VOL. XIX.] PHILOSOPHICAL TRANSACTIONS. ig 



tion and applause : and the great geometricians of this age have not been want- 

 ing to cultivate this subject with all the accuracy and subtilty a matter of that 

 consequence deserves ; and they have demonstrated several admirable proper- 

 ties of these artificial numbers, which have rendered their construction much 

 more facile than by those operose methods at first used by their truly noble 

 inventor, the Lord Napier, and our worthy countryman Mr. Briggs, 



But notwithstanding all their endeavours, I find very few of those who make 

 constant use of logarithms, to have attained an adequate notion of them, to 

 know how to make or examine them : or to understand the extent of the 

 use of them ; contenting themselves with the tables of them as they find them, 

 without daring to question them, or caring to know how to rectify them, should 

 they be found amiss ; being, I suppose, under the apprehension of some great 

 difficulty therein. For the sake of such, the following tract is principally 

 intended, but not without hopes however to produce something that may be 

 acceptable to the most knowing in these matters. 



But first, it may be requisite to premise a definition of logarithms, in order 

 to render the ensuing discourse more clear, the rather because the old one 

 numerorum proportionalium eequi difFerentes comites seems too scanty to 

 define them fully. They may much more properly be said to be, numeri 

 rationum exponentes: wherein we consider ratio as a quantitas sui generis, 

 beginning from the ratio of equality, or 1 to 1 = O ; being affirmative when 

 the ratio is increasing, as of unity to a greater number, but negative when 

 decreasing ; and these rationes we suppose to be measured by the number of 

 ratiunculae contained in each. Now these ratiunculfc are so to be understood, 

 as in a continued scale of proportionals infinite in number, between the two 

 terms of the ratio, which infinite number of mean proportionals, is to that 

 infinite number of the like and equal ratiunculae between any other two terms, 

 as the logarithm of the one ratio, is to the logarithm of the other. Thus, if 

 there be supposed between 1 and 1 an infinite scale of mean proportionals, 

 whose number is 100000 &c. in infinitum ; between 1 and 2 there shall be 

 30102 &CC. of such proportionals, and between 1 and 3 there will be 47712 

 &c. of them, which numbers therefore are the logarithms of the rationes of 

 1 to 10, 1 to 2, and I to 3 ; and not so properly to be called the logarithms of 

 10, 2 and 3. 



This being laid down, it is obvious that if between unity and any number 

 proposed, there be taken any infinity of mean proportionals, the infinitely little 

 augment or decrement of the first of those means from unity, will be a ratiun 

 cula, that is, the momentum or fluxion of the ratio of unity to the said num- 



D 2 



