Page:Philosophical Transactions - Volume 004.djvu/17

 Means, and Doubling the Cube, in Prop. 3. to 6,

3. And albeit all Cubick Æquations may be solved, either by the finding of two Means, or the Trisection of an Angle, yet he shews the Extent of his Method, in finding out other Infinite ways for the doing thereof, from Prop. 7. to 12.

4. The Trisection of an Angle by a Circle and Hyperbola, Prop, 13, and by a Parabola instead thereof, Prop. 15. And the finding of two Means by a Circle and Parabola, Prop. 14.

In the Second part of his Book De Analysis, the Author first gives you the Analysis or Algebra, whereby all his General Methods of finding two Means were invented. And afterwards, for the advancement of Geometry, gives you the Analysis, that relates to his particular Methods, as in case you would find but one of those Means, and afterwards by an easie operation the other. After that, he comes to shew, how the Effections or Delineations for Cubick Æquations were invented; And then, how those Constructions for the Trisection of an Angle were found out: the use whereof is, to give Lines in a known measure, equal to the quantity's sought, whereby either to give aid in the easie obtaining the first and second figures of the root, or controul the same.

Lastly, he comes to treat of General Constructions for the resolving of all solid Problems, without reduction of the Æquations proposed; and sheweth a general Construction for all Cubick and Bi-quadratick Æquations by ayd of a Circle and a Parabola, letting Ordinates fall from the points of Intersection on some Diameter of the Parabola (which is always parallel to the Axis,) whereas Des Chartes letting those Ordinates always fall upon the Axis, was forced to prepare and alter the Æquations by driving out or taking away the second term (which is next the highest,) that the sum of the Negative roots might be equal to the sum of the Affirmative ones, as his Constructions always require.

But how to find out all the variety's of solving all Solid Problems by the Conick Sections, hear the Author to the Reader: Methodum non adscripsi, tum quad gratius ac utilius futurum arbitratus sum, si eam ipse privato Studio, ex hisce Speciminibus eliceres, tum etiam quad judicium tuum de tota re præstolarer, Decrevi enim, Rh