WEBVTT 1 00:00:01.040 --> 00:00:04.120 Chapter sixteen, Part one of a short account of the 2 00:00:04.240 --> 00:00:09.359 history of mathematics. This is a LibriVox recording. All LibriVox 3 00:00:09.400 --> 00:00:13.119 recordings are in the public domain. For more information or 4 00:00:13.199 --> 00:00:18.039 to volunteer, please visit LibriVox dot org. This is a 5 00:00:18.079 --> 00:00:21.399 reading by Paul King p j K dot scripts dot 6 00:00:21.559 --> 00:00:26.559 mit dot edu forward slash p kJ. A short account 7 00:00:26.640 --> 00:00:31.920 of the history of mathematics by W. W. Rowseball. Chapter sixteen, 8 00:00:32.439 --> 00:00:37.960 The Life and Works of Newton. The mathematicians considered in 9 00:00:38.000 --> 00:00:41.960 the last chapter commenced the creation of those processes which 10 00:00:42.079 --> 00:00:48.359 distinguish modern mathematics. The extraordinary abilities of Newton enabled him, 11 00:00:48.399 --> 00:00:51.719 within a few years to perfect the more elementary of 12 00:00:51.759 --> 00:00:56.439 these processes, and to distinctly advance every branch of mathematical 13 00:00:56.479 --> 00:01:01.320 science then studied, as well as to create some new subjects. 14 00:01:02.560 --> 00:01:06.280 Newton was the contemporary and friend of Wallace, Huygens, and 15 00:01:06.400 --> 00:01:09.840 others of those mentioned in the last chapter. But though 16 00:01:10.040 --> 00:01:12.920 most of his mathematical work was done in between the 17 00:01:13.000 --> 00:01:17.680 years sixteen sixty five and sixteen eighty six, the Bulkovit 18 00:01:17.760 --> 00:01:21.079 was not printed at any rate in book form till 19 00:01:21.120 --> 00:01:24.719 some years later. I propose to discuss the works of 20 00:01:24.799 --> 00:01:28.760 Newton somewhat more fully than those of other mathematicians, partly 21 00:01:28.799 --> 00:01:33.439 because of the intrinsic importance of his discoveries, and partly 22 00:01:33.480 --> 00:01:37.280 because this book is mainly intended for English readers, and 23 00:01:37.319 --> 00:01:40.159 the development of mathematics in Great Britain was for a 24 00:01:40.280 --> 00:01:45.760 century entirely in the hands of the Newtonian school. Isaac 25 00:01:45.799 --> 00:01:50.760 Newton was born in Lincolnshire, near Grantham, on December twenty fifth, 26 00:01:51.239 --> 00:01:56.280 sixteen forty two, and died at Kensington, London, on March twentieth, 27 00:01:56.439 --> 00:02:01.280 seventeen twenty seven. He was educated at Trinity CID College, Cambridge, 28 00:02:01.319 --> 00:02:05.519 and lived there from sixteen sixty one till sixteen ninety six, 29 00:02:05.640 --> 00:02:08.599 during which time he produced the bulk of his work 30 00:02:08.639 --> 00:02:12.840 in mathematics. In sixteen ninety six he was appointed to 31 00:02:12.879 --> 00:02:16.719 a valuable government office and moved to London, where he 32 00:02:16.800 --> 00:02:21.080 resided till his death. His father, who had died shortly 33 00:02:21.120 --> 00:02:25.000 before Newton was born, was a Yeoman farmer, and it 34 00:02:25.120 --> 00:02:29.599 was intended that Newton should carry on the paternal farm. 35 00:02:30.159 --> 00:02:33.080 He was sent to school at Grantham, where his learning 36 00:02:33.120 --> 00:02:38.360 and mechanical proficiency excited some attention, and as one instance 37 00:02:38.400 --> 00:02:41.599 of his ingenuity. I may mention that he constructed a 38 00:02:41.639 --> 00:02:47.080 clock worked by water, which kept very fair time. In 39 00:02:47.199 --> 00:02:50.719 sixteen fifty six he returned home to learn the business 40 00:02:50.800 --> 00:02:55.479 of a farmer under the guidance of an old family servant. Newton, however, 41 00:02:55.560 --> 00:03:00.479 spent most of his time studying problems, making experiments, or 42 00:03:00.520 --> 00:03:05.599 devising mechanical models. His mother, noticing this, sensibly resolved to 43 00:03:05.639 --> 00:03:09.840 find some more congenial occupation for him, and his uncle, 44 00:03:10.240 --> 00:03:15.199 having been himself educated at Trinity College, Cambridge, recommended that 45 00:03:15.280 --> 00:03:19.680 he should be sent there. In sixteen sixty one, Newton 46 00:03:19.759 --> 00:03:24.400 accordingly entered as a subsidiar at Trinity College, where for 47 00:03:24.439 --> 00:03:27.800 the first time he found himself among surroundings which were 48 00:03:27.960 --> 00:03:32.120 likely to develop his powers. He seemed somewhat to have 49 00:03:32.199 --> 00:03:35.319 but little interest for the general society or for any 50 00:03:35.400 --> 00:03:39.240 pursuit save science and mathematics, and he complained to his 51 00:03:39.319 --> 00:03:44.439 friends that he found the other undergraduates disorderly. Luckily, he 52 00:03:44.560 --> 00:03:47.199 kept a diary, and we can thus form a fair 53 00:03:47.280 --> 00:03:50.639 idea of the course of education of the most advanced 54 00:03:50.680 --> 00:03:54.360 students at at English university at that time. He had 55 00:03:54.400 --> 00:03:58.439 not read any mathematics before coming into residence, but was 56 00:03:58.520 --> 00:04:02.800 acquainted with Sanderson's logic, which was then frequently read as 57 00:04:02.879 --> 00:04:08.319 preliminary to mathematics. At the beginning of his first October term, 58 00:04:08.800 --> 00:04:12.680 he happened to stroll down to Southridge Fair and there 59 00:04:12.759 --> 00:04:16.199 picked up a book on astrology, but could not understand 60 00:04:16.279 --> 00:04:21.800 it on account of the geometry and trigonometry. He therefore 61 00:04:22.240 --> 00:04:25.959 bought a Euclid and was surprised to find how obvious 62 00:04:26.000 --> 00:04:32.720 the preposition seemed. He thereupon read Aughtred's Clavis and Descartes's geometry, 63 00:04:33.879 --> 00:04:37.639 and the latter of which he managed to master by himself, 64 00:04:38.199 --> 00:04:42.319 though it's some difficulty. The interest he felt in the 65 00:04:42.319 --> 00:04:45.639 subject led him to take up mathematics rather than chemistry 66 00:04:45.720 --> 00:04:51.920 as serious study. His subsequent mathematical reading as an undergraduate 67 00:04:52.199 --> 00:04:56.800 was found on Kepler's Optics, the works of Vieta van 68 00:04:56.839 --> 00:05:04.560 Schutten's Miscellanies and Dascartes Geometre three, and Wallace's Arithmetica Infinitorum. 69 00:05:04.920 --> 00:05:08.720 He also attended Barrow's lectures at a later time. On 70 00:05:08.800 --> 00:05:12.399 reading Euclid more carefully, he formed a high opinion of 71 00:05:12.439 --> 00:05:16.000 it as an instrument of education, and he used to 72 00:05:16.040 --> 00:05:19.160 express his regret that he had not applied himself to 73 00:05:19.240 --> 00:05:26.279 geometry before proceeding to algebraic analysis. There is a manuscript 74 00:05:26.560 --> 00:05:30.839 of his dated May twenty eighth, sixteen sixty five, written 75 00:05:30.920 --> 00:05:33.199 in the same year as that which he took his 76 00:05:33.279 --> 00:05:36.519 BA degree, which is the earliest documentary proof of his 77 00:05:36.639 --> 00:05:40.560 invention of fluxions. It was about the same time that 78 00:05:40.639 --> 00:05:44.560 he discovered the binomial theorem. On account of the plague, 79 00:05:44.600 --> 00:05:47.360 the college was shut down in the summer of sixteen 80 00:05:47.439 --> 00:05:50.639 sixty five, and for a large part of the next 81 00:05:50.720 --> 00:05:54.120 year and a half Newton lived at home. This period 82 00:05:54.240 --> 00:05:58.439 was crowded with brilliant discoveries. He thought out the fundamental 83 00:05:58.480 --> 00:06:02.399 principles of his theory of gret gravitation, namely that every 84 00:06:02.439 --> 00:06:07.079 particle of matter attracts every other particle, and he suspected 85 00:06:07.399 --> 00:06:10.480 that the attraction varied as the product of their masses, 86 00:06:10.519 --> 00:06:14.040 and inversely as the square of the distance between them. 87 00:06:14.600 --> 00:06:20.120 He also worked out the fluctional calculus tolerably completely. Thus, 88 00:06:20.160 --> 00:06:24.360 in a manuscript dated November thirteenth, sixteen sixty five, he 89 00:06:24.519 --> 00:06:28.279 used fluxions to find the tangent and the radius of 90 00:06:28.399 --> 00:06:31.959 curvature at any point on a curve, and in October 91 00:06:32.079 --> 00:06:35.759 sixteen sixty six, he applied them to several problems in 92 00:06:35.800 --> 00:06:40.720 the theory of equations. Newton communicated these results to his 93 00:06:40.800 --> 00:06:45.519 friends and pupils from and after sixteen sixty nine, but 94 00:06:45.600 --> 00:06:48.319 they were not published in print till many years later. 95 00:06:49.480 --> 00:06:52.600 It was also while staying at home at this time 96 00:06:53.120 --> 00:06:57.439 that he devised some instruments for grinding lenses to particular 97 00:06:57.560 --> 00:07:02.399 forms other than spherical, and perhaps he decomposed solar light 98 00:07:03.120 --> 00:07:08.720 into different colors, leaving out details and taking round numbers only. 99 00:07:09.560 --> 00:07:12.519 His reasoning at this time on the theory of gravitation 100 00:07:12.759 --> 00:07:16.839 seems to have been as follows. He suspected that the 101 00:07:16.879 --> 00:07:19.680 force which retained the Moon in its orbit about the 102 00:07:19.759 --> 00:07:24.120 Earth was the same as terrestrial gravity, and to verify 103 00:07:24.160 --> 00:07:28.800 this hypothesis he proceeded thus. He knew that if a 104 00:07:28.839 --> 00:07:31.519 stone were allowed to fall near the surface of the Earth, 105 00:07:31.560 --> 00:07:33.839 the attraction of the Earth, that is, the weight of 106 00:07:33.879 --> 00:07:37.920 the stone, caused it to move through sixteen feet in 107 00:07:37.959 --> 00:07:42.079 one second. The Moon's orbit relative to the Earth is 108 00:07:42.160 --> 00:07:46.120 nearly a circle, and as a rough approximation, taking it 109 00:07:46.160 --> 00:07:48.600 to be so, he knew that the distance of the Moon, 110 00:07:48.680 --> 00:07:52.240 and therefore the length of its path. He also knew 111 00:07:52.519 --> 00:07:54.839 that time the Moon took to go once round it, 112 00:07:55.040 --> 00:08:00.279 namely a month. Hence he could easily find its velocity 113 00:08:00.319 --> 00:08:04.240 at any point such as m. He could therefore find 114 00:08:04.279 --> 00:08:08.600 the distance Mt through which he would move in the 115 00:08:08.639 --> 00:08:12.000 next second if it were not pulled by the Earth's attraction. 116 00:08:13.040 --> 00:08:15.560 At the end of that second, it was, however, at 117 00:08:15.720 --> 00:08:19.319 M prime, and therefore the Earth must have pulled it 118 00:08:19.399 --> 00:08:23.560 through the distance TM prime in one second, assuming the 119 00:08:23.600 --> 00:08:27.399 direction of the Earth pull to be constant. Now, he 120 00:08:27.519 --> 00:08:31.079 and several physicists of the time had conjectured from Kepler's 121 00:08:31.120 --> 00:08:33.879 third law that the attraction of the Earth on a 122 00:08:33.919 --> 00:08:37.480 body would be found to decrease as the body was 123 00:08:37.559 --> 00:08:41.440 removed further away from the Earth, in a proportion inversely 124 00:08:41.519 --> 00:08:44.240 as the square of the distance from the center of 125 00:08:44.279 --> 00:08:49.240 the Earth. If this were the actual law, and gravity 126 00:08:49.279 --> 00:08:52.039 were the sole force which retained the Moon in its orbit, 127 00:08:52.919 --> 00:08:56.279 then t M prime should be to sixteen feet in 128 00:08:56.320 --> 00:08:59.320 a proportion which inversely is the square of the distance 129 00:08:59.360 --> 00:09:01.759 of the Moon from the center of the Earth to 130 00:09:01.840 --> 00:09:05.639 the square the radius of the Earth. In sixteen seventy nine, 131 00:09:05.679 --> 00:09:09.279 when he repeated the investigation, t M prime was found 132 00:09:09.320 --> 00:09:12.679 to have the value which was required by the hypothesis, 133 00:09:13.000 --> 00:09:16.759 and the verification was complete. But in sixteen sixty six 134 00:09:17.200 --> 00:09:20.039 his estimate of the distance of the Moon was inaccurate, 135 00:09:20.679 --> 00:09:22.840 and when he made the calculation he found that t 136 00:09:23.039 --> 00:09:26.320 m prime was about one eighth less than it ought 137 00:09:26.360 --> 00:09:30.960 to have been on his hypothesis. This discrepancy does not 138 00:09:31.080 --> 00:09:33.519 seem to have shaken his faith in the belief that 139 00:09:33.559 --> 00:09:37.600 gravity extended to the Moon and varied inversely as the 140 00:09:37.639 --> 00:09:41.159 square of the distance. But from Wiston's notes of a 141 00:09:41.240 --> 00:09:44.799 conversation with Newton, it would seem that Newton inferred that 142 00:09:44.919 --> 00:09:49.639 some other force, probably Descartes's vortices, acted on the Moon 143 00:09:49.720 --> 00:09:55.000 as well as gravity. This statement is confirmed by Pemberton's 144 00:09:55.000 --> 00:09:59.879 account of the investigation. It seems, moreover that Newton all 145 00:10:00.000 --> 00:10:04.480 already believed firmly in the principle of universal gravitation, that is, 146 00:10:04.639 --> 00:10:08.840 every particle of matter attracts every other particle, and suspected 147 00:10:08.840 --> 00:10:12.039 that the attraction varied as the product of their masses 148 00:10:12.320 --> 00:10:15.519 and inversely as the square of the distance between them. 149 00:10:15.919 --> 00:10:18.480 But it is certain that he did not then know 150 00:10:18.759 --> 00:10:21.759 what the attraction of a spherical mass on any external 151 00:10:21.799 --> 00:10:24.840 point would be, and he did not think it likely 152 00:10:25.159 --> 00:10:27.679 that a particle would be attracted by the Earth, as 153 00:10:27.759 --> 00:10:31.159 if the latter were concentrated into a single particle at 154 00:10:31.200 --> 00:10:37.000 its center. On his return to Cambridge in sixteen sixty seven, 155 00:10:37.399 --> 00:10:40.759 Newton was elected to a fellowship at his college and 156 00:10:40.879 --> 00:10:45.360 permanently took up his residence there. In the early part 157 00:10:45.399 --> 00:10:48.720 of sixteen sixty nine, or perhaps in sixteen sixty eight, 158 00:10:49.279 --> 00:10:52.840 he revised Barrow's lectures for him. The end of Lecture 159 00:10:52.919 --> 00:10:55.559 fourteen is known to have been written by Newton, but 160 00:10:55.720 --> 00:10:58.360 how much of the rest is due to his suggestions 161 00:10:58.720 --> 00:11:02.399 cannot now be determined. As soon as this was finished, 162 00:11:02.480 --> 00:11:04.960 he was asked by Barrow and Collins to edit and 163 00:11:05.080 --> 00:11:10.879 add notes to a translation of Kinghusen's Algebra, which he 164 00:11:10.960 --> 00:11:13.600 consented to do, but on condition that his name should 165 00:11:13.639 --> 00:11:18.200 not appear in the matter. In sixteen seventy he also 166 00:11:18.279 --> 00:11:22.360 began a systematic exposition of his analysis by infinite series, 167 00:11:22.919 --> 00:11:25.559 the object of which was to express the ordinate of 168 00:11:25.600 --> 00:11:30.120 a curve in an infinite algebraical series, every term of 169 00:11:30.200 --> 00:11:34.679 which could be integrated by Wallace's rule. His results on 170 00:11:34.720 --> 00:11:38.600 the subject have been communicated to Barrow, Collins, and others. 171 00:11:38.639 --> 00:11:43.120 In sixteen sixty nine, this was never finished. The fragment 172 00:11:43.240 --> 00:11:46.480 was published in seventeen eleven, but the substance of it 173 00:11:46.559 --> 00:11:49.679 had been printed as an appendix to the Optics in 174 00:11:49.759 --> 00:11:53.440 seventeen o four. These works were only the first of 175 00:11:53.480 --> 00:11:57.320 Newton's leisure, most of the time during these two years 176 00:11:57.759 --> 00:12:02.960 being given up to optical researches. In October sixteen sixty nine, 177 00:12:03.080 --> 00:12:09.039 Barrow resigned the Lucasian Chair in favor of Newton. During 178 00:12:09.080 --> 00:12:12.159 his tenure of the professorship, it was Newton's practice to 179 00:12:12.279 --> 00:12:16.000 lecture publicly once a week, from half an hour to 180 00:12:16.080 --> 00:12:19.360 an hour at a time, in one term of each year, 181 00:12:20.159 --> 00:12:23.360 probably dictating his lectures as rapidly as they could be 182 00:12:23.440 --> 00:12:27.200 taken down, and in the week following the lecture, to 183 00:12:27.279 --> 00:12:31.480 devote four hours to appointments, which he gave to students 184 00:12:31.480 --> 00:12:34.080 who wished to come to his rooms to discuss the 185 00:12:34.120 --> 00:12:39.039 results of the previous lecture. He never repeated a course, 186 00:12:39.440 --> 00:12:43.200 which usually consisted of nine or ten lectures, and generally 187 00:12:43.240 --> 00:12:46.399 the lectures of one course began from the point at 188 00:12:46.399 --> 00:12:50.480 which the preceding course had ended. The manuscripts of his 189 00:12:50.600 --> 00:12:54.320 lectures for seventeen of the first eighteen years of his 190 00:12:54.440 --> 00:12:59.720 tenure are extant. When first appointed, Newton chose optics for 191 00:12:59.759 --> 00:13:03.240 the subject of his lectures and researches, and before the 192 00:13:03.360 --> 00:13:05.879 end of sixteen sixty nine he had worked out the 193 00:13:05.919 --> 00:13:09.799 details of his discovery on the decomposition of a ray 194 00:13:09.919 --> 00:13:13.200 of white light into rays of different colors by means 195 00:13:13.200 --> 00:13:17.480 of a prism. The complete explanation of the theory of 196 00:13:17.600 --> 00:13:22.840 the rainbow followed from this discovery. These discoveries formed the 197 00:13:22.840 --> 00:13:26.720 subject matter of the lectures which he delivered as Lucasian 198 00:13:26.799 --> 00:13:31.639 Professor in the years sixteen sixty nine, sixteen seventy and 199 00:13:31.759 --> 00:13:36.840 sixteen seventy one. The chief new results were embodied in 200 00:13:36.919 --> 00:13:41.480 a paper communicated to the Royal Society in February sixteen 201 00:13:41.600 --> 00:13:47.840 seventy two, and subsequently published in the Philosophical Transactions. The 202 00:13:47.840 --> 00:13:51.519 manuscript of his original lectures was printed in seventeen twenty 203 00:13:51.600 --> 00:13:56.399 nine under the title Leccioni's Optiquet. This work is divided 204 00:13:56.440 --> 00:13:59.679 into two books, the first of which contains four sections, 205 00:13:59.679 --> 00:14:02.879 and the s second five. The first section of the 206 00:14:02.879 --> 00:14:06.039 first book deals with the decomposition of solar light by 207 00:14:06.039 --> 00:14:09.840 a prism in consequence of the unequal refragibility of the 208 00:14:09.919 --> 00:14:14.360 rays that compose it, and a description of his experiments 209 00:14:14.399 --> 00:14:18.360 is added. The second section contains an account of the 210 00:14:18.399 --> 00:14:23.279 method which Newton invented for the determining of the coefficients 211 00:14:23.320 --> 00:14:27.639 of refraction of different bodies. This is done by making 212 00:14:27.679 --> 00:14:31.200 a ray pass through a prism of the material so 213 00:14:31.240 --> 00:14:35.480 that the deviation is a minimum, and he proves that 214 00:14:35.600 --> 00:14:38.200 if the angle of the prism be i and the 215 00:14:38.240 --> 00:14:43.320 deviation of the ray be delta, then the refractive index 216 00:14:43.360 --> 00:14:46.279 will be the sign of one half ee plus delta 217 00:14:46.879 --> 00:14:51.440 multiplied by the cosec and half i. The third section 218 00:14:51.759 --> 00:14:56.879 is on refractions at plane surfaces. Here he sews that 219 00:14:56.919 --> 00:15:00.000 if a ray pass through a prism with minimum debas, 220 00:15:01.240 --> 00:15:04.639 the angle of incidents is equal to the angle of emergence. 221 00:15:05.320 --> 00:15:08.679 Most of this section is devoted to geometrical solutions of 222 00:15:08.759 --> 00:15:13.879 different problems. The fourth section contains a discussion of refractions 223 00:15:13.879 --> 00:15:17.960 at curved surfaces. The second book treats of the theory 224 00:15:18.039 --> 00:15:23.159 of colors of the rainbow. By a curious chapter of accidents, 225 00:15:24.000 --> 00:15:28.000 Newton failed to correct the chromatic aberration of two colors 226 00:15:28.000 --> 00:15:31.600 by means of a couple of prisms. He therefore abandoned 227 00:15:31.639 --> 00:15:35.919 the hope of making a refracting telescope which should be achromatic, 228 00:15:36.000 --> 00:15:40.080 and instead designed a reflecting telescope, probably on the model 229 00:15:40.080 --> 00:15:43.039 of a small one which he had made in sixteen 230 00:15:43.080 --> 00:15:46.879 sixty eight. The form he used is that still known 231 00:15:46.919 --> 00:15:52.320 by his name. The idea of it was naturally suggested 232 00:15:52.679 --> 00:15:58.159 by Gregory's telescope. In sixteen seventy two he invented a 233 00:15:58.200 --> 00:16:02.720 reflecting microscope, and some years later he invented the sixtent, 234 00:16:03.279 --> 00:16:08.600 which was rediscovered by Hadley in seventeen thirty one. His 235 00:16:08.720 --> 00:16:12.879 professorial lectures from sixteen seventy three to sixteen eighty three 236 00:16:13.320 --> 00:16:17.320 were on algebra and the theory of equations, and are described below. 237 00:16:18.279 --> 00:16:21.240 But much of his time during these years was occupied 238 00:16:21.279 --> 00:16:25.159 with other investigations, and I may remark that throughout his 239 00:16:25.279 --> 00:16:28.360 life Newton must have devoted at least as much attention 240 00:16:28.480 --> 00:16:33.440 to the chemistry and theology as to mathematics, though his 241 00:16:33.600 --> 00:16:37.240 conclusions are not of sufficient interest to require mention here. 242 00:16:38.279 --> 00:16:41.600 His theory of colors and his deductions from his optical 243 00:16:41.639 --> 00:16:46.240 experiments were attacked with considerable vehemence by parodies of France, 244 00:16:46.919 --> 00:16:51.279 Linus and Lucas at lieges Hooke in England and Huygens 245 00:16:51.279 --> 00:16:56.159 in Paris, but his opponents were finally refuted. The correspondence 246 00:16:56.240 --> 00:17:00.000 which this entailed on Newton occupied nearly all his life 247 00:17:00.000 --> 00:17:04.119 pleasure in the years sixteen seventy two to sixteen seventy five, 248 00:17:04.200 --> 00:17:08.920 and proved extremely distasteful to him. Writing on December ninth, 249 00:17:09.000 --> 00:17:12.279 sixteen seventy five, he says, I was so persecuted with 250 00:17:12.400 --> 00:17:15.799 discussions arising out of my theory of light that I 251 00:17:15.839 --> 00:17:19.759 blame my own imprudence for parting with so substantial a 252 00:17:19.799 --> 00:17:25.359 blessing as my quiet to run after a shadow again. 253 00:17:25.559 --> 00:17:30.039 After November eighteenth, sixteen seventy six, he observes, I see 254 00:17:30.160 --> 00:17:34.319 I have made myself a slave to philosophy. But if 255 00:17:34.319 --> 00:17:37.839 I get rid of mister Linus's business, I will resolutely 256 00:17:37.960 --> 00:17:41.720 bid adieu to it eternally, accepting what I do for 257 00:17:41.880 --> 00:17:45.000 my private satisfaction, or leave to come out after me. 258 00:17:45.920 --> 00:17:48.440 For I see a man must either resolve to put 259 00:17:48.519 --> 00:17:51.559 out nothing new or to become a slave to defend it. 260 00:17:52.559 --> 00:17:57.319 The unreasonable dislike to have his conclusions doubted or to 261 00:17:57.720 --> 00:18:01.599 be involved in any correspondence about was a prominent trait 262 00:18:01.720 --> 00:18:06.759 in Newton's character. He next set himself to examine the 263 00:18:06.799 --> 00:18:10.119 problem as to how light was really produced, and by 264 00:18:10.160 --> 00:18:12.720 the end of sixteen seventy five he had worked out 265 00:18:12.759 --> 00:18:16.519 the corpuscular or emission theory, a theory to which he 266 00:18:16.640 --> 00:18:19.920 was perhaps led by his researches on the theories of attraction. 267 00:18:21.039 --> 00:18:23.960 Only three ways have been suggested in which light can 268 00:18:24.000 --> 00:18:27.839 be produced mechanically. Either the eye may be supposed to 269 00:18:27.920 --> 00:18:31.839 send out something which, so to speak, feels the object, 270 00:18:32.119 --> 00:18:35.640 as the Greeks believed, or the object perceived may send 271 00:18:35.640 --> 00:18:39.000 out something which hits or affects the eye, as assumed 272 00:18:39.400 --> 00:18:43.319 in the emission theory. Or there may be some medium 273 00:18:43.359 --> 00:18:45.799 between the eye and the object, and the object may 274 00:18:45.920 --> 00:18:49.759 cause some change in the form or nature of this 275 00:18:49.880 --> 00:18:53.480 intervening medium, and thus affect the eye, as Hook and 276 00:18:53.559 --> 00:18:59.599 Huygen's supposed in the wave or undulatory theory. It will 277 00:18:59.599 --> 00:19:02.079 be enough here to say that on either of the 278 00:19:02.119 --> 00:19:06.720 two latter theories, all the obvious phenomena of geometrical optics, 279 00:19:06.759 --> 00:19:10.920 such as reflection, refraction, et cetera, can be accounted for. 280 00:19:11.799 --> 00:19:15.599 Within the present century, crucial experiments have been devised which 281 00:19:15.640 --> 00:19:19.039 give different results. According as one or the other theory 282 00:19:19.119 --> 00:19:22.799 is adopted. All these experiments agree with the results of 283 00:19:22.839 --> 00:19:26.480 the undulatory theory and differ from the results of the 284 00:19:26.880 --> 00:19:31.680 Newtonian theory. The latter is therefore untenable, but whether the 285 00:19:31.720 --> 00:19:34.599 former represents the whole truth and nothing but the truth 286 00:19:34.680 --> 00:19:38.559 is still an open question. Until, however, the theory of 287 00:19:38.640 --> 00:19:42.519 interference suggested by Young was worked out by Fresnel. The 288 00:19:42.640 --> 00:19:46.279 hypothesis of Huygens failed to account for all the facts 289 00:19:46.720 --> 00:19:49.279 and was open to more questions than that of Newton. 290 00:19:50.119 --> 00:19:53.400 It should be noted that Newton nowhere expresses an opinion 291 00:19:53.759 --> 00:19:57.319 that the corpuscular theory is true, but always treats it 292 00:19:57.440 --> 00:20:01.640 as a hypothesis from which, if true, certain results would follow. 293 00:20:02.559 --> 00:20:05.200 It would, moreover seem that he believed that the wave 294 00:20:05.279 --> 00:20:09.039 theory to be intrinsically more probable, and it was only 295 00:20:09.079 --> 00:20:12.799 the difficulty of explaining diffraction on that theory that led 296 00:20:12.839 --> 00:20:16.240 him to reject it. His remarks on the other physical 297 00:20:16.279 --> 00:20:22.400 subjects show a similar caution. Newton's corpuscular theory was expounded 298 00:20:22.440 --> 00:20:26.720 in memoirs communicated to the Royal Society in December sixteen 299 00:20:26.839 --> 00:20:31.759 seventy five, which are substantially reproduced in his Optics, published 300 00:20:32.200 --> 00:20:35.920 in seventeen o four. In the latter work he dealt 301 00:20:35.920 --> 00:20:39.680 in detail with his theory of fits of easy reflection 302 00:20:39.880 --> 00:20:43.559 and transmission, and the colors of thin plates, to which 303 00:20:43.559 --> 00:20:47.039 he added an explanation of the colors of thick plates 304 00:20:47.079 --> 00:20:51.720 and observations on the inflection of light. Two letters written 305 00:20:51.720 --> 00:20:55.000 by Newton in the year sixteen seventy six are sufficiently 306 00:20:55.079 --> 00:21:00.720 interesting to justify an allusion to them. Liibinets, who had 307 00:21:00.759 --> 00:21:04.599 been in London in sixteen seventy three, communicated some results 308 00:21:04.640 --> 00:21:08.079 to the Royal Society which he had supposed to be new, 309 00:21:08.240 --> 00:21:10.400 but which, it was pointed out to him, had been 310 00:21:10.440 --> 00:21:15.319 previously proved by Mouton. This led to a correspondence with oldenburg, 311 00:21:15.640 --> 00:21:20.200 the secretary of the Society, in sixteen seventy four. Leibniz 312 00:21:20.240 --> 00:21:24.200 wrote saying that he possessed general analytical methods depending on 313 00:21:24.279 --> 00:21:28.759 the infinite series. Oldenburgen reply told him that Newton and 314 00:21:28.799 --> 00:21:32.559 Gregory had used such series in their work. In answer 315 00:21:32.599 --> 00:21:36.799 to her request for information, Newton wrote on June thirteenth, 316 00:21:36.839 --> 00:21:39.880 sixteen seventy six, giving a brief account of his method, 317 00:21:40.240 --> 00:21:43.160 but adding the expansions of a binomial i e. The 318 00:21:43.160 --> 00:21:47.119 binomial theorem and of the arc sign of x, from 319 00:21:47.119 --> 00:21:49.440 the latter of which he deduced that of sign of x. 320 00:21:50.119 --> 00:21:53.240 This seems to be the earliest known incense of the 321 00:21:53.319 --> 00:21:57.559 reversion of a series. He also inserted an expression for 322 00:21:57.640 --> 00:22:01.319 the rectification of an elliptic arc in an infinite series. 323 00:22:02.079 --> 00:22:06.400 Leibnitz wrote on August twenty seventh asking for fuller details, 324 00:22:06.759 --> 00:22:09.640 and Newton, in a long but interesting reply, day on 325 00:22:09.759 --> 00:22:14.160 October twenty fourth, sixteen seventy six, and sent through Oldenburg, 326 00:22:14.720 --> 00:22:17.519 gives an account of the way in which he had 327 00:22:17.559 --> 00:22:21.279 been led to some of his results. In this letter, 328 00:22:21.599 --> 00:22:25.039 Newton begins by saying that altogether he had used three 329 00:22:25.079 --> 00:22:29.240 methods for expansion in a series. His first was arrived 330 00:22:29.240 --> 00:22:32.480 at from the study of the method of interpolation, by 331 00:22:32.759 --> 00:22:35.960 which Wallace had found expressions for the area of a 332 00:22:36.000 --> 00:22:41.039 circle and hyperbola. Thus, by considering the series of expressions 333 00:22:41.200 --> 00:22:44.880 one minus x squared quantity to the power zero over two, 334 00:22:45.960 --> 00:22:49.400 one minus x squad quantity to the powers two over two, 335 00:22:50.400 --> 00:22:53.880 and one minus x gred quantity to the four over two, 336 00:22:53.960 --> 00:22:58.359 and so on, he deduced by interpolations the law which 337 00:22:58.400 --> 00:23:02.680 connects the successive coefficial in the expansions of one minus 338 00:23:02.839 --> 00:23:06.359 x squared to the power of a half, one minus 339 00:23:06.519 --> 00:23:08.880 x square to the power of three halves, and so on, 340 00:23:09.599 --> 00:23:12.720 and then by analogy obtained the expression for the general 341 00:23:12.799 --> 00:23:16.079 term in the expansion of a binomial i e. The 342 00:23:16.119 --> 00:23:20.000 binomial theorem. He says that he proceeded to test this 343 00:23:20.200 --> 00:23:23.000 by forming the square of the expansion one minus x 344 00:23:23.079 --> 00:23:26.039 squared to the power of a half, which reduced to 345 00:23:26.519 --> 00:23:30.799 one minus x squared, and he proceeded in a similar 346 00:23:30.839 --> 00:23:35.079 way with other expansions. He next tested the theorem in 347 00:23:35.119 --> 00:23:38.119 the case of one minusx squared quantity to the power 348 00:23:38.119 --> 00:23:41.480 of one half by extracting the square root of one 349 00:23:41.559 --> 00:23:46.559 minus x squared mare arithmetico. He also used the series 350 00:23:46.599 --> 00:23:49.480 to determine the areas of the circle and the hyperbola 351 00:23:49.559 --> 00:23:52.160 in infinite series, and found that the results were the 352 00:23:52.200 --> 00:23:55.359 same as those he had arrived at by other means. 353 00:23:57.039 --> 00:24:00.319 Having established this result, he then discarded the method of 354 00:24:00.359 --> 00:24:04.480 interpolation in series and employed his binomial theorem to express, 355 00:24:04.519 --> 00:24:07.960 when possible, the ordinate of a curve in an infinite 356 00:24:08.000 --> 00:24:12.640 series in ascending powers of the abscissa, and thus, by 357 00:24:12.680 --> 00:24:16.799 Wallace's method he obtained expressions in an infinite series for 358 00:24:16.880 --> 00:24:20.400 the areas and arcs of curves, in the manner described 359 00:24:20.440 --> 00:24:24.720 in the Appendix to his Optics and his di anuisi 360 00:24:24.839 --> 00:24:31.440 perre equessiones numero to menorum infinitorum. He states that he 361 00:24:31.599 --> 00:24:36.200 had employed his second method before the plague in sixteen 362 00:24:36.240 --> 00:24:39.680 sixty five sixty six, and goes on to say that 363 00:24:39.720 --> 00:24:43.680 he was then obliged to leave Cambridge, and subsequently i e. 364 00:24:43.839 --> 00:24:47.359 Presumably on his return to Cambridge, he ceased to pursue 365 00:24:47.400 --> 00:24:51.400 these ideas, as he found that Nicholas Mercat had employed 366 00:24:51.440 --> 00:24:57.240 some of them in his Logarithmimotechnica, published in sixteen sixty eight, 367 00:24:57.799 --> 00:25:00.680 and he supposed that the remainder had been or would 368 00:25:00.680 --> 00:25:03.920 be found out before he himself was likely to publish 369 00:25:03.960 --> 00:25:08.640 his discoveries. Newton next explains that he had also a 370 00:25:08.680 --> 00:25:12.079 third method, of which he says he had about sixteen 371 00:25:12.160 --> 00:25:15.519 sixty nine sent an account to Barrow and Collins, illustrated 372 00:25:15.559 --> 00:25:21.680 by applications to areas rectification, qubature, and et cetera. This 373 00:25:21.880 --> 00:25:24.960 was the method of fluxions, but Newton gives no description 374 00:25:25.039 --> 00:25:28.640 of it here, though he adds some illustrations of its use. 375 00:25:29.279 --> 00:25:32.200 The first illustration is on the quadrature of the curve 376 00:25:32.279 --> 00:25:36.279 represented by the equation y equals ax to the power 377 00:25:36.440 --> 00:25:40.039 m multiplied by the quantity b plus c x to 378 00:25:40.079 --> 00:25:43.880 the power n quantity to the power p, which he 379 00:25:43.960 --> 00:25:48.079 says can be affected as a sum of quantity m 380 00:25:48.119 --> 00:25:52.240 plus one divided by n terms if quantity m plus 381 00:25:52.279 --> 00:25:55.359 one divided by m be a positive integer, and which 382 00:25:55.519 --> 00:25:59.920 he thinks cannot otherwise be affected except by an infinite series. 383 00:26:01.079 --> 00:26:03.400 He also gives a list of other forms which are 384 00:26:03.440 --> 00:26:07.920 immediately integrable, of which g f are x is raised 385 00:26:07.920 --> 00:26:11.559 to the power of m times n minus one multiplied 386 00:26:11.599 --> 00:26:14.480 by open bracket a plus b x the power n 387 00:26:14.640 --> 00:26:17.720 plus c x to the power of two n close 388 00:26:17.799 --> 00:26:22.319 bracket raise the power of minus one. Next x to 389 00:26:22.400 --> 00:26:26.640 the power of open bracket m plus one half close 390 00:26:26.720 --> 00:26:31.279 bracket n minus one multiplied by open bracket a plus 391 00:26:31.319 --> 00:26:33.640 b x the power of n plus c x of 392 00:26:33.680 --> 00:26:36.279 the power of two n close bracket raised to the 393 00:26:36.319 --> 00:26:40.559 power of minus one. The next term x to the 394 00:26:40.599 --> 00:26:45.039 power of m times n minus one, multiplied by open 395 00:26:45.119 --> 00:26:47.680 bracket a plus b x to the power of n 396 00:26:47.839 --> 00:26:50.119 plus c x of the power of two en close 397 00:26:50.240 --> 00:26:53.680 bracket raise to the power of plus or minus a half. 398 00:26:55.119 --> 00:26:58.279 Next term x to the power of M times n 399 00:26:58.359 --> 00:27:02.960 minus one multiplied open bracket A plus b x to 400 00:27:03.039 --> 00:27:06.640 the power of n close bracket plus or minus a 401 00:27:06.720 --> 00:27:10.279