8 



continuation are as follow : firfl:, tlje indices of e in the fcveral terms 

 are the natural numhers in their order : fecondly, the numeral co-effi- 

 cient of any term is i divided by the index of e in that term : and 

 thirdly, when the quantity is i + f , the terms are alternately affirma- 

 tive and negative; but when it is i — e, the terms are all negative. 



And that the omiffion -of the fraftion — in the third and fol- 



n 



lowing terms of the feries A (in Lemma i) will not caufe any error 

 here, within the prefcribed ^number of places, will fippear by confider- 

 ipg the value of the quantity that is omitted in any term of the fqries 



of this Lemma : for it appears to be always lefs than — , the greatefl 

 poffible value ,of which is a decimal confiding of i preceded by as 

 many cyphers except two as there are places of figures in n ; and there- 

 fore the omiilions in i oo terms will not amount to ac unit in the laft 

 of the prefcribed places. 



l^Qte, That Doftor Halley has derived the fubftance of thefe lem-, 

 mas in .a fliorter manner, from a fuppofition of n and ?;/, the deno- 

 minators of the indices, being infinite numbers. But as there may be 

 fpme difficulty in conceiving this, I have propofed as much of the do.c- 

 trine as was wanted here, without making that fuppofition. 



Art. I. By ratiuncula I underfland a ratio of inequality, but which 

 is very near to a ratio of equality. Thus if r be a decimal frac- 

 tion having many cyphers (as i o, 20, 50, 1 00) before the firfl figni- 

 ficant figure, the ratio of i to i + r, or of i to i — r may be called 

 a ratiuncula. 



£. Logarithims are the exponents (or numeral meafures) of ratios. 



3. Now ratios arc meafured by the numbers of equal ratiuncula; of 

 which they are compounded, or into which they may be fuppofed to 

 be refolved. Thus, if between i and 10 be placed 999999999 mean 

 proportionals, then will the ratio of i to to be refolved into ipoooooooo 

 ratlunculcE, each equal to the ratio of i to the firfl or leaft of thofe 

 jpeans: and if of thefe means 30102999,5 Hand between i and 2 



the 



