Page:Philosophical Transactions - Volume 003.djvu/14

 that Science: that in it the Author hath delivered a new Method Analytical for giving the Aggregate of an Infinite or Indefinite converging Series: and that from that ground he teaches a Method of Squaring the Circle, Ellipsis, and Hyperbole, by an Infinite Series, thence calculating the true dimensions as near as you please: And lastly, that by the same method from the Hyperbola he calculateth both the Logarithms of any Natural Number assign'd, and vice versa, the Natural Number of any Logarithm given.

Only a few of these Books were printed by the Author for his own use, and that of his Friends, and a Copy sent over whereby to reprint it here, which is now a doing.

The Mathematical Mr. John Collins, upon a more particular examination of this Book, communicated what follows concerning the same.

The Author's Computation of the Area of a Circle agrees with the Numbers of Van Ceulen; and his computation of the supplemental spaces between the Hyperbola and its Asymptote by Parallels to the other Asymptote, is correspondent to what Gregory of St. Vincent and his Commentators Francis Aynscomb and Alphonse de Sarasa have demonstrated concerning the Logorithms, as represented by those spaces, viz. That if one Asymptote be divided into a rank of continual Proportionals, and if parallels to the other Asymptote be drawn passing through the said rank, and be terminated at the Hyperbola, the spaces contain'd between each such pair of Parallels, are equal to each other, and so added or conceived to be one continued space, may represent the Logarithms; or the said Proportionals, fitted in parallel to the divided Asymptote, do the like, by reason that a Rectangle apply'd to the several Terms of a Geometrical Progression increasing, renders another in the same Ratio decreasing. And both performed by the above-mentioned Analytical method of conveying complicated Polygons circumscrib'd and inscrib'd in the sector of a Circle, Ellipsis, or Hyperbola, which he asserts to be quantities like Surds, not absolutely to be express'd in Numbers.

And it being manifest, that the making of the Table of Logarithms is in effect the same thing as the computing of Area's of those supplemental spaces, the Author accordingly applies it thereto