328 On the Doclnne of Fluxions. [May, 



seem to think it extremely difficult to obtain even the first prin- 

 ciples of that important branch of mathematical science. In what 

 follows, (which, if you think proper, you may insert in the Annals 

 of Philosophy,) I intend to solve inductively the fluxional problem 

 as extensively as Newton has demonstrated it in the second lemma 

 of the second book ; then to demonstrate that problem rigorouslv 

 in the manner of the ancients, independently of infinitesimals, of 

 motion, or of vanishing quantities ; and, lastly, to subjoin some 

 observations. 



I am, my dear Sir, yours faithfully, 



Eftnbwgh, itfarcA 20, 1815. AlEX. ChRISTISON. 



Of Fluxions. 



In consequence of repeated trials, I have long thought that a 

 boy, duly prepared, passes from common algebra into fluxions as 

 easily as he does from multiplication into division. 



In solving the fluxional problem three things are to be distin- 

 guished — the conception, the notation, the demonstration. 



1. With regard to the conception, I shall, in order to be easily 

 understood, proceed as if I were questioning a learner. Ask him 

 thus — If a straight line, A D, fig. 1, move parallel to itself at 

 right angles along D C, blackening 

 the parallelogram A F, whose side 

 A E or B F is 5 ; and reddening 

 the parallelogram E C, whose side 

 ED or F C is unit ; at what rate 

 does it always blacken the one 

 parallelogram and redden the 

 other? He will answer — As 5 to 

 1. The conception is much aided, 

 at first, by his imagining that the 

 two rail elograms are generated of 

 difleit It colours. 



Ask iiim now thus — If a straight line move parallel to itself, at 

 right angles along E D, fig. 2, so 

 that it can generate only the 

 paralleio!;rani A D, whose side 

 A E is unit, and the triangle ABC 

 half a square, while all the rest of 

 the b^pace is covered ; at what rate 

 does it, at H F = G, generate the 

 triangle and the parallelogram ? 

 He will answer — As 5 to 1 ; or as 

 5 X J : 1x1; that is, as the 

 area of the |x»rallelogram H M is 

 to the area of the parallelogram 



G L ; but lie will probably add, that the instant before the rate 

 was less, and that the instant after it will be greater. He may be 

 toldj that it is not the rate tlie instant before, nor the instant after. 



D 



