WEBVTT 1 00:00:05.799 --> 00:00:08.960 Chapter sixteen, Part two of A Short Account of the 2 00:00:09.080 --> 00:00:14.199 History of Mathematics. This is a LibriVox recording. All LibriVox 3 00:00:14.240 --> 00:00:17.920 recordings are in the public domain. For more information or 4 00:00:17.960 --> 00:00:22.399 to volunteer, please visit LibriVox dot org. This is a 5 00:00:22.440 --> 00:00:26.039 reading by Paul King p J K dot scripts dot 6 00:00:26.199 --> 00:00:28.839 MI I T dot E d U forward slash p 7 00:00:29.039 --> 00:00:33.240 K J. A Short Account of the History of Mathematics 8 00:00:33.640 --> 00:00:38.759 by W. W. Rowseball. Chapter sixteen, The Life and Works 9 00:00:38.759 --> 00:00:44.359 of Newton, Part two, instigated by Halley. Newton now returned 10 00:00:44.359 --> 00:00:47.520 to the problem of gravitation, and before the autumn of 11 00:00:47.560 --> 00:00:50.439 sixteen eighty four he had worked out the substance of 12 00:00:50.520 --> 00:00:55.880 propositions one through nineteen twenty one thirty thirty two through 13 00:00:55.920 --> 00:01:00.479 thirty five in the first book of the Principia. These, 14 00:01:00.560 --> 00:01:02.719 together with the notes on the laws of motion and 15 00:01:02.840 --> 00:01:06.400 various lemmas, were read for his lectures in the Michailmas 16 00:01:06.519 --> 00:01:12.879 Term sixteen eighty four. In November Halley received Newton's promised communication, 17 00:01:13.519 --> 00:01:17.840 which probably consisted of the substance of propositions one eleven 18 00:01:17.920 --> 00:01:23.159 and either seventeen or corollaries one of thirteen, and thereupon 19 00:01:23.319 --> 00:01:26.840 he again went to Cambridge, where he saw a curious 20 00:01:26.879 --> 00:01:32.400 treatise Desmo Too, drawn up since August. Most likely this 21 00:01:32.519 --> 00:01:37.000 contained Newton's manuscript notes of the lectures above alluded to. 22 00:01:37.799 --> 00:01:40.640 These notes are now in the University Library and are 23 00:01:40.640 --> 00:01:46.400 headed de motu cooporum. Halley begged that the results might 24 00:01:46.439 --> 00:01:49.719 be published, and finally secured a promise that they should 25 00:01:49.760 --> 00:01:54.040 be sent to the Royal Society. They were accordingly communicated 26 00:01:54.079 --> 00:01:57.959 to the Society not later than February sixteen eighty five, 27 00:01:58.439 --> 00:02:02.120 in the paper desmo Too, which contained the substance of 28 00:02:02.159 --> 00:02:11.400 the following propositions in the Principia Book I propositions one, four, six, seven, ten, eleven, fifteen, 29 00:02:11.599 --> 00:02:16.879 seventeen thirty two and Book two Propositions two, three and four. 30 00:02:17.840 --> 00:02:20.599 It seems also to have been due to the influence 31 00:02:20.680 --> 00:02:23.680 in tact of Halley at this visit in November sixteen 32 00:02:23.719 --> 00:02:28.000 eighty four, that Newton undertook to attack the whole problem 33 00:02:28.039 --> 00:02:33.319 of gravitation and practically pledged himself to publish his results. 34 00:02:34.879 --> 00:02:37.960 As yet, Newton had not determined the attraction of a 35 00:02:37.960 --> 00:02:42.360 spherical body on an external point, nor had he calculated 36 00:02:42.400 --> 00:02:46.039 the details of the planetary motions. Even if the members 37 00:02:46.039 --> 00:02:49.919 of the Solar system could be regarded as points. The 38 00:02:50.000 --> 00:02:53.639 first problem was solved in sixteen eighty five, probably either 39 00:02:53.680 --> 00:02:57.719 in January or February. No sooner, to quote from doctor 40 00:02:57.759 --> 00:03:01.240 Galicier's address on the Bison tenorary of the publication of 41 00:03:01.280 --> 00:03:05.680 the Principia, had Newton proved the superb theorem. And we 42 00:03:05.759 --> 00:03:08.240 know from his own words that he had no expectation 43 00:03:08.400 --> 00:03:11.080 of so beautiful a result till it emerged from his 44 00:03:11.199 --> 00:03:15.840 mathematical investigation that all the mechanism of the universe at 45 00:03:15.879 --> 00:03:20.400 once lay spread before him. When he discovered the theorems 46 00:03:20.439 --> 00:03:23.879 that from the first three sections of Book one, when 47 00:03:23.879 --> 00:03:26.919 he gave them in his lectures of sixteen eighty four, 48 00:03:27.719 --> 00:03:30.360 he was unaware that the Sun and Earth exerted their 49 00:03:30.400 --> 00:03:34.280 attractions as if they were but points. How different must 50 00:03:34.280 --> 00:03:38.039 these propositions have seen to Newton's eyes when he realized 51 00:03:38.080 --> 00:03:41.000 that these results, which he had believed to be only 52 00:03:41.039 --> 00:03:47.400 approximately true when applied to the Solar system, were really exact. Hitherto, 53 00:03:47.919 --> 00:03:50.520 they had been only true in so far as he 54 00:03:50.560 --> 00:03:53.199 could regard the Sun as a point compared to the 55 00:03:53.240 --> 00:03:56.280 distance of the planets, or the Earth as a point 56 00:03:56.319 --> 00:03:59.639 compared to the distance of the Moon, a distance amounting 57 00:03:59.680 --> 00:04:03.400 to all about sixty times the Earth's radius. But now 58 00:04:03.879 --> 00:04:09.360 they were mathematically true, excepting only for the slight deviation 59 00:04:10.000 --> 00:04:13.800 from a perfectly spherical form of the Sun, Earth, and planets. 60 00:04:14.960 --> 00:04:18.199 We can imagine the effect of this sudden transition from 61 00:04:18.240 --> 00:04:23.879 approximation to exactitude in stimulating Newton's mind to still greater efforts. 62 00:04:24.519 --> 00:04:27.720 It was now in his power to apply the mathematical 63 00:04:27.759 --> 00:04:32.759 analysis with absolute precision to the actual problems of astronomy. 64 00:04:34.040 --> 00:04:37.879 Of the three fundamental principles applied in the principia, we 65 00:04:37.959 --> 00:04:40.920 may say that the idea that every particle attracts every 66 00:04:40.920 --> 00:04:43.720 other particle in the universe was formed at least as 67 00:04:43.800 --> 00:04:48.800 early as sixteen sixty six. The law of equable distribution 68 00:04:48.920 --> 00:04:52.199 of areas its consequences, and the fact that if the 69 00:04:52.319 --> 00:04:54.759 law of attraction were that of the inverse square of 70 00:04:54.800 --> 00:04:57.800 the orbit of a particle about a center of force 71 00:04:58.279 --> 00:05:01.680 would be a conic where proved in sixteen seventy nine. 72 00:05:01.839 --> 00:05:05.480 And lastly, the discovery that a sphere whose density at 73 00:05:05.519 --> 00:05:08.240 any point depends only on the distance from the center 74 00:05:09.000 --> 00:05:12.680 attracts an external point as if the whole mass were 75 00:05:13.079 --> 00:05:16.680 collected at its center were made in sixteen eighty five. 76 00:05:18.040 --> 00:05:21.360 It was this last discovery that enabled him to apply 77 00:05:21.439 --> 00:05:25.040 the first two principles to the phenomenon of bodies of 78 00:05:25.120 --> 00:05:30.000 finite size. The draft of the first Book of the 79 00:05:30.040 --> 00:05:34.079 Principia was finished before the summer of sixteen eighty five, 80 00:05:34.560 --> 00:05:38.079 but the corrections and editions took some time, and the 81 00:05:38.120 --> 00:05:41.800 book was not presented to the Royal Society until April 82 00:05:41.839 --> 00:05:47.600 twenty eighth, sixteen eighty six. This book is given up 83 00:05:47.639 --> 00:05:50.839 to the consideration of the motion of particles or bodies 84 00:05:50.879 --> 00:05:54.560 in free space, either in known orbits, or under the 85 00:05:54.600 --> 00:05:59.680 action of known forces, or under their mutual attraction. In it, 86 00:06:00.040 --> 00:06:03.279 Uton generalizes the law of attraction into a statement that 87 00:06:03.360 --> 00:06:06.480 every particle of matter in the universe attracts every other 88 00:06:06.519 --> 00:06:10.040 particle with a force which varies directly as the product 89 00:06:10.120 --> 00:06:13.319 of their masses and inversely as a square of the 90 00:06:13.360 --> 00:06:16.879 distance between them, and he thence deduces the law of 91 00:06:16.959 --> 00:06:22.040 attraction for spherical shells of constant density. The book is 92 00:06:22.120 --> 00:06:27.560 prefaced by an introduction on the science of dynamics. The 93 00:06:27.639 --> 00:06:30.879 second Book of the Principia was completed by the summer 94 00:06:30.959 --> 00:06:35.199 of sixteen eighty six. This book treats of motion in 95 00:06:35.279 --> 00:06:40.000 a resting medium, and of hydrostatics and hydrodynamics, with a 96 00:06:40.040 --> 00:06:44.839 special application to waves, tides, and acoustics. He concludes it 97 00:06:44.879 --> 00:06:49.639 by shewing that the Cartesian theory of vortices is inconsistent 98 00:06:49.839 --> 00:06:52.519 both with the known facts and with the laws of motion. 99 00:06:53.639 --> 00:06:56.199 The next nine or ten months were devoted to the 100 00:06:56.279 --> 00:07:01.360 third book, probably for this he had originally no materials 101 00:07:01.399 --> 00:07:04.879 ready in it. The theorems obtained in the first book 102 00:07:04.920 --> 00:07:08.000 are applied to the chief phenomenon of the Solar System. 103 00:07:08.240 --> 00:07:12.319 The masses and distances of the planets, and, wherever sufficient 104 00:07:12.399 --> 00:07:17.439 data existed, of their satellites are determined. In particular, the 105 00:07:17.480 --> 00:07:21.839 motion of the Moon, the various inequalities therein, and the 106 00:07:21.839 --> 00:07:25.319 theory of the tides are worked out in detail. He 107 00:07:25.399 --> 00:07:28.720 also investigates the theory of comets shoes that they belong 108 00:07:28.800 --> 00:07:32.560 to the Solar system, explains how from three observations the 109 00:07:32.720 --> 00:07:36.639 orbit can be determined, and illustrates his results by considering 110 00:07:36.720 --> 00:07:41.199 certain special commets. The third book, as we have it, 111 00:07:41.279 --> 00:07:43.480 is but little more than a sketch of what Newton 112 00:07:43.560 --> 00:07:47.800 had finally proposed to himself to accomplish His original scheme 113 00:07:47.920 --> 00:07:52.399 is among the Portsmouth papers and his notes shoe that 114 00:07:52.519 --> 00:07:55.439 he continued to work at it for some years after 115 00:07:55.480 --> 00:07:58.920 the publication of the first edition of the Principia. The 116 00:07:58.959 --> 00:08:02.839 most interesting of his memoranda are those in which, by 117 00:08:02.920 --> 00:08:06.439 means of fluxions, he has carried out his results beyond 118 00:08:06.439 --> 00:08:09.360 the point at which he was able to translate them 119 00:08:09.399 --> 00:08:15.639 into geometry footnote. I take this opportunity of saying that 120 00:08:15.720 --> 00:08:18.720 I hope shortly to publish a memoir on the history 121 00:08:18.759 --> 00:08:23.079 and compilation of the Principia. The following brief summary of 122 00:08:23.120 --> 00:08:25.879 the contents of the work will give the reader a 123 00:08:25.879 --> 00:08:30.160 general idea of its arrangement. The Principia is preceded by 124 00:08:30.160 --> 00:08:33.120 a preface in which Newton says that his object is 125 00:08:33.159 --> 00:08:38.159 to apply mathematics to the phenomena of nature. Among these phenomena, 126 00:08:38.440 --> 00:08:42.320 motion is one of the most important. Now, motion is 127 00:08:42.360 --> 00:08:46.000 the effective force, and though he does not know what 128 00:08:46.120 --> 00:08:50.080 is the nature or origin of force, still many of 129 00:08:50.120 --> 00:08:53.000 its effects can be measured, and it is these that 130 00:08:53.120 --> 00:08:58.360 form the subject matter of the work. The work begins, therefore, naturally, 131 00:08:58.440 --> 00:09:02.519 with an introduction on dynamics, the science of motion, This 132 00:09:02.600 --> 00:09:06.440 commences with the eight definitions of the various terms such 133 00:09:06.440 --> 00:09:10.399 as mass, momentum, and so on. Newton then lays down 134 00:09:10.480 --> 00:09:13.799 three laws of motion, which are incapable of exact proof, 135 00:09:14.480 --> 00:09:18.559 but are confirmed partly by direct experiments, partly by the 136 00:09:18.840 --> 00:09:23.840 agreement with observation of the deductions from them. From these 137 00:09:23.919 --> 00:09:28.080 he deduces six fundamental principles of mechanics and adds an 138 00:09:28.120 --> 00:09:34.480 appendix on the motion of falling bodies, projectiles, oscillations, impact, 139 00:09:34.840 --> 00:09:38.639 and the mutual attractions of two bodies. The most important 140 00:09:38.639 --> 00:09:44.559 deduction is that of the parallelogram of velocities, accelerations, and forces. 141 00:09:45.720 --> 00:09:48.679 The first book of the Principia is on the motion 142 00:09:48.799 --> 00:09:52.919 of bodies in free space, and is divided into fourteen sections. 143 00:09:54.440 --> 00:09:59.039 The first section consists of eleven preliminary lemmas treated by 144 00:09:59.039 --> 00:10:02.679 the method of prime and ultimate ratios and not by 145 00:10:02.720 --> 00:10:07.879 that of indivisibles. The second section commences by shewing that 146 00:10:07.960 --> 00:10:10.759 if a body such as a planet, revolve in an 147 00:10:10.879 --> 00:10:13.840 orbit subject to a force tending to affix points such 148 00:10:13.879 --> 00:10:17.559 as the Sun, the areas swept out by radii drawn 149 00:10:17.720 --> 00:10:20.120 from the body to the point are in one plane 150 00:10:20.720 --> 00:10:25.200 and are proportional to the times of describing them, and conversely, 151 00:10:25.360 --> 00:10:29.000 if the areas be proportional to the times, the force 152 00:10:29.039 --> 00:10:31.480 acting on the body must be directed to the point. 153 00:10:32.279 --> 00:10:35.399 Newton then chows how if the orbit be known and 154 00:10:35.480 --> 00:10:38.080 the center of force be given, the law of force 155 00:10:38.120 --> 00:10:41.320 can be determined, and he finds the law for various curves. 156 00:10:42.519 --> 00:10:45.759 In the third section, he applies these propositions to a 157 00:10:45.799 --> 00:10:49.840 body which describes a conic section about a focus, and 158 00:10:50.000 --> 00:10:53.120 proves that the force must vary inversely as a square 159 00:10:53.159 --> 00:10:56.960 of the distance, and that Kepler's third law would necessarily 160 00:10:57.039 --> 00:11:00.799 be true of such a system. He proves that if 161 00:11:00.799 --> 00:11:04.000 a body were projected in any way and subject to 162 00:11:04.039 --> 00:11:07.559 a central force which varied according to this law, then 163 00:11:07.559 --> 00:11:10.240 it must move in a conic section having the center 164 00:11:10.759 --> 00:11:16.240 of force in a focus. He concludes proposition seventeen corollaries 165 00:11:16.279 --> 00:11:20.200 three and four with a suggestion as to how the 166 00:11:20.240 --> 00:11:24.480 effects of distributing forces should be calculated. This was first 167 00:11:24.519 --> 00:11:28.840 done by the brilliant investigations of Laplace and Lagrange, and 168 00:11:28.960 --> 00:11:33.200 Laplace says that Lagrange's paper in the Berlin Memoirs for 169 00:11:33.320 --> 00:11:36.720 seventeen eighty six, on which the modern treatment of the 170 00:11:36.759 --> 00:11:40.360 subject is founded, was suggested by these remarks of Newton. 171 00:11:41.360 --> 00:11:44.600 The fourth and fifth sections are devoted to the geometry 172 00:11:44.600 --> 00:11:49.159 of conic sections, especially to the construction of conics, which 173 00:11:49.200 --> 00:11:54.159 satisfy five conditions. In section four. One of the conditions 174 00:11:54.200 --> 00:11:57.039 is that the focus is given. This includes the problem 175 00:11:57.039 --> 00:11:59.799 of finding the path of a comet from three observations, 176 00:11:59.799 --> 00:12:03.080 with which Newton says he found the most difficult problem 177 00:12:03.200 --> 00:12:07.240 of any which he had to solve curiously enough. He 178 00:12:07.279 --> 00:12:10.200 gave a second solution of the problem in Book three, 179 00:12:10.360 --> 00:12:14.639 proposition forty one, in which he recommended as more simple, 180 00:12:14.720 --> 00:12:19.519 but which is inapplicable in practice. The sixth section is 181 00:12:19.559 --> 00:12:22.879 devoted to determining what at any given time is the 182 00:12:22.960 --> 00:12:26.240 velocity and what is the position of a body, which 183 00:12:26.279 --> 00:12:29.639 is describing a given conic about a center of attraction 184 00:12:29.759 --> 00:12:35.039 in a focus, together with various converse problems. To effect this, 185 00:12:35.200 --> 00:12:37.440 Newton had to find the area of a sector of 186 00:12:37.480 --> 00:12:41.240 a conic. This is easily done for the parabola. He 187 00:12:41.320 --> 00:12:44.240 then endeavors to shew that the exact quadrature of any 188 00:12:44.279 --> 00:12:48.120 closed oval curve having no infinite branches, such as the ellipse, 189 00:12:48.639 --> 00:12:52.279 is impossible. The proof is not correct, as it stands 190 00:12:52.440 --> 00:12:54.879 since the result is not true for ovals of the 191 00:12:54.960 --> 00:12:58.159 form why to the power of two M equals the 192 00:12:58.240 --> 00:13:00.879 quantity A to the power of t two n minus 193 00:13:01.080 --> 00:13:04.320 x to the power of two n all multiplied by 194 00:13:05.200 --> 00:13:08.399 two to the power of two m times n to 195 00:13:08.480 --> 00:13:13.120 the power of two m multiplied by x to the 196 00:13:13.159 --> 00:13:17.120 power of two M times two n minus one, where 197 00:13:17.320 --> 00:13:21.200 M and n are positive integers. Newton seems himself to 198 00:13:21.279 --> 00:13:24.440 have felt some doubt about inserting it, though he believed 199 00:13:24.440 --> 00:13:28.679 the result to be true, an exact quadrature being impossible. 200 00:13:29.080 --> 00:13:33.279 He proceeds to give three ways, two arithmetical and one geometrical, 201 00:13:33.720 --> 00:13:37.639 of approximating to the sectorial area of an ellipse as 202 00:13:37.679 --> 00:13:42.279 closely as is desired. The seventh section is given up 203 00:13:42.320 --> 00:13:45.360 to the discussion of motion in a straight line under 204 00:13:45.360 --> 00:13:48.480 a force which varies inversely as a square of the 205 00:13:48.519 --> 00:13:52.000 distance and its comparison with the motion, and a conic 206 00:13:52.120 --> 00:13:56.399 under the same force. He concludes by giving a general 207 00:13:56.440 --> 00:13:59.840 solution for all the problems considered in this section. For 208 00:13:59.960 --> 00:14:04.440 X any law of force. He here determines geometrically, what 209 00:14:04.559 --> 00:14:08.399 is equivalent of finding the integral of X divided by 210 00:14:08.440 --> 00:14:11.519 the square root of the quantity ax minus x squared. 211 00:14:13.000 --> 00:14:16.480 The eighth section contains a general solutions for any orbit 212 00:14:16.559 --> 00:14:19.759 described under any central force of some of the problems 213 00:14:19.799 --> 00:14:24.519 previously considered. In proposition forty he states that the kinetic 214 00:14:24.679 --> 00:14:27.399 energy acquired by a body and moving from one point 215 00:14:27.440 --> 00:14:30.120 to another is equal to the total work done by 216 00:14:30.120 --> 00:14:34.360 the force between those two points. In the ninth section, 217 00:14:34.519 --> 00:14:37.600 he discusses the case or the orbit is in motion 218 00:14:37.799 --> 00:14:40.679 in its own plane round the center of force, and 219 00:14:40.759 --> 00:14:44.559 treats in detail of the motion of the apse line 220 00:14:45.039 --> 00:14:48.000 and the forces by which a given motion would be produced. 221 00:14:48.840 --> 00:14:53.279 Newton applied this reasoning proposition forty five corollary IWO to 222 00:14:53.360 --> 00:14:56.279 the case of the moon, but the resulting motion of 223 00:14:56.320 --> 00:14:59.279 the apses only came out about one half of the 224 00:14:59.320 --> 00:15:03.279 actual amount. The approximation was, in fact not carried to 225 00:15:03.279 --> 00:15:07.480 a sufficiently high order. Newton was aware of the discrepancy, 226 00:15:08.000 --> 00:15:11.960 and as he explained the similar difficulty in the case 227 00:15:12.039 --> 00:15:17.679 of the nodes, it had been long suspected Goldfrays Lunar Theory, 228 00:15:17.960 --> 00:15:22.000 second Edition, Article sixty eight that this scolium of the 229 00:15:22.039 --> 00:15:26.799 first edition to book three, Proposition thirty five meant that 230 00:15:26.840 --> 00:15:32.080 he had found the explanation. Nowhere in the principia does he, however, 231 00:15:32.200 --> 00:15:35.559 give any hint as to how this was affected, and 232 00:15:35.600 --> 00:15:38.279 that the true explanation of a difference which had long 233 00:15:38.440 --> 00:15:42.519 formed an obstacle to the universal acceptance of the Newtonian system, 234 00:15:43.559 --> 00:15:48.919 was first given by claireaut in seventeen fifty two. The 235 00:15:48.960 --> 00:15:53.159 Portsmouth Papers contained Newton's original work, and shew that he 236 00:15:53.200 --> 00:15:56.799 had obtained the true value by carrying the approximation to 237 00:15:56.879 --> 00:16:00.919 a sufficiently high order. It also seemed clear from these 238 00:16:00.960 --> 00:16:04.759 papers that Newton gave the corollary to Book one proposition 239 00:16:04.879 --> 00:16:08.080 forty five as a mere illustration to the motion of 240 00:16:08.120 --> 00:16:11.639 the apses and orbits, which are nearly circular, and did 241 00:16:11.639 --> 00:16:14.279 not mean it to apply to the moon. But by 242 00:16:14.320 --> 00:16:18.399 an inadvertence in the second and third editions, a reference 243 00:16:18.440 --> 00:16:21.840 to it as an authenticity for a result connected with 244 00:16:21.919 --> 00:16:25.840 the moon was added, which would naturally deceive any reader. 245 00:16:27.039 --> 00:16:29.840 Newton left most of the revision to the second edition 246 00:16:30.320 --> 00:16:34.679 to Cotes, and it is probable that the mistake is 247 00:16:34.759 --> 00:16:39.519 due to a blunder of the editor. Other questions connected 248 00:16:39.559 --> 00:16:44.320 with lunar and planetary irregularities are also discussed in this proposition, 249 00:16:44.799 --> 00:16:48.840 but the extreme conclusions of Newton misled all the early 250 00:16:48.960 --> 00:16:52.960 commentators and even Laplace in his se stem de Monde, 251 00:16:53.240 --> 00:16:56.919 published in seventeen ninety six, speaks of Newton as having 252 00:16:56.960 --> 00:17:00.480 only roughly sketched out this part of the subject, leaving 253 00:17:00.519 --> 00:17:04.279 it to be completed when the calculus should be further perfected. 254 00:17:05.039 --> 00:17:08.559 But in the last volume of his Mechanique Cerest, published 255 00:17:08.599 --> 00:17:12.160 in eighteen twenty five, he says that on more careful reading, 256 00:17:12.480 --> 00:17:15.359 he has no hesitation in regarding it as among the 257 00:17:15.400 --> 00:17:20.160 most profound parts of the work. The tenth section is 258 00:17:20.200 --> 00:17:24.799 devoted to the consideration of motion of bodies along given surfaces, 259 00:17:24.839 --> 00:17:27.599 but not in planes passing through the center of force, 260 00:17:28.200 --> 00:17:32.079 with special reference to the vibration of pendulums and the 261 00:17:32.119 --> 00:17:36.720 determination of the accelerating effect of gravity. In connection with 262 00:17:36.759 --> 00:17:43.119 the latter problem, Newton investigates the chief geometrical properties of cycloids, epicycloids, 263 00:17:43.160 --> 00:17:48.720 and hypocycloids. In the eleventh section are considered the problems 264 00:17:48.759 --> 00:17:52.200 connected with motions in orbits where the center of forces 265 00:17:52.279 --> 00:17:56.319 disturbed or the moving body is disturbed by other forces. 266 00:17:56.920 --> 00:18:01.079 Until calculus of variations was invented by lug in seventeen 267 00:18:01.160 --> 00:18:04.599 fifty five, it was impossible to do more than sketch 268 00:18:04.640 --> 00:18:07.400 out the principles on which the problem should be solved, 269 00:18:07.880 --> 00:18:11.000 and laplace in his mechanique. Cerest was the first to 270 00:18:11.039 --> 00:18:14.680 work out most of the questions in any detail. Newton 271 00:18:14.759 --> 00:18:18.960 commences by considering the disturbance produced by the mutual action 272 00:18:19.079 --> 00:18:23.640 of two bodies revolving around one another. He then proceeds 273 00:18:23.680 --> 00:18:26.839 to consider the problem of three or more bodies which 274 00:18:26.960 --> 00:18:31.279 mutually attract one another. He first solves the question completely 275 00:18:31.440 --> 00:18:34.839 if the force of attraction varies directly as the distance. 276 00:18:35.559 --> 00:18:38.640 He next takes the case of the three bodies moving 277 00:18:38.759 --> 00:18:42.720 under their mutual attractions as in nature. This problem has 278 00:18:42.799 --> 00:18:46.000 not been yet solved generally, but in Newton's day it 279 00:18:46.079 --> 00:18:48.920 was beyond any analysis of which he had the command. 280 00:18:49.680 --> 00:18:53.279 He contrived, however, to work out roughly the chief effects 281 00:18:53.279 --> 00:18:55.920 of the disturbing action of the Sun on the motion 282 00:18:56.079 --> 00:19:01.279 of the moon proposition sixty six. Position was singled out 283 00:19:01.319 --> 00:19:04.960 by Lagrange as the most striking single illustration of the 284 00:19:05.000 --> 00:19:09.839 genius of Newton. To this proposition, twenty two corollaries are appended, 285 00:19:09.839 --> 00:19:13.160 in which it is applied to determine the motion in longitude. 286 00:19:13.240 --> 00:19:17.400 In latitude, the annual equation, the motion of the apse 287 00:19:17.480 --> 00:19:21.119 line and of the nodes, the evection, the change and 288 00:19:21.200 --> 00:19:24.680 inclination of the plane of the lunar orbit, the precession 289 00:19:24.720 --> 00:19:28.559 of the equinoxes, and the theory of the tides. The 290 00:19:28.599 --> 00:19:31.519 greater part of the third book consists of the numerical 291 00:19:31.559 --> 00:19:34.839 application of these principles to the case of the moon 292 00:19:34.880 --> 00:19:38.519 in the Earth. Lastly, Newton showed how from the motion 293 00:19:38.720 --> 00:19:41.960 of the nodes of the interior, constitution of the body 294 00:19:42.400 --> 00:19:46.160 could be roughly determined. Up to this point, Newton had 295 00:19:46.240 --> 00:19:49.559 generally treated the bodies with which he dealt as if 296 00:19:49.640 --> 00:19:53.160 they were particles. He now proceeds in section twelve to 297 00:19:53.240 --> 00:19:57.240 consider the attraction of spherical masses, which are either of 298 00:19:57.359 --> 00:20:01.119 uniform density or whose density at an point is a 299 00:20:01.160 --> 00:20:04.079 single valued function of the distance of the point from 300 00:20:04.079 --> 00:20:06.960 the center of the sphere. These are worked out for 301 00:20:07.039 --> 00:20:11.880 any law of attraction. In section thirteen he gives some 302 00:20:12.039 --> 00:20:15.440 general theorems on the theory of attractions and on some 303 00:20:15.720 --> 00:20:20.160 propositions dealing with the attractions of solids of revolution. But 304 00:20:20.240 --> 00:20:23.759 these problems are almost insoluble without the aid of the 305 00:20:23.799 --> 00:20:29.640 infinitesimal calculus and the Newtonian account of them is incomplete. 306 00:20:31.559 --> 00:20:34.960 The fourteenth section contains a statement of some theories and 307 00:20:35.079 --> 00:20:39.240 experiments in physical optics, and a solution by geometry of 308 00:20:39.319 --> 00:20:42.759 some problems in geometrical optics, particularly on the form of 309 00:20:42.839 --> 00:20:48.880 aplanatic refracting surfaces of revolution. The second book of the 310 00:20:48.920 --> 00:20:54.000 Principia is concerned with hydromechanics, and especially with motion in 311 00:20:54.079 --> 00:20:57.839 a resisting medium. These questions are not worked out so 312 00:20:58.000 --> 00:21:01.279 completely as those treated in the first book, and though 313 00:21:01.319 --> 00:21:04.359 this book provided the basis on which much of the 314 00:21:04.400 --> 00:21:09.440 subsequent work of Danielle Bernoulli, clairout De, Lambert, Euler and 315 00:21:09.559 --> 00:21:13.519 Laplace were erected, it is not of the same epoch 316 00:21:13.599 --> 00:21:17.839 making character as the first book. This book is divided 317 00:21:17.880 --> 00:21:21.640 into nine sections. The motion of bodies in a medium 318 00:21:21.680 --> 00:21:25.839 where the resistance varies directly as the velocity is considered 319 00:21:25.880 --> 00:21:29.519 in the first section. The motions where resistance varies as 320 00:21:29.559 --> 00:21:32.759 the square of the velocity is discussed in the second section. 321 00:21:33.519 --> 00:21:37.240 The motion where the resistance can be expressed as a 322 00:21:37.240 --> 00:21:40.799 sum of two terms, one of which varies as the velocity, 323 00:21:40.839 --> 00:21:43.960 and the others the square of the velocity, is dealt 324 00:21:43.960 --> 00:21:49.440 with in the third section. The second section contains proposition 325 00:21:49.559 --> 00:21:53.079 twenty five, a construction for the shape of the solid 326 00:21:53.160 --> 00:21:57.359 of least resistance. No proof is given, and it had 327 00:21:57.440 --> 00:22:00.400 been long somewhat of a mystery to know how Newton 328 00:22:00.400 --> 00:22:03.160 had contrived to solve the problem without the use of 329 00:22:03.279 --> 00:22:08.200 calculus of variations. Newton's demonstrations, there are two of them, 330 00:22:08.559 --> 00:22:13.359 have been recently discovered in the Portsmouth collection. The fourth 331 00:22:13.440 --> 00:22:17.680 section is devoted to a spiral motion in a resisting medium, 332 00:22:19.079 --> 00:22:23.799 the fifth to the theory of hydrostatics and elastic fluids, 333 00:22:24.319 --> 00:22:28.240 the sixth to the motion of pendulums in a resisting medium, 334 00:22:28.680 --> 00:22:33.000 the seventh to hydrodynamics, especially to the motion of projectiles 335 00:22:33.039 --> 00:22:37.920 in air and other fluids. The eighth to the theory 336 00:22:37.960 --> 00:22:41.240 of waves, including the principles from which the chief effects 337 00:22:41.279 --> 00:22:45.440 of the wave hypotheses in light and sound are calculated, 338 00:22:45.839 --> 00:22:50.359 and in particular the velocity of sound is determined. In 339 00:22:50.400 --> 00:22:54.440 a ninth section, Newton discusses the Cartesian theory of vortices. 340 00:22:55.119 --> 00:22:58.400 He begins by shewing that if there were no internal friction, 341 00:22:59.039 --> 00:23:02.880 the motion would be impossible. He must therefore assume some 342 00:23:03.079 --> 00:23:07.680 law of friction, and as a working hypothesis he supposes 343 00:23:07.759 --> 00:23:11.759 that the resistance arising from want of lubricity in the 344 00:23:11.759 --> 00:23:17.000 parts of a fluid is catius peribus proportional to the 345 00:23:17.079 --> 00:23:21.000 velocity of which the parts of the fluid are separated 346 00:23:21.000 --> 00:23:25.359 from each other. This hypothesis, as he himself remarks, is 347 00:23:25.400 --> 00:23:28.079 probably not altogether correct, but he thinks that it will 348 00:23:28.119 --> 00:23:31.480 give a general idea of the motion. He next proves 349 00:23:31.480 --> 00:23:35.359 that on this hypothesis that the motion would be unstable. 350 00:23:35.880 --> 00:23:40.480 He must therefore suppose that some constraining force prevents this catastrophe, 351 00:23:40.960 --> 00:23:44.279 and he then chows that in that case, Kepler's third 352 00:23:44.359 --> 00:23:47.880 law could not be true. Lastly, he chows, by independent 353 00:23:47.960 --> 00:23:52.200 reasoning that the hypothesis must lead to results which are 354 00:23:52.200 --> 00:23:56.160 inconsistent with Kepler's other two laws, and that both the 355 00:23:56.279 --> 00:24:01.200 vortices and the motion of the planets would necessarily be unstable. 356 00:24:01.920 --> 00:24:06.839 Several continental mathematicians made attempts to modify the Cartesian hypothesis 357 00:24:06.880 --> 00:24:10.079 so as to avoid these conclusions, but they could never 358 00:24:10.200 --> 00:24:14.880 explain one phenomenon without introducing fresh difficulties. It may be 359 00:24:14.960 --> 00:24:19.519 taken that by seventeen fifty the Cartesian theory was finally abandoned. 360 00:24:21.000 --> 00:24:24.160 The third book is headed on the system of the world, 361 00:24:24.319 --> 00:24:27.480 and is concerned chiefly with the application of the results 362 00:24:27.519 --> 00:24:30.880 of the first book to the Solar System. It is 363 00:24:30.960 --> 00:24:36.000 introduced by certain rules of philosophizing and as a list 364 00:24:36.079 --> 00:24:42.440 of certain data obtained from astronomical observations. The rules are one, 365 00:24:42.720 --> 00:24:46.279 we may only assume as the possible causes of phenomenon, 366 00:24:46.839 --> 00:24:50.119 such causes as are sufficient to explain them, and are 367 00:24:50.160 --> 00:24:55.240 also verai kausi, a vera causa being one which is 368 00:24:55.319 --> 00:24:58.599 capable of detection and such that its connection with the 369 00:24:58.599 --> 00:25:03.960 phenomenon can be ultimate shewn by independent evidence. Two effects 370 00:25:03.960 --> 00:25:08.079 of a similar kind must have similar causes. Three, whatever 371 00:25:08.160 --> 00:25:11.799 properties of bodies are found by experience to be invariable 372 00:25:12.279 --> 00:25:15.480 should be assumed to be so in places where direct 373 00:25:15.519 --> 00:25:21.680 experiments cannot be made. Newton commences by illustrating the universality 374 00:25:21.759 --> 00:25:25.039 of the law of gravitation and sketches out the principles 375 00:25:25.039 --> 00:25:27.240 which lead him to think that the Solar system is 376 00:25:27.319 --> 00:25:31.240 necessarily stable. He determines that the mass of the Moon, 377 00:25:31.319 --> 00:25:34.440 the masses of the planets, their distances from the Sun, 378 00:25:35.119 --> 00:25:40.759 and their figures. In the first edition, he estimated proposition 379 00:25:40.880 --> 00:25:43.160 thirty seven that the ratio of the mass of the 380 00:25:43.200 --> 00:25:45.680 Moon to that of the Earth was approximately that of 381 00:25:45.680 --> 00:25:48.960 one to twenty six. In the second and third editions 382 00:25:48.960 --> 00:25:51.400 this was altered to a ratio which is nearly that 383 00:25:51.480 --> 00:25:54.440 of one in forty. But except for the mass of 384 00:25:54.480 --> 00:25:57.599 the moon, he approximates to the results now known with 385 00:25:57.720 --> 00:26:02.319 astonishing closeness. He finds the disturbing force exerted by the 386 00:26:02.359 --> 00:26:06.039 Sun and the Moon, and considers the five chief irregularities 387 00:26:06.160 --> 00:26:09.160 in the orbit of the Moon. He next discusses the 388 00:26:09.200 --> 00:26:13.920 solar and lunar tides, determines the precession of the equinoxes, 389 00:26:14.400 --> 00:26:17.079 and finally shows how the elements of a comet can 390 00:26:17.119 --> 00:26:21.839 be determined by these observations, and applies his results to 391 00:26:21.920 --> 00:26:26.519 certain comets. Before this time it had been commonly believed 392 00:26:26.519 --> 00:26:29.200 that the comets had nothing to do with the Solar system, 393 00:26:29.799 --> 00:26:33.759 though in sixteen eighty one d'orfel had shewn that the 394 00:26:33.799 --> 00:26:37.079 path of the Great Comet of sixteen eighty was a 395 00:26:37.119 --> 00:26:43.440 parabola having the Sun at its focus. Lastly, the precipia 396 00:26:43.599 --> 00:26:47.759 is concluded by a general scolium containing the reflection on 397 00:26:47.799 --> 00:26:51.319 the constitution of the universe and on the eternal, the 398 00:26:51.359 --> 00:26:55.640 infinite and perfect being by whom it is governed. The 399 00:26:55.759 --> 00:27:00.319 chief alterations in the second edition, published in seventeen thirteen 400 00:27:00.559 --> 00:27:03.960 were the substitutions of simpler proofs for some of the 401 00:27:04.000 --> 00:27:07.599