The Way To Geometry

by Petrus Ramus

Because by the [1 e vj], like plains have a doubled reasó of their homologall sides. But in rectilineals inscribed the diameters are the homologall sides, or they are proportionall to their homologall sides. As let the like rectangled triangles be aei, and ouy; Here because ae and ou, are the diameters, the matter appeareth to be plaine at the first sight. But in the Obliquangled triangles, sei, and ruy, alike also, the diameters are proportionall to their homologall sides, to wit, ei and uy. For by the grant, as se is to ru: so is ei to uy, And therefore, by the former, as the diameter ea and uo.

In like Triangulates, seeing by the [4 e x], they may be resolved into like triangles, the same will fall out.

Therefore

4. If it be as the diameter of the circle is unto the side of rectilineall inscribed, so the diameter of the second circle be unto the side of the second rectilineall inscribed, and the severall triangles of the inscripts be alike and likely situate, the rectilinealls inscribed shall be alike and likely situate.

This Euclide did thus assume at the 2 p xij, and indeed as it seemeth out of the 18 p vj. Both which are conteined in the [23 e iiij]. And therefore we also have assumed it.

Adscription of a Circle is with any triangle: But with a triangulate it is with that onely which is ordinate: And indeed adscription of a Circle is common to all.

5. If two right lines doe cut into two equall parts two angles of an assigned rectilineall, the circle of the ray from their meeting perpendicular unto the side, shall be inscribed unto the assigned rectilineall. 4 and 8. p. iiij.

As in the Triangle aei, let the right lines ao, and eu, halfe the angles a and e: And from y, their meeting, let the perpendiculars unto the sides be yo, yu, ys; I say that the center y, with the ray yo, or ya, or ys, is the circle inscribed, by the [17 e xv]. Because the halfing lines with the perpendiculars shall make equilater triangles, by the [2 e vij]. And therefore the three perpendiculars, which are the bases of the equilaters, shall be equall.