(99) 
Affirniativs ^hen the ratio is increafing, as of Unity to a 
greater Number, but Negative when decreafing j and thefe 
rationes we fuppofe to be nieafured by the Number of ratim- 
cula contained in each. Now thefe ratiuncula are (b to be 
underftood as in a continued Scale of Proportionals infinite 
in Number between the two terms of thcratio^ which infinite 
Number of mean Proportionals is to that infinite Number of 
the like and equal ratiuncula between any other two terms, 
as the Logarithm of the one ratio is to the Logarithm of the 
other. Thus if there be fuppofed between i and lo an infi- 
nite Scale of mean Proportionals, whofe Number is looooo &e. 
in infinitum I between i and 2 there fliall be 30102 &c. of 
fiich Proportionals, and between i and 3 there will be 
47712 &c. of them, which Numbers therefore are the Lo- 
garithms of the rationes of i to 10, i to 2, and i to 5 ; and 
not fo properly to be called the Logarithms of 10, 2 and 
This being laid down, it is obvious that if between Uni- 
ty and any Number propofed, there be taken any infinity 
of mean Proportionals, the infinitely little augment or decre- 
ment of the firft of thofe means from Unity, will be a ra- 
tiuncula^ that is, the momentum or Fluxion of the ratio of U^ 
nity to the faid Number : And feeing that in thefe continual ^ 
Proportionals all the ratiuncula are equal, their Sum, or the 
whole ratio will be as the faid momentum is diredly ; that is, 
the Logarithm of each ratio will be as the Fluxion thereof. 
Wterefore if the Root of any Infaiite Power be extraded ^ 
out of any Number, the difftrentiola of the faid Root from 
Unity, fliall be as the Logarithm of that Number. So that 
Logarithms thus produced may be of as many forms as you 
pleafe to aflUme infinite Indices of the Power whofe Root yoa 
feek : as if xhQ Index ba fuppofed looooo &c, infinitely, the 
Roots (hall be the Logarithms inven-ed by the Lord Napeir ; 
but if the faid Index- were 23025-85 &c Mr. Briggs's Loga- 
rithms would immediately be p odaced. And if yoa pleale • 
to flop at any number of Figures, and nor to continue chem - 
on, it will fuffice to alSame an Index of a Figure or two more ; 
than your intended Logarithm is to h as Mr. Briggs did, 
who to have his Log . richms true tc places, by continual i 
extraction of the Sq ia^e Root, at Vic ne to hive the Root 
of the i4o757488g5'yg28 f^. Powc ^ oat ho^ operofe that 
Extradion'was, will be cafiiy judged by whofb lhall under- 
take, to examine his Qakulm, Now 
