WEBVTT 1 00:00:01.159 --> 00:00:04.919 Chapter sixteen, Part three of A Short Account of the 2 00:00:05.040 --> 00:00:10.880 History of Mathematics. This is a LibriVox recording. All LibriVox 3 00:00:10.919 --> 00:00:14.759 recordings are in the public domain. For more information or 4 00:00:14.800 --> 00:00:20.239 to volunteer, please visit LibriVox dot org. This reading is 5 00:00:20.280 --> 00:00:23.719 by Paul King p J K dot scripts dot mit 6 00:00:23.960 --> 00:00:28.199 dot ed U forward slash p k J. A Short 7 00:00:28.199 --> 00:00:32.240 Account of the History of Mathematics by W. W. Rowseball, 8 00:00:33.520 --> 00:00:38.000 Chapter sixteen, The Life and Works of Newton, Part three. 9 00:00:39.679 --> 00:00:44.039 In seventeen o four, Newton published his Optics, which contains 10 00:00:44.079 --> 00:00:48.880 the results of papers already mentioned the first edition of 11 00:00:48.960 --> 00:00:52.320 this book were appended to two minor works which have 12 00:00:52.399 --> 00:00:56.960 no special connection with Optics, one being on cubic curves, 13 00:00:57.159 --> 00:01:00.799 the other on the quadrature of curves and u on fluxions. 14 00:01:01.920 --> 00:01:05.000 Both of them were old manuscripts with which his friends 15 00:01:05.000 --> 00:01:08.680 and pupils were familiar, but they were here published ubi 16 00:01:08.760 --> 00:01:13.120 et orbi for the first time. The first of these 17 00:01:13.159 --> 00:01:19.159 appendices is entitled Umeratio linierm terti rorubdinis. The object seems 18 00:01:19.200 --> 00:01:23.120 to be to illustrate the use of analytical geometry, and 19 00:01:23.200 --> 00:01:27.040 as the application to conics was well known, Newton selected 20 00:01:27.040 --> 00:01:31.560 the theory of cubics. He begins with some general theorems 21 00:01:31.599 --> 00:01:35.680 and classifies curves according as to whether their equations are 22 00:01:35.719 --> 00:01:40.319 algebraical or transcendental, the former being cut by a straight 23 00:01:40.400 --> 00:01:44.120 line in a number of points real or imaginary equal 24 00:01:44.200 --> 00:01:47.760 to the degree of the curve, the latter being cut 25 00:01:47.799 --> 00:01:50.799 by a straight line and an infinite number of points. 26 00:01:52.319 --> 00:01:55.760 Newton then choose that many of the most important properties 27 00:01:55.760 --> 00:01:59.519 of conics have their analogs in the theory of cubics, 28 00:02:00.239 --> 00:02:05.560 and he discusses the theory of asymptotes and curve linear diameters. 29 00:02:07.000 --> 00:02:11.360 After these general theorems, he commences his detailed examination of 30 00:02:11.400 --> 00:02:14.599 cubics by pointing out that a cubic must have at 31 00:02:14.680 --> 00:02:18.840 least one real point at infinity. If the asymptote or 32 00:02:18.879 --> 00:02:22.360 tangent at this point be a finite distance, it may 33 00:02:22.400 --> 00:02:26.400 be taken for the axis of y. This asymptote will 34 00:02:26.439 --> 00:02:30.000 cut the curve in three points altogether, of which at 35 00:02:30.080 --> 00:02:34.080 least two are an infinity. If the third point be 36 00:02:34.120 --> 00:02:37.599 a finite distance, then by one of his general theorems 37 00:02:37.599 --> 00:02:40.840 on asymptotes, the equation can be written in the form 38 00:02:40.919 --> 00:02:45.919 of x y squared plus h y equals ax cubed 39 00:02:46.120 --> 00:02:50.479 plus b x squared plus c x plus d, where 40 00:02:50.520 --> 00:02:53.120 the axes of X and Y are the asymptotes of 41 00:02:53.159 --> 00:02:56.960 the hyperbola, which is the locus of the middle points 42 00:02:57.080 --> 00:03:01.240 of all chords drawn parallel to the axis of While 43 00:03:01.680 --> 00:03:04.840 if on the third point in which this asymptote cuts 44 00:03:04.879 --> 00:03:08.479 the curve be also at infinity, the equation can be 45 00:03:08.520 --> 00:03:12.639 written in the form x y equals ax cube plus 46 00:03:12.680 --> 00:03:17.199 b x squared plus c x plus d. Next, he 47 00:03:17.280 --> 00:03:19.800 takes the case where the tangent at the real point 48 00:03:19.800 --> 00:03:24.240 at infinity is not at a finite distance. A line 49 00:03:24.240 --> 00:03:27.280 parallel to the direction in which the curve goes to 50 00:03:27.360 --> 00:03:32.240 infinity may be taken as the axis of Y. Any 51 00:03:32.280 --> 00:03:35.080 such line will cut the curve in three points altogether, 52 00:03:36.159 --> 00:03:40.479 of which one is by hypothesis at infinity and one 53 00:03:40.560 --> 00:03:45.319 is necessarily at a finite distance. He then choose that 54 00:03:45.439 --> 00:03:48.639 if the remaining point in which this line cuts the 55 00:03:48.680 --> 00:03:51.719 curve be at a finite distance, the equation can be 56 00:03:51.759 --> 00:03:55.639 written in the form hy squared equals ax cube plus 57 00:03:55.680 --> 00:03:59.000 b x squared plus c x plus d, while if 58 00:03:59.120 --> 00:04:02.319 it be at an infinite distance, the equation can be 59 00:04:02.400 --> 00:04:06.800 written in the form why equals axqube plus bx squared 60 00:04:06.840 --> 00:04:12.479 plus cx plus d. Any cubic is therefore reducible to 61 00:04:12.560 --> 00:04:16.879 one of the four characteristic forms. Each of these forms 62 00:04:17.199 --> 00:04:20.519 is then discussed in detail, and the possibility of the 63 00:04:20.560 --> 00:04:25.000 existence of double points, isolated ovals, et cetera is worked out. 64 00:04:25.879 --> 00:04:29.600 The final result is that in all, there are seventy 65 00:04:29.600 --> 00:04:33.600 eight possible forms which a cubic may take. Of these, 66 00:04:33.959 --> 00:04:38.199 Newton enumerated only seventy two. Four of the remainder were 67 00:04:38.240 --> 00:04:41.959 mentioned by Sterling in seventeen seventeen, and one by Nicole 68 00:04:42.120 --> 00:04:46.360 in seventeen thirty one, and one by Nicholas Bernoulli at 69 00:04:46.360 --> 00:04:49.560 about the same time. In the course of the work, 70 00:04:49.680 --> 00:04:53.439 Newton states the remarkable theorem that just as the shadow 71 00:04:53.480 --> 00:04:56.199 of a circle cast by a luminous point on a 72 00:04:56.240 --> 00:05:01.879 plane gives rise to all the conics, so the shadows 73 00:05:01.920 --> 00:05:06.000 of the curves represented by the equation y squared equals 74 00:05:06.040 --> 00:05:09.120 ax cube plus bx squared plus c x plus d 75 00:05:09.800 --> 00:05:13.879 give rise to all the cubics. This remained an unsolved 76 00:05:13.920 --> 00:05:18.959 puzzle until seventeen thirty one, when Nicole and Clairot gave 77 00:05:19.120 --> 00:05:22.839 demonstrations of it. A better proof is that given by 78 00:05:22.920 --> 00:05:27.240 Murdoch in seventeen forty, which depends on the classification of 79 00:05:27.279 --> 00:05:31.439 these curves into five species, according to as to whether 80 00:05:31.920 --> 00:05:34.759 their points of intersection with the axis of X are 81 00:05:34.839 --> 00:05:38.879 real and unequal, real and two of them equal, two 82 00:05:38.959 --> 00:05:44.040 cases real and all equal, or finally two imaginary and 83 00:05:44.120 --> 00:05:50.120 one real. In this tract, Newton also discusses double points 84 00:05:50.120 --> 00:05:54.199 in the plane and at infinity, and the description of 85 00:05:54.319 --> 00:05:59.879 curves satisfying given conditions, and the graphical solution of pro 86 00:06:00.000 --> 00:06:04.759 problems by the use of curves. The second appendix to 87 00:06:04.839 --> 00:06:11.000 the Optics is entitled des Quadratura corvorum. Most of it 88 00:06:11.120 --> 00:06:14.560 had been communicated to Barrow in sixteen sixty eight or 89 00:06:14.600 --> 00:06:18.360 sixteen sixty nine, and probably was familiar to Newton's pupils 90 00:06:18.399 --> 00:06:23.040 and friends from that time onwards. It consists of two parts. 91 00:06:23.839 --> 00:06:26.360 The bulk of the first part is a statement of 92 00:06:26.399 --> 00:06:30.480 Newton's method of affecting the quadrature and the rectification of 93 00:06:30.560 --> 00:06:35.480 curves by means of infinite series. It is noticeable as 94 00:06:35.560 --> 00:06:40.759 containing the earliest use in print of literal indices, and 95 00:06:40.959 --> 00:06:45.439 also the first printed statement of the binomial theorem, but 96 00:06:45.519 --> 00:06:49.920 these are introduced only Incidentally, the main object is to 97 00:06:49.959 --> 00:06:53.160 give rules for developing a function of X in a 98 00:06:53.279 --> 00:06:56.680 series of ascending powers of X, so as to enable 99 00:06:56.759 --> 00:07:00.680 mathematicians to effect the quadrature of any curve in which 100 00:07:00.839 --> 00:07:04.639 the ordinate y can be expressed as an explicit algebraical 101 00:07:04.720 --> 00:07:10.959 function of the ebscissa X. Wallace had shewn how this 102 00:07:11.079 --> 00:07:14.040 quadrature could be found when y was given as a 103 00:07:14.079 --> 00:07:17.120 sum of a number of multiples of powers of X, 104 00:07:17.800 --> 00:07:23.199 and Newton's rules of expansion here established rendered possible the 105 00:07:23.279 --> 00:07:27.439 similar quadrature of any curve whose ordinate can be expressed 106 00:07:27.439 --> 00:07:30.279 as a sum of an infinite number of such terms. 107 00:07:31.439 --> 00:07:34.519 In this way, he affects the quadrature of the curves 108 00:07:34.720 --> 00:07:39.000 y equals a square divided by the quantity b plus x, 109 00:07:40.319 --> 00:07:44.240 Y equals the square root of a squared plus or 110 00:07:44.240 --> 00:07:49.040 minus x squared, or y equals the square root of 111 00:07:49.439 --> 00:07:54.800 x minus x squared, or y equals the square root 112 00:07:55.879 --> 00:08:00.199 of one minus a x squared divided by the square 113 00:08:00.279 --> 00:08:04.360 root of one minus b x squared. But the results 114 00:08:04.360 --> 00:08:08.720 are of course expressed as infinite series. He then proceeds 115 00:08:08.759 --> 00:08:11.800 to curves whose ordinate is given as an implicit function 116 00:08:11.879 --> 00:08:14.639 of the obsessor, and he gives a method by which 117 00:08:14.800 --> 00:08:17.839 y can be expressed as an infinite series and ascending 118 00:08:17.920 --> 00:08:21.040 powers of x. But the application of the rule to 119 00:08:21.120 --> 00:08:26.199 any curve demands, in general such complicated numerical calculations as 120 00:08:26.199 --> 00:08:30.680 to render it of little value. He concludes this part 121 00:08:30.759 --> 00:08:33.879 by shewing that the rectification of a curve can be 122 00:08:33.919 --> 00:08:38.399 effected in a somewhat similar way. His process is equivalent 123 00:08:38.440 --> 00:08:41.879 to finding the integral with regard to x of the 124 00:08:41.879 --> 00:08:45.080 square root of one plus y squared in the form 125 00:08:45.120 --> 00:08:49.519 of an infinite series. I should add that Newton indicates 126 00:08:49.559 --> 00:08:53.720 the importance of determining whether the series are convergent, an 127 00:08:53.799 --> 00:08:57.799 observation far in advance of his time, But he knew 128 00:08:57.879 --> 00:09:00.879 of no general test for the purpose, and in fact 129 00:09:00.960 --> 00:09:04.200 it was not until Gaussenkoschi took up the question that 130 00:09:04.279 --> 00:09:09.320 the necessity of such limitations were commonly recognized. The part 131 00:09:09.360 --> 00:09:12.320 of the appendix which I have just described is practically 132 00:09:12.360 --> 00:09:18.440 the same as Newton's manuscript Diannealisi periqueisiones numero to menora minfinitas, 133 00:09:19.120 --> 00:09:23.679 which was subsequently printed in seventeen eleven. It is said 134 00:09:23.720 --> 00:09:26.519 that this was originally intended to form an appendix to 135 00:09:26.600 --> 00:09:31.639 Kinkhousen's algebra. The substance of it was communicated to Brow 136 00:09:31.960 --> 00:09:34.799 and by him to Collins and letters of July thirty 137 00:09:34.840 --> 00:09:39.080 first and August twelfth, sixteen sixty nine, and a summary 138 00:09:39.240 --> 00:09:41.960 of part of it was included in a letter of 139 00:09:42.039 --> 00:09:46.960 October twenty fourth, sixteen seventy six, sent to Leibniz. It 140 00:09:47.039 --> 00:09:51.480 should be read in connection with Newton's methodis differentialis published 141 00:09:51.559 --> 00:09:56.360 in seventeen thirty six. Some additional theorems are there given, 142 00:09:56.799 --> 00:10:00.279 and he discusses his method of interpolation, which which had 143 00:10:00.279 --> 00:10:03.919 been briefly described in the letter of October twenty fourth, 144 00:10:04.039 --> 00:10:08.759 sixteen seventy six. The principle is this, if y equals 145 00:10:08.799 --> 00:10:11.440 fi x be a function of x, and if when 146 00:10:11.799 --> 00:10:15.200 x is successively put equal to a one, a, two, 147 00:10:15.240 --> 00:10:19.279 and so on, the values of y be known, and b, 148 00:10:19.399 --> 00:10:23.039 b one, b two, and so on, then a parabola 149 00:10:23.039 --> 00:10:26.639 whose equation is y equals p plus q x plus 150 00:10:26.759 --> 00:10:31.120 r x squared plus so on can be drawn through 151 00:10:31.159 --> 00:10:33.559 the points A one, b one, a two, b two, 152 00:10:33.759 --> 00:10:37.279 and so on, And the ordinate of this parabola may 153 00:10:37.320 --> 00:10:40.320 be taken as an approximation to the ordinate of the 154 00:10:40.399 --> 00:10:43.919 curve the degree of the parabola will, of course be 155 00:10:44.000 --> 00:10:47.399 one less than the number of the given points. Newton 156 00:10:47.600 --> 00:10:51.039 points out that in this way the areas of any 157 00:10:51.159 --> 00:10:55.399 curves can be approximately determined. The second part of this 158 00:10:55.480 --> 00:10:59.320 appendix to the optics, contained in a description of Newton's method, 159 00:10:59.399 --> 00:11:03.320 is fluction. This is best considered in connection with Newton's 160 00:11:03.360 --> 00:11:06.320 manuscript on the same subject, which was published by John 161 00:11:06.399 --> 00:11:10.080 Coulson in seventeen thirty six, and of which it is 162 00:11:10.120 --> 00:11:15.279 a summary. The fluctional calculus is one form of the 163 00:11:15.360 --> 00:11:20.519 infinitesimal calculus expressed in a certain notation, just as the 164 00:11:20.559 --> 00:11:24.840 differential calculus is another aspect of the same calculus expressed 165 00:11:24.879 --> 00:11:30.799 in a different notation. Newton assumed that all geometrical magnitudes 166 00:11:30.879 --> 00:11:35.559 must be conceived as generated by continuous motion. Thus a 167 00:11:35.639 --> 00:11:38.559 line may be considered as generated by the motion of 168 00:11:38.600 --> 00:11:42.159 a point, a surface by that of a line, a 169 00:11:42.360 --> 00:11:45.840 solid by that of a surface, a plane angle by 170 00:11:45.840 --> 00:11:49.559 the rotation of a line, and so on. The quantity 171 00:11:49.639 --> 00:11:53.559 thus generated was defined by him as the fluent or 172 00:11:53.639 --> 00:11:58.759 flowing quantity. The velocity of the moving magnitude was defined 173 00:11:58.759 --> 00:12:02.120 as the fluxion of the f fluent. This seems to 174 00:12:02.159 --> 00:12:05.840 be the earliest definite recognition of the idea of a 175 00:12:05.919 --> 00:12:10.120 continuous function, though it had been foreshadowed in some of 176 00:12:10.440 --> 00:12:15.440 Napier's papers. The following is a summary of Newton's treatment 177 00:12:15.480 --> 00:12:19.240 of fluctions. There are two kinds of problems. The object 178 00:12:19.279 --> 00:12:21.720 of the first is to find the fluxion of a 179 00:12:21.759 --> 00:12:25.720 given quantity, or more generally, the relation of the fluence 180 00:12:25.759 --> 00:12:30.360 being given, to find the relation of their fluctions. This 181 00:12:30.399 --> 00:12:34.919 is equivalent to differentiation. The object of the second, or 182 00:12:34.919 --> 00:12:38.159 inverse method of fluctions is from the fluxion or some 183 00:12:38.399 --> 00:12:43.600 relations involving it, to determine the fluent or more generally, 184 00:12:43.960 --> 00:12:48.039 an equation being proposed exhibiting the relation of the fluxions 185 00:12:48.120 --> 00:12:51.639 of quantities to find the relations of those quantities or 186 00:12:51.679 --> 00:12:56.440 fluence to one another. This is equivalent either to integration, 187 00:12:56.559 --> 00:13:01.360 which Newton termed the method of quadrature, or to the 188 00:13:01.399 --> 00:13:04.759 solution of a differential equation, which was called by Newton 189 00:13:05.200 --> 00:13:10.080 the inverse method of tangents. The methods for solving these 190 00:13:10.080 --> 00:13:13.919 problems are discussed at considerable length. Newton then went on 191 00:13:14.000 --> 00:13:17.960 to apply these results to questions connected with the maxima 192 00:13:18.000 --> 00:13:22.200 and minima of quantities. The method of drawing tangents to 193 00:13:22.279 --> 00:13:27.240 curves and the curvature of curves, namely the determination of 194 00:13:27.240 --> 00:13:31.080 the center of curvature, the radius of curvature, and the 195 00:13:31.200 --> 00:13:35.559 rate at which the radius of curvature increases. He next 196 00:13:35.639 --> 00:13:40.000 considered the quadrature of curves and the rectification of curves. 197 00:13:40.840 --> 00:13:45.399 In finding the maximum and minimum of functions of one variable, 198 00:13:45.480 --> 00:13:48.480 we regard the change of sign of the difference between 199 00:13:48.519 --> 00:13:53.240 two consecutive values of the function as the true criterion. 200 00:13:53.919 --> 00:13:58.200 But his argument is that when a quantity increasing has 201 00:13:58.240 --> 00:14:02.840 attained its maximum, it can have no further increment, or 202 00:14:02.919 --> 00:14:05.919 when decreasing, it has attained its minimum and can have 203 00:14:06.000 --> 00:14:11.600 no further decrement. Consequently, the fluction must be equal to nothing. 204 00:14:12.320 --> 00:14:16.360 It has been remarked that neither Newton nor Lobbyingits produced 205 00:14:16.399 --> 00:14:20.960 a calculus that is a classified collection of rules, and 206 00:14:21.080 --> 00:14:25.559 that the problems they discussed were treated from first principles. 207 00:14:26.240 --> 00:14:28.919 That no doubt is the usual sequence in the history 208 00:14:28.960 --> 00:14:33.039 of such discoveries, though the fact is frequently forgotten by 209 00:14:33.080 --> 00:14:37.039 subsequent writers. In this case, I think the statement, so 210 00:14:37.159 --> 00:14:40.279 far as Newton's treatment of the differential or fluctional part 211 00:14:40.279 --> 00:14:44.240 of the calculus is concerned, is incorrect, as the foregoing 212 00:14:44.279 --> 00:14:50.480 account sufficiently shoes. If a flowing quantity or fluent were 213 00:14:50.519 --> 00:14:56.039 represented by x, Newton denoted its fluxion by x dot, 214 00:14:56.480 --> 00:15:00.200 and the fluxions of x dot or second flunction of 215 00:15:00.480 --> 00:15:04.279 x by x dot dot, and so on. Similarly, the 216 00:15:04.320 --> 00:15:07.440 fluent of x was denoted by x written in a box, 217 00:15:07.639 --> 00:15:12.080 or sometimes by x prime or x inside square brackets. 218 00:15:13.480 --> 00:15:16.679 The infinitely small part by which a fluence such as 219 00:15:16.919 --> 00:15:19.759 x increased in a small interval of time measured by 220 00:15:19.799 --> 00:15:23.360 omicron was called the moment of the fluent, and its 221 00:15:23.440 --> 00:15:28.320 value was shown to be x dot omicron. Newton adds 222 00:15:28.360 --> 00:15:31.840 the important remark that thus we may, in any problem 223 00:15:32.240 --> 00:15:35.759 neglect the terms multiplied by the second and higher powers 224 00:15:35.759 --> 00:15:39.279 of omicron, and we can always find an equation between 225 00:15:39.360 --> 00:15:42.960 the coordinates of x y of a point on a 226 00:15:43.080 --> 00:15:48.360 curve and their fluxions x dot y dot. It is 227 00:15:48.399 --> 00:15:51.679 an application of this principle which constitutes one of the 228 00:15:51.759 --> 00:15:56.320 chief values of calculus. For if we desire to find 229 00:15:56.360 --> 00:15:59.960 the effect produced by several causes of a system, then 230 00:16:00.000 --> 00:16:02.279 when and if we can find the effect produced by 231 00:16:02.360 --> 00:16:06.080 each cause when acting alone in a very small time, 232 00:16:06.480 --> 00:16:09.360 the total effect produced in that time will be equal 233 00:16:09.399 --> 00:16:12.840 to the sum of the separate effects. I should here 234 00:16:12.960 --> 00:16:16.600 note the fact that Vince and other English writers in 235 00:16:16.639 --> 00:16:20.120 the eighteenth century used x dot to denote the increment 236 00:16:20.200 --> 00:16:23.919 of x, and not the velocity with which it increased. 237 00:16:25.039 --> 00:16:28.159 That is, x dot in their writing stands for what 238 00:16:28.240 --> 00:16:31.919 Newton would have expressed by x dot omicron, and what 239 00:16:32.080 --> 00:16:38.120 Leibnitz would have written as dx. I need not discuss 240 00:16:38.679 --> 00:16:42.159 in detail the manner in which Newton treated the problem 241 00:16:42.279 --> 00:16:45.960 above mentioned. I will only add that, in spite of 242 00:16:46.000 --> 00:16:50.240 the form of his definition, the introduction into geometry of 243 00:16:50.320 --> 00:16:53.879 the idea of time was evaded by supposing that some 244 00:16:54.000 --> 00:16:56.960 quantity e g. The absissa of a point on a 245 00:16:57.000 --> 00:17:02.480 curve increased equably, and the required results then depend on 246 00:17:02.600 --> 00:17:06.160 the rate at which other quantities e g. The ordinate 247 00:17:06.279 --> 00:17:09.839 or the radius of curvature increased relatively to the one 248 00:17:09.920 --> 00:17:13.880 so chosen. The fluence so chosen is what we now 249 00:17:13.960 --> 00:17:20.160 call the independent variable. Its fluction was termed the principal fluction, 250 00:17:20.400 --> 00:17:23.119 and of course if it were denoted by x, then 251 00:17:23.359 --> 00:17:28.240 x dot was constant, and consequently x dot dot equalled zero. 252 00:17:29.440 --> 00:17:31.920 There is no question that Newton used the method of 253 00:17:31.960 --> 00:17:36.039 Fluctions in sixteen sixty six, and it is practically certain 254 00:17:36.119 --> 00:17:40.079 that accounts of it were communicated in manuscript to friends 255 00:17:40.079 --> 00:17:44.960 and pupils from and after sixteen sixty nine. The manuscript 256 00:17:45.000 --> 00:17:47.720 from which most of the above summary has been taken 257 00:17:48.279 --> 00:17:51.160 is believed to have been written between sixteen seventy one 258 00:17:51.240 --> 00:17:54.920 and sixteen seventy seven, and to have been in circulation 259 00:17:55.079 --> 00:18:00.240 at Cambridge from that time onward. It was unfortunate that 260 00:18:00.319 --> 00:18:03.799 it was not published at once. Strangers at a distance 261 00:18:03.960 --> 00:18:07.240 naturally judged of the method by the letter to Wallace 262 00:18:07.319 --> 00:18:12.119 in sixteen ninety two or by the Tractatus did Quadratura crverum, 263 00:18:12.319 --> 00:18:14.759 and were not aware that it had been so completely 264 00:18:14.799 --> 00:18:18.079 developed at an earlier date. This was the cause of 265 00:18:18.160 --> 00:18:23.000 numerous misunderstandings. At the same time, it must be added 266 00:18:23.039 --> 00:18:26.880 that all mathematical analysis was leading up to the idea 267 00:18:27.039 --> 00:18:32.359 and methods of the infinitesimal calculus. Foreshadowings of the principles 268 00:18:32.359 --> 00:18:34.839 and even of the language of that calculus can be 269 00:18:34.880 --> 00:18:41.920 found in the writings of Napier, Kepler, Cavalieri, Pascal Fermat, Wallace, 270 00:18:42.160 --> 00:18:45.759 and Berrow. It was Newton's good luck to come at 271 00:18:45.799 --> 00:18:49.519 a time when everything was ripe for the discovery, and 272 00:18:49.640 --> 00:18:52.880 his ability enabled him to construct almost at once a 273 00:18:52.960 --> 00:18:58.519 complete calculus. The notation of the fluctional calculuses, for most 274 00:18:58.559 --> 00:19:03.799 purposes less convenient than that of the differential calculus. The 275 00:19:03.920 --> 00:19:07.480 latter was invented by Libaneates in sixteen seventy five and 276 00:19:07.599 --> 00:19:11.240 published in sixteen eighty four, some nine years before the 277 00:19:11.279 --> 00:19:15.920 earliest printed account of Newton's method of fluxis But the 278 00:19:16.000 --> 00:19:20.960 question whether the general idea of the calculus expressed in 279 00:19:21.000 --> 00:19:24.680 that notation was obtained by Libyaneths from Newton, or whether 280 00:19:24.720 --> 00:19:28.480 it was invented independently, gave rise to a long and 281 00:19:28.559 --> 00:19:33.880 bitter controversy. The leading facts are given in the next chapter. 282 00:19:34.720 --> 00:19:38.279 The question is one of considerable difficulty, But I will 283 00:19:38.319 --> 00:19:40.480 hear only say that from what I have read of 284 00:19:40.519 --> 00:19:43.880 the voluminous literature on the question, I think on the 285 00:19:43.920 --> 00:19:46.640 whole it points to the fact that Lebanets obtained the 286 00:19:46.680 --> 00:19:51.000 idea of the differential calculus from a manuscript of Newton's 287 00:19:51.000 --> 00:19:55.359 which he saw in sixteen seventy five. I believe, however, 288 00:19:55.759 --> 00:19:59.920 that the prevalent opinion is that the inventions were independent. 289 00:20:02.240 --> 00:20:05.799 The remaining events of Newton's life require little or no comment. 290 00:20:06.720 --> 00:20:10.960 In seventeen o five he was knighted. From this time onward, 291 00:20:11.079 --> 00:20:14.640 he devoted much of his leisure to theology, and wrote 292 00:20:14.880 --> 00:20:19.119 at great length on the prophecies and predictions, subjects which 293 00:20:19.160 --> 00:20:23.799 had always been of interest to him. His Universal arithmetic 294 00:20:23.960 --> 00:20:27.759 was published by Wiston in seventeen o seven, and his 295 00:20:27.960 --> 00:20:32.640 Analysis by Infinite Series in seventeen eleven, but Newton had 296 00:20:32.680 --> 00:20:35.640 nothing to do with the preparation of either these or 297 00:20:35.680 --> 00:20:39.799 the press. His evidence before the House of Commons in 298 00:20:39.920 --> 00:20:44.599 seventeen fourteen on the determination of longitude at sea marks 299 00:20:44.640 --> 00:20:49.200 an important epoch in the history of navigation. The dispute 300 00:20:49.359 --> 00:20:52.680 with Leibnitz as to whether he had derived the ideas 301 00:20:52.759 --> 00:20:56.720 of the differential calculus from Newton or invented it independently, 302 00:20:57.359 --> 00:21:02.119 originated about seventeen o eight and occupied much of Newton's time, 303 00:21:02.319 --> 00:21:06.319 especially between the years seventeen o nine and seventeen sixteen. 304 00:21:07.759 --> 00:21:11.680 In seventeen o nine, Newton was persuaded to allow Cotes 305 00:21:11.759 --> 00:21:16.000 to prepare the long talked of second edition of the Principia. 306 00:21:16.799 --> 00:21:21.319 It was issued in March of seventeen thirteen. A third 307 00:21:21.480 --> 00:21:25.000 edition was published in seventeen twenty six, under the direction 308 00:21:25.119 --> 00:21:29.799 of Henry Pemberton. In seventeen twenty five, Newton's health began 309 00:21:29.920 --> 00:21:34.079 to fail. He died on March twentieth, seventeen twenty seven, 310 00:21:34.599 --> 00:21:37.480 and eight days later was buried with great state in 311 00:21:37.559 --> 00:21:43.000 Westminster Abbey. His chief works, taking them in their order 312 00:21:43.039 --> 00:21:47.400 of publication, are the Principia published in sixteen eighty seven, 313 00:21:48.000 --> 00:21:52.880 The Optics with Appendices on cubic curves, the Quadrature and 314 00:21:53.000 --> 00:21:57.119 Rectification of Curves by use of the Infinite Series, and 315 00:21:57.359 --> 00:22:01.839 the Method of Fluxions published in seventeen four, and the 316 00:22:02.000 --> 00:22:06.519 Universal Arithmetic published in seventeen o seven. The Analysis per 317 00:22:06.640 --> 00:22:12.160 series Fluxions et cetera published in seventeen eleven, Leccioni's Optique 318 00:22:12.440 --> 00:22:16.720 published in seventeen twenty nine, Method of Fluxions et cetera 319 00:22:17.160 --> 00:22:21.400 I E. Newton's Manuscript on Fluxions, translated by J. Colson 320 00:22:21.440 --> 00:22:26.480 and published in seventeen thirty six, and the methodis differentialis 321 00:22:26.920 --> 00:22:33.319 also published in seventeen thirty six. In appearance, Newton was short, and, 322 00:22:33.400 --> 00:22:37.400 towards the close of his life, rather stout, but well set, 323 00:22:37.559 --> 00:22:41.240 with a square lower jaw, brown eyes and a very 324 00:22:41.240 --> 00:22:45.559 broad forehead and a rather sharp features. His hair turned 325 00:22:45.599 --> 00:22:49.359 gray before he was thirty, and remained thick and white 326 00:22:49.480 --> 00:22:53.160 and silver till his death. As to his manners, he 327 00:22:53.279 --> 00:22:58.319 dressed slovenly, was rather languid, and was often so absorbed 328 00:22:58.359 --> 00:23:00.480 in his own thoughts as to be any but a 329 00:23:00.519 --> 00:23:05.359 lively companion. Many anecdotes of his extreme absence of mind 330 00:23:05.880 --> 00:23:10.640 when engaged in any investigation have been preserved. Thus Once, 331 00:23:10.839 --> 00:23:13.599 when riding home from Granthem, he dismounted to lead his 332 00:23:13.720 --> 00:23:16.359 horse up a steep hill. When he turned at the 333 00:23:16.359 --> 00:23:18.920 top to remount, he found that he had the bridle 334 00:23:18.960 --> 00:23:22.079 in his hand, while his horse had slipped it and 335 00:23:22.200 --> 00:23:26.839 gone away again. On the few occasions when he sacrifices 336 00:23:27.039 --> 00:23:29.880 time to entertain his friends, if he left him to 337 00:23:29.920 --> 00:23:33.160 get more wine or for any similar reason, he would 338 00:23:33.240 --> 00:23:35.920 as often as not be found after the lapse of 339 00:23:36.000 --> 00:23:39.319 some time working out a problem. Oblivious alike of his 340 00:23:39.519 --> 00:23:44.000 expectant guests and of his errand, he took no exercise, 341 00:23:44.119 --> 00:23:49.000 indulged in no amusements, and worked incessantly, often spending eighteen 342 00:23:49.079 --> 00:23:52.680 or nineteen hours out of the twenty four in writing. 343 00:23:54.119 --> 00:23:58.680 In character, he was religious and conscientious, with an exceptionally 344 00:23:58.759 --> 00:24:03.160 high standard of morales, having, as Bishop BRUNEI said, the 345 00:24:03.240 --> 00:24:08.400 whitest soul he ever knew. Newton was always perfectly straightforward 346 00:24:08.400 --> 00:24:12.839 and honest, but in his controversies with Leibanez's Hook and others, 347 00:24:13.279 --> 00:24:17.440 though scrupulously just, he was not generous, and it would 348 00:24:17.519 --> 00:24:20.599 seem that he had frequently took offense at a chance 349 00:24:20.640 --> 00:24:25.960 of expression when none was intended. He modestly attributed his 350 00:24:26.079 --> 00:24:30.000 discoveries largely to the admirable work done by his predecessors, 351 00:24:30.559 --> 00:24:33.640 and once explained that if he had seen farther than 352 00:24:33.720 --> 00:24:36.960 other men, it was only because he stood on the 353 00:24:37.000 --> 00:24:41.079 shoulders of giants. He summed up his own estimate of 354 00:24:41.119 --> 00:24:44.039 his work in the sentence, I do not know what 355 00:24:44.200 --> 00:24:47.480 I may appear to the world, but to myself I 356 00:24:47.599 --> 00:24:50.519 seemed to have been only like a boy playing on 357 00:24:50.599 --> 00:24:53.680 the sea shore and diverting myself in now and then 358 00:24:53.880 --> 00:24:57.200 finding a smoother pebble or a prettier shell than ordinary, 359 00:24:58.079 --> 00:25:02.240 whilst the great ocean of truth lay all undiscovered before me. 360 00:25:03.240 --> 00:25:06.920 He was morbidly sensitive to being involved in any discussions. 361 00:25:07.640 --> 00:25:10.920 I believe that, with the exception of his papers on optics, 362 00:25:11.440 --> 00:25:14.400 every one of his works was published only under pressure 363 00:25:14.559 --> 00:25:18.759 from his friends and against his own wishes. There are 364 00:25:18.839 --> 00:25:22.880 several instances of his communicating papers and results on condition 365 00:25:23.200 --> 00:25:27.039 that his name should not be published. Thus, when in 366 00:25:27.160 --> 00:25:31.920 sixteen sixty nine he had, at Collier's requests, solved some 367 00:25:32.079 --> 00:25:36.319 problems on the harmonic series and on annuities which had 368 00:25:36.400 --> 00:25:41.759 previously baffled investigation, he only gave permission that his results 369 00:25:41.759 --> 00:25:45.480 should be published. So it be as he says, without 370 00:25:45.519 --> 00:25:48.240 my name to it. For I see not what there 371 00:25:48.359 --> 00:25:52.400 is desirable in public esteem. Were I able to acquire 372 00:25:52.440 --> 00:25:56.960 and maintain it, it would perhaps increase my acquaintance the 373 00:25:57.000 --> 00:26:01.359 thing which I chiefly studied, to decline in intellect. He 374 00:26:01.440 --> 00:26:06.319 has never been surpassed, and probably never been equalled. Of this. 375 00:26:06.640 --> 00:26:10.559 His extant works are the only proper test. Perhaps the 376 00:26:10.599 --> 00:26:14.559 most wonderful single illustration of his powers was the composition 377 00:26:15.200 --> 00:26:18.920 in seven months of the first book of the Principia. 378 00:26:20.279 --> 00:26:24.160 As specific illustrations of his ability, I may mention his 379 00:26:24.279 --> 00:26:29.119 solutions of the problems of Pappus, of John Bernoulli's challenge, 380 00:26:29.400 --> 00:26:33.599 and of the question of orthogonal trajectories. The problem of 381 00:26:33.680 --> 00:26:37.200 Pappus is to find the locus of a point such 382 00:26:37.240 --> 00:26:41.119 that the rectangle under its distances from two different straight 383 00:26:41.160 --> 00:26:44.240 lines shall be in a given ratio to the rectangle 384 00:26:44.400 --> 00:26:48.839 under its distances from two other given straight lines. Many 385 00:26:48.920 --> 00:26:52.599 geometricians from the time of Apollonius had tried to find 386 00:26:52.599 --> 00:26:56.920 a geometrical solution and had failed. But what had proved 387 00:26:56.920 --> 00:27:00.359 insuperabowl to his predecessors seemed to have presented life little 388 00:27:00.359 --> 00:27:05.640 difficulty to Newton, who gave an elegant demonstration as that 389 00:27:05.680 --> 00:27:11.759 the locust was a conic geometry, said Lagrange was when 390 00:27:11.880 --> 00:27:15.079 recommending the study of analysis to his pupils. Is a 391 00:27:15.160 --> 00:27:19.880 strong bow, but it is one which only a Newton 392 00:27:19.960 --> 00:27:25.039 could fully utilize. As another example, I may mention that 393 00:27:25.119 --> 00:27:29.799 in sixteen ninety six John Bernoulli challenged mathematicians one to 394 00:27:29.920 --> 00:27:34.519 determine the brachistochrone, and two to find a curve such 395 00:27:34.559 --> 00:27:37.039 that if any line drawn from a fixed point O 396 00:27:37.920 --> 00:27:40.920 cut it in P and Q, then O P to 397 00:27:41.000 --> 00:27:44.240 the power n plus q o q to the power 398 00:27:44.359 --> 00:27:47.440 n would be a constant leiben It solved the first 399 00:27:47.480 --> 00:27:50.640 of these questions after an interval of rather more than 400 00:27:50.720 --> 00:27:53.640 six months, and then suggested that they should be sent 401 00:27:53.680 --> 00:27:57.160