. The Elements of Euclid : viz. the first six books, together with the eleventh and twelfth : the errors, by which Theon, or others, have long ago vitiated these books, are corrected, and some of Euclid's demonstrations are restored : also, the book of Euclid's Data, in like manner corrected. a ir. 12. THE ELEMENTS PROP. XVIII. THEOR. SPHERES have to one another the triplicate ra-tio of that which their diameters have. Let ABC, DEF be two spheres, of which the diameters areBC, EF. The sphere ABC has to the spliere DEF the triplicateratio of that which BC has to EF. For, if it has not, the sphe
. The Elements of Euclid : viz. the first six books, together with the eleventh and twelfth : the errors, by which Theon, or others, have long ago vitiated these books, are corrected, and some of Euclid's demonstrations are restored : also, the book of Euclid's Data, in like manner corrected. a ir. 12. THE ELEMENTS PROP. XVIII. THEOR. SPHERES have to one another the triplicate ra-tio of that which their diameters have. Let ABC, DEF be two spheres, of which the diameters areBC, EF. The sphere ABC has to the spliere DEF the triplicateratio of that which BC has to EF. For, if it has not, the sphere ABC shall have to a sphere ei-ther less or greater than UEF, the triplicate ratio of that whichBC has to EF. First, let it have that ratio to a less, the sphere GHK ; and let the sphere DEF have the samecentre with GHK; and in the greater sphere DEF describe ^. bCor17. 12. c 14. 5. a solid polyhedron, the superficies of which does not meet thelesser sphere GHK; and in the sphere ABC describe anothersimilar to that in the sphere DEF: therefore the solid polyhe-dron in the sphere ABC has to the solid polyhedron in thesphere DEF, the triplicate ratio ^ of that which BC has to the sphere ABC has to the sphere GHK the triplicate ra-tio of that which BC has to EF; therefore, as the sphere ABCto the sphere GHK, so is the solid polyhedron in the sphere ABCto the solid polyhedron in the sphere DEF: but the sphereABC is greater than the solid polyhedron in it; therefore ^ al-so the sphere GHK is greater than the solid polyhedron in thesphere DEF: but it is also less, because it is contained withinit, which is impossible: therefore the sphere ABC has not toany sphere less than DEF the triplicate ratio of that whichBC has to EF. In the same manner it may be demonstrated,that the sphere DEF has not to any sphere less than ABC thetriplicate ratio o
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