WEBVTT 1 00:00:01.199 --> 00:00:04.040 Chapter nineteen, Part two of A short account of the 2 00:00:04.160 --> 00:00:08.640 history of mathematics by W. W. Rowse Ball. This is 3 00:00:08.640 --> 00:00:12.919 a LibriVox recording. All LibriVox recordings are in the public domain. 4 00:00:13.720 --> 00:00:17.679 For more information or to volunteer, please visit LibriVox dot org. 5 00:00:18.480 --> 00:00:21.000 This is a reading by Paul King p J K 6 00:00:21.199 --> 00:00:23.280 dot scripts that I might t dot e d U 7 00:00:23.440 --> 00:00:27.079 forward slash p k J A short account of the 8 00:00:27.199 --> 00:00:31.920 history of mathematics by W. W. Rous Ball, Chapter nineteen, 9 00:00:32.560 --> 00:00:39.719 Mathematics of Recent Times, Part two, di Grey. One of 10 00:00:39.799 --> 00:00:43.719 Gauss's pupils to whom I may here allude is lejeun Derichree, 11 00:00:44.320 --> 00:00:47.799 who is generally known for his exposition of the discoveries 12 00:00:47.840 --> 00:00:51.439 of Jacoby, who was his father in law and of Gauss, 13 00:00:51.560 --> 00:00:55.880 rather than for his own original investigations, valuable, though some 14 00:00:55.920 --> 00:01:00.119 of these are Peter Gustave le Jeundr Crey was born 15 00:01:00.119 --> 00:01:04.439 at Dhurin on February thirteenth, eighteen o five and died 16 00:01:04.519 --> 00:01:09.079 at Guttingen on May fifth, eighteen fifty nine. He held 17 00:01:09.120 --> 00:01:14.599 successively professorships at Breslau and Berlin, and on Gauss's death 18 00:01:14.640 --> 00:01:18.239 in eighteen fifty five, was appointed to succeed him as 19 00:01:18.319 --> 00:01:23.000 professor of the Higher Mathematics at Gottingen. He intended to 20 00:01:23.000 --> 00:01:27.239 finish Gauss's incomplete works, for which he was admirably fitted, 21 00:01:27.560 --> 00:01:31.439 but his early death prevented this. He produced, however, several 22 00:01:31.480 --> 00:01:35.680 memoirs which have considerably facilitated the comprehension of some of 23 00:01:35.680 --> 00:01:41.200 Gauss's more abstruse methods. Of de Regree's original work, the 24 00:01:41.239 --> 00:01:45.079 most celebrated is that on the Determination of Means with 25 00:01:45.280 --> 00:01:50.719 applications to the distribution of prime numbers. The researches of 26 00:01:50.840 --> 00:01:54.640 Gauss on the theory of numbers were continued or supplemented 27 00:01:55.040 --> 00:01:59.959 by Jacobi, who firsts proved the law of cubic reciproc 28 00:02:00.760 --> 00:02:04.840 discussed on the theory of residues, and in his Canon 29 00:02:04.920 --> 00:02:11.159 Arithmeticus gave a table of residues of prime roots Eisenstein. 30 00:02:12.560 --> 00:02:16.599 This subject was next taken up by Ferdinand Gotthold Eisenstein, 31 00:02:17.000 --> 00:02:21.080 a professor at the University of Berlin, who was born 32 00:02:21.120 --> 00:02:24.240 at Berlin on April sixteenth, eighteen twenty three and died 33 00:02:24.240 --> 00:02:28.439 there on October eleventh, eighteen fifty two. The theory of 34 00:02:28.520 --> 00:02:32.520 numbers may be divided into two main divisions, namely the 35 00:02:32.520 --> 00:02:36.919 theory of congruences and the theory of forms. The solution of 36 00:02:37.000 --> 00:02:40.639 the problem of the representation of numbers by binary quadratic 37 00:02:40.719 --> 00:02:44.520 forms is one of the great achievements of Gauss, and 38 00:02:44.560 --> 00:02:49.120 the fundamental principles upon which the treatment of such questions 39 00:02:49.159 --> 00:02:53.000 rest were given by him in the Disquisition Arithmetic k. 40 00:02:54.159 --> 00:02:58.199 Guss there added some results relating to ternary quadratic forms, 41 00:02:58.520 --> 00:03:02.360 but the general extension from two to three indeterminates was 42 00:03:02.400 --> 00:03:06.360 the work of Eisenstein, who, in his memoir Nua Theorem 43 00:03:06.479 --> 00:03:12.240 de Hoharen Arithmetic, defined the ordinal and generic characters of 44 00:03:12.400 --> 00:03:17.319 ternary quadratic forms of an uneven determinant, and in the 45 00:03:17.360 --> 00:03:20.479 case of definite forms, assigned the weight of any order 46 00:03:20.599 --> 00:03:23.800 or genus. But he did not consider forms of an 47 00:03:23.840 --> 00:03:29.680 even determinant, nor give any demonstration of his work. Eisenstein 48 00:03:29.919 --> 00:03:34.080 also considered the theorems relating to the possibility of representing 49 00:03:34.120 --> 00:03:37.080 a number as a sum of squares. En shewed that 50 00:03:37.159 --> 00:03:42.120 the general theorem was limited to eight squares. The solutions 51 00:03:42.159 --> 00:03:45.759 in the cases of two, four, and six squares may 52 00:03:45.800 --> 00:03:49.879 be obtained by means of elliptic functions, but the cases 53 00:03:49.919 --> 00:03:52.879 in which the numbers of squares is uneven involved the 54 00:03:52.919 --> 00:03:58.120 special processes peculiar to the theory of numbers. Eisenstein gave 55 00:03:58.159 --> 00:04:01.439 the solution in the case of three square. He also 56 00:04:01.520 --> 00:04:04.000 left a statement of the solution he had obtained in 57 00:04:04.039 --> 00:04:07.400 the case of five squares, but his results were published 58 00:04:07.439 --> 00:04:10.560 without proofs and apply only to numbers which are not 59 00:04:10.639 --> 00:04:16.000 divisible by a square. Among Eisenstein's other investigations, I single 60 00:04:16.040 --> 00:04:20.399 out for special mention the remarkable rule he enunciated by 61 00:04:20.439 --> 00:04:23.480 means of which it is possible to distinguish whether a 62 00:04:23.519 --> 00:04:30.720 given series represents an algebraical or transcendental function. Henry Smith. 63 00:04:31.879 --> 00:04:35.000 One of the most original and powerful mathematicians of the 64 00:04:35.000 --> 00:04:39.160 school founded by Gauss was Henry Smith. Henry John Stephen 65 00:04:39.240 --> 00:04:43.240 Smith was born in London on November two, eighteen twenty six, 66 00:04:43.319 --> 00:04:46.519 and died at Oxford on February ninth, eighteen eighty three. 67 00:04:47.759 --> 00:04:51.439 He was educated at Rugby and at Balliol College, Oxford, 68 00:04:51.480 --> 00:04:55.000 of which a latter society he was a fellow, and 69 00:04:55.079 --> 00:04:58.279 in eighteen sixty one he was elected civilian Professor of 70 00:04:58.319 --> 00:05:03.360 Geometry at Oxford. He resided till his death. The subject 71 00:05:03.399 --> 00:05:07.160 in connection with which Smith's name will be always specially 72 00:05:07.199 --> 00:05:10.560 remembered is the theory of numbers, and to this he 73 00:05:10.639 --> 00:05:14.399 devoted the years from eighteen fifty four to eighteen sixty four. 74 00:05:15.000 --> 00:05:19.519 The results of his historical researches were given in his report, 75 00:05:19.639 --> 00:05:24.160 published in parts in the Transactions of the British Association 76 00:05:24.279 --> 00:05:28.519 from eighteen fifty nine to eighteen sixty five. This report 77 00:05:28.560 --> 00:05:30.720 contains an account of what had been done on the 78 00:05:30.759 --> 00:05:35.879 subject to that time, together with some additional matter. The 79 00:05:36.000 --> 00:05:39.199 chief outcome of his own original work on the subjects 80 00:05:39.240 --> 00:05:43.480 is included in two memoirs printed in the Philosophical Transactions 81 00:05:43.519 --> 00:05:47.319 for eighteen sixty one and eighteen sixty seven, the first 82 00:05:47.360 --> 00:05:51.800 being on linear indeterminate equations and Congruences, and the second 83 00:05:51.839 --> 00:05:55.720 on the orders and genera of ternary quadratic forms. In 84 00:05:55.759 --> 00:05:59.920 the latter memoir, demonstrations of Eisenstein's results on their extension 85 00:06:00.079 --> 00:06:04.160 to ternary quadratic forms of an even determinant were supplied, 86 00:06:05.319 --> 00:06:10.720 and a complete classification of ternary quadratic forms was given. Smith, however, 87 00:06:11.079 --> 00:06:14.360 did not confine himself to the case of three indeterminates, 88 00:06:14.759 --> 00:06:19.000 but succeeded in establishing the principles on which the extension 89 00:06:19.199 --> 00:06:23.800 to the general case of n indeterminates depends and obtained 90 00:06:23.959 --> 00:06:28.120 the general formulae, thus affecting the greatest advance made in 91 00:06:28.160 --> 00:06:31.959 the subject since the publication of Gauss's work. In the 92 00:06:32.000 --> 00:06:35.240 account of his Methods and Results, which appeared in the 93 00:06:35.279 --> 00:06:39.240 Proceedings of the Royal Society, Smith remarked that the theorems 94 00:06:39.279 --> 00:06:42.920 relating to the representation of numbers by four squares and 95 00:06:43.079 --> 00:06:47.360 other simple quadratic forms are deducible by a uniform method 96 00:06:47.680 --> 00:06:53.040 from the principles there indicated, as also are the theorems relating 97 00:06:53.079 --> 00:06:56.560 to the representation of numbers by six and eight squares. 98 00:06:58.040 --> 00:07:01.319 He then proceeded to say that as the series of 99 00:07:01.399 --> 00:07:05.040 theorems relating to the representation of numbers by sums of 100 00:07:05.160 --> 00:07:10.560 square's ceases for the reason assigned by Eisenstein, when the 101 00:07:10.680 --> 00:07:15.519 number of squares surpasses eight, it was desirable to complete it. 102 00:07:15.600 --> 00:07:18.959 The results for even squares were known. The principal theorems 103 00:07:19.000 --> 00:07:21.720 relating to this case of five squares had been given 104 00:07:21.759 --> 00:07:25.639 by Eisenstein, but he had considered only those numbers which 105 00:07:25.639 --> 00:07:28.639 are not divisible by a square, and he had not 106 00:07:28.720 --> 00:07:32.519 considered the case of seven squares. Smith here completed the 107 00:07:32.560 --> 00:07:35.600 annunciation of the theorems for the case of five squares, 108 00:07:36.040 --> 00:07:39.560 and added the corresponding theorems for the case of seven squares. 109 00:07:40.680 --> 00:07:43.920 This paper was the occasion of a dramatic incident in 110 00:07:43.959 --> 00:07:48.319 the history of mathematics. Fourteen years later, in ignorance of 111 00:07:48.319 --> 00:07:52.920 Smith's work, the demonstration and completion of Eisenstein's theorems for 112 00:07:53.000 --> 00:07:55.759 five squares were set by the French Academy as a 113 00:07:55.800 --> 00:08:00.480 subject of their Grand Prix des Science Mathematique. Wrote out 114 00:08:00.480 --> 00:08:03.680 the demonstration of his general theorem so far as was 115 00:08:03.720 --> 00:08:06.720 required to prove the results in the special case of 116 00:08:06.759 --> 00:08:09.439 five squares, and only a month after his death in 117 00:08:09.480 --> 00:08:12.800 March eighteen eighty three, the prize was awarded to him, 118 00:08:13.240 --> 00:08:17.399 another prize being awarded to H. Minkowski of Bonn. No 119 00:08:17.560 --> 00:08:20.839 episode could bring out in a more striking light the 120 00:08:20.920 --> 00:08:24.240 extent of Smith's researches than that a question of which 121 00:08:24.600 --> 00:08:27.639 he had given in the solution in eighteen sixty seven, 122 00:08:27.720 --> 00:08:31.040 as a corollary from the general formulae which governed the 123 00:08:31.079 --> 00:08:34.360 whole class of investigations to which it belonged, should have 124 00:08:34.399 --> 00:08:37.240 been regarded by the French Academy as one whose solution 125 00:08:37.360 --> 00:08:40.799 which is of such difficulty and importance as to be 126 00:08:40.879 --> 00:08:44.320 worthy of their great prize. It has also been a 127 00:08:44.320 --> 00:08:47.840 matter of comment that they should have known so little 128 00:08:47.840 --> 00:08:51.480 of contemporary English and German researches on the subject as 129 00:08:51.519 --> 00:08:53.600 to be unaware that the result of the problem they 130 00:08:53.639 --> 00:08:57.639 were proposing was then lying in their own library. Among 131 00:08:57.759 --> 00:09:02.279 Smith's other investigations, I may specially mentioned his geometrical memoir 132 00:09:02.960 --> 00:09:08.320 cirquecure problem quebique a bi quadratique, for which in eighteen 133 00:09:08.399 --> 00:09:11.480 sixty eight he was awarded the Steiner Prize of the 134 00:09:11.480 --> 00:09:15.159 Berlin Academy. In a paper which he contributed to the 135 00:09:15.200 --> 00:09:19.440 Atti of the Academia de Linsei for eighteen seventy seven, 136 00:09:19.519 --> 00:09:24.399 he established a very remarkable analytical relation connecting the modular 137 00:09:24.440 --> 00:09:28.120 equation of order N and the theory of binary quadratic 138 00:09:28.200 --> 00:09:32.759 forms belonging to the positive determinant N. In this paper, 139 00:09:33.120 --> 00:09:36.440 the modular curve is represented analytically by a curve in 140 00:09:36.480 --> 00:09:40.200 such a manner as to present an actual geometrical image 141 00:09:40.200 --> 00:09:44.399 of the complete systems of the reduced quadratic forms belonging 142 00:09:44.399 --> 00:09:48.320 to the determinant, and a geometrical interpretation is given to 143 00:09:48.360 --> 00:09:53.159 the ideas of class equivalence and reduced form. He was 144 00:09:53.200 --> 00:09:56.879 also the author of important papers in which he succeeded 145 00:09:56.960 --> 00:10:01.840 in extending to complex quadratic forms. Many of Gauss's investigations 146 00:10:02.200 --> 00:10:06.519 relating to real quadratic forms. He was led by researches 147 00:10:06.519 --> 00:10:09.879 on the theory of numbers to the theory of elliptic functions, 148 00:10:10.240 --> 00:10:14.559 and the results he arrived at, especially on the theory 149 00:10:14.639 --> 00:10:19.879 of the theta and omega functions, are of importance. The 150 00:10:19.879 --> 00:10:23.200 theory of numbers as treated today may be said to 151 00:10:23.279 --> 00:10:27.000 originate with Gauss. I have already mentioned very briefly the 152 00:10:27.039 --> 00:10:32.240 subject of the subsequent investigations of Jacobi Derichree Eisenstein and 153 00:10:32.360 --> 00:10:36.679 Henry Smith, among other mathematicians who have written on it. 154 00:10:36.799 --> 00:10:41.720 I may allude to the following Riemann, who investigated the 155 00:10:41.759 --> 00:10:47.720 distribution of primes. James Joseph Sylvester, civilian professor in the 156 00:10:47.879 --> 00:10:52.399 University of Oxford born in London on September third, eighteen fourteen, 157 00:10:53.039 --> 00:10:57.399 who has also written on the distribution of primes. Koshi, 158 00:10:58.360 --> 00:11:03.360 who in particular to discussed the expression of quadratic binomials. 159 00:11:04.360 --> 00:11:09.039 Joseph Lieuville, the editor from eighteen thirty six to eighteen 160 00:11:09.120 --> 00:11:12.559 seventy four of the well known journal, who was born 161 00:11:12.799 --> 00:11:15.919 at Saint Omer on March twenty fourth, eighteen o nine 162 00:11:15.960 --> 00:11:19.840 and died in eighteen eighty two, most of whose numerous 163 00:11:19.840 --> 00:11:24.240 investigations dealt with the representation of numbers by special forms. 164 00:11:25.240 --> 00:11:30.320 Ernest Edward Coomer born at Soroux on January twenty ninth, 165 00:11:30.360 --> 00:11:34.399 eighteen ten, and until recently professor at Berlin, to whom 166 00:11:34.399 --> 00:11:38.039 we owe the conception of the so called ideal primes 167 00:11:38.080 --> 00:11:41.480 which are required in the treatment of complexes, and which 168 00:11:41.519 --> 00:11:45.600 he applied to the problem of Fermat's equation, and whose 169 00:11:45.679 --> 00:11:50.080 paper on hypergeometric series may rank with that Bygaus is 170 00:11:50.120 --> 00:11:55.679 a classical memoir on the subject. Leopold Kronecker, professor in Berlin, 171 00:11:56.080 --> 00:11:59.679 born at lie Niitz on December seventh, eighteen twenty three 172 00:12:00.080 --> 00:12:03.720 and died at Berlin on December twenty ninth, eighteen ninety one, 173 00:12:04.480 --> 00:12:07.759 most of whose investigations on this branch of mathematics were 174 00:12:07.799 --> 00:12:12.320 on nary on quadratic forms. On his investigations generally see 175 00:12:12.360 --> 00:12:16.480 the Bulletin of the New York Mathematical Society, Volume one, 176 00:12:16.600 --> 00:12:19.720 pages one seventy three to one eighty four. See also 177 00:12:19.799 --> 00:12:25.279 below page four sixty nine. Charles Hermite, professor in Paris, 178 00:12:25.720 --> 00:12:29.279 born in Lorraine on December twenty fourth, eighteen twenty two, 179 00:12:29.799 --> 00:12:35.960 who wrote on ternary forms. Julius Wilhelm Richard Dedekind born 180 00:12:36.000 --> 00:12:40.320 at Brunswick on October sixth, eighteen thirty one, whose more 181 00:12:40.399 --> 00:12:44.919 important researches, given in an appendix to his edition of 182 00:12:45.000 --> 00:12:49.639 Dirichley's writings, are on ideal primes. See also below page 183 00:12:49.639 --> 00:12:56.279 four ninety three. Panotige Chebyshev, formerly professor at the University 184 00:12:56.320 --> 00:12:59.159 of Saint Petersburg, born in Russia in eighteen twenty one, 185 00:12:59.440 --> 00:13:02.440 who has written on the number of primes between given limits, 186 00:13:03.480 --> 00:13:07.200 a problem also considered by Legendre de Richley and Riemonn 187 00:13:08.399 --> 00:13:13.279 and James Whitebread. Lee Glacier, a fellow and tutor of 188 00:13:13.360 --> 00:13:18.039 Trinity College in Cambridge, born at Lewisham on November fifth, 189 00:13:18.080 --> 00:13:21.519 eighteen forty eight, from whose numerous papers I may single 190 00:13:21.559 --> 00:13:25.480 out those relating to prime numbers, those on functions of 191 00:13:25.480 --> 00:13:29.399 a number which are formed from its real or complex divisors, 192 00:13:29.960 --> 00:13:32.960 and on those on the possible divisors of numbers of 193 00:13:32.960 --> 00:13:36.320 a given form. Finally, I may mention that the problem 194 00:13:36.440 --> 00:13:40.240 of the partition of numbers, to which Euler paid considerable attention, 195 00:13:41.000 --> 00:13:44.279 has in recent times attracted the attention of Arthur Cayley, 196 00:13:44.759 --> 00:13:48.639 Sadleian professor in the University of Cambridge, born in Richmond, 197 00:13:49.279 --> 00:13:55.360 Surrey on August sixteenth, eighteen twenty one, of Sylvester and 198 00:13:55.440 --> 00:14:00.440 of Perry Percy. Alexander McMahon, professor at Woolwich and a 199 00:14:00.480 --> 00:14:04.559 major in the English Artillery, born at Malta on September 200 00:14:04.559 --> 00:14:09.440 twenty sixth, eighteen fifty four. Interest and problems connected with 201 00:14:09.480 --> 00:14:12.799 the theory of numbers seems recently to have flagged, and 202 00:14:13.000 --> 00:14:16.679 possibly it may be found hereafter that the subject is 203 00:14:16.720 --> 00:14:23.320 approached better on other lines. The theory of functions of 204 00:14:23.480 --> 00:14:29.000 double and multiple periodicity is another subject to which much 205 00:14:29.000 --> 00:14:32.600 attention has been paid during this century. I have already 206 00:14:32.639 --> 00:14:35.639 mentioned that as early as eighteen o eight Gauss had 207 00:14:35.639 --> 00:14:40.600 discovered the theta functions and their chief properties, but his 208 00:14:40.720 --> 00:14:44.759 investigations remained for many years concealed in note books, and 209 00:14:44.799 --> 00:14:47.559 it was to the researches made between eighteen twenty and 210 00:14:47.600 --> 00:14:51.519 eighteen thirty by Abel and Jacoby that the modern development 211 00:14:51.559 --> 00:14:55.000 of the subject is due. The treatment of it has 212 00:14:55.080 --> 00:15:00.720 been completely superseded that used by Legendre, and they are 213 00:15:00.759 --> 00:15:04.480 justly reckoned as the creators of this branch of mathematics. 214 00:15:05.960 --> 00:15:12.240 Abel Nils Henrik Abel born at Findo in Norway on 215 00:15:12.320 --> 00:15:15.679 August fifth, eighteen o two and died at Arundel on 216 00:15:16.039 --> 00:15:20.840 April sixth, eighteen twenty nine, at the age of twenty six. 217 00:15:22.480 --> 00:15:25.759 His memoirs on elliptic functions, which were originally published in 218 00:15:25.840 --> 00:15:28.360 Krell's Journal, treat the subject from the point of view 219 00:15:28.440 --> 00:15:32.080 of the theory of equations and algebraic forms, a treatment 220 00:15:32.120 --> 00:15:35.759 to which his research has naturally led him. The important 221 00:15:35.759 --> 00:15:39.200 and very general result known as Abel's theorem, which was 222 00:15:39.240 --> 00:15:43.120 subsequently applied by Riemann to the theory of transcendental functions, 223 00:15:43.600 --> 00:15:46.759 was sent to the French Academy in eighteen twenty eight, but, 224 00:15:47.039 --> 00:15:50.519 mainly through the action of Koshi, was not published for 225 00:15:50.600 --> 00:15:55.440 several years. The name of Abelian function has been given 226 00:15:55.480 --> 00:16:00.559 to the higher transcendence of multiple periodicity, which were first 227 00:16:00.679 --> 00:16:04.559 discussed by Abel. He criticized the use of infinite series, 228 00:16:04.600 --> 00:16:07.519 but I do not know that the results lead to 229 00:16:07.600 --> 00:16:12.799 any definite rules for testing convergency. As illustrating his fertility 230 00:16:12.840 --> 00:16:17.720 of ideas, I May in passing notice his celebrated demonstration 231 00:16:18.279 --> 00:16:21.840 that it is impossible to solve a quintic equation by 232 00:16:21.919 --> 00:16:26.279 means of radicals. This theorem was the more important since 233 00:16:26.320 --> 00:16:31.519 it definitely limited a field of mathematics which had previously 234 00:16:31.600 --> 00:16:36.879 attracted numerous writers. I should add that this theorem had 235 00:16:36.919 --> 00:16:41.399 been enunciated as early as seventeen ninety eight by Pallo Raffini, 236 00:16:42.080 --> 00:16:46.480 an Italian physician practicing at Modena, but I believe that 237 00:16:46.519 --> 00:16:52.320 the proof he gave was deficient in generality. Jacoby Karl 238 00:16:52.399 --> 00:16:56.679 Gustav Jacob Jacobi born of Jewish parents at Potsdam on 239 00:16:56.720 --> 00:17:00.440 December tenth, eighteen o four and dine at Berlin on 240 00:17:00.480 --> 00:17:04.759 February eighteenth, eighteen fifty one. Was educated at the University 241 00:17:04.759 --> 00:17:07.039 of Berlin, where he obtained a degree of Doctor of 242 00:17:07.119 --> 00:17:11.599 Philosophy in eighteen twenty five. In eighteen twenty seven he 243 00:17:11.720 --> 00:17:16.839 became extraordinary professor of mathematics at Knigsburg, and in eighteen 244 00:17:16.880 --> 00:17:20.720 twenty nine was promoted to be an ordinary professor. This 245 00:17:20.920 --> 00:17:24.440 chair he occupied till eighteen forty two, when the Prussian 246 00:17:24.440 --> 00:17:27.240 government gave him a pension and he moved to Berlin, 247 00:17:27.359 --> 00:17:29.920 where he continued to live until his death in eighteen 248 00:17:29.960 --> 00:17:36.359 fifty one. Jacobi's most celebrated investigations are those on elliptic functions, 249 00:17:36.720 --> 00:17:39.880 the modern notation in which is due to him, and 250 00:17:39.960 --> 00:17:43.440 the theory of which he established simultaneously with Abel, but 251 00:17:43.559 --> 00:17:48.240 independently of him. These are given in his treatise Fundamental 252 00:17:48.359 --> 00:17:55.240 Nova theire Functionionum Ellipticarum, Kanigsburg, eighteen twenty nine and in 253 00:17:55.279 --> 00:17:59.759 some later papers in Krell's journal. The correspondence between la 254 00:17:59.799 --> 00:18:04.000 Jeh and Jacobi on elliptic functions has been reprinted in 255 00:18:04.039 --> 00:18:08.160 the first volume of Jacobi's collected works. Jacob, like Abel, 256 00:18:08.559 --> 00:18:12.400 recognized elliptic functions were not merely a group of theorems 257 00:18:12.440 --> 00:18:15.599 on integration, but that there were types of a new 258 00:18:15.720 --> 00:18:19.839 kind of function, namely one of double periodicity. Hence, he 259 00:18:19.920 --> 00:18:23.400 paid particular attention to the theory of the theta function. 260 00:18:24.640 --> 00:18:28.440 The following passage in which he explains this view is 261 00:18:28.759 --> 00:18:35.480 sufficiently interesting to deserve textual reproduction. Equo communi versam k 262 00:18:35.720 --> 00:18:44.839 fingipo test amplicatur perio datticatam and eleticam elusit functiones ellipticas 263 00:18:44.880 --> 00:18:51.759 non alis ad numerari de ber transendantibus k quibusdam god 264 00:18:51.880 --> 00:19:00.640 dant eligantis fortas pluribus elas otte marioribus said speciam quadnam 265 00:19:01.079 --> 00:19:07.799 is ines perfecti et absoluti. Among Jacobi's other investigations, I 266 00:19:07.880 --> 00:19:12.000 may specially single out his papers on determinants, which did 267 00:19:12.039 --> 00:19:14.640 a great deal to bring them into general use, and 268 00:19:14.799 --> 00:19:18.799 particularly his invention of the Jacobian, that is, of the 269 00:19:18.920 --> 00:19:23.920 functional determinant formed by the n squared partial differential coefficients 270 00:19:23.960 --> 00:19:28.240 of the first order of n given functions of n 271 00:19:28.400 --> 00:19:32.559 independent variables. I ought also to mention his papers on 272 00:19:32.680 --> 00:19:37.680 Abelian transcendance, his investigations on the theory of numbers, his 273 00:19:37.839 --> 00:19:41.519 important work on the theory of partial differential equations, his 274 00:19:41.680 --> 00:19:46.119 development of the calculus of variations, and his numerous memoirs 275 00:19:46.240 --> 00:19:51.119 on the planetary theory and other particular dynamical problems, in 276 00:19:51.200 --> 00:19:54.119 the course of which he also extended the theory of 277 00:19:54.240 --> 00:19:58.279 differential equations. Most of the results of the researches last 278 00:19:58.400 --> 00:20:05.039 named are included in his Valusungen uber Dynamic, edited by Klebsch, Berlin, 279 00:20:05.400 --> 00:20:13.319 eighteen sixty six. Riemann Jeorg Friedrich Bernhard Riemann was born 280 00:20:13.400 --> 00:20:18.640 at Bersens on September seventeenth, eighteen twenty six, and died 281 00:20:18.720 --> 00:20:24.000 at Selaska on July twentieth, eighteen sixty six. He studied 282 00:20:24.000 --> 00:20:28.480 at Gattigen under Gauss and subsequently at Berlin under Jacobi 283 00:20:28.799 --> 00:20:33.640 dirichle Steiner and Eisenstein, all of whom were professors there 284 00:20:33.680 --> 00:20:37.079 at the same time. His earliest paper, written in eighteen 285 00:20:37.160 --> 00:20:41.000 fifty was on algebraic functions of a complex variable and 286 00:20:41.160 --> 00:20:44.960 on at The recent investigations of Schwartz, Klein, and Pointerrey 287 00:20:45.079 --> 00:20:48.519 are largely based to those that refer very briefly below. 288 00:20:49.559 --> 00:20:52.960 In eighteen fifty four, Riemann wrote his celebrated memoir on 289 00:20:53.039 --> 00:20:57.279 the Hypotheses on which geometry is founded. This was succeeded 290 00:20:57.359 --> 00:21:00.599 by his memoirs on elliptic functions and the theory of numbers. 291 00:21:01.119 --> 00:21:05.400 He also wrote on physical subjects. The question of the 292 00:21:05.599 --> 00:21:09.519 truth of the assumptions usually made in our geometry had 293 00:21:09.559 --> 00:21:13.839 been considered by J. Sacheri as long ago as seventeen 294 00:21:13.920 --> 00:21:17.279 thirty three, and in more recent times had been discussed 295 00:21:17.319 --> 00:21:22.559 by Nikolai Avenovich Lobachevsky, professor at Khan born at Ninji 296 00:21:23.319 --> 00:21:27.880 Novogorod in seventeen ninety three, and died at Kassen on 297 00:21:28.039 --> 00:21:32.119 February twelfth, eighteen fifty six, in eighteen twenty six and 298 00:21:32.200 --> 00:21:35.559 again in eighteen forty, by Gauss in eighteen thirty one 299 00:21:35.640 --> 00:21:39.839 and in eighteen forty six, and by Johann Bolyai born 300 00:21:39.920 --> 00:21:43.680 at Klausenberg in eighteen o two and died at Masmaros 301 00:21:43.799 --> 00:21:47.759 Wazerheli in eighteen sixty and in eighteen thirty two in 302 00:21:47.880 --> 00:21:51.920 the appendix to the first volume of his father's Tentamen. 303 00:21:52.680 --> 00:21:56.279 But Riemon's memoir of eighteen fifty four attracted general attention 304 00:21:56.440 --> 00:21:59.920 to the subject of hypergeometry, and the theory has been 305 00:22:00.200 --> 00:22:05.960 since extended and simplified by various writers, notably by Eugenio Beltrami, 306 00:22:06.440 --> 00:22:10.799 professor at Pavia, born at Cremonia in eighteen thirty five, 307 00:22:11.480 --> 00:22:16.200 and by Hermann Ludwig Ferdinand von Helmholtz, professor at Berlin, 308 00:22:16.680 --> 00:22:20.680 born at Potsdam on August thirty one, eighteen twenty one. 309 00:22:21.359 --> 00:22:24.759 The subject is so technical that I confine myself to 310 00:22:24.839 --> 00:22:27.799 a bare sketch of the argument from which the idea 311 00:22:27.960 --> 00:22:31.839 is derived that a space of two dimensions should have 312 00:22:32.000 --> 00:22:35.759 the geometrical properties with which we are made familiar in 313 00:22:35.960 --> 00:22:40.000 the study of elementary geometry, it is necessary that it 314 00:22:40.039 --> 00:22:43.440 should be possible at any place to construct a figure 315 00:22:43.559 --> 00:22:46.920 congruoned to a given figure, and this is so only 316 00:22:46.960 --> 00:22:50.039 if the product of the principal radii of curvature at 317 00:22:50.079 --> 00:22:54.880 every point in space or surface be a constant. There 318 00:22:54.920 --> 00:22:59.480 are three species of surfaces which possess this property, namely, 319 00:22:59.680 --> 00:23:04.680 one spherical surfaces where the product is positive, two plane 320 00:23:04.799 --> 00:23:08.640 surfaces which lead to Euclidean geometry where it is zero, 321 00:23:09.319 --> 00:23:14.480 and three what Beryltrami has called pseudospherical surfaces where it 322 00:23:14.599 --> 00:23:19.160 is negative. Moreover, if any service be bent without dilation 323 00:23:19.400 --> 00:23:24.240 or contraction, the measure of curvature remains unaltered. Thus, these 324 00:23:24.319 --> 00:23:28.160 three species of surfaces are types of three kinds on 325 00:23:28.240 --> 00:23:32.240 which congruent figures can be constructed. For instance, a plane 326 00:23:32.279 --> 00:23:35.599 can be rolled into a cone, and the system of 327 00:23:35.720 --> 00:23:38.720 geometry on a conical surface is similar to that on 328 00:23:38.799 --> 00:23:43.599 a plane. These kinds of space of two dimensions are 329 00:23:43.680 --> 00:23:48.119 distinguished from one another by a simple test through a 330 00:23:48.200 --> 00:23:53.720 point of spherical space. No geodedic line a geodetic line 331 00:23:53.759 --> 00:23:57.400 being defined as the shortest distance between two points, can 332 00:23:57.480 --> 00:24:02.160 be drawn parallel to a given gen decline through a 333 00:24:02.240 --> 00:24:05.400 point of Euclidean or plane space, one and only one 334 00:24:05.519 --> 00:24:09.440 geoddic line i e. Is straight line can be drawn 335 00:24:09.640 --> 00:24:13.240 parallel to a given geod decline. Through a point of 336 00:24:13.279 --> 00:24:17.480 pseudospherical space, more than one geodedic line can be drawn 337 00:24:17.599 --> 00:24:21.000 parallel to a given geod decline, But all these lines 338 00:24:21.039 --> 00:24:24.799 form a pencil whose vertical angle is constant. It may 339 00:24:24.880 --> 00:24:27.720 be thought that we have a demonstration that our space 340 00:24:27.839 --> 00:24:32.000 is plain, since through a given point we can draw 341 00:24:32.119 --> 00:24:35.000 only one straight line parallel to a given straight line. 342 00:24:35.480 --> 00:24:38.400 This is not so, for it is conceivable that our 343 00:24:38.440 --> 00:24:41.440 means of observation do not permit us to say with 344 00:24:41.519 --> 00:24:45.960 absolute accuracy whether two lines are parallel. Hence, we cannot 345 00:24:46.119 --> 00:24:48.720 use this as a means to tell whether our space 346 00:24:48.920 --> 00:24:52.519 is plain or not. A better test can be deduced 347 00:24:52.599 --> 00:24:56.119 from the proposition that in any two dimensional space of 348 00:24:56.279 --> 00:24:59.759 uniform curvature, the sum of the angles of a triangle, 349 00:25:00.079 --> 00:25:03.079 if it differ from two right angles, will differ by 350 00:25:03.160 --> 00:25:07.160 a quantity proportional to the area of the triangle. Hence, 351 00:25:07.240 --> 00:25:11.759 it may happen possibly that although for triangles such as 352 00:25:11.880 --> 00:25:16.640 we can measure, the difference is imperceptible, yet for triangles 353 00:25:16.680 --> 00:25:19.039 which are millions of times bigger. There would be a 354 00:25:19.160 --> 00:25:25.319 sensible difference. If space be spherical or pseudospherical, its extent 355 00:25:25.519 --> 00:25:29.640 is finite. If it be plain, its extent is infinite. 356 00:25:30.480 --> 00:25:34.599 In regard to pseudospherical space, I should add that its 357 00:25:34.720 --> 00:25:38.039 extent may be infinite if it be constructed in space 358 00:25:38.200 --> 00:25:42.559 of four dimensions. In the preceding sketch of the foundations 359 00:25:42.599 --> 00:25:46.960 of non Euclidean geometry, I have assumed tacitly that the 360 00:25:47.039 --> 00:25:51.000 measure of a distance remains the same everywhere cline assume 361 00:25:51.119 --> 00:25:54.160 that if this be not the case, and if the 362 00:25:54.279 --> 00:25:58.599 law of the measurement of distance be properly chosen, we 363 00:25:58.759 --> 00:26:02.279 can obtain three systems of plane geometry analogous to the 364 00:26:02.400 --> 00:26:08.359 three systems mentioned above. These are called, respectively, elliptic, parabolic, 365 00:26:08.559 --> 00:26:14.519 and hyperbolic geometries. The above refers to only to hyper 366 00:26:14.599 --> 00:26:19.960 space of two dimensions. Naturally, there arises the question whether 367 00:26:20.079 --> 00:26:24.119 there are different kinds of hyperspace of three or more dimensions. 368 00:26:24.799 --> 00:26:29.599 Remontchute that there are three kinds of hyperspace of three 369 00:26:29.680 --> 00:26:34.720 dimensions having properties analogous to the three kinds of hyperspace 370 00:26:34.799 --> 00:26:39.759 of two dimensions already discussed. These are differentiated by the 371 00:26:39.920 --> 00:26:45.000 test whether at every point no geodetical surfaces, or one 372 00:26:45.079 --> 00:26:50.440 geodetical surface or a facisculus of geodetical surfaces can be 373 00:26:50.640 --> 00:26:55.400 drawn parallel to a given surface, a geodiitical surface being 374 00:26:55.519 --> 00:26:59.799 defined as such that every geodetic line joining two points 375 00:27:00.279 --> 00:27:04.640 on it lies wholly on the surface. I returned now 376 00:27:04.720 --> 00:27:10.079 to Riemon's other investigations in multiply periodic functions. It is 377 00:27:10.200 --> 00:27:13.359 hardly too much to say that he and his memoir 378 00:27:13.640 --> 00:27:18.359 in Bouchardt's Journal for eighteen fifty seven did for the 379 00:27:18.480 --> 00:27:22.559 Abelian functions what abel had done for the elliptic functions. 380 00:27:23.240 --> 00:27:26.279 And it is this, perhaps that will constitute one of 381 00:27:26.319 --> 00:27:30.720 his chief claims to future distinction. In the theory of numbers, 382 00:27:31.160 --> 00:27:33.799 Riemond's short tract of eight pages on the number of 383 00:27:33.960 --> 00:27:37.720 primes which lie between two given numbers affords a striking 384 00:27:37.880 --> 00:27:43.279 instance of his analytical powers. Legendre had previously shewn that 385 00:27:43.519 --> 00:27:46.880 the number of primes less than n is very approximately 386 00:27:47.240 --> 00:27:51.119 n divided by the quantity log n minus one point 387 00:27:51.480 --> 00:27:55.519 eight three sixty six. But Riemon went further, and this 388 00:27:55.759 --> 00:28:00.279 tract and a memoir by Chebyshev contains nearly all that 389 00:28:00.720 --> 00:28:04.039 had been done yet in connection with a problem of 390 00:28:04.200 --> 00:28:07.799 so obvious of a character that has suggested itself to 391 00:28:07.920 --> 00:28:11.400 all who has considered the theory of numbers, and yet 392 00:28:11.960 --> 00:28:16.599 which overtacks the powers even of Lagrange and Gauss. Among 393 00:28:16.680 --> 00:28:20.920 others than those already named, I may mention the following 394 00:28:21.200 --> 00:28:26.480 who have written on elliptic and Abelian functions. Johann Jorg Rosenheian, 395 00:28:26.720 --> 00:28:32.119 professor in Knigsburg, born there on June ten, eighteen sixteen 396 00:28:32.200 --> 00:28:34.720 and died in eighteen eighty seven, who wrote in eighteen 397 00:28:34.880 --> 00:28:39.440 eighty four on the hyper elliptic double theta function and 398 00:28:39.680 --> 00:28:45.319 functions of two variables with four periods. Adolf Goeppel born 399 00:28:45.519 --> 00:28:49.119 at Rostock in September eighteen twelve and died at Berlin 400 00:28:49.400 --> 00:28:53.720 in March eighteen forty seven, who discussed hyper elliptic functions 401 00:28:54.000 --> 00:28:58.680