*

away.

and Vertices the Point A. I fay, the faid Polyhedron does not touch the Superficies of the Sphere, wherein the Circle FGH is. Let AZ be drawn ‡‡ 11. 11. from the Point A, perpendicular to the Plane of the quadrilateral Figure KBSO, meeting it in the Point Z, and join BZ, ZK. Then fince AZ is perpendicular to the Plane of the quadrilateral Figure KBSO, it fhall alfo be perpendicular to all Right Lines that * Def. 3. 11. touch it, and are in the fame Plane. Wherefore AZ is perpendicular to BZ and ZK. And because AB is equal to AK, the Square of AB fhall be alfo equal to the Square of AK: And the Squares of AZ, ZŻB are † equal to the Square of AB. For the Angle at† 47. I. Z is a Right Angle. And the Squares of A Z, ZK, are equal to the Square of AK. Therefore the Squares of AZ, ZB, are equal to the Squares of AZ, ZK. Let the common Square of AZ be taken a And then the Square of BZ remaining, is equal to the Square of ZK remaining: And fo the Right Line BZ is equal to the Right Line ZK. After the fame Manner we demonftrate that Right Lines drawn from the Point Z to the Points O, S, are each equal to BZ, ZK. Therefore a Circle defcribed about the Center Z, with either of the Distances ZB, ZK, will also pafs thro' the Points O, S. And becaufe BKSO is a quadrilateral Figure in a Circle, and OB, BK, KS, are equal, and OS is lefs than BK; the Angle BZK fhall be obtufe; and fo BK greater than BZ. But GL alfo is much greater than BK. Therefore GL is greater than B Z. And the Square of GL is greater than the Square of BZ. And fince AL is equal to AB, the Square of AL fhall be equal to the Square of AB. But the Squares of AG, GL, together, are equal to the Square of AL, and the Squares of BZ, ZA, together, equal to the Square of AB: Therefore the Squares of AG, GL, together, are equal to the Squares of B Z, ZA, together: But the Square of BZ is lefs than the Square of GL: Therefore the Square of ZA is greater than the Square of AG; and fo the Right Line ZA will be greater than the Right Line AG. But AZ is perpendicular to one Bafe of the Polyhedron, and AG to the Superficies. Wherefore the Polyhedron does not touch the Superficies of the leffer Sphere. There

fore,

fore, there is defcribed a folid Polyhedron in the greater, of two Spheres having the fame Center, which doth not touch the Superficies of the leffer Sphere; which was to be demonstrated.

Coroll. Alfo if a folid Polyhedron be described in fome other Sphere, fimilar to that which is defcribed in the Sphere BCDE; the folid Polyhedron described in the Sphere BCDE, to the folid Polyhedron described in that other Sphere, fhall have a triplicate Proportion of that which the Diameter of the Sphere BCDE hath to the Diameter of that other Sphere. For the Solids being divided into Pyramids, equal in Number and of the fame Order, the fame Pyramids fhall be fimilar. But fimilar Pyramids are to each other in a triplicate Proportion of their homologous Sides. Therefore the Pyramid whofe Bafe is the quadrilateral Figure KBOS, and Vertex the Point A, to the Pyramid of the fame Order into the other Sphere, has a triplicate Proportion of that which the homologous Side of one, has to the homologous Side of the other; that is, which AB, drawn from the Center A of the Sphere, to that Line which is drawn from the Center of the other Sphere. In like Manner, every one of the Pyramids, that are in the Sphere whofe Center is A, to every one of the Pyramids of the fame Order in the other Sphere, hath a triplicate Proportion of that which AB has to that Line drawn from the Center of the other Sphere. And as one of the Antecedents is to one of the Confequents, fo are all the Antecedents to all the Confequents. Wherefore the whole folid Polyhedron, which is in the Sphere described about the Center A, to the whole folid Polyhedron that is in the other Sphere, hath a triplicate Proportion of that which AB hath to the Line drawn from the Center of the other Sphere; that is, which the Diameter BD has to the Diameter of the other Sphere.

PRO

PROPOSITION XVIII.

THEOREM.

Spheres are to one another in a triplicate Proportion of their Diameters.

S

Uppofe ABC, DEF, are two Spheres, whofe Diameters are BC, EF. I fay, the Sphere ABC to the Sphere DEF has a triplicate Proportion of that which BC has to EF.

*

*

Cor. to the laft Prop

For if it be not fo, the Sphere ABC to a Sphere either leffer or greater than DEF, will have a triplicate Proportion of that which BC has to EF. First, let it be to a leffer as GHK. And fuppofe the Sphere DEF to be defcribed about the Sphere GHK; and let there be defcribed a folid Polyhedron in the great- 17 of this. er Sphere DEF, not touching the Superficies of the leffer Sphere GHK; alfo let a folid Polyhedron bet described in the Sphere ABC, fimilar to that which is defcribed in the Sphere DEF. Then the folid Polyhedron in the Sphere ABC, to the folid Polyhedron in the Sphere DEF, will have † a triplicate Proportion of that which BC has to EF: But the Sphere' ABC to the Sphere GHK, hath a triplicate Proportion of that which B C hath to EF. Therefore as the Sphere ABC is to the Sphere GHK, fo is the folid Polyhedron in the Sphere A B C to the folid Polyhedron in the Sphere DEF; and (by Inverfion) as the Sphere ABC is to the folid Polyhedron that is in it, fo is the Sphere GHK to the folid Polyhedron that is in the Sphere DEF; but the Sphere ABC is greater than the folid Polyhedron that is in it. Therefore the Sphere GHK is alfo greater than the folid Polyhedron that is in the Sphere DEF, and alfo lefs than it, as being comprehended thereby, which is abfurd. Therefore the Sphere ABC to a Sphere less than the Sphere DE F, hath not a triplicate Proportion of that which BC has to EF. After the fame Manner it is demonftrated that the Sphere DEF to a Sphere less than ABC, has not a triplicate Proportion of that which EF has to BC. I fay, moreover, that the Sphere ABC to a Sphere greater than DE F, hath not a triplicate

plicate Proportion of that which BC has to EF; for, if it be poffible, let it have to the Sphere LMN greater than DEF. Then (by Inverfion) the Sphere LMN to the Sphere ABC, fhall have a triplicate Proportion of that which the Diameter EF has to the Diameter BC; but as the Sphere LMN is to the Sphere ABC, fo is 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, hath a triplicate Proportion to that which EF has to BC, which is abfurd, and has been before proved. Therefore the Sphere ABC to a Sphere greater than DEF, has not a triplicate Proportion of that which BC has to EF. But it has alfo been demonftrated, that the Sphere ABC to a Sphere less than DEF, has not a triplicate Proportion of that which BC has to EF. Therefore the Sphere ABC to the Sphere DEF, has a triplicate Proportion of that which BC has to EF; which was to be demonftrated,

FINI C.

THE

THE

ELEMENTS

Of Plain and Spherical

TRIGONOMETRY.

DEFINITIONS.

T

HE Bufinefs of Trigonometry is to find the Angles when the Sides are given, and the Sides, or the Ratio's of the Sides, when the Angles are given, and to find Sides and Angles, when Sides and Angles are given: In order to which, it is necessary that not only the Peripheries of Circles, but also certain Right Lines in and about Circles be fuppofed divided into fome determined Number of

Parts.

And fo the ancient Mathematicians thought fit to divide the Periphery of a Circle into 360 Parts (which they call Degrees;) and every Degree into 60 Minutes, and every Minute into 60 Seconds: And again, every Second into 60 Thirds, and fo on. And every Angle is faid to be of fuch a Number of Degrees and Minutes, as there are in the Arc measuring that Angle.

There are fome that would have a Degree divided into centefimal Parts, rather than fexagefimal ones: And it would perhaps be more useful to divide, not only a Degree, but even the whole Circle in a decuple Ratio; which Divifion may fome time or other gain Place. Now, if a Circle contains 360 Degrees, a Quadrant thereof, which is the Measure of a Right Angle, will T

be

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