ZK=ZB. In like manner we demonstrate = a are * AZ + BZ. But BZ GL, whence ZAAG, and so ZA - AG. But AZ is perpend. to one Base of the Polyhedron, and AG to the Superficies. Therefore the Superficies BOSK does not touch the Superficies of the lesser Sphere; and the same may be demonstrated of the other Planes of the Polyhedron. Q. E. D. COROL. Also if a folid 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 folid 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 said Pyramids shall be similar. But similar Pyramids are to each other in a triplicate Propor- tion of their homologous Sides. Therefore the Pyramid whose Base is the quadrilateral Figure KBOS, and Vertex the Point A, to the Pyramid of the same Order in 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 Centre 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 same 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, fo 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 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. PROP. XVIII. Spheres BAC, EDF are to one another in the Triplicate Proportion of their Diameters BC, EF. For if you deny it, let the Sphere BAC be to the Sphere G in the triplicate Proportion of the Diameter BC to the Diameter EF. I say G = EDF. For if possible let G EDF. And Q P 12. byp. d 14. 5. And suppose the Sphere G to be Concentrical 17. 12. with the Sphere EDF. Inscribe a Polyhedron in the Sphere EDF, not touching the Sphere G, and a similar Polyhedron in the Sphere BAC. cor. 17. These Polyhedrons are b in the triplicate Proportion of the Diameters BC, EF, that is of the Sphere BAC to G. And confequently the Sphere G is greater than the Polyhedron infcrib'd within the Sphere EDF; the Part to the Whole. Again, if possible, let the Sphere GEDF. And suppose as the Sphere EDF to some other Sphere H, so is G to BAC. That is, in the triplicate Proportion of the Diameters EF to BC; therefore since BACH, we fall into the Absurdity of the former part, and so the Sphere G=EDF. Q. E. D. e byp. In vers. € 14.5. e COROL. Hence, as one Sphere to another, so is a Po lyhedron described in the one, to a similar Poly hedron described in the other. FINIS, THE HE Merchant's Magazine; or Tradesman's Treasury: Con1. Arithmetick in Whole Numbers and Fractions, Vuigar and Decimal, &c. 2. Merchants Accounts, &c. 3. Book• Keeping, after a plain, easy, and natural Method, &c. 4. Maxims concerning Bills of Exchange, Factors and Factorage, &c. 5. The Port of Letters to and from foreign Countries, &c. 6. An Account of the Commodities produc'd by all Countries, c. 7. 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