Page:Philosophical Transactions - Volume 004.djvu/16

 which troubled all Greece. The Solution of which Probleme in Geometry may be compared to that with the giving of the Cube-root of any Number proposed in Arithmetick: For, in Arithmetick, the first of two continual Proportionals between an Unit and any Number proposed, is the Cube-root of that Number, and the Unit in Arithmetick is represented by a Line in Geometry, which is one of the Extreams.

Concerning this Probleme, the Author declares himself to be none of those, that search for that which cannot be found, to wit, to perform it by Right Lines and a Circle. 'Tis true indeed, it may be so done, to wit, by tryals and profers; as, who cannot in that manner divide an Arch into three Equal parts? But such Mechanismes are accounted ageometrick; and such operations may be well resembled to the vulgar Rule of False Position in Arithmetick, which cannot give an absolute true Resolution of one of the meanest of Questions, when the thing fought is Multiplex of it self, or Involved; for instance, what Number is that, which multiplyed in it self makes 9; who knoweth it not to be 3? But who can find it to be absolutely so by the aid of the ordinary rules of False Position, wherein the Extraction of a Square Root is not prescribed?

The Author observes; that amongst those, that solve this Probleme by the Conick Sections, they seem to have afforded fewer Effections thereof, than there have been Ages, since it was first proposed. Very few by ayd of a Circle and an Hyperbola or Parabola: by a Circle and Ellipsis none, that he could observe to have been published,

The which the Author considering, and studying how to supply, he found out not onely one, but infinite such Effections, and that not in one Method, but many; following the guidance of which Methods, by the like felicity he hath constructed all solid Problems infinite ways, by a Circle and an Ellipsis or Hyperbola.

1. His general Methods for finding two Means, by a Circle and either an Hyperbola or Ellipsis, are laid down in Prop. 1, 2, 16, and in this 16 Prop. he sheweth to do it with any Ellipsis and a Circle.

2. Particular Effections for finding but one or both of the Rh