WEBVTT 1 00:00:01.399 --> 00:00:04.440 Chapter eighteen, Part one of a short account of the 2 00:00:04.559 --> 00:00:09.839 history of mathematics. This is a LibriVox recording. All LibriVox 3 00:00:09.880 --> 00:00:13.839 recordings are in the public domain. For more information or 4 00:00:13.880 --> 00:00:18.519 to volunteer, please visit LibriVox dot org. This reading is 5 00:00:18.559 --> 00:00:22.559 by Paul King PJK dot scripts dot mit dot edu 6 00:00:22.839 --> 00:00:27.399 forward slash p kJ. A short account of the history 7 00:00:27.399 --> 00:00:34.359 of mathematics by W. W. Rousbaul. Chapter eighteen Lagrange, Laplace 8 00:00:35.079 --> 00:00:41.880 and their Contemporaries circa seventeen forty to eighteen thirty. The 9 00:00:42.000 --> 00:00:46.399 last chapter contains the history of two separate schools, the 10 00:00:46.520 --> 00:00:50.119 Continental and the British. In the early years of the 11 00:00:50.159 --> 00:00:54.759 eighteenth century, the English school appeared vigorous and fruitful, but 12 00:00:55.000 --> 00:00:59.520 decadence rapidly set in and after the death of McLaurin 13 00:00:59.600 --> 00:01:03.240 and Sennas Ubson. No British mathematician appeared who is at 14 00:01:03.280 --> 00:01:07.480 all comparable to the Continental mathematicians of the latter half 15 00:01:07.519 --> 00:01:11.959 of the eighteenth century. This fact is partly explicable by 16 00:01:12.000 --> 00:01:15.599 the isolation of the school, partly by its tendency to 17 00:01:15.680 --> 00:01:22.439 rely too exclusively on geometrical and fleuctional methods. Some attention was, however, 18 00:01:22.480 --> 00:01:26.799 given to practical science, But except for a few remarks 19 00:01:26.799 --> 00:01:30.239 on English physicists, I do not think it necessary to 20 00:01:30.280 --> 00:01:35.480 discuss English mathematicians further until about eighteen twenty, when analytical 21 00:01:35.519 --> 00:01:40.319 methods again come into vogue on the continent under the 22 00:01:40.359 --> 00:01:44.040 influence of John Bernoulli, the calculus had become an instrument 23 00:01:44.120 --> 00:01:49.480 of great analytical power, expressed in an admirable notation, and 24 00:01:49.680 --> 00:01:54.920 for practical applications it is impossible to overestimate the value 25 00:01:55.040 --> 00:01:59.719 of a good notation. The subject of mechanics remained, however, 26 00:01:59.760 --> 00:02:02.560 in much the condition in which Newton had left it, 27 00:02:03.159 --> 00:02:06.879 until de la Maire, in putting Newton's results into the 28 00:02:06.959 --> 00:02:11.080 language of the differential calculus, did something to extend it. 29 00:02:12.039 --> 00:02:17.199 Universal gravitation, as enunciated in the Principia, was accepted as 30 00:02:17.240 --> 00:02:21.680 an established fact, but the geometrical methods adopted in proving 31 00:02:21.719 --> 00:02:25.360 it were difficult to follow or use, or to use 32 00:02:25.400 --> 00:02:31.680 in analogous problems. McLaurin, Simpson and Clairot may be regarded 33 00:02:31.759 --> 00:02:37.159 as the last mathematicians of distinction who employed them. Lastly, 34 00:02:37.560 --> 00:02:42.400 the Newtonian theory of light was generally received as correct. 35 00:02:44.800 --> 00:02:48.560 The leading mathematicians of the era of which we are 36 00:02:48.759 --> 00:02:55.520 now entering are Euler, Lagrange, Laplace, and la Genre. Briefly, 37 00:02:55.919 --> 00:02:59.879 we may say that Euler extended, summed up, and completed 38 00:03:00.400 --> 00:03:05.240 the work of his predecessors, while Lagrange, with almost unrivaled skill, 39 00:03:05.840 --> 00:03:11.639 developed the infinitesimal calculus and theoretical mechanics into the form 40 00:03:11.680 --> 00:03:15.280 in which we now know them. At the same time, 41 00:03:15.599 --> 00:03:20.400 Laplace made some additions to the infinitesimal calculus and applied 42 00:03:20.599 --> 00:03:24.960 that calculus to the theory of universal gravitation. He also 43 00:03:25.039 --> 00:03:30.639 created a calculus of probabilities. Le Gendre invented spherical harmonic 44 00:03:30.680 --> 00:03:35.000 analysis and elliptic integrals, and added to the theory of numbers. 45 00:03:35.680 --> 00:03:39.280 The works of these writers are still standard authorities, and 46 00:03:39.360 --> 00:03:42.960 are hardly yet the subject matter of history. I shall 47 00:03:42.960 --> 00:03:46.520 therefore content myself with a mere sketch of their discoveries, 48 00:03:46.840 --> 00:03:52.319 referring anyone who wishes to know more to the works themselves. Lagrange, Laplace, 49 00:03:52.400 --> 00:03:56.240 and le Gendre created a French school of mathematics, of 50 00:03:56.280 --> 00:04:00.000 which the younger members are divided into two groups. Include 51 00:04:00.240 --> 00:04:06.360 Poissant and Fourier began to apply the mathematical analysis to physics, 52 00:04:06.400 --> 00:04:12.120 and the other including Monte Carnot and Poncelette created modern geometry. 53 00:04:12.680 --> 00:04:16.680 Strictly speaking, some of the great mathematicians of recent times, 54 00:04:16.680 --> 00:04:21.079 such as Gauss and Abel, were contemporaries of the mathematicians 55 00:04:21.160 --> 00:04:24.959 last named, But except for this remark, I think it 56 00:04:25.040 --> 00:04:28.600 convenient to defer any consideration of them to the next chapter, 57 00:04:30.240 --> 00:04:37.040 the development of analysis and mathematics. Euler Leonard Euler was 58 00:04:37.079 --> 00:04:41.639 born in Bail on April fifteenth, seventeen o seven and 59 00:04:41.680 --> 00:04:45.959 died at Saint Petersburg on September seventh, seventeen eighty three. 60 00:04:46.800 --> 00:04:49.240 He was a son of a Lutheran minister who had 61 00:04:49.240 --> 00:04:52.720 settled at Bail, and was educated in his native town 62 00:04:52.920 --> 00:04:57.040 under the direction of John Bernoulli, with whose sons Daniel 63 00:04:57.079 --> 00:05:01.600 and Nicholas he formed a lifelong friendship. When in seventeen 64 00:05:01.639 --> 00:05:04.879 twenty five the younger Bernoullis went to Russia on the 65 00:05:04.920 --> 00:05:08.839 invitation of the Empress, they procured a place there for Euler, 66 00:05:09.319 --> 00:05:12.199 which in seventeen thirty three he exchanged for the share 67 00:05:12.319 --> 00:05:17.399 of mathematics then vacated by Daniel Bernoulli. The severity of 68 00:05:17.439 --> 00:05:20.759 the climate affected his eyesight, and in seventeen thirty five 69 00:05:21.199 --> 00:05:24.600 he lost the use of one eye completely. In seventeen 70 00:05:24.639 --> 00:05:27.160 forty one, he moved to Berlin at the request or 71 00:05:27.240 --> 00:05:31.439 rather command, of Frederic the Great. Here he stayed till 72 00:05:31.480 --> 00:05:35.519 seventeen sixty six, when he returned to Russia and was 73 00:05:35.560 --> 00:05:39.800 succeeded at Berlin by Lagrange. Within two or three years 74 00:05:39.839 --> 00:05:43.079 of his going back to Saint Petersburg he became blind. 75 00:05:43.759 --> 00:05:46.639 But in spite of this, and although his house, together 76 00:05:47.360 --> 00:05:50.720 with many of his papers, were burnt in seventeen seventy one, 77 00:05:51.240 --> 00:05:54.839 he recast and improved most of his earlier works. He 78 00:05:54.920 --> 00:05:58.839 died of ape apoplexy in seventeen eighty three. He was 79 00:05:58.920 --> 00:06:02.920 married twice. I think we may sum up Oiler's work 80 00:06:02.959 --> 00:06:07.000 by saying that he created analysis and revised almost all 81 00:06:07.079 --> 00:06:10.759 the branches of pure mathematics which were then known, filling 82 00:06:10.839 --> 00:06:14.160 up the details, adding proofs, and arranging the whole in 83 00:06:14.240 --> 00:06:18.000 consistent form. Such work is very important, and it is 84 00:06:18.079 --> 00:06:21.360 fortunate for science when it falls into the hands as 85 00:06:21.480 --> 00:06:26.480 competent as those of Euler. Euler wrote an immense number 86 00:06:26.480 --> 00:06:31.199 of memoirs on all kinds of mathematical subjects. His chief work, 87 00:06:31.279 --> 00:06:35.079 in which many of the results of earlier memoirs are embodied, 88 00:06:35.120 --> 00:06:38.600 are as follows in the first place. He wrote in 89 00:06:38.680 --> 00:06:43.680 seventeen forty eight, as in his Introductio in analyisim infiniteorum, 90 00:06:44.519 --> 00:06:47.720 which was intended to serve as an introduction to pure 91 00:06:47.759 --> 00:06:52.560 analytical mechanics. This is divided into two parts. The first 92 00:06:52.639 --> 00:06:57.920 part of the Analysis infinitorum contains the bulk of the 93 00:06:58.199 --> 00:07:00.639 matter which is to be found in the modern textbooks 94 00:07:00.680 --> 00:07:06.360 on algebra, theory of equations, and trigonometry. In the algebra, 95 00:07:06.519 --> 00:07:11.079 he paid particular attention to the expansion of various functions 96 00:07:11.079 --> 00:07:15.240 in series and to the summation of given series, and 97 00:07:15.439 --> 00:07:19.160 pointed out explicitly that an infinite series cannot be safely 98 00:07:19.199 --> 00:07:24.800 employed unless it is convergent. In the trigonometry, much of 99 00:07:24.839 --> 00:07:28.920 which is founded on FC. Mayer's Arithmetic of signs, which 100 00:07:28.920 --> 00:07:32.639 has been published in seventeen twenty seven, Euler developed the 101 00:07:32.680 --> 00:07:35.959 idea of John Bernoulli that the subject was a branch 102 00:07:36.000 --> 00:07:39.959 of analysis and not a mere appendage of astronomy or geometry. 103 00:07:40.519 --> 00:07:46.199 He also introduced, contemporaneously with Simpson the current abbreviations for 104 00:07:46.240 --> 00:07:51.920 the trigonometrical functions. Ensued that the trigonometrical and exponential functions 105 00:07:52.199 --> 00:07:56.639 were connected by the relation cosine theta plus isigin theta 106 00:07:57.199 --> 00:08:01.959 equals e to the power of eye times scasta. Here 107 00:08:02.079 --> 00:08:05.759 too we meet the symbol E used to denote the 108 00:08:05.800 --> 00:08:12.319 base of the Nypurian logarithms, namely the incommensurable number two 109 00:08:12.360 --> 00:08:15.879 point seven one a two eight, and the symbol PI 110 00:08:16.399 --> 00:08:20.160 used to denote the incommensurable number three point one four 111 00:08:20.160 --> 00:08:23.279 one five nine. The use of a single symbol to 112 00:08:23.360 --> 00:08:26.120 denote the number two point seven one eight two eight 113 00:08:26.240 --> 00:08:31.079 seems to be due to Coats, who denoted it by M. Newton, was, 114 00:08:31.120 --> 00:08:33.000 as far as I know, the first to employ the 115 00:08:33.080 --> 00:08:37.720 literal exponential notation. An Euler using the form A to 116 00:08:37.799 --> 00:08:40.960 the z has taken A as the base of any 117 00:08:40.960 --> 00:08:45.080 system of logarithms. It is probable that the choice of 118 00:08:45.159 --> 00:08:48.000 E for a particular base was determined by its being 119 00:08:48.039 --> 00:08:52.559 the vowel consecutive to A. The use of a single 120 00:08:52.639 --> 00:08:55.559 symbol to denote the number three point one four one 121 00:08:55.679 --> 00:08:59.759 five nine appears to have been introduced by John Bernoulli, 122 00:09:00.159 --> 00:09:04.600 who represented it by C. Euler in seventeen thirty four, 123 00:09:04.759 --> 00:09:08.360 denoted it by P and in a letter of seventeen 124 00:09:08.440 --> 00:09:11.120 thirty six in which he enunciated the theorem that the 125 00:09:11.159 --> 00:09:14.399 sum of the squares of the reciprocals of the natural 126 00:09:14.480 --> 00:09:18.679 numbers is one over six times pie squared. He used 127 00:09:18.679 --> 00:09:24.519 the letter C, Christian Goldbach in seventeen forty two used 128 00:09:24.519 --> 00:09:29.960 the Greek letter PI, and after the publication of Euler's analysis, 129 00:09:30.240 --> 00:09:35.000 the symbol pie was generally employed. The numbers E and 130 00:09:35.120 --> 00:09:39.080 PI would enter into the mathematical analysis from whatever side 131 00:09:39.120 --> 00:09:43.440 the subject was approached. The latter represents, among other things, 132 00:09:43.440 --> 00:09:46.720 the ratio of the circumference of a circle to the diameter. 133 00:09:47.399 --> 00:09:50.759 But it is a mere accident that it is taken 134 00:09:50.879 --> 00:09:55.840 for its definition. De Morgan, in The Budget of Paradoxes, 135 00:09:56.519 --> 00:10:01.200 tells an anecdote which illustrates how little the use usual 136 00:10:01.279 --> 00:10:06.600 definition suggests its real use. He was explaining to an actuary, 137 00:10:07.120 --> 00:10:09.919 what was the chance that at the end of a 138 00:10:09.960 --> 00:10:13.200 given time a certain proportion of some group of people 139 00:10:13.240 --> 00:10:18.879 would be alive? And quoted the actuarial formula involving pie, which, 140 00:10:18.919 --> 00:10:22.159 in answer to a question he explained, stood for the 141 00:10:22.279 --> 00:10:27.399 ratio the circumference of a circle to its diameter. His acquaintance, 142 00:10:27.399 --> 00:10:31.679 who so far listened to the explanation with interest interrupted 143 00:10:31.720 --> 00:10:36.000 of men, explained, my dear friend that must be a delusion. 144 00:10:36.440 --> 00:10:38.639 What can a circle have to do with the number 145 00:10:38.720 --> 00:10:41.120 of people alive at the end of a given time. 146 00:10:43.240 --> 00:10:48.000 The second part of the analysis infinitetorum is on analytical geometry. 147 00:10:48.519 --> 00:10:52.200 Euler commenced this part by dividing curves into algebraical and 148 00:10:52.279 --> 00:10:56.519 transcendental and established a variety of propositions which are true 149 00:10:56.559 --> 00:11:00.919 for all algebraical curves. He then applied to the general 150 00:11:00.960 --> 00:11:05.320 equations of the second degree into in two dimensions, shewed 151 00:11:05.320 --> 00:11:09.240 that it represents the various conic sections and deduced most 152 00:11:09.320 --> 00:11:13.320 of their properties from the general equation. He also considered 153 00:11:13.320 --> 00:11:18.279 the classification of cubic, cortic and other algebraical curves. He 154 00:11:18.360 --> 00:11:22.360 next discussed the question as to what surfaces are represented 155 00:11:22.399 --> 00:11:25.840 by the general equation of the second degree in three dimensions, 156 00:11:26.360 --> 00:11:29.320 and how they may be discriminated one from the other. 157 00:11:29.919 --> 00:11:33.799 Some of these surfaces had not been previously investigated. In 158 00:11:33.840 --> 00:11:36.480 the course of this analysis, he laid down the rules 159 00:11:36.480 --> 00:11:41.000 for the transformation of coordinates in space. Here also we 160 00:11:41.120 --> 00:11:44.720 find the earliest attempt to bring the curvature of services 161 00:11:44.759 --> 00:11:49.000 within the domain of mathematics, and the first complete discussion 162 00:11:49.120 --> 00:11:54.320 of tortuous curves. The Analysis Infinitorum was followed in seventeen 163 00:11:54.399 --> 00:11:59.720 fifty five by the Instituciones Calculi Differentialis, to which it 164 00:11:59.799 --> 00:12:03.720 was intended as an introduction. This was the first textbook 165 00:12:03.759 --> 00:12:07.200 on the differential calculus which has any claim to be 166 00:12:07.279 --> 00:12:10.759 regarded as complete, and it may be said that most 167 00:12:10.799 --> 00:12:13.919 modern treatises on the subject are based on it. At 168 00:12:13.919 --> 00:12:16.879 the same time, it should be added that the exposition 169 00:12:17.039 --> 00:12:20.759 of the principles of the subject is often prolux and obscure, 170 00:12:21.200 --> 00:12:25.240 and sometimes not altogether accurate. This series of works was 171 00:12:25.279 --> 00:12:29.360 completed by the publication in three volumes in seventeen sixty 172 00:12:29.360 --> 00:12:35.000 eight to seventeen seventy of the Institutiononi's Calculi Integralis, in 173 00:12:35.080 --> 00:12:38.799 which the results of several of Euler's earlier memoirs on 174 00:12:38.879 --> 00:12:43.679 the same subject and on differential equations are included. This, 175 00:12:44.039 --> 00:12:47.960 like the similar Treatise on the differential Calculus, summed up 176 00:12:48.080 --> 00:12:50.879 what was then known on the subject, but many of 177 00:12:50.879 --> 00:12:54.519 the theorems were recast and the proofs improved. The beta 178 00:12:54.600 --> 00:12:58.480 and gamma functions were invented by Euler and discussed here, 179 00:12:58.879 --> 00:13:01.759 but only as illis sstrations of the methods of reduction 180 00:13:01.919 --> 00:13:07.039 and integration. His treatment of elliptical integrals as superficial. It 181 00:13:07.120 --> 00:13:09.519 was due to a theorem given by John Landon, a 182 00:13:09.600 --> 00:13:14.559 writer who was suggestive rather than powerful, in the Philosophical 183 00:13:14.600 --> 00:13:18.759 Transactions for seventeen fifty five, connecting the arcs of a 184 00:13:18.840 --> 00:13:23.799 hyperbola and an ellipse. Euler's works that form this trilogy 185 00:13:23.919 --> 00:13:29.919 have gone through numerous subsequent editions. The classic problems on 186 00:13:30.120 --> 00:13:36.320 isoperimetrical curves, the brachistochrone in a resulting medium, and the theory 187 00:13:36.240 --> 00:13:39.519 of geodesics, all of which had been suggested by his master, 188 00:13:39.639 --> 00:13:43.399 John Bernoulli, had engaged Euler's attention at an early date, 189 00:13:43.919 --> 00:13:46.480 and in solving them he was led to the calculus 190 00:13:46.480 --> 00:13:50.399 of variations. The general idea of this was laid down 191 00:13:50.440 --> 00:13:56.360 in his Corvera maxima minieve proprietate Gaudentium in ventio nova Akfasilis, 192 00:13:56.879 --> 00:14:00.759 published in seventeen forty four, but the complete development of 193 00:14:00.799 --> 00:14:04.200 the new calculus was first affected by Lagrange in seventeen 194 00:14:04.240 --> 00:14:07.679 fifty nine. The method used by Lagrange as described in 195 00:14:07.759 --> 00:14:11.080 Euler's Integral Calculus and is the same as that given 196 00:14:11.120 --> 00:14:15.120 in most modern textbooks on the subject. In seventeen seventy, 197 00:14:15.200 --> 00:14:20.360 Euler published the aneleisiic Zur Algebra in two volumes. The 198 00:14:20.440 --> 00:14:25.000 first volume treats of determinate Algebra. This contains one of 199 00:14:25.000 --> 00:14:28.039 the earliest attempts to place the fundamental process on a 200 00:14:28.080 --> 00:14:32.879 scientific basis. The same subject had attracted d. Lambert's attention. 201 00:14:33.879 --> 00:14:36.879 This work also includes the proof of the binomial theorem 202 00:14:37.039 --> 00:14:40.759 for an unrestricted index, which is still known by Euler's name. 203 00:14:41.799 --> 00:14:45.039 The proof is founded on the principle of the permanence 204 00:14:45.039 --> 00:14:48.759 of equivalent forms, but Euler made no attempt to investigate 205 00:14:48.799 --> 00:14:52.759 the convergency of the series that he should have omitted. 206 00:14:52.799 --> 00:14:57.159 This essential step is the more curious, as he had 207 00:14:57.240 --> 00:15:01.519 himself recognized the necessity of concus littering the convergency of 208 00:15:01.559 --> 00:15:05.879 the infinite series. The second volume treats of the indeterminate 209 00:15:06.000 --> 00:15:10.120 or Diophantine algebra. This contains a solution of some of 210 00:15:10.120 --> 00:15:15.519 the problems proposed by Fermont, which had hitherto remained unsolved. 211 00:15:16.039 --> 00:15:19.879 A French translation of the algebra with numerous and valuable 212 00:15:20.000 --> 00:15:23.879 editions by Lagrange was brought out in seventeen ninety four, 213 00:15:24.360 --> 00:15:27.840 and a treatise on arithmetic by Euler was appended to it. 214 00:15:29.320 --> 00:15:32.799 These four works comprise most of what Euler produced in 215 00:15:32.879 --> 00:15:37.039 pure mathematics. He also wrote numerous memoirs on nearly all 216 00:15:37.080 --> 00:15:42.039 subjects of applied mathematics and mathematical physics, then studied the 217 00:15:42.159 --> 00:15:48.240 chief results in them are as follows. In the mechanics 218 00:15:48.240 --> 00:15:51.240 of a grid system, he determined the general equation of 219 00:15:51.279 --> 00:15:54.000 a motion of a body about a fixed point, which 220 00:15:54.039 --> 00:15:57.960 are ordinarily written of the form A multiplied by di 221 00:15:58.120 --> 00:16:02.840 omega i by d t minus in brackets b minus 222 00:16:02.919 --> 00:16:08.159 c closed bracket times omega two times omega three equals L. 223 00:16:08.799 --> 00:16:11.080 And he gave the general equation of the motion of 224 00:16:11.120 --> 00:16:14.480 a free body, which are usually presented in the form 225 00:16:14.919 --> 00:16:19.559 d y d t times mu minus m v theta 226 00:16:19.639 --> 00:16:26.879 three plus m w theta two equals x, and dh 227 00:16:27.000 --> 00:16:31.240 one prime by d t equals h two prime theta 228 00:16:31.320 --> 00:16:36.440 three plus h three prime theta two equals L. He 229 00:16:36.519 --> 00:16:40.679 also defended and elaborated the theory of least action, which 230 00:16:40.720 --> 00:16:45.840 had been propounded by Marpetuis in seventeen fifty one. In 231 00:16:45.960 --> 00:16:51.440 his Sa de cosmal g In hydrodynamics, Euler established the 232 00:16:51.480 --> 00:16:54.639 general equations of motion which are commonly expressed in the 233 00:16:54.720 --> 00:16:58.360 form one over row d row by d x equals 234 00:16:58.600 --> 00:17:02.480 x minus d u by d s minus u times 235 00:17:02.600 --> 00:17:07.160 du by dx minus v times du by d y 236 00:17:07.400 --> 00:17:12.440 minus w times du by d z. At the time 237 00:17:12.480 --> 00:17:14.799 of his death he was engaged in writing a treatise 238 00:17:14.799 --> 00:17:18.640 on hydromechanics, in which the treatment of the subject would 239 00:17:18.680 --> 00:17:23.319 have been completely recast. His most important works on astronomy 240 00:17:23.759 --> 00:17:29.960 are as Theorea Mortrum Planetarum a comatarum, published in seventeen 241 00:17:30.039 --> 00:17:35.200 forty four, is Theoria Motus Lunarius, published in seventeen fifty three, 242 00:17:35.759 --> 00:17:40.400 and is Theoria Motum lune published in seventeen seventy two. 243 00:17:41.119 --> 00:17:44.400 In these he attacked the problem of three bodies. He 244 00:17:44.519 --> 00:17:47.839 supposed the body considered e g. The Moon, to carry 245 00:17:47.839 --> 00:17:51.839 three rectangular axes with it in its motion, the axes 246 00:17:51.880 --> 00:17:55.920 moving parallel to themselves, and to these axes all the 247 00:17:55.920 --> 00:17:59.680 motions were referred. This method is not convenient, but it 248 00:17:59.720 --> 00:18:03.440 was for from Euler's results that Meyer constructed the lunar tables, 249 00:18:03.759 --> 00:18:07.920 from which his widow in seventeen seventy received five thousand pounds, 250 00:18:08.359 --> 00:18:11.440 being the prize offered by the English Parliament, and in 251 00:18:11.519 --> 00:18:14.839 recognition of Euler's services, A sum of three hundred pounds 252 00:18:14.920 --> 00:18:19.279 was voted as an honorarium to him. Euler was much 253 00:18:19.319 --> 00:18:23.160 interested in optics. In seventeen forty six he discussed the 254 00:18:23.240 --> 00:18:28.079 relative merits of the omission and undulatory theories of light. He, 255 00:18:28.160 --> 00:18:31.559 on the whole, preferred the latter. In seventeen seventy to 256 00:18:31.640 --> 00:18:35.480 seventeen seventy one he published his optical researches in three 257 00:18:35.559 --> 00:18:41.599 volumes under the title Dioptrica. He also wrote an elementary 258 00:18:41.599 --> 00:18:46.279 work on physics and Fundamental Principles of mathematical philosophy. This 259 00:18:46.400 --> 00:18:49.559 originated from an invitation he received when he first went 260 00:18:49.599 --> 00:18:52.480 to Berlin to give the Lessons on Physics to the 261 00:18:52.519 --> 00:18:58.279 Princess of an Halt Desso. These lectures were published in 262 00:18:58.359 --> 00:19:01.279 seventeen sixty eight to seventy teen seventy two in three 263 00:19:01.359 --> 00:19:06.319 volumes under the title Lettres ser carcurg suget de physique, 264 00:19:06.839 --> 00:19:09.680 and for half a century remained a standard treatise on 265 00:19:09.720 --> 00:19:13.960 the subject. Of course, Euler's magnificent works were not the 266 00:19:14.000 --> 00:19:19.319 only textbooks containing original matter produced at this time. Amongst 267 00:19:19.440 --> 00:19:22.559 numerous writers, I would be I would specially single out 268 00:19:22.640 --> 00:19:28.880 Daniel Bernoulli, Simpson, Lambert Bizou, Tremblay, and abrogast As having 269 00:19:28.960 --> 00:19:33.200 influenced the development of mathematics. To the first two mentioned, 270 00:19:33.279 --> 00:19:39.079 I have already alluded in the last chapter, end of 271 00:19:39.160 --> 00:19:43.279 section thirty. Recording by Paul King, Oakville, Ontario, p j 272 00:19:43.519 --> 00:19:47.759 K dot scripts dot mit dot edu. Forward slash p 273 00:19:47.960 --> 00:19:48.319 kJ