Page:Philosophical Transactions - Volume 003.djvu/63

 be accounted a-geometrical, because they are not perform'd by the Sole aid of Ruler and Compass; which he suggests to be well observ'd à Subtilissimo Mathemetico D. Carolo Reinaldino in Geometra suo promoto, dum tractat de novis illis Lineis, quas Mediceas appellat: Concerning which Author, he faith thus, p, 132. Qui autem desiderat plenum Analyseos & Æquationum doctrinam, expectet absolutissimum Caroli Reinaldini Opus de Resolutione & Compositione Mathematica, quod nunc sub prælo est. It seems, that the Book will be three large Volums in folio; the first whereof, being Introductory, and containing the Algebra of the Antients, is already in England.

And for the Confirmation of what hath been asserted, the Author thus demonstrates, that no Cubick Æquation (that is irreducible to a Quadratick) can be resolv'd by the sole aid of Ruler and Compass. For every Cubick Æquation hath either but one or three real roots, which if they could be found by the said sole aid of Rule and Compass, or by the Intersection of a Circle and a Right Line, then a Right Line should cut a Circle either in one Point or three; either of which is most absurd. And for the like reason a Cubick Æquation, having three real roots, can never be reduced to a pure Æquation, which hath but one onely root; for in these Æquations, Reduction shall no wise profit, forasmuch as 'tis impossible, by aid thereof to change an Imaginary root into areal one, and the Converse.

As to the Argument of the Book it self, it contains these several Heads.

1. The Mensuration of sundry Solids, with General Methods to that purpose; concerning which the Author faith, p. 123. ''Totus namque Archimedis Tractatus de Sphæra & Cylindro facilè demonstratur ex hujus 3. ad modum hujus 46. & aliquot sequentium. Liber de Conoidibus & Sphæeroidibus, & tota Lucæ Valerii doctrina, ex hujus 21. Tota Guldini, Johannis de la Faille, & AndreæTacqueti doctrina, ex hujus 35. & aliquot sequentium''. And as a Corollary of Prop. 62. he Cubeth or measureth either of the Segments of a Parabolical Conoid cut with a Plain, parallel to the Axis. Hence we observe, that supposing such a Segment, again cut with a Plain, erect to the former Plain, the Proposition may be well apply'd to the Gauging of Cask part out, Rh