68 On the PRINCIPLES of the 



which has to i, or unit, a ratio having to the ratio of A+N to i 

 the ratio of /■ to q. In it A, N, r, ^, may be any numerical or 

 arithmetical magnitudes whatever, whole, fradlional, furd or 

 mixed. This formula, or antecedent, is exacflly what is com- 

 monly called the Binomial Theorem. 



If we fuppofe B to be reprefented by 2, we derive immedi- 

 ately from this geometrical antecedent or formula, the following 

 aridimetical one : 



Ail + ^,Aii:L.N-f-^.^:=L.Al=^.N'±-L.I=X.IZ:!f.Al=3?.NH + &c. 



9 11 1 H 1 1 H ii 1 



which has to 2 a ratio having to the ratio of A+N to 2, the ra- 

 tio of r to q. 



To fuch arithmetical formuls there is no end or limit. And 

 this I take to be the true and fyflematic method of deriving 

 them, viz. from geometrical antecedents or formula, when 

 they are fuppofed to become numerical. 



When i or unit is the flandard of comparifon, its various 

 combinations with itfelf and the other numerical magnitudes, 

 do not appear in tlie formula or antecedent. This circumftance 

 renders it of all others the moft commodious for common ufe 

 in algebra and arithmetic, though the leaft calculated of any 

 for fliewing the rationalia or ground-work of the various opera- 

 tions in tliefe two fciences. For when the formvila or antece- 

 dent fhows the dijEFerent combinations of the confequent or 

 ftandard of comparifon with itfelf and the other numerical 

 magnitudes, it is a fort of language announcing or exhibiting 

 the reafons of its formation. 



It is evident, that half the excefs of the two geometrical 

 expreflions taken together, wliich have refpedlively to B ratios, 

 having to the ratios of A-|-N to B and A — N to B, the ratio of 



aA 

 R to Q^ above twice — rZIo ' °^ twice the magnitude, which 



has 



