fore, there is described a solid Polyhedron in the greater, of two Spheres having the same Center, which doth not touch the Superficies of the lefser Sphere; which was to be demonstrated. Coroll. Also if a solid Polyhedron be described in some other Sphere, fimilar to that which is described in the Sphere BCDE; the folid Polyhedron described in the Sphere BCDE, to the solid Polyhedron described in that other Sphere, shall 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 same Order, the same Pyramids shall be similar. But similar Pyramids are to each other in a triplicate Proportion of their homologous Sides. Therefore the Pyramid whose Base 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 whose 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 Consequents, so are all the Antecedents to all the Consequents. Wherefore the whole folid Polyhedron, which is in the Sphere described about the Center A, to the whole solid 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 B D has to the Diameter of the other Sphere. PROPOSITION XVIII, THEOREM. tion of their Diameters. Uppose ABC, DEF, are two Spheres, 'whose Diameters are BC, EF. I say, the Sphere ABC to the Sphere DEF has a triplicate Proportion of that which B C has to EF. For if it be not for 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 lesser as GHK. And suppose the Sphere DEF to be described about the Sphere GHK; and let there be described * a solid Polyhedron in the great- * 17 of this. er Sphere DEF, not touching the Superficies of the lesser Sphere GHK; also let a solid Polyhedron be described in the Sphere ABC, similar to that which is described in the Sphere DEF. Then the solid Polyhedron in the Sphere ABC, to the folid Polyhedron in the Sphere DE F, will have t a triplicate Propor- Cor. to the tion of that which BC has 'to EF: But the Sphere laft Prop. 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, so is the solid Polyhedron in the Sphere A B C to the solid Polyhedron in the Sphere DEF; and (by Inverfion) as the Sphere ABC is to the solid 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 also greater than the folid Polyhedron that is in the Sphere DEF, and also less than it, as being comprehended thereby, which is absurd. Therefore the Sphere ABC to a Sphere less than the Sphere DEF, hath not a triplicate Proportion of that which BC has to EF. After the same Manner it is demonstrated that the Sphere DEF to a Sphere less than ABC, has not a triplicate Proportion of that which EF has to BC. I say, moreover, that the Sphere ABC to a Sphere greater than DEF, hath not a tri plicate plicate Proportion of that which BC has to EF; for, if it be possible, let it have to the Sphere LMN greater than DEF. Then (by Inversion) the Sphere LMN to the Sphere ABC, shall 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, so is the Sphere DEF to some 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 absurd, 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 also been demonstrated, 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 B C has to EF; which was to be demonstrated, 275 Τ Η Ε ELEMENTS Of Plain and Spherical TRIGONOMETRY. DEFINITIONS. T HE Business of Trigonometry is to find 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 supposed divided into some determined Number of Parts. And to the ancient Mathematicians thought fit to divide the Periphery of a Circle into 360 Perts (which they call Degrees;) and every Degree into 60 Minutes, and every Minute into 60 Seconds : And again, every Second into 60 Thirds, and so on. And every Angle is said to be of such a Number of Degrees and Minutes, as there are in the Arc measuring that Angle. There are some that would have a Degree divided into centesimal Parts, rather than sexagesimal ones : And it would perhaps be more useful to divide, not only a Degree, but even the whole Circle in a decuple Ratio ; which Division may some time or other gain Place. Now, if a Circle contains 360 Degrees, a Quadrant shereof, which is the Measure of a Right Angle, will T be Mecalul be go of those Parts : And if it contains 100 Parts, a Quadrant will be 25 of these Parts. The Complement of an Arc is the Difference thereof from a Quadrant. A Chord, or Subtense, is a Right Line drawn from one End of the Arc to the other. ' The Right Sine of any Arc, which also is commonly called only a Sine, is a Perpendicular falling from one End of an Arc, to the Radius drawn through the other End of the said Arc. And is therefore the Semisubtense of double the Arc, viz. DE=DO, and the Arc DO is double of the Arc DB. Hence, the Sine of an Arc of 30 Degrees, is equal to the one half of the Radius. For (by 15. El. 4.) the side of an Hexagon infcribd in a Circle, that is, the Subtense of 60 Degrees is equal to the Radius. A Sine divides the Radius into two Segments CE, EB; one of which, CE, which is intercepted between the Center and the Right Sine, is the Sine Complement of the Arc D B to a Quadrant, (for CE=FD which is the Sine of the Arc DH,) and is called the Cofine. The other Segment BE, which is intercepted between the Right Sine and the Periphery, is called a versed Sine, and sometimes a Sagitta. And if the Right Line CG be produced from the Center C, thro' one End D of the Arc, until it meets the Right Line BG, which is perpendicular to the Diameter drawn thro' the other End B of the Ars, then CG is called the Secant, and BG the Tangent of the Arc D B. The Cosecant and Cotagent of an Arc is the Secant or Tangent of that Arc, which is the Complement of the former Arc to a Quadrant. Note, As the Chord of an Arc, and of its Complement to a Circle, is the Jame; fo likewise is the Sine, Tangent, and Secant of 18 'an Are the same as the Sine, Tangent, and Secant of its Complement to a Semicircle. The Sinus Totus is the greatest Sine, or, the Sine of 90 Degrees, which is equal to the Radius of the Circle. A Trigonometrical Canon is a Table, which, beginning from one Minute, orderly expresses the Lengths that every Sine, Tangent, and Secant, have in respect of the Radius, which is supposed Unity, and is conceived to be divided 10,000,000, or more decimal Parts. And fo |