200 



♦ KNOWLEDGE ♦ 



[July 2, 1888. 



Lesson IV. is headed " Differentiating Simple Functions " ; Lesson V. 

 the same; Lesson VI., "Differentiating Composite Functions"; 

 and that Lessons VIII., X , XI , XII., and XIII. are all devoted to 

 " Illustrations of the Dse of the Calcnlas." However, it suited him 

 better to describe Lesson III., preceding even the simplest differen- 

 tiations, as though it were the only lesson relating to the use of the 

 CalculusJ. When we turn to see how Mr. Proctor applies this 

 great engine, we find only two little problems, neitlier of which 

 requires anything of the sort. The tir=t is a question about a 

 straight line. In this there is nothing wanted beyond the iiflli pro- 

 position of Euclid's second book to do easily in two lines or three, 

 what Mr. Proctor occupies a printed page with. Need one have a 

 telescope fetched, tripod and all [strange idea this man has about 

 an astronomer's telescope] to look at the clock on the chimney- 

 piece ? The second jiroblem or " illustration " is equally simple, and 

 equally undeserving of having so great an engine set in motion ; in 

 fact, it is easily shown to depend on the same proposition of Euclid 

 when a rectangle is interpreted algebraically, and can thus be done 

 more quickly than by Mr. Proctor's mode [meaning " method," it 

 may be presumed]. 



Asa matter of fact, the geometrical problem would not 

 require the second book of Euclid at <^ll, thonjrb it could be 

 given as a corollary to the problem mentioned b^' the 

 Saturday Revievjei-. The geometrical solution might, for 

 example, rim as follows, without going beyond Book I. : — • 



Let A B (tig. 1) be the line, C its bi.ser-tion, D a point in 

 AC. Complete the square A C F E, and the rectangle 



I 



F 



FK4. 



D B H G, having the side B H equal to A D. Produce 

 D CI to meet E F in K. Then the rectangle A K is equal 

 to the rectangle C H, since A D=B H and A E=AC=C B). 

 Add G C. Then the rectangle G B is equal to the fin-ure 

 E D L, or is less than the square E C. Q.E.D. 



Algebraically, the proof of the proposition that (^ax—a-) 



is greatest when x=^^ can hardly escape anyone who has 

 ever solved a quadratic equation. It runs simply thus : 



, a- 



a x—x-^ 



4 



which is manifestly greatest when .<■ 



QED. 



Now my page of printed matter (really two-thirds of 

 a page, but that is near enough for the ,S'. /,'.) gives not only 

 the solution of the problem, but an explanation of the prin- 

 ciple on which the application of the differential calculus to 

 such problems depends — this being what I desired to show, 

 not how to solve a problem which involves in itself no diffi- 

 culty whatever. " Mr. Proctor's mode " or method con- 

 sisted in showing at the very beginning of his little book 

 how the differential calculus is applieil to problems of a 

 certain class. (For this a page would not have been too 

 much, though little more than half a page sufficed.) 



Albeit, if one can imagine a mathematician in any 

 momentary doubt as to the value of .<■ which will make 

 {ax — x-) a maximum, one would certainly expect him to 

 apply the differential calculus, because this would save him 

 all trouble. Antecedently it would be doubtful in most 

 cases of the kind whether a geometrical or algebraical 

 method could be lit upon easily — and it may be remarked 

 in passing that whea we are told that this or that problem, 

 readily solved by the differential calcuhis, can be solved 



geometrically or algebraically, we may generally be sure (if it 



has any difficulty in it) that the geometrician or algebraist 



has first solved it by the calculus, or seen some one else's 



solution, and has then inquired how the known result could 



be obtained geometrically. 



A mathematician proceeding on the sensible plan of 



applying the only method which is sure and easy, without 



troubling himself about its being a "great engine"" (which is 



sheer nonsense), would have easier work than either the 



geometrician or algebraist, easy though (as we have seen) 



their task would be. The problem is :— " If ?/=« a; — a--, 



when is ya maximum?" The "great engine" does the 



work thus (the engineer knowing that for any expression to 



be a maximum its diflereutial coefficient must vanish) : 



d V .T a 



-^=a—2x = o- .-. x=^. 

 dx -J, 



Of course the easier the example selected, the less favourably 

 the work of the " great engine " compares with the work of 

 any smaller engine capable of effecting it. Still, the above 

 is not very hard work ; and be it noticed that it involves 

 much less thought than even the simple geometrical and 

 algebraical methods presented e.arlier, little though the 

 thought be that they require. (We see that even & Satur- 

 day lleviev) mathematician can puzzle out one of those solu- 

 tions, though, to be sure, he makes a large fuss over his 

 small achievement, and his solution is f\u- from being even 

 the best of its kind — alwa3-s supposing it original.) 



This particular example illustrates the use of the "great 

 engine" ("I thank thee," S.R., "for teaching me ^hat 

 w-ord") quite as well as one I give further on— in Lesson 

 ^"flL — which, treated as a geometrical problem, would be 

 more difficult. (I would back it to beat the Saturdai/ 

 lieviewer, though I think— but cannot be sure till I have 

 tried — that a geometrician would not take long over it. I 

 will try presentl.v.) It is the problem of dete'rmining the 

 greatest cone which can ba inscrilied in a sphere of radius r. 

 Here the solution by the dilferenti.al calculus runs easily 

 and smoothly as follows : — 



Taking x for the height of the cone, we get for its volume 

 ((/) the expression 



y=|(2r.r'2-,r3); 



whence ~=^ (4r.x- — 3a;'-) —o (for a maximum or 



minimum). 



So that either x=o, giving obviously a minimum, 



4: y . . 

 or 0!=-:^, giving the required maximum. 

 o 



As some readers may like to see how this problem is to 



be solved geometrically, I give the solution in that way, 



fearing lest the weakling mathematician of the Saturday 

 Revieiv should waste a month of his valuable time in vainly 

 attemptinjr to find one : — 



Let ABC (fig. 2) be a section of the maximum cone 

 through O, the sphere's centre; B D, the cone's height. 



