WEBVTT 1 00:00:01.280 --> 00:00:04.160 Chapter eighteen, Part two of A short account of the 2 00:00:04.280 --> 00:00:09.560 history of mathematics. This is a LibriVox recording. All LibriVox 3 00:00:09.599 --> 00:00:13.160 recordings are in the public domain. For more information or 4 00:00:13.199 --> 00:00:17.399 to volunteer, please visit LibriVox dot org. This is a 5 00:00:17.480 --> 00:00:21.120 reading by Paul King PJK dot scripts thought I might 6 00:00:21.359 --> 00:00:25.640 dot edu forward slash p kJ. A short account of 7 00:00:25.679 --> 00:00:30.519 the history of mathematics by W. W. Rousbaul Chapter eighteen, 8 00:00:31.199 --> 00:00:33.840 Libaniz and the Mathematicians of the first half of the 9 00:00:33.880 --> 00:00:40.399 eighteenth century, Part three. Laplace Pierre Simon la Place was 10 00:00:40.439 --> 00:00:44.359 born at Beaumont an Ogue in Normandy on March twenty third, 11 00:00:44.479 --> 00:00:48.200 seventeen forty nine, and died at Paris on March fifth, 12 00:00:48.240 --> 00:00:51.719 eighteen twenty seven. He was the son of a small 13 00:00:51.799 --> 00:00:55.759 cottager or perhaps a farm laborer, and owed his education 14 00:00:56.240 --> 00:00:59.240 to the interest excited in some wealthy neighbors by his 15 00:00:59.359 --> 00:01:04.040 abilities and engaging presence. Very little is known of his 16 00:01:04.159 --> 00:01:07.920 early years, for when he became distinguished, he held himself 17 00:01:07.959 --> 00:01:10.920 aloof from both his relatives and from those who had 18 00:01:10.959 --> 00:01:15.680 assisted him. A similar pettiness of character marked many of 19 00:01:15.719 --> 00:01:19.599 his actions. It would seem that from a pupil he 20 00:01:19.680 --> 00:01:23.079 became an usher in the school at Beaumont, but having 21 00:01:23.120 --> 00:01:26.599 procured a letter of introduction to Lette de Lambert, he 22 00:01:26.640 --> 00:01:29.920 went to Paris to push his fortune. A paper on 23 00:01:29.959 --> 00:01:33.959 the principles of mechanics excited de Lambert's interest, and on 24 00:01:34.000 --> 00:01:37.359 his recommendations, a place in the military school was offered 25 00:01:37.400 --> 00:01:43.359 to Laplace. Secure of a competency, Laplace now threw himself 26 00:01:43.400 --> 00:01:47.040 onto the original research, and in the next seventeen years 27 00:01:47.079 --> 00:01:51.599 seventeen seventy one to seventeen eighty seven he produced much 28 00:01:51.640 --> 00:01:56.760 of his original work in astronomy. This commenced with a 29 00:01:56.799 --> 00:02:00.480 memoir read before the French Academy in seventeen seven three 30 00:02:00.959 --> 00:02:04.280 which he shewed that the planetary motions were stable and 31 00:02:04.439 --> 00:02:07.239 carried the proof as far as the cubes of the 32 00:02:07.319 --> 00:02:13.159 eccentricities and inclinations. This was followed by several papers on 33 00:02:13.280 --> 00:02:18.960 points in the integral calculus, finite differences, differential equations, and astronomy. 34 00:02:20.879 --> 00:02:24.039 During the years seventeen eighty four to seventeen eighty seven 35 00:02:24.439 --> 00:02:29.280 he produced some memoirs of exceptional power. Prominent among those 36 00:02:29.879 --> 00:02:34.199 is one read in seventeen eighty four and reprinted in 37 00:02:34.240 --> 00:02:37.639 the third volume of the mechanic Celeste, in which he 38 00:02:37.719 --> 00:02:41.159 completely determined the attraction of a spheroid on a particle 39 00:02:41.199 --> 00:02:45.080 outside it. This is memorable for the introduction into an 40 00:02:45.000 --> 00:02:50.000 analysis of spherical harmonics or Laploce's coefficients, and also for 41 00:02:50.080 --> 00:02:53.599 the development of the use of potential, a name first 42 00:02:53.639 --> 00:02:57.919 given by Green in eighteen twenty eight. If the coordinates 43 00:02:57.919 --> 00:03:01.120 of two points be r meure u omega and our 44 00:03:01.240 --> 00:03:06.479 prime uprime omega prime, and our prime is not less 45 00:03:06.520 --> 00:03:10.000 than r, the reciprocal of the distance between them can 46 00:03:10.039 --> 00:03:13.800 be expanded in powers of R divided by our prime, 47 00:03:14.560 --> 00:03:20.080 and the respected coefficients are Laplace coefficients. Their utility arises 48 00:03:20.120 --> 00:03:22.400 from the fact that every function of the coordinates of 49 00:03:22.439 --> 00:03:25.319 a point on a sphere can be expanded in a 50 00:03:25.479 --> 00:03:28.479 series of them. It should be stated that the similar 51 00:03:28.560 --> 00:03:32.000 coefficients for space of two dimensions, together with some of 52 00:03:32.039 --> 00:03:35.840 their properties, had been previously been given by Legendre in 53 00:03:35.879 --> 00:03:39.120 a paper sent to the French Academy in seventeen eighty three. 54 00:03:39.800 --> 00:03:42.599 Legentre had good reason to complain of the way in 55 00:03:42.639 --> 00:03:47.280 which he was treated in this manner. This paper is 56 00:03:47.319 --> 00:03:51.599 also remarkable for the development of the idea of the potential, 57 00:03:52.120 --> 00:03:56.120 which was appropriated from Lagrange, who had used it in 58 00:03:56.120 --> 00:03:59.879 his memoirs of seventeen seventy three seventeen seventy seven n. 59 00:04:00.199 --> 00:04:05.560 Seventeen eighty. Laplastiwd that the potential always satisfies the differential 60 00:04:05.599 --> 00:04:09.680 equation Dell squared V equals the second partial derivative of 61 00:04:09.719 --> 00:04:13.400 the voltage with respect to X plus the second partial 62 00:04:13.439 --> 00:04:16.720 derivative of the voltage with respect to y plus the 63 00:04:16.759 --> 00:04:20.040 second partial derivative of the voltage with respect to z 64 00:04:20.240 --> 00:04:24.000 equals zero, And on this result his subsequent work on 65 00:04:24.040 --> 00:04:28.319 attractions was based. The quantity Dell squared v had been 66 00:04:28.439 --> 00:04:31.720 termed the concentration of V, and its value at any 67 00:04:31.759 --> 00:04:34.759 point indicates the excess of the value of V over 68 00:04:34.800 --> 00:04:39.439 its mean value in the neighborhood of the point. Laplace's equation, 69 00:04:40.199 --> 00:04:43.560 or the more general form Dell squared v equals negative 70 00:04:43.639 --> 00:04:47.959 four pi row appears in all branches of mathematical physics. 71 00:04:48.879 --> 00:04:51.639 According to some writers, this follows at once from the 72 00:04:51.720 --> 00:04:56.279 fact that Dell squared is a scalar operator, or the 73 00:04:56.360 --> 00:05:00.360 equation may represent analytically some general law of nature which 74 00:05:00.360 --> 00:05:04.079 has not yet been reduced to words, or possibly I 75 00:05:04.160 --> 00:05:08.279 have sometimes thought it might be regarded by a Kantian 76 00:05:08.680 --> 00:05:12.360 as the outward sign of one of the necessary forms 77 00:05:12.399 --> 00:05:18.040 through which all phenomena are perceived. This memoir was followed 78 00:05:18.040 --> 00:05:23.240 by another unplanetary Inequalities, which was presented in three sections 79 00:05:23.279 --> 00:05:27.800 in seventeen eighty four, seventeen eighty five, and seventeen eighty six. 80 00:05:28.519 --> 00:05:31.879 This deals mainly with the explanation of the great inequality 81 00:05:32.000 --> 00:05:37.199 of Jupiter and Saturn. Laplace chewed by general considerations that 82 00:05:37.240 --> 00:05:41.279 the mutual action of two planets could never largely affect 83 00:05:41.279 --> 00:05:46.319 the eccentricities and inclinations of their orbits, and that the 84 00:05:46.360 --> 00:05:50.160 peculiarities of the Jovian system were due to the near 85 00:05:50.240 --> 00:05:55.399 approach to commensurability of the mean motions of Jupiter and Saturn. 86 00:05:56.040 --> 00:05:59.560 Further developments of these theorems on planetary motion were given 87 00:05:59.600 --> 00:06:03.240 in his two memoirs of seventeen eighty eight and seventeen 88 00:06:03.319 --> 00:06:07.920 eighty nine. It was on these data that de Lambre 89 00:06:08.439 --> 00:06:13.480 computed his astronomical tables. The year seventeen eighty seven was 90 00:06:13.480 --> 00:06:18.000 rendered memorable by Laplace's explanation and analysis of the relation 91 00:06:18.639 --> 00:06:22.560 between the lunar acceleration and the secular changes in the 92 00:06:22.600 --> 00:06:27.680 eccentricity of the Earth's orbit. This investigation completed the proof 93 00:06:27.759 --> 00:06:30.680 of the stability of the whole Solar system on the 94 00:06:30.720 --> 00:06:33.720 assumption that it consists of a collection of rigid bodies. 95 00:06:34.959 --> 00:06:38.079 All the memoirs above alluded to were presented to the 96 00:06:38.120 --> 00:06:42.040 French Academy, and they are printed in the memoir Presente 97 00:06:42.040 --> 00:06:46.319 pals di versavant. Laplace now set himself the task to 98 00:06:46.360 --> 00:06:49.160 write a work which should offer a complete solution of 99 00:06:49.199 --> 00:06:52.879 the great mechanical problem presented by the Solar system and 100 00:06:53.000 --> 00:06:57.360 bring theory to coincide so closely with observation that empirical 101 00:06:57.399 --> 00:07:01.000 equations should no longer find a place in astronomical tables. 102 00:07:01.560 --> 00:07:06.399 The result is embodied in the Exposition System DuMond and 103 00:07:06.519 --> 00:07:11.199 the Mechanique Arrest. The former was published in seventeen ninety 104 00:07:11.240 --> 00:07:14.480 six and gives a general explanation of the phenomena with 105 00:07:14.560 --> 00:07:18.439 a summary of the history of astronomy, but omits all details. 106 00:07:19.079 --> 00:07:24.519 The nebular hypothesis was here enunciated. According to this hypothesis, 107 00:07:25.040 --> 00:07:28.439 the Solar system had spin evolved from a globular mass 108 00:07:28.439 --> 00:07:32.279 of incandescent gas rotating around an axis through its center 109 00:07:32.360 --> 00:07:36.360 of mass as it cooled, this mass contracted and successive 110 00:07:36.439 --> 00:07:39.879 rings broke off from its outer edge. These rings, in 111 00:07:39.959 --> 00:07:43.720 their turn cooled and finely condensed into the planets, while 112 00:07:43.759 --> 00:07:46.800 the sun represents the central core, which was still left. 113 00:07:47.839 --> 00:07:52.000 Certain corrections required by modern science were added by M. Rosch, 114 00:07:52.639 --> 00:07:56.399 and recently the theory has been discussed critically by R. Wolf. 115 00:07:57.480 --> 00:08:01.600 The arguments against the hypothesis are some in phase Regin 116 00:08:01.720 --> 00:08:07.040 Dumonde Paris, eighteen eighty four, where an ingenious modification of 117 00:08:07.079 --> 00:08:10.720 the hypothesis is proposed, by which the author attempts to 118 00:08:10.800 --> 00:08:14.759 explain the peculiarities of the axial rotation of Neptune and 119 00:08:14.920 --> 00:08:18.839 Uranus and the retrograde motion of the satellites of the 120 00:08:18.920 --> 00:08:23.720 latter planet. Perhaps modern opinion is inclined to attribute the 121 00:08:23.759 --> 00:08:27.399 separation of the various members of the planetary system to 122 00:08:27.519 --> 00:08:31.879 tidal friction rather than to the successive separation and condensation 123 00:08:32.039 --> 00:08:35.360 of nebular rings. But the subject is one of great 124 00:08:35.399 --> 00:08:39.840 difficulty according to the rule published by Titus of Wittenberg 125 00:08:40.000 --> 00:08:44.240 in seventeen sixty six, but generally known as Bode's law 126 00:08:45.159 --> 00:08:48.039 from the fact that attention was called to it by 127 00:08:48.399 --> 00:08:52.960 Johann elt Bode in seventeen seventy eight. The distances of 128 00:08:53.000 --> 00:08:55.480 the planets from the Sun are nearly in the ratio 129 00:08:55.519 --> 00:08:58.919 of the numbers zero plus four, three plus four, six 130 00:08:58.960 --> 00:09:02.039 plus four to five, twelve plus four, et cetera, and 131 00:09:02.080 --> 00:09:04.759 the n minus tooth term being two to the power 132 00:09:04.919 --> 00:09:08.480 n times three with four added. It would be an 133 00:09:08.480 --> 00:09:11.399 interesting fact if this could be deduced from either the 134 00:09:11.480 --> 00:09:14.399 nebula or the title hypothesis, but so far as I 135 00:09:14.399 --> 00:09:17.480 am aware, only one serious attempt to do so has 136 00:09:17.519 --> 00:09:20.919 been made, and the conclusion was that the law was 137 00:09:20.960 --> 00:09:24.360 not sufficiently exact to be more than a convenient means 138 00:09:24.399 --> 00:09:29.159 of remembering the general result. The substance of Laplace's hypothesis 139 00:09:29.200 --> 00:09:32.399 had been published by Kant in seventeen fifty five in 140 00:09:32.480 --> 00:09:37.080 his Algamina nat Chogaschista, but it is probable that Laplace 141 00:09:37.240 --> 00:09:40.960 was not aware of this. This historical summary procured for 142 00:09:41.039 --> 00:09:44.320 its author the honor of admission to the forty of 143 00:09:44.360 --> 00:09:48.120 the French Academy. It is commonly esteemed one of the 144 00:09:48.159 --> 00:09:52.639 masterpieces of French literature, though it is not altogether reliable 145 00:09:52.720 --> 00:09:57.000 for the later periods of which it treats. The full 146 00:09:57.039 --> 00:10:01.120 analytical discussion of the solar system is given in Mechaniques Arrest, 147 00:10:01.600 --> 00:10:05.039 published in five volumes, Volumes one and two in seventeen 148 00:10:05.120 --> 00:10:09.720 ninety nine, volume three in eighteen oh two, Volume four 149 00:10:09.840 --> 00:10:13.720 in eighteen oh five, and volume five in eighteen twenty five. 150 00:10:15.159 --> 00:10:18.919 An analysis of the contents is given in the English Cyclopedia. 151 00:10:19.919 --> 00:10:23.240 The first two volumes contain methods for calculating the motions 152 00:10:23.240 --> 00:10:27.559 of the planets, determining their figures, and resolving title problems. 153 00:10:28.559 --> 00:10:32.000 The third and fourth volumes contain the application of these methods, 154 00:10:32.039 --> 00:10:37.240 and also several astronomical tables. The fifth volume is mainly historical, 155 00:10:37.720 --> 00:10:42.240 but it gives as appendices the results of Laplace's latest researches. 156 00:10:42.879 --> 00:10:46.559 Laplace's own investigations embodied in it are so numerous and 157 00:10:46.639 --> 00:10:49.679 valuable that it is regrettable to have to add that 158 00:10:49.720 --> 00:10:53.960 many results are appropriated from writers with scanty or no acknowledgment, 159 00:10:54.600 --> 00:10:57.519 and the conclusions, which have been described as the organized 160 00:10:57.519 --> 00:11:01.399 result of a century of patent toli oil are generally 161 00:11:01.440 --> 00:11:04.399 mentioned as if they were due to Laplace. And it 162 00:11:04.440 --> 00:11:06.639 is said, for I have not looked up into the 163 00:11:06.639 --> 00:11:11.120 matter myself, that the praise which he lavishes on Newton 164 00:11:11.200 --> 00:11:14.559 and Clairout is only the cloak under which he appropriates 165 00:11:14.600 --> 00:11:18.759 the work of other less known writers. The Mechanique seirest 166 00:11:18.919 --> 00:11:23.279 is by no means easy reading. Billotte, who assisted Laplace 167 00:11:23.360 --> 00:11:26.720 in revising it for the press, says that Laplace himself 168 00:11:26.799 --> 00:11:29.799 was frequently unable to recover the details in the chain 169 00:11:29.879 --> 00:11:33.879 of reasoning, and if satisfied that the conclusions were correct, 170 00:11:34.360 --> 00:11:39.000 he was content to insert the constantly recurring formula iritiz avoir. 171 00:11:40.399 --> 00:11:43.279 The best tribute to the excellency of the work is 172 00:11:43.320 --> 00:11:46.639 that it left very little for his successors to add. 173 00:11:47.519 --> 00:11:50.360 It is not only the translation of the Principia into 174 00:11:50.360 --> 00:11:54.080 the language of the differential calculus, but it also completes 175 00:11:54.120 --> 00:11:57.000 part of which Newton had been unable to fill in 176 00:11:57.039 --> 00:12:01.919 the details. M Tisseran's recent work may be considered as 177 00:12:01.960 --> 00:12:08.159 a continuation of Laplace's treatise. Laplace went in state to 178 00:12:08.200 --> 00:12:11.519 beg Napoleon to accept a copy of his work, and 179 00:12:11.600 --> 00:12:15.519 the following account of the interview is well authenticated and 180 00:12:15.600 --> 00:12:19.320 so characteristic of all parties. Concerned that I quoted in full. 181 00:12:20.360 --> 00:12:23.159 Someone had told Napoleon that the book contained no mention 182 00:12:23.240 --> 00:12:26.320 of the name of God. Napoleon, who was fond of 183 00:12:26.320 --> 00:12:30.639 putting embarrassing questions received it with the remark, Monsieur la Place, 184 00:12:31.080 --> 00:12:33.320 they tell me you have written this large book on 185 00:12:33.440 --> 00:12:36.200 the system of the universe and have never even mentioned 186 00:12:36.240 --> 00:12:42.799 its creator. Laplace, who though most supple of politicians, was 187 00:12:42.840 --> 00:12:45.679 as stiff as a martyr on every point of his philosophy, 188 00:12:46.320 --> 00:12:51.799 drew himself up and answered bluntly, Geneve Pavessois de setre pothees. 189 00:12:51.919 --> 00:12:57.039 La Napoleon, greatly amused, told this reply to La Grange, 190 00:12:57.039 --> 00:13:02.799 who exclaimed, ah, sit in beer, hypothe expric bo could chous. 191 00:13:03.759 --> 00:13:08.240 In eighteen twelve, Laplace issued is theory analytique de probabilities. 192 00:13:09.159 --> 00:13:12.240 The theory is stated to be only common sense expressed 193 00:13:12.279 --> 00:13:16.000 in mathematical language. The method of estimating the ratio of 194 00:13:16.039 --> 00:13:19.080 the number of favorable cases to the whole number of 195 00:13:19.279 --> 00:13:23.120 possible cases has been indicated by Laplace in a paper 196 00:13:23.159 --> 00:13:27.320 written in seventeen seventy nine. It consists in treating the 197 00:13:27.399 --> 00:13:30.639 successive values of any function as the coefficients in the 198 00:13:30.679 --> 00:13:34.639 expansion of another function with reference to a different variable. 199 00:13:35.399 --> 00:13:39.120 The latter is therefore called the generating function of the former. 200 00:13:40.679 --> 00:13:45.519 Laplace then choose how by means of interpolation, these coefficients 201 00:13:45.559 --> 00:13:50.519 may be determined from the generating function. Next, he attacks 202 00:13:50.519 --> 00:13:54.320 the converse problem, and from the coefficiency he finds the 203 00:13:54.440 --> 00:13:57.799 generating function. This is offered by the solution of an 204 00:13:57.799 --> 00:14:03.279 equation in finite differences. The method is cumbersome, and, in 205 00:14:03.399 --> 00:14:07.679 consequence of the increased power of analysis, is now rarely used. 206 00:14:09.399 --> 00:14:12.600 A summary of Laplace's reasoning is given in the article 207 00:14:12.679 --> 00:14:21.840 on probability in his Encyclopedia Metropolitana. This treatise includes an 208 00:14:21.840 --> 00:14:24.960 exposition of the method of lease squares, which is a 209 00:14:25.000 --> 00:14:29.200 remarkable testimony to Laplace's command over the process of analysis. 210 00:14:30.679 --> 00:14:34.039 The method of lease squares for the combination of numerous 211 00:14:34.080 --> 00:14:38.159 observations had been given empirically by Gauss and Legendre, but 212 00:14:38.440 --> 00:14:41.840 the fourth chapter of this work contains a formal proof 213 00:14:41.840 --> 00:14:44.320 of it, on which the whole of the theory of 214 00:14:44.519 --> 00:14:48.600 errors has been since spaced. This was effected only by 215 00:14:48.639 --> 00:14:52.799 a most intricate analysis specially invented for the purpose, but 216 00:14:53.039 --> 00:14:55.320 the form in which it is presented as so meager 217 00:14:55.399 --> 00:14:59.120 and unsatisfactory that, in spite of the uniform accuracy of 218 00:14:59.120 --> 00:15:02.320 the results, it was at one time question whether Laplace 219 00:15:02.360 --> 00:15:05.519 had actually gone through the difficult work he so briefly 220 00:15:05.600 --> 00:15:11.279 and often incorrectly indicates. In eighteen nineteen, Laplace published a 221 00:15:11.320 --> 00:15:15.240 popular account of his work on probability. This book bears 222 00:15:15.240 --> 00:15:19.399 the same relation to his theory the probabilities that the 223 00:15:19.440 --> 00:15:25.320 Si stem DuMond does to the mechanique Cereste. Amongst the 224 00:15:25.399 --> 00:15:29.039 minor discoveries of Laplace in pure mathematics, I may mention 225 00:15:29.200 --> 00:15:34.080 his discussion, simultaneously with Van Dermonde, of the general theory 226 00:15:34.159 --> 00:15:38.000 of determinants in seventeen seventy two, his proof that every 227 00:15:38.039 --> 00:15:40.720 equation of an even degree must have at least one 228 00:15:40.799 --> 00:15:45.159 real quadratic factor, his reduction of the solution of linear 229 00:15:45.240 --> 00:15:49.879 differential equations to definite integrals, and his solution to the 230 00:15:49.960 --> 00:15:54.679 linear partial equation of the second order. He was also 231 00:15:54.960 --> 00:15:57.960 the first to consider the difficult problems involved in the 232 00:15:58.000 --> 00:16:01.480 equations of mixed differences, and to prove that the solution 233 00:16:01.600 --> 00:16:04.639 of an equation and finite differences of the first degree 234 00:16:04.879 --> 00:16:08.240 and the second order might always be obtained in the 235 00:16:08.279 --> 00:16:13.240 form of a continued fraction. Besides these original discoveries, he 236 00:16:13.360 --> 00:16:17.039 determined in his theory of probabilities the values of a 237 00:16:17.120 --> 00:16:20.720 number of the more common definite integrals, and in the 238 00:16:20.759 --> 00:16:24.279 same book gave the general proof of the theorem enunciated 239 00:16:24.320 --> 00:16:28.080 by Lagrange for the development of any implicit function in 240 00:16:28.159 --> 00:16:34.120 a series by means of differential coefficients. In theoretical physics, 241 00:16:34.159 --> 00:16:38.159 the theory of capillary attraction is due to Laplace, who 242 00:16:38.159 --> 00:16:43.159 accepted the idea propounded by Hawkesby in the Philosophical Transactions 243 00:16:43.200 --> 00:16:46.799 for seventeen o nine that the phenomenon was due to 244 00:16:46.840 --> 00:16:51.240 a force of attraction which was insensible at sensible distances. 245 00:16:51.879 --> 00:16:54.080 The part which deals with the action of a solid 246 00:16:54.120 --> 00:16:56.840 on a liquid and the mutual actions of two liquids 247 00:16:56.960 --> 00:17:00.799 was not worked out thoroughly, but ultimately was completed by Gauss. 248 00:17:01.200 --> 00:17:04.480 Newman later filled in a few details. In eighteen sixty two, 249 00:17:04.599 --> 00:17:08.599 Lord Kelvin Sir William Thompson shoot that if we assume 250 00:17:08.680 --> 00:17:12.240 the molecular constitution of matter, the laws of capillary attraction 251 00:17:12.400 --> 00:17:16.759 can be deduced from the Newtonian law of gravitation. Laplace, 252 00:17:16.839 --> 00:17:19.839 in eighteen sixteen was the first to point out explicitly 253 00:17:19.880 --> 00:17:23.839 why Newton's theory of vibratory motion gave an incorrect value 254 00:17:23.880 --> 00:17:27.640 for the velocity of sound. The actual velocity is greater 255 00:17:27.720 --> 00:17:31.000 than that calculated by Newton in consequence of the heat 256 00:17:31.039 --> 00:17:34.640 developed by the sudden compression of air, which increases the 257 00:17:34.680 --> 00:17:40.920 elasticity and therefore the velocity of sound transmitted. Laplace's investigations 258 00:17:40.960 --> 00:17:44.680 and practical physics were confined to those carried on by 259 00:17:44.799 --> 00:17:48.759 him jointly with Lovoisier in the year seventeen eighty two 260 00:17:49.119 --> 00:17:52.640 and seventeen eighty four on the specific heat of various bodies. 261 00:17:54.039 --> 00:17:57.400 Laplace seems to have regarded analysis as merely a means 262 00:17:57.480 --> 00:18:01.559 of attacking physical problems, though though the ability with which 263 00:18:01.559 --> 00:18:06.640 he invented the necessary analysis is almost phenomenal, As long 264 00:18:06.680 --> 00:18:09.880 as his results were true, he took but little trouble 265 00:18:09.960 --> 00:18:12.400 to explain the steps by which he arrived at them. 266 00:18:12.920 --> 00:18:16.799 He never studied elegance or symmetry in his processes, and 267 00:18:17.279 --> 00:18:20.319 it was sufficient for him if he could, by any 268 00:18:20.319 --> 00:18:25.000 means solve the particular question he was discussing. In these 269 00:18:25.039 --> 00:18:31.319 respects he stands in marked contrast to his great contemporary Lagrange. 270 00:18:32.559 --> 00:18:35.720 It would have been well for Laplace's reputation if he 271 00:18:35.759 --> 00:18:39.200 had been content with his scientific work, But above all 272 00:18:39.279 --> 00:18:43.759 things he coveted social fame. The skill and rapidity with 273 00:18:43.799 --> 00:18:47.519 which he managed to change his politics as occasion required 274 00:18:48.000 --> 00:18:50.680 would be amusing if they had not been so servile. 275 00:18:51.359 --> 00:18:58.079 As Napoleon's power increased, Laplace abandoned his Republican principles, which, 276 00:18:58.440 --> 00:19:01.319 since they had faithfully read reflected the opinions of the 277 00:19:01.359 --> 00:19:06.000 party in power, had themselves gone through numerous changes, and 278 00:19:06.079 --> 00:19:09.000 begged for the First Council to give him the post 279 00:19:09.000 --> 00:19:13.680 of Minister of the Interior. Napoleon, who desired the support 280 00:19:13.759 --> 00:19:17.559 of men of science, accepted the offer, but a little 281 00:19:17.640 --> 00:19:22.400 less than six weeks saw the close of Laplace's political career. 282 00:19:23.039 --> 00:19:27.319 Napoleon's memorandum on the subject as as follows. Geometre de 283 00:19:27.480 --> 00:19:33.599 premier range la Place natal da asse montre administraturplus cux 284 00:19:33.680 --> 00:19:39.559 mediocre des sent premier travaill news raconneum cunu zetien trompe 285 00:19:40.160 --> 00:19:43.720 la place ne sees santon coun question sus saint vere 286 00:19:43.839 --> 00:19:48.000 table pronte de vieux charche de sabtili te partu nave 287 00:19:48.279 --> 00:19:52.880 cux desi de problematique er porte enfant la sprite's infinima 288 00:19:52.960 --> 00:19:59.400 petit juscud dans la administracion. Although Laplace was expelled from office, 289 00:19:59.599 --> 00:20:03.680 it was desirable to retain his allegiance. He was accordingly 290 00:20:03.839 --> 00:20:06.839 raised to the Senate, and to the third volume of 291 00:20:06.920 --> 00:20:10.000 the Mechanique Celeste he prefixed a note that, of all 292 00:20:10.039 --> 00:20:13.160 the truths therein contained, the most precious to the author 293 00:20:13.759 --> 00:20:17.000 was the declaration he thus made of his devotion toward 294 00:20:17.079 --> 00:20:21.519 the peacemaker of Europe in copies sold after the restoration. 295 00:20:21.960 --> 00:20:25.880 This was struck out in eighteen fourteen. It was evident 296 00:20:25.960 --> 00:20:29.559 that the Empire was falling. Laplace hastened to tender his 297 00:20:29.720 --> 00:20:34.759 services to the Bourbons, and on the restoration was rewarded 298 00:20:34.839 --> 00:20:39.359 with the title of Marquis. The contempt that his more 299 00:20:39.440 --> 00:20:42.960 honest colleagues felt for his conduct in the matter may 300 00:20:43.039 --> 00:20:47.319 be read in the pages of Pauluis Courier. His knowledge 301 00:20:47.400 --> 00:20:50.960 was useful on the numerous scientific commissions on which he served, 302 00:20:51.480 --> 00:20:54.359 and probably accounts for the manner in which his political 303 00:20:54.480 --> 00:20:58.920 insincerity was overlooked. But the pettiness of his character must 304 00:20:59.000 --> 00:21:02.319 not make us forget how great were his services to science. 305 00:21:04.079 --> 00:21:07.279 That Laplace was vain and selfish is not denied by 306 00:21:07.400 --> 00:21:11.279 his warmest admirers. His conduct to the benefactors of his 307 00:21:11.440 --> 00:21:15.720 youth and his political friends was ungrateful and contemptible, while 308 00:21:15.759 --> 00:21:19.680 his appropriation of the results of those who were comparatively 309 00:21:19.799 --> 00:21:24.400 unknown seems to be well established and is absolutely indefensible. 310 00:21:25.119 --> 00:21:28.319 Of those whom he thus treated, three subsequently rose to 311 00:21:28.400 --> 00:21:33.000 distinction le Gentre and Froyer in France and Young in England, 312 00:21:33.640 --> 00:21:36.319 and never forgot the injustice of which they had been 313 00:21:36.359 --> 00:21:39.720 the victims. On the other side, it may be said 314 00:21:40.079 --> 00:21:43.960 that on some questions he shoot independence of character, and 315 00:21:44.200 --> 00:21:48.279 he never concealed his views on religion, philosophy, or science, 316 00:21:48.400 --> 00:21:51.640 however distasteful they might be to the authorities in power. 317 00:21:52.400 --> 00:21:55.039 It should be also added that, towards the close of 318 00:21:55.079 --> 00:21:57.880 his life, and especially to the work of his pupils, 319 00:21:58.440 --> 00:22:02.759 Laplace was both generous and appreciative, and in one case 320 00:22:03.440 --> 00:22:06.400 suppressed a paper of his own in order that a 321 00:22:06.480 --> 00:22:11.119 pupil might have the sole credit of the discovery. Le 322 00:22:11.240 --> 00:22:17.799 Gendre Atrienne Marie Legendre was born at Toulouse on September eighteenth, 323 00:22:18.119 --> 00:22:21.960 seventeen fifty two and died at Paris on January tenth, 324 00:22:22.200 --> 00:22:25.799 eighteen thirty three. The leading events of his life are 325 00:22:25.920 --> 00:22:29.160 very simple, and may be summed up briefly. He was 326 00:22:29.359 --> 00:22:33.400 educated at the Mazarin College in Paris. Appointed the professor 327 00:22:34.200 --> 00:22:37.599 at the Military School in Paris in seventeen seventy seven. 328 00:22:38.640 --> 00:22:41.279 Was a member of the Anglo French Commission of seventeen 329 00:22:41.359 --> 00:22:45.640 eighty seven to connect Greenwich and Paris. Geodetically served on 330 00:22:45.799 --> 00:22:48.880 several of the public commissions from seventeen ninety two to 331 00:22:49.000 --> 00:22:53.240 eighteen ten. Was made a professor at the Normal School 332 00:22:53.680 --> 00:22:57.839 in seventeen ninety five, and subsequently held a few minor 333 00:22:57.920 --> 00:23:03.039 government appointments. The influence of Laplace was steadily exerted against 334 00:23:03.079 --> 00:23:08.160 his obtaining office or public recognition, and Legendre, who was 335 00:23:08.319 --> 00:23:12.400 a timid student, accepted the obscurity to which the hostility 336 00:23:12.480 --> 00:23:16.880 of his colleague condemned him. Le Gentre's analysis is of 337 00:23:16.960 --> 00:23:19.920 a high order of excellence, and is second only to 338 00:23:20.039 --> 00:23:23.400 that produced by Lagrange and Laplace, though it is not 339 00:23:23.559 --> 00:23:27.799 so original. His chief works are his geometry, his Theery 340 00:23:27.960 --> 00:23:32.640 de Numbre, and his calcul integral and his functions elliptique. 341 00:23:33.799 --> 00:23:37.279 These include the results of his various papers on these subjects. 342 00:23:37.920 --> 00:23:41.480 Besides these, he wrote a treatise which gave the rule 343 00:23:41.559 --> 00:23:44.960 for the method of least squares and two groups of memoirs, 344 00:23:45.519 --> 00:23:47.839 one on the theory of attractions and the other on 345 00:23:48.000 --> 00:23:54.079 geodetical operations. The memoirs on attractions are analyzed and discussed 346 00:23:54.440 --> 00:23:58.160 in Todd Hunter's History of the Theory of Attraction. The 347 00:23:58.319 --> 00:24:02.440 earliest of these memoirs, presented in seventeen eighty three, was 348 00:24:02.519 --> 00:24:06.400 on the attraction of spheroids. This contain the introduction of 349 00:24:06.519 --> 00:24:11.680 la Gendre's coefficients, which are sometimes called circular horizonal harmonics, 350 00:24:12.200 --> 00:24:18.119 and which are particular cases of Laplace's coefficients. It also 351 00:24:18.200 --> 00:24:21.160 includes the solution of a problem in which the potential 352 00:24:21.319 --> 00:24:25.559 is used. The second memoir was communicated in seventeen eighty 353 00:24:25.640 --> 00:24:29.079 four and is of the form of equilibrium of a 354 00:24:29.200 --> 00:24:34.200 mass of rotating liquid which is approximately spherical. The third, 355 00:24:34.279 --> 00:24:36.960 written in seventeen eighty six, is on the attraction of 356 00:24:37.160 --> 00:24:41.359 confocal elliptoids. The fourth is on the figure which a 357 00:24:41.440 --> 00:24:45.799 fluid planet would assume and its law of density. His 358 00:24:45.960 --> 00:24:49.160 papers on geodes are three in number and were presented 359 00:24:49.279 --> 00:24:52.839 to the Academy in seventeen eighty seven and seventeen eighty eight. 360 00:24:53.680 --> 00:24:56.440 The most important result is that by which a spherical 361 00:24:56.559 --> 00:24:59.839 triangle may be treated as a plane, provided certain corrections 362 00:25:00.160 --> 00:25:03.680 applied to the angles. In connection with this subject, he 363 00:25:03.799 --> 00:25:08.839 paid considerable attention to geodesics. The method of Lee squares 364 00:25:08.960 --> 00:25:13.200 was enunciated in his Nouvelle Methode, published in eighteen o six, 365 00:25:14.400 --> 00:25:17.680 to which supplements were added in eighteen ten and eighteen twenty. 366 00:25:18.440 --> 00:25:22.720 Gauss independently had arrived at the same result and had 367 00:25:22.839 --> 00:25:26.039 used it in seventeen ninety five, and published it in 368 00:25:26.119 --> 00:25:29.920 the Law of Facility in eighteen o nine. Laplace was 369 00:25:30.039 --> 00:25:32.680 the earlier writer to give a proof of it. This 370 00:25:32.920 --> 00:25:37.200 was in eighteen twelve. On the other books produced by 371 00:25:37.279 --> 00:25:40.519 Le Gendre, the one most widely known is his Ede 372 00:25:40.680 --> 00:25:44.519 Monte Gianmetri, which was published in seventeen ninety four and 373 00:25:44.720 --> 00:25:49.400 was generally adopted on the continent as a substitute for Euclid. 374 00:25:50.160 --> 00:25:54.519 The later editions contain the elements of trigonometry and proofs 375 00:25:54.559 --> 00:26:00.440 of the irrationality of pie and pie squared. Indix on 376 00:26:00.519 --> 00:26:03.319 the difficult question of the theory of parallel lines was 377 00:26:03.359 --> 00:26:06.279 issued in eighteen o three and is bound up with 378 00:26:06.519 --> 00:26:10.799 most of the subsequent editions. Is The Ree de Numbre 379 00:26:11.359 --> 00:26:14.920 was published in seventeen ninety eight and appendices were added 380 00:26:14.960 --> 00:26:18.839 in eighteen sixteen and eighteen twenty five. The third edition, 381 00:26:19.400 --> 00:26:23.079 issued in two volumes in eighteen thirty, includes the results 382 00:26:23.119 --> 00:26:26.079 of his various later papers, and still remains a standard 383 00:26:26.160 --> 00:26:28.960 work on the subject. It may be said that he 384 00:26:29.240 --> 00:26:32.440 here carried the subject as far as was possible by 385 00:26:32.480 --> 00:26:36.680 the application of ordinary algebra, but he did not realize 386 00:26:36.720 --> 00:26:39.599 that it might be regarded as a higher arithmetic and 387 00:26:39.799 --> 00:26:44.359 so form a distinctive subject in mathematics. The law of 388 00:26:44.519 --> 00:26:49.880 quadratic reciprocity, which connects any two odd primes, is first 389 00:26:50.000 --> 00:26:53.960 proved in this book, but the result had been enunciated 390 00:26:54.039 --> 00:26:57.680 in a memoir of seventeen eighty five. Gauss called the 391 00:26:57.759 --> 00:27:02.079 proposition the gem of arithmetic, and no less than six 392 00:27:02.200 --> 00:27:05.160 separate proofs are to be found in his works. The 393 00:27:05.279 --> 00:27:08.440 theorem is as follows. If P be a prime and 394 00:27:08.680 --> 00:27:10.799 N be a prime to P, then we know that 395 00:27:10.920 --> 00:27:13.039 the remainder when the square root of nd to the 396 00:27:13.160 --> 00:27:15.839 power of P minus one is divided by P, is 397 00:27:15.920 --> 00:27:20.759 either plus one or minus one. Legendre denoted this remainder 398 00:27:20.920 --> 00:27:24.720 by N over P. When the remainder is plus one, 399 00:27:24.839 --> 00:27:27.599 it is possible to find a square number which, when 400 00:27:27.680 --> 00:27:30.880 divided by P leaves a remainder N. That is, N 401 00:27:31.079 --> 00:27:34.680