Page:Philosophical Transactions - Volume 004.djvu/42

 by pairs, you shall create a rank of Numbers whose 2d. differences are equal; and if by ternaries, then the 3d. differences of those Products shall be equal. And how to find the greatest Product of an Arithmetical Progression of any number of terms having any common difference assign'd, contain'd in any Number proposed, is shew'd by Pascal in his Tract du Triangle Aritbmetique, where he apply's it to the Extraction of the Roots of simple powers.

4. It appears, How this rank may be caried easily by Addition, till you have a Resolvend either equall or greater or lesse, than that proposed.

5. When you have a Majus and Minus, you may interpole as many more termes in the Arithmetical Progression as you will, that is to say, Subdivide the Common difference in the Arithmetical Progression, and render it lesse; and then renew, and find the Resolvends, which are easily obtain'd out of the Powers and their Coefficients, which are suppos'd knowe, and may be readily rais'd from a Table of Squares and Cubes, &c. with which kind the Reader may be furnisht in Guldini Centrobaryca and Babingtons Fireworks: By this means you may obtain divers Figures of the Root; and then the General Method of Vieta and Harriot runs away more easily, and is so far improv'd, that after any figure is plac'd in the Root, most certain Characters are given to know by aide of the subsequent Dividend and Divisor, Whether the figure before assum'd be too great or too small: or lastly it may well be concluded, that, as in Logarithmes, when you propose, such an one as is not absolutely given in the Canon, you doe by Proportional Work, using the aid of their first differences (when their Absolute Numbers differ by Unit) find the absolute Number true to 5. or 6. places further than the Canoon gives it (the reason whereof is, that the first Differences doe likewise agree to about the same Number of places;) that I say, the like maybe done in, Æquations, after divers of the first figures of the root are found; provided there be the like agreement in the first differences of the interpoled Resolvends.

Moreover we ought here remake notice of a more subtle kind of Interpolation, common to all gradual Ranks or Progressions of Numbers, wherein Differences happen to be equal: Of which Rh