Page:The method of fluxions and infinite series.djvu/26

2 presented, the first being general and indefinite, and the other definite and particular: I cannot but wonder that no body has thought of accommodating the lately-discover'd Doctrine of Decimal Fractions in like manner to Species, (unless you will except the Quadrature of the Hyberbola by Mr. Nicolas Mercator;) especially since it might have open'd a way to more abstruse Discoveries. But since this Doctrine of Species, has the same relation to Algebra, as the Doctrine of Decimal Numbers has to common Arithmetick; the Operations of Addition, Subtraction, Multiplication, Division, and Extraction of Roots, may easily be learned from thence, if the Learner be but skill'd in Decimal Arithmetick, and the Vulgar Algebra, and observes the correspondence that obtains between Decimal Fractions and Algebraick Terms infinitely continued. For as in Numbers, the Places towards the right-hand continually decrease in a Decimal or Subdecuple Proportion; so it is in Species respectively, when the Terms are disposed, (as is often enjoin'd in what follows,) in an uniform Progression infinitely continued, according to the Order of the Dimensions of any Numerator or Denominator. And as the convenience of Decimals is this, that all vulgar Fractions and Radicals, being reduced to them, in some measure acquire the nature of Integers, and may be managed as such; so it is a convenience attending infinite Series in Species, that all kinds of complicate Terms, (such as Fractions whose Denominators are compound Quantities, the Roots of compound Quantities, or of affected Equations, and the like,) may be reduced to the Class of simple Quantities; that is, to an infinite Series of Fractions, whose Numerators and Denominators are simple Terms; which will no longer labour under those difficulties, that in the other form seem'd almost insuperable. First therefore I shall shew how these Reductions are to be perform'd, or how any compound Quantities may be reduced to such simple Terms, especially when the Methods of computing are not obvious. Then I shall apply this Analysis to the Solution of Problems.

3. Reduction by Division 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