WEBVTT 1 00:00:01.080 --> 00:00:04.519 Chapter nineteen, Part three of a short account of the 2 00:00:04.599 --> 00:00:09.279 history of mathematics by W. W. Rowseball. This is a 3 00:00:09.320 --> 00:00:13.720 LibriVox recording. All LibriVox recordings are in the public domain. 4 00:00:14.359 --> 00:00:18.640 For more information or to volunteer, please visit LibriVox dot org. 5 00:00:19.440 --> 00:00:23.000 This is a reading by Paul King PJK dot Scripts 6 00:00:23.039 --> 00:00:26.920 dot mit dot edu. A short account of the history 7 00:00:26.960 --> 00:00:33.000 of mathematics by W. W. Rowsball, Chapter nineteen, Mathematics of 8 00:00:33.119 --> 00:00:40.560 Recent Times, Part three. The consideration of algebraical, trigonometrical, elliptic, 9 00:00:40.759 --> 00:00:44.679 hyper elliptic and other special kinds of functions pave the 10 00:00:44.719 --> 00:00:48.079 way for a theory of functions which promises to prove 11 00:00:48.119 --> 00:00:52.520 a most important and far reaching branch of mathematics. To 12 00:00:52.600 --> 00:00:56.479 a large extent, this is the work of living mathematicians, 13 00:00:57.320 --> 00:01:01.200 and therefore outside the limits of this chapter. I will 14 00:01:01.240 --> 00:01:05.400 content myself by referring to the following writers. First, I 15 00:01:05.439 --> 00:01:10.879 may mention Kosche, who gave the general elementary theory of functions, 16 00:01:10.959 --> 00:01:15.879 and Lieuville, who wrote chiefly on doubly periodic functions. Their 17 00:01:15.920 --> 00:01:19.239 investigations were extended and connected in the work of Brio 18 00:01:19.359 --> 00:01:25.040 and Bouquet, and further been developed by Hermite Next, I 19 00:01:25.079 --> 00:01:28.719 may refer to the researches on the theory of algebraic functions, 20 00:01:28.760 --> 00:01:32.599 which have their origin in Riemond's paper of eighteen fifty. 21 00:01:33.159 --> 00:01:37.920 Schwartz has established accurately certain theorems of which proofs given 22 00:01:37.959 --> 00:01:44.079 by Riemond were open to objection. Klein has connected Riemond's 23 00:01:44.120 --> 00:01:47.359 theory of functions with the theory of groups and have 24 00:01:47.400 --> 00:01:53.799 written on automorphic functions. Henri point Carey, professor in Paris, 25 00:01:53.879 --> 00:01:58.200 born at Nancy in eighteen fifty four, has also written 26 00:01:58.200 --> 00:02:02.719 on automorphic functions and on the general theory with special 27 00:02:02.760 --> 00:02:08.199 applications to differential equations. Finally, I may refer to the 28 00:02:08.240 --> 00:02:15.319 work of Weierstrauss and Mittaghlefer. Of these, Karl Weierstrauss has 29 00:02:15.400 --> 00:02:18.360 created a large part of the modern theory of functions, 30 00:02:18.360 --> 00:02:23.000 and in particular has constructed the theory of uniform analytical functions. 31 00:02:24.000 --> 00:02:29.159 And Magnus Gustav mitteg Leffler born at Stockholm in eighteen 32 00:02:29.240 --> 00:02:33.120 forty six and now professor there has greatly developed the 33 00:02:33.120 --> 00:02:37.240 theory of analytical functions, a subject on which Hermite has 34 00:02:37.280 --> 00:02:42.159 also written. In connection with these researches, Paul Emil Appel 35 00:02:42.439 --> 00:02:46.039 professor in Paris, born at Strasburg in eighteen fifty eight. 36 00:02:46.800 --> 00:02:52.800 C Emir Picard of Paris Edouard Goursat of Paris has 37 00:02:52.840 --> 00:02:57.360 written on special branches of the theory. As textbooks. I may 38 00:02:57.400 --> 00:03:03.319 mention Doctor Forsiss Theory of Functions of a complex variable, Cambridge, 39 00:03:03.360 --> 00:03:09.800 eighteen ninety three and Karl Numont's Vorslungen Uberriman's Theory der 40 00:03:10.080 --> 00:03:18.280 Absersion integral second edition, Leipezig, eighteen eighty four. The theory 41 00:03:18.360 --> 00:03:21.520 of numbers may be considered as a higher arithmetic, and 42 00:03:21.639 --> 00:03:25.879 the theory of elliptical and Abelian functions as a higher trigonometry. 43 00:03:26.639 --> 00:03:30.080 The theory of higher algebra, including the theory of equations, 44 00:03:30.479 --> 00:03:34.759 has also attracted considerable attention and was a favorite subject 45 00:03:34.800 --> 00:03:38.800 of study of the three mathematicians Kosche, Hamilton, and de Morgan, 46 00:03:39.360 --> 00:03:43.120 whom I propose to mention next. Though the interests of 47 00:03:43.159 --> 00:03:47.240 these writers were by no means limited to this subject. Koschi, 48 00:03:48.479 --> 00:03:52.000 the first of these mathematicians, is the best representative of 49 00:03:52.080 --> 00:03:56.960 the French school of analysis in this century. Augustine Luis Kosche, 50 00:03:57.319 --> 00:04:00.919 who was born in Paris on August twenty first, seventeen 51 00:04:01.000 --> 00:04:05.039 eighty nine, and died at Sioux on May twenty fifth, 52 00:04:05.280 --> 00:04:09.479 eighteen fifty seven. Was educated at the Polytechnic School, the 53 00:04:09.639 --> 00:04:13.680 nursery of so many French mathematicians of that time, and 54 00:04:13.840 --> 00:04:18.439 adopted the profession of a civil engineer. His earliest mathematical 55 00:04:18.519 --> 00:04:23.199 paper was won on polyhedra in eighteen eleven. Legendre thought 56 00:04:23.240 --> 00:04:25.920 so highly of it that he asked Koshi to attempt 57 00:04:25.959 --> 00:04:30.879 the solution of an analogous problem which had baffled previous investigators, 58 00:04:31.360 --> 00:04:34.319 and his advice was justified by the success of Kochi 59 00:04:34.439 --> 00:04:38.920 in eighteen twelve. Memoirs on Analysis and the Theory of Numbers, 60 00:04:39.000 --> 00:04:43.480 presented in eighteen thirteen, eighteen fourteen and eighteen fifteen, shewed 61 00:04:43.519 --> 00:04:47.120 that his ability was not confined to geometry alone. In 62 00:04:47.240 --> 00:04:49.920 one of these papers he generalized some results which had 63 00:04:49.959 --> 00:04:53.480 been established by Gauss and Legendre. In another of them, 64 00:04:53.560 --> 00:04:56.040 he gave a theorem on the number of values which 65 00:04:56.079 --> 00:05:00.160 an algebraical function can assume when the literal constance it 66 00:05:00.360 --> 00:05:04.240 contains are interchanged. It was the latter theorem that enabled 67 00:05:04.319 --> 00:05:08.560 Abel to shew that in general, an algebraic equation of 68 00:05:08.639 --> 00:05:12.120 a degree higher than the fourth cannot be solved by 69 00:05:12.160 --> 00:05:17.560 the use of purely algebraical expressions. To Koshe and Gauss, 70 00:05:17.639 --> 00:05:21.160 we owe the scientific treatment of series, which have an 71 00:05:21.199 --> 00:05:25.279 infinite number of terms, and the former established general rules 72 00:05:25.920 --> 00:05:31.000 for investigating the convergency and divergency of such series. It 73 00:05:31.160 --> 00:05:33.279 is only a few works of an earlier date that 74 00:05:33.480 --> 00:05:37.399 contain any discussion as to the limitations of the series employed. 75 00:05:38.439 --> 00:05:41.959 It is said that Laplace, who was present when Koshi 76 00:05:42.040 --> 00:05:45.279 read his first paper on the subject, was so impressed 77 00:05:45.319 --> 00:05:48.839 by the illustrations of the danger of employing such series 78 00:05:48.920 --> 00:05:53.000 without a rigorous investigation of their convergency, that he put 79 00:05:53.079 --> 00:05:56.360 on side on one side the work on which he 80 00:05:56.560 --> 00:06:00.079 was then engaged, and denied himself to all visitors, in 81 00:06:00.240 --> 00:06:03.000 order to see if any of the demonstrations given in 82 00:06:03.079 --> 00:06:07.879 the earlier volumes of the Mechanique Celeste were invalid, and 83 00:06:08.040 --> 00:06:11.279 he was fortunate enough to find that no material errors 84 00:06:11.360 --> 00:06:16.079 had been thus introduced. The treatment of series and of 85 00:06:16.199 --> 00:06:19.319 the fundamental conceptions of the calculus in most of the 86 00:06:19.399 --> 00:06:24.800 textbooks then current, was based on Euler's works, and to 87 00:06:25.000 --> 00:06:29.279 any one trained to accurate habits of thought, was not 88 00:06:29.480 --> 00:06:32.639 free from objection. It is one of the chief merits 89 00:06:32.639 --> 00:06:35.959 of Koshe that he placed those subjects on a logical foundation. 90 00:06:37.199 --> 00:06:41.439 On the restoration. In eighteen sixteen, the French Academy was purged, 91 00:06:41.639 --> 00:06:44.600 and in spite of the indignation and scorn of French 92 00:06:44.639 --> 00:06:49.000 scientific society, Kosche accepted a seat which was procured for 93 00:06:49.199 --> 00:06:54.120 him by the expulsion of Munge. He was also at 94 00:06:54.199 --> 00:06:57.879 the same time made professor at the Polytechnic, and his 95 00:06:58.040 --> 00:07:01.360 lectures there on algebraic and elisons, the calculus and the 96 00:07:01.480 --> 00:07:06.079 theory of curves were published as textbooks on the Revolution. 97 00:07:06.279 --> 00:07:09.680 In eighteen thirty he went into exile and was first 98 00:07:09.720 --> 00:07:14.199 appointed professor at Tehennes, where he soon moved to Prague 99 00:07:14.279 --> 00:07:19.720 to undertake the education of the Comte de Chambaud. He 100 00:07:19.839 --> 00:07:22.720 returned to France in eighteen thirty seven, and in eighteen 101 00:07:22.839 --> 00:07:25.879 forty eight, and again in eighteen fifty one, by special 102 00:07:26.000 --> 00:07:29.720 dispensation of the Emperor, was allowed to occupy a chair 103 00:07:29.959 --> 00:07:34.639 of mathematics without taking the oath of allegiance. His activity 104 00:07:34.759 --> 00:07:37.959 was prodigious, and from eighteen thirty to eighteen fifty nine 105 00:07:38.079 --> 00:07:40.800 he published in the transactions of the Academy of the 106 00:07:40.879 --> 00:07:45.279 Comptes Rendeus over six hundred original memoirs and about one 107 00:07:45.360 --> 00:07:48.920 hundred and fifty reports. In most of them the feverish 108 00:07:49.000 --> 00:07:51.720 haste with which they were thrown off as too visible, 109 00:07:52.399 --> 00:07:56.160 and many are marred by obscurity, repetition of old results, 110 00:07:56.240 --> 00:08:00.439 and blunders. Among the more important of his resist searches 111 00:08:00.480 --> 00:08:03.879 are the discussion of tests for the convergency of series 112 00:08:04.439 --> 00:08:07.480 and the determination of the number of real and imaginary 113 00:08:07.560 --> 00:08:11.920 roots of any algebraic equation, his method of calculating these 114 00:08:12.040 --> 00:08:16.040 roots approximately, his theory of the symmetric functions of the 115 00:08:16.120 --> 00:08:20.639 coefficiens of equations of any degree, his a priori valuation 116 00:08:20.839 --> 00:08:23.879 of quantity less than the least difference between the roots 117 00:08:23.920 --> 00:08:28.040 of an equation, and his papers on determinants in eighteen 118 00:08:28.120 --> 00:08:30.319 forty one, which did a great deal to bring them 119 00:08:30.360 --> 00:08:35.159 into general use. Koshi also did something to reduce the 120 00:08:35.320 --> 00:08:39.519 art of determining definite integrals to a science, but this 121 00:08:39.720 --> 00:08:43.080 branch of the integral calculus still remains without much system 122 00:08:43.240 --> 00:08:46.639 or method. The rule for finding the principal values of 123 00:08:46.720 --> 00:08:50.919 integrals was enunciated by him, and the calculus of residues 124 00:08:51.440 --> 00:08:55.159 was his invention. His proof of Taylor's theorem seems to 125 00:08:55.240 --> 00:08:58.919 have originated from a discussion of the double periodicity of 126 00:08:59.000 --> 00:09:02.879 elliptic functions. The means of shewing a connection between the 127 00:09:02.960 --> 00:09:07.000 different branches of a subject by giving imaginary values to 128 00:09:07.200 --> 00:09:11.519 independent variables is largely due to him. He also gives 129 00:09:11.559 --> 00:09:15.879 a direct analytical method for determining planetary inequalities of long 130 00:09:16.000 --> 00:09:19.320 period and to physics, he contributed a memoir on the 131 00:09:19.440 --> 00:09:23.279 quantity of light reflected from the surfaces of metals, as 132 00:09:23.360 --> 00:09:29.120 well as other papers on optics. Argande I may mention 133 00:09:29.360 --> 00:09:33.559 here the name of Jean Robert Argaunde, who was born 134 00:09:33.600 --> 00:09:37.799 at Geneva on July twenty seconds, seventeen sixty eight and 135 00:09:38.039 --> 00:09:42.919 died circa eighteen twenty five. In his essay issued in 136 00:09:43.200 --> 00:09:46.960 eighteen o six, he gave a geometrical representation of a 137 00:09:47.039 --> 00:09:50.639 complex number and applied it to shew that every algebraic 138 00:09:50.720 --> 00:09:54.279 equation has a root. This was prior to the memoirs 139 00:09:54.320 --> 00:09:57.639 of Gaussen Kosche on the same subject, but the essay 140 00:09:57.759 --> 00:10:00.519 did not attract much attention when it was first published. 141 00:10:00.960 --> 00:10:03.919 An earlier demonstration that the square root of minus one 142 00:10:04.120 --> 00:10:09.480 indicates perpendicularity, due to Beuet, was published in the Philosophical 143 00:10:09.600 --> 00:10:13.960 Transactions for eighteen o six, and the idea was foreshadowed 144 00:10:14.159 --> 00:10:18.080 in a memoir by H. Kuhn in the Transactions for 145 00:10:18.279 --> 00:10:25.639 seventeen fifty of the Saint Petersburg Academy. Hamilton, in the 146 00:10:25.720 --> 00:10:29.279 opinion of some writers, the theory of quaternions will be 147 00:10:29.440 --> 00:10:32.559 ultimately esteemed by one of the great discoveries of this century. 148 00:10:33.159 --> 00:10:37.480 The discovery is due to Sir William Rowan Hamilton, who 149 00:10:37.600 --> 00:10:40.399 was born of Scotch parents in Dublin on August fourth, 150 00:10:40.559 --> 00:10:44.240 eighteen o five, and died there on September second, eighteen 151 00:10:44.360 --> 00:10:48.240 sixty five. His education, which was carried out at home, 152 00:10:48.919 --> 00:10:52.519 seems to have been singularly discursive. Under the influence of 153 00:10:52.519 --> 00:10:55.440 an uncle who was a good linguist, he first devoted 154 00:10:55.480 --> 00:10:58.440 himself to linguistic studies. By the time he was seven, 155 00:10:58.519 --> 00:11:01.600 he could read Latin, Greek, Friends and German with facility, 156 00:11:02.360 --> 00:11:04.320 and when thirteen he was able to boast that he 157 00:11:04.480 --> 00:11:07.639 was familiar with as many languages as he had lived years. 158 00:11:08.440 --> 00:11:10.440 It was about this time that he came across a 159 00:11:10.559 --> 00:11:14.799 copy of Newton's Universal Arithmetic. This was his introduction to 160 00:11:14.919 --> 00:11:18.720 modern analysis, and he soon mastered the elements of analytical 161 00:11:18.799 --> 00:11:23.360 geometry and the calculus. He next read the Principia and 162 00:11:23.480 --> 00:11:27.879 the four volumes then published of Laplace's Mechanique Celeste. In 163 00:11:27.960 --> 00:11:30.679 the latter he detected a mistake, and his paper on 164 00:11:30.720 --> 00:11:33.919 the subject, written in eighteen twenty three, placed him at 165 00:11:34.000 --> 00:11:37.960 once in the front rank of mathematicians. In the following 166 00:11:38.039 --> 00:11:42.279 year he entered at Trinity College, Dublin. His university careers 167 00:11:42.399 --> 00:11:45.840 unique for the chair of Astronomy, becoming vacant in eighteen 168 00:11:45.879 --> 00:11:49.600 twenty seven while he was yet an undergraduate. He was 169 00:11:49.679 --> 00:11:52.519 asked by the electors to stand for it, and was 170 00:11:52.600 --> 00:11:56.519 elected unanimously, it being understood that he should be left 171 00:11:56.600 --> 00:12:00.759 free to pursue his own line of study. Earliest paper, 172 00:12:00.840 --> 00:12:04.039 written in eighteen twenty three was on optics, and was 173 00:12:04.080 --> 00:12:07.360 published in eighteen twenty eight under the title of a 174 00:12:07.519 --> 00:12:11.960 Theory of Systems of Rays, which two supplements written in 175 00:12:12.039 --> 00:12:15.559 eighteen thirty one in eighteen thirty two were afterwards added. 176 00:12:16.159 --> 00:12:19.360 In the latter of these, the phenomenon of conical refraction 177 00:12:19.559 --> 00:12:22.399 is predicted. This was followed by a paper in eighteen 178 00:12:22.440 --> 00:12:25.399 twenty seven on the principle of varying action, and in 179 00:12:25.519 --> 00:12:28.639 eighteen thirty four and eighteen thirty five by memoirs on 180 00:12:28.960 --> 00:12:33.399 a general method in dynamics, the subject of theoretical dynamics 181 00:12:33.440 --> 00:12:37.360 being properly treated as a branch of pure mathematics. His 182 00:12:37.559 --> 00:12:42.639 Lectures on Quaternions was published in eighteen fifty two. Amongst 183 00:12:42.799 --> 00:12:46.000 his other papers, I may specially mention one on the 184 00:12:46.320 --> 00:12:49.519 form of the solution of the general algebraic equation of 185 00:12:49.600 --> 00:12:54.159 the fifth degree, which confirmed the conclusion arrived at by 186 00:12:54.320 --> 00:12:57.279 Abel that it cannot be expressed in terms of the 187 00:12:57.360 --> 00:13:02.960 more elementary operations and functions, one on fluctuating functions, one 188 00:13:03.039 --> 00:13:08.159 on the hodograph, and lastly on the numerical solution of 189 00:13:08.240 --> 00:13:12.960 differential equations e. His Elements of Quaternions were issued in 190 00:13:13.039 --> 00:13:17.039 eighteen sixty six. Of this, a competent authority says that 191 00:13:17.240 --> 00:13:20.600 the methods of analysis there given shoe as great an 192 00:13:20.639 --> 00:13:24.080 advance over those of analytical geometry, as the latter shued 193 00:13:24.320 --> 00:13:28.440 over those of Euclidean geometry. In more recent times a 194 00:13:28.799 --> 00:13:33.320 subject has been further developed by Tait. Hamilton was painfully 195 00:13:33.440 --> 00:13:37.919 fastidious on what he published, and he left an immense 196 00:13:38.000 --> 00:13:40.720 collection of manuscript books, which are now in the library 197 00:13:40.759 --> 00:13:44.159 of Trinity College, Dublin, and some of which it is hoped, 198 00:13:44.240 --> 00:13:50.000 will be ultimately printed Grassman. The idea of noncommutative algebras 199 00:13:50.039 --> 00:13:53.759 and of quaternions seemed to have occurred to Grassmann about 200 00:13:53.840 --> 00:13:58.799 the same time as to Hamilton. Hermann Gunther Grassmann born 201 00:13:58.879 --> 00:14:02.120 at stettin An April fifteenth, eighteen o nine and died 202 00:14:02.159 --> 00:14:05.960 there in eighteen seventy seven. He was a professor at 203 00:14:06.000 --> 00:14:10.519 the Gymnasium at Stettin. His researches on noncommutative algebras are 204 00:14:10.600 --> 00:14:14.480 contained in his Austio Anglire, first published in eighteen forty 205 00:14:14.559 --> 00:14:19.320 four and enlarged in eighteen sixty two. The scientific treatment 206 00:14:19.440 --> 00:14:23.000 of the fundamental principles of algebra initiated by Hamilton and 207 00:14:23.080 --> 00:14:27.840 Grassmann were continued by De Morgan and Buell, and subsequently 208 00:14:28.000 --> 00:14:30.639 was further developed by H. Henkel in his work on 209 00:14:30.799 --> 00:14:35.320 Complexes eighteen sixty seven and by G. Cantor on his 210 00:14:35.480 --> 00:14:38.960 Memoirs on the Theory of Irrationals eighteen seventy one. The 211 00:14:39.080 --> 00:14:42.840 discussion is, however, so technical, that I am unable to 212 00:14:42.960 --> 00:14:46.759 do more than allude to it. Grassmann also investigated the 213 00:14:46.799 --> 00:14:53.240 properties of homoloidal hyperspace. De Morgan Augustus de Morgan born 214 00:14:53.320 --> 00:14:57.039 in Madura, Madras in June eighteen o six, and died 215 00:14:57.080 --> 00:15:01.279 in London on March eighteenth, eighteen seventy one. Was educated 216 00:15:01.320 --> 00:15:04.120 at Trinity College, Cambridge, but in the then state of 217 00:15:04.159 --> 00:15:07.960 the law, was as a Unitarian ineligible to a fellowship. 218 00:15:08.919 --> 00:15:11.679 In eighteen twenty eight he became professor at the newly 219 00:15:12.039 --> 00:15:15.919 established University of London, which is the same institution as 220 00:15:16.000 --> 00:15:20.159 that now known as University College. There, except for five 221 00:15:20.279 --> 00:15:23.399 years from eighteen thirty one to eighteen thirty five, he 222 00:15:23.480 --> 00:15:27.360 taught continuously till eighteen sixty seven, and through his works 223 00:15:27.519 --> 00:15:31.679 and pupils, exercised a wide influence on English mathematicians of 224 00:15:31.720 --> 00:15:36.679 the present day. The London Mathematical Society was largely his creation, 225 00:15:37.279 --> 00:15:39.639 and he took a prominent part in the proceedings of 226 00:15:39.720 --> 00:15:44.000 the Royal Astronomical Society. He was perhaps more deeply read 227 00:15:44.039 --> 00:15:47.679 in the philosophy and history of mathematics than any of 228 00:15:47.799 --> 00:15:51.600 his contemporaries, but the results are given in scattered articles 229 00:15:51.639 --> 00:15:56.879 as well, which well deserved collection and republication. A list 230 00:15:56.960 --> 00:15:59.600 of these is given in his life, and I have 231 00:15:59.720 --> 00:16:02.559 made considerable use of some of them in this book. 232 00:16:03.519 --> 00:16:05.639 The best known of his works are the Memoirs of 233 00:16:05.679 --> 00:16:11.039 the Foundation of Algebra, Cambridge Philosophical Transactions, Volumes eight and 234 00:16:11.240 --> 00:16:15.279 nine is treatise on the differential calculus, published in eighteen 235 00:16:15.320 --> 00:16:18.720 forty two, a work of great ability and noticeable for 236 00:16:18.840 --> 00:16:22.480 the rigorous treatment of infinite series, and his articles on 237 00:16:22.559 --> 00:16:25.960 the calculus of functions on the theory of probabilities in 238 00:16:26.039 --> 00:16:31.080 the Encyclopedia Metropolitana. The article on the calculus of functions 239 00:16:31.159 --> 00:16:35.840 contains an investigation of the principles of symbolic reasoning, but 240 00:16:35.960 --> 00:16:39.600 the applications deal with the solution of functional equations rather 241 00:16:39.720 --> 00:16:42.919 than with the general theory of functions. The article on 242 00:16:43.000 --> 00:16:45.879 the theory of probabilities gives a clear analysis of the 243 00:16:45.960 --> 00:16:48.919 mathematics of the subject to the time at which it 244 00:16:49.039 --> 00:16:53.240 was written. Besides those above named, I may mention the 245 00:16:53.360 --> 00:16:56.240 following who have written on the subjects of higher algebra, 246 00:16:56.799 --> 00:17:01.600 the theory of forms and the theory of equations. Joseph 247 00:17:01.720 --> 00:17:05.720 Ludwig Rabbe, who in eighteen thirty two discussed the tests 248 00:17:05.759 --> 00:17:09.640 for the convergency of series, a subject also discussed later 249 00:17:09.720 --> 00:17:14.359 by Joseph Louis Francois Bertrand, Secretary of the French Academy 250 00:17:14.799 --> 00:17:20.440 born in Paris in eighteen eight twenty two. Couhmer URIs 251 00:17:20.640 --> 00:17:26.559 Deni of Pisa and A. Pringsheim of Munich, and on 252 00:17:26.640 --> 00:17:29.279 the researches of the above writers see the Bulletin of 253 00:17:29.400 --> 00:17:33.359 the New York Mathematical Society, October eighteen ninety two, pages 254 00:17:33.440 --> 00:17:39.160 one to ten. George Boule born at Lincoln in November second, 255 00:17:39.200 --> 00:17:43.119 eighteen fifteen, died at Cork on December eighth, eighteen sixty four, 256 00:17:43.559 --> 00:17:47.680 who invented a system of noncommunative algebra, and from whose 257 00:17:47.759 --> 00:17:52.720 memoirs on linear transformations, part of the theory of covariance 258 00:17:52.759 --> 00:17:58.319 has developed. Everist galois one of the most original and 259 00:17:58.480 --> 00:18:03.000 powerful mathematicians of the century, born at Paris on October 260 00:18:03.039 --> 00:18:05.920 twenty six, eighteen eleven, and then killed in a duel 261 00:18:06.519 --> 00:18:09.799 on May thirtieth, eighteen thirty two, at the early age 262 00:18:09.839 --> 00:18:13.519 of twenty, whose writings are mainly concerned with the theory 263 00:18:13.559 --> 00:18:19.960 of equations and substitution groups. On his investigations see Liauville's 264 00:18:20.039 --> 00:18:25.359 Journal for eighteen forty six, Volume eleven, pages three eighty one, 265 00:18:25.599 --> 00:18:28.839 four forty four and the American Journal of Mathematics for 266 00:18:28.960 --> 00:18:33.400 eighteen ninety one, Volume thirteen pages one hundred nine to 267 00:18:33.559 --> 00:18:40.079 one hundred forty two. Carl Wilhelm Borchardt, Professor in Berlin, 268 00:18:40.680 --> 00:18:45.160 born there on February twenty second, eighteen seventeen, died there 269 00:18:45.400 --> 00:18:49.480 eighteen eighty, who in particular discussed generating functions in the 270 00:18:49.559 --> 00:18:54.720 theory of equations and er arithmetic geometric means. A collected 271 00:18:54.880 --> 00:18:58.359 edition of his works, edited by g Hettner, was issued 272 00:18:58.359 --> 00:19:03.680 at Berlin in eighteen eighty one. Eight Kayley, whose ten 273 00:19:03.799 --> 00:19:09.119 classical memoirs on quantics, binary and ternary forms, and researches 274 00:19:09.200 --> 00:19:13.599 on noncommutative algebras, especially on matrices, will be found in 275 00:19:13.680 --> 00:19:19.160 the collected edition of his works. Sylvester from among those 276 00:19:19.240 --> 00:19:23.480 whose numerous memoirs I may in particular single out those 277 00:19:23.559 --> 00:19:28.200 on canonical forms, on the theory of contravariance, reciprocants of 278 00:19:28.440 --> 00:19:32.960 differential invariants, on the theory of equations, and that on 279 00:19:33.160 --> 00:19:35.839 Newton's rule, to which I may add that he has 280 00:19:35.960 --> 00:19:39.160 created the language and notation of considerable parts of the 281 00:19:39.279 --> 00:19:44.000 subjects on which he has written. Camille Jordan, who has 282 00:19:44.039 --> 00:19:47.240 written on the theory of substitutions in general and with 283 00:19:47.359 --> 00:19:54.759 special applications to differential equations. Sir George Gabriel Stokes, Lucasian 284 00:19:54.839 --> 00:20:00.960 professor in the University of Cambridge, born near Sligo, August thirteenth, 285 00:20:01.160 --> 00:20:04.359 eighteen nineteen, who has written on the critical values of 286 00:20:04.400 --> 00:20:08.160 the sums of the periodic series and on the summation 287 00:20:08.359 --> 00:20:13.400 of series Cambridge Philosophical Transactions, eighteen forty seven, Volume eight, 288 00:20:13.839 --> 00:20:17.119 pages five thirty three to five eighty three. See also 289 00:20:17.240 --> 00:20:22.279 below pages four ninety two and four ninety six. Eugene 290 00:20:22.359 --> 00:20:28.160 Netto of Strasburg, who has written on substitutions Hermite, who 291 00:20:28.319 --> 00:20:32.160 has in particular discussed the theory of associated contravariance in 292 00:20:32.240 --> 00:20:36.440 binary quantics, the theory of Turnery quantics, and who has 293 00:20:36.519 --> 00:20:40.519 applied elliptic functions to the solution of the quintic equation. 294 00:20:42.400 --> 00:20:45.559 Enrico Betty of Pisa, who died in eighteen ninety two 295 00:20:46.039 --> 00:20:52.559 and Brioschi, both of whom discussed binary quantics. Siegfried Heinrich 296 00:20:52.720 --> 00:20:57.759 Hnhold born at Engerberg on July sixteenth, eighteen nineteen, who 297 00:20:57.839 --> 00:21:02.759 developed symbolic methods, especially in connection with turnery quantics. This 298 00:21:03.000 --> 00:21:07.519 was done concurrently with, but independently of Keiley's work on 299 00:21:07.640 --> 00:21:12.920 the same subject. Paul Gorden, professor at Erlingen, who had 300 00:21:13.200 --> 00:21:15.880 discussed the theory of forms enchewed that there were only 301 00:21:16.079 --> 00:21:20.279 finite number of concomitants of quantics. An edition of his 302 00:21:20.400 --> 00:21:25.480 work on invariants, determinants and binary forms, edited by G. Kersteiner, 303 00:21:26.200 --> 00:21:29.599 was issued at Leipzig in three volumes eighteen eighty five, 304 00:21:29.839 --> 00:21:35.759 eighteen eighty seven and eighteen ninety three. Rudolph Friedrich Alfred 305 00:21:36.000 --> 00:21:40.440 Klebsch born at Knigsburg in eighteen thirty three, died in Gautigen, 306 00:21:41.119 --> 00:21:44.599 where he was professor in eighteen seventy two, who also 307 00:21:44.759 --> 00:21:48.599 independently investigated the theory of binary forms in some papers 308 00:21:49.039 --> 00:21:52.880 collected and published in eighteen seventy one. An account of 309 00:21:52.920 --> 00:21:56.279 his life and works as printed in The Mathematist Annelen 310 00:21:57.240 --> 00:22:01.279 eighteen seventy three, volume six, pages one ninety seven to 311 00:22:01.400 --> 00:22:05.960 two oh two and eighteen seventy four volume seven, pages 312 00:22:06.039 --> 00:22:12.480 one to fifty five. See also below. McMahon, who has 313 00:22:12.599 --> 00:22:16.480 written on the connection of symmetric functions, the derivation of 314 00:22:16.559 --> 00:22:21.200 invariants and the covariance from elementary algebra, and the concomitance 315 00:22:21.240 --> 00:22:26.759 of binary forms. Sophis Lae, professor at Leipzig, who has 316 00:22:26.839 --> 00:22:31.599 written on groups of continuous substitutions, differential invariants and complexes 317 00:22:31.680 --> 00:22:36.920 of lines. Klein, who has investigated the problem of discontinuous 318 00:22:36.960 --> 00:22:43.160 substitutions and polyhedra groups. And lastly Andrew Russell Forsyth, Fellow 319 00:22:43.240 --> 00:22:47.680 and lecturer of Trinity College, Cambridge, born at Glasgow on 320 00:22:47.839 --> 00:22:50.720 June eighteenth, eighteen fifty eight, who has developed a theory 321 00:22:50.759 --> 00:22:58.000 of invariants of differential equations, ternarians and quaternarians. No account 322 00:22:58.079 --> 00:23:02.160 of contemporary writings of this subject would be complete without 323 00:23:02.160 --> 00:23:05.559 a reference to the admirable textbooks produced by George Salmon, 324 00:23:05.720 --> 00:23:10.680 Provost at Trinity Trinity College, Dublin, born in eighteen nineteen 325 00:23:11.480 --> 00:23:15.559 in his Higher algebra, and by Joseph Alfred Serret, professor 326 00:23:15.680 --> 00:23:18.880 at the Sorbonne in Paris on August thirtieth, eighteen nineteen 327 00:23:19.119 --> 00:23:23.319 died in eighteen eighty five, in his Coups delchebre Superieur, 328 00:23:23.519 --> 00:23:27.240 in which the chief discoveries of their respective of authors 329 00:23:27.400 --> 00:23:31.960 are embodied. An admirable historical summary of the theory of 330 00:23:32.000 --> 00:23:36.720 the complex variable is given in the vassolingen Uberg Dies 331 00:23:36.839 --> 00:23:41.839 complex in Zalen, Leipzig, eighteen sixty seven by H. Henkel, 332 00:23:42.000 --> 00:23:45.920 professor at Tubingen, born at Halle in eighteen thirty nine 333 00:23:46.000 --> 00:23:50.359 and died at Schremburg in eighteen seventy three. Before mentioning 334 00:23:50.440 --> 00:23:54.400 the creators of modern synthetic geometry, it will be convenient 335 00:23:54.480 --> 00:23:58.079 to call attention to two other divisions of pure mathematics, 336 00:23:58.119 --> 00:24:01.880 which have been greatly developed in recent years. But any 337 00:24:01.920 --> 00:24:04.599 sketch of the results arrived at or the methods by 338 00:24:04.640 --> 00:24:07.279 which they have been attained, would be so closely connected 339 00:24:07.359 --> 00:24:10.240 with the work of living mathematicians that I shall do 340 00:24:10.359 --> 00:24:14.799 little more than mention the names of the subjects. Analytical 341 00:24:14.880 --> 00:24:18.440 geometry has been studied by a host of modern writers, 342 00:24:18.960 --> 00:24:22.240 but I do not propose to describe their investigations, and 343 00:24:22.359 --> 00:24:25.680 I shall content myself by merely mentioning the names of 344 00:24:25.759 --> 00:24:30.880 the following mathematicians. James Booth born in the country of 345 00:24:30.960 --> 00:24:34.599 Leetrim on August twenty fifth, eighteen o six and died 346 00:24:34.640 --> 00:24:38.680 at Buckinghamshire on April fifteenth, eighteen seventy eight was one 347 00:24:38.720 --> 00:24:41.880 of the earliest writers in this century to devote himself 348 00:24:41.920 --> 00:24:45.920 to the development of analytical geometry. His chief results are 349 00:24:45.960 --> 00:24:48.960 embodied in his work entitled A treatise on some new 350 00:24:49.079 --> 00:24:54.079 geometrical methods. The researches of James McCulloch, professor in Dublin, 351 00:24:54.160 --> 00:24:57.559 born near Starbane in eighteen o nine and died in 352 00:24:57.680 --> 00:25:01.759 Dublin in October twenty fourth eve eighteen forty six, which 353 00:25:01.839 --> 00:25:05.839 includes some valuable discoveries in the theory of quadrics, will 354 00:25:05.920 --> 00:25:10.920 be found in this collected works edited by Jellet and Houghton, Dublin, 355 00:25:11.039 --> 00:25:15.079 eighteen eighty See also below eighteen page four ninety six. 356 00:25:16.319 --> 00:25:20.680 Julius Plucker, professor after eighteen thirty six in Bonn, born 357 00:25:20.799 --> 00:25:23.960 at Elberfield on July sixteenth, eighteen o one and died 358 00:25:24.000 --> 00:25:27.680 at Bonn on May twenty second, eighteen sixty eight, devoted 359 00:25:27.759 --> 00:25:30.680 himself chiefly to the study of algebraic curves of a 360 00:25:30.759 --> 00:25:33.640 geometry in which the line is the element in space, 361 00:25:34.359 --> 00:25:38.799 and the theory of congruences and complexes. His equations connecting 362 00:25:38.880 --> 00:25:42.359 the singularities of curves are well known. In eighteen forty 363 00:25:42.440 --> 00:25:45.160 seven he exchanged his chair for one of physics, and 364 00:25:45.480 --> 00:25:49.799 his subsequent researches were on spectra and magnetism. An account 365 00:25:49.839 --> 00:25:54.920 of his works was published by clubsch gelschafter Wizenschaften, Goettingen, 366 00:25:55.119 --> 00:25:59.960 eighteen seventy two, Volume sixteen. The majority of the memoirs 367 00:26:00.160 --> 00:26:04.599 on analytical geometry by Kayley and by Henry Smith's deal 368 00:26:04.720 --> 00:26:07.599 with the theory of curves and services. The most remarkable 369 00:26:07.680 --> 00:26:11.279 of those of Ludwig otto Hess born at Knigsburg on 370 00:26:11.440 --> 00:26:14.680 eight Pril twenty second, eighteen eleven and died at Munich, 371 00:26:14.799 --> 00:26:17.960 where he was professor in eighteen seventy four, or on 372 00:26:18.079 --> 00:26:21.960 the plain geometry of curves see the notice of these 373 00:26:22.039 --> 00:26:26.359 by FC. Klein, of those of Jean Gaston d'arboux, professor 374 00:26:26.440 --> 00:26:31.240 in Paris born at Nines at eighteen forty two, on 375 00:26:31.359 --> 00:26:35.039 the geometry of services, and those of Halfen on the 376 00:26:35.200 --> 00:26:40.480 singularities of surfaces and on tortuous curves. The singularities of 377 00:26:40.559 --> 00:26:44.960 curves and surfaces have also been considered by Hieronymus George Suthen, 378 00:26:45.559 --> 00:26:49.880 professor at Copenhagen born in eighteen thirty nine, and by 379 00:26:50.000 --> 00:26:55.240 Hermann Keesar Hannibal Schubert, professor at Hamburg, born at Potsdam 380 00:26:55.319 --> 00:26:58.160 in eighteen forty eight. The lectures of the latter have 381 00:26:58.319 --> 00:27:03.079 been published by F. Lindemann two volumes, Leipezig, eighteen seventy 382 00:27:03.119 --> 00:27:06.759 five and eighteen ninety one. Notether has discussed the theory 383 00:27:06.880 --> 00:27:11.880 of tortuous curves, and Klebsch has applied Abel's theorem to geometry. 384 00:27:12.640 --> 00:27:17.559 Among more recent textbooks are Klebsch's vordusunjen ubur Geometry edited 385 00:27:17.640 --> 00:27:22.000 by F. Lindemann, and Selman's conic Sections Geometry of three 386 00:27:22.079 --> 00:27:26.240 dimensions and Higher Plane Curves, in which the chief discoveries 387 00:27:26.279 --> 00:27:29.720 of these writers are embodied. Finally, I may allude to 388 00:27:29.839 --> 00:27:33.160 the extension of the subject matter of analytical geometry by 389 00:27:33.200 --> 00:27:36.599 the introduction of the ideas of space of n dimensions 390 00:27:36.640 --> 00:27:39.960 and the writings of Grassmann in eighteen forty four. In 391 00:27:40.000 --> 00:27:43.480 eighteen sixty two, among those who have extended the range 392 00:27:43.480 --> 00:27:47.799 of analysis, including the calculus and differential equations are whom 393 00:27:47.839 --> 00:27:50.480 it is difficult to place in any of the preceding 394 00:27:50.599 --> 00:27:54.359 categories are the following, whom I place in alphabetical order, 395 00:27:55.119 --> 00:28:01.160 Appel bertrand Buel Koshi d'arbou scythe who has written on 396 00:28:01.440 --> 00:28:04.400 Faff's problem and is also the author of the standard 397 00:28:04.480 --> 00:28:10.240 English treatise on differential equations. Robinius Lazarus Fuchs, Professor at 398 00:28:10.279 --> 00:28:15.519 Berlin born in Prussia eighteen thirty three, Halfen Jacobi, Jordan Kanigsberger, 399 00:28:15.720 --> 00:28:20.640 Sophie Kuloweski, professor at Stockholm born on December twenty seventh, 400 00:28:20.720 --> 00:28:25.400 eighteen fifty three and died eighteenth, eighteen ninety one. See 401 00:28:25.480 --> 00:28:30.119