WEBVTT 1 00:00:01.399 --> 00:00:05.599 Chapter fifteen of A Short account of the History of mathematics. 2 00:00:06.240 --> 00:00:10.519 This is a LibriVox recording. All LibriVox recordings are in 3 00:00:10.560 --> 00:00:15.240 the public domain. For more information or to volunteer, please 4 00:00:15.359 --> 00:00:19.839 visit LibriVox dot org. This is a reading by Paul 5 00:00:19.960 --> 00:00:24.600 King PJK dot scripts dot mit dot edu forward slash 6 00:00:24.679 --> 00:00:29.519 p kJ. A Short account of the history of mathematics 7 00:00:29.559 --> 00:00:36.079 by W. W. Rowse Ball, Chapter fifteen, History of Mathematics 8 00:00:36.119 --> 00:00:41.039 from Descartes to Huygens circa sixteen thirty five to sixteen 9 00:00:41.200 --> 00:00:47.240 seventy five. I propose in this chapter to consider the 10 00:00:47.359 --> 00:00:51.079 history of mathematics during the forty years in the middle 11 00:00:51.079 --> 00:00:57.039 of the seventeenth century. I regard Descartes, Cavalieri, Pascal, Wallace, 12 00:00:57.240 --> 00:01:01.960 Fermat and Huygens as the leading mathematicians of this time. 13 00:01:02.719 --> 00:01:05.799 I shall treat them in that order, and I shall 14 00:01:05.840 --> 00:01:09.079 conclude with a brief list of the more eminent remaining 15 00:01:09.120 --> 00:01:13.840 mathematicians of the same date. I have already stated that 16 00:01:13.879 --> 00:01:17.239 the mathematicians of this period, and the remark applies more 17 00:01:17.239 --> 00:01:22.879 particularly to Descartes, Pascal, and Fermont, were largely influenced by 18 00:01:22.959 --> 00:01:26.280 the teaching of Kepler and de Rogues. And I would 19 00:01:26.359 --> 00:01:30.959 repeat again that I regard these latter and Galileo as 20 00:01:31.079 --> 00:01:34.439 forming a connecting link between the writers of the Renaissance 21 00:01:34.480 --> 00:01:38.400 and those of modern times. I should also add that 22 00:01:38.439 --> 00:01:43.359 the mathematicians considered in this chapter were contemporaries, and although 23 00:01:43.400 --> 00:01:45.879 I have tried to place them roughly in such an 24 00:01:46.040 --> 00:01:49.920 order that their chief works shall come in chronological arrangement, 25 00:01:50.760 --> 00:01:53.439 it is essential to remember that they were in relation 26 00:01:53.640 --> 00:01:57.040 with one another, and in general, were acquainted with one 27 00:01:57.040 --> 00:02:03.560 another's researches as soon as they were published Descartes. Subject 28 00:02:03.640 --> 00:02:06.280 to the above remarks, we may consider Descartes as the 29 00:02:06.319 --> 00:02:10.120 first of the modern school of mathematics. Renee Descout was 30 00:02:10.199 --> 00:02:13.800 born near Tours on March thirty first, fifteen ninety six, 31 00:02:14.400 --> 00:02:19.520 and died at Stockholm on February eleventh, sixteen fifty. He 32 00:02:19.639 --> 00:02:24.120 was thus a contemporary of Galileo and des Ragues. His father, 33 00:02:24.280 --> 00:02:26.840 who as the name implies, was of a good family, 34 00:02:27.439 --> 00:02:30.199 was accustomed to spend half the year at Rennes when 35 00:02:30.360 --> 00:02:33.080 the local parliament, in which he held a commission as 36 00:02:33.080 --> 00:02:36.080 a councilor, was in session, and the rest of the 37 00:02:36.159 --> 00:02:42.159 time on his family estate at Lake Harte at La Haye. Renee, 38 00:02:42.319 --> 00:02:45.080 the second of a family of two sons and one daughter, 39 00:02:45.599 --> 00:02:47.479 was sent at the age of eight years to the 40 00:02:47.560 --> 00:02:51.520 Jesuit school at La Fleche, and of the admirable discipline 41 00:02:51.560 --> 00:02:55.599 and education there given he speaks most highly. On account 42 00:02:55.599 --> 00:02:58.000 of his delicate health, he was permitted to lie in 43 00:02:58.120 --> 00:03:01.080 bed till late in the morning. This was a custom 44 00:03:01.560 --> 00:03:05.080 which he had always followed, and when he visited Pascal 45 00:03:05.159 --> 00:03:08.199 in sixteen forty seven, he told him that the only 46 00:03:08.240 --> 00:03:11.039 way to do good work in mathematics and to preserve 47 00:03:11.080 --> 00:03:14.199 his health was never to allow anyone to make him 48 00:03:14.240 --> 00:03:16.759 get up in the morning before he felt inclined to 49 00:03:16.840 --> 00:03:20.439 do so, an opinion which I chronicle for the benefit 50 00:03:20.479 --> 00:03:23.840 of any schoolboy into whose hands this work may fall. 51 00:03:25.400 --> 00:03:29.560 On leaving school in sixteen twelve, Descartes went to Paris 52 00:03:29.919 --> 00:03:34.199 to be introduced to the world of fashion. Here, through 53 00:03:34.240 --> 00:03:37.360 the medium of the Jesuits, he made the acquaintance of 54 00:03:37.439 --> 00:03:42.599 Medorges and renewed his schoolboy friendship with father Mersenne, and 55 00:03:42.759 --> 00:03:46.719 together with them he devoted the two years of sixteen 56 00:03:46.800 --> 00:03:51.639 fifteen and sixteen sixteen to the study of mathematics. At 57 00:03:51.680 --> 00:03:55.520 the time, a man of position usually entered either the 58 00:03:55.639 --> 00:03:59.360 army or the church. Des Cartes chose the former profession, 59 00:03:59.759 --> 00:04:03.919 and in sixteen seventeen joined the army of Prince Marie 60 00:04:03.960 --> 00:04:08.319 of Orange. Then at Breda. Walking through the streets, he 61 00:04:08.360 --> 00:04:12.599 saw a placard in Dutch, which excited his curiosity, and, 62 00:04:12.759 --> 00:04:16.560 stopping the first passer, asked him to translate it into 63 00:04:16.560 --> 00:04:20.319 either French or Latin. The stranger, who happened to be 64 00:04:20.399 --> 00:04:23.680 Isaac Beekman, the head of the Dutch college at Dort, 65 00:04:23.800 --> 00:04:27.040 offered to do so if Descartes would answer it, the 66 00:04:27.120 --> 00:04:30.240 placard being in fact a challenge to all the world 67 00:04:30.439 --> 00:04:34.600 to solve a geometrical problem there given. Descartes worked it 68 00:04:34.639 --> 00:04:37.399 out within a few hours, and a warm friendship between 69 00:04:37.480 --> 00:04:41.399 him and Beakman was the result. This unexpected test of 70 00:04:41.399 --> 00:04:45.639 his mathematical attainments made the uncongenial life of the army 71 00:04:45.720 --> 00:04:49.600 distasteful to him, but under family influence and tradition, he 72 00:04:49.680 --> 00:04:53.079 remained a soldier and was persuaded at the commencement of 73 00:04:53.120 --> 00:04:56.279 the Thirty Years War to volunteer under the Court at 74 00:04:56.279 --> 00:05:00.279 Boucquoi in the Army of Bavaria. He continued sank youed 75 00:05:00.399 --> 00:05:04.240 all this time to occupy his leisure with mathematical studies, 76 00:05:04.560 --> 00:05:08.000 and was accustomed to date the first ideas of his 77 00:05:08.079 --> 00:05:12.519 new philosophy and of his analytical geometry from three dreams 78 00:05:12.519 --> 00:05:16.399 he experienced on the night of November tenth, sixteen nineteen, 79 00:05:16.439 --> 00:05:21.240 at Newburgh, when campaigning on the Danube. He regarded this 80 00:05:21.319 --> 00:05:24.199 as the critical day of his life, and one which 81 00:05:24.360 --> 00:05:28.480 determined his whole future. He resigned his commission in the 82 00:05:28.480 --> 00:05:31.399 spring of sixteen twenty one and spent the next five 83 00:05:31.480 --> 00:05:34.920 years in travel, during most of which time he continued 84 00:05:34.920 --> 00:05:39.480 to study pure mathematics. In sixteen twenty six we find 85 00:05:39.560 --> 00:05:43.360 him settled at Paris, a little well built figure, modest 86 00:05:43.560 --> 00:05:46.920 clad in green taffety, and only wearing a sword and 87 00:05:47.000 --> 00:05:51.480 feather in token of his quality as a gentleman. During 88 00:05:51.519 --> 00:05:55.600 the first two years there he interested himself in general 89 00:05:55.680 --> 00:06:00.160 society and spent his leisure in the construction of optical instruments. 90 00:06:00.759 --> 00:06:04.279 But these pursuits were merely the relaxations of one who 91 00:06:04.360 --> 00:06:08.120 failed to find in philosophy that theory of the universe, 92 00:06:08.680 --> 00:06:12.879 which he was convinced finally awaited him. In sixteen twenty eight, 93 00:06:13.040 --> 00:06:17.079 Cardinal Debray, the founder of the Oratorians, met Descartes, and 94 00:06:17.240 --> 00:06:21.040 was so much impressed by his conversation that he urged 95 00:06:21.079 --> 00:06:24.000 on him the duty of devoting his life to the 96 00:06:24.040 --> 00:06:28.920 examination of truth. Descartes agreed, and, the better to secure 97 00:06:29.040 --> 00:06:33.000 himself from interruption, moved to Holland, then, at the height 98 00:06:33.079 --> 00:06:37.800 of its power. There for twenty years he lived, giving 99 00:06:37.879 --> 00:06:43.000 up all his time to philosophy and mathematics. Science, he says, 100 00:06:43.319 --> 00:06:46.279 may be compared to a tree. Metaphysics is the root, 101 00:06:46.839 --> 00:06:52.839 physics is the trunk, and the three chief branches are mechanics, medicine, 102 00:06:53.000 --> 00:06:57.600 and morals, these forming the three applications of our knowledge, 103 00:06:58.199 --> 00:07:01.319 namely to the external world, to the human body, and 104 00:07:01.360 --> 00:07:05.199 to the conduct of life. And with these subjects alone 105 00:07:05.439 --> 00:07:10.000 his writings are concerned. He spent the first four years 106 00:07:10.040 --> 00:07:13.040 sixteen twenty nine to sixteen thirty three of his stay 107 00:07:13.079 --> 00:07:17.399 in Holland in writing Lemonde, which embodies an attempt to 108 00:07:17.439 --> 00:07:20.680 give a physical theory of the universe, but finding that 109 00:07:20.759 --> 00:07:23.959 its publication was likely to bring on him the hostility 110 00:07:23.959 --> 00:07:27.000 of the church, and having no desire to pose as 111 00:07:27.040 --> 00:07:31.439 a martyr, he abandoned it. The complete manuscript was published 112 00:07:31.480 --> 00:07:35.920 in sixteen sixty four. He then devoted himself to composing 113 00:07:35.920 --> 00:07:39.920 a treatise on universal science. This was published at Leyden 114 00:07:40.000 --> 00:07:45.600 in sixteen thirty seven, titled The Discourse on Method, and 115 00:07:45.839 --> 00:07:51.560 was accompanied with three appendices entitled Optics, Meteorology, and Geometry. 116 00:07:52.360 --> 00:07:55.000 It is from the last of these that the invention 117 00:07:55.160 --> 00:08:00.000 of analytical geometry dates. In sixteen forty one he published 118 00:08:00.199 --> 00:08:03.920 to work called Meditations, in which he explained at some 119 00:08:04.040 --> 00:08:08.199 length his views of philosophy as sketched out in the Discourse. 120 00:08:09.639 --> 00:08:14.560 In sixteen forty four he issued the Principia philosophy as 121 00:08:14.600 --> 00:08:18.199 a greater part of which was devoted to physical science, 122 00:08:18.319 --> 00:08:21.800 especially the laws of motion and the theory of vortices. 123 00:08:22.759 --> 00:08:25.959 In sixteen forty seven he received a pension from the 124 00:08:25.959 --> 00:08:29.839 French court in honor of his discoveries. He went to 125 00:08:29.879 --> 00:08:33.360 Sweden on the invitation of the Queen in sixteen forty nine, 126 00:08:33.799 --> 00:08:36.879 and died a few months later of inflammation of the lungs. 127 00:08:38.399 --> 00:08:41.600 In appearance, Descartes was a small man, with a large head, 128 00:08:41.679 --> 00:08:45.360 projecting brow, prominent nose, and black hair coming down to 129 00:08:45.399 --> 00:08:49.919 his eyebrows. His voice was feeble. Considering the range of 130 00:08:49.919 --> 00:08:53.000 his studies, he was by no means widely read. And 131 00:08:53.080 --> 00:08:57.200 he despised both learning and art unless something tangible could 132 00:08:57.200 --> 00:09:03.200 be extracted. Therefrom physician. He was cold and selfish. He 133 00:09:03.279 --> 00:09:06.519 never married and left no descendants, though he had one 134 00:09:06.600 --> 00:09:11.000 illegitimate daughter who died young. As to his philosophical theories, 135 00:09:11.000 --> 00:09:13.279 it would be sufficient to say that he discussed the 136 00:09:13.320 --> 00:09:16.759 same problems which have been debated for the last two 137 00:09:16.840 --> 00:09:21.440 thousand years. It is hardly necessary to say that the 138 00:09:21.519 --> 00:09:25.399 problems themselves are of great interest. But from the nature 139 00:09:25.559 --> 00:09:29.279 of the case, no solution ever offered is capable either 140 00:09:29.320 --> 00:09:33.320 of proof or disproof. And whenever a philosopher like Descartes 141 00:09:33.399 --> 00:09:36.960 believes that he has at last finally settled a question, 142 00:09:37.600 --> 00:09:40.080 it has been easy for his successors to point out 143 00:09:40.080 --> 00:09:43.919 the fallacy and his assumptions. All that can be effected 144 00:09:44.159 --> 00:09:47.600 is to make one explanation somewhat more probable than another. 145 00:09:48.720 --> 00:09:52.120 I have read somewhere that philosophy has always been chiefly 146 00:09:52.159 --> 00:09:57.559 engaged with the interrelations of God, nature, and man. The 147 00:09:57.600 --> 00:10:02.120 earliest philosophers were Greeks, who occupied themselves mainly with the 148 00:10:02.200 --> 00:10:06.399 relations between God and Nature, and dealt with man separately. 149 00:10:07.320 --> 00:10:10.440 The Christian Church was so absorbed in the relation of 150 00:10:10.519 --> 00:10:15.519 God to man as to entirely neglect nature. Finally, modern 151 00:10:15.559 --> 00:10:20.879 philosophers concern themselves chiefly with the relations between man and nature. 152 00:10:21.559 --> 00:10:25.279 Whether this is a correct historical generalization of the views 153 00:10:25.279 --> 00:10:28.519 which have been successfully prevalent, I do not care to 154 00:10:28.600 --> 00:10:32.000 discuss here. But the statement as to the scope of 155 00:10:32.120 --> 00:10:37.200 modern philosophy marks the limitations of Descartes's writings, and they 156 00:10:37.240 --> 00:10:41.759 may be taken as the commencement of the modern school. 157 00:10:41.919 --> 00:10:47.279 Descartes's chief contributions to mathematics were his analytical geometry and 158 00:10:47.320 --> 00:10:51.080 his theory of vortices, and it is on his researches 159 00:10:51.159 --> 00:10:54.320 in connection with the former of these subjects that his 160 00:10:54.440 --> 00:11:00.679 reputation rests. Analytical geometry does not consist merely as a sometimes, 161 00:11:00.679 --> 00:11:04.919 as loosely said in the application of algebra to geometry, 162 00:11:05.080 --> 00:11:07.840 that had been done by Archimytes and many others, and 163 00:11:07.919 --> 00:11:11.080 had become the usual method of procedure in the works 164 00:11:11.080 --> 00:11:15.519 of the mathematicians of the sixteenth century. The great advance 165 00:11:15.600 --> 00:11:18.320 made by Descartes was that he saw that a point 166 00:11:18.360 --> 00:11:21.919 in a plane could be completely determined if its distances, 167 00:11:21.960 --> 00:11:25.600 say x and y form two fixed lines drawn at 168 00:11:25.720 --> 00:11:29.480 right angles in the plane were given with the convention 169 00:11:29.679 --> 00:11:32.960 familiar to us as to the interpretation of positive and 170 00:11:33.039 --> 00:11:37.639 negative values, and that though an equation f of xy 171 00:11:37.799 --> 00:11:41.440 equals zero was indeterminate and could be satisfied by an 172 00:11:41.519 --> 00:11:45.000 infinite number of values of x and y, yet these 173 00:11:45.120 --> 00:11:48.360 values of x and y determine the coordinates of a 174 00:11:48.480 --> 00:11:51.480 number of points which form a curve of which the 175 00:11:51.519 --> 00:11:56.600 equation f of xy equals zero expresses some geometrical property, 176 00:11:56.679 --> 00:12:00.000 that is, a property true of the curve at every point. 177 00:12:01.600 --> 00:12:05.279 Descartes assured that a point in space could be similarly 178 00:12:05.399 --> 00:12:09.519 determined by three coordinates, but he confined his attention to 179 00:12:09.600 --> 00:12:14.080 plain curves. It was at once seen by Descartes and 180 00:12:14.159 --> 00:12:17.519 his successors that in order to investigate the properties of 181 00:12:17.559 --> 00:12:22.200 a curve, it was sufficient to select any characteristic geometrical 182 00:12:22.240 --> 00:12:26.279 property as a definition, and to express it by means 183 00:12:26.320 --> 00:12:29.720 of an equation between the current coordinates of any point 184 00:12:29.759 --> 00:12:33.600 in the curve. That is, to translate the definition into 185 00:12:33.639 --> 00:12:39.600 a language of analytical geometry. The equation so obtained contains 186 00:12:39.679 --> 00:12:44.440 implicitly every property of the curve, and any particular property 187 00:12:44.480 --> 00:12:48.559 can be deduced from it by ordinary algebra without troubling 188 00:12:48.559 --> 00:12:52.679 about the geometry of the figure. The points in which 189 00:12:52.720 --> 00:12:56.320 the two curves intersect can be determined by finding the 190 00:12:56.440 --> 00:12:59.919 roots common to their two equations. I need not go 191 00:13:00.279 --> 00:13:04.000 further into details, for nearly everyone to whom the above 192 00:13:04.080 --> 00:13:08.320 is intelligible will have read analytical Geometry and be able 193 00:13:08.360 --> 00:13:13.480 to appreciate the value of its invention. Descartes Geometry is 194 00:13:13.559 --> 00:13:17.799 divided into three books. The first two of these treat 195 00:13:18.000 --> 00:13:21.840 of analytical geometry, and the third includes an analysis of 196 00:13:21.879 --> 00:13:25.360 the algebra then current. It is somewhat difficult to follow 197 00:13:25.399 --> 00:13:29.159 the reasoning, but the obscurity was intentional and due to 198 00:13:29.200 --> 00:13:33.799 the jealousy of Descartes. Translated, I omitted nothing says he 199 00:13:34.000 --> 00:13:38.240 except design. I had expected that people who have any 200 00:13:38.320 --> 00:13:41.320 math skill would not fail to say that. I had 201 00:13:41.320 --> 00:13:44.039 not said anything that has not already been said about 202 00:13:44.039 --> 00:13:47.960 the coordinate system. But I have hopefully made this topic 203 00:13:48.080 --> 00:13:53.559 more intelligible. The first book commences with an explanation of 204 00:13:53.559 --> 00:13:57.399 the principles of analytic geometry, and contains a discussion of 205 00:13:57.440 --> 00:14:00.519 a certain problem which has been propounded by Paswers in 206 00:14:00.559 --> 00:14:03.960 the seventh book of his Synagogue, and of which some 207 00:14:04.039 --> 00:14:09.039 particular cases have been considered by Euclid and Apollonius. The 208 00:14:09.120 --> 00:14:13.960 general theorem has baffled previous geometricians, and it was in 209 00:14:14.000 --> 00:14:16.440 the attempt to solve it that Descartes was led to 210 00:14:16.480 --> 00:14:20.639 the invention of analytical geometry. The full nunciation of the 211 00:14:20.679 --> 00:14:24.559 problem is rather involved, but the most important case is 212 00:14:24.639 --> 00:14:26.759 to find the locus of a point such that the 213 00:14:26.799 --> 00:14:31.000 product of the perpendiculars on m given straight lines shall 214 00:14:31.039 --> 00:14:33.279 be in a constant ratio to the product of the 215 00:14:33.320 --> 00:14:38.679 perpendiculars on n given straight lines. The ancients had solved 216 00:14:38.679 --> 00:14:42.480 this geometrically for the case em equals on n equals one, 217 00:14:42.960 --> 00:14:47.399 and the case m equals one, n equals two. Pappus 218 00:14:47.399 --> 00:14:51.360 had further stated that if m equals n equals to, 219 00:14:51.759 --> 00:14:54.519 the locus was a conic but he gave no proof. 220 00:14:55.000 --> 00:14:58.879 Daycartes also failed to prove this by pure geometry, but 221 00:14:58.960 --> 00:15:01.759 he shewed that the curve was represented by an equation 222 00:15:01.960 --> 00:15:06.399 of the second degree, that is, it was a conic section. Subsequently, 223 00:15:06.960 --> 00:15:10.679 Newton gave an elegant solution of the problem by pure geometry. 224 00:15:12.039 --> 00:15:16.039 In the second book, Descartes divides curves into two classes, 225 00:15:16.519 --> 00:15:21.759 namely geometrical and mechanical curves. He defines geometrical curves as 226 00:15:21.840 --> 00:15:25.240 those which can be generated by the intersection of two lines, 227 00:15:25.279 --> 00:15:30.600 each moving parallel to one coordinate axis, with commensurable velocities, 228 00:15:30.720 --> 00:15:33.360 by which he meant that dy by dx was an 229 00:15:33.399 --> 00:15:38.080 algebraical function, as for example is the case in the 230 00:15:38.120 --> 00:15:42.200 ellipse and the cissoid. He calls the curve mechanical when 231 00:15:42.200 --> 00:15:45.679 the ratio of the velocities of these lines is incommensurable, 232 00:15:46.240 --> 00:15:48.399 by which he meant that dy by dx was a 233 00:15:48.440 --> 00:15:52.639 transcendental function, as for example is the case in the 234 00:15:52.679 --> 00:15:58.840 cycloid and the quadratris. Descartes confined his discussion to algebraical 235 00:15:58.919 --> 00:16:02.600 curves das not treat of the theory of mechanical curves. 236 00:16:03.240 --> 00:16:09.039 The classification into algebraical and transcendental curves now usual is 237 00:16:09.120 --> 00:16:13.200 due to Newton. Daycart also paid particular attention to the 238 00:16:13.200 --> 00:16:16.840 theory of the tangents to curves, as perhaps might be 239 00:16:16.879 --> 00:16:21.399 inferred from his system of classification just alluded to. Then, 240 00:16:21.440 --> 00:16:24.279 the current definition of a tangent at a point was 241 00:16:24.320 --> 00:16:27.480 a straight line through the point such that between it 242 00:16:27.600 --> 00:16:30.480 and the curve no other straight line could be drawn, 243 00:16:30.879 --> 00:16:35.919 i e. The straight line of closest contact. Daycartes proposed 244 00:16:35.919 --> 00:16:39.120 to substitute for this that the tangent was the limiting 245 00:16:39.120 --> 00:16:42.440 position of the secin formatt and at a later date 246 00:16:42.519 --> 00:16:47.279 McLaurin and Lagrange adopted this definition. Barrow, followed by Newton 247 00:16:47.360 --> 00:16:50.159 and Leebanitz, considered the curve as the limit of an 248 00:16:50.159 --> 00:16:55.000 inscribed polygon when the sides become indefinitely small, and stated 249 00:16:55.000 --> 00:16:58.120 that the side of the polygon, when produced, became in 250 00:16:58.200 --> 00:17:02.519 the limit a tengent to the curve. Roberval, on the 251 00:17:02.559 --> 00:17:05.880 other hand, defined a tangent at a point as the 252 00:17:05.960 --> 00:17:09.799 direction of the motion at that instant of a point 253 00:17:09.839 --> 00:17:13.839 which was describing the curve. The results are the same 254 00:17:13.880 --> 00:17:17.440 whichever definition is selected, but the controversy as to which 255 00:17:17.519 --> 00:17:23.319 definition was the correct one was nonetheless lively. Descartes illustrated 256 00:17:23.319 --> 00:17:26.960 his theory by giving the general rule for drawing tangents 257 00:17:27.000 --> 00:17:31.519 and normals to a roulette. The method used by Descartes 258 00:17:31.599 --> 00:17:35.119 for finding the tangent or normal at any point of 259 00:17:35.160 --> 00:17:39.279 a given curve was substantially as follows. He determined the 260 00:17:39.319 --> 00:17:42.359 center and radius of a circle which should cut the 261 00:17:42.440 --> 00:17:47.119 curve into consecutive points. There, the tangent to the circle 262 00:17:47.480 --> 00:17:50.720 at that point will be the required tangent to the curve. 263 00:17:51.680 --> 00:17:55.160 In modern textbooks, it is usual to express the condition 264 00:17:55.440 --> 00:17:57.920 that two of the points in which a straight line 265 00:17:58.039 --> 00:18:01.839 such as y equals mx plas c cuts the curve 266 00:18:02.240 --> 00:18:06.640 shall coincide with the given point. This enables us to 267 00:18:06.680 --> 00:18:09.839 determine M and C, and thus the equation of the 268 00:18:09.920 --> 00:18:14.759 tangent there is determined. Descartes, however, did not venture to 269 00:18:14.839 --> 00:18:18.039 do this. But selecting a circle as the simplest curve 270 00:18:18.200 --> 00:18:20.839 and one to which we know how to draw a tangent, 271 00:18:20.960 --> 00:18:23.599 he so fixed his circle as to make it touch 272 00:18:23.720 --> 00:18:26.839 the given curve at the point in question, and thus 273 00:18:26.880 --> 00:18:29.720 reduce the problem to drawing a tangent to a circle. 274 00:18:31.079 --> 00:18:33.599 I should note, in passing that he only applied this 275 00:18:33.759 --> 00:18:37.759 method to curves which are symmetrical about an axis, and 276 00:18:37.839 --> 00:18:40.799 he took the center of the circle on the axis. 277 00:18:42.160 --> 00:18:44.559 Much of the reasoning in these two books is not 278 00:18:44.720 --> 00:18:48.119 easy to follow, but a Latin translation of them with 279 00:18:48.279 --> 00:18:53.400 explanatory notes was prepared by F. De Bone, and an 280 00:18:53.400 --> 00:18:57.759 addition of this with commentary by F. Van Schutten, was 281 00:18:57.799 --> 00:19:01.680 issued in sixteen fifty nine and had a wide circulation. 282 00:19:03.279 --> 00:19:06.920 The third book of Geometrie contains an analysis of the 283 00:19:07.000 --> 00:19:10.319 algebra then current, and it has affected the language of 284 00:19:10.359 --> 00:19:13.799 the subject. By fixing the custom of employing the letters 285 00:19:13.839 --> 00:19:17.440 at the beginning of the alphabet to denote known quantities 286 00:19:18.039 --> 00:19:20.599 and those at the end of the alphabet to denote 287 00:19:20.960 --> 00:19:26.400 unknown quantities. Descartes further introduced the system of indices now 288 00:19:26.400 --> 00:19:29.319 in use. But I would here remind the reader that 289 00:19:29.359 --> 00:19:33.119 the suggestion had been made by previous writers, though it 290 00:19:33.160 --> 00:19:36.640 had not been generally adopted, but very likely it was 291 00:19:36.680 --> 00:19:41.119 the original on the part of Descartes. I think also 292 00:19:41.200 --> 00:19:43.880 that Descartes was the first to realize that his letters 293 00:19:43.960 --> 00:19:48.400 might represent any quantities positive or negative, and that it 294 00:19:48.480 --> 00:19:51.759 was sufficient to prove a proposition for one general case 295 00:19:52.359 --> 00:19:56.319 compared to the old procedure. In this book, he made 296 00:19:56.559 --> 00:19:58.960 use of the rule for determining a limit to the 297 00:19:59.039 --> 00:20:02.920 number of pausesative and negative roots of an algebraical equation, 298 00:20:03.680 --> 00:20:06.559 which is still known by his name, and introduce the 299 00:20:06.599 --> 00:20:11.160 method of indeterminate coefficients for the solution of equations. He 300 00:20:11.200 --> 00:20:14.240 believed that he had given a method by which algebraical 301 00:20:14.279 --> 00:20:17.039 equations of any order could be solved, but in this 302 00:20:17.359 --> 00:20:20.200 he was mistaken. He made use of the method of 303 00:20:20.200 --> 00:20:25.359 indeterminate coefficients. Of the other two appendices to the discourse 304 00:20:25.519 --> 00:20:29.039 was one devoted to optics. The chief interest of this 305 00:20:29.599 --> 00:20:32.680 consists in the statement given of the law of refraction. 306 00:20:33.400 --> 00:20:36.400 This appears to have been taken from Snell's work, but 307 00:20:36.559 --> 00:20:39.440 not only is there no acknowledgment of the source from 308 00:20:39.519 --> 00:20:42.839 which it was obtained, but it is enunciated in such 309 00:20:42.880 --> 00:20:45.039 a way so as to lead a careless reader to 310 00:20:45.119 --> 00:20:47.960 suppose that it is due to the researches of Daycartes. 311 00:20:49.119 --> 00:20:53.240 Daiscartes would seem to have repeated Snell's experiments when in 312 00:20:53.359 --> 00:20:57.200 Paris in sixteen twenty six or sixteen twenty seven, and 313 00:20:57.279 --> 00:21:00.759 it is possible that he subsequently forgot how much he 314 00:21:00.839 --> 00:21:05.079 owed to the earlier investigations of Snell. A large part 315 00:21:05.119 --> 00:21:08.359 of the optics is devoted to determining the best shape 316 00:21:08.400 --> 00:21:12.119 for the lenses of a telescope, but the mechanical difficulties 317 00:21:12.160 --> 00:21:15.200 in grinding a surface of glass to a required form 318 00:21:15.240 --> 00:21:18.839 are so great as to render these investigations of little 319 00:21:18.880 --> 00:21:23.400 practical use. Daycartes seemed to have been doubtful whether to 320 00:21:23.519 --> 00:21:26.640 regard the rays of light as proceeding from the eye, 321 00:21:26.680 --> 00:21:30.039 and so to speak, touching the object as the Greeks 322 00:21:30.039 --> 00:21:32.839 had done, or as proceeding from the object and so 323 00:21:32.960 --> 00:21:36.720 affecting the eye. But since he considered the velocity of 324 00:21:36.799 --> 00:21:39.039 light to be infinite, he did not deem the point 325 00:21:39.200 --> 00:21:45.440 particularly important. The other appendix on meteors contains an explanation 326 00:21:45.519 --> 00:21:51.400 of numerous atmospheric phenomenon, including the rainbow. Daycart was unacquainted 327 00:21:51.400 --> 00:21:55.359 with the unequal refrangibility of rays of light of different colors, 328 00:21:55.680 --> 00:22:01.079 and the explanation of the latter is necessarily incomplete. Descartes's 329 00:22:01.079 --> 00:22:04.480 physical theory of the universe, embodying most of the results 330 00:22:04.519 --> 00:22:08.480 contained in his earlier and unpublished Le Monde, was given 331 00:22:08.559 --> 00:22:13.000 in his Principia sixteen forty four, and rests on a 332 00:22:13.039 --> 00:22:17.720 metaphysical basis. He commences with a discussion on motion, and 333 00:22:17.759 --> 00:22:20.839 then lays down ten laws of nature, of which the 334 00:22:20.880 --> 00:22:24.039 first two are almost identical with the first two laws 335 00:22:24.039 --> 00:22:28.200 of motion as given by Newton. The remaining eight laws 336 00:22:28.240 --> 00:22:32.200 are inaccurate. He next proceeds to discuss the nature of matter, 337 00:22:32.599 --> 00:22:36.480 which he regards as uniform in kind, though there are 338 00:22:36.519 --> 00:22:40.359 three forms of it. He assumes that the matter of 339 00:22:40.400 --> 00:22:43.960 the universe must be in motion and that the motion 340 00:22:44.119 --> 00:22:48.079 must result in a number of vortices. He states that 341 00:22:48.160 --> 00:22:50.720 the Sun is the center of an immense whirlpool of 342 00:22:50.759 --> 00:22:54.519 this matter, in which the planets float and are swept 343 00:22:54.640 --> 00:22:58.599 round like straws in a whirlpool of water. Each planet 344 00:22:58.720 --> 00:23:02.000 is supposed to be the center of a secondary whirlpool 345 00:23:02.160 --> 00:23:06.359 by which its satellites are carried. These secondary whirlpools are 346 00:23:06.400 --> 00:23:10.359 supposed to produce variations of density in the surrounding medium 347 00:23:10.759 --> 00:23:14.920 which constitute the primary whirlpool, and so cause the planets 348 00:23:14.960 --> 00:23:18.920 to move on in ellipses and not circles. All these 349 00:23:18.960 --> 00:23:24.400 assumptions are arbitrary and unsupported by any investigation. It is 350 00:23:24.440 --> 00:23:28.200 not difficult to prove that on his hypothesis, the Sun 351 00:23:28.519 --> 00:23:31.039 would be in the center of these ellipses, and not 352 00:23:31.160 --> 00:23:34.000 at a focus, as Kepler had shewn was the case, 353 00:23:34.519 --> 00:23:36.680 and that the weight of a body at every place 354 00:23:36.720 --> 00:23:40.119 on the surface of the Earth except the equator, would 355 00:23:40.119 --> 00:23:43.920 act in a direction which was not vertical. But it 356 00:23:43.960 --> 00:23:46.440 will be sufficient here to say that Newton, in the 357 00:23:46.480 --> 00:23:50.440 second book of his Principia in sixteen eighty seven, considered 358 00:23:50.440 --> 00:23:54.279 the theory in detail, enchewed that its consequences are not 359 00:23:54.319 --> 00:23:57.440 only inconsistent with each of Kepler's laws and with the 360 00:23:57.480 --> 00:24:00.880 fundamental laws of mechanics, but are all also at variance 361 00:24:00.920 --> 00:24:04.599 with the ten laws of nature assumed by Descartes. Still, 362 00:24:04.880 --> 00:24:08.519 in spite of its crudeness and its inherent defects, the 363 00:24:08.519 --> 00:24:12.440 theory of vortices marks a fresh era in astronomy, for 364 00:24:12.519 --> 00:24:14.880 it was an attempt to explain the phenomenon of the 365 00:24:14.920 --> 00:24:19.400 whole universe by the same mechanical laws which experiments shoes 366 00:24:19.440 --> 00:24:25.200 to be true here on Earth. Cavalieri, almost contemporaneously with 367 00:24:25.279 --> 00:24:30.359 the publication in sixteen thirty seven of Descartes's geometry, the 368 00:24:30.440 --> 00:24:33.200 principles of the integral calculus, so far as they are 369 00:24:33.240 --> 00:24:37.279 concerned with summation, were being worked out in Italy. This 370 00:24:37.480 --> 00:24:40.720 was affected by what was called the principle of indivisibles, 371 00:24:41.279 --> 00:24:45.160 and was the invention of Cavalieri. It was applied to 372 00:24:45.279 --> 00:24:49.079 numerous problems connected with the quadrature of curves and surfaces, 373 00:24:49.599 --> 00:24:53.839 the determination of volumes, and the positions of centers of mass, 374 00:24:53.880 --> 00:24:57.880 to the complete exclusion of the tedious method of exhaustions 375 00:24:57.960 --> 00:25:02.240 used by the Greeks. In principle, the methods are the same, 376 00:25:02.400 --> 00:25:05.960 but the notation of indivisibles is more concise and convenient. 377 00:25:06.680 --> 00:25:09.359 It was in its turn superseded at the beginning of 378 00:25:09.400 --> 00:25:13.799 the eighteenth century by the integral calculus, but its use 379 00:25:13.839 --> 00:25:16.759 will be familiar to all mathematicians who have read any 380 00:25:16.799 --> 00:25:20.039 commentary on the first section of the first book of 381 00:25:20.039 --> 00:25:23.640 Newton's Principia, in the application of lemmas two and three 382 00:25:23.960 --> 00:25:29.359 to the determination of areas, volumes, and so on. Bonaventura 383 00:25:29.440 --> 00:25:33.480 Calvieri was born at Milan in fifteen ninety eight and 384 00:25:33.559 --> 00:25:37.440 died at Bologna in November twenty seventh, sixteen forty seven. 385 00:25:38.079 --> 00:25:41.079 He became a Jesuit at an early age, and on 386 00:25:41.160 --> 00:25:44.559 the recommendation of the order, he was in sixteen twenty 387 00:25:44.640 --> 00:25:49.319 nine made professor of mathematics at Bologna, and he continued 388 00:25:49.559 --> 00:25:52.720 to occupy the chair there until his death. I have 389 00:25:52.799 --> 00:25:56.240 already mentioned Cavalieri's name for the part of the book 390 00:25:56.480 --> 00:25:59.920 that he took in introducing the use of logarithms into 391 00:26:00.039 --> 00:26:03.960 to Italy. He was one of the most influential mathematicians 392 00:26:04.000 --> 00:26:08.000 of his time, but his subsequent reputation rests mainly on 393 00:26:08.079 --> 00:26:13.200 his invention of the principle of indivisibles. The principle of 394 00:26:13.240 --> 00:26:17.319 indivisibles has been used by Kepler in sixteen o four 395 00:26:17.359 --> 00:26:21.200 and sixteen fifteen in somewhat crude form. It was first 396 00:26:21.279 --> 00:26:24.480 stated by Cavalieri in sixteen twenty nine, but he did 397 00:26:24.480 --> 00:26:28.559 not publish his results till sixteen thirty five. In his 398 00:26:28.680 --> 00:26:32.960 early annunciation of the principle in sixteen thirty five, Cavalieri 399 00:26:33.079 --> 00:26:35.640 asserted that a line was made up of an infinite 400 00:26:35.720 --> 00:26:40.319 number of points, each without magnitude, and a surface of 401 00:26:40.400 --> 00:26:44.480