April 19, 1900] 



NATURE 



5«3 



that elusive abstraction, the Exponential Theorem, 

 which, as these Lectures show, may be omitted and dis- 

 regarded in a course of Practical Mathematics. We 

 remember the elaborate and majestically slow overture 

 of Todhunter's Differential Calculus, wherein the initial 

 motive of the Exponential Theorem is developed at 

 such length, pausing to investigate the influence of 

 fractional as well as infinite steps in the neighbourhood 

 of the limiting infinity. No wonder the subject of the 

 Calculus was a sealed book to all but a few of our students. 

 Contrast this with the good fortune of the French 

 schoolboy, who is introduced to the notions of the flow 

 of variable quantities, and to the sacred symbols of the 

 little d and the long / in a course of elementary algebra, 

 such as "Cours d'algebre el^mentaire, conforme aux 

 derniers programmes." 



Omitting all reference to the Exponential Theorem, 

 and dealing only with the common logarithms employed 

 in ordinary calculations, the number x is defined as the 

 logarithm of/ from the relation \<y=^y, x = \<:>^ y. 



Then, by employing the ordinary arithmetical opera- 

 tion of square root, Mr. Edser has given a ready method 

 of starting a calculation of the logarithms, or rather the 

 antilogarithms, as follows • — 



The cube root of lo, worked out by ordinary arith- 

 metical methods, and the fifth root of lo, obtainable by 

 Home's method, will provide additional antilogarithms ; 

 and thence by multiplication, we find y for x — ^, -^, 

 . . ., 7^, , . . ; and plotting these relations by a curve on 

 squared paper (Nature, p. 415, March i, 1900, Dr. A. 

 Dufton), we can arrive at the simple results of io°'3°'°3 = 2, 



3, 10 



,0-8451. 



= 7, 



and thence construct the 



chief divisions of the Slide Rule, and by further inter- 

 polation calculate the Table of four-figure antilogarithms 

 and logarithms, which suffice for all practical purposes. 



Even the theoretical student, who cultivates mathe- 

 matics as a subject which he will be called upon to teach 

 in his turn, but never to employ on vulgar practical 

 applications, would profit by approaching the study of 

 logarithms in the same way. Afterwards he may open 

 a page of seven-figure logarithms, and consider how 

 much calculation has been expended on it ; he will be 

 surprised to find that, after he has written down the 

 logarithms of a few of the composite numbers to serve 

 as bench marks, the Arithmometer set at a constant 

 difference will run out the intermediate logarithms as 

 fast as the handle can be turned, with an occasional 

 change in the eighth decimal of the difference, wherever 

 a bench mark shows that it is required. A little ele- 

 mentary drill of this kind will soon substantiate Prof. 

 Perry's complaint (p. 38) : — 



" Some friends of mine assert that no man or boy 

 ought to be allowed to use logarithms until he knows 

 how to calculate them. They say this, knowing that the 

 calculation is a branch of Higher Mathematics, and that 



NO. 1590, VOL. 61] 



i the average schoolboy, after six years of mathematics, 

 finds it hopeless to even begin the study of the Expo- 

 riential Theorem. It is a hard saying ! It is exactly 

 like saying that a boy must not wear a watch or a pair 

 of trousers until he is able to make a watch or a pair 

 of trousers. It is the sort of unfeeling statement which 

 so well illustrates the attitude of the superior person." 



The essentials of the subject of Analytical Geometry, 

 which blocks the way in our present system of mathe- 

 matical instruction, are given here under the title of 

 Squared Paper. After an encomium on its practical vir- 

 tues. Prof Perry has the audacity to follow the Continental 

 lead, and carry his audience straight up to the Calculus. 

 Accurate diagrams on this squared paper are given of 

 the elementary curves, the graphs of x>\ x~", sin x, cos x, 

 tan X, e^, . . ., as well as of the ellipse and hyperbola, 

 practically all the mathematical principles required for 

 the graphical representation of the variation of physical 

 quantities. 



The exponential curve, the graph of ^'^, represents a 

 quantity which grows or dwindles at constant compound 

 interest or discount like a row of organ pipes, the rate 

 of extinction of light, or the diminution of the density ot 

 the atmosphere in a balloon ascent. 



The graphs of y=x" and x~"- are the representation 

 of quantities, such that i 7o increase in x causes n "/^ 

 increase or decrease in y. This is the familiar state- 

 ment of a complicated mechanical law, such as Froude's 

 Law, which asserts that in similar steamers over a given 

 voyage i % increase in speed requires 6 % increase in 

 tonnage and coal capacity, and 7 % increase in engine 

 power. 



When the practical man is compelled to invoke theory 

 to his aid, it is generally in some such manner as the 

 above ; from a known performance, say, of a steamer, he 

 has to argue the requisite alterations, expressed in 7o> 

 for slight differences in a new design. 



The empirical formulas of internal ballistics and of 

 armour-piercing are all examples of the same theory ; 

 the index n, sometimes carefully guarded as an ofllicial 

 secret, is at once revealed by plotting on Human's 

 logarithmic co-ordinate sheets, which the possessor of 

 a Slide Rule can readily construct for himself ; and Mr. 

 Vincent has shown, in his Report to the British As- 

 sociation, 1898, that the semi-logarithmic co-ordinates 

 can be employed usefully, in which a combination is 

 made of the Slide Rule Graduations with the equidistant 

 graduations of the foot or metre rule. 



In answer to his question on p. 53, for a method of 

 finding a", where a and n are any numbers, the author 

 may be referred to Mr. Lanchester's radial cursor attach- 

 ment, as well as to the double-logarithmic scale, which 

 he will find described in the Catalogue of the Mathe- 

 matical Exhibition at Munich, 1893, as the invention of 

 Blanc, of Hamburg; and M. d'Ocagne's "Traite de 

 nomographie," recently reviewed in Nature, will pro- 

 vide a complete description of all such methods of 

 graphical calculation. 



Euclid supplies the place in our schools of the study 

 of Formal Logic, so far as the essentials of strict demon- 

 stration ; but as no one employs the syllogism, not even 

 Euclid, and time is limited now there is no longer the 

 lack of the Middle Ages in subjects of scientific interest^ 



