'The Method of FLUXIONS,

prefented, the firft being general and indefinite, and the other definite and particular : I cannot but wonder that no body has thought of accommodating the lately-difcover'd Doctrine of Decimal Frac- tions in like manner to Species, (unlels you will except the Qua- drature of the Hyberbola by Mr. Nicolas Mercator ;) efpecially fince it might have open'd a way to more abftrufe Discoveries. But iince this Doctrine of Species, has the fame relation to Algebra, as the Doctrine of Decimal Numbers has to common Arithmetick ; the Operations of Addition, Subtraction, Multiplication, Divifion, and Extraction of Roots, may eafily be learned from thence,, if the Learner be but fk.ill'd in Decimal Arithmetick, and the Vulgar Algebra, and obferves the correfpondence that obtains be- tween Decimal Fractions and Algebraick Terms infinitely continued. For as in Numbers, the Places towards the right-hand continually decreafe in a Decimal or Subdecuple Proportion ; fo it is in Species refpedtively, when the Terms are difpofed, (as is often enjoin 'd in what follows,) in an uniform Progreflion infinitely continued, according to the Order of the Dimenfions of any Numerator or Denominator. And as the convenience of Decimals is this, that all vulgar Fractions and Radicals, being reduced to them, in fome meafure acquire the nature of Integers, and may be managed as fuch ; fo it is a convenience attending infinite Series in Species, that all kinds of complicate Terms, ( fuch as Fractions whofe Denominators are compound Quantities, the Roots of compound Quantities, or of affected Equations, and the like,) may be reduced to the Clafs of fimple Quantities ; that is, to an infinite Series of Fractions, whofe Numerators and Denominators are fimple Terms ; which will no longer labour under thofe difficulties, that in the other form feem'd almoft infuperable. Firft therefore I mail fhew how thefe Re- ductions are to be perform'd, or how any compound Quantities may be reduced to fuch fimple Terms, efpecially when the Methods of computing are not obvious. Then I fhall apply this Analyfis to the Solution of Problems. 3. Reduction by Divifion and Extraction of Roots will be plain from the following Examples, when you compare like Methods of Operation in Decimal and in Specious Arithmetick. Examples