i8o LIMITS AND FLUXIONS 



163. We quote Sir William Rowan Hamilton's 

 remarks on the lemma of the anonymous Explana- 

 tion of Fluxions (1741) and the derivation of the 

 fluxion of xy^ based upon it. Hamilton knew this 

 proof as it is given in a later edition of Simpson's 

 fluxions. Says Hamilton :^ *' I notice that Thomas 

 Simpson treats fluxions as finite . . . Thomas 

 Simpson's conceptions appear to have been very 

 clear and distinct, and I do not venture to say that 

 the geometrical investigation which he gives of the 

 fluxion of a rectangle, avowedly supplied to him by 

 a young but unnamed friend, is insufficient in itself, 

 but it fails to convince me, perhaps because I was 

 not early accustomed to fluxions. Certainly there 

 is no neglecting of ab^ or .1 j> ^'^ small ; for in fact 

 that rectangle of the fluxions is not represented at 

 ali in his Figure ... He conceives the varying 

 rectangle xy to be the suin of two mixtilinear triangles, 

 of which the two separate fluxions are yx and xy. 

 This is very ingenious, but I do not feel sure to 

 what degree I could rely on it and build upon it any 

 superstructure, if I were now coming, for the first 

 time, as a learner^ to the subject. However, I 

 suppose that a pupil, if reasonably modest or even 

 prudent, vvill take, for a while, his teacher's state- 

 ments upon trust ; reserving to himself to return 

 upon them, and to examine closely their truth and 

 logie when he shall have acquired some degree of 

 familiarity with the subject taught." 



^ Life of Sir Williain Rowan Hamillon, by R. P. Graves, voi. iii, p. 571. 



