WEBVTT 1 00:00:01.240 --> 00:00:04.799 Chapter seventeen, Part two of A short account of the 2 00:00:04.919 --> 00:00:10.439 History of mathematics. This is a LibriVox recording. All LibriVox 3 00:00:10.480 --> 00:00:14.320 recordings are in the public domain. For more information or 4 00:00:14.359 --> 00:00:19.519 to volunteer, please visit LibriVox dot org. This is a 5 00:00:19.559 --> 00:00:23.359 reading by Paul King PJK dot scripts dot mit dot 6 00:00:23.559 --> 00:00:27.359 ed U forward slash p k J. A Short account 7 00:00:27.399 --> 00:00:30.879 of the history of mathematics by W. W. Rowse Ball, 8 00:00:31.559 --> 00:00:35.960 Chapter seventeen, Libinitz and the Mathematicians of the first half 9 00:00:36.000 --> 00:00:40.679 of the eighteenth century. Part two, the development of analysis 10 00:00:40.719 --> 00:00:45.000 on the continent. Leaving for a moment the English mathematicians 11 00:00:45.000 --> 00:00:48.159 of the first half of the eighteenth century, we come 12 00:00:48.200 --> 00:00:52.439 to a number of continental writers who barely escape mediocrity, 13 00:00:53.039 --> 00:00:55.359 and to whom it will be necessary to devote but 14 00:00:55.439 --> 00:01:00.840 few words. Their writings mark the steps by which analytics, geometry, 15 00:01:01.320 --> 00:01:04.840 and the differential and integral calculus were perfected and made 16 00:01:04.840 --> 00:01:09.159 familiar to mathematicians. Nearly all of them were pupils of 17 00:01:09.239 --> 00:01:12.920 one or other of the two elder Bernoullis, and they 18 00:01:12.920 --> 00:01:16.359 were so nearly contemporaries that it is difficult to arrange 19 00:01:16.400 --> 00:01:21.920 them chronologically. The most eminent of them are Cramer, Degois 20 00:01:22.159 --> 00:01:29.599 de Montemret, Fagagnon Lo Pital, NiCoT, parent Riccouti, Sarin and 21 00:01:29.760 --> 00:01:37.319 Verignon lo Pital, Guillon, Francois Antoine lo Pital. Marquis de 22 00:01:37.439 --> 00:01:41.439 Saint Mes born at Paris in sixteen sixty one and 23 00:01:41.519 --> 00:01:45.840 died there on February second, seventeen o four, was among 24 00:01:45.879 --> 00:01:49.599 the earliest pupils of John Bernoulli, who in sixteen ninety 25 00:01:49.599 --> 00:01:53.920 one spent some months at Lopital's house in Paris for 26 00:01:53.959 --> 00:01:58.840 the purpose of teaching them the new calculus. It seems strange, 27 00:01:58.959 --> 00:02:01.640 but it is substantial true that a knowledge of the 28 00:02:01.680 --> 00:02:05.599 infinitesimal calculus and the power of using it was then 29 00:02:05.680 --> 00:02:10.280 confined to Newton, Leibnitz and the two elder Bernoulli's, and 30 00:02:10.400 --> 00:02:13.960 it will be noticed that they were the only mathematicians 31 00:02:14.280 --> 00:02:18.039 who solve the more difficult problems then proposed as challenges. 32 00:02:20.280 --> 00:02:23.520 There was at the time no textbook on the subject, 33 00:02:23.879 --> 00:02:27.039 and the credit of putting together the first treatise, which 34 00:02:27.039 --> 00:02:29.680 explained the principles and use of the method, is due 35 00:02:29.680 --> 00:02:33.680 to Lopitale. It was published in sixteen ninety six under 36 00:02:33.680 --> 00:02:38.840 the title Annelis des infinitemon petit. This contains a partial 37 00:02:38.879 --> 00:02:43.360 investigation of the limiting value of the ratio of functions which, 38 00:02:43.439 --> 00:02:47.639 for a certain value of the variable take the indeterminate 39 00:02:47.759 --> 00:02:52.719 form zero over zero, a problem solved by John Bernoulli 40 00:02:52.800 --> 00:02:56.520 in seventeen o four. This work had a wide circulation. 41 00:02:57.159 --> 00:03:00.960 It brought the differential notation into universally use in France 42 00:03:01.439 --> 00:03:04.479 and helped to make it generally known in Europe. A 43 00:03:04.520 --> 00:03:09.319 supplement containing a similar treatment of the integral calculus, together 44 00:03:09.360 --> 00:03:12.680 with the editions of the differential calculus which had been 45 00:03:12.719 --> 00:03:17.000 made in the following half century, was published at Paris 46 00:03:17.039 --> 00:03:20.759 seventeen fifty four to seventeen fifty six by L. A. 47 00:03:21.240 --> 00:03:26.120 De Bourganville. Lopitale took part in most of the challenges 48 00:03:26.159 --> 00:03:30.439 by Leibniz, the Bernoullis, and other continental mathematicians of the time. 49 00:03:30.759 --> 00:03:34.840 In particular, he gave a solution of the burchistochrone and 50 00:03:34.919 --> 00:03:38.360 investigated the form of the solid of least resistance of 51 00:03:38.400 --> 00:03:42.599 which Newton and the Principia had stated the result. He 52 00:03:42.680 --> 00:03:46.000 also wrote a treatise on analytical conics, which was published 53 00:03:46.000 --> 00:03:49.159 in seventeen o seven and for nearly a century was 54 00:03:49.199 --> 00:03:55.479 deemed a standard work. On the subject. Verignon Pierre Verignon 55 00:03:56.000 --> 00:03:59.159 born at Caine in sixteen fifty four and died in 56 00:03:59.280 --> 00:04:03.560 Paris on dad December twenty second, seventeen twenty two was 57 00:04:03.599 --> 00:04:08.159 an intimate friend of Newton, Leibniz and the Bernoullis, and 58 00:04:08.199 --> 00:04:12.199 after Lo Pitale, was the earliest and most powerful advocate 59 00:04:12.240 --> 00:04:15.719 in France of the use of the differential calculus. He 60 00:04:15.800 --> 00:04:19.160 realized the necessity of obtaining a test for examining the 61 00:04:19.199 --> 00:04:25.040 convergency of series, but the analytical difficulties were beyond his powers. 62 00:04:25.800 --> 00:04:29.240 He simplified the proofs of many of the leading propositions 63 00:04:29.240 --> 00:04:33.480 in mechanics, and in sixteen eighty seven recast the treatment 64 00:04:33.519 --> 00:04:37.160 of the subject, basing it on the composition of forces. 65 00:04:37.639 --> 00:04:41.079 His works were published at Paris in seventeen twenty five. 66 00:04:41.759 --> 00:04:46.240 For further details see the erouge by b. De Fontinelles, Paris, 67 00:04:46.399 --> 00:04:53.120 seventeen sixty six. Des Montemur Pierre Raymond de Montemur born 68 00:04:53.199 --> 00:04:57.399 at Paris on October twenty seventh, sixteen seventy eight and 69 00:04:57.480 --> 00:05:02.000 died there on October seventh, seventeen nineteen was interested in 70 00:05:02.040 --> 00:05:07.319 the subject of finite differences. He determined in seventeen thirteen, 71 00:05:07.600 --> 00:05:10.480 the sum of n terms of a finite series of 72 00:05:10.519 --> 00:05:14.120 the form en choose one times a plus, en choose 73 00:05:14.160 --> 00:05:17.360 two times delta a plus, en choose three times delta 74 00:05:17.399 --> 00:05:20.879 squared day, and so on, a theorem which seems to 75 00:05:20.920 --> 00:05:26.160 have been independently rediscovered by Christian Goldbach in seventeen eighteen. 76 00:05:27.680 --> 00:05:31.759 Nicole Francois Nicole, who was born at Paris on December 77 00:05:31.800 --> 00:05:36.279 twenty third, sixteen eighty three and died there on January eighteenth, 78 00:05:36.399 --> 00:05:40.600 seventeen fifty eight, was the first to publish a systematic 79 00:05:40.720 --> 00:05:45.600 treatise on the finite differences. Taylor had regarded the differential 80 00:05:45.639 --> 00:05:50.399 coefficient as the ratio of two infinitesimal differences as the 81 00:05:50.439 --> 00:05:54.120 limiting value of the ratio of two finite differences, a 82 00:05:54.199 --> 00:05:57.680 method which is still used by many English writers, though 83 00:05:57.720 --> 00:06:01.199 it has been generally abandoned on the DVD continent, and 84 00:06:01.279 --> 00:06:03.800 thus had been led to give a sketch of the 85 00:06:03.839 --> 00:06:09.639 subject in his Methodists, published in seventeen fifteen. Nicoles Treete 86 00:06:09.800 --> 00:06:14.439 de Quercuur de di France Finice was published in seventeen seventeen. 87 00:06:15.560 --> 00:06:18.439 It is a well arranged book and contains rules both 88 00:06:18.480 --> 00:06:22.800 for forming differences and for affecting the summation of given series. 89 00:06:23.680 --> 00:06:27.120 Besides this, in seventeen o six he wrote a book 90 00:06:27.160 --> 00:06:32.639 on Roulette's especially spherical epicycloids, and in seventeen twenty nine 91 00:06:32.759 --> 00:06:36.480 and seventeen thirty one he published memoirs on Newton's essay 92 00:06:36.519 --> 00:06:42.639 on curves of the third degree. Parents Antoine Parents born 93 00:06:42.720 --> 00:06:47.199 at Paris on September sixteenth, sixteen sixty six, and died 94 00:06:47.199 --> 00:06:51.879 there on September twenty six, seventeen sixteen. Wrote in seventeen 95 00:06:52.000 --> 00:06:56.439 hundred on analytical geometry of three dimensions. His works were 96 00:06:56.439 --> 00:06:59.959 collected and published in three volumes at Paris in Sea 97 00:07:00.000 --> 00:07:07.120 seventeen thirteen. Sarrans Joseph Sarran, born at Corteizan in sixteen 98 00:07:07.160 --> 00:07:10.759 fifty nine and died at Paris on December twenty ninth, 99 00:07:10.800 --> 00:07:14.160 seventeen thirty seven, was the first to show how the 100 00:07:14.199 --> 00:07:17.839 tangents at the multiple points of curves could be determined 101 00:07:17.879 --> 00:07:22.639 by analysis. De Nois Jean Paul de GUIs d'ar Mauves 102 00:07:22.920 --> 00:07:26.639 was born at Carcasson in seventeen thirteen and died at 103 00:07:26.639 --> 00:07:31.040 Paris on June second, seventeen eighty five. He published in 104 00:07:31.120 --> 00:07:34.839 seventeen forty a work on analytical geometry, in which he 105 00:07:34.920 --> 00:07:38.360 applied it without the aid of the differential calculus, to 106 00:07:38.360 --> 00:07:42.439 find the tangents, asymptotes, and various singular points of an 107 00:07:42.480 --> 00:07:47.639 algebraical curve, and he further shoed how singular points and 108 00:07:47.759 --> 00:07:52.240 isolated loops were affected by conical projection. He gave the 109 00:07:52.240 --> 00:07:55.279 proof of Descartes's rule of signs, which is to be 110 00:07:55.360 --> 00:07:58.639 found in most modern works. It is not clear whether 111 00:07:58.720 --> 00:08:02.000 Descartes ever proved its own strictly, and Newton seems to 112 00:08:02.000 --> 00:08:08.040 have regarded it as obvious. Kramer Gabrielle Kramer born at 113 00:08:08.040 --> 00:08:11.519 Geneva in seventeen o four and died at Bagnol in 114 00:08:11.600 --> 00:08:16.920 seventeen fifty two, was professor at Geneva. The work by 115 00:08:16.959 --> 00:08:21.040 which he is best known is his Treatise on Algebraical Curves, 116 00:08:21.519 --> 00:08:25.000 published in seventeen fifty, which, as far as it goes, 117 00:08:25.120 --> 00:08:29.120 is fairly complete. It contains the earliest demonstration that a 118 00:08:29.199 --> 00:08:32.720 curve of the nth degree is in general determined if 119 00:08:33.559 --> 00:08:36.399 one half end times en plus three points on it 120 00:08:36.480 --> 00:08:40.759 to be given. This work is still sometimes read. Besides this, 121 00:08:40.919 --> 00:08:43.480 he edited that the works of the two elder Bernoullis, 122 00:08:43.600 --> 00:08:46.639 and wrote on the physical cause of the spheroidal shapes 123 00:08:46.679 --> 00:08:50.360 of the planets and the motion of their apses seventeen 124 00:08:50.480 --> 00:08:57.039 thirty and on Newton's treatment of cubic curves seventeen forty six. Riccati, 125 00:08:57.320 --> 00:09:02.159 Jacopo Francesco Count Riccati, born at Venice on May twenty eighth, 126 00:09:02.240 --> 00:09:07.159 sixteen seventy six and died at Treves on April fifteenth, 127 00:09:07.240 --> 00:09:11.120 seventeen fifty four, did a great deal to disseminate a 128 00:09:11.240 --> 00:09:15.799 knowledge of the Newtonian philosophy in Italy. Besides the equation 129 00:09:15.960 --> 00:09:19.840 known by his name, certain cases of which he succeeded 130 00:09:19.879 --> 00:09:24.039 in integrating, he discussed the question of the possibility of 131 00:09:24.159 --> 00:09:28.440 lowering the order of a given differential equation. His works 132 00:09:28.480 --> 00:09:32.360 were published at Treves in four volumes in seventeen fifty eight. 133 00:09:33.039 --> 00:09:35.799 He had two sons who wrote on several minor points 134 00:09:35.840 --> 00:09:40.320 connected with the integral calculus and differential equations, and applied 135 00:09:40.320 --> 00:09:45.799 the calculus to several mechanical questions. These were Vincenzo, who 136 00:09:45.960 --> 00:09:48.840 was born in seventeen o seven and died in seventeen 137 00:09:48.919 --> 00:09:52.600 seventy five, and Giordano who was born in seventeen o 138 00:09:52.759 --> 00:09:59.000 nine and died in seventeen ninety. Fagnano Giulio Carlo, Count 139 00:09:59.039 --> 00:10:05.320 Fagnano and Marquise de Toci, born at Senegaglia on December sixth, 140 00:10:05.480 --> 00:10:09.240 sixteen eighty two and died on September twenty sixth, seventeen 141 00:10:09.360 --> 00:10:12.240 sixty six, may be said to have been the first 142 00:10:12.240 --> 00:10:16.200 writer who directed attention to the theory of elliptic functions. 143 00:10:16.720 --> 00:10:21.200 Failing to rectify the ellipse or hyperbolea, Faniano attempted to 144 00:10:21.279 --> 00:10:26.080 determine arcs whose difference should be rectifiable. He also pointed 145 00:10:26.080 --> 00:10:30.240 out the remarkable analogy existing between the integrals which represent 146 00:10:30.320 --> 00:10:32.519 the arc of a circle and the arc of a 147 00:10:32.600 --> 00:10:37.600 Lemnas gate. Finally, he proved the formula pie equals two 148 00:10:37.679 --> 00:10:41.440 times i times the log of one minus i divided 149 00:10:41.480 --> 00:10:44.519 by one plus i, where I stands for the square 150 00:10:44.600 --> 00:10:48.440 root of minus one. His works were collected and published 151 00:10:48.480 --> 00:10:54.320 in two volumes in Persero in seventeen fifty. It was 152 00:10:54.360 --> 00:10:58.200 inevitable that some mathematicians should object to methods of analysis 153 00:10:58.200 --> 00:11:02.200 founded on the infinitesimal calcis. The most prominent of these 154 00:11:02.840 --> 00:11:08.399 were Viviani de la Here and roll chronologically they come here, 155 00:11:09.120 --> 00:11:12.000 but they flourished half a century after the date to 156 00:11:12.080 --> 00:11:19.080 which their writings properly belong. Viviani Vincenzo Viviani, a pupil 157 00:11:19.120 --> 00:11:24.279 of Galileo and Torricelli, born at Florence on April fifth, 158 00:11:24.320 --> 00:11:28.279 sixteen twenty two, and died there on September twenty second. 159 00:11:28.519 --> 00:11:32.240 Seventeen o three, brought out in sixteen fifty nine a 160 00:11:32.320 --> 00:11:36.799 restoration of the lost Book of Apollonius on conic sections, 161 00:11:36.840 --> 00:11:40.039 and in seventeen oh one a restoration of the work 162 00:11:40.080 --> 00:11:44.799 of Erestius. He explained in sixteen seventy seven how an 163 00:11:44.840 --> 00:11:48.000 angle could be trisected by the aid of an echilateral 164 00:11:48.080 --> 00:11:52.720 hyperbola or the conchoid. In sixteen ninety two he proposed 165 00:11:52.720 --> 00:11:56.840 the problem to construct four windows in a hemispherical vault 166 00:11:57.000 --> 00:12:01.000 so that the remainder of the surface can be accurate determined, 167 00:12:01.639 --> 00:12:06.080 a celebrated problem of which analytical solutions were given by 168 00:12:06.120 --> 00:12:12.200 Wallace Leibnitz, David Gregory, and James Bernoulli. De la Here 169 00:12:12.440 --> 00:12:16.080 Philippe de la Here or Lahire born in Paris on 170 00:12:16.200 --> 00:12:20.159 March eighteen, sixteen forty and died there on April twenty first, 171 00:12:20.320 --> 00:12:25.399 seventeen nineteen, wrote on graphical methods sixteen seventy three, on 172 00:12:25.440 --> 00:12:30.240 the conic sections sixteen eighty five, a treatise on epicycloids 173 00:12:30.240 --> 00:12:34.320 sixteen ninety four, one on roulettes seventeen o two, and 174 00:12:34.480 --> 00:12:39.360 lastly another on conchoids seventeen oh eight. His works on 175 00:12:39.440 --> 00:12:42.879 conic sections and epicycloids were founded on the teaching of 176 00:12:42.960 --> 00:12:47.960 de Rogues, whose favorite pupil he was. He also translated 177 00:12:48.000 --> 00:12:52.600 the essay of Muscopolis on magic squares and collected many 178 00:12:52.639 --> 00:12:56.360 of the theorems on them which were previously known. This was 179 00:12:56.399 --> 00:13:01.679 published in seventeen oh five. Roule Michel Rol born at 180 00:13:01.720 --> 00:13:06.240 Ambert on April twenty first, sixteen fifty two and died 181 00:13:06.279 --> 00:13:10.679 in Paris on November eighth, seventeen nineteen. Wrote an algebra 182 00:13:10.919 --> 00:13:14.399 in sixteen eighty nine which contains the theorem on the 183 00:13:14.440 --> 00:13:17.600 position of roots of an equation, which is known by 184 00:13:17.639 --> 00:13:21.559 his name. He published in sixteen ninety six a treatise 185 00:13:21.559 --> 00:13:26.559 on the solution of equations whether determinate or indeterminate, and 186 00:13:26.720 --> 00:13:30.799 he produced several other minor works. He taught that differential 187 00:13:30.840 --> 00:13:36.679 calculus was nothing but a collection of ingenious fallacies so far, 188 00:13:37.279 --> 00:13:39.519 no one of the school of Leibnitz and the two 189 00:13:39.559 --> 00:13:43.639 elder Bernoullis had shewn any exceptional ability. But by the 190 00:13:43.679 --> 00:13:47.240 action of a number of second rate writers, the methods 191 00:13:47.240 --> 00:13:50.799 and language of analytical geometry and the differential calculus had 192 00:13:50.799 --> 00:13:54.519 become well known. By about seventeen forty. The close of 193 00:13:54.519 --> 00:13:57.600 this school is marked by the appearance of Cleirot de 194 00:13:57.759 --> 00:14:03.159 l'embert and Danielle Bernoulli. Their lives overlap the period considered 195 00:14:03.200 --> 00:14:06.000 in the next chapter. But though it is difficult to 196 00:14:06.080 --> 00:14:08.879 draw a sharp dividing line which shall separate by a 197 00:14:08.919 --> 00:14:12.879 definite date, the mathematicians there considered from those whose writings 198 00:14:12.879 --> 00:14:16.159 are discussed in this chapter, I think that on the whole, 199 00:14:16.240 --> 00:14:19.600 the works of these three writers are best treated here. 200 00:14:20.559 --> 00:14:26.159 Claiaut Alexis Claude Clat was born at Paris on May thirteenth, 201 00:14:26.240 --> 00:14:31.480 seventeen thirteen and died there on May seventeenth, seventeen sixty five. 202 00:14:32.159 --> 00:14:35.000 He belongs to the small group of children who, though 203 00:14:35.039 --> 00:14:40.480 of exceptional precaucity, survive and maintain their powers when grown up. 204 00:14:40.919 --> 00:14:43.000 As early as the age of twelve, he wrote a 205 00:14:43.039 --> 00:14:47.159 memoir on four geometrical curves, but his first important work 206 00:14:47.279 --> 00:14:50.879 was a treatise on tortuous curves, published when he was eighteen, 207 00:14:51.519 --> 00:14:55.559 a work which first procured for him immediate admission to 208 00:14:55.600 --> 00:14:59.039 the French Academy. In seventeen thirty one, he gave a 209 00:14:59.080 --> 00:15:02.679 demonstration of the fact noted by Newton that all curves 210 00:15:02.720 --> 00:15:06.279 of the third order were projections of one of five parabolas. 211 00:15:07.440 --> 00:15:12.240 In seventeen forty one Clairot went on a scientific expedition 212 00:15:12.320 --> 00:15:15.440 to measure the length of a meridian degree on the 213 00:15:15.440 --> 00:15:19.840 Earth's surface, and on his return in seventeen forty three 214 00:15:19.960 --> 00:15:24.279 he publishes THII de la figure de lataire. This is 215 00:15:24.320 --> 00:15:28.000 founded on a paper by Maclaurin, where it has been 216 00:15:28.039 --> 00:15:31.360 shown that a mass of homogeneous fluids set in rotation 217 00:15:31.519 --> 00:15:34.480 about a line through its center of mass, would, under 218 00:15:34.519 --> 00:15:37.399 the mutual attraction of its particles, take the form of 219 00:15:37.399 --> 00:15:43.000 a spheroid. This work of cliat treated of heterogeneous spheroids 220 00:15:43.320 --> 00:15:46.679 and contains the proof of his formula for the accelerating 221 00:15:46.879 --> 00:15:50.759 effect of gravity in a place of latitude L namely, 222 00:15:51.399 --> 00:15:57.080 G equals big g multiplied by one plus five halves 223 00:15:57.240 --> 00:16:02.799 m minus epsilon multiplied by sience squaredel where g is 224 00:16:02.799 --> 00:16:06.720 the value of equatorial gravity, m the ratio of the 225 00:16:06.759 --> 00:16:11.399 centrifugal force to gravity at the equator, and epsilon the 226 00:16:11.440 --> 00:16:15.440 epelicity of a meridian section of the Earth. In eighteen 227 00:16:15.519 --> 00:16:18.840 forty nine, Professor Stokes shewed that the same result was 228 00:16:18.919 --> 00:16:22.519 true whatever was the interior construction or density of the Earth, 229 00:16:23.080 --> 00:16:27.559 provided the surface was a spheroid of equilibrium of small 230 00:16:27.720 --> 00:16:33.279 ellipticity impressed by it the power of geometry, as shewn 231 00:16:33.360 --> 00:16:38.440 in the writing of Newton and McLaurin. Clarat abandoned analysis, 232 00:16:38.679 --> 00:16:42.240 and his next work, The Theore de l Lune, published 233 00:16:42.240 --> 00:16:46.600 in seventeen fifty two, is strictly Newtonian in character. This 234 00:16:46.759 --> 00:16:50.480 contains the explanation of the motion of the apse, which 235 00:16:50.519 --> 00:16:55.759 had previously puzzled astronomers, and which Clarat had at first 236 00:16:55.759 --> 00:16:58.879 deemed so inexplicable that he was on the point of 237 00:16:58.879 --> 00:17:01.480 publishing a new hype apothesis as to the law of 238 00:17:01.480 --> 00:17:05.400 attraction when it occurred to him to carry the approximation 239 00:17:05.559 --> 00:17:08.880 to the third order, and he thereupon found that the 240 00:17:08.960 --> 00:17:14.200 result was in accordance with observations. This was followed in 241 00:17:14.279 --> 00:17:18.960 seventeen fifty four by some lunar tables. Claiat subsequently wrote 242 00:17:19.200 --> 00:17:22.519 various papers on the orbit of the Moon and on 243 00:17:22.599 --> 00:17:25.559 the motion of the comets as affected by the perturbation 244 00:17:25.720 --> 00:17:29.400 of the planets, particularly on the path of Halley's comet. 245 00:17:31.920 --> 00:17:36.519 His growing popularity in society hindered his scientific work on Gaget, 246 00:17:36.599 --> 00:17:41.359 says beaussu a de supei a de viller entarnaier parent 247 00:17:41.480 --> 00:17:48.720 gudvive poulefame Voulain alais brasier, a stravau denaire a padire 248 00:17:48.880 --> 00:17:56.960 pou la sante enfan villellages de Saint Conden de Lambert. 249 00:17:58.079 --> 00:18:01.039 Jean de ren d' lambert was born at Paris on 250 00:18:01.079 --> 00:18:06.240 November sixteenth, seventeen seventeen, and died there on October twenty ninth, 251 00:18:06.319 --> 00:18:11.680 seventeen thirteen. He was the illegitimate child of the Chevalier Destouche, 252 00:18:12.160 --> 00:18:14.759 being abandoned by his mother on the steps of the 253 00:18:14.759 --> 00:18:18.839 little church of Saint Jean Leran, which then nestled under 254 00:18:18.839 --> 00:18:21.759 the great porch of nottal Dame, he was taken to 255 00:18:21.839 --> 00:18:26.440 the parish commissary, who, following the usual practice in such cases, 256 00:18:26.759 --> 00:18:30.119 gave him the Christian name of Jean laran I. Do 257 00:18:30.200 --> 00:18:33.200 not know by what title. He subsequently assumed the right 258 00:18:33.279 --> 00:18:36.759 to prefix DA to his name. He was boarded out 259 00:18:36.799 --> 00:18:39.839 by the parish with the wife of a glazier in 260 00:18:39.920 --> 00:18:43.000 a small way of business who lived near the cathedral, 261 00:18:43.480 --> 00:18:45.759 and here he seems to have found a real home, 262 00:18:46.200 --> 00:18:49.599 though a very humble one. His father appears to have 263 00:18:49.640 --> 00:18:52.000 looked after him and paid for his going to a 264 00:18:52.160 --> 00:18:56.720 school where he obtained a fair mathematical education. An essay 265 00:18:56.720 --> 00:19:00.079 written in seventeen thirty eight on the integral calculus, and 266 00:19:00.119 --> 00:19:04.160 another in seventeen forty on ducks and drakes or ricochets, 267 00:19:04.279 --> 00:19:07.400 attracted some attention, and in the same year he was 268 00:19:07.440 --> 00:19:11.480 elected a member of the French Academy. This was probably 269 00:19:11.559 --> 00:19:14.680 due to the influence of his father. It is to 270 00:19:14.759 --> 00:19:18.640 his credit that he absolutely refused to leave his adopted mother, 271 00:19:19.079 --> 00:19:21.799 with whom he continued to live until her death in 272 00:19:21.880 --> 00:19:26.519 seventeen fifty seven. It cannot be said that she sympathized 273 00:19:26.519 --> 00:19:28.960 with his success, for at the height of his fame 274 00:19:29.599 --> 00:19:33.440 she remonstrated with him for wasting his talents on such work. 275 00:19:34.400 --> 00:19:41.200 Vuniserreemeh kun Philissov said she e keskun Philissov Situnfu quissitour 276 00:19:41.279 --> 00:19:46.920 Monte padas Seville pour compard de luis LUs kilne serre plus. 277 00:19:48.480 --> 00:19:51.960 Nearly all his mathematical works were produced within the years 278 00:19:52.079 --> 00:19:56.880 seventeen forty three to seventeen fifty four. The first of 279 00:19:57.000 --> 00:20:00.839 these was his Treete de di neemik public in seventeen 280 00:20:00.960 --> 00:20:04.359 forty three, in which he enunciates the principle known by 281 00:20:04.440 --> 00:20:08.160 his name, namely that the internal forces of inertia i e. 282 00:20:08.359 --> 00:20:12.519 The forces which resist acceleration must be equal and opposite 283 00:20:12.559 --> 00:20:16.519 to the forces which produce the acceleration. This is a 284 00:20:16.640 --> 00:20:20.119 particular case of Newton's second reading of his third Law 285 00:20:20.200 --> 00:20:23.440 of motion, but the full consequence of it had not 286 00:20:23.599 --> 00:20:28.559 been realized previously. The application of this principle enables us 287 00:20:28.640 --> 00:20:32.680 to obtain the differential equations of motion of any rigid system. 288 00:20:33.400 --> 00:20:37.960 In seventeen forty four, Dalembert publishest t rete de EquiLibre 289 00:20:38.119 --> 00:20:41.759 in de movement de fluids, in which he applies his 290 00:20:41.920 --> 00:20:46.559 principle to fluids. This led to partial differential equations, which 291 00:20:46.599 --> 00:20:50.160 he was then unable to solve. In seventeen forty five 292 00:20:50.279 --> 00:20:52.799 he developed that part of the Subjects, which dealt with 293 00:20:52.920 --> 00:20:56.359 the motion of air in his Thera General de Van, 294 00:20:57.279 --> 00:21:01.559 and this again led him to partial difference equations. A 295 00:21:01.680 --> 00:21:05.559 second edition of this in seventeen forty six was dedicated 296 00:21:05.640 --> 00:21:09.359 to Frederick the Great of Prussia, and procured an invitation 297 00:21:09.559 --> 00:21:13.119 to Berlin and the offer of a pension. He declined 298 00:21:13.200 --> 00:21:17.759 the former, but subsequently, after some pressing, pocketed his pride 299 00:21:18.000 --> 00:21:22.240 and the latter. In seventeen forty seven he applied the 300 00:21:22.319 --> 00:21:26.079 differential calculus to the problem of a vibrating string, and 301 00:21:26.200 --> 00:21:30.640 again arrived at a partial differential equation. His analysis had 302 00:21:30.720 --> 00:21:33.359 three times brought him to an equation of the form 303 00:21:33.920 --> 00:21:37.319 the second derivative of you with respect to T equals 304 00:21:37.359 --> 00:21:41.079 the second derivative of U with respect to X, and 305 00:21:41.279 --> 00:21:44.319 he now succeeded in shewing that it was satisfied by 306 00:21:44.480 --> 00:21:50.480 U equals Fi of x plus T plus Sci of 307 00:21:51.039 --> 00:21:55.839 x minus t, Where Pi and Si are arbitrary functions. 308 00:21:56.319 --> 00:21:59.480 It may be interesting to give his solution, which was 309 00:21:59.559 --> 00:22:02.920 published in the Transactions of the Berlin Academy for seventeen 310 00:22:03.079 --> 00:22:07.319 forty seven. He begins by saying that if the partial 311 00:22:07.440 --> 00:22:10.359 derivative of U with respect to X be denoted by P, 312 00:22:10.680 --> 00:22:13.079 and the partial derivative of u with respect to t 313 00:22:13.319 --> 00:22:19.200 by q, then du equals pdx plus qdt. But by 314 00:22:19.240 --> 00:22:22.799 the given equation, the partial derivative of q with respect 315 00:22:22.880 --> 00:22:25.799 to T equals the partial derivative of p with respect 316 00:22:25.880 --> 00:22:30.519 to x, and therefore pdt plus qdx is also an 317 00:22:30.599 --> 00:22:36.799 exact differential. Denote it by DV. Therefore, dv equals pdt 318 00:22:37.000 --> 00:22:41.839 plus qdx. Hence du plus dv equals pdx plus qdt 319 00:22:42.079 --> 00:22:46.799 plus pdt plus qdx equals p plus q multiplied by 320 00:22:46.880 --> 00:22:52.000 dx plus dt, and du minus dv equals pdx plus 321 00:22:52.119 --> 00:22:57.759 qdt minus. The quantity pdt plus qdx equals the quantity 322 00:22:58.079 --> 00:23:01.960 p minus q multiplied the quantity d x minus d T. 323 00:23:03.079 --> 00:23:05.599 Thus U plus v must be a function of x 324 00:23:05.680 --> 00:23:09.279 plus t, and U minus v must be a function 325 00:23:09.359 --> 00:23:12.640 of x minus t. We may therefore put U plus 326 00:23:12.759 --> 00:23:17.400 v equals two times fi of x plus t, and 327 00:23:17.920 --> 00:23:21.319 U minus v equals to ci of x minus t, 328 00:23:22.559 --> 00:23:26.720 Hence U equals fi of x plus t plus ci 329 00:23:26.880 --> 00:23:30.559 of x minus t. De Lambert added that the conditions 330 00:23:30.599 --> 00:23:33.920 of the physical problem of a vibrating string demand that 331 00:23:34.039 --> 00:23:37.279 when x equals zero, U should vanish for all values 332 00:23:37.319 --> 00:23:42.480 of T, Hence, identically fi of T plus Si of 333 00:23:42.599 --> 00:23:47.240 negative T equals zero. Assuming that both functions can be 334 00:23:47.359 --> 00:23:51.119 expanded in integral powers of t, this requires that they 335 00:23:51.160 --> 00:23:55.839 should contain only odd powers. Hence si of negative T 336 00:23:56.119 --> 00:23:59.240 equals negative, Fi of T equals fi of negative T. 337 00:24:00.200 --> 00:24:05.480 Therefore U equals pi of x plus T plus fi 338 00:24:05.640 --> 00:24:09.200 of x minus t. Euler now took the matter up 339 00:24:09.279 --> 00:24:11.359 and shewed that the equation of the form of the 340 00:24:11.440 --> 00:24:15.000 string was the second derivative of U with respect to 341 00:24:15.119 --> 00:24:19.640 T equals a squared multiplied by the second derivative of 342 00:24:19.799 --> 00:24:23.359 U with respect to x, and that the general integral 343 00:24:23.960 --> 00:24:29.599 was U equals pi of x minus at plus pi 344 00:24:29.839 --> 00:24:34.519 of x plus at, where pi and pi are arbitrary functions. 345 00:24:35.400 --> 00:24:39.599 The chief remaining contributions of de Lambert to mathematics were 346 00:24:39.680 --> 00:24:44.559 on physical astronomy, especially on the precession of the equinoxes, 347 00:24:45.119 --> 00:24:48.799 and on variations of the obliquity of the ecliptic. These 348 00:24:48.880 --> 00:24:52.599 were collected on his Sistem des Mondes, published in three 349 00:24:52.720 --> 00:24:57.400 volumes in seventeen fifty four. During the latter part of 350 00:24:57.480 --> 00:25:00.000 his life he was mainly occupied with the d S 351 00:25:00.119 --> 00:25:04.519 Great French Encyclopedia. For this he wrote the introduction, and 352 00:25:04.720 --> 00:25:09.039 numerous philosophical and mathematical articles. The best are those on 353 00:25:09.200 --> 00:25:14.279 geometry and on probabilities. His style is brilliant but not polished, 354 00:25:14.680 --> 00:25:18.680 and faithfully reflects his character, which was bold, honest and frank. 355 00:25:19.319 --> 00:25:23.200 He defended a severe criticism which he had offered on 356 00:25:23.359 --> 00:25:28.119 some mediocre work by the remark Jeans musettin Seville Connuier, 357 00:25:28.759 --> 00:25:31.680 and with his dislike of sickophants and boors, it is 358 00:25:31.759 --> 00:25:35.000 not surprising that during his life he had more enemies 359 00:25:35.079 --> 00:25:42.880 than friends. End of section twenty eight. Recording by Paul King, Oakville, Ontario, 360 00:25:44.039 --> 00:25:46.400 p j K dot scripts dot m I T dot 361 00:25:46.480 --> 00:25:48.839 e edu forward slash p kJ