WEBVTT 1 00:00:00.520 --> 00:00:03.720 Chapter fifteen, Part three of a short account of the 2 00:00:03.879 --> 00:00:09.039 history of mathematics. This is a LibriVox recording. All LibriVox 3 00:00:09.119 --> 00:00:12.919 recordings are in the public domain. For more information or 4 00:00:12.919 --> 00:00:17.839 to volunteer, please visit LibriVox dot org. This reading is 5 00:00:17.879 --> 00:00:21.559 by Paul King p J K dot scripts dot mi 6 00:00:21.600 --> 00:00:24.679 I T dot E d U forward slash p K J. 7 00:00:25.760 --> 00:00:30.519 A short account of the history of mathematics by W. W. Rowseball. 8 00:00:31.160 --> 00:00:35.280 History of Mathematics from Descartes to Huygens circa sixteen thirty 9 00:00:35.280 --> 00:00:43.200 five to sixteen seventy five, Part three. Huygens Christian Huygens 10 00:00:43.359 --> 00:00:47.159 was born at the Hague on April fourteenth, sixteen twenty nine, 11 00:00:47.799 --> 00:00:51.880 and died in the same town on June eighth, sixteen 12 00:00:51.960 --> 00:00:55.960 ninety five. He generally wrote his name is Hougens h 13 00:00:56.079 --> 00:00:59.799 u g E N S, but I follow the usual 14 00:00:59.840 --> 00:01:06.760 cust In spelling it as Huygens. It is also sometimes 15 00:01:06.760 --> 00:01:14.719 written as Huyghens. His life was uneventful and is a 16 00:01:14.760 --> 00:01:19.040 mere record of the dates of his various works. In 17 00:01:19.120 --> 00:01:22.680 sixteen fifty one he published an essay in which he 18 00:01:22.760 --> 00:01:26.959 shewed the fallacy in a system of quadratures. Proposed by 19 00:01:27.000 --> 00:01:30.599 Gregois de Saint Veisant, who was well versed in the 20 00:01:30.640 --> 00:01:34.400 geometry of the Greeks, but had not grasped the essential 21 00:01:34.480 --> 00:01:39.879 points in the more modern methods. This essay was followed 22 00:01:39.879 --> 00:01:42.959 by tracts on the quadrature of the conics and the 23 00:01:43.040 --> 00:01:48.920 approximate rectification of the circle. In sixteen fifty four, his 24 00:01:49.040 --> 00:01:53.719 attention was directed to the improvement of the telescope. In 25 00:01:53.840 --> 00:01:57.400 conjunction with his brother, he devised a new and better 26 00:01:57.439 --> 00:02:01.719 way of grinding and polishing lenses. As a result of 27 00:02:01.760 --> 00:02:07.079 these improvements, he was able during the following two years 28 00:02:07.120 --> 00:02:11.759 sixteen fifty five and sixteen fifty six to resolve numerous 29 00:02:12.000 --> 00:02:17.680 astronomical questions, as, for example, the nature of Saturn's appendage. 30 00:02:18.639 --> 00:02:23.319 His astronomical observations required some exact means of measuring time, 31 00:02:24.080 --> 00:02:27.719 and he was thus led in sixteen fifty six to 32 00:02:27.759 --> 00:02:31.199 invent the pendulum clock, as described in his tract on 33 00:02:31.360 --> 00:02:36.840 Holurogium sixteen fifty eight. The time pieces previously in use 34 00:02:36.879 --> 00:02:41.879 had been balanced clocks. In the year sixteen fifty seven, 35 00:02:42.039 --> 00:02:45.879 Huygens wrote a small work on the calculus of probabilities, 36 00:02:46.400 --> 00:02:51.039 founded on the correspondence of Pascal and Format. He spent 37 00:02:51.120 --> 00:02:54.560 a couple of years in England. About this time. His 38 00:02:54.719 --> 00:02:58.560 reputation was now so great that in sixteen sixty five 39 00:02:58.639 --> 00:03:01.639 Louis the fourteenth offered him a pension if he would 40 00:03:01.680 --> 00:03:05.919 live in Paris, which accordingly then became his place of residence. 41 00:03:07.199 --> 00:03:10.360 In sixteen sixty eight he sent to the Royal Society 42 00:03:10.360 --> 00:03:13.479 of London in answer to a problem. They had proposed 43 00:03:13.960 --> 00:03:18.000 a memoir in which, simultaneously with Wallace and Wren, he 44 00:03:18.120 --> 00:03:21.199 proved by experiment that the momentum in a certain direction 45 00:03:21.360 --> 00:03:24.439 before the collision of two bodies is equal to the 46 00:03:24.439 --> 00:03:28.719 momentum in that direction after the collision. This was one 47 00:03:28.800 --> 00:03:32.280 of the points in mechanics on which Descartes had been mistaken. 48 00:03:33.360 --> 00:03:38.240 The most important of Huygens's work was in his Horologium Oscilatorium, 49 00:03:38.360 --> 00:03:43.560 published in Paris in sixteen seventy three. The first chapter 50 00:03:43.719 --> 00:03:48.120 is devoted to the pendulum clocks. The second chapter contains 51 00:03:48.159 --> 00:03:51.840 a complete account of the descent of heavy bodies under 52 00:03:51.879 --> 00:03:54.879 their own weights in a vacuum, either vertically down or 53 00:03:54.919 --> 00:03:59.719 on smooth curves. The second chapter contains a complete account 54 00:03:59.759 --> 00:04:02.080 of the us the descent of heavy bodies under their 55 00:04:02.080 --> 00:04:05.240 own weights in a vacuum, either vertically down or on 56 00:04:05.360 --> 00:04:10.280 smooth curves. Amongst other propositions, he shows that the cycloid 57 00:04:10.319 --> 00:04:16.360 is tautochronous. In the third chapter, he defines evolutes and involutes, 58 00:04:16.839 --> 00:04:20.680 proves some of their more elementary properties, and illustrates his 59 00:04:20.800 --> 00:04:24.600 methods by finding the evolutes of the cycloid and the parabola. 60 00:04:25.480 --> 00:04:29.079 These are the earliest instances in which the envelope of 61 00:04:29.120 --> 00:04:33.360 a moving line is determined. In the fourth chapter, he 62 00:04:33.480 --> 00:04:37.439 solves the problem of the compound pendulum and chows that 63 00:04:37.480 --> 00:04:42.639 the centers of oscillation and suspension are interchangeable. In the 64 00:04:42.680 --> 00:04:46.519 fifth and last chapter, he discusses again the theory of clocks, 65 00:04:47.120 --> 00:04:49.560 points out that if the bob of the pendulum were 66 00:04:49.600 --> 00:04:54.240 made by means of cycloidal checks to oscillate in a cycloid, 67 00:04:54.800 --> 00:04:59.279 the oscillations would be isochronous, and finishes by shewing that 68 00:04:59.319 --> 00:05:02.560 the centrifia force on a body which moves in a 69 00:05:02.600 --> 00:05:07.439 circle of radius R with uniform velocity V varies directly 70 00:05:07.560 --> 00:05:12.519 as V squared and inversely as R. This work contains 71 00:05:12.560 --> 00:05:17.720 the first attempt to apply dynamics to bodies of finite size, 72 00:05:17.800 --> 00:05:22.600 and not merely to particles. In sixteen seventy five, Huygens 73 00:05:22.639 --> 00:05:25.959 proposed to regulate the motion of watches by use of 74 00:05:26.000 --> 00:05:29.279 the balanced spring, in the theory of which he had 75 00:05:29.319 --> 00:05:33.920 been perhaps anticipated in a somewhat ambiguous and incomplete statement 76 00:05:34.399 --> 00:05:39.319 made by Hook in sixteen fifty eight. Watches or portable 77 00:05:39.399 --> 00:05:43.759 clocks had been invented early in the sixteenth century, and 78 00:05:43.839 --> 00:05:46.839 by the end of that century were not very uncommon, 79 00:05:47.439 --> 00:05:51.279 but they were clumsy and unreliable, being driven by a 80 00:05:51.319 --> 00:05:56.720 main spring and regulated by a conical pulley and verge escapement. Moreover, 81 00:05:56.879 --> 00:06:02.639 until sixteen eighty seven, they had only one hand. The 82 00:06:02.759 --> 00:06:06.160 first watch whose motion was regulated by a balanced spring 83 00:06:06.279 --> 00:06:10.360 was made at Paris under Huygen's directions and presented by 84 00:06:10.480 --> 00:06:14.680 him to Louis the fourteenth The increasing intolerance of the 85 00:06:14.759 --> 00:06:18.759 Catholics led to his return to Holland in sixteen eighty one, 86 00:06:19.319 --> 00:06:22.839 and after the revocation of the Edict of Nantes, he 87 00:06:22.959 --> 00:06:27.959 refused to hold any further communication with France. He now 88 00:06:28.000 --> 00:06:32.199 devoted himself to the construction of lenses of enormous focal length. 89 00:06:32.839 --> 00:06:36.240 Of these three of focal length one hundred twenty three 90 00:06:36.319 --> 00:06:40.120 feet one hundred eighty feet and two hundred ten feet 91 00:06:40.240 --> 00:06:44.600 was subsequently given by him to the Royal Society in London, 92 00:06:44.639 --> 00:06:49.560 in whose possession they still remain. It was about this 93 00:06:49.639 --> 00:06:53.399 time that he discovered the achromatic eyepiece for a telescope, 94 00:06:53.480 --> 00:06:57.000 which is known by his name. In sixteen eighty nine 95 00:06:57.040 --> 00:06:59.720 he came from Holland to England in order to make 96 00:07:00.079 --> 00:07:04.160 the acquaintance of Newton, whose Principia had been published in 97 00:07:04.279 --> 00:07:09.040 sixteen eighty seven, the extraordinary merits of which Huygens had 98 00:07:09.079 --> 00:07:14.600 at once recognized. On his return. In sixteen ninety Huygens 99 00:07:14.639 --> 00:07:19.120 published as Treatise on Light, in which the undulatory theory 100 00:07:19.639 --> 00:07:23.879 was expounded and explained. Most of this had been written 101 00:07:23.920 --> 00:07:27.560 as early as sixteen seventy eight. The general idea of 102 00:07:27.600 --> 00:07:30.839 the theory had been suggested by Robert Hook in sixteen 103 00:07:30.920 --> 00:07:34.959 sixty four, but he had not investigated its consequences in 104 00:07:35.040 --> 00:07:40.120 any detail. This publication falls outside the years considered in 105 00:07:40.120 --> 00:07:43.399 this chapter, but here it may be briefly said that, 106 00:07:43.519 --> 00:07:47.600 according to the wave or undulatory theory, space is filled 107 00:07:47.639 --> 00:07:51.560 with an extremely rare ether, and light is caused by 108 00:07:51.560 --> 00:07:54.839 a series of waves or vibrations in this ether, which 109 00:07:54.839 --> 00:07:58.920 are set in motion by the pulsations of the luminous body. 110 00:08:00.079 --> 00:08:03.759 From this hypothesis, Huygens deduced that the laws of reflection 111 00:08:03.920 --> 00:08:08.920 and refraction explained the phenomenon a double refraction, and gave 112 00:08:08.959 --> 00:08:13.879 a construction for the extraordinary ray in biaxial crystals. While 113 00:08:14.000 --> 00:08:20.000 he found by experiment the chief phenomenon of polarization, the 114 00:08:20.040 --> 00:08:24.480 immense reputation and unrivaled powers of Newton led to the 115 00:08:24.560 --> 00:08:27.800 disbelief in the theory, which he rejected, and to the 116 00:08:27.920 --> 00:08:31.839 general adoption of Newton's emission theory. But it should be 117 00:08:31.959 --> 00:08:35.519 noted that Huygen's explanation of some phenomenon, such as the 118 00:08:35.559 --> 00:08:40.399 colors of thin plates, was inconsistent with the results of experiments, 119 00:08:40.840 --> 00:08:43.720 Nor was it until Young and Wallaston at the beginning 120 00:08:43.759 --> 00:08:48.320 of this century revived the undulatory theory and modified some 121 00:08:48.399 --> 00:08:52.960 of its details, and Fresnel elaborated their views that its 122 00:08:53.039 --> 00:08:58.399 acceptance could be fully justified. Besides these works, Huygens took 123 00:08:58.480 --> 00:09:01.639 part in most of the contrary verses and challenges which 124 00:09:01.679 --> 00:09:04.639 then played so large a part in the mathematical world, 125 00:09:05.159 --> 00:09:08.639 and wrote several minor tracts in one of these, he 126 00:09:08.720 --> 00:09:12.840 investigated the form and properties of the caternary. In another, 127 00:09:13.000 --> 00:09:16.559 he stated in general terms the rule for finding maxima 128 00:09:16.600 --> 00:09:20.159 and minima, of which format had made use, and shewed 129 00:09:20.200 --> 00:09:23.879 that the subtangent of an algebraical curve fx y equal 130 00:09:24.039 --> 00:09:28.080 zero was equal to y f y by fx, where 131 00:09:28.279 --> 00:09:32.600 f y is the derivative of fx y, regarded as 132 00:09:32.639 --> 00:09:36.840 a function of y. In some posthumous works issued at 133 00:09:36.919 --> 00:09:40.639 Leyden in seventeen o three, he further shewed how from 134 00:09:40.679 --> 00:09:44.559 the focal lengths of the component lenses the magnifying power 135 00:09:44.600 --> 00:09:47.639 of a telescope could be determined, and explains some of 136 00:09:47.639 --> 00:09:52.120 the phenomenon connected with halos and parhelia. I should add 137 00:09:52.159 --> 00:09:55.440 that almost all of its demonstrations, like those of Newton, 138 00:09:55.919 --> 00:09:59.840 are rigidly geometrical, and he would seem to have made 139 00:09:59.840 --> 00:10:03.679 no no use of the differential or fluctional calculus, though 140 00:10:03.679 --> 00:10:08.200 he admitted the validity of the methods used therein. Thus, 141 00:10:08.240 --> 00:10:11.320 even when first written, his works were expressed in an 142 00:10:11.399 --> 00:10:16.399 archaic language and perhaps received less attention than their intrinsic 143 00:10:16.480 --> 00:10:21.039 merits deserved. I have now traced the development of mathematics 144 00:10:21.039 --> 00:10:24.039 for a period which we may take roughly as dating 145 00:10:24.080 --> 00:10:27.879 from sixteen thirty five to sixteen seventy five, under the 146 00:10:27.919 --> 00:10:34.840 influence of Descartes, Cavalieri, Pascal Wallace, Fermat and Huygens. The 147 00:10:34.919 --> 00:10:39.000 life of Newton partly overlaps this period. His works and 148 00:10:39.080 --> 00:10:43.399 influence are considered in the next chapter. I may dismiss 149 00:10:43.480 --> 00:10:47.080 the remaining mathematicians of this time, whom I dare to 150 00:10:47.200 --> 00:10:52.799 mention with comparatively slight notice. The following is an alphabetical 151 00:10:52.879 --> 00:10:56.440 list of the more remarkable among them. The dates given 152 00:10:56.480 --> 00:10:59.679 to those of the birth and death of the mathematician, 153 00:10:59.720 --> 00:11:04.440 to who whose name they are appended. Bauchet fifteen eighty 154 00:11:04.480 --> 00:11:09.480 one to sixteen thirty eight, Barrow sixteen thirty to sixteen 155 00:11:09.559 --> 00:11:17.279 seventy seven, Brokner sixteen twenty to sixteen eighty four, Collins 156 00:11:17.320 --> 00:11:23.000 sixteen twenty five to sixteen eighty three, Coursier sixteen oh 157 00:11:23.080 --> 00:11:27.679 four to sixteen ninety two, Des Bonnes sixteen oh one 158 00:11:28.080 --> 00:11:32.639 to sixteen fifty two, De la Lubert sixteen hundred to 159 00:11:32.759 --> 00:11:39.360 sixteen sixty four, Frenikre sixteen o five to sixteen seventy, 160 00:11:40.279 --> 00:11:46.799 Gregory sixteen thirty eight to sixteen seventy five, Hook sixteen 161 00:11:46.879 --> 00:11:52.279 thirty five to seventeen oh three, Hood sixteen thirty three 162 00:11:52.600 --> 00:11:57.320 to seventeen oh four, kink Housens sixteen thirty to sixteen 163 00:11:57.440 --> 00:12:03.919 seventy nine, Mercadix sixteen twenty to sixteen eighty seven, Marsenne 164 00:12:04.639 --> 00:12:10.679 fifteen eighty eight to sixteen forty eight, Midorge fifteen eighty 165 00:12:10.759 --> 00:12:16.519 five to sixteen forty seven, Pell sixteen ten to sixteen 166 00:12:16.559 --> 00:12:22.919 eighty five, Richie sixteen nineteen to sixteen ninety two, Roberval 167 00:12:23.519 --> 00:12:29.279 sixteen oh two to sixteen seventy five, Rohmer sixteen forty 168 00:12:29.320 --> 00:12:35.200 four to seventeen ten, Saint Vincent fifteen eighty four to 169 00:12:35.320 --> 00:12:41.159 sixteen sixty seven, Sleuz sixteen twenty two to sixteen eighty five, 170 00:12:41.840 --> 00:12:48.559 Torricelli sixteen oh eight to sixteen forty seven, Chernhausen sixteen 171 00:12:48.639 --> 00:12:53.159 thirty one to seventeen oh eight, Van Schutten died in 172 00:12:53.279 --> 00:12:58.960 sixteen sixty one, and Renn sixteen thirty two to seventeen 173 00:12:59.039 --> 00:13:03.279 twenty three. In the following notes, I have arranged the 174 00:13:03.320 --> 00:13:07.279 above mentioned mathematicians so that, as far as possible, their 175 00:13:07.360 --> 00:13:13.960 chief contributions shall come in chronological order. Bachet Claude Gaspar 176 00:13:14.120 --> 00:13:17.320 Baluchet de mess Iraq was born in Bourg in fifteen 177 00:13:17.360 --> 00:13:20.879 eighty one and died in sixteen thirty eight. He wrote 178 00:13:20.919 --> 00:13:26.200 the problem pleisante sixteen twelve, second and enlarged editions sixteen 179 00:13:26.279 --> 00:13:31.039 twenty four, which contains an interesting collection of arithmetical tricks 180 00:13:31.039 --> 00:13:34.720 and questions, many of which are quoted in chapter one 181 00:13:34.919 --> 00:13:40.200 of My Mathematical Recreations and Problems. Also lesali Monte Arithmetique, 182 00:13:40.480 --> 00:13:46.200 which exists in manuscript, and a translation of arithmetic by Diafontis. 183 00:13:47.360 --> 00:13:50.799 Bachet was the earliest writer who discussed a solution of 184 00:13:50.840 --> 00:13:58.200 indetermined equations by means of continued fractions. Midorge Claude Mid'orge, 185 00:13:58.320 --> 00:14:01.120 born in Paris in fifteen eight eighty five and died 186 00:14:01.159 --> 00:14:04.960 in sixteen forty seven, belonged to a distinguished family of 187 00:14:05.000 --> 00:14:09.320 the Robe, and was himself a councilor at Chatellers, then 188 00:14:09.440 --> 00:14:14.279 treasurer to the local parliament at Amiens. He published some 189 00:14:14.320 --> 00:14:17.799 works on optics, of which one issued in sixteen thirty 190 00:14:17.840 --> 00:14:21.279 one as extant, and in sixteen forty one a treatise 191 00:14:21.320 --> 00:14:25.879 and conic sections. He also left a manuscript containing solutions 192 00:14:25.919 --> 00:14:29.519 of over one thousand geometrical problems, many of which are 193 00:14:29.559 --> 00:14:33.480 said to be ingenious. The annunciations were published by M. 194 00:14:33.559 --> 00:14:40.279 Charles Henry in eighteen eighty two. Mersennes Marin Merceinne born 195 00:14:40.360 --> 00:14:43.399 in fifteen eighty eight and died in Paris in sixteen 196 00:14:43.480 --> 00:14:47.200 forty eight was a Franciscan friar who made it his 197 00:14:47.320 --> 00:14:51.120 business to be acquainted and correspond with the French mathematicians 198 00:14:51.159 --> 00:14:55.000 of that date and many of their foreign contemporaries. In 199 00:14:55.080 --> 00:14:59.440 sixteen thirty four he published a translation of Galileo's Mechanics. 200 00:14:59.679 --> 00:15:04.799 In Si sixteen forty four he issued is Kojitata Physico Mathematica, 201 00:15:04.799 --> 00:15:07.960 by which he is best known, containing an account of 202 00:15:08.039 --> 00:15:12.679 some experiments in physics. He also wrote a synopsis of mathematics, 203 00:15:12.720 --> 00:15:16.639 which was printed in sixteen sixty four. The preface to 204 00:15:16.679 --> 00:15:20.919 the Cogitata contains a statement, probably due to fermat that 205 00:15:21.039 --> 00:15:23.639 in order that two P minus one may be prime, 206 00:15:24.200 --> 00:15:27.639 the only values of P not greater than two hundred 207 00:15:27.639 --> 00:15:35.879 and fifty seven which are possible are one, two, three, five, seven, thirteen, seventeen, 208 00:15:36.000 --> 00:15:40.399 nineteen thirty one, sixty seven, one hundred twenty seven, and 209 00:15:40.480 --> 00:15:45.279 two hundred fifty seven, to which list her Silhoff has shewn, 210 00:15:45.679 --> 00:15:49.960 we must add the number sixty one. With this addition, 211 00:15:50.159 --> 00:15:53.320 the statement has been verified for all except twenty three 212 00:15:53.399 --> 00:15:58.600 values of P, namely sixty one, seventy one, eighty nine, 213 00:15:58.799 --> 00:16:01.759 one hundred and one, one hundred and three, one hundred 214 00:16:01.759 --> 00:16:05.279 and seven, one hundred and nine, one hundred twenty seven, 215 00:16:05.440 --> 00:16:08.480 one hundred thirty seven, one hundred thirty nine, one hundred 216 00:16:08.559 --> 00:16:12.200 forty nine, one hundred fifty seven, one hundred sixty three, 217 00:16:12.399 --> 00:16:16.000 one hundred sixty seven, one hundred seventy three, one hundred 218 00:16:16.080 --> 00:16:19.679 eighty one, one ninety three, one ninety seven, one ninety nine, 219 00:16:19.720 --> 00:16:22.320 two twenty seven, two twenty nine, two forty one, and 220 00:16:22.320 --> 00:16:27.279 two fifty seven. Of these values, Mersenne asserted that P 221 00:16:27.399 --> 00:16:31.559 equals sixty seven, P equals one twenty seven, and P 222 00:16:31.679 --> 00:16:36.000 equals two fifty seven make two to the exponent P 223 00:16:36.120 --> 00:16:40.120 minus one a prime, and that the other values make 224 00:16:40.360 --> 00:16:44.120 two P minus one a composite number. It is most 225 00:16:44.360 --> 00:16:48.159 likely that these results are particular cases of some general 226 00:16:48.200 --> 00:16:51.879 theorem on the subject, which remains to be discovered. The 227 00:16:52.000 --> 00:16:55.399 number two to the power sixty one minus one contains 228 00:16:55.559 --> 00:16:59.639 nineteen digits and is the highest number at present known 229 00:16:59.679 --> 00:17:05.160 to be prime. Its value in digits is two three, o, 230 00:17:05.519 --> 00:17:09.279 five eight, four, three, zero, zero nine two one three 231 00:17:09.480 --> 00:17:13.839 six nine, three nine five one. The theory of perfect 232 00:17:13.960 --> 00:17:18.400 numbers depends directly on that of Mersenne's numbers. It is 233 00:17:18.519 --> 00:17:21.759 probable that all perfect numbers are included in the formula 234 00:17:21.839 --> 00:17:25.279 two to the power of P minus one, as opposed 235 00:17:25.279 --> 00:17:28.880 to negative one plus two to the P. Where negative 236 00:17:28.880 --> 00:17:31.960 one plus two to the P is prime. You could 237 00:17:32.079 --> 00:17:35.559 prove that any number of this form is perfect. Euler 238 00:17:35.640 --> 00:17:39.240 showed that the formula includes all even perfect numbers, and 239 00:17:39.319 --> 00:17:43.279 there is reason to believe though a rigid demonstration is wanting, 240 00:17:43.680 --> 00:17:47.079 then a odd number cannot be perfect. If we assume 241 00:17:47.119 --> 00:17:50.079 that the last of these statements is true, then every 242 00:17:50.119 --> 00:17:54.000 perfect number is of the above form. Thus, if P 243 00:17:54.160 --> 00:17:58.160 equals two, three, five, seven, thirteen, seventeen, nineteen, thirty one, 244 00:17:58.279 --> 00:18:02.759 sixty one, then my ver sends rule. The corresponding values 245 00:18:02.799 --> 00:18:05.839 of negative one plus two to the P are prime. 246 00:18:06.480 --> 00:18:10.640 They are three, seven, thirty one, one hundred, twenty seven, 247 00:18:11.400 --> 00:18:14.359 eight thousand, one hundred ninety one, one hundred and thirty, 248 00:18:14.359 --> 00:18:18.519 one thousand and seventy one, five hundred and twenty four thousand, 249 00:18:18.559 --> 00:18:23.079 two hundred eighty seven two billion, one hundred forty seven million, 250 00:18:23.279 --> 00:18:27.559 four hundred eighty three thousand, six hundred forty seven, and finally, 251 00:18:27.680 --> 00:18:32.039 two quintillion, three hundred five quadrillion, eight hundred and forty 252 00:18:32.079 --> 00:18:36.400 three trillion, nine billion, two hundred and thirteen million, six 253 00:18:36.559 --> 00:18:41.079 hundred ninety three thousand, nine hundred fifty one, and the 254 00:18:41.119 --> 00:18:47.000 corresponding perfect numbers are six twenty eight, four hundred ninety six, 255 00:18:47.599 --> 00:18:51.200 eight thousand, one hundred and twenty eight, thirty three million, 256 00:18:51.359 --> 00:18:56.279 five hundred and fifty thousand, three hundred thirty six, eight billion, 257 00:18:56.480 --> 00:19:00.000 five hundred eighty nine million, eight hundred sixty nine thousand, 258 00:19:00.440 --> 00:19:04.720 fifty six, one hundred and thirty seven billion, four hundred 259 00:19:04.759 --> 00:19:08.039 and thirty eight million, six hundred ninety one thousand and 260 00:19:08.079 --> 00:19:12.720 three hundred twenty eight. Next is two quintillion, three hundred 261 00:19:12.799 --> 00:19:17.880 and five quadrillion, eight hundred and forty three trillion, eight billion, 262 00:19:18.319 --> 00:19:21.359 one hundred thirty nine million, nine hundred and fifty two 263 00:19:21.480 --> 00:19:26.799 thousand and one hundred twenty eight, and finally two undecillion, 264 00:19:27.119 --> 00:19:31.079 six hundred and fifty eight decillion, four hundred and fifty five, 265 00:19:31.200 --> 00:19:35.400 none million, nine hundred ninety one octillion, five hundred and 266 00:19:35.440 --> 00:19:40.160 sixty nine septillion, eight hundred and thirty one heccillion, seven 267 00:19:40.240 --> 00:19:45.400 hundred and forty four, quintillion, six hundred fifty four quadrillion, 268 00:19:45.759 --> 00:19:50.440 six hundred ninety two trillion, six hundred fifteen billion, nine 269 00:19:50.519 --> 00:19:53.960 hundred and fifty three million, eight hundred and forty two thousand, 270 00:19:54.400 --> 00:20:00.720 one hundred and seventy six. De Beaume Coone de Beaume 271 00:20:01.039 --> 00:20:04.319 was born in Blois in sixteen oh one and died 272 00:20:04.359 --> 00:20:09.039 in sixteen fifty two. Wrote explanatory notes on the obscure 273 00:20:09.200 --> 00:20:14.359 and difficult analytical geometry of Descartes. He also discussed as 274 00:20:14.400 --> 00:20:18.000 superior and inferior limits to the roots of an equation. 275 00:20:18.400 --> 00:20:24.680 This was not published till sixteen fifty nine. Roberval Jill 276 00:20:24.920 --> 00:20:29.480 Persigner de Roberval, born at Roberval in sixteen o two 277 00:20:29.559 --> 00:20:34.079 and died at Paris in sixteen seventy five, described himself 278 00:20:34.240 --> 00:20:37.480 from the place of his birth as de Roberval, a 279 00:20:37.599 --> 00:20:42.000 seigniorial title to which he had no right. He discussed 280 00:20:42.039 --> 00:20:45.759 the nature of the tangents to the curves, solved some 281 00:20:45.920 --> 00:20:50.880 of the easier questions connected with the cycloid, generalized Archimedes's 282 00:20:51.000 --> 00:20:54.759 theorems on the spiral and wrote on mechanics and the 283 00:20:54.799 --> 00:20:58.799 method of indivisibles, which he rendered more precise and logical. 284 00:21:00.079 --> 00:21:03.240 He was a professor in the University of Paris and 285 00:21:03.319 --> 00:21:06.680 in correspondence with nearly all of the leading mathematicians of 286 00:21:06.720 --> 00:21:10.599 his time. A complete edition of his works was included 287 00:21:10.640 --> 00:21:14.200 in the Old Memoir of the Academy of Sciences, published 288 00:21:14.240 --> 00:21:20.359 in sixteen ninety three. Van Schutten Franz Vanschuten, to whom 289 00:21:20.359 --> 00:21:24.240 we owe an edition of Vieta's works, succeeded his father, 290 00:21:24.319 --> 00:21:28.319 who had taught mathematics to Huygen's hood, and slews as 291 00:21:28.400 --> 00:21:32.400 a professor at Leyden in sixteen forty six. He brought 292 00:21:32.440 --> 00:21:37.200 out in sixteen fifty nine a Latin translation of Descartes geometry, 293 00:21:37.799 --> 00:21:41.920 and in sixteen fifty seven a collection of mathematical exercises, 294 00:21:42.279 --> 00:21:45.200 in which he recommended the use of coordinates in space 295 00:21:45.319 --> 00:21:50.640 of three dimensions. He died in sixteen sixty one. Saint 296 00:21:50.720 --> 00:21:55.720 Vincent Gregoire de Saint Vincent, a Jesuit born at Bruges 297 00:21:55.759 --> 00:21:58.960 in fifteen eighty four and died at Ghent in sixteen 298 00:21:59.039 --> 00:22:03.039 sixty seven, discovered the expansion of the logarithm of one 299 00:22:03.079 --> 00:22:08.039 plus x in ascending powers of X. Although a circle squarer, 300 00:22:08.200 --> 00:22:10.960 he is worthy of mention for the numerous theorems of 301 00:22:11.039 --> 00:22:14.920 interest which he discovered in his search after the impossible, 302 00:22:15.640 --> 00:22:19.599 and Montucla ingeniously remarks that no one ever squared the 303 00:22:19.680 --> 00:22:24.200 circle with so much ability, except for his principal object, 304 00:22:24.400 --> 00:22:28.920 with so much success. He wrote two books on the subject, 305 00:22:29.079 --> 00:22:32.599 one published in sixteen forty seven and the other in 306 00:22:32.680 --> 00:22:36.440 sixteen sixty eight, which covers some two or three thousand 307 00:22:36.519 --> 00:22:41.160 closely printed pages. The fallacy in the quadrature was pointed 308 00:22:41.160 --> 00:22:45.480 out by Huygens in the former work, he had used indivisibles. 309 00:22:45.799 --> 00:22:51.839 An earlier work entitled Theoremata Mathematica, published in sixteen twenty four, 310 00:22:52.240 --> 00:22:55.359 contains a clear account of the method of exhaustions, which 311 00:22:55.400 --> 00:22:59.160 is applied to several quadratures, notably that of the hyperbola. 312 00:22:59.720 --> 00:23:03.960 For further details of Saint Vincent's life and works, see 313 00:23:04.119 --> 00:23:08.400 La j Quaterie histois des science che le Bleges, Brussels, 314 00:23:08.720 --> 00:23:16.599 eighteen sixty six. Torricelli Evangelista Torricelli born at Fenza in 315 00:23:16.720 --> 00:23:21.279 October fifteenth, sixteen oh eight, died at Florence in sixteen 316 00:23:21.359 --> 00:23:25.440 forty seven, wrote on the quadrature of the cycloid and conix, 317 00:23:25.519 --> 00:23:28.839 the theory of the barometer, the value of gravity found 318 00:23:28.880 --> 00:23:31.519 by observing the motion of two weights connected by a 319 00:23:31.559 --> 00:23:35.599 string passing over a fixed pulley, the theory of projectiles, 320 00:23:35.640 --> 00:23:39.799 and the motion of fluids. His mathematical writings were published 321 00:23:39.839 --> 00:23:46.759 in sixteen forty four. Hood Johann Hood, a burgomaster of Amsterdam, 322 00:23:47.400 --> 00:23:50.359 was born there in sixteen thirty three and died in 323 00:23:50.400 --> 00:23:53.960 the same town in seventeen oh four. He wrote two 324 00:23:54.160 --> 00:23:57.720 tracts in sixteen fifty nine. One was on the reduction 325 00:23:58.200 --> 00:24:02.200 of equations which have equal roots, and the other he stated, 326 00:24:02.240 --> 00:24:05.240 what is equivalent to the proposition that if f of 327 00:24:05.480 --> 00:24:09.240 xy equals zero be the algebraical equation of a curve, 328 00:24:09.759 --> 00:24:14.079 then that subtangent is negative y of the partial derivative 329 00:24:14.279 --> 00:24:17.319 of x with respect to y, divided by the partial 330 00:24:17.359 --> 00:24:21.559 derivative of x with respect to x. But being ignorant 331 00:24:21.559 --> 00:24:25.319 of the notation of the calculus, his annunciation is involved. 332 00:24:27.240 --> 00:24:31.960 Frenich Bernard Franich de Bessi born in Paris circa sixteen 333 00:24:32.000 --> 00:24:35.920 o five and died in sixteen seventy wrote numerous papers 334 00:24:35.960 --> 00:24:39.640 on combinations and on the theory of numbers, and also on 335 00:24:39.759 --> 00:24:43.680 magic squares. It may be interesting to add that he 336 00:24:43.839 --> 00:24:47.519 challenged Huygens to solve the following system of equations in 337 00:24:47.519 --> 00:24:52.079 integers x squared plus y squared equals z squared, x 338 00:24:52.119 --> 00:24:56.799 squared equals U squared plus v squared, and x minus 339 00:24:56.960 --> 00:25:00.640 y equals U minus v. A solution was given by m. 340 00:25:00.680 --> 00:25:06.400 Pepin in eighteen eighty. Frenko's misscellaneous works, edited by de 341 00:25:06.519 --> 00:25:09.960 la Haire, was published in the Memoir de l Academie, 342 00:25:10.200 --> 00:25:16.640 volume five, sixteen ninety one. De la Lubert Antoine de 343 00:25:16.759 --> 00:25:21.119 la Lubert, a Jesuit born in Languedoc in sixteen hundred 344 00:25:21.599 --> 00:25:26.039 died at Toulouse in sixteen sixty four, is chiefly celebrated 345 00:25:26.079 --> 00:25:29.839 for an incorrect solution of Pascal's problems on the cycloid, 346 00:25:30.359 --> 00:25:33.480 which he gave in sixteen sixty, but he has a 347 00:25:33.480 --> 00:25:36.799 better claim to distinction in having been the first mathematician 348 00:25:37.279 --> 00:25:44.119 to study the properties of the helix. Kinkhusen Gerrard Kinkhusen 349 00:25:44.240 --> 00:25:47.599 born in Holland in sixteen thirty and died in sixteen 350 00:25:47.680 --> 00:25:51.880 seventy nine, wrote in sixteen sixty a textbook on analytical 351 00:25:51.920 --> 00:25:56.839 conics in sixteen sixty one an algebra, and in sixteen 352 00:25:56.920 --> 00:26:00.599 sixty nine formed a collection of geometrical problems solved by 353 00:26:00.599 --> 00:26:08.279 analytical geometry. Corsier Pierre Coursier, a Jesuit, born at Troy's 354 00:26:08.559 --> 00:26:12.279 in sixteen o four and died at Auxeire in sixteen 355 00:26:12.359 --> 00:26:15.680 ninety two, wrote on the curves of intersection of a 356 00:26:15.759 --> 00:26:20.759 sphere with a cylinder or cone, also on spherical polygons. 357 00:26:21.240 --> 00:26:25.759 The latter work was published in sixteen sixty three. Richie 358 00:26:26.359 --> 00:26:31.640 Michel Angrichi born in sixteen nineteen, made a cardinal in 359 00:26:31.720 --> 00:26:35.480 sixteen eighty one and died in Rome in sixteen ninety two, 360 00:26:36.240 --> 00:26:40.079 wrote in sixteen sixty six a treatise in which he 361 00:26:40.240 --> 00:26:44.039 solved by Greek geometry those problems on maxima and minima, 362 00:26:44.240 --> 00:26:47.519 and on tangents to curves which had been considered by 363 00:26:47.519 --> 00:26:56.319 Descartes Pascal informat. N mcketter Nicholas Macketter, sometimes known as Kaufmann, 364 00:26:57.160 --> 00:27:01.119 born in Holstein about sixteen twenty, resided most of his 365 00:27:01.279 --> 00:27:05.279 life in England. He went to France in sixteen eighty three, 366 00:27:05.400 --> 00:27:09.440 where he designed and constructed the fountains at Versailles, but 367 00:27:09.559 --> 00:27:13.119 when they were finished, Louis the fourteenth refused to make 368 00:27:13.200 --> 00:27:16.880 him the payment agreed on unless he would turn Catholic. 369 00:27:17.640 --> 00:27:21.400 He died of vexation and poverty in Paris in sixteen 370 00:27:21.440 --> 00:27:27.079 eighty seven. He wrote a treatise on logarithms entitled Logarithmotechnia, 371 00:27:27.519 --> 00:27:31.960 published in sixteen sixty eight, and discovered the series log 372 00:27:32.079 --> 00:27:36.680 of one plus x equals x minus x squared over 373 00:27:36.720 --> 00:27:39.759 two plus x squared over three minus x to the 374 00:27:39.799 --> 00:27:43.519 power four over four, and so on. He proved this 375 00:27:43.640 --> 00:27:46.559 by writing the equation of a hyperbola in the form 376 00:27:46.680 --> 00:27:50.279 of y equals one over one plus x, which would 377 00:27:50.319 --> 00:27:54.240 be equal to one minus x plus x squared minus 378 00:27:54.400 --> 00:27:59.160 xque plus and so on, to which Wallace's formula stated 379 00:27:59.200 --> 00:28:03.240 before could be applied. The same series had been independently 380 00:28:03.319 --> 00:28:08.640 discovered by Saint Vincent Barrow. Isaac Barrow was born in 381 00:28:08.720 --> 00:28:12.079 London in sixteen thirty and died at Cambridge in sixteen 382 00:28:12.160 --> 00:28:16.440 seventy seven. He went to school first at Charterhouse, where 383 00:28:16.480 --> 00:28:19.519 he was so troublesome that his father was heard to 384 00:28:19.559 --> 00:28:21.680 pray that if it pleased God to take any of 385 00:28:21.720 --> 00:28:27.880 his children, he could best spare Isaac, and subsequently to Felstedt. 386 00:28:28.240 --> 00:28:33.000 He completed his education at Trinity College, Cambridge, after taking 387 00:28:33.000 --> 00:28:35.519 his degree in sixteen forty eight. He was elected to 388 00:28:35.559 --> 00:28:39.680 a fellowship in sixteen forty nine. He then resided for 389 00:28:39.759 --> 00:28:42.599 a few years in college, but in sixteen fifty five 390 00:28:42.680 --> 00:28:45.839 he was driven out by the persecution of the Independence. 391 00:28:46.480 --> 00:28:49.279 He spent the next four years in the East of Europe, 392 00:28:49.720 --> 00:28:54.319 and after many adventures, returned to England in sixteen fifty nine. 393 00:28:54.400 --> 00:28:56.640 He was ordained the next year and appointed to the 394 00:28:56.680 --> 00:29:01.119 professorship of Greek at Cambridge. In sixty two he was 395 00:29:01.160 --> 00:29:04.680 made professor of Geometry at Gresham College, and in sixteen 396 00:29:04.720 --> 00:29:08.240 sixty three was selected as the first occupier of the 397 00:29:08.319 --> 00:29:13.279 Lucasian Chair at Cambridge. He resigned the latter to his 398 00:29:13.440 --> 00:29:18.000 pupil Newton in sixteen sixty nine, whose superior abilities he 399 00:29:18.119 --> 00:29:22.240 recognized and frankly acknowledged. For the remainder of his life 400 00:29:22.279 --> 00:29:25.759 he devoted himself to the study of divinity. He was 401 00:29:25.799 --> 00:29:29.160