BINOMIAL THEOREM. 



4^9 



Afcxcus n«yx?^?5f (whicli he fo calls on account of its frequent and cxceU 

 •en?: '\^f, and of v;hicl> fmall fpecimen is here annexed,) that the num- 

 ^^n in the diagooa] directions, afcending from right to left. 







Abacus nArXPH2T02. 







n I 



('- 



F 



E 



D 



C 



B 



A 



! -(H), 1 ~ 



— (7^; 



f(6) 



+(5) 



-(4) 



-(3) 



+(2) 



(1) 



, » 



8 



7 



6 



6 



4 



3 



2 





3c> : 



28 



21 



15 



10 



6 



3 





a 



84 



56 



35 



20 



10 



4 









126 



70 



35 



15 



5 









126 



56 



21 



6 







, 



84 



28 



7 



k# ■. 







36 



8 

















9 



are the coefEcients of the powers of Binomials, the indices being the figures 

 in the firfl perpendicular column A, which are alfo the coefficients of 

 the 2d terms of each power, (those of the firfl; terms being i, are here 

 omitted) ; and that any one of thefe diagonal numbers is in proportion 

 to the next higher in the diagonal, as the vertical of the former is to the 

 marginal of the latter; that is, as the uppermost number in the column 

 of the former is to the first or right hand number in the line of the 

 latter. Having (hewn thefe things, I fay, he thereby teaches the genera- 

 tion of the coefficients of any power, independently of all other powers, 

 by the very fame law or rule which we now ufe in the Binomial Theo- 

 rem. Thus, for the 9th power; 9 being the coefficient of the 2d term, 

 and 1 always that of the ift, to find the 3d coefficient, we have 2:8:: 

 9: 36; for the 4th term, 3:7**36:84; for the 5th term, 4:6:: 84: 

 126; and fo on for the reft. That is to fay. the coefficients in the terms 

 in any power m, are inverfely as the vertical numbers or firfl line 1, 2, 



