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 r to q. In it A, N, r, q^ may be any numerical or 

 arithmetical magnitvides whateA^er, whole, fradlional, furd or 

 mixed. This formula, or antecedent, is exadlly 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 

 arithmetical one : 



A — + — A i-.N-f- i-.A ^.N'± — . i-. 1I.A-— ^.N3+ + &c. 



q q q q n q ^ H 11 a 



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

 tio of /" to q. 



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

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

 them, viz. from geometrical antecedents or formulas, 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 the formula or antecedent. This circumftance 

 renders it of all others the mofl commodious for common ufe 

 in algebra and arithmetic, though the leaft calculated of any 

 for (hewing the rationalia or ground-work of the various opera- 

 tions in thefe two fciences. For when the formula or antece- 

 dent fliows the different combinations of the confequent or 

 flandard of comparifon w^ith 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, which have refpedlively to B ratios, 

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



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



has 



