[ 3H ] 



ticular cafe to the legality of fnch an afiumption not being felf- 

 cvident, often requires perplexing confiderations to avoid intro- 

 ducing unnecclTary terms. Indeed the greateft difficulty often 

 occurs in that part. In this method no feries is to be affumed. 

 The feries derived follows from the nature of the problem. Often- 

 times its law even in very complex cafes can be derived, in which 

 by the method of afTuming a feries it would be almoft impolfible 

 to demonftrate it, Thus the truth of the law of the Mul- 

 tinomial Theorem, when the power is negative or fractional, is de- 

 monftrated by this method. It was done by De Moivre for in- 

 tegral powers, and I know of no author who has generally de- 

 monflrated it for all powers. The examples given to illuftrate 

 "the method are moft of them fuch as are well knov/n, and may 

 be compared with the fame as done by other methods. Among 

 them are two feries firft given by Mr. James Gregory (See Comm. 

 Epift.) the invefligation of the latter of which has been confidered 

 by mathematicians as very difficult. 



Demonflration of Dr. Brooke Taylor's Theorem*. 

 Theo. If ar and x be cotemporaneous values of two quantities 

 any how related, and z and x = flux, of .v, cotemporaneous incre- 

 ments, of which .V is uniformly generated, then will 



. .* • 



z + z =t z -r --I 1 4- - "T, i&c. 



I I. 2. I. 2. 3. I. 2....OT ^ . 



when this feries terminates or converges. 



Demonstration. ' 



♦ Authors who have given this theorem have not been fo attentive to accuracy of 

 demonftration as the importance of the theorem fcems to require. 



1 



