A Theory of Fluxions. 



331 



Fig. 2. 



In the triangle CED, sup- 

 pose the generating line AB, 

 variable in length, to move 

 from C to D, the area gen- 

 erated by this motion is term- 

 j ed the fluent. If the gener- 



.U i& ating line should still flow 



I, the magnitude it had at the moment in which the 

 fluent CED is completed, the rectangle Ea&D, generated by 

 this motion, is called the fluxion. 



Fig- 3 - . cl ■ x u 



We may imagine a superficies to be 

 xy '"":" generated thus: let the dimension of 



• length be represented by x, Fig. 3, and 

 that of breadth by y ; let these two quan- 

 tities be supposed to commence their 

 existence and motion together, at the cor- 

 ner A, and let them so proceed in their 

 motion, as to preserve the intended pro- 

 portion between them. When they ar- 

 rive at the situation CD, DB, the fluent 

 CDBA equal to xy is generated. Con- 

 ceive the lines x and y to move still on- 

 ward with the same motion they had at the instant when the 

 fluent was completed, for any assigned time, great or small; 

 and they will generate the parallelograms yx' and xy, termed 

 the fluxion. When the length and breadth are equal, the 

 figure becomes a square, the fluent xx being represented by 

 CDBA, and the fluxion by the two equal parallelograms xx' 

 -\-xx', or *2xx\ 



Again we may conceive a 

 solid to be generated in the 

 following manner. Let x be 

 the variable quantity rep- 

 resenting the dimension of 

 length, y that of breadth, 

 and z that of thickness. Let 

 each of these be supposed to 

 commence its existence and 

 motion at O, and let each 

 dimension increase in such a 

 manner as to preserve the in- 

 tended proportion between 

 the lines x, y, z. The lines 

 x and y, by their motion, 



■ it tA-'tAj • 





C D 

 A x B 



XX' i 

 X' \ 



