A Theory of Fluxions. 337 



lowing theorems. Th. I. Any two terms in a series of flu- 

 ents, will have their corresponding fluxions in the same ratio 

 with themselves. Th. II. Any two terms in a series of flux- 

 ions, will have their corresponding fluents in the same ratio 

 with those fluxions. These two theorems lay the founda- 

 tion upon which the whole doctrine of fluxions is built. 



Sec. 4. In the series of fluents, Fig. 5, take IVm the fluent 

 sought, together with any one of the remaining terms, as 

 N Wr, then from what has been said concerning the grounds 

 of fluxions, it appears, that these fluents and their corres- 

 ponding fluxions, are in the same ratio, that is, Wuwr : NWr 

 ; '.Vxsm : IVm. For the purpose of illustration, admit that 

 the second term NWr, is by some means obtained, and that 



13. J. 



it is — — — - — — , &c. The first and third terms can 

 3 ba\ 



readily be had ; for, from the nature of fluxions, Wuwr— 



3 



WrX#*= a 2 x 2 x' — — , &c, and Vxsm=YmX n — = 



1 5. 



n x — — , &c. Now the product of the second 



«! 2af 



j A f j x • 2n 2 x 2 x' 8n 2 x 3 x' , n 2 x l x' , . , , . 



and third term is — + , which being 



3a 15a 2 10a 3 s 



3 5. 



divided by the first term, the result is n x __ w x . £_ 



3af 5«f 



=IVm the fluent sought. Hence if any easy method of ob» 

 taining the second term can be had, the object will be at- 

 tained : but here a difficulty is presented, which in the ordi- 

 nary method of getting the last term of four proportional 

 quantities, would be insurmountable. This property, then, 

 consisting in the identity of ratios in fluxions and their flu- 

 ents, would have been wholly useless, had it not been far a 

 remarkable peculiarity in these proportional quantities. It 

 is this, that fluxions and their fluents mutually arise, and as 

 it were, grow out of each other ; that is, a fluent can be made 

 out from certain known parts of its fluxion, and converselv, 

 by means of a few rules, embracing what is called the direct 

 and inverse method of fluxions. To illustrate the manner in 

 which this is done, let the four terms of the proportion be 

 Vol. XIV.— No. 2. 1.7 



