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they may much moreproperly be said to be Numeri Ultionum Exsonent es ; w he reinRatio is confider’d as a Quantitas fui generis, beginning from the Ratio of equality,or i to i — o ; being affirmative when the Ratio is increasmg, as of Unity toa greater Number, but Negative when decreasing; and these Rationes wesopposc to be measored by the Number of Ratiunculae contained in each.Now these Ratiunculae are so to be understood as in a continued Scaleof Proportionals, infinite in Number between the two terms of the Ratio, which in-finite Number of mean Proportionais is to that infinite Number of the likeand equal Ratiuncula between any othertwo terms, as the Logarithm of theone Ratio is to the Logarithm of the other. Thus if there be sopposed be.tween i and io an infinite Scale of mean Proportionais, whose Number isiooooo, &c. in infinitum ; between i and 2 there {hall be joioa,&c. of suchProportionais, and between i and z there will be 47712, &c. of them,which Numbers therefore are the Logarithms of the Rationes, of 1 to 10,1 to2, and i to z ; and not so properly to be called the Logarithms of 1 o, 2,and z. _

This being laid down, it is obvious that if between Unity and anyNumber propol'ed, there be taken any infinity of mean Proportionais, the in-finitely lrttle Augment or Decrement of the first of those Means from Unity,will be a Ratiuncula, that is, the Momentum or Fluxion of the Ratio of U-nity to the seid Number : And seeing that in these Continual Proportionais allthe Ratiunculae are equal, their Sum, or the whole Ratio will be as the seidMomentum is directly ; that is, the Logarithm of each Ratio will be as theFluxion thereof. Wherefore if the Root of any Infinite Power be extractedout of any Number, the DisserentiqJa of the seid Root from Unity, {hall beas the Logarithm of that Number/',So that Logarithms thus produced, maymay be of as many forms as you please, to aflume Infinite Indices of thePower whose Root youseek: as ifthe Index be supposed iooooo, &c. infinite-ly, the Roots {hall be the Logarithms invented by the Lord Napicr ; but iftheseid Index were 2302787, See. Mr. Briggs Logarithms would immedfatelybe produced. And if you plcase to stop at any Number of Figures, and notto continue them on, it will suffice to aflume an Index of a Fi-gvire or two more than your intended Logarithm is-to have, as - Mr. Briggsdid, who to have his Logarithms true to i4.places, by continual extraction ofthe Square Root, at last came to have the Root ofthe 1407374883773285/,.Power; but how operose that extraction was, will be easily judged by whose»{hall undertake to examine his Calculus.

Now,though the Notion of an infinite Power mayßem very stränge,and tothosethat know the difficulty of the Extraction of the Roots of High Powers, perhapsimpracticable; yetby t'nehelp of that admirablelnvention of Mr. Newton, where-by he determines the Unciae or numbers prefixt to the members composing Pow-ers, (on which chiefly depends the DoRrinc of Serics) the Infinity of the In-dex contributes to render the Expression much more easy : For if the InfinitePower to bc resolved be put (aftcr Mr. Newtons Method)

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