WEBVTT 1 00:00:01.199 --> 00:00:04.679 Chapter fifteen, Part two of a short account of the 2 00:00:04.839 --> 00:00:10.080 history of mathematics. This is a LibriVox recording. All LibriVox 3 00:00:10.119 --> 00:00:13.679 recordings are in the public domain. For more information or 4 00:00:13.720 --> 00:00:18.199 to volunteer, please visit LibriVox dot org. This reading is 5 00:00:18.239 --> 00:00:20.920 by Paul King p J K Dot scripts dot MI 6 00:00:20.960 --> 00:00:27.120 I T dot e edu forward slash p K J. Pascal. 7 00:00:28.399 --> 00:00:33.840 Among the contemporaries of Descartes, none displayed greater natural genius 8 00:00:33.880 --> 00:00:37.679 than Pascal, But his reputation rests more on what he 9 00:00:37.799 --> 00:00:41.159 might have done than on what he actually affected, as 10 00:00:41.240 --> 00:00:43.880 during a considerable part of his life he deemed it 11 00:00:43.920 --> 00:00:48.200 his duty to devote his whole time to religious exercises. 12 00:00:49.320 --> 00:00:54.079 Blaise Pascal was born at Clermont on June nineteenth, sixteen 13 00:00:54.119 --> 00:00:58.719 twenty three, and died at Paris on August nineteenth, sixteen 14 00:00:58.799 --> 00:01:03.119 sixty two. His father, a local jucture at Claremont and 15 00:01:03.200 --> 00:01:08.200 himself of some scientific reputation, moved to Paris in sixteen 16 00:01:08.280 --> 00:01:13.280 thirty one, partly to prosecute his own scientific studies, partly 17 00:01:13.640 --> 00:01:16.359 to carry on the education of his only son, who 18 00:01:16.400 --> 00:01:21.480 had already displayed exceptional ability. Pascal was kept at home 19 00:01:21.560 --> 00:01:25.599 in order to assure his not being overworked, and with 20 00:01:25.719 --> 00:01:29.079 the same object, it was directed that his education should 21 00:01:29.120 --> 00:01:32.760 be at first confined to the study of languages, and 22 00:01:32.799 --> 00:01:38.200 should not include any mathematics. This naturally excited the boy's curiosity, 23 00:01:38.400 --> 00:01:41.560 and one day, being then twelve years old, he asked 24 00:01:42.079 --> 00:01:46.599 in what geometry consisted. His tutor replied that it was 25 00:01:46.680 --> 00:01:51.640 the science of constructing exact figures and of determining the 26 00:01:51.719 --> 00:01:57.040 proportions between their different parts. Pascal, stimulated no doubt by 27 00:01:57.040 --> 00:02:00.560 the injunction against reading, it, gave up his place to 28 00:02:00.640 --> 00:02:03.359 the new study, and in a few weeks had discovered 29 00:02:03.359 --> 00:02:07.760 for himself many properties of figures, and in particular the 30 00:02:07.840 --> 00:02:10.719 proposition that the sum of the angles of a triangle 31 00:02:11.280 --> 00:02:14.919 is equal to two right angles. I have read somewhere, 32 00:02:14.919 --> 00:02:17.879 but I cannot lay my hand on the authority that 33 00:02:17.960 --> 00:02:21.319 his proof merely consisted in turning the angular points of 34 00:02:21.360 --> 00:02:24.680 a triangular piece of paper over so as to meet 35 00:02:24.719 --> 00:02:28.719 in the center of the inscribed circle. A similar demonstration 36 00:02:28.919 --> 00:02:31.879 can be got by turning the angular points over so 37 00:02:31.919 --> 00:02:34.879 as to meet at the foot of the perpendicular drawn 38 00:02:34.919 --> 00:02:38.560 from the biggest angle to the opposite side. His father, 39 00:02:38.719 --> 00:02:41.520 struck by this display of ability gave him a copy 40 00:02:41.560 --> 00:02:45.680 of Euclid's Elements, a book which Pascal read with avidity 41 00:02:46.199 --> 00:02:51.520 and soon mastered. At the age of fourteen, he was 42 00:02:51.560 --> 00:02:56.199 admitted to the weekly meetings of Robervelle, Mersenne Midorge and 43 00:02:56.360 --> 00:03:01.120 other French geometricians, from which the French Academy he ultimately sprung, 44 00:03:01.639 --> 00:03:05.439 being created by ordinance of Louis the fourteenth on December 45 00:03:05.479 --> 00:03:10.879 twenty second, sixteen sixty six. At sixteen, Pascal wrote an 46 00:03:11.000 --> 00:03:14.599 essay on conic sections, and in sixteen forty one, at 47 00:03:14.639 --> 00:03:18.479 the age of eighteen, he constructed the first arithmetical machine, 48 00:03:18.919 --> 00:03:23.199 an instrument which eight years later he further improved and patented. 49 00:03:24.000 --> 00:03:27.479 His correspondence with Fermat About this time Shews that he 50 00:03:27.599 --> 00:03:31.280 was then turning his attention to analytical geometry and physics. 51 00:03:31.919 --> 00:03:35.800 He repeated Torcelli's experiments by which the pressure of the 52 00:03:35.879 --> 00:03:39.599 atmosphere could be estimated as a weight, and he confirmed 53 00:03:39.639 --> 00:03:44.120 his theory of the cause of barometrical variations by obtaining 54 00:03:44.159 --> 00:03:47.439 at the same instant readings at different altitudes on the 55 00:03:47.560 --> 00:03:52.199 hill of Puis di Domme. In sixteen fifty, when in 56 00:03:52.240 --> 00:03:56.159 the midst of these researches, Pascal suddenly abandoned his favorite 57 00:03:56.159 --> 00:03:59.599 pursuits to study religion, or, as he says in his 58 00:03:59.680 --> 00:04:04.120 Ponte Say, to contemplate the greatness and the misery of man. 59 00:04:04.599 --> 00:04:07.120 And about the same time he persuaded the younger of 60 00:04:07.159 --> 00:04:12.080 his two sisters to enter the port Royal Society. In 61 00:04:12.159 --> 00:04:15.919 sixteen fifty three, he had to administer his father's estate. 62 00:04:17.319 --> 00:04:20.240 He now took up his old life again and made 63 00:04:20.319 --> 00:04:24.600 several experiments on pressure exerted by gases and liquids. It 64 00:04:24.680 --> 00:04:28.920 was also about this period that he invented the arithmetical triangle, 65 00:04:29.000 --> 00:04:34.360 and together with Framat, created the calculus of probabilities. He 66 00:04:34.439 --> 00:04:38.120 was meditating marriage when an accident again turned the current 67 00:04:38.240 --> 00:04:41.600 of his thoughts to a religious life. He was driving 68 00:04:41.600 --> 00:04:45.240 a foreign hand on November twenty third, sixteen fifty four, 69 00:04:45.800 --> 00:04:49.519 when the horses ran away. The two leaders dashed over 70 00:04:49.560 --> 00:04:53.319 the parapret of the bridge at Nui, and Pascal was 71 00:04:53.360 --> 00:04:57.879 only saved by the traces. Breaking. Always somewhat of a mystic, 72 00:04:58.000 --> 00:05:01.920 he considered this a special summon to abandon the world. 73 00:05:02.279 --> 00:05:04.720 He wrote an account of the accident on a small 74 00:05:04.759 --> 00:05:07.560 piece of parchment, which for the rest of his life 75 00:05:07.600 --> 00:05:10.800 he wore next to his heart to perpetually remind him 76 00:05:11.360 --> 00:05:15.000 of his covenant, and shortly moved to Port Royal, where 77 00:05:15.040 --> 00:05:18.600 he continued to live until his death in sixteen sixty two. 78 00:05:19.279 --> 00:05:23.160 Always delicate, he had injured his health by his incessant study. 79 00:05:23.759 --> 00:05:27.120 From the age of seventeen or eighteen, he suffered from insomnia, 80 00:05:27.240 --> 00:05:30.519 an acute dyspepsia, and at the time of his death 81 00:05:30.839 --> 00:05:36.680 was completely worn out. His famous provincial letters directed against 82 00:05:36.720 --> 00:05:40.639 the Jesuits and his Ponset were written towards the close 83 00:05:40.680 --> 00:05:43.800 of his life and are the first example of that 84 00:05:43.920 --> 00:05:48.360 finished form which is characteristic of the best French literature. 85 00:05:49.680 --> 00:05:53.120 The only mathematical work that he produced after retiring to 86 00:05:53.199 --> 00:05:57.399 Port Royal was the Essay of the Cycloid in sixteen 87 00:05:57.480 --> 00:06:01.600 fifty eight. He was suffering from leaplessness and a toothache 88 00:06:01.639 --> 00:06:04.639 when the idea occurred to him, and to his surprise, 89 00:06:04.759 --> 00:06:08.839 his teeth immediately ceased to ache. Regarding this as a 90 00:06:08.879 --> 00:06:13.480 divine intimation to proceed with the problem, he worked incessantly 91 00:06:13.600 --> 00:06:17.000 for eight days at it and completed a tolerably full 92 00:06:17.000 --> 00:06:21.319 account of the geometry of the cycloid. I now proceed 93 00:06:21.360 --> 00:06:25.959 to consider his mathematical works in rather greater detail. His 94 00:06:26.199 --> 00:06:29.639 early essay of the geometry of conics, written in sixteen 95 00:06:29.720 --> 00:06:33.959 thirty nine but not published until seventeen seventy nine, seems 96 00:06:33.959 --> 00:06:37.439 to have been founded on the teachings of Derog. Two 97 00:06:37.519 --> 00:06:40.439 of the results are important as well as interesting. The 98 00:06:40.480 --> 00:06:44.600 first of these is the theorem known now as Pascal's theorem, 99 00:06:45.240 --> 00:06:48.639 namely that if a hexagon is inscribed in a conic, 100 00:06:49.160 --> 00:06:52.240 the points of the intersection of the opposite sides will 101 00:06:52.279 --> 00:06:55.439 lie in a straight line. The second, which is really 102 00:06:55.519 --> 00:06:59.560 due to Derague, is that if a quadrilateral be inscribed 103 00:06:59.560 --> 00:07:02.360 in a and a straight line be drawn cutting the 104 00:07:02.480 --> 00:07:05.759 sides taken in order in the points abcn, d and 105 00:07:05.800 --> 00:07:11.199 the conic in P and Q, then PA times pc 106 00:07:11.560 --> 00:07:17.040 over PD times pd equals Qa times QC over QB 107 00:07:17.199 --> 00:07:23.399 times QD. Pascal's arithmetical triangle, known today as Pascal's triangle, 108 00:07:23.959 --> 00:07:27.399 was written in sixteen fifty three but not printed until 109 00:07:27.480 --> 00:07:32.120 sixteen sixty five. The triangle is constructed as in the 110 00:07:32.160 --> 00:07:36.680 annexed figure. By the way, the annexed figure shows Pascal's 111 00:07:36.800 --> 00:07:40.920 triangle as being kind of on its side by today's standards, 112 00:07:41.879 --> 00:07:44.759 A one at the apex, followed by a one to one, 113 00:07:45.279 --> 00:07:47.680 and then in the third row one two to one, 114 00:07:48.000 --> 00:07:50.600 and in the fourth row one three three one, and 115 00:07:50.680 --> 00:07:54.480 in the fifth row one four six four one. So 116 00:07:54.720 --> 00:07:59.800 they will refer to these lines as diagonal lines. The 117 00:07:59.800 --> 00:08:03.879 whole horizontal lines, the way they're depicted are one one one, one, one, 118 00:08:04.240 --> 00:08:07.439 one two, three, four, five, one three six, ten, fifteen, 119 00:08:07.959 --> 00:08:11.399 one four ten, twenty thirty five, and one five fifteen 120 00:08:11.480 --> 00:08:17.160 thirty five, seventy. And they have that written in actual 121 00:08:17.319 --> 00:08:22.199 rows and columns. So, taking the latter description, each horizontal 122 00:08:22.199 --> 00:08:24.639 line being formed from the one above it by making 123 00:08:24.680 --> 00:08:27.199 every number in it equal to the sum of those 124 00:08:27.240 --> 00:08:29.480 above it and to the left of it in the 125 00:08:29.600 --> 00:08:33.799 row immediately above. For example, in the fourth line, twenty 126 00:08:34.519 --> 00:08:37.000 is equal to one plus three plus six plus ten. 127 00:08:38.320 --> 00:08:42.519 Then Pascal's arithmetical triangle to any required order is got 128 00:08:42.559 --> 00:08:45.840 by drawing a diagonal downwards from the right to left. 129 00:08:45.960 --> 00:08:49.039 Is in the figure. The numbers are what are now 130 00:08:49.080 --> 00:08:53.159 called the figure it numbers. Those in the first line 131 00:08:53.159 --> 00:08:55.279 are called the numbers of the first order. Those in 132 00:08:55.320 --> 00:08:58.559 the second line one, two, three, four, five, and so 133 00:08:58.679 --> 00:09:01.159 on are called natural none numbers or numbers of the 134 00:09:01.240 --> 00:09:04.320 second order, those in the third line numbers of the 135 00:09:04.360 --> 00:09:08.679 third order, and so on. It is easily shown that 136 00:09:08.759 --> 00:09:12.000 the m number in the n th row is M 137 00:09:12.039 --> 00:09:16.200 plus n minus two factorial divided by m minus one 138 00:09:16.279 --> 00:09:21.279 factorial times n minus one factorial. The numbers in any 139 00:09:21.320 --> 00:09:25.000 diagonal give the coefficients of the expansion of a binomial. 140 00:09:25.559 --> 00:09:31.159 For example, the figures in the fifth diagonal, namely one, four, six, 141 00:09:31.200 --> 00:09:34.799 four one, are the coefficients in the expansion of a 142 00:09:35.000 --> 00:09:39.000 plus b raised to the power four. Pascal use the 143 00:09:39.039 --> 00:09:42.840 triangle partly for this purpose, and partly to find the 144 00:09:42.919 --> 00:09:46.720 numbers of combinations of m things taken n at a time, 145 00:09:47.200 --> 00:09:50.759 which you stated correctly to be n plus one times 146 00:09:50.879 --> 00:09:53.559 n plus two times n plus three dot dot dot 147 00:09:53.720 --> 00:09:59.399 times m divided by m minus n factorial. Perhaps as 148 00:09:59.440 --> 00:10:02.840 a mathematic Pascal is best known in connection with his 149 00:10:02.960 --> 00:10:07.480 correspondence with Fermont in sixteen fifteen four, in which he 150 00:10:07.600 --> 00:10:11.440 laid down the principles of the theory of probabilities. This 151 00:10:11.559 --> 00:10:15.639 correspondence arose from a problem proposed by a gamester, the 152 00:10:15.759 --> 00:10:20.240 Chevalier de Maire, to Pascal, who communicated it to Fermat. 153 00:10:21.120 --> 00:10:25.039 The problem was this, two players of equal skill want 154 00:10:25.120 --> 00:10:29.440 to leave the table before finishing the game. Their scores 155 00:10:29.480 --> 00:10:32.840 and the number of points which constitute the game being given, 156 00:10:33.360 --> 00:10:36.600 it is desired to find in what proportion should they 157 00:10:36.639 --> 00:10:41.080 divide the stakes. Pascal and Fermat agreed on the answer, 158 00:10:41.159 --> 00:10:45.080 but gave different proofs. The following is a translation of 159 00:10:45.159 --> 00:10:50.480 Pascal's solution. That of Fermat is given later. The following 160 00:10:50.720 --> 00:10:54.039 is my method for determining the share of each player. When, 161 00:10:54.120 --> 00:10:57.440 for example, two players play a game of three points 162 00:10:57.480 --> 00:11:02.279 and each player has staked thirty two piece fo suppose 163 00:11:02.320 --> 00:11:05.600 that the first player had gained two points and then 164 00:11:05.639 --> 00:11:09.240 the second player one point. They now have to play 165 00:11:09.519 --> 00:11:12.679 for a point on this condition that if the first 166 00:11:12.679 --> 00:11:15.559 player gain, he takes all the money which is at stake, 167 00:11:15.679 --> 00:11:19.679 namely sixty four pietols, while if the second player gain, 168 00:11:19.759 --> 00:11:22.600 each player has two points, so that they are on 169 00:11:22.720 --> 00:11:27.240 terms of equality, and if they leave off playing, each 170 00:11:27.360 --> 00:11:31.240 ought to take thirty two piecetoles. Thus, if the first 171 00:11:31.240 --> 00:11:34.840 player gain, then sixty four pistoles belonged to him, and 172 00:11:34.919 --> 00:11:39.200 if he loses, then thirty two piecetoles belong to him. 173 00:11:40.000 --> 00:11:42.960 If therefore, the players do not wish to play this game, 174 00:11:43.000 --> 00:11:46.519 but to separate without playing it. The first player would 175 00:11:46.559 --> 00:11:49.600 say to the second I am certain of thirty two 176 00:11:49.639 --> 00:11:53.039 piece stoles, even if I lose this game, and for 177 00:11:53.120 --> 00:11:56.360 the other thirty two piece stoles, perhaps I shall have them, 178 00:11:56.399 --> 00:11:59.799 and perhaps you will have them. The chances are equal. 179 00:12:00.639 --> 00:12:03.840 Let us divide these thirty two pistoles equally and give 180 00:12:03.879 --> 00:12:07.000 me also the thirty two pistoles of which I am certain. 181 00:12:08.240 --> 00:12:11.000 Thus the first player will have forty eight pistoles in 182 00:12:11.039 --> 00:12:16.360 the second sixteen pistoles. Next, suppose that the first player 183 00:12:16.440 --> 00:12:19.840 has gained two points in the second player none, and 184 00:12:19.960 --> 00:12:22.879 they are about to play for a point. The condition 185 00:12:23.080 --> 00:12:25.440 is that if the first player gain this point, he 186 00:12:25.519 --> 00:12:29.320 secures the game and takes sixty four pistols, And if 187 00:12:29.320 --> 00:12:32.039 the second player gain this point, then the players will 188 00:12:32.039 --> 00:12:36.120 be in the situation already examined, in which the first 189 00:12:36.120 --> 00:12:39.039 player is entitled the forty eight pistoles and the second 190 00:12:39.120 --> 00:12:43.360 to sixteen pistos. Thus, if they do not wish to play, 191 00:12:43.440 --> 00:12:46.240 the first player would say to the second if I 192 00:12:46.320 --> 00:12:49.600 gain this point, I gain sixty four pistos. If I 193 00:12:49.679 --> 00:12:53.639 lose it, I am titled to forty eight pistos. Give 194 00:12:53.679 --> 00:12:56.399 me then the forty eight pistoles of which I am certain, 195 00:12:56.480 --> 00:12:59.799 and divide the other sixteen. Since our chances of gaining 196 00:12:59.840 --> 00:13:03.399 the point are equal. Thus the first player will have 197 00:13:03.480 --> 00:13:08.480 fifty six pistoles and the second player eight pistos. Finally, 198 00:13:08.879 --> 00:13:12.000 suppose that the first player had gained one point and 199 00:13:12.039 --> 00:13:15.480 the second player none. If they proceed to play for 200 00:13:15.559 --> 00:13:18.639 a point, the condition is that if the first player 201 00:13:18.759 --> 00:13:22.879 gain it, the players will be in the situation first examined, 202 00:13:23.320 --> 00:13:26.799 in which the first player is entitled to fifty six pistos. 203 00:13:27.519 --> 00:13:30.639 If the first player lose the point, each player has 204 00:13:30.720 --> 00:13:34.759 then a point, and each is entitled to thirty two pistoles. 205 00:13:35.399 --> 00:13:38.399 Give me the thirty two pistoles of which I am certain, 206 00:13:38.559 --> 00:13:42.120 and divide the remainder of the fifty six pistoles equally, 207 00:13:42.639 --> 00:13:47.840 that is, divide the twenty four pistoles equally. Thus the 208 00:13:47.879 --> 00:13:50.639 first player will have the sum of thirty two and 209 00:13:50.799 --> 00:13:55.120 twelve pistoles, that is, forty four pistoles, and consequently the 210 00:13:55.200 --> 00:14:01.399 second will have twenty pistoles. Pascal proceeds next to consider 211 00:14:01.480 --> 00:14:04.919 the similar problem when the game is won by whomever 212 00:14:05.440 --> 00:14:09.600 first obtains M plus en points, and one player has 213 00:14:09.879 --> 00:14:13.639 m while the other has end points. The answer is 214 00:14:13.679 --> 00:14:20.320 obtained by using the arithmetical triangle or Pascal's triangle. The 215 00:14:20.399 --> 00:14:24.559 general solution, in which the skill of the players is unequal, 216 00:14:24.679 --> 00:14:28.279 is given in many modern textbooks on algebra, and agrees 217 00:14:28.360 --> 00:14:32.120 with Pascal's result, though of course the notation of the 218 00:14:32.200 --> 00:14:36.960 latter is different and less convenient. Pascal made a most 219 00:14:36.960 --> 00:14:40.120 illegitimate use of the new theory in the seventh chapter 220 00:14:40.240 --> 00:14:44.759 of his Ponce. He practically puts his argument that, as 221 00:14:44.799 --> 00:14:48.799 the value of eternal happiness must be infinite, then even 222 00:14:48.840 --> 00:14:52.919 if the probability of a religious life ensuring eternal happiness 223 00:14:52.919 --> 00:14:56.919 be very small, still the expectation, which is measured by 224 00:14:56.960 --> 00:15:00.000 the product of the two must be of sufficient MEAs 225 00:15:00.000 --> 00:15:05.320 magnitude to make it worth while to be religious. The argument, 226 00:15:05.480 --> 00:15:09.559 if worth anything, would apply equally to any religion which 227 00:15:09.559 --> 00:15:13.480 has promised eternal happiness to those who accepted its doctrines. 228 00:15:14.159 --> 00:15:17.519 If any conclusion may be drawn from the statement, it 229 00:15:17.600 --> 00:15:21.960 is the undesirability of applying mathematics to questions of morality, 230 00:15:22.480 --> 00:15:25.720 of which some of the data are necessarily outside the 231 00:15:25.799 --> 00:15:29.720 range of an exact science. It is only fair to 232 00:15:29.840 --> 00:15:34.440 add that no one had more contempt than Pascal for 233 00:15:34.519 --> 00:15:38.240 those who changed their opinions according to the prospect of 234 00:15:38.399 --> 00:15:43.759 material benefit, and this isolated passage is at variance with 235 00:15:43.840 --> 00:15:48.480 the spirit of his writings. The last mathematical work of 236 00:15:48.559 --> 00:15:51.960 Pascal was that on the Cycloid of sixteen fifty eight. 237 00:15:52.600 --> 00:15:55.399 The cycloid is the curve traced out by the point 238 00:15:55.480 --> 00:15:58.480 on the circumference of a circular hoop which rolls along 239 00:15:58.480 --> 00:16:02.759 a straight line. Galileo in sixteen thirty had been the 240 00:16:02.799 --> 00:16:06.559 first to call attention to this curve, and had suggested 241 00:16:06.879 --> 00:16:09.759 that the arches of bridges should be built in the 242 00:16:09.759 --> 00:16:13.440 form of it. It is a graceful curve, but the 243 00:16:13.480 --> 00:16:17.039 only bridge with a cycloidal arch of which I have 244 00:16:17.200 --> 00:16:20.120 heard is the one built by Essex in the grounds 245 00:16:20.120 --> 00:16:24.519 of Trinity College at Cambridge. Four years later, in sixteen 246 00:16:24.559 --> 00:16:29.879 thirty four, Roberval found the area of the cycloid. Descartes 247 00:16:29.960 --> 00:16:33.399 thought little of this solution and defied him to find 248 00:16:33.440 --> 00:16:37.240 its tangents, the same challenge also being sent to Fermont, 249 00:16:37.720 --> 00:16:42.360 who at once solved the problem. Several questions connected with 250 00:16:42.480 --> 00:16:45.720 the curve and with the surface and volume generated by 251 00:16:45.759 --> 00:16:49.159 its revolution, about its axis base or the tangent at 252 00:16:49.159 --> 00:16:54.720 its vertex, were then proposed by various mathematicians. These and 253 00:16:54.840 --> 00:16:58.080 some analogous questions, as well as the positions of the 254 00:16:58.159 --> 00:17:02.080 centers of mass of the solids there formed, were solved 255 00:17:02.080 --> 00:17:05.640 by Pascal in sixteen fifty eight, and the results were 256 00:17:05.680 --> 00:17:09.960 issued as a challenge to the world. Wallace succeeded in 257 00:17:10.039 --> 00:17:13.480 solving all the questions except those connected with the center 258 00:17:13.559 --> 00:17:18.160 of mass. Pascal's own solutions were affected by the method 259 00:17:18.160 --> 00:17:21.480 of indivisibles, and are similar to those which a modern 260 00:17:21.480 --> 00:17:25.000 mathematician would give. By the aid of the integral calculus. 261 00:17:26.119 --> 00:17:30.519 He obtained by summation what are equivalent to the following integrals, 262 00:17:31.240 --> 00:17:36.839 the integral of sign phi de phi, also the integral 263 00:17:36.960 --> 00:17:40.839 of sign squared phi de phi, and also the integral 264 00:17:41.200 --> 00:17:46.920 of Phi times sign phi d phi, one limit being 265 00:17:46.960 --> 00:17:51.920 either zero or half Pi. He also investigated the geometry 266 00:17:51.960 --> 00:17:56.359 of the Archimedean spiral. These researches, according to de Lambert, 267 00:17:56.599 --> 00:18:00.599 form a connecting link between the geometry of archimytes and 268 00:18:00.640 --> 00:18:07.960 the infinitesimal calculus of Newton Wallace. John Wallace was born 269 00:18:08.039 --> 00:18:12.079 at Ashford on November twenty second, sixteen sixteen, and died 270 00:18:12.119 --> 00:18:17.200 ah Oxford on October twenty eighth seventeen o three, when 271 00:18:17.240 --> 00:18:19.559 fifteen years old, he happened to see a book on 272 00:18:19.640 --> 00:18:23.799 arithmetic in the hands of his brother. Struck with curiosity 273 00:18:23.880 --> 00:18:26.759 at the odd signs and symbols in it, he borrowed 274 00:18:26.799 --> 00:18:31.039 the book and in a fortnight mastered the subject. It 275 00:18:31.160 --> 00:18:33.799 was intended that he should be a doctor, and he 276 00:18:33.920 --> 00:18:37.799 was sent to Emmanuel College, Cambridge. While there he kept 277 00:18:37.839 --> 00:18:41.160 an act on the doctrine of the circulation of the blood. 278 00:18:41.759 --> 00:18:44.200 This is said to have been the first occasion in 279 00:18:44.279 --> 00:18:48.880 Europe on which this theory was publicly maintained in disputation. 280 00:18:50.839 --> 00:18:55.440 His interests, however, centered on mathematics. He was elected to 281 00:18:55.519 --> 00:18:59.799 a fellowship at Queen's College, Cambridge and subsequently took orders, 282 00:19:00.279 --> 00:19:04.000 but on the whole adhered to the Puritan Party, whom 283 00:19:04.039 --> 00:19:09.240 he rendered great assistance in deciphering the Royalist despatches. He however, 284 00:19:09.359 --> 00:19:13.319 joined the Moderate Presbyterians in signing the Remonstrance against the 285 00:19:13.359 --> 00:19:17.519 Execution of Charles the First, by which he incurred the 286 00:19:17.599 --> 00:19:23.000 lasting hostility of the Independence. In spite of their opposition, 287 00:19:23.720 --> 00:19:27.000 he was appointed in sixteen forty nine to the civilian 288 00:19:27.119 --> 00:19:30.880 chair of Geometry at Oxford, where he lived until his 289 00:19:31.039 --> 00:19:36.279 death on October twenty eighth, seventeen o three. Besides his 290 00:19:36.440 --> 00:19:41.559 mathematical works, he wrote on theology, logic, and philosophy, and 291 00:19:41.720 --> 00:19:44.880 was the first to devise a system for teaching deaf mutes. 292 00:19:45.559 --> 00:19:48.400 I confine myself to a few notes on his more 293 00:19:48.440 --> 00:19:53.240 important mathematical writings. They are notable partly for the introduction 294 00:19:53.440 --> 00:19:56.400 of the use of infinite series as an ordinary part 295 00:19:56.400 --> 00:19:59.799 of analysis, and partly for the fact that they revealed, 296 00:20:00.160 --> 00:20:04.039 explained to all students the principles of those new methods 297 00:20:04.079 --> 00:20:09.599 which distinguish modern from classical mathematics. The most important of 298 00:20:09.680 --> 00:20:15.720 Wallace's works was his Arithmetica Infinitorum, which was published in 299 00:20:15.799 --> 00:20:20.079 sixteen fifty six. In this treatise, the methods of analysis 300 00:20:20.119 --> 00:20:25.599 of Descartes and Cavalieri were systematized and greatly extended, but 301 00:20:25.680 --> 00:20:30.079 their logical exposition is open to criticism. It at once 302 00:20:30.160 --> 00:20:33.119 became the standard book on the subject, and is constantly 303 00:20:33.160 --> 00:20:36.799 referred to by subsequent writers. It is prefaced by a 304 00:20:36.839 --> 00:20:41.359 short tract on conic sections, which was subsequently expanded into 305 00:20:41.519 --> 00:20:45.640 a separate treatise. He commences by proving the law of 306 00:20:45.680 --> 00:20:49.400 indices chews that X to the zero x to the 307 00:20:49.440 --> 00:20:53.160 minus one, x to the minus two, and so on 308 00:20:53.319 --> 00:20:56.799 represent one one over x, one over x squared, and 309 00:20:56.960 --> 00:20:59.960 so on. That x to the power of one half 310 00:21:00.279 --> 00:21:03.799 represents the square root of X, that x to the 311 00:21:03.839 --> 00:21:07.480 power of two thirds represents the qb root of x squared, 312 00:21:08.000 --> 00:21:12.319 and that generally that x to the exponent minus n 313 00:21:12.440 --> 00:21:15.960 represents the reciprocal of x to the power n, and 314 00:21:16.000 --> 00:21:18.599 that x to the p over q represents the q 315 00:21:18.839 --> 00:21:22.279 th root of x to the p. Leaving the numerous 316 00:21:22.319 --> 00:21:27.559 algebraical applications of this discovery, he next proceeds defined by 317 00:21:27.559 --> 00:21:32.119 the method of indivisibles, the area inclosed between the curve 318 00:21:32.400 --> 00:21:34.960 y equals x to the m the axis of x 319 00:21:35.759 --> 00:21:39.119 and any ordinate x equals h, and he proves that 320 00:21:39.160 --> 00:21:42.720 the ratio of this area is equal to that of 321 00:21:42.759 --> 00:21:46.359 the parallelogram on the same base and of the same altitude, 322 00:21:46.440 --> 00:21:49.839 and is equal to the ratio one over M plus one. 323 00:21:51.440 --> 00:21:54.279 He apparently assumed that the same result would be true 324 00:21:54.400 --> 00:21:57.279 also for the curve y equals a, x rays to 325 00:21:57.319 --> 00:21:59.960 the power m, where A is any constant an n 326 00:22:00.119 --> 00:22:03.240 m is any number positive or negative. But he only 327 00:22:03.279 --> 00:22:07.599 discusses the case of which the parabola of m equals two, 328 00:22:08.400 --> 00:22:12.920 and that of the hyperbola in which m equals minus one. 329 00:22:13.559 --> 00:22:17.119 In the latter case, his interpretation of the result is incorrect. 330 00:22:17.880 --> 00:22:21.000 He then chows that similar results might be written down 331 00:22:21.079 --> 00:22:24.160 for any curve of the form Y equals the sum 332 00:22:24.880 --> 00:22:29.079 of a x rays to the power m, and hence 333 00:22:29.279 --> 00:22:32.440 that if the ordinate Y on a curve can be 334 00:22:32.519 --> 00:22:36.920 expanded in powers of the obsisa x, its quadrature can 335 00:22:37.000 --> 00:22:40.359 be determined. Thus, he said that if the equation of 336 00:22:40.400 --> 00:22:43.079 a curve where y equals x to the zero plus 337 00:22:43.200 --> 00:22:45.240 x to the one plus x squared, and so on, 338 00:22:45.880 --> 00:22:49.200 its area would be x plus one half x squared 339 00:22:49.240 --> 00:22:53.359 plus one third x cubed, and so on. He then 340 00:22:53.480 --> 00:22:57.519 applies this to the quadrature of the curves. Y equals 341 00:22:57.599 --> 00:23:01.640 quantity x minus x squared to the power zero. Y 342 00:23:01.799 --> 00:23:05.480 equals the quantity x minus x squared to the power one. 343 00:23:06.480 --> 00:23:11.319 Y equals x minus x squared quantity to the power two, 344 00:23:12.519 --> 00:23:15.759 Y equals the quantity x minus x square to the 345 00:23:15.799 --> 00:23:18.799 power three, and so on. Taken between the limits x 346 00:23:18.839 --> 00:23:21.799 equals zero and x equals one, and shows that the 347 00:23:21.839 --> 00:23:27.319 areas are respectively one one over six, one over thirty 348 00:23:27.720 --> 00:23:31.599 one over one, forty, and so on. He next considers 349 00:23:31.640 --> 00:23:35.039 the curve of the form y equals x to the 350 00:23:35.079 --> 00:23:38.599 power one over M, and establishes the theorem that the 351 00:23:38.640 --> 00:23:41.920 area bounded by the curve the axis of x and 352 00:23:41.960 --> 00:23:44.759 the ordinate x equals one is to the area of 353 00:23:44.799 --> 00:23:47.880 the rectangle on the same base and the same altitude 354 00:23:47.920 --> 00:23:52.119 as M over M plus one. This is equivalent of 355 00:23:52.200 --> 00:23:56.319 finding the value of the integral from zero to one 356 00:23:56.880 --> 00:23:59.480 of x rays to the power one over md x. 357 00:24:00.759 --> 00:24:04.119 He illustrates this by the parabola in which M equals two. 358 00:24:05.039 --> 00:24:08.160 He states, but does not prove, that the corresponding result 359 00:24:08.240 --> 00:24:10.920 of the curve of the form y equals x rays 360 00:24:11.000 --> 00:24:14.440 to the p over q. This work contains also one 361 00:24:14.480 --> 00:24:17.880 of the earliest investigations of the formation and properties of 362 00:24:17.960 --> 00:24:24.000 continued fractions, a discussion that was suggested by Bruckner's use 363 00:24:24.079 --> 00:24:29.119 of these fractions. Wallace shewed considerable ingenuity in reducing the 364 00:24:29.200 --> 00:24:32.119 equations of curves to the forms given above, but as 365 00:24:32.160 --> 00:24:35.279 he was unacquainted with the binomial theorem, he could not 366 00:24:35.319 --> 00:24:38.599 affect the quadrature of the circle whose equation is y 367 00:24:38.799 --> 00:24:42.880 equals quantity x minus x square to the half. Since 368 00:24:42.920 --> 00:24:45.880 he was unable to expand this in powers of x, 369 00:24:46.519 --> 00:24:51.680 he laid down. However, the principle of interpolation. Thus, as 370 00:24:51.720 --> 00:24:54.599 the ordinate of the circle y equals x minus x 371 00:24:54.640 --> 00:24:59.279 squared raised to the half, is a geometrical mean between 372 00:24:59.359 --> 00:25:03.359 the ordinate of the curve y equals quantity x minus 373 00:25:03.480 --> 00:25:07.039 x squared rays to the zero, and y equals quantity 374 00:25:07.200 --> 00:25:09.839 x minus x squared rays to the one. It might 375 00:25:09.880 --> 00:25:13.599 be supposed that, as an approximation the area of the semicircle, 376 00:25:14.039 --> 00:25:17.119 the integral from zero to one of quantity x minus 377 00:25:17.240 --> 00:25:21.200 x squared dx, which is one eighth Pi, might be 378 00:25:21.279 --> 00:25:24.279 taken as the geometrical mean between the values of the 379 00:25:24.319 --> 00:25:27.480 integral from zero to one of quantity x minus x 380 00:25:27.480 --> 00:25:31.920 squared rays to the zero dx, and the integral of 381 00:25:32.079 --> 00:25:36.279 zero to one of quantity x minus x squared rays 382 00:25:36.319 --> 00:25:39.319 to the one dx. That is, the values one and 383 00:25:39.519 --> 00:25:44.039 one sixth. This is equivalent to taking four multiplied by 384 00:25:44.039 --> 00:25:47.880 the root of two thirds or three point two six 385 00:25:47.920 --> 00:25:53.359 approximately as the value of pi. But Wallace argued, we 386 00:25:53.480 --> 00:25:57.240 have in fact a series one one sixth one over thirty, 387 00:25:57.240 --> 00:26:00.039 one over one forty and so on, and therefore the 388 00:26:00.160 --> 00:26:03.599 term interpolated between one and one sixth ought to be 389 00:26:03.720 --> 00:26:07.720 so chosen as to obey the law of the series. This, 390 00:26:07.920 --> 00:26:11.279 by an elaborate method which I need not describe in detail, 391 00:26:12.119 --> 00:26:15.039 leads to a value for the interpolated term, which is 392 00:26:15.119 --> 00:26:22.039 equivalent to taking pie equals two times two over one 393 00:26:22.160 --> 00:26:25.920 times two thirds times four thirds times four fifths times six, 394 00:26:26.039 --> 00:26:28.960 fifth times six over seven times eight over seven times 395 00:26:28.960 --> 00:26:34.880 eight over nine, and so on. The subsequent mathematicians of 396 00:26:34.920 --> 00:26:40.839 the seventeenth century constantly used interpolation to obtain results which 397 00:26:40.880 --> 00:26:44.960 we should attempt to obtain by direct analysis. A few 398 00:26:45.039 --> 00:26:49.160 years later, in sixteen fifty nine, Wallace published a tract 399 00:26:49.839 --> 00:26:53.599 containing the solution of the problems on the cycloid which 400 00:26:53.599 --> 00:26:58.640 had been proposed by Pascal. In this he incidentally explained 401 00:26:58.720 --> 00:27:03.319