COR. 2. Prisms of equal altitudes are to one another as their bases; because the pyramids upon the same bases, and of the same altitude, are to one another as their bases.

Similar pyramids, having triangular bases, are one to another in the triplicate ratio of that of their homologous sides.

COR. From this it is evident, that similar pyramids which have multangular bases, are likewise to one another in the triplicate ratio of their homologous sides: for they may be divided into similar pyramids having triangular bases, because the similar polygons, which are their bases, may be divided into the same number of similar triangles homologous to the whole polygons: therefore, as one of the triangular pyramids in the first multangular pyramid is to one of the triangular pyramids in the other, so are all the triangular pyramids in the first to all the triangular pyramids in the other; that is, so is the first multangular pyramid to the other: but one triangular pyramid is to its similar triangular pyramid, in the triplicate ratio of their homologous sides; and therefore the first multangular pyramid has to the other, the triplicate ratio of that which one of the sides of the first has to the homologous side of the other.

PROP. IX. THEOREM.

The bases and altitudes of equal pyramids having triangular bases are reciprocally proportional: and, conversely, triangular pyramids of which the bases and altitudes are reciprocally proportionals, are equal to one another.

Every cone is the third part of a cylinder which has the same base, and is of an equal altitude with it.

Cones and cylinders of the same altitude are to one another as their bases.

PROP. XII. THEOREM.

Similar cones and cylinders have to one another the triplicate ratio of that which the diameters of their bases have.

PROP. XIII. THEOREM.

If a cylinder be cut by a plane parallel to its opposite planes, or bases; it divides the cylinder into two cylinders, one of which is to the other as the axis of the first to the axis of the other.

PROP. XIV. THEOREM.

Cones and cylinders upon equal bases are to one another as their altitudes.

PROP. XV. THEOREM.

The bases and altitudes of equal cones and cylinders, are reciprocally proportional; and, conversely, if the bases and altitudes be reciprocally proportional, the cones and cylinders are equal to one another.

PROP. XVI. PROBLEM.

In the greater of two circles that have the same centre, to inscribe a polygon of an even number of equal sides, that shall not meet the less circle.

LEMMA II.

If two trapeziums ABCD, EFGH be inscribed in the circles, the centres of which are the points K, L; and if the sides AB, DC be parallel, as also EF, HG; and the other four sides AD, BC, EH, FG, be all equal to one another; but the side AB greater than EF, and DC greater than HG; the straight line KA from the centre of the circle in which the greater sides are, is greater than the straight line LE drawn from the centre to the circumference of the other circle.

PROP. XVII. PROBLEM.

In the greater of two spheres which have the same centre, to describe a solid polyhedron, the superficies of which shall not meet the less sphere.

COR. And if in the less sphere there be inscribed a solid polyhedron, by drawing straight lines betwixt the points in which the straight lines from the centre of the sphere drawn to all the angles of the solid polyhedron in the greater sphere meet the superficies of the less; in the same order in which are joined the points in which the same lines from the centre meet the superficies of the greater sphere; the solid polyhedron in the sphere BCDE has to this other solid polyhedron the triplicate ratio of that which the diameter of the sphere BCDE has to the diameter of the other sphere. For if these two solids be divided into the same number of pyramids, and in the same order, the pyramids shall be similar to one another, each to each: because they have the solid angles at their common vertex, the centre of the sphere, the same in each pyramid, and their other solid angles at the bases equal to one another, each to each, because they are contained by three plane angles, each equal to each; and the pyramids are contained by the same number of similar planes; and are therefore similar to one another, each to each: but similar pyramids have to one another the triplicate ratio of their homologous sides. Therefore the pyramid of which the base is the quadrilateral KBOS, and vertex A, has to the pyramid in the other sphere of the same order, the triplicate ratio of their homologous sides, that is, of that ratio which AB from the centre of the greater sphere has to the straight line from the same centre to the superficies of the less sphere, and in like manner, each pyramid in the greater sphere has to each of the same order in the less, the triplicate ratio of that which AB has to the semidiameter of the less sphere. And as one antecedent is to its consequent, so are all the antecedents to all the consequents. Wherefore the whole solid polyhedron in the greater sphere has to the whole solid polyhedron in the other, the triplicate ratio of that which AB the semidiameter of the first has to the semidiameter of the other; that is, which the diameter BD of the greater has to the diameter of the other sphere.

PROP. XVIII. THEOREM.

Spheres have to one another the triplicate ratio of that which their diameters have.

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