A Theory of Fluxions. 



339 



ygons, it is proved that circ. NOPQ,: circ. IKLM: :WX 2 : 

 VZ 2 . 



That this is the peculiar proportion, that fluents bear to 

 their fluxions, can be made to appear in the following man- 

 ner, in Fig. 5. The fluent NWr ": IVra! :WX 2 : VZ 2 , be- 

 cause these are similar portions of similar figures, and are as 

 the squares of their homologous sides, and in circles as 

 squares of their diameters. Hence NOPQ, : IKLM : WX 2 



: VZ 2 : : fluent NWr : fluent IVm'. : fluxion (ax - x 2 )^ x - : 



n 2 — 



fluxion - T .(ax — x 2 ) 2 x\ Although the principle of fluxions 

 a 2 



was thus touched upon, it was (as it were) unwittingly. None 

 of the antients, as it seems, understood its nature, and ex- 

 tent ; but they proceeded but little farther than to investi- 

 gate some of the properties of the conic sections. 



Demonstration of the identity of ratios in fluents and their 

 fluxions. 



Lemma. 



Fig. 6. 



Let AHNB be any 

 K curve whatever, and sup- 

 pose DFME to be anoth- 

 er, drawn parallel to it, 

 and consequently similar. 

 u Draw the lines CM, CF 

 R from the determined point 

 C to the curve DFME, and 

 -6 from the points, where 

 these lines intersect the 

 curves, draw the ordinates 

 NT, MX, HO, FP. Now, 

 if the line CQ, be drawn 

 from C to any proposed 

 CJ point Q in the arc FM, and 

 ordinates be drawn from the points of intersection ; the parts, 

 into which the curvilineal spaces HNTO, FMXP are divi- 

 ded, are proportional; that is, WNTY : QMXS>:HWYO 

 : FQSP. 



» E 



A OS TtX T 



