Page:Philosophical Transactions - Volume 004.djvu/45

 relate as projected, and proceed to the revolution of the Problem propos'd: And how to fit in those Sections, see the 7. books of Apollonius, Mydorgius, the 3d, Volume of Des Cartes's Letters, ''Leotaudi Geometrica practica, Andersonii Exerciat. Geometricæ.''

As to the Problem it self, it is determin'd on the Sphere by the Intersections of the two lesser Circles of Distance, whole Poles are the known Starrs, And this Problem hath divers Geometrick ways of revolution.

1. By Plain Geometry (in the sense before mentioned;) Supposing a Plain to touch the Sphere at the North-pole: if the Eye be at the South-pole, projecting those Circles into the said Plain, they are still Circles (by reason of the sub-contrary Sections of the Visual Cones) whose Centers fall in the sides of the Right-lin'd Angle, made by the Projected Meridians, that pass through the known Starrs; and thus the Problem is easily solv'd in this manner.

2, If it be required to be performed by Conick Geometry; in one case it may be done, by placing the Ey at the Center of the Sphere, and projecting as before; to wit, when the longer Axes of the figures being produced concur above the Vertex; Here the Problem is determined by the Intersections of two Conick Sections (whereof a Circle cannot be one, unless its Center be in the Axis of the other figure.) And in this second Case these points of Intersection fall in the same right line or projected Meridian, they did before, but at a more remote distance from the Pole-point, to wit, in the former Supposition, the Solar distance was measur'd by a Right line, that was the double Tangent of half the Arch; here it is the Tangent of the whole Arch. Hence it is evident, how one Projection may beget another, yea infinite others, altering the Scale; and how the lesser Circles in the Stereographick Projection help to describe the Conick Sections in the Gnomonick Projection: But (to reduce the matter to one common radius) if we suppose two Spheres equal, and so placed about the same Axis, that the Pole-point of the one shall pass through the Center of the other, and the Touch-plain to pass through the said Center or Pole-point, and that a lesser Circle hath the same position in the one as in the other; Then, if the Ey be at the South-Pole of the one, it is at the Center of the other, and any projected Meridian drawn from the projected Pole-point to pass through both, the projections of these Rh