WEBVTT 1 00:00:01.480 --> 00:00:04.960 Chapter seventeen, Part three of A Short Account of the 2 00:00:05.120 --> 00:00:10.039 History of Mathematics. This is a LibriVox recording. All LibriVox 3 00:00:10.080 --> 00:00:13.640 recordings are in the public domain. For more information or 4 00:00:13.679 --> 00:00:19.280 to volunteer, please visit LibriVox dot org. This reading is 5 00:00:19.320 --> 00:00:22.359 by Paul King p J K dot scripts dot MI 6 00:00:22.519 --> 00:00:26.399 t dot ed U forward slash p k J. A 7 00:00:26.480 --> 00:00:29.719 Short Account of the History of Mathematics by W. W. 8 00:00:29.920 --> 00:00:35.320 Rowse Ball. Chapter eighteen Leibniz and the Mathematicians of the 9 00:00:35.320 --> 00:00:42.119 first half of the eighteenth century. Daniel Bernoulli. Daniel Bernoulli, 10 00:00:42.119 --> 00:00:45.200 whose name I mentioned above, and who was by far 11 00:00:45.439 --> 00:00:49.039 the ablest of the younger Bernoulli's, was a contemporary and 12 00:00:49.200 --> 00:00:52.240 intimate friend of Euler, whose works are mentioned in the 13 00:00:52.280 --> 00:00:56.520 next chapter. Daniel Bernoulli was born on February ninth, seventeen 14 00:00:56.600 --> 00:00:59.600 hundred and died at Bail, where he was professor of 15 00:00:59.679 --> 00:01:04.400 natural philosophy, on March seventeenth, seventeen eighty two. He went 16 00:01:04.439 --> 00:01:08.439 to Saint Petersburg in seventeen twenty four as professor of mathematics, 17 00:01:08.719 --> 00:01:11.879 but the roughness of the social life was distasteful to him, 18 00:01:12.159 --> 00:01:14.519 and he was not sorry when a temporary illness in 19 00:01:14.599 --> 00:01:17.439 seventeen thirty three allowed him to plead his health as 20 00:01:17.480 --> 00:01:21.400 an excuse for leaving. He then returned to bail, where 21 00:01:21.480 --> 00:01:26.719 he held successively chairs of medicine, metaphysics, and natural philosophy. 22 00:01:26.760 --> 00:01:32.560 There his earliest mathematical work was on the Excreciones, published 23 00:01:32.560 --> 00:01:35.920 in seventeen twenty four. These contained a theory of the 24 00:01:35.959 --> 00:01:40.200 oscillations of rigid bodies and a solution to the differential 25 00:01:40.239 --> 00:01:44.920 equations proposed by Riccatti. Two years later, he pointed out 26 00:01:45.000 --> 00:01:48.920 for the first time the frequent desirability of resolving a 27 00:01:49.000 --> 00:01:53.560 compound motion into motions of translation and motions of rotation. 28 00:01:54.400 --> 00:01:57.959 His chief work is his hydro Damonique, published in seventeen 29 00:01:58.000 --> 00:02:02.560 thirty eight. It resembled is Lagrange's Mechanique an arritique in 30 00:02:02.680 --> 00:02:06.319 being arranged so that all the results are consequences of 31 00:02:06.359 --> 00:02:11.199 a single principle, namely, in this case, the conservation of energy. 32 00:02:12.120 --> 00:02:14.639 This was followed by a memoir on the theory of 33 00:02:14.719 --> 00:02:18.639 the tides, to which, conjointly with memoirs by Euler and maclaurin, 34 00:02:19.199 --> 00:02:22.479 a prize was awarded by the French Academy. These three 35 00:02:22.560 --> 00:02:26.400 memoirs contain all that was done on the subject between 36 00:02:26.479 --> 00:02:32.039 publication of Newton's Principia and the investigations of Laplace. Bernoulli 37 00:02:32.159 --> 00:02:36.000 also wrote a large number of papers on various mechanical questions, 38 00:02:36.639 --> 00:02:41.159 especially on problems connected with vibrating strings and the solutions 39 00:02:41.199 --> 00:02:44.879 given by Taylor and d' lambert. He is the earliest 40 00:02:44.879 --> 00:02:48.719 writer who attempted to formulate a kinetic theory of gases, 41 00:02:49.319 --> 00:02:52.520 and he applied the idea to explain the law associated 42 00:02:52.719 --> 00:02:58.039 with the names of Boyle and Mariotte, the English mathematicians 43 00:02:58.080 --> 00:03:02.599 of the eighteenth century. I have reserved a notice of 44 00:03:02.680 --> 00:03:06.759 the English mathematicians who succeeded Newton, in order that the 45 00:03:06.800 --> 00:03:09.439 members of the English school may all be treated together. 46 00:03:10.319 --> 00:03:13.639 It was almost a matter of course that the English should, 47 00:03:13.680 --> 00:03:16.919 at least at first have adopted the notation of Newton 48 00:03:17.400 --> 00:03:21.400 in the infinitesimal calculus in preference to that of Libanates, 49 00:03:21.919 --> 00:03:24.919 and the English school would consequently, in any case have 50 00:03:25.000 --> 00:03:29.039 developed on somewhat different lines to that of the continent, 51 00:03:29.120 --> 00:03:32.879 where a knowledge of the infinitesimal calculus was derived solely 52 00:03:33.479 --> 00:03:37.479 from Lebants and the Bernoullis. But this separation into two 53 00:03:37.479 --> 00:03:40.360 distinct schools became very marked owing to the action of 54 00:03:40.439 --> 00:03:44.560 Libanates and John Bernoulli, which was naturally resented by Newton's friends, 55 00:03:45.120 --> 00:03:48.439 and so for forty or fifty years, to the mutual 56 00:03:48.479 --> 00:03:52.759 disadvantage of both sides, the quarrel raged. The leading members 57 00:03:52.759 --> 00:03:57.879 of the English School were Coats, des Mois, Detont, David Gregory, 58 00:03:58.199 --> 00:04:03.319 Halle McLaurin, Simpson and Taylor. I may, however, again remind 59 00:04:03.360 --> 00:04:06.560 my readers that as we approach the modern times, the 60 00:04:06.680 --> 00:04:10.240 number of capable mathematicians in Britain, France, Germany and Italy 61 00:04:10.639 --> 00:04:15.000 becomes very considerable. But that in a popular sketch like 62 00:04:15.039 --> 00:04:17.279 this book, it is only the leading men whom I 63 00:04:17.360 --> 00:04:23.120 propose to mention to David Gregory, Halley and Detonte. I 64 00:04:23.160 --> 00:04:28.399 need to devote but a few words. David Gregory. David Gregory, 65 00:04:28.480 --> 00:04:32.079 the nephew of James Gregory, mentioned above in page three 66 00:04:32.079 --> 00:04:35.879 point fifteen, born at Aberdeen on June twenty fourth, sixteen 67 00:04:35.959 --> 00:04:39.800 sixty one, and died at Maidenhead on October tenth, seventeen 68 00:04:39.839 --> 00:04:44.160 oh eight. Was appointed professor at Edinburgh in sixteen eighty four, 69 00:04:44.279 --> 00:04:49.360 and in sixteen ninety one was on Newton's recommendation elected 70 00:04:49.399 --> 00:04:55.120 Civilian Professor at Oxford. His chief works are one on geometry, 71 00:04:55.199 --> 00:04:59.199 issued in sixteen eighty four, one on optics published in 72 00:04:59.279 --> 00:05:03.040 sixteen ninety five, which contains the earliest suggestion of the 73 00:05:03.079 --> 00:05:07.839 possibility of making an achromatic combinations of lenses, and one 74 00:05:07.920 --> 00:05:12.120 on the Newtonian geometry, Physics and Astronomy, issued in seventeen 75 00:05:12.160 --> 00:05:18.319 o two. Halle Edmund Halley born in London in sixteen 76 00:05:18.399 --> 00:05:22.040 fifty six and died at Greenwich in seventeen forty two. 77 00:05:22.800 --> 00:05:26.319 Was educated at Saint Paul's School, London and Queen's College, Oxford. 78 00:05:26.360 --> 00:05:31.160 In seventeen o three succeeded Wallace as civilian professor, and subsequently, 79 00:05:31.199 --> 00:05:36.279 in seventeen twenty was appointed Astronomer Royal in succession to Flamsteed, 80 00:05:36.560 --> 00:05:41.600 whose Historia Cerestis Britannica he edited in seventeen twelve first 81 00:05:41.720 --> 00:05:45.439 and imperfect edition. Halley's name will be recollected for the 82 00:05:45.480 --> 00:05:49.240 generous manner in which he secured the immediate publication of 83 00:05:49.360 --> 00:05:53.879 Newton's Principia in sixteen eighty seven. Most of his original 84 00:05:53.879 --> 00:05:57.319 work on was on astronomy and allied subjects, and lies 85 00:05:57.360 --> 00:06:00.199 outside the limits of this book. It may be, however, 86 00:06:00.319 --> 00:06:03.319 said that the work is of excellent quality, and both 87 00:06:03.399 --> 00:06:06.759 Lalande and Mehran speak of it in the highest terms. 88 00:06:07.519 --> 00:06:11.040 Halley conjecturally restored the eighth and lost Book of the 89 00:06:11.120 --> 00:06:15.279 Conics of Apollonius, and in seventeen ten brought out a 90 00:06:15.319 --> 00:06:18.839 magnificent edition of the whole work. He also edited the 91 00:06:18.839 --> 00:06:22.480 works of Serenus and those of Mennelos, and some of 92 00:06:22.560 --> 00:06:26.040 the minor works of Apollonius. He was, in his turn 93 00:06:26.160 --> 00:06:32.319 succeeded at Greenwich as an astronomer Royal by Bradley Diton. 94 00:06:32.759 --> 00:06:37.040 Humphrey Dieton born at Salisbury in May twenty ninth, sixteen 95 00:06:37.079 --> 00:06:40.759 seventy five and died in London in seventeen fifteen at 96 00:06:40.800 --> 00:06:45.160 Christ's Hospital, where he was mathematical master. He does not 97 00:06:45.279 --> 00:06:48.120 seem to have paid much attention to mathematics until he 98 00:06:48.160 --> 00:06:51.720 came to London about seventeen o five, and his early 99 00:06:51.800 --> 00:06:55.920 death was a distinct loss to English science. He published 100 00:06:55.920 --> 00:06:59.839 in seventeen o six a textbook on fluxions. This and 101 00:07:00.079 --> 00:07:03.560 another similar work by William Jones, which was issued in 102 00:07:03.600 --> 00:07:07.240 seventeen eleven, occupied in England much the same place as 103 00:07:07.519 --> 00:07:12.879 Laptel's treatise did in France. In seventeen oh nine, Detont 104 00:07:13.040 --> 00:07:17.959 issued an algebra and in seventeen twelve a treatise on perspective. 105 00:07:18.759 --> 00:07:23.279 He also wrote numerous papers on the philosophical transactions. He 106 00:07:23.399 --> 00:07:26.439 was the earliest writer to attempt to explain the phenomenon 107 00:07:26.560 --> 00:07:32.240 of capillary on mathematical principles, and he invented a method 108 00:07:32.279 --> 00:07:35.680 of finding the longitude which has been since used on 109 00:07:35.800 --> 00:07:44.439 various occasions. Taylor Brooke Taylor born at Edmonton on August eighteenth, 110 00:07:44.600 --> 00:07:48.879 sixteen eighty five and died in London on December twenty ninth, 111 00:07:48.959 --> 00:07:53.680 seventeen thirty one. Was educated at Saint John's College, Cambridge, 112 00:07:53.879 --> 00:07:58.519 and was among the most enthusiastic of Newton's admirers. From 113 00:07:58.519 --> 00:08:01.959 the year seventeen twelve onwards he wrote numerous papers on 114 00:08:02.040 --> 00:08:07.040 philosophical transactions, in which, among other things, he discussed the 115 00:08:07.040 --> 00:08:10.959 motion of projectiles, the center of oscillation, and the forms 116 00:08:10.959 --> 00:08:15.759 of liquids raised by capillarity. In seventeen nineteen he resigned 117 00:08:16.079 --> 00:08:19.879 the secretaryship of the Royal Society and abandoned the study 118 00:08:19.879 --> 00:08:23.680 of mathematics. His earliest work, that by which he is 119 00:08:23.759 --> 00:08:29.759 generally known, is his methodis incrematorum directa et inversa, published 120 00:08:29.800 --> 00:08:33.960 in London in seventeen fifteen. This contains a proof of 121 00:08:34.000 --> 00:08:37.440 the well known theorem f x plus h equals f 122 00:08:37.440 --> 00:08:42.360 of x plus h, f primed of x plus h 123 00:08:42.480 --> 00:08:46.279 squared over two factorial multiplied by f double primed x 124 00:08:46.639 --> 00:08:49.399 plus and so on, by which any function of a 125 00:08:49.440 --> 00:08:53.720 single variable can be expanded in powers of it. He 126 00:08:53.759 --> 00:08:57.200 does not consider the convergency of the series, and the proof, 127 00:08:57.279 --> 00:09:03.279 which involves numerous assumptions, is not worth reproducing. The work 128 00:09:03.320 --> 00:09:07.720 also includes several theorems on interpolation. Taylor was the earliest 129 00:09:07.759 --> 00:09:10.200 writer to deal with theorems on the change of the 130 00:09:10.240 --> 00:09:13.759 independent variable. He was perhaps the first to realize the 131 00:09:13.799 --> 00:09:18.039 possibility of a calculus of operation, and just as he 132 00:09:18.120 --> 00:09:22.960 denotes the n differential coefficient of why by why sub n, 133 00:09:23.720 --> 00:09:28.559 so he uses why subnegative one to represent the integral 134 00:09:28.639 --> 00:09:33.039 of why. Lastly, he is usually recognized as the creator 135 00:09:33.120 --> 00:09:37.240 of the theory of finite differences. The applications of the 136 00:09:37.240 --> 00:09:40.720 calculus to various questions given in the Methodists have hardly 137 00:09:40.759 --> 00:09:44.440 received that attention they deserve. The most important of them 138 00:09:44.639 --> 00:09:48.360 is the theory of the transverse vibration of strings, a 139 00:09:48.440 --> 00:09:54.360 problem which had baffled previous investigators. In this investigation, Taylor 140 00:09:54.399 --> 00:09:57.799 showes that the number of half vibrations executed in a 141 00:09:57.879 --> 00:10:01.480 second is by multi applied by the square root of 142 00:10:01.519 --> 00:10:05.879 the quantity dp divided by l n, Where l is 143 00:10:05.919 --> 00:10:08.240 the length of the string, n is its weight, p 144 00:10:08.440 --> 00:10:11.039 is the weight which stretches it, and D is the 145 00:10:11.159 --> 00:10:15.679 length of a second's pendulum. This is correct, but in 146 00:10:15.840 --> 00:10:18.519 arriving at it, he assumes that every point on the 147 00:10:18.559 --> 00:10:21.840 string will pass through its position of equilibrium at the 148 00:10:21.879 --> 00:10:26.440 same instant, a restriction which de la Bern subsequently shewed 149 00:10:26.480 --> 00:10:32.519 to be unnecessary. Taylor also found the form which the 150 00:10:32.759 --> 00:10:36.720 string assumes at any instant. This work also contains the 151 00:10:36.759 --> 00:10:40.759 earliest determination of the differential equation of the path of 152 00:10:40.799 --> 00:10:45.000 a ray of light when traversing a heterogeneous medium and 153 00:10:45.120 --> 00:10:48.080 assuming that the density of the air depends only on 154 00:10:48.159 --> 00:10:52.639 its distance from the Earth's surface tailor obtained by means 155 00:10:52.639 --> 00:10:57.720 of quadratures, The approximate form of the curve, the form 156 00:10:57.840 --> 00:11:01.480 of the caternary, and the deters termination of the centers 157 00:11:01.480 --> 00:11:06.799 of oscillation and percussion are also discussed. A treatise on Perspective, 158 00:11:06.919 --> 00:11:11.600 published in seventeen nineteen, contains the earliest general annunciation of 159 00:11:11.639 --> 00:11:15.919 the principle of vanishing points, though the idea of vanishing 160 00:11:15.960 --> 00:11:20.080 points for horizontal and parallel lines in a picture hung 161 00:11:20.120 --> 00:11:24.360 in a vertical plane had been enunciated by Guido Ubaldi 162 00:11:24.440 --> 00:11:29.759 in his perspective Libri Pisa sixteen hundred. Coats Roger Coates 163 00:11:29.840 --> 00:11:33.440 was born near Leicester on July tenth, sixteen eighty two, 164 00:11:34.000 --> 00:11:38.320 and died at Cambridge on June fifth, seventeen sixteen. He 165 00:11:38.399 --> 00:11:42.039 was educated at Trinity College, Cambridge, of which society he 166 00:11:42.120 --> 00:11:45.279 was a fellow, and in seventeen o six was elected 167 00:11:45.320 --> 00:11:48.399 to the newly created Plumian Chair of Astronomy in the 168 00:11:48.519 --> 00:11:53.000 University of Cambridge. From seventeen oh nine to seventeen thirteen, 169 00:11:53.440 --> 00:11:56.679 his time was mainly occupied in editing the second edition 170 00:11:56.759 --> 00:11:59.960 of the Principia. The remark of Newton that if only 171 00:12:00.159 --> 00:12:04.000 Coats had lived, we should have learnt something indicates the 172 00:12:04.039 --> 00:12:07.440 opinion of his abilities held by most of his contemporaries. 173 00:12:08.120 --> 00:12:11.679 Coates's writings were collected and published in seventeen twenty two 174 00:12:11.919 --> 00:12:18.279 under the titles Harmonia Mensurarum and Opera Miscellania. His lectures 175 00:12:18.279 --> 00:12:22.600 on hydrostatics were published in seventeen thirty eight. A large 176 00:12:22.639 --> 00:12:26.240 part of the Harmonium Mensuratum was given up to the 177 00:12:26.279 --> 00:12:32.960 decomposition and integration of rational algebraical expressions. That part, which 178 00:12:33.000 --> 00:12:37.399 deals with the theory of partial fractions, was left unfinished, 179 00:12:38.159 --> 00:12:43.440 but was completed by des Maire. Coates's theorem in trigonometry, 180 00:12:43.799 --> 00:12:47.120 which depends on forming the quadratic factors of x to 181 00:12:47.159 --> 00:12:51.080 the power n minus one, is well known. The proposition 182 00:12:51.200 --> 00:12:53.639 that if from a fixed point ozh a line be 183 00:12:53.799 --> 00:12:57.399 drawn cutting a curve in q one, q two, and 184 00:12:57.480 --> 00:12:59.799 so on to q n, and a point p b 185 00:12:59.879 --> 00:13:02.240 s taken on the line, so that the reciprocal of 186 00:13:02.360 --> 00:13:06.360 op is the arithmetic mean of the reciprocals of oq one, 187 00:13:06.440 --> 00:13:09.080 oq two, oq three, and so on all the way 188 00:13:09.120 --> 00:13:13.759 to oq n, then the locus of p will be 189 00:13:13.799 --> 00:13:18.279 a straight line. Is also due to Coats, the title 190 00:13:18.279 --> 00:13:21.320 of the book was derived from the latter theorem. The 191 00:13:21.320 --> 00:13:25.080 Opera Miscellania contains a paper on the method for determining 192 00:13:25.120 --> 00:13:28.840 the most probable result from a number of observations. This 193 00:13:29.000 --> 00:13:31.639 was the earliest attempt to frame a theory of errors. 194 00:13:32.279 --> 00:13:37.320 It also contains essays on Newton's methodist differentiatis, the construction 195 00:13:37.440 --> 00:13:40.639 of tables by the method of differences, on the descent 196 00:13:40.720 --> 00:13:44.759 of a body under gravity, on the cycloidal pendulum, and 197 00:13:44.960 --> 00:13:53.120 on projectiles. Demoivre Abraham Demarra, more correctly written des Moirvre 198 00:13:53.399 --> 00:13:57.600 in two words, was born at Vitri in May twenty sixth, 199 00:13:57.639 --> 00:14:02.200 sixteen sixty seven, and died in Lund on November twenty seventh, 200 00:14:02.320 --> 00:14:05.840 seventeen fifty four. His parents came to England when he 201 00:14:05.919 --> 00:14:10.080 was a boy, and his education and friends were alike English. 202 00:14:11.080 --> 00:14:14.279 His interest in the higher mathematics is said to have originated, 203 00:14:14.360 --> 00:14:17.120 and is coming by chance across a copy of Newton's 204 00:14:17.159 --> 00:14:21.639 Principia from the Eleugee on Him, delivered in seventeen fifty 205 00:14:21.639 --> 00:14:24.679 four before the French Academy. It would seem that as 206 00:14:24.720 --> 00:14:27.559 a young fellow, his work as a teacher of mathematics 207 00:14:27.879 --> 00:14:30.039 had led him to the house of the Earl of 208 00:14:30.120 --> 00:14:34.159 Devonshire at the instant when Newton, who had asked permission 209 00:14:34.200 --> 00:14:36.159 to present a copy of his work to the Earl, 210 00:14:36.320 --> 00:14:40.080 was coming up taking up the book and charmed by 211 00:14:40.120 --> 00:14:44.159 the far reaching conclusions and the apparent simplicity of the reasoning, 212 00:14:44.639 --> 00:14:47.879 des Mauvre thought nothing would be easier than to master 213 00:14:47.960 --> 00:14:51.279 the subject, but to his surprise, found that to follow 214 00:14:51.320 --> 00:14:56.440 the argument overtaxed his powers. He however, bought a copy, and, 215 00:14:57.480 --> 00:15:00.480 as he had but little leisure, he tol or out 216 00:15:00.480 --> 00:15:03.200 the pages in order to carry one or two of 217 00:15:03.240 --> 00:15:05.919 them loose in his pocket, so that he could study 218 00:15:05.919 --> 00:15:10.039 them in the intervals of his work as a teacher. Subsequently, 219 00:15:10.120 --> 00:15:14.039 he joined the Royal Society, and he became intimately connected 220 00:15:14.159 --> 00:15:18.320 with Newton, Halley and other mathematicians of the English School. 221 00:15:19.960 --> 00:15:23.679 The manner of his death has a curious interest for psychologists. 222 00:15:24.360 --> 00:15:27.559 Shortly before it, he declared that it was necessary for 223 00:15:27.679 --> 00:15:30.039 him to sleep some ten minutes or a quarter of 224 00:15:30.080 --> 00:15:33.480 an hour longer each day than the preceding one. The 225 00:15:33.600 --> 00:15:36.120 day after he had thus reached a total of something 226 00:15:36.840 --> 00:15:39.399 over twenty three hours, he slept up to the limit 227 00:15:39.480 --> 00:15:41.879 of twenty four hours, and then died in his sleep. 228 00:15:43.159 --> 00:15:46.399 He is best known for having, together with Lambert, created 229 00:15:46.440 --> 00:15:51.120 that part of trigonometry which deals with imaginary quantities. Two 230 00:15:51.159 --> 00:15:53.720 theorems on this part of the subject are still connected 231 00:15:53.759 --> 00:15:57.639 with his name, namely that which asserts that sign n 232 00:15:57.960 --> 00:16:01.840 x plus ikos nx is one of the values of 233 00:16:02.240 --> 00:16:05.879 the quantity sin x plus KOs x all to the 234 00:16:05.919 --> 00:16:10.200 power n, and that which gives the various quadratic factors 235 00:16:10.320 --> 00:16:14.720 of x to the power two nus two times p 236 00:16:14.960 --> 00:16:18.440 x to the power n plus one. His chief works 237 00:16:18.480 --> 00:16:23.039 other than numerous papers in the Philosophical Transactions where the 238 00:16:23.080 --> 00:16:27.879 Doctrine of Chances published in seventeen eighteen and the Miscellania 239 00:16:27.879 --> 00:16:31.879 Analytica published in seventeen thirty. In the former, the theory 240 00:16:31.919 --> 00:16:35.080 of recurring series was first given, and the theory of 241 00:16:35.120 --> 00:16:39.759 partial fractions, which Coates's premature death had left unfinished, was completed, 242 00:16:40.320 --> 00:16:43.200 while the rule for finding the probability of a compound 243 00:16:43.240 --> 00:16:49.559 event was enunciated. The latter, besides the trigonometrical propositions mentioned above, 244 00:16:50.000 --> 00:16:53.240 contains some theorems in astronomy, but they are treated as 245 00:16:53.320 --> 00:16:59.080 problems in analysis. Maclaurin Colin maclaurin, who was born in 246 00:16:59.200 --> 00:17:03.919 Kilmoden in Argyllshire in February sixteen ninety eight and died 247 00:17:03.960 --> 00:17:08.519 at York on June fourteenth, seventeen forty six, was educated 248 00:17:08.559 --> 00:17:12.599 at the University of Glasgow. In seventeen seventeen, he was 249 00:17:12.640 --> 00:17:16.599 elected at the early age of nineteen Professor of Mathematics 250 00:17:16.599 --> 00:17:20.480 at Aberdeen, and in seventeen twenty five he was appointed 251 00:17:20.799 --> 00:17:25.559 the deputy of the mathematical Professor at Edinburgh, and ultimately 252 00:17:25.599 --> 00:17:29.240 succeeded him. There was some difficulty in securing a stipend 253 00:17:29.319 --> 00:17:33.119 for deputy, and Newton privately wrote offering to bear the 254 00:17:33.240 --> 00:17:36.640 cost so as to enable the university to secure the 255 00:17:36.680 --> 00:17:41.079 services of mclauren. McLaurin took an active part in opposing 256 00:17:41.079 --> 00:17:44.240 the advance of the young Pretender. In seventeen forty five, 257 00:17:44.720 --> 00:17:47.160 on the approach of the Highlanders, he fled to York, 258 00:17:48.000 --> 00:17:51.400 but the exposure in the trenches at Edinburgh and the 259 00:17:51.400 --> 00:17:55.400 privations he endured in his escape proved fatal to him. 260 00:17:56.000 --> 00:18:00.680 His chief works are Geometrica Organica, London, seventeen teen nineteen, 261 00:18:01.200 --> 00:18:06.440 his Treatise on Fluxions, Edinburgh, seventeen forty two, His Algebra, London, 262 00:18:06.559 --> 00:18:10.960 seventeen forty eight, and his Account of Newton's Discoveries, London, 263 00:18:11.079 --> 00:18:17.200 seventeen forty eight. The Geometrica Organica is on the extension 264 00:18:17.279 --> 00:18:21.519 of a theorem given by Newton. Newton had shewn that 265 00:18:21.559 --> 00:18:24.440 if two angles bounded by straight lines turned round their 266 00:18:24.480 --> 00:18:28.240 respective summits, so that the point of intersection of two 267 00:18:28.279 --> 00:18:31.519 of these line moves along a straight line. The other 268 00:18:31.640 --> 00:18:35.079 point of intersection will describe a conic, and if the 269 00:18:35.119 --> 00:18:38.720 first point move along a conic, the second will describe 270 00:18:38.839 --> 00:18:43.279 a cortic. Maclaurin gave an analytical discussion on the general 271 00:18:43.319 --> 00:18:47.279 theorem and shewed by how this method various curves could 272 00:18:47.319 --> 00:18:51.440 be practically traced. This work contains an elaborate discussion on 273 00:18:51.559 --> 00:18:55.039 curves and their pedals, a branch of geometry which he 274 00:18:55.079 --> 00:18:59.400 had created in two papers published in the Philosophical Transactions 275 00:18:59.720 --> 00:19:04.799 for s. Seventeen eighteen n seventeen nineteen. In the following year, 276 00:19:04.920 --> 00:19:09.519 seventeen twenty, Maclaurin issued a supplement which is practically the 277 00:19:09.559 --> 00:19:16.920 same as his Delinearum Geometricarum Propritatibus. It is divided into 278 00:19:16.920 --> 00:19:20.759 three sections and an appendix. The first section contains a 279 00:19:20.799 --> 00:19:24.279 proof of Coats's theorem as alluded to, and also the 280 00:19:24.319 --> 00:19:28.160 analogous theorem discovered by himself that if a straight line 281 00:19:28.319 --> 00:19:30.799 O P one P two is drawn through a fixed point, 282 00:19:30.920 --> 00:19:34.720 oh cut a curve of the n degree in n 283 00:19:34.799 --> 00:19:37.319 points P one, P two, and so on. And if 284 00:19:37.359 --> 00:19:39.680 the tangents at P one, P two cut in a 285 00:19:39.720 --> 00:19:43.680 fixed line ox in points A one, A, two, and 286 00:19:43.720 --> 00:19:46.440 so on, then the sum of the receptacles of the 287 00:19:46.480 --> 00:19:49.720 distances A one, O, A two, and so forth is 288 00:19:49.799 --> 00:19:52.559 constant for all positions of the line O, P one, 289 00:19:52.680 --> 00:19:56.599 P two, and so on. These two theorems are generalizations 290 00:19:56.640 --> 00:20:00.480 of those given by Newton on diameters and asymptotes. Either 291 00:20:00.640 --> 00:20:03.960 is deducible from the other. In the second section, these 292 00:20:04.000 --> 00:20:07.640 theorems are applied to conics. Most of these harmonic properties 293 00:20:07.680 --> 00:20:12.119 connected with an inscribed quadrilateral are determined, and in particular, 294 00:20:12.200 --> 00:20:15.279 the theorem on an inscribed hexagon, which is known by 295 00:20:15.279 --> 00:20:19.440 the name of Pascal, is deduced. Pascal's essay was not 296 00:20:19.480 --> 00:20:24.119 published until seventeen seventy nine, and the earliest printed denunciation 297 00:20:24.279 --> 00:20:29.400 of his theorem was that given by maclaurin. In the 298 00:20:29.440 --> 00:20:35.039 third section, these theorems are applied to cubic curves. Amongst 299 00:20:35.079 --> 00:20:39.559 other propositions, he shews that if if a quadrilateral be 300 00:20:39.640 --> 00:20:44.680 inscribed in a cubic and if the points of intersection 301 00:20:44.839 --> 00:20:48.000 of the opposite sides also lie on the curve, then 302 00:20:48.079 --> 00:20:51.400 the tangents to the cubic at any two opposite angles 303 00:20:51.400 --> 00:20:55.559 of the quadrilateral will meet on the curve. The appendix 304 00:20:55.599 --> 00:20:59.960 contained some general theorems. One of these, which includes Pascal's 305 00:21:00.039 --> 00:21:02.960 as a particular case, is that if a polygon be 306 00:21:03.039 --> 00:21:06.000 deformed so that, while each of its sides passes through 307 00:21:06.000 --> 00:21:10.279 a fixed point, its angles save one describe respectively curves 308 00:21:10.279 --> 00:21:13.640 of the m F, N, P, F, and so forth degrees. 309 00:21:14.160 --> 00:21:17.599 Then shall the remaining angle describe a curve of the 310 00:21:17.680 --> 00:21:22.359 degree to mnp. But if the given points be collinear, 311 00:21:22.440 --> 00:21:26.079 the resulting curve will be only of the degree mnp. 312 00:21:27.079 --> 00:21:31.400 This essay was reprinted with editions on the Philosophical Transactions 313 00:21:31.440 --> 00:21:35.759 for seventeen thirty five. The Treatise of Fluxions, published in 314 00:21:35.839 --> 00:21:40.599 seventeen forty two, was the first logical and systematic exposition 315 00:21:40.799 --> 00:21:44.440 of the method of fluxions. The cause of its publication 316 00:21:44.680 --> 00:21:47.759 was an attack by Berkeley on the principles of the 317 00:21:47.799 --> 00:21:53.000 infinitesimal calculus. In it, article seven fifty one, page six 318 00:21:53.079 --> 00:21:56.519 hundred and ten, McLaurin gave a proof of the theorem 319 00:21:56.599 --> 00:22:00.640 that fivx equals f of zero plus five x f 320 00:22:00.839 --> 00:22:05.480 prime of zero plus x squared over two factorial f 321 00:22:05.559 --> 00:22:09.599 double prime of zero plus so on, and so forth. 322 00:22:10.279 --> 00:22:13.960 This was obtained in the manner given in many modern textbooks, 323 00:22:13.960 --> 00:22:17.240 by assuming that fivx can be expanded in a form 324 00:22:17.480 --> 00:22:22.559 like fivx equals a not plus a one x plus 325 00:22:22.559 --> 00:22:25.519 a two x squared plus and so on and so on. 326 00:22:26.200 --> 00:22:30.200 Then on differentiating and putting x equals zero in the 327 00:22:30.319 --> 00:22:33.880 successive results, the values of a zero, a, one, and 328 00:22:33.960 --> 00:22:37.599 so on are obtained. But he did not investigate the 329 00:22:37.599 --> 00:22:41.680 convergency of the series. The result had been previously given 330 00:22:41.720 --> 00:22:46.759 in seventeen thirty by James Sterling in his methodis differentialis, 331 00:22:47.279 --> 00:22:50.880 and of course is at once deducible from Taylor's theorem, 332 00:22:51.400 --> 00:22:55.240 on which the proofs by Sterling and McLaurin are admittedly founded. 333 00:22:56.160 --> 00:23:01.279 McLaurin also here enunciated article three fifty, page two nine, 334 00:23:01.400 --> 00:23:05.319 the important theorem that if pix be positive and decreases 335 00:23:05.519 --> 00:23:08.759 x increases from x equels a to x equals infinity, 336 00:23:09.400 --> 00:23:12.400 then the series five A plus five, A plus one 337 00:23:12.559 --> 00:23:16.000 plus five, a plus two and so on is convergent 338 00:23:16.160 --> 00:23:21.599 or divergent, as the integral going from a to infinity 339 00:23:22.000 --> 00:23:27.480 of five x dx is finite or infinite. He also 340 00:23:27.559 --> 00:23:31.359 gives the correct theory of maxima and minima, and rules 341 00:23:31.359 --> 00:23:36.000 for finding and discriminating multiple points. This treatise is, however, 342 00:23:36.119 --> 00:23:39.960 especially valuable for the solutions that contains of numerous problems 343 00:23:40.000 --> 00:23:45.640 in geometry, statics, the theory of attractions, and astronomy. To 344 00:23:45.680 --> 00:23:50.039 solve these, he reverted to classical methods, and so powerful 345 00:23:50.079 --> 00:23:53.759 did these processes seem when used by him that Claat, 346 00:23:54.200 --> 00:23:57.480 after reading the work, abend an analysis and attack the 347 00:23:57.559 --> 00:24:00.200 problem of the figure of the Earth again by pure 348 00:24:00.279 --> 00:24:03.400 geometry at a later time. This part of the book 349 00:24:03.519 --> 00:24:06.680 was described by Lagrange as the chef de ro de 350 00:24:06.799 --> 00:24:13.640 giometri compeu capere atus qiachimde nousalasse de plubo e des 351 00:24:13.759 --> 00:24:19.599 plus angeneux. McLaurin also determined the attraction of a homogeneous 352 00:24:19.640 --> 00:24:23.960 ellipsoid at an internal point and gave some theorems on 353 00:24:24.079 --> 00:24:27.640 its attraction at an external point. In effecting this, he 354 00:24:27.759 --> 00:24:32.720 introduced the conception of level surfaces, i e. Surfaces at 355 00:24:32.759 --> 00:24:36.839 every point of which the resultant attraction is perpendicular to 356 00:24:36.880 --> 00:24:40.720 the surface. No further advances in the theory of attractions 357 00:24:40.839 --> 00:24:44.960 was made until lagrange. In seventeen seventy three introduced the 358 00:24:45.000 --> 00:24:50.000 idea of the potential sea below. McLaurin also shewed that 359 00:24:50.079 --> 00:24:53.440 a spheroid was a possible form of equilibrium of a 360 00:24:53.519 --> 00:24:58.079 mass of homogeneous liquid rotating about an axis passing through 361 00:24:58.119 --> 00:25:02.920 its center of mass. Finally, he discussed the tides. This 362 00:25:03.039 --> 00:25:06.839 part had been previously published in seventeen forty and had 363 00:25:06.880 --> 00:25:11.759 received a prize from the French Academy. Among maclaurin's minor 364 00:25:11.839 --> 00:25:15.920 works is his Algebra, published in seventeen forty eight and 365 00:25:16.079 --> 00:25:20.599 founded on Newton's universal arithmetic. It contains the results of 366 00:25:20.640 --> 00:25:24.880 some early papers of maclaurin, notably of two written in 367 00:25:25.000 --> 00:25:29.160 seventeen twenty six and seventeen twenty nine on the number 368 00:25:29.240 --> 00:25:33.319 of imaginary roots of an equation suggested by Newton's theorems 369 00:25:33.359 --> 00:25:36.720 see above, and of one written in seventeen twenty nine, 370 00:25:36.880 --> 00:25:39.839 containing the well known rule for finding equal roots by 371 00:25:39.960 --> 00:25:44.079 means of the derived equation. To this, a treatise entitled 372 00:25:44.160 --> 00:25:51.680 Delinearum Geometricarum propeitatibus generalibus was added as an appendix. Besides 373 00:25:51.799 --> 00:25:55.480 the paper of seventeen twenty above alluded to, it contained 374 00:25:55.480 --> 00:25:59.960 some additional and elegant theorems. McLaurin also produced in Stea 375 00:26:00.000 --> 00:26:04.240 seventeen twenty eight, an exposition of the Newtonian philosophy, in 376 00:26:04.279 --> 00:26:08.160 which is incorporated in the posthumous work printed in seventeen 377 00:26:08.240 --> 00:26:12.160 forty eight. Almost the last paper he wrote was one 378 00:26:12.759 --> 00:26:17.119 printed in the Philosophical Transactions in seventeen forty three, in 379 00:26:17.160 --> 00:26:20.319 which he discussed, from a mathematical point of view, the 380 00:26:20.400 --> 00:26:25.839 form of abees cell. McLaurin was succeeded in his chair 381 00:26:25.880 --> 00:26:30.480 at Edinburgh by pupil Matthew Stewart born at Rothsay in 382 00:26:30.559 --> 00:26:35.119 seventeen seventeen and died in Edinburgh on January twenty third, 383 00:26:35.640 --> 00:26:40.920 seventeen eighty five. A mathematician of considerable power, to whom 384 00:26:40.920 --> 00:26:43.799 I allude in passing for his theorems on the problem 385 00:26:44.160 --> 00:26:48.119 of three bodies, and for his discussion treated by transversals 386 00:26:48.119 --> 00:26:52.319 and involution of the properties of the circle and straight line. 387 00:26:53.119 --> 00:26:56.000 McLaurin was one of the most able mathematicians of the 388 00:26:56.039 --> 00:26:59.720 eighteenth century, but his influence on the progress of British 389 00:26:59.720 --> 00:27:04.359 mathod thematics was on the whole unfortunate, by himself abandoning 390 00:27:04.400 --> 00:27:09.000 the use of both analysis and of the infinitesimal calculus. 391 00:27:09.039 --> 00:27:13.519 He induced Newton's countrymen to confine themselves to Newton's methods, 392 00:27:13.880 --> 00:27:17.720 and I remarked before it was not until about eighteen twenty, 393 00:27:18.079 --> 00:27:22.160 when the differential calculus was introduced into the Cambridge curriculum, 394 00:27:22.319 --> 00:27:25.960 that English mathematicians made any general use of the more 395 00:27:26.000 --> 00:27:32.799 powerful methods of modern analysis. Simpson, the last member of 396 00:27:32.839 --> 00:27:36.880 the English school whom I need mention here is Thomas Simpson, 397 00:27:37.599 --> 00:27:42.359 who was born in Leicestershire on August twentieth, seventeen ten 398 00:27:42.599 --> 00:27:47.559 and died on May fourteenth, seventeen sixty one. His father 399 00:27:47.720 --> 00:27:50.279 was a weaver, and he owed his education to his 400 00:27:50.359 --> 00:27:54.759 own efforts. His mathematical interests were first aroused by the 401 00:27:54.799 --> 00:27:58.519