Biographical Notice

width and alertness of scientific interest in Cambridge at this time ; papers of George Green were read at the Society in 1832, 1833, 1837 and 1839 ; James Gumming, whose chemical lectures Sylvester attended, Sir John Herschel, De Morgan, and Whewell are aiiiong the early contributors. Sir John Herschel's Preliminary Discourse on the Study of Natural Philo- sophy is dated 1831. The third meeting of the British Association was in Cambridge, on 24 June 1833. Whewell's History of the Inductive Sciences was published at Cambridge in 1837, the Philosophy of the Inductive Sciences in 1840. But we find* that in 1818 Sedgwick gave up his assistant tutor- ship, whose duties were mainly those of teaching the mathematical students of Trinity College, on the ground that "as far as the improvement of the mind is considered, I am at this moment doing nothing....! am... very sensibly approximating to that state of fatuity to which we must all come if we remain here long enough." This was before Sylvester's student time, and while mathematics at Cambridge was still suffering, partly from the long consequences of the controversy in regard to Leibniz and Newton, and more immediately from the loss of communication with the mathematicians of the Continent due to the war. Yet Sir John Herschelf, writing in 1833, feels compelled to speak very decidedly of the long-subsisting superiority of foreign mathematics to our own, as he phrases it, and there seems to be no doubt that mathematics, as distinct from physics, was then at a very low ebb in Cambridge, notwithstanding the success of the struggle, about a quarter of a century before, to introduce the analytical methods then in use on the Continent. C. Babbage, in his amusing Passages from the Life of a Philo- sopher, describes how he went (about 1812) to his public tutor to ask the solution of one of his mathematical difficulties and received the answer that it would not be asked in the Senate House, and was of no sort of con- sequence, with the advice to get up the earlier subjects of the university studies ; and how, after two further attempts and similar replies from other teachers, he acquired a distaste for the routine of the place. His connexion with the translation of Lacroix's Elementary Differential Calculus (1816), and his association with George Peacock, Sir John Herschel and others in the Analytical Society, is well known ; the title proposed by him for a volume of their Transactions, " The principles of pure D-ism in opposition to the Dot-age of the University," has often been quoted.

In addition to the better known accounts, there is an echo of what is usually said about Cambridge in this connexion in an Eloge on Sir John Herschel, read at the Royal Astronomical Society, 9 February 1872, by a writer who compares the work of Lagrange on the theory of equations with that of Waring, who was born in the same year, and was Senior Wrangler at Cambridge in 1757. We may add to this the bare titles of two continental

* Life of Adam Sedgwick, by J. W. Clark, i, p. 154. t Collected Essays, Longmans, 1857, pp. SO 39.

Biographical Notice xxi

publications of 1837, the year of Sylvester's Tripos Examination : C. Lejeune Dirichlet, Beweis des Satzes, doss jede unbegrenzte arithmetische Progression, deren erstes Glied und Differem game Zahlen ohne gemeinschaftlichen Factor sind, unendlich viele Primzahlen enthdlt ; E. Kummer, De aequatione a?*- + y^ = z^ per numeros integros resolvenda. Augustus De Morgan, who was fourth Wrangler in 1827, speaking in 1865, at the inaugural meeting of the London Mathematical Society, pronounces that "The Cambridge Examination is nothing but a hard trial of what we must call problems since they call them so between the Senior Wrangler that is to be of this present January, and the Senior Wrangler of some three or four years ago. The whole object seems to be to produce problems or, as I should prefer to call them, hard ten-minute conundrums.... It is impossible in such an examination to propose a matter that would take a competent mathematician two or three hours to solve, and for the consideration of which it would be necessary for him to draw his materials from different sources, and see how he can put together his previous knowledge, so as to bring it to bear most effectually on this particular subject." This is the mathematician's criticism of the system then, and, to a large extent, still in vogue. A criticism from another point of view is found in a letter* of Sir Frederick Pollock, written in 1869, to De Morgan : " I believe the most valuable qualities for practical life cannot be got at by any examination such as steadiness and perse- verance....! think a Cambridge education has for its object to make good members of society not to extend science and make profound mathema- ticians " These criticisms appear to agree in one implication, the dominance

of the examination in the training offered by the University ; and they are necessary to a right appreciation of Sylvester's university life and subsequent work. Accordingly, we do not hear, as frequently we do in the case of young students at continental universities, of Sylvester being led to study for himself the great masters in Mathematics. We find him, in 1839 (i 39), disclaiming a first-hand knowledge of Gauss's works ; there is no anecdote, known to me, to put with that he himself tells of Riemann. In a sheet of verses issued by himself, in February 1896 one of many such sheets, I believe there is a footnote containing the following : " ...the hotel on the river at Nuremberg, where I conversed outside with a Berlin bookseller, bound, like myself, for Prague.... He told me he was formerly a fellow pupil of Riemann, at the University, and that, one day, after receipt of some numbers of the Comptes rendus from Paris, the latter shut himself up for some weeks, and when he returned to the society of his friends, said (referring to newly-published papers of Cauchy), 'This is a new mathematic.'" We find Sylvester, how- ever, writing in 1839 of " the reflexions which Sturm's memorable theorem had originally excited " (I 44), and we know how much of his subsequent thought was given to this matter. Whether he read Sturm's paper of W. W. B. Ball, HUtory of Mathematict at Cambridge, 1889, p. 113.

xxii Biographical Notice

23 May 1829 {Bulletin de Firmsac, xi, 1829, p. 419 ; Mimoires par divers Savans, Vl, 1835, pp. 273 318), or in what way he learnt of the theorem, there seems to be no record. It is not referred to in the Report on Analysis by George Peacock, Cambridge British Association Report, 1833, pp. 185 352, which deals at length with Fourier's method. Sylvester records (ii 655 6) that Sturm told him that the theorem originated in the theory of compound pendulums, but he makes no reference to Sturm's recognition of the applica- tion of his principles to certain differential equations of the second order.

Another aspect of Sylvester's time at Cambridge must be referred to. At this time, and indeed until 1871, it was necessary, in order to obtain the Cambridge degree, to subscribe to the Articles of the Church of England ; one of the attempts, in 1834, to remove the restriction, is recorded in the Life of Adam Sedgwick, already referred to (i 418 ; Sedgwick writes a letter to the Times, 8 April 1834). Sylvester was, in his own subsequent bitter phrase (ill 81), one of the first holding "the faith in which the Founder of Christianity was educated " to compete for high honours in the Mathematical Tripos ; not only could he not obtain a degree, but he was excluded from the examination for Dr Smith's mathematical prizes, which, founded in 1769, was usually taken by those who had been most successful in the Mathematical Tripos. Most probably, too, had the facts been otherwise, he would have been shortly elected to a Fellowship at St John's College. To obtain a degree he removed to Trinity College, Dublin, from which, it appears, he received in turn the B.A. and the M.A. (1841). He finally received the B.A. degree at Cambridge, 29 February 1872, the M.A. (honoris caiisa) following 25 May of the same year. 1838 In the year succeeding his Tripos examination at Cambridge, he was elected to the Professorship of Natural Philosophy at (what is now) University College, London, and so became a colleague of Professor De Morgan. The list of the supporters of his candidature includes the names of Dr Olinthus Gregory, who had examined him in Algebra when a schoolboy of eleven, of Dr Richard Wilson, who had taught him before his entrance at St John's College, of the Senior Moderator and Senior Examiner in his Tripos examina- tion, of Philip Kelland, of Queens' College, Senior Wrangler in 1834, after- wards Professor at Edinburgh, and of J. W. Colenso, afterwards Bishop of Natal ; the two last had been private tutors of Sylvester at some portions of his career at Cambridge. He held the post of Professor of Natural Philosophy for a few years only; Professor G. B. Halsted (Science, 11 April 1897) makes a statement suggesting that the examination papers set by him during his tenure of the office are of a nature to indicate that he did not find his subject congenial. During these years he was elected a Fellow of the Royal Society (25 April 1839), at the early age of twenty-five. About this time also an oil- painting of him was made by Patten, of the Royal Scottish Academy, from the recorded description of which it appears that he had dark curly hair and

Biographical Notice xxiii

wore spectacles. It has been said that he took his Tripos examination in January 1837 ; he at once began to publish, in the Philosophical Magazine of 1837 38. The first four of his papers are on the analytical develop- ment of Fresnel's optical theory of crystals, and on the motion and rest of fluids and rigid bodies ; but the papers immediately following contain the dialytic method of elimination, and the expression of Sturm's functions in terms of the roots of the equation, as well as many results afterwards included in the considerable memoir on the theory of the syzygetic relations of two polynomials, publi-shed in the Philosophical Transactions of 1853.

Leaving University College in the session of 1840 41, he proceeded 1841 as Professor of Mathematics across the Atlantic, to the University of Virginia, founded in 1824 at Charlottesville, Albemarle Co., where* his colleague, Key, of University College, had previously occupied the chair of Mathematics. Such a considerable change deserved a better fate than befell ; in Virginia at this time the question of slavery was a subject of bitter con- tention, and Sylvester had a horror of slavery. The outcome was his almost immediate return ; apparently he had intervened vigorously in a quarrel between two of his students.

On his return from America Sylvester seems to have abandoned mathe- 1844 matics for a time. In 1844 he accepted the post of Actuary to the Legal and Equitable Life Assurance Company, and threw himself into the work with great energy. He did not accept another teaching post for ten years, until 1854, but seems to have given some private instruction, as it is related f that he had, what was unusual at that time, a lady among his pupils whose name was afterwards famous Miss Florence Nightingale. He entered at the Inner Temple 29 July 1846, and was called to the Bar 22 November 1850. He also founded the Law Reversionary Interest Society. It was in 1846 184& that Cayley, who had been Senior Wrangler in 1842, left Cambridge and became a pupil of the famous conveyancer, Mr Christie, entering at Lincoln's Inn. He was already an author, and had in fact entered upon one of the main activities of his life; for in 1845 he had published his fundamental paper "On the Theory of Linear Transformations," in which he discusses Boole's discovery of the invariance of a discriminant. To us, knowing how pregnant with consequences the meeting was, it would be interesting to have some details of the introduction of Cayley and Sylvester; the latter lived, then or soon after, in Lincoln's Inn Fields, and we are told+ that during the following years they might often be found walking together round the Courts of Lincoln's Inn, discussing no doubt many things but among them assuredly the Theory of Invariants. Perhaps it was particularly of this time that Sylvester was thinking when he described Cayley (l 376) as " habitually

* J. J. Walker, Proe. Land. Math. Soc. «viu (1896—97), p. 582.

+ The EagU, «x (1897), p. .597.

J Biographical notice ol Arthur Cayley, Cayley's Collected Papers, Volume viii.

8. IV. c

xxiv Biographical Notice

1846 discoursing pearls and rubies," or when, much later (iv 300), he spoke of " Cayley, who, though younger than myself, is my spiritual progenitor who first opened my eyes and purged them of dross so that they could see and accept the higher mysteries of our common mathematical faith." It is in a paper published in 1851 (i 246) that we find him saying, "The theorem above enunciated was in part suggested in the course of a conversation with Mr Cayley (to whom I am indebted for my restoration to the enjoyment of mathematical life) " ; and Sylvester's productiveness during the latter part of this period is remarkable. In particular there are seven papers whose date of publication is 1850, including the paper on the intersections, contacts and other correlations of two conies, wherein he was on the way to establish the properties of the invariant factors of a determinant, afterwards recog- nised by Weierstrass; and there are thirteen papers whose date is 1851, including the sketch of a memoir on elimination, transformation and canonical forms, in which the remarkable expression of a cubic surface by five cubes is given, the essay on Canonical Forms, and the paper on the relation between the minor determinants of linearly equivalent quadratic functions, in which the notion of invariant factors is implicit ; while in 1852 is dated the first of the papers " On the principles of the Calculus of Forms." Dr Noether remarks* how important for the history of mathematics these years were in other respects ; Kummer's memoir, " Ueber die Zerlegung der aus Wurzeln der Einheit gebildeten complexen Zahlen in ihre Primfactoren," appeared in 1847 {Crelle, xxxv); Weierstrass's " Beitrag zur Theorie der Abel'schen Integrale " (Beilage zum Jahresbericht iiber das Gymnasium zu Braunsberg) is dated 1849 ; Riemann's Inaugural-dissertation, " Grundlagen fur eine allgemeine Theorie der Functionen eiuer veranderlichen complexen Grosse," is dated 1851. Referring to the discovery of the Canonical Forms in order to enforce the statement that observation, induction, invention and experi- mental verification all play a part in mathematical discovery (ii 714), Sylvester tells an anecdote which has a personal interest : " I discovered and developed the whole theory of canonical binary forms for odd degrees, and, as far as yet made out, for even degrees too, at one evening sitting, with a decanter of port wine to sustain nature's flagging energies, in a back office in Lincoln's Inn Fields. The work was done, and well done, but at the usual cost of racking thought a brain on fire, and feet feeling, or feelingless, as if plunged in an ice-pail. That night we slept