1

Let ABC, DEF, be two spheres, and BC, EF, their diameters, Book XII the sphere ABC is to the sphere DEF, in the triplicate ratio of BC to EF. If not, let the sphere ABC be to a sphere GHK less than the sphere DEF, in the triplicate ratio of BC to EF. Let this sphere GHK be inscribed within the sphere DEF; likewife, in DEF, infcribe a polyhedron, which shall not touch the fuperficies of the lesser sphere GHK 2. In the sphere ABC inscribe a a 17. polyhedron, fimilar and alike fituate to that in DEF; then these fimilar polyhedrons are to one another in the triplicate ratio of their diameters BC, EFb; but the sphere ABC, to the sphere & cor. 17. GHK, hath a triplicate ratio of BC to EF ; therefore the sphere c hyp. ABC is to the sphere GHK. as the polyhedron ABC to the fimilar polyhedron in DEF; but the sphere AB is greater than the polyhedron in it; therefore the sphere GHK is likewife greater than the polyhedron in DEF; but it is less, as contained in it; which is abfurd; therefore the sphere ABC, to the sphere less than DEF, has not a triplicate ratio of BC to EF. For the fame reafon, the sphere DEF, to a sphere less than ABC, has not a triplicate ratio of EF to BC. Again, the sphere ABC, to a fphere of LMN, greater than DEF, has not a triplicate ratio of BC to EF. If it can, then, by inverf. the sphere LMN, to the sphere ABC, shall have a triplicate ratio of the diameters EF to the diameter BC; but the sphere LMN is to the sphere ABC as the sphere DEF to fome sphere less than ABC, because the sphere LMN is greater than DEF; therefore the sphere DEF, to a sphere less than ABC, has a triplicate ratio of what EF has to BC; which is proved absurd; therefore the sphere ABC, to a sphere greater or less than DEF, has not a triplicate ratio of what BC has to EF. Therefore ABC has to the sphere DEF a triplicate ratio of what BC has to EF: Which was to be demonstrated.

THE

THE

PLAIN TRIGONOMETRY.

E business of trigonometry is to find the angles when

the fides are given, and the fides, or ratio of the fides, when the angles are given; and to find fides and angles, when fides and angles are given. For which, it is neceffary, that, not only the periphery of the circle, but likewise certain right lines in it, be fuppofed divided into fome determinate number of parts. The ancient geometers have supposed the periphery divided into 360 parts or degrees, and every degree into 60 minutes, and every minute into 60 seconds, &c.; and every angle is faid to be of fuch a number of degrees and minutes as there are in that part of the periphery measuring the angle.

I.

An arch is any part of the periphery or circumference, and is the measure of the angle at the center which it fubtends.

The quadrant of a circle is one fourth part of the circumference; the difference of an arch from a quadrant or go degrees, is called the complement of that arch.

III.

A chord or fubtense, is a right line drawn from one part of