WEBVTT 1 00:00:01.240 --> 00:00:04.360 Chapter eighteen, Part two of A Short Account of the 2 00:00:04.480 --> 00:00:09.880 History of Mathematics. This is a LibriVox recording. All LibriVox 3 00:00:09.960 --> 00:00:13.919 recordings are in a public domain. For more information or 4 00:00:13.960 --> 00:00:17.960 to volunteer, please visit LibriVox dot org. This is a 5 00:00:18.000 --> 00:00:21.199 reading by Paul King p J K dot scrips dot 6 00:00:21.359 --> 00:00:23.920 m I T dot E d U forward slash p 7 00:00:24.120 --> 00:00:27.920 K J. A Short Account of the History of mathematics 8 00:00:27.960 --> 00:00:32.399 by W. W. Rouse Baul. Chapter eighteen Leibniz and the 9 00:00:32.439 --> 00:00:36.159 Mathematicians of the first half of the eighteenth century, Part two. 10 00:00:37.159 --> 00:00:44.359 Lambert Johann Heinrich Lambert was born at Mulhausen on August 11 00:00:44.439 --> 00:00:48.280 twenty eighth, seventeen twenty eight and died at Berlin on 12 00:00:48.359 --> 00:00:52.520 September twenty fifth, seventeen seventy seven. He was the son 13 00:00:52.560 --> 00:00:55.200 of a small tailor and had to rely on his 14 00:00:55.280 --> 00:00:58.719 own efforts for his education. From a clerk in some 15 00:00:58.920 --> 00:01:02.399 iron works, he got a place in a newspaper office, 16 00:01:02.640 --> 00:01:06.280 and subsequently, on the recommendation of the editor, he was 17 00:01:06.319 --> 00:01:09.560 appointed tutor in a private family, which secured him the 18 00:01:09.680 --> 00:01:13.400 use of a good library and sufficient leisure to use it. 19 00:01:14.239 --> 00:01:17.879 In seventeen fifty nine he settled in Augsburg, and in 20 00:01:17.959 --> 00:01:21.760 seventeen thirty three removed to Berlin, where he was given 21 00:01:21.799 --> 00:01:25.079 a small pension and finally made editor of the Prussian 22 00:01:25.159 --> 00:01:30.799 Astronomical Almanac. Lambel's most important work were one on optics, 23 00:01:30.920 --> 00:01:35.079 issued in seventeen fifty nine, which suggested to arago the 24 00:01:35.159 --> 00:01:40.359 lines of investigation. He subsequently pursued a treatise on perspective, 25 00:01:40.439 --> 00:01:44.439 published in seventeen fifty nine, to which in seventeen sixty 26 00:01:44.480 --> 00:01:49.239 eight an appendix given practical applications was added, and a 27 00:01:49.280 --> 00:01:53.519 treatise on comets printed in seventeen sixty one, containing the 28 00:01:53.560 --> 00:01:56.640 well known expression for the area of a focal sector 29 00:01:56.719 --> 00:01:59.359 of a conic in terms of the cord and the 30 00:01:59.359 --> 00:02:04.480 bounding race. Besides these, he communicated numerous papers to the 31 00:02:04.480 --> 00:02:09.439 Berlin Academy. Of the most important are his memoir in 32 00:02:09.520 --> 00:02:14.000 seventeen sixty eight on transcendental magnitudes, in which he proved 33 00:02:14.000 --> 00:02:19.599 that pi is incommensurable. The proof is given in Legendre's geometry, 34 00:02:20.400 --> 00:02:26.240 and is there extended to pi squared. His paper on trigonometry, 35 00:02:26.319 --> 00:02:30.759 read in seventeen sixty eight, in which he developed Demoffra's 36 00:02:30.840 --> 00:02:35.159 theorems on the trigonometry of complex variables, and introduced the 37 00:02:35.280 --> 00:02:39.840 hyperbolic sign and co sign, denoted by the symbols sign 38 00:02:40.080 --> 00:02:46.879 h x cohx. His essay entitled the Analytical Observations, published 39 00:02:46.919 --> 00:02:50.560 in seventeen seventy one, which is the earliest attempt to 40 00:02:50.599 --> 00:02:54.840 form the functional equations by expressing the given properties in 41 00:02:54.919 --> 00:02:59.840 the language of the differential calculus and then integrating. Lastly, 42 00:03:00.360 --> 00:03:05.159 his paper on viz Viva published in seventeen eighty three, 43 00:03:05.360 --> 00:03:08.400 in which for the first time he expressed Newton's second 44 00:03:08.479 --> 00:03:14.199 law of motion in the notation of the differential calculus. 45 00:03:14.199 --> 00:03:17.960 Of the other mathematicians above mentioned, I hear add a 46 00:03:18.000 --> 00:03:24.080 few words. Etienne Bezous born in Nemours on March thirty first, 47 00:03:24.479 --> 00:03:28.599 seventeen thirty and died on September twenty seventh, seventeen eighty three. 48 00:03:29.240 --> 00:03:35.520 Besides numerous minor works, wrote to Therrie Generreal de sequeigion Algebraque, 49 00:03:35.960 --> 00:03:40.080 published at Paris in seventeen seventy nine, which in particular 50 00:03:40.159 --> 00:03:43.120 contained much new and valuable matter on the theory of 51 00:03:43.199 --> 00:03:46.759 elimination and symmetrical functions of the roots of an equation. 52 00:03:47.680 --> 00:03:51.120 He used his determinants in a paper in the Histoire 53 00:03:51.199 --> 00:03:55.759 de leca Demi Royale seventeen sixty four, but did not 54 00:03:55.840 --> 00:04:01.000 treat of the general theory. Jean Trembrey born at Geneva 55 00:04:01.120 --> 00:04:06.000 in seventeen forty nine and died at September eighteenth, eighteen eleven, 56 00:04:06.280 --> 00:04:11.439 contributed to the development of differential equations, finite differences, and 57 00:04:11.520 --> 00:04:17.720 the calculus of probabilities. Louis Francois d'artoine Abregaste born a 58 00:04:17.800 --> 00:04:22.319 dal Sas on October fourth, seventeen fifty nine and died 59 00:04:22.360 --> 00:04:26.360 at Strasbourg, where he was professor, on April eighth, eighteen 60 00:04:26.399 --> 00:04:30.680 o three, wrote on series and the derivatives. Known by 61 00:04:30.680 --> 00:04:33.879 his name, he was the first writer to separate the 62 00:04:33.920 --> 00:04:37.800 symbols of operation from those of quantity. I do not 63 00:04:37.879 --> 00:04:40.519 wish to crowd my pages with an account of those 64 00:04:41.079 --> 00:04:44.439 who have not distinctly advanced the subject, but I have 65 00:04:44.560 --> 00:04:48.439 mentioned the above writers because their names are still well known. 66 00:04:48.839 --> 00:04:51.959 We may, however, say that the discoveries of Euler and 67 00:04:52.079 --> 00:04:56.279 Lagrange in the subjects which they treated, were so complete 68 00:04:56.279 --> 00:05:00.240 and far reaching that what their less gifted contemporary he's 69 00:05:00.279 --> 00:05:03.800 added is not of sufficient importance to require mention in 70 00:05:03.839 --> 00:05:09.639 a book of this nature. Lagrange. Joseph Louis Lagrange, the 71 00:05:09.639 --> 00:05:14.000 greatest mathematician of the eighteenth century, was born at Turin 72 00:05:14.560 --> 00:05:18.279 on January twenty fifth, seventeen thirty six, and died at 73 00:05:18.279 --> 00:05:23.439 Paris on April tenth, eighteen thirteen. His father, who had 74 00:05:23.600 --> 00:05:28.560 the charge of the Sardinian Military Chest, was of good 75 00:05:28.600 --> 00:05:32.160 social position and wealthy, but before his son grew up, 76 00:05:32.160 --> 00:05:36.079 he had lost most of his property in speculations, and 77 00:05:36.160 --> 00:05:39.079 young Lagrange had to rely for his position on his 78 00:05:39.120 --> 00:05:43.000 own abilities. He was educated at the College of Tyrann, 79 00:05:43.519 --> 00:05:46.560 but was not until he was seventeen that he chewed 80 00:05:46.600 --> 00:05:50.439 any taste for mathematics, his interest in the subject being 81 00:05:50.519 --> 00:05:53.759 first excited by a memoir by Halley across which he 82 00:05:53.839 --> 00:05:58.079 came by accident, alone and unaided. He threw himself into 83 00:05:58.160 --> 00:06:02.040 mathematical studies, and at the end of a year's incessant toil, 84 00:06:02.199 --> 00:06:06.000 he was already an accomplished mathematician, and was made a 85 00:06:06.079 --> 00:06:10.240 lecturer in the artillery school. The first of these labors 86 00:06:10.600 --> 00:06:13.800 was his letter, written when he was still only nineteen 87 00:06:13.920 --> 00:06:18.319 to Euler, in which he solved the isoparametrical problem, which 88 00:06:18.360 --> 00:06:20.839 for more than half a century had been the subject 89 00:06:20.839 --> 00:06:24.319 of discussion to effect a solution in which he sought 90 00:06:24.319 --> 00:06:27.199 to determine the form of a function so that a 91 00:06:27.240 --> 00:06:30.560 formula in which it entered should satisfy a certain condition. 92 00:06:31.199 --> 00:06:35.879 He enunciated the principles of the calculus of variations. Euler 93 00:06:35.920 --> 00:06:40.399 recognized the generality of the method adopted and its superiority 94 00:06:40.519 --> 00:06:44.240 to that used by himself, and with rare courtesy he 95 00:06:44.279 --> 00:06:47.519 withheld a paper that he had previously written, which covered 96 00:06:47.639 --> 00:06:49.720 some of the same ground, in order that the young 97 00:06:49.800 --> 00:06:53.480 Italian might have time to complete his work and claim 98 00:06:53.680 --> 00:06:59.360 the undisputed invention of the new calculus. The name of 99 00:06:59.399 --> 00:07:03.319 this branch of analysis was suggested by Euler. This memoir 100 00:07:03.360 --> 00:07:06.839 at once placed Lagrange in the front rank of mathematicians. 101 00:07:06.879 --> 00:07:12.079 Then living. In seventeen fifty eight, Lagrange established, with the 102 00:07:12.120 --> 00:07:16.439 aid of his pupils, a society which was subsequently incorporated 103 00:07:16.560 --> 00:07:20.079 as the Turan Academy, and in the five volumes of 104 00:07:20.120 --> 00:07:25.120 its transactions, usually known as the Miscellanea Tourernensia, most of 105 00:07:25.120 --> 00:07:28.600 his early writings are to be found. Many of these 106 00:07:28.600 --> 00:07:32.399 are elaborate works. The first volume contains a memoir of 107 00:07:32.439 --> 00:07:35.759 the theory of the propagation of sound and in this 108 00:07:35.920 --> 00:07:39.439 he indicates a mistake made by Newton, obtains a general 109 00:07:39.439 --> 00:07:43.839 differential equation for the motion, and integrates it for motion 110 00:07:44.040 --> 00:07:47.720 in a straight line. This volume also contains the complete 111 00:07:47.759 --> 00:07:51.439 solution of the problem of a string vibrating transversely. In 112 00:07:51.519 --> 00:07:54.399 this paper, he points out a lack of generality in 113 00:07:54.439 --> 00:07:58.399 the solutions previously given by Taylor, de Lambert and Euler, 114 00:07:59.000 --> 00:08:01.519 and arrives at the con inclusion that form of a 115 00:08:01.639 --> 00:08:04.160 curve at any time T is given by the equation 116 00:08:04.480 --> 00:08:10.240 y equals a sign mx sign nt. The article concludes 117 00:08:10.279 --> 00:08:14.680 with a masterly discussion of echoes, beats, and compound sounds. 118 00:08:16.040 --> 00:08:20.079 Other articles in this volume are on recurring series probabilities 119 00:08:20.120 --> 00:08:24.480 and the calculus of variations. The second volume containing a 120 00:08:24.519 --> 00:08:27.800 long paper embodying the results of several memoirs in the 121 00:08:27.839 --> 00:08:30.759 first volume on the theory and notation of the calculus 122 00:08:30.759 --> 00:08:35.360 of variations, and he illustrates its use by deducing the 123 00:08:35.399 --> 00:08:39.519 principle of least action and also by solutions of various 124 00:08:39.559 --> 00:08:44.519 problems in dynamics. The third volume includes a solution of 125 00:08:44.639 --> 00:08:48.639 several dynamical problems by means of the calculus of variations. 126 00:08:49.200 --> 00:08:53.080 Some papers on the integral calculus, a solution of Fermat's 127 00:08:53.120 --> 00:08:57.600 problem mentioned above, and on the general differential equations of 128 00:08:57.720 --> 00:09:03.240 motion for three bodies moved under their mutual attractions. In 129 00:09:03.320 --> 00:09:07.240 seventeen sixty one, Lagrange stood without arrival as the foremost 130 00:09:07.240 --> 00:09:11.679 mathematician living, But the unceasing labour of the preceding nine 131 00:09:11.720 --> 00:09:15.799 years had seriously affected his health, and the doctors refused 132 00:09:15.840 --> 00:09:19.159 to be responsible for his reason or life unless he 133 00:09:19.159 --> 00:09:24.600 would take rest and exercise. Although his health was temporarily restored, 134 00:09:24.759 --> 00:09:28.919 his nervous system never quite recovered its tone, and henceforth 135 00:09:28.960 --> 00:09:33.679 he constantly suffered from attacks of profound melancholy. The next 136 00:09:33.720 --> 00:09:36.720 work he produced was in seventeen sixty four on the 137 00:09:36.759 --> 00:09:40.360 libration of the Moon and an explanation as to why 138 00:09:40.399 --> 00:09:43.360 the same face was always turned to the Earth, a 139 00:09:43.440 --> 00:09:48.200 problem which he treated by aid of virtual work. His 140 00:09:48.320 --> 00:09:52.000 solution is especially interesting as containing the germ of the 141 00:09:52.080 --> 00:09:57.320 idea of generalized equations of motion equations, which he first 142 00:09:57.519 --> 00:10:02.039 formally proved in seventeen eighty. He now started to go 143 00:10:02.159 --> 00:10:05.799 on a visit to London, but on the way fell 144 00:10:05.879 --> 00:10:09.320 ill at Paris. There he was received with the most 145 00:10:09.399 --> 00:10:12.879 marked honour, and it was with regret that he left 146 00:10:12.919 --> 00:10:16.039 the brilliant society of that city to return to his 147 00:10:16.080 --> 00:10:20.279 provincial life at Tehran. His further stay in Piedmont was, 148 00:10:20.320 --> 00:10:24.639 however short. In seventeen sixty six, Euler left Berlin, and 149 00:10:24.759 --> 00:10:28.120 Frederick the Great immediately wrote expressing the wish of the 150 00:10:28.120 --> 00:10:31.399 greatest king in Europe to have the greatest mathematician in 151 00:10:31.440 --> 00:10:36.720 Europe resident in his court. Lagrange accepted the offer and 152 00:10:37.279 --> 00:10:41.440 spent the next twenty years at Prussia, where he produced 153 00:10:41.600 --> 00:10:44.519 not only the long series of memoirs published at the 154 00:10:44.519 --> 00:10:50.840 Berlin and Tyran Transactions, but his monumental work Mechanique Anarritique. 155 00:10:50.879 --> 00:10:56.360 His residence at Berlin commenced with an unfortunate mistake, finding 156 00:10:56.440 --> 00:10:59.879 most of his colleagues married and assured by their wives 157 00:11:00.000 --> 00:11:03.399 that it was the only way to be happy. He married, 158 00:11:03.879 --> 00:11:06.879 his wife soon died, but the union was not a 159 00:11:06.960 --> 00:11:11.559 happy one. Lagrange was a favorite of the King, who 160 00:11:11.679 --> 00:11:15.120 used frequently to discourse to him on the advantage of 161 00:11:15.200 --> 00:11:19.759 perfect regularity of life. The lesson went home, and thenceforth 162 00:11:19.879 --> 00:11:23.360 Lagrange studied his mind and body as though they were machines, 163 00:11:23.840 --> 00:11:27.399 and found by experiment the exact amount of work which 164 00:11:27.440 --> 00:11:30.720 he was able to do without breaking down. Every night 165 00:11:30.759 --> 00:11:33.440 he set himself a definite task for the next day, 166 00:11:33.960 --> 00:11:36.799 and on completing any branch of a subject, he wrote 167 00:11:37.200 --> 00:11:40.679 a short analysis to see what points in the demonstrations 168 00:11:41.159 --> 00:11:45.000 or in the subject matter were capable of improvement. He 169 00:11:45.080 --> 00:11:48.159 always thought out the subject of his papers before he 170 00:11:48.200 --> 00:11:51.759 began to compose them, and usually wrote them straight off 171 00:11:51.799 --> 00:11:56.159 without a single erasure or correction. His mental activity during 172 00:11:56.240 --> 00:11:59.720 these twenty years was amazing. Not only did he produce 173 00:11:59.759 --> 00:12:04.159 his splendid mechanique and rietique, but he contributed between one 174 00:12:04.159 --> 00:12:07.639 and two hundred papers to the academies of Berlin, Tyrann 175 00:12:07.759 --> 00:12:11.919 and Paris. Some of these are complete treatises, and all, 176 00:12:11.960 --> 00:12:16.759 without exception, are of Hahaigh order of excellence, except for 177 00:12:16.799 --> 00:12:19.960 a short time when he was ill, and produced on 178 00:12:20.080 --> 00:12:23.679 an average about one memoir a month. Of these, I 179 00:12:23.720 --> 00:12:27.639 note the following as among the most important. First, his 180 00:12:27.759 --> 00:12:31.960 contributions to the fourth and fifth volumes seventeen sixty six 181 00:12:32.000 --> 00:12:37.320 to seventeen seventy three of the Miscellanea Taurernensia, of which 182 00:12:37.399 --> 00:12:40.720 the most important was the one in seventeen seventy one, 183 00:12:41.320 --> 00:12:45.080 in which he discussed how numerous astronomical observations should be 184 00:12:45.159 --> 00:12:49.159 combined so as to give the most probable result, and 185 00:12:49.320 --> 00:12:53.200 later his contributions to the first two volumes seventeen eighty 186 00:12:53.240 --> 00:12:56.879 four to seventeen eighty five of the Transactions of the 187 00:12:56.919 --> 00:13:01.519 Turin Academy, to the first of which he contributed a 188 00:13:01.559 --> 00:13:05.279 paper on the pressure exerted by fluids in motion, and 189 00:13:05.360 --> 00:13:09.399 to the second an article on integration by infinite series 190 00:13:09.840 --> 00:13:12.600 and the kind of problems for which it is suitable. 191 00:13:14.279 --> 00:13:18.279 Most of the memoirs sent to Paris were on astronomical questions, 192 00:13:18.320 --> 00:13:22.240 and among these I ought particularly to mention his memoir 193 00:13:22.360 --> 00:13:26.679 on the Jovian System in seventeen sixty six, his essay 194 00:13:26.759 --> 00:13:30.279 on the problem of the three bodies in seventeen seventy two, 195 00:13:30.919 --> 00:13:34.000 his work on the secular equation of the Moon in 196 00:13:34.120 --> 00:13:39.159 seventeen seventy three, and his treatise on cometary perturbations in 197 00:13:39.279 --> 00:13:43.879 seventeen seventy eight. These were all written on subjects proposed 198 00:13:43.919 --> 00:13:46.960 by the French Academy, and in each case the prize 199 00:13:47.039 --> 00:13:50.879 was awarded to him. The greater number of his papers 200 00:13:50.960 --> 00:13:54.639 during the same time were, however, contributed to the Berlin Academy. 201 00:13:55.320 --> 00:13:59.519 Several of them deal with questions on algebra. In particular. 202 00:13:59.679 --> 00:14:02.679 I may mention one his discussion of the solution of 203 00:14:02.720 --> 00:14:07.679 indeterminate equations on integers in seventeen seventy with special notice 204 00:14:07.720 --> 00:14:13.000 of indeterminate quadratics in seventeen sixty nine, two, His tract 205 00:14:13.039 --> 00:14:17.639 on the Theory of Elimination seventeen seventy three, His Memoirs 206 00:14:17.639 --> 00:14:21.039 on the General Process for solving in algebraical equations of 207 00:14:21.080 --> 00:14:26.679 any degree seventeen seventy and seventeen seventy one. This method 208 00:14:26.720 --> 00:14:30.320 fails for equations of an order above the fourth, because 209 00:14:30.360 --> 00:14:33.200 then it involves a solution of an equation of higher 210 00:14:33.200 --> 00:14:36.519 dimensions than the one proposed, but it gives all the 211 00:14:36.559 --> 00:14:40.720 solutions of his predecessors as modifications of a single principle. 212 00:14:41.480 --> 00:14:45.039 He found, however, the complete solution of the binomial equations 213 00:14:45.039 --> 00:14:49.000 of any degree and fourth. Lastly, in seventeen seventy three 214 00:14:49.159 --> 00:14:52.440 he treated of determinants of the second and third order. 215 00:14:53.840 --> 00:14:57.080 Several of his early papers also deal with questions connected 216 00:14:57.120 --> 00:15:00.840 with the neglected but singularly fascinating suchbject of the theory 217 00:15:00.879 --> 00:15:04.639 of numbers. Among these are one his proof of the 218 00:15:04.639 --> 00:15:08.519 theorem that every integer which is not a square can 219 00:15:08.559 --> 00:15:11.360 be expressed as a sum of either two, three, or 220 00:15:11.399 --> 00:15:17.279 four integral squares seventeen seventy His proof of Wilson's theorem 221 00:15:17.919 --> 00:15:22.240 the off n b prime, then n minus one factorial 222 00:15:22.360 --> 00:15:26.720 plus one is always a multiple of nds seventeen seventy one. Three, 223 00:15:27.200 --> 00:15:30.919 His memoirs of seventeen seventy three, seventeen seventy five, and 224 00:15:31.000 --> 00:15:35.799 seventeen seventy seven, which give the demonstrations of several results 225 00:15:35.960 --> 00:15:41.120 enunciated by format and not previously proved. Four And lastly, 226 00:15:41.279 --> 00:15:45.080 his method for determining the factors of numbers of the 227 00:15:45.120 --> 00:15:49.600 form x squared plus a y squared. There are also 228 00:15:49.679 --> 00:15:54.320 numerous articles on various points of analytical geometry. In two 229 00:15:54.360 --> 00:15:57.080 of them, in seventeen ninety two and seventeen ninety three, 230 00:15:57.600 --> 00:16:01.559 he reduced the equations of the quadrix or conochoids to 231 00:16:01.639 --> 00:16:06.360 their canonical forms. During the years from seventeen seventy two 232 00:16:06.440 --> 00:16:09.639 to seventeen eighty five he contributed a long series of 233 00:16:09.679 --> 00:16:14.600 memoirs which created the science of deferential equations at any rate, 234 00:16:14.639 --> 00:16:18.320 as far as partial differential equations are concerned, I do 235 00:16:18.440 --> 00:16:21.879 not think that any previous writer had done anything beyond 236 00:16:21.960 --> 00:16:26.159 considering equations of some particular form. A large part of 237 00:16:26.200 --> 00:16:29.399 these results were collected in the second edition of Euler's 238 00:16:29.399 --> 00:16:34.120 Integral Calculus, which was published in seventeen ninety four. His 239 00:16:34.279 --> 00:16:38.320 papers on mechanics require no separate mention here, as the 240 00:16:38.360 --> 00:16:43.639 results arrived at are embodied in the mechanique analytique, which 241 00:16:43.679 --> 00:16:48.679 is described below. Lastly, there are numerous memoirs on problems 242 00:16:48.679 --> 00:16:53.000 in astronomy. Of these, the most important are the following one, 243 00:16:53.240 --> 00:16:57.200 The Attraction of ellipsoids seventeen seventy three. This is founded 244 00:16:57.240 --> 00:17:01.960 on mclaurin's work two on the secular equation of the 245 00:17:02.000 --> 00:17:07.240 Moon seventeen seventy three. Also noticeable for the earliest introduction 246 00:17:07.440 --> 00:17:11.400 of the idea of the potential. The potential of a 247 00:17:11.440 --> 00:17:14.000 body at any point is the sum of the mass 248 00:17:14.039 --> 00:17:16.960 of every element of the body when divided by its 249 00:17:17.000 --> 00:17:21.480 distance from the point. Lagrant should that if the potential 250 00:17:21.519 --> 00:17:24.559 of a body at an external point were known, the 251 00:17:24.640 --> 00:17:28.960 attraction in any direction could be at once found. The 252 00:17:28.960 --> 00:17:32.119 theory of the potential was elaborated in a paper sent 253 00:17:32.160 --> 00:17:37.240 to Berlin in seventeen seventy seven three on the motion 254 00:17:37.440 --> 00:17:40.680 of the nodes of a planet's orbit seventeen seventy four. 255 00:17:41.519 --> 00:17:47.839 Four on the Stability of Planetary orbits seventeen seventy six, five. 256 00:17:48.599 --> 00:17:51.759 Two memoirs in which the method of determining the orbit 257 00:17:51.839 --> 00:17:55.160 of a comet from three observations is completely worked out 258 00:17:55.279 --> 00:17:59.640 seventeen seventy eight and seventeen eighty three. This is not 259 00:17:59.640 --> 00:18:03.440 indeed proved practically available, but his system of calculating the 260 00:18:03.480 --> 00:18:07.759 perturbations by means of mechanical quadratures has formed the basis 261 00:18:07.839 --> 00:18:13.200 of most subsequent researches on the subject. Six His Determination 262 00:18:13.440 --> 00:18:16.440 of the secular and periodic variations of the elements of 263 00:18:16.480 --> 00:18:20.119 the planets seventeen eighty one to seventeen eighty four. The 264 00:18:20.200 --> 00:18:24.039 upper limits assigned for these agree closely with those obtained 265 00:18:24.119 --> 00:18:27.839 later by Leverrier, and he proceeded as far as the 266 00:18:27.920 --> 00:18:33.200 knowledge then possessed of the masses of the planets permitted seven. 267 00:18:33.519 --> 00:18:37.920 Three Memoirs on the method of interpolation seventeen eighty three, 268 00:18:38.279 --> 00:18:42.079 seventeen ninety two, and seventeen ninety three. The part of 269 00:18:42.119 --> 00:18:45.640 finite differences dealing Therewith is now in the same stage 270 00:18:45.759 --> 00:18:49.160 as that in which Lagrange left it. Over and above 271 00:18:49.200 --> 00:18:53.599 these various papers, he composed his great treatise Mechanique Analytique. 272 00:18:54.319 --> 00:18:56.759 In this he lays down the law of virtual work, 273 00:18:56.960 --> 00:19:00.200 and from that one fundamental principle, by the age of 274 00:19:00.240 --> 00:19:03.839 the calculus of variations, he deduces the whole of mechanics, 275 00:19:03.960 --> 00:19:07.960 both of solids and fluids. The object of the book 276 00:19:08.039 --> 00:19:10.880 is to shew that the subject is implicitly included in 277 00:19:10.960 --> 00:19:14.400 a single principle, and to give the general formulae from 278 00:19:14.640 --> 00:19:18.720 which any particular result can be obtained. The method of 279 00:19:18.799 --> 00:19:22.519 generalized coordinates by which he obtained this result is perhaps 280 00:19:22.519 --> 00:19:26.519 the most brilliant result of his analysis. Instead of following 281 00:19:26.559 --> 00:19:29.759 the motion of each individual part of a material system, 282 00:19:31.039 --> 00:19:34.440 as de Lambert and Euler had done, he shewed that 283 00:19:34.480 --> 00:19:37.759 if we determine its configuration by a sufficient number of 284 00:19:37.839 --> 00:19:40.200 variables whose number is the same as that of the 285 00:19:40.240 --> 00:19:43.759 degrees of freedom possessed by the system, then the connecticut 286 00:19:43.839 --> 00:19:46.720 potential energies of the system can be expressed in terms 287 00:19:46.759 --> 00:19:50.640 of these, and the differential equations of motions then deduced 288 00:19:50.680 --> 00:19:55.200 by simple differentiation, for example, in dynamics of a rigid system. 289 00:19:55.279 --> 00:19:58.519 He replaces the consideration of the particular problem by the 290 00:19:58.559 --> 00:20:01.319 general equation, which is no now usually written in the 291 00:20:01.359 --> 00:20:05.480 form of D by dt times the partial derivative of 292 00:20:05.839 --> 00:20:09.480 capital T with respect to theta minus, the partial derivative 293 00:20:09.519 --> 00:20:12.880 of capital T with respect to theta plus the partial 294 00:20:12.920 --> 00:20:19.160 derivative of V with respect to theta equals zero, amongst 295 00:20:19.599 --> 00:20:23.359 other minor theorems. Here given, I may mention the proposition 296 00:20:23.440 --> 00:20:27.119 that the kinetic energy imparted by given impulses to a 297 00:20:27.160 --> 00:20:32.240 material system under given constraints is a maximum, and the 298 00:20:32.240 --> 00:20:36.640 principle of least action. All the analysis is so elegant 299 00:20:36.680 --> 00:20:40.440 that Sir William Rowan Hamilton said the work could be 300 00:20:40.559 --> 00:20:45.279 only described as a scientific poem. It may be interesting 301 00:20:45.359 --> 00:20:48.359 to note that Legrange remarked that the mechanics was really 302 00:20:48.440 --> 00:20:51.880 a branch of pure mathematics, analogous to a geometry of 303 00:20:51.920 --> 00:20:56.839 four dimensions, namely the time and the three coordinates of 304 00:20:56.880 --> 00:20:59.799 the point in space. And it is said that he 305 00:21:00.000 --> 00:21:03.559 he prided himself that from the beginning to the end 306 00:21:03.599 --> 00:21:07.160 of the work there was not a single diagram. At first, 307 00:21:07.160 --> 00:21:09.359 no printer could be found who had published the book. 308 00:21:09.480 --> 00:21:13.240 Be Legendre at last persuaded a Paris firm to undertake it, 309 00:21:13.759 --> 00:21:17.599 and it was issued under his supervision in seventeen eighty eight. 310 00:21:18.319 --> 00:21:22.319 In seventeen eighty seven, Frederick died, and Lagrange, who had 311 00:21:22.359 --> 00:21:26.839 found the climate of Berlin trying, gladly accepted the offer 312 00:21:26.880 --> 00:21:30.680 of Louis the sixteenth to migrate to Paris. He received 313 00:21:30.880 --> 00:21:34.720 similar invitations from Spain and Naples. In France, he was 314 00:21:34.799 --> 00:21:38.559 received with every mark of distinction, and special apartments in 315 00:21:38.680 --> 00:21:42.799 the Louverre were prepared for his reception. For the first 316 00:21:42.880 --> 00:21:46.079 two years of his residence here he was seized with 317 00:21:46.200 --> 00:21:50.079 an attack of melancholy, and even the printed copy of 318 00:21:50.160 --> 00:21:53.000 his mechanique on which he had worked for a quarter 319 00:21:53.000 --> 00:21:56.039 of a century, lay for more than two years unopened 320 00:21:56.079 --> 00:22:00.240 on his desk. Curiosity as to the results so the 321 00:22:00.279 --> 00:22:03.839 French Revolution first stirred him out of his lethargy, a 322 00:22:04.119 --> 00:22:08.200 curiosity which soon turned to alarm as the revolution developed. 323 00:22:08.920 --> 00:22:12.640 It was about the same time, seventeen ninety two, that 324 00:22:12.880 --> 00:22:17.160 the unaccountable sadness of his life and timidity moved the 325 00:22:17.759 --> 00:22:21.000 compassion of a young girl, who insisted on marrying him 326 00:22:21.519 --> 00:22:25.200 and proved a devoted wife to whom he became warmly attached. 327 00:22:26.279 --> 00:22:29.839 Although the decree of October seventeen ninety three, which ordered 328 00:22:29.880 --> 00:22:34.359 all foreigners to leave France, specially exempted him by name. 329 00:22:35.680 --> 00:22:38.759 He was preparing to escape when he was offered the 330 00:22:38.799 --> 00:22:42.720 presidency of the Commission for the Reform of Weights and Measures. 331 00:22:43.400 --> 00:22:46.599 The choice of the units finally selected was largely due 332 00:22:46.640 --> 00:22:50.640 to him, and it was mainly owing to his influence 333 00:22:50.680 --> 00:22:54.480 that the decimal subdivision was accepted by the Commission of 334 00:22:54.599 --> 00:22:59.200 seventeen ninety nine. The general idea of the decimal system 335 00:22:59.319 --> 00:23:03.039 was taken from a work by Thomas Williams entitled Method 336 00:23:03.119 --> 00:23:08.000 for Fixing a Universal Standard for Weights and Measures, published 337 00:23:08.000 --> 00:23:11.960 in London in seventeen eighty eight. This almost unknown writer 338 00:23:12.160 --> 00:23:17.680 has hardly received the credit due to his suggestion. Though 339 00:23:17.759 --> 00:23:21.799 Lagrange had determined to escape from France while there was 340 00:23:21.880 --> 00:23:25.559 yet time, he was never in any danger, and the 341 00:23:25.559 --> 00:23:30.400 different revolutionary governments and at a later time Napoleon loaded 342 00:23:30.480 --> 00:23:34.319 him with honors and distinctions. A striking testimony to the 343 00:23:34.359 --> 00:23:37.319 respect in which he was held was shewn in seventeen 344 00:23:37.359 --> 00:23:40.920 ninety six when the French Commissary in Italy ordered to 345 00:23:40.960 --> 00:23:44.640 attend in full state on Lagrange's father and tender the 346 00:23:44.680 --> 00:23:48.119 congratulations of the Republic on the achievements of his son, 347 00:23:48.680 --> 00:23:52.200 who had done honor to all mankind by his genius, 348 00:23:52.240 --> 00:23:54.880 and whom it was a special glory of Piedmont to 349 00:23:54.960 --> 00:23:59.240 have produced. In seventeen ninety five, Lagrange was appointed to 350 00:23:59.279 --> 00:24:03.759 a mathematic chair at the newly established Echol Noomal which 351 00:24:03.920 --> 00:24:08.319 only enjoyed a brief existence of four months. His lectures 352 00:24:08.359 --> 00:24:12.880 here were quite elementary and contain nothing of any special importance. 353 00:24:13.160 --> 00:24:16.720 But they were published because the professors had to pledge 354 00:24:16.759 --> 00:24:20.599 themselves to the representatives of the people and to each other, 355 00:24:21.119 --> 00:24:24.680 neither to read nor to repeat from memory, and the 356 00:24:24.759 --> 00:24:28.759 discourses were ordered to be taken down in shorthand in 357 00:24:28.960 --> 00:24:32.319 order to enable the deputies to see how the professors 358 00:24:32.319 --> 00:24:37.680 acquitted themselves. On the establishment of the Ecole Polytechnique in 359 00:24:37.839 --> 00:24:42.240 seventeen ninety seven, Lagrange was made a professor, and his 360 00:24:42.359 --> 00:24:45.440 lectures there are described by mathematicians who had the good 361 00:24:45.519 --> 00:24:48.799 fortune to be able to attend them, as almost perfect, 362 00:24:48.960 --> 00:24:52.519 both in form and matter. Beginning with the merest elements, 363 00:24:52.559 --> 00:24:55.759 he led his hearers on until almost unknown to themselves, 364 00:24:55.839 --> 00:24:59.640 they were themselves extending the bounds of the subject. Above all, 365 00:24:59.680 --> 00:25:02.799 he pressed on his pupils the advantage of always using 366 00:25:02.880 --> 00:25:07.599 general methods expressed in a symmetrical notation. His lectures on 367 00:25:07.640 --> 00:25:11.200 the differential calculus form the basis of his Theire de 368 00:25:11.440 --> 00:25:16.440 Functio Analytique, which was published in seventeen ninety seven. This 369 00:25:16.599 --> 00:25:19.039 work is the extension of an idea contained in a 370 00:25:19.079 --> 00:25:22.559 paper he had sent to the Berlin Memoirs in seventeen 371 00:25:22.640 --> 00:25:26.039 seventy two, and its object is to substitute for the 372 00:25:26.079 --> 00:25:29.480 differential calculus as a group of theorems based on the 373 00:25:29.519 --> 00:25:34.119 development of algebraic functions in series. A somewhat similar method 374 00:25:34.200 --> 00:25:38.119 had been previously used by John Landon in his Residual Analysis, 375 00:25:38.160 --> 00:25:42.559 published in London in seventeen fifty eight. Lagrange believed that 376 00:25:42.640 --> 00:25:46.200 he could thus get rid of those difficulties connected with 377 00:25:46.240 --> 00:25:49.920 the use of infinitely large or infinitely small quantities which 378 00:25:49.960 --> 00:25:54.279 philosophers professed to see in the usual treatment of the 379 00:25:54.279 --> 00:25:59.200 differential calculus. The book is divided into three parts. Of these, 380 00:25:59.640 --> 00:26:03.160 the first treats of the general theory of functions and 381 00:26:03.279 --> 00:26:06.920 gives an algebraic proof of Taylor's theorem, the validity of 382 00:26:06.960 --> 00:26:10.119 which is, however, to open the question. The second deals 383 00:26:10.119 --> 00:26:14.880 with applications to geometry, and the third with applications to mechanics. 384 00:26:15.319 --> 00:26:18.559 Another treatise on the same lines was his lessence Le 385 00:26:18.680 --> 00:26:23.359 Calcure de function, issued in eighteen o four. These works 386 00:26:23.400 --> 00:26:26.359 may be considered as the starting point for the researches 387 00:26:26.400 --> 00:26:30.799 of Koshi and Jacoby. At a later period, Lagrange reverted 388 00:26:30.839 --> 00:26:34.440 to the use of infinitesimals in preference to the founding 389 00:26:34.519 --> 00:26:38.119 the differential calculus on a study of algebraic forms, and 390 00:26:38.240 --> 00:26:40.960 in the preface to the second edition of the Mechanique, 391 00:26:41.240 --> 00:26:44.759 which was issued in eighteen eleven, he justifies their use 392 00:26:44.799 --> 00:26:48.079 and concludes by saying that when we have grasped the 393 00:26:48.160 --> 00:26:52.359 spirit of the infinitesimal method, and have verified the exactness 394 00:26:52.359 --> 00:26:55.759 of its results, either by the geometrical method of prime 395 00:26:55.799 --> 00:27:01.039 and ultimate ratios or by the analytical method of derived functions, 396 00:27:01.480 --> 00:27:05.759 we may employ infinitely small quantities as a sure and 397 00:27:05.920 --> 00:27:11.400 valuable means of shortening and simplifying our proofs. His resolution 398 00:27:11.480 --> 00:27:15.680 des equasion nu meric, published in seventeen ninety eight, was 399 00:27:15.759 --> 00:27:19.720 also the fruit of his lectures at the Polytechnic. In 400 00:27:19.759 --> 00:27:23.400 this he gives the method of approximating to the real 401 00:27:23.480 --> 00:27:27.279