Application of the Fluxional Ratio, fyc. 



303 



by multiplying the second and third terms together, and dividing 

 the product by the first term according to the rule of proportion in 

 common arithmetic. But it will lead to the same result, if we mul- 



b 



tiply the third term Da bv the ratio — 

 r J a 



? 



Fig. 3. 



a 



b 



Here a relation manifestly 



exists between the quantities Ba, B6, and the quantities Da, D6, 

 which is that of proportion. 



Mathematicians, in explaining the nature of fluxions, have pro- 

 ceeded on the tacit acknowledgment, that a circle is a polygon of 

 an infinite number of sides, (Brewster's Encyclopaedia, Art. Flux- 

 ions) and have considered a fluxion to be an elementary part of its 

 fluent. But, if we proceed ever so far in making the fluxion small 

 since it is a rectilineal quantity, a part, Cba (Fig. 1.) being the dif- 

 ference between the increment CHI6, and the corresponding fluxion 

 CHIa, will still be left behind. In theory, therefore, something is 

 lost. The rejection of this part renders the theory imperfect, and 

 unsatisfactory, and creates a suspicion, that it may lead to some er- 

 ror. But in the application of the method no such error is ever 

 found. This practical correctness depends on the fact, that fluxions 

 are not the elementary parts of their fluents, 

 but merely proportional quantities. Let the 

 lines EF, FG, (Fig. 3.) commence an existence 

 at the corner C, and proceed with a uniform 

 motion generating the square ABDC- Now it is 

 manifest, that, since the generating lines EF, 

 FG are continually increasing, the superficies 

 EFGC increases in a ratio, which is momently varying. Suppose 

 that the lines MN, NO, (Fig. 4.) moving with a uniform velocity, gen- 

 erate the square HILK in the same time, 

 that the lines EF, EG, generate the 

 square ABDC. We have now the two 

 fluents ABDC and HILK. Now if these 

 fluents are produced by the lines AC (Fig. 

 3.) and HK, (Fig. 4.) moving with such 

 velocity, that the area generated by AC 

 is always equal to the area generated by 

 EF, FG, and the area generated by 

 HK is always equal to the area generated by MN, NO, it is 

 evident that AC and HK must move with a velocity, which is 

 accelerated. Hence ABDC and HTLK may be considered as 



h 



Fig. 4. 



