WEBVTT 1 00:00:01.000 --> 00:00:05.000 Chapter seventeen of A Short Account of the History of Mathematics. 2 00:00:05.599 --> 00:00:09.640 This is a LibriVox recording. All LibriVox recordings are in 3 00:00:09.679 --> 00:00:14.000 the public domain. For more information or to volunteer, please 4 00:00:14.119 --> 00:00:18.440 visit LibriVox dot org. This is a reading by Paul 5 00:00:18.600 --> 00:00:22.199 King p J K dot scripts dot mi t dot 6 00:00:22.399 --> 00:00:26.960 edu forward slash p k J. A Short Account of 7 00:00:27.039 --> 00:00:33.280 the History of Mathematics by W. W. Rowse Ball. Chapter seventeen. 8 00:00:33.880 --> 00:00:37.119 Leibniz and the Mathematicians of the first half of the 9 00:00:37.159 --> 00:00:42.840 eighteenth century, Part one. I have briefly traced in the 10 00:00:42.920 --> 00:00:48.000 last chapter the nature and extent of Newton's contributions to science. 11 00:00:49.079 --> 00:00:52.840 Modern analysis is, however, derived directly from the works of 12 00:00:52.920 --> 00:00:57.399 Leibniz and the elder Bernoulli's, and it is immaterial to 13 00:00:57.520 --> 00:01:00.679 us whether the fundamental ideas of it were by them 14 00:01:00.679 --> 00:01:05.680 from Newton or discovered independently. The English mathematicians of the 15 00:01:05.799 --> 00:01:09.680 years considered in this chapter continued to use the language 16 00:01:09.680 --> 00:01:13.319 and notation of Newton. They are thus somewhat distinct from 17 00:01:13.359 --> 00:01:17.760 their Continental contemporaries, and I have therefore grouped them together 18 00:01:18.159 --> 00:01:23.760 in a section by themselves. Leibnitz and the Bernoullis Leibnitz. 19 00:01:24.439 --> 00:01:28.120 Gottfried Wilhelm Leibnitz was born at Leipzig on June the 20 00:01:28.159 --> 00:01:32.359 twenty first, sixteen forty six and died at Hanover on 21 00:01:32.439 --> 00:01:37.560 November fourteenth, seventeen sixteen. His father died before he was six, 22 00:01:38.040 --> 00:01:40.040 and the teaching at the school to which he was 23 00:01:40.120 --> 00:01:45.159 then sent was inefficient, but his industry triumphed over all difficulties. 24 00:01:45.599 --> 00:01:47.680 By the time he was twelve, he had taught himself 25 00:01:47.719 --> 00:01:52.519 to read Latin easily and had begun Greek, and before 26 00:01:52.599 --> 00:01:57.400 he was twenty he had mastered the ordinary textbooks on mathematics, philosophy, 27 00:01:57.519 --> 00:02:01.799 theology and law. Refused the degree of Doctor of Laws 28 00:02:01.840 --> 00:02:04.239 at Leipzig by those who were jealous of his youth 29 00:02:04.280 --> 00:02:08.639 and learning, he moved to Nuremberg. An essay which he 30 00:02:08.759 --> 00:02:11.759 there wrote on the study of law was dedicated to 31 00:02:11.840 --> 00:02:15.000 the Elector of Mainz and led to his appointment by 32 00:02:15.000 --> 00:02:18.919 the Elector on a commission for the revision of some statutes, 33 00:02:18.960 --> 00:02:22.639 from which he was subsequently promoted to the diplomatic service. 34 00:02:23.479 --> 00:02:27.759 In the latter capacity, he supported unsuccessfully the claims of 35 00:02:27.759 --> 00:02:31.520 the German candidate for the Crown of Poland. The violent 36 00:02:31.639 --> 00:02:35.960 seizure of various small places in Alsas in sixteen seventy 37 00:02:36.560 --> 00:02:40.039 excited universal alarm in Germany as to the designs of 38 00:02:40.120 --> 00:02:44.400 Louis the fourteenth, and Leibnitz drew up a scheme by 39 00:02:44.400 --> 00:02:48.319 which it was proposed to offer German cooperation if France 40 00:02:48.560 --> 00:02:51.000 liked to take Egypt and use the possession of that 41 00:02:51.159 --> 00:02:54.840 country as a basis for attack against Holland and Asia, 42 00:02:55.000 --> 00:02:58.039 on the condition that Germany was to be left undisturbed 43 00:02:58.080 --> 00:03:02.080 by France. This bears a curious resemblance to the similar 44 00:03:02.159 --> 00:03:05.599 plan by which Napoleon the First proposed to attack England 45 00:03:06.800 --> 00:03:10.560 in sixteen seventy two. Leibnitz went to Paris on the 46 00:03:10.599 --> 00:03:13.960 invitation of the French government to explain the details of 47 00:03:14.000 --> 00:03:18.919 the scheme, but nothing came of it. At Paris he 48 00:03:19.000 --> 00:03:23.400 met Huygens, who was then residing there, and their conversation 49 00:03:23.560 --> 00:03:26.719 led him to study geometry, which he described as opening 50 00:03:26.759 --> 00:03:29.800 a new world to him, though he had, as a 51 00:03:29.800 --> 00:03:33.599 matter of fact, previously written some tracts on various minor 52 00:03:33.639 --> 00:03:37.759 points in mathematics, the most important of them being a 53 00:03:37.800 --> 00:03:42.080 paper on combinations written in sixteen sixty eight and a 54 00:03:42.120 --> 00:03:46.879 description of a new calculating machine. In January sixteen seventy 55 00:03:46.919 --> 00:03:49.560 three he was sent on a political mission to London, 56 00:03:49.639 --> 00:03:53.080 where he stopped some months and made the acquaintance of Oldenburg, 57 00:03:53.199 --> 00:03:56.479 Collins and others. It was at this time that he 58 00:03:56.520 --> 00:04:00.520 communicated the memoir to the Royal Society, in which he 59 00:04:00.599 --> 00:04:05.240 was found to have been forestalled by Mouton. In sixteen 60 00:04:05.280 --> 00:04:08.639 seventy three, the Elector of Mainz died, and in the 61 00:04:08.680 --> 00:04:12.319 following year Leibnitz entered the service of the Brunswick family. 62 00:04:13.000 --> 00:04:16.800 In sixteen seventy six he again visited London and then 63 00:04:16.879 --> 00:04:20.319 moved to Hanover, where till his death he occupied the 64 00:04:20.319 --> 00:04:25.199 well paid post of librarian in the Ducal Library. His 65 00:04:25.319 --> 00:04:28.759 pen was thenceforth employed in all the political matters which 66 00:04:28.800 --> 00:04:33.839 affected the Hanoverian family, and his services were recognized by 67 00:04:33.879 --> 00:04:37.920 honors and distinctions of various kinds. His memoranda on the 68 00:04:38.000 --> 00:04:42.800 various political, historical and theological questions which concerned the dynasty 69 00:04:43.399 --> 00:04:46.639 during the forty years from sixteen seventy three to seventeen 70 00:04:46.759 --> 00:04:51.560 thirteen form a valuable contribution to the history of that time. 71 00:04:52.399 --> 00:04:55.759 His appointment in the Hanoverian service gave him an increased 72 00:04:55.839 --> 00:05:00.680 leisure for his favorite pursuits. Leibnets used to assert that 73 00:05:00.759 --> 00:05:03.959 as the first fruit of his increased leisure, he invented 74 00:05:04.000 --> 00:05:08.360 the differential and integral calculus in sixteen seventy four, but 75 00:05:08.480 --> 00:05:11.040 the earliest traces of the use of it in his 76 00:05:11.199 --> 00:05:15.199 extant notebooks do not occur till sixteen seventy five, and 77 00:05:15.279 --> 00:05:18.399 it was not till sixteen seventy seven that we find 78 00:05:18.439 --> 00:05:22.839 it developed into a consistent system. He was not published 79 00:05:22.959 --> 00:05:27.839 until sixteen eighty four. Nearly all his mathematical papers were 80 00:05:27.879 --> 00:05:31.920 produced within the ten years from sixteen eighty two to 81 00:05:32.040 --> 00:05:35.279 sixteen ninety two, most of them in a journal called 82 00:05:35.319 --> 00:05:39.920 the Acta Eruditorum, which he and Ottomenkad founded in sixteen 83 00:05:40.000 --> 00:05:43.839 eighty two and which had a wide circulation on the continent. 84 00:05:46.000 --> 00:05:49.040 Leibnitz occupies at least as large a place in the 85 00:05:49.160 --> 00:05:53.079 history of philosophy as he does in the history of mathematics. 86 00:05:53.199 --> 00:05:56.399 Most of his philosophical writings were composed in the last 87 00:05:56.480 --> 00:06:00.240 twenty or twenty five years of his life, and the 88 00:06:00.279 --> 00:06:03.079 point as to whether his views were original or whether 89 00:06:03.120 --> 00:06:07.240 they were appropriated from Spinoza, whom he visited in sixteen 90 00:06:07.279 --> 00:06:11.160 seventy six, is still in question among philosophers, though the 91 00:06:11.240 --> 00:06:16.279 evidence seems to point to the originality of Leibnitz. As 92 00:06:16.319 --> 00:06:19.279 to Leibnitz's system of philosophy, it will be enough to 93 00:06:19.360 --> 00:06:22.759 say that he regarded the ultimate elements of the universe 94 00:06:23.279 --> 00:06:29.680 as individual precipient beings, whom he called monads. According to him, 95 00:06:29.720 --> 00:06:34.920 the monads are centers of force and substances force, while space, matter, 96 00:06:35.000 --> 00:06:39.199 and motion are merely phenomenal. Finally, the existence of God 97 00:06:39.319 --> 00:06:44.040 is inferred from the existing harmony among the monads. His 98 00:06:44.160 --> 00:06:48.759 services to literature were almost as considerable as those to philosophy. 99 00:06:49.240 --> 00:06:52.800 In particular, I may single out his overthrow of the 100 00:06:52.800 --> 00:06:57.160 then prevalent belief that Hebrew was the primeval language of 101 00:06:57.199 --> 00:07:03.160 the entire human race. In seventeen hundred, the Academy of 102 00:07:03.160 --> 00:07:06.839 Berlin was created on his advice, and he drew up 103 00:07:06.839 --> 00:07:10.560 the first body of statutes for it. On the accession 104 00:07:10.920 --> 00:07:15.040 in seventeen fourteen of his master, George First to the 105 00:07:15.120 --> 00:07:18.959 throne of England, Leibnitz was practically thrown aside as a 106 00:07:19.079 --> 00:07:22.920 useless tool. He was forbidden to come to England, and 107 00:07:23.439 --> 00:07:25.680 the last two years of his life were spent in 108 00:07:25.800 --> 00:07:30.560 neglect and dishonor. He died at Hanover in seventeen sixteen. 109 00:07:31.439 --> 00:07:37.000 He was over fond of money and personal distinctions, was unscrupulous, 110 00:07:37.160 --> 00:07:40.759 as might be expected of a professional diplomat of that time, 111 00:07:41.560 --> 00:07:46.199 but possessed singularly attractive manners, and all who once came 112 00:07:46.279 --> 00:07:49.879 under the charm of his personal presence remained sincerely attached 113 00:07:49.920 --> 00:07:54.879 to him. His mathematical reputation was largely augmented by the 114 00:07:54.920 --> 00:07:59.839 eminent position that he occupied in diplomacy, philosophy, and literature, 115 00:08:00.319 --> 00:08:04.279 and the power thence derived was considerably increased by his 116 00:08:04.439 --> 00:08:08.800 influence in the management of the Acta Rodatorum, which I 117 00:08:08.879 --> 00:08:12.240 believe was the only private scientific journal of the time. 118 00:08:13.399 --> 00:08:16.199 The last years of his life, from seventeen o nine 119 00:08:16.240 --> 00:08:20.680 to seventeen sixteen were embittered by the long controversy with 120 00:08:20.839 --> 00:08:24.079 John Kyle Newton and others as to whether he had 121 00:08:24.120 --> 00:08:29.600 discovered the differential calculus independently of Newton's previous investigations, or 122 00:08:29.639 --> 00:08:32.639 whether he had derived the fundamental idea from Newton and 123 00:08:32.720 --> 00:08:38.519 merely invented another notation for it. The controversy occupies a 124 00:08:38.559 --> 00:08:41.320 place in the scientific history of the early years of 125 00:08:41.360 --> 00:08:46.720 the eighteenth century quite disproportionate to its true importance, but 126 00:08:47.039 --> 00:08:50.679 it so materially affected the history of mathematics in Western 127 00:08:50.720 --> 00:08:54.159 Europe that I feel obliged to give the leading facts, 128 00:08:54.720 --> 00:08:57.440 though I am reluctant to take up so much space 129 00:08:57.639 --> 00:09:02.200 with questions of a personal character. The case in favor 130 00:09:02.240 --> 00:09:06.120 of the independent invention by Leibnitz is stated in Gerhaltz 131 00:09:06.200 --> 00:09:12.000 Leibenismus mathematische Schriften in Biatt and Leefort's edition of the 132 00:09:12.080 --> 00:09:18.159 Commercium Epistocolum Paris eighteen fifty six. The arguments on the 133 00:09:18.200 --> 00:09:23.840 other side are given in H. Sloman's Lebanistan's unspruckhofti Erfindong 134 00:09:23.960 --> 00:09:29.240 der di frienzia Rechnung Leipezig, eighteen fifty seven, of which 135 00:09:29.240 --> 00:09:32.879 an English translation with the editions by doctor Sloman was 136 00:09:32.879 --> 00:09:37.879 published at Cambridge in eighteen sixty. The history of the 137 00:09:37.919 --> 00:09:41.120 invention of the calculus is given in an article of 138 00:09:41.159 --> 00:09:45.240 it in the ninth edition of the Encyclopedia Britannica, and 139 00:09:45.519 --> 00:09:51.399 in P. Manson's is Quiste lestraal t Calcula Infinities mal 140 00:09:51.720 --> 00:09:57.200 Gand eighteen eighty seven, end of footnote. The ideas of 141 00:09:57.279 --> 00:10:01.480 the infinitesimal calculus can be expressed either in the notation 142 00:10:01.720 --> 00:10:06.039 of fluxions or in that of differentials. The former was 143 00:10:06.120 --> 00:10:09.240 used by Newton in sixteen sixty six and communicated in 144 00:10:09.360 --> 00:10:13.320 manuscript to his friends and pupils from sixteen sixty nine onwards, 145 00:10:13.960 --> 00:10:17.120 but no distinct account of it was printed until sixteen 146 00:10:17.200 --> 00:10:20.519 ninety three. The earliest use of the latter in the 147 00:10:20.559 --> 00:10:24.480 notebooks of Leibnitz is dated sixteen seventy five, and was 148 00:10:24.519 --> 00:10:27.960 employed in the letters sent to Newton in sixteen seventy seven, 149 00:10:28.360 --> 00:10:30.759 and an account of it was printed in the Memoir 150 00:10:31.080 --> 00:10:35.120 of sixteen eighty four described below. There is no question 151 00:10:35.840 --> 00:10:39.440 that the differential notation is due to Libanates, and that 152 00:10:39.480 --> 00:10:42.559 the sole question is as to whether the general idea 153 00:10:42.639 --> 00:10:46.759 of the calculus was taken from Newton or discovered independently. 154 00:10:48.480 --> 00:10:52.039 The case in favor of the independent invention by Libnates 155 00:10:52.120 --> 00:10:54.519 rests on the ground that he published a description of 156 00:10:54.559 --> 00:10:58.399 his methods some years before Newton printed anything on fluctions, 157 00:10:59.440 --> 00:11:02.080 that he always alluded to the discovery as being his 158 00:11:02.159 --> 00:11:07.039 own invention, and that for many years this statement was unchallenged, 159 00:11:08.080 --> 00:11:11.200 While of course there must be a strong presumption that 160 00:11:11.240 --> 00:11:14.759 he acted in good faith to Rebut this case, it 161 00:11:14.840 --> 00:11:17.879 is necessary to shew that one he saw some of 162 00:11:17.919 --> 00:11:21.799 Newton's papers on the subject in or before sixteen seventy five, 163 00:11:21.879 --> 00:11:25.200 or at least sixteen seventy seven, and two that E 164 00:11:25.240 --> 00:11:29.440 thence derived the fundamental ideas of the calculus. The fact 165 00:11:29.519 --> 00:11:32.919 that his claim was unchallenged for some years, in my opinion, 166 00:11:33.519 --> 00:11:38.960 in the particular circumstances of the case, is immaterial. That 167 00:11:39.080 --> 00:11:43.919 Leibnitz saw some of Newton's manuscripts was always intrinsically probable. 168 00:11:44.320 --> 00:11:47.879 But when in eighteen forty nine C. J. Gerhardt examined 169 00:11:47.919 --> 00:11:51.759 Leibnitz's papers, he found among them a manuscript copy, the 170 00:11:51.799 --> 00:11:56.320 existence of which had been previously unsuspected, in Leibanez's handwriting 171 00:11:56.360 --> 00:12:02.720 of extracts from Newton's Di Analisi periquezion is numero timenurum infinitas, 172 00:12:03.360 --> 00:12:07.879 which was printed in the Di Quatrata Corverum in seventeen 173 00:12:07.919 --> 00:12:12.600 oh four, together with notes on their expression in the 174 00:12:12.600 --> 00:12:16.960 differential notation. The question of the date at which these 175 00:12:17.000 --> 00:12:20.799 extracts were made is therefore all important. It is known 176 00:12:21.000 --> 00:12:24.360 that a copy of Newton's manuscript had been sent to 177 00:12:24.759 --> 00:12:28.759 Chernhausen in May sixteen seventy five, and that in that 178 00:12:28.960 --> 00:12:32.080 year he and Leibnitz were engaged together in a piece 179 00:12:32.120 --> 00:12:36.120 of work. It is not impossible that these extracts were 180 00:12:36.120 --> 00:12:39.759 made then. It is also possible that they may have 181 00:12:39.799 --> 00:12:43.919 been made in sixteen seventy six, for Leibnitz discuss the 182 00:12:44.000 --> 00:12:47.399 question of analysis by infinite series with Collins and Oldenburg 183 00:12:47.480 --> 00:12:51.759 in that year, and it is a priori probable that 184 00:12:51.960 --> 00:12:54.960 they would have then shewn him the manuscript of Newton 185 00:12:55.039 --> 00:12:58.279 on that subject, a copy of which was possessed by 186 00:12:58.279 --> 00:13:01.000 one or both of them. On the other hand, it 187 00:13:01.080 --> 00:13:04.799 may have been supposed that Leibanetz made the extracts from 188 00:13:04.840 --> 00:13:09.320 the printed copy in or after seventeen o four. Libanutz, 189 00:13:09.360 --> 00:13:12.240 shortly before his death, admitted in a letter to Conti 190 00:13:12.799 --> 00:13:16.399 that that in sixteen seventy six Collins had shown him 191 00:13:16.440 --> 00:13:20.159 some Newtonian papers, but implied that they were of little 192 00:13:20.279 --> 00:13:23.919 or no value. Presumably he referred to Newton's letters of 193 00:13:24.000 --> 00:13:28.240 June thirteenth and October twenty fourth, sixteen seventy six, and 194 00:13:28.399 --> 00:13:31.720 to the letter of December tenth, sixteen seventy two, on 195 00:13:31.799 --> 00:13:35.399 the method of tangents extracts from which accompanied the letter 196 00:13:35.559 --> 00:13:38.480 of June thirteenth. But it was curious that on the 197 00:13:38.519 --> 00:13:41.360 receipt of these letters Lybanuts should have made no further 198 00:13:41.440 --> 00:13:45.519 inquiries unless he was already aware from other sources of 199 00:13:45.559 --> 00:13:50.039 the method followed by Newton. Whether Leibanetz made no use 200 00:13:50.080 --> 00:13:52.679 of the manuscript from which he had copied extracts, or 201 00:13:52.679 --> 00:13:57.360 whether he had previously invented the calculus, are questions on which, 202 00:13:57.559 --> 00:14:01.279 at this distance of time, no direct evidence is available. 203 00:14:02.039 --> 00:14:06.120 It is, however, worth noting that the unpublished Portsmouth papers 204 00:14:06.200 --> 00:14:10.080 schue that when in seventeen eleven Newton went carefully into 205 00:14:10.080 --> 00:14:14.720 the whole dispute, he picked out this manuscript as the 206 00:14:14.759 --> 00:14:18.679 one which had probably somehow fallen into the hands of Leibniz. 207 00:14:19.639 --> 00:14:22.840 At that time, there was no direct evidence that Leibnitz 208 00:14:22.919 --> 00:14:26.159 had seen this manuscript before it was printed in seventeen 209 00:14:26.200 --> 00:14:30.440 o four, and accordingly Newton's conjecture was not published. But 210 00:14:30.559 --> 00:14:34.200 Gerhardt's discovery of the copy made by Leibanez tends to 211 00:14:34.240 --> 00:14:38.039 confirm the accuracy of Newton's judgment in the matter. It 212 00:14:38.120 --> 00:14:41.159 is said by some that to a man of Leibnitz's ability, 213 00:14:41.240 --> 00:14:45.399 the manuscript, especially if supplemented by the letter of December tenth, 214 00:14:45.519 --> 00:14:49.240 sixteen seventy two, would supply sufficient hints to give him 215 00:14:49.279 --> 00:14:52.360 a clue to the methods of the calculus, though as 216 00:14:52.440 --> 00:14:56.399 fluctional notation is not employed in it, anyone who used 217 00:14:56.440 --> 00:14:59.279 it would have to invent a notation, but this is 218 00:14:59.320 --> 00:15:03.039 denied by others. There was at first no reason to 219 00:15:03.080 --> 00:15:06.279 suspect the good faith of Leibnitz, and it was not 220 00:15:06.480 --> 00:15:09.679 until the appearance in seventeen o four of an anonymous 221 00:15:09.679 --> 00:15:13.320 review of Newton's Tract on Quadrature, in which it was 222 00:15:13.360 --> 00:15:16.279 implied that Newton had borrowed the idea of the flectional 223 00:15:16.360 --> 00:15:21.480 calculus from Leibnitz, that any responsible mathematician questioned the statement 224 00:15:21.919 --> 00:15:26.360 that Leibnitz had invented the calculus independently of Newton. It 225 00:15:26.399 --> 00:15:30.600 is universally admitted that there was no justification or authority 226 00:15:31.159 --> 00:15:34.399 for the statements made in this review, which was rightly 227 00:15:34.440 --> 00:15:38.360 attributed to Leibnitz, But the subsequent discussion led to a 228 00:15:38.399 --> 00:15:42.440 critical examination of the whole question, and doubt was expressed 229 00:15:42.440 --> 00:15:45.879 as to whether Leibanates had not derived the fundamental idea 230 00:15:45.919 --> 00:15:49.240 from Newton. The case against Libyanates, as it appeared to 231 00:15:49.320 --> 00:15:54.360 Newton's friends, was summed up in the Commercium Epistilicolum, issued 232 00:15:54.399 --> 00:15:59.360 in seventeen twelve. The evidence there collected may be inconclusive, 233 00:15:59.440 --> 00:16:02.919 but at any rate, detailed references are given for all 234 00:16:03.000 --> 00:16:07.720 facts mentioned. No such summary with fact states and references 235 00:16:07.759 --> 00:16:11.320 of the case for Leibniz was issued by his friends, 236 00:16:11.799 --> 00:16:15.720 But John Berneulli attempted to indirectly weaken the evidence by 237 00:16:15.720 --> 00:16:19.159 attacking the personal character of Newton. This was in a 238 00:16:19.240 --> 00:16:24.080 letter dated June seventh, seventeen thirteen. The charges were false, 239 00:16:24.480 --> 00:16:27.759 and when pressed for an explanation of them, Bernoulli most 240 00:16:28.320 --> 00:16:32.600 solemnly denied having written the letter. In accepting the denial, 241 00:16:32.759 --> 00:16:36.840 Newton added in a private letter to him the following remarks, 242 00:16:36.840 --> 00:16:40.360 which are interesting as giving Newton's account of why he 243 00:16:40.480 --> 00:16:44.039 was last induced to take any part in the controversy. 244 00:16:45.200 --> 00:16:48.120 I have never said he grasped at the fame among 245 00:16:48.240 --> 00:16:52.159 foreign nations. But I am very desirous to preserve my 246 00:16:52.360 --> 00:16:56.960 character for honesty for which the author of that epistle, 247 00:16:58.080 --> 00:17:01.000 as if by the authority of a great judge, has 248 00:17:01.080 --> 00:17:04.440 endeavored to wrest from me. Now that I am old 249 00:17:05.039 --> 00:17:07.880 I have little pleasure in mathematical studies, and I have 250 00:17:08.000 --> 00:17:11.559 never tried to propagate my opinions over the world. But 251 00:17:11.640 --> 00:17:15.559 I have rather taken care not to involve myself in 252 00:17:15.680 --> 00:17:21.240 disputes on account of them. Leibnitz's defense or explanation of 253 00:17:21.319 --> 00:17:26.200 his silence is given in the following letter, dated April nineteenth, 254 00:17:26.319 --> 00:17:33.079 seventeen sixteen, from him to Conti poor ris pondre de 255 00:17:33.200 --> 00:17:37.839 point a point aluvreche pour bles, contremis il fulois, un 256 00:17:37.880 --> 00:17:42.640 ottre jos souis grand pro le, moile sluillo, il ferois, 257 00:17:42.880 --> 00:17:47.640 entre danze grand details de conte te de minis, passe ille, 258 00:17:47.759 --> 00:17:53.000 trene carn don jeon, meer sauvignois, guerre, il me felix 259 00:17:53.079 --> 00:17:59.119 cherche mer vielle lettre d'ant plusier ser sant perdu utre cole, pleu, 260 00:17:59.279 --> 00:18:05.279 su vange, pointe garde, le mignet, demens, eluzotto, sona salvier, 261 00:18:05.640 --> 00:18:10.240 don zun grand has pepier qujeune, pavois de bourier, cuevic 262 00:18:10.319 --> 00:18:14.039 dutamp e de la pacience, mei gen en avois, guerre, 263 00:18:14.200 --> 00:18:21.839 loisier etints charges presentmento capassion juneutres otto natur. The death 264 00:18:21.880 --> 00:18:25.720 of Leibyanitz in seventeen sixteen only put a temporary stop 265 00:18:25.759 --> 00:18:29.640 to the controversy, which was bitterly debated for many years later. 266 00:18:30.359 --> 00:18:33.839 The question is one of great difficulty. The evidence is 267 00:18:33.960 --> 00:18:39.079 conflicting and circumstantial, and everyone must form for themselves the 268 00:18:39.200 --> 00:18:43.519 opinion which seems most probable. I think the majority of 269 00:18:43.640 --> 00:18:47.440 modern writers would accept the view that probably Leibane's invention 270 00:18:47.680 --> 00:18:51.200 of the calculus was independent of that of Newton, and 271 00:18:51.440 --> 00:18:56.519 everyone will hope that they are right. For myself, I cannot, however, 272 00:18:56.599 --> 00:19:00.119 but think it probable that Leibyanez read Newton's manuscript d 273 00:19:00.279 --> 00:19:05.039 Analsi before sixteen seventy seven, and was materially assisted by it. 274 00:19:05.720 --> 00:19:08.880 His unacknowledged possession of a copy of part of one 275 00:19:08.920 --> 00:19:13.319 of Newton's manuscripts may be explicable. But the admitted fact 276 00:19:13.559 --> 00:19:17.039 that on more than one occasion he deliberately altered or 277 00:19:17.119 --> 00:19:21.319 added to important documents before publishing them seems to me 278 00:19:22.000 --> 00:19:25.839 to make his own testimony of little value. In mitigation 279 00:19:26.039 --> 00:19:28.359 of his conduct, I can only say that it must 280 00:19:28.400 --> 00:19:31.680 be recollected that what he is alleged to have received 281 00:19:31.880 --> 00:19:34.319 was rather a series of hints than an account of 282 00:19:34.359 --> 00:19:37.359 the calculus. And it seems to me that the facts 283 00:19:37.440 --> 00:19:40.759 that he did not publish his results of sixteen seventy 284 00:19:40.839 --> 00:19:44.359 seven until sixteen eighty four, and that the notation and 285 00:19:44.519 --> 00:19:47.759 subsequent development of it were all of his own invention, 286 00:19:48.880 --> 00:19:52.359 may have led him thirty years later to minimize any 287 00:19:52.400 --> 00:19:56.119 assistance which he obtained originally, and finally, consider that it 288 00:19:56.240 --> 00:20:01.000 was immaterial. If we must confine in ourselves to one 289 00:20:01.119 --> 00:20:04.759 system of notation, then there can be no doubt that 290 00:20:05.000 --> 00:20:08.200 that which was invented by Leibniz is better fitted for 291 00:20:08.359 --> 00:20:11.920 most of the purposes to which the infinitesimal calculus is 292 00:20:11.960 --> 00:20:16.839 applied than that of fluxions, and for some as such 293 00:20:16.880 --> 00:20:20.720 as the calculus of variations, it is indeed almost essential. 294 00:20:21.319 --> 00:20:24.200 It should be remembered, however, that at the beginning of 295 00:20:24.279 --> 00:20:28.359 the eighteenth century, the methods of the infinitesimal calculus had 296 00:20:28.400 --> 00:20:33.200 not been systematized, and either notation was equally good. The 297 00:20:33.319 --> 00:20:36.160 development of that calculus was the main work of the 298 00:20:36.279 --> 00:20:40.319 mathematician of the first half of the eighteenth century. The 299 00:20:40.440 --> 00:20:45.559 differential form was adopted by continental mathematicians, the application of 300 00:20:45.640 --> 00:20:49.000 it by Euler, Lagrange and Laplace to the principles of 301 00:20:49.079 --> 00:20:53.559 mechanics laid down in the Principia was the great achievement 302 00:20:53.640 --> 00:20:57.160 of the last half of that century, and finally demonstrated 303 00:20:57.200 --> 00:21:01.640 the superiority of the differential to the Fluck calculus. The 304 00:21:01.759 --> 00:21:05.640 translation of the Principia into the language of modern analysis, 305 00:21:05.759 --> 00:21:08.640 and the filling in of the details of the Newtonian 306 00:21:08.759 --> 00:21:12.920 theory by the aid of that analysis, were affected by Laplace. 307 00:21:14.279 --> 00:21:18.039 The controversy with Leibniz was regarded in England as an 308 00:21:18.039 --> 00:21:21.480 attempt by foreigners to defraud Newton of the credit of 309 00:21:21.559 --> 00:21:25.599 his invention, and the question was complicated on both sides 310 00:21:25.920 --> 00:21:30.200 by national jealousies. It was therefore natural, though it was 311 00:21:30.319 --> 00:21:34.640 unfortunate that in England the geometrical and fluctional methods as 312 00:21:34.759 --> 00:21:38.839 used by Newton were alone studied and employed for more 313 00:21:38.920 --> 00:21:41.839 than a century. The English school was thus out of 314 00:21:42.000 --> 00:21:47.039 touch with continental mathematicians. The consequence was that, in spite 315 00:21:47.079 --> 00:21:50.359 of the brilliant band of scholars formed by Newton, the 316 00:21:50.519 --> 00:21:53.880 improvements in the methods of analysis gradually effected on the 317 00:21:53.960 --> 00:21:58.000 continent were almost unknown in Britain. It was not until 318 00:21:58.039 --> 00:22:00.880 eighteen twenty that the value of an Olye methods was 319 00:22:00.960 --> 00:22:05.839 fully recognized in England, and that Newton's countrymen again took 320 00:22:05.960 --> 00:22:10.400 any large share in the development of mathematics. Leaving now 321 00:22:10.480 --> 00:22:13.319 this long controversy, I come to the discussion of the 322 00:22:13.400 --> 00:22:17.319 mathematical papers produced by Libanitz, all the more important of 323 00:22:17.400 --> 00:22:21.640 which were published in the Acta Eruditorum. They are mainly 324 00:22:21.759 --> 00:22:26.119 concerned with the applications of the infinitesimal calculus and with 325 00:22:26.440 --> 00:22:30.920 various questions on mechanics. The only papers of first rate 326 00:22:31.000 --> 00:22:35.079 importance which he produced are those on the differential calculus. 327 00:22:35.759 --> 00:22:38.400 The earliest of these was one published in the Acta 328 00:22:38.599 --> 00:22:43.720 Eruditorum for October sixteen eighty four, in which he enunciated 329 00:22:43.759 --> 00:22:47.200 a general method for finding maxima and minima and for 330 00:22:47.720 --> 00:22:52.640 drawing tangents to curves. One inverse problem, namely to find 331 00:22:52.680 --> 00:22:57.519 the curve whose subtangent is constant, was also discussed. The 332 00:22:57.640 --> 00:23:01.039 notation is the same as that with which we are familiar, 333 00:23:01.519 --> 00:23:04.599 and the differential coefficients of x to the power N 334 00:23:05.400 --> 00:23:10.039 and of products and quotients are determined. In sixteen eighty 335 00:23:10.119 --> 00:23:13.319 six he wrote a paper on the principles of the 336 00:23:13.400 --> 00:23:17.680 new calculus. In both of these papers, the principle of 337 00:23:17.799 --> 00:23:22.759 continuity is explicitly assumed, while his treatment of the subject 338 00:23:22.880 --> 00:23:25.680 is based on the use of infinitesimals and not on 339 00:23:25.799 --> 00:23:29.960 that of the limiting value of ratios. In answer to 340 00:23:30.079 --> 00:23:33.119 some objections which are raised in sixteen ninety four by 341 00:23:33.279 --> 00:23:36.920 Bernard Neuantite, who asserted that d y by dx stood 342 00:23:37.039 --> 00:23:42.359 for an unmeaning quantity like zero over zero, Leibnitz explained, 343 00:23:42.599 --> 00:23:45.960 in the same way as Barrow had previously done, that 344 00:23:46.119 --> 00:23:49.440 the value of d y by dx in geometry could 345 00:23:49.440 --> 00:23:54.240 be expressed as the ratio of two finite quantities. I 346 00:23:54.400 --> 00:23:57.440 think leibnitz a statement of the objects and methods of 347 00:23:57.519 --> 00:24:02.039 the infinitesimal calculus, as can contained in these papers, which 348 00:24:02.119 --> 00:24:04.720 are the three most important memoirs on it, that he 349 00:24:04.839 --> 00:24:08.640 had produced, is somewhat obscure, and his attempt to place 350 00:24:08.720 --> 00:24:12.440 the subject on a metaphysical basis did not tend to clearness. 351 00:24:13.240 --> 00:24:16.759 But the notation he introduced is superior to that of Newton, 352 00:24:17.200 --> 00:24:20.200 and the fact that all the results of modern mathematics 353 00:24:20.240 --> 00:24:23.680 are expressed in the language invented by Leibnitz has proved 354 00:24:23.720 --> 00:24:27.759 to be the best monument of his work. In sixteen 355 00:24:27.839 --> 00:24:31.240 eighty six, and sixteen ninety two he wrote papers on 356 00:24:31.839 --> 00:24:37.920 osculating curves. These, however, contained some bad blunders, as for example, 357 00:24:38.000 --> 00:24:42.000 the assertion that an osculating circle will necessarily cut a 358 00:24:42.119 --> 00:24:45.880 curve in four consecutive points. This error was pointed out 359 00:24:45.920 --> 00:24:49.519 by John Bernoulli, but in his article of sixteen ninety 360 00:24:49.559 --> 00:24:53.680 two Leibnitz defended his original assertion and insisted that a 361 00:24:53.759 --> 00:24:57.039 circle could never cross a curve it touched it. In 362 00:24:57.160 --> 00:25:00.319 sixteen ninety two, Leibnitz wrote a memoir and which he 363 00:25:00.480 --> 00:25:03.880 laid the foundation of the theory of envelopes. This was 364 00:25:04.000 --> 00:25:07.480 further developed in another paper in sixteen ninety four, in 365 00:25:07.559 --> 00:25:11.000 which he introduced for the first time the terms coordinates 366 00:25:11.160 --> 00:25:15.960 and axes of coordinates. Leibnitz also published a good many 367 00:25:16.079 --> 00:25:20.079 papers on mechanical subjects, but some of them contain mistakes 368 00:25:20.119 --> 00:25:22.880 which shew that he did not understand the principles of 369 00:25:22.960 --> 00:25:26.880 the subject. Thus, in sixteen eighty five he wrote a 370 00:25:26.960 --> 00:25:29.920 memoir to find the pressure exerted by a sphere of 371 00:25:30.000 --> 00:25:34.519 wight w placed between two inclined planes of complimentary inclinations, 372 00:25:35.240 --> 00:25:37.599 placed so that the lines of the greatest slope are 373 00:25:37.680 --> 00:25:41.240 perpendicular to the line of the intersection of the plains. 374 00:25:41.799 --> 00:25:45.319 He asserted that the pressure on each plane must consist 375 00:25:45.400 --> 00:25:49.920 of two components unum coot di crevetere di santre tensit 376 00:25:50.559 --> 00:25:56.200 al teremko planum di crive premith. He further said that 377 00:25:56.680 --> 00:25:59.680 for metaphysical reasons, that the sum of the two pressures 378 00:25:59.759 --> 00:26:03.279 must be equal to W. Hence, if R and r 379 00:26:03.440 --> 00:26:07.200 prime be the required pressures, and alpha and one half 380 00:26:07.359 --> 00:26:11.759 prime minus alpha the inclination of the planes, he finds 381 00:26:11.799 --> 00:26:16.000 that R minus one half W multiplied by the quantity 382 00:26:16.200 --> 00:26:21.960 one minus sign alpha plus cos alpha, and our prime 383 00:26:22.119 --> 00:26:26.599 equals one half W multiplied by one minus coas alpha 384 00:26:27.240 --> 00:26:30.720 plus sign alpha. The true values are R equals w 385 00:26:31.079 --> 00:26:36.440 cos alpha and r prime equals w sign alpha. Nevertheless, 386 00:26:36.680 --> 00:26:40.160 some of his papers on mechanics are valuable. Of these, 387 00:26:40.240 --> 00:26:43.079 the most important were two in sixteen eighty nine and 388 00:26:43.200 --> 00:26:45.960 sixteen ninety four, in which he solved the problem of 389 00:26:46.079 --> 00:26:50.279 finding an isochronous curve one in sixteen ninety seven on 390 00:26:50.359 --> 00:26:54.200 the curve of quickest descent. This was the problem sent 391 00:26:54.279 --> 00:26:57.680 as a challenge to Newton, and two in sixteen ninety 392 00:26:57.720 --> 00:27:00.160 one and sixteen ninety two, in which he stated the 393 00:27:00.240 --> 00:27:03.480 intrinsic equation of the curve assumed by a flexible rope 394 00:27:03.519 --> 00:27:07.039 suspended from two points i e. The caternary, but gave 395 00:27:07.119 --> 00:27:12.519 no proof. This last problem had been originally proposed by Galileo. 396 00:27:13.880 --> 00:27:16.960 In sixteen eighty nine, that as two years after the 397 00:27:17.079 --> 00:27:20.359 Principia has been published, he wrote on the movements of 398 00:27:20.440 --> 00:27:23.319 the planets, which he stated were produced by a motion 399 00:27:23.480 --> 00:27:26.880 of the ether. Not only were the equations of motion 400 00:27:27.119 --> 00:27:30.519 which he obtained wrong, but his deductions from them were 401 00:27:30.559 --> 00:27:34.240