ERRATA for Book XI. and XII. B. XI. pr. 2. for def. 4.11. r. def. 4.1. pr. 21. for note a2 r. ażo. pr. 240 for note, b 23.1 r. b 33. I p. 123. l. 13. for P. r. PL. B. XII. pr. 7. 1. 5. for ABCD. r. ABED. p, 145. 1. 24. for note 6.11. r. 7.1Io 1, 25. for ZV r. QV. and 1. 31. for note 7.11 r. 6.11. DE A F M 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 fphere ABC be to a fphere GHK lefs than the sphere DEF, in the triplicate ratio of BC to EF. Let this fphere GHK be infcribed within the fphere DEF; likewife, in DEF, infcribe a polyhedron, which fhall not touch the fuperficies of the leffer fphere GHK 2. In the fphere ABC inscribe a a 17. polyhedron, fimilar and alike fituate to that in DEF; then thefe fimilar polyhedrons are to one another in the triplicate ratio of their diameters BC, EF ; but the sphere ABC, to the sphere b cor. 17. GHK, hath a triplicate ratio of BC to EF; therefore the sphere c hyp. ABC is to the fphere GHK. as the polyhedron ABC to the fimilar polyhedron in DEF; but the fphere AB is greater than the polyhedron in it; therefore the fphere GHK is likewife greater than the polyhedron in DEF; but it is lefs, as contained in it; which is abfurd; therefore the sphere ABC, to the sphere lefs than DEF, has not a triplicate ratio of BC to EF. For the fame reafon, the fphere DEF, to a fphere lefs 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 fphere LMN, to the sphere ABC, fhall have a triplicate ratio of the diameters EF to the diameter BC; but the fphere LMN is to the fphere ABC as the sphere DEF to fome fphere lefs than ABC, because the sphere LMN is greater than DEF; therefore the fphere DEF, to a sphere less than ABC, has a triplicate ratio of what EF has to BC; which is proved abfurd; therefore the sphere ABC, to a fphere greater or less than DEF, has not a triplicate ratio of what BC has to EF. Therefore ABC has to the fphere DEF a triplicate ratio of what BC has to EF: Which was to be demonftrated. THE THE ELEMENTS O F PLAIN AND SPHERICAL TRIGONOMETRY. TH PLAIN TRIGONOMETRY. HE bufinefs 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 likewife certain right lines in it, be fuppofed divided into fome determinate number of parts. The ancient geometers have fuppofed the periphery divided into 36 parts or degrees, and every degree into 60 minutes, and every minute into 60 feconds, &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. II. 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 fubtenfe, is a right line drawn from one part of an arch to another. IV. The |