the three parallelograms DEFP, HEFR, DEHX, and those opposite to them. And because the pyramid

ABCG is similar to the pyramid DEFH, the angle

*

11.

ABC is equal to the angle DEF, and the angle GBC * 11 Def. to the angle HEF, and ABG to DEH: and AB is 1 Def.

to BC, as DE to EF; that is, the sides about the equal 6.

* 24. 11.

angles are proportionals; wherefore the parallelogram BM is similar to EP: for the same reason, the parallelogram BN is similar to ER, and BK to EX: therefore the three parallelograms BM, BN, BK are similar to the three EP, ER, EX: but the three BM, BN, BK, are equal and similar to the three which are opposite to them, and the three EP, ER, EX equal and similar to the three opposite to them: wherefore the solids BGML, EHPO are contained by the same number of similar planes; and their solid angles* are * B. 11. equal; and therefore the solid BGML is similar* to * 11 Def. the solid EHPO: but similar parallelopipeds have the 11.

*

triplicate ratio of that which their homologous sides 33. 11. have: therefore the solid BGML has to the solid EHPO the triplicate ratio of that which the side BC

has to the homologous side EF. But as the solid

BGML is to the solid EHPO, so is the pyramid * 15. 5. ABCG to the pyramid DEFH; because the pyramids

* 7. 12.

are the sixth part of the solids, since the prism, which is the half of the parallelopiped, is triple* of the 28. 11. pyramid. Wherefore likewise the pyramid ABCG has to the pyramid DEFH, the triplicate ratio of that which BC has to the homologous side EF. Q. E. D.

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 trian12. 5. gular 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 mu tangular 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. THEOR.

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

Let the pyramids of which the triangles ABC, Def are the bases, and which have their vertices in the points G, H, be equal to one another: the bases and altitudes of the pyramids ABCG, DEFH shall be reciprocally proportional, viz. the base ABC shall be to the base DEF, as the altitude of the pyramid DEFH to the altitude of the pyramid ABCG.

Complete the parallelograms AC, AG, GC, DF, DH, HF; and the parallelopipeds BGML, EHPO contained by these planes and those opposite to them.

5.

& 7. 12.

And because the pyramid ABCG is equal to the pyramid DEFH, and that the solid BGML is sextuple* of * 28. 11. the pyramid ABCG, and the solid EHPO sextuple of the pyramid DEFH; therefore the solid BGML is equal to the solid EHPO. But the bases and alti-1 Ax. tudes of equal parallelopipeds are reciprocally proportional; therefore as the base BM to the base EP, * 34. 11. so is the altitude of the solid EHPO to the altitude of the solid BGML: but as the base BM to the base EP, so is the triangle ABC to the triangle DEF; there- 15. 5. fore as the triangle ABC to the triangle DEF, so is

*

the altitude of the solid EHPO to the altitude of the solid BGML: but the altitude of the solid EHPO is the same with the altitude of the pyramid DEFH; and the altitude of the solid BGML is the same with the altitude of the pyramid ABCG: therefore, as the base ABC to the base DEF, so is the altitude of the pyramid DEFH to the altitude of the pyramid ABCG: therefore the bases and altitudes of the pyramids ABCG, DEFH are reciprocally proportional.

Again, let the bases and altitudes of the pyramids ABCG, DEFH be reciprocally proportional, viz. the base ABC to the base DEF, as the altitude of the pyramid DEFH to the altitude of the pyramid ABCG: the pyramid ABCG shall be equal to the pyramid DEFH.

The same construction being made, because as the base ABC to the base DEF, so is the altitude of the

pyramid DEFH to the altitude of the pyramid ABCG: and as the base ABC to the base DEF, so is the parallelogram BM to the parallelogram EP; therefore the parallelogram BM is to EP, as the altitude of the pyramid DEFH to the altitude of the pyramid ABCG: but the altitude of the pyramid DEFH is the same with the altitude of the parallelopiped EHPO; and the altitude of the pyramid ABCG is the same with the altitude of the parallelopiped BGML: therefore, as the base BM to the base EP, so is the altitude of the parallelopiped EHPO to the altitude of the parallelopiped BGML. But parallelopipeds having their bases and altitudes reciprocally proportional, are * 34. 11. equal to one another: therefore the parallelopiped BGML is equal to the parallelopiped EHPO. And the pyramid ABCG is the sixth part of the solid BGML, and the pyramid DEFH is the sixth part of the solid EHPO: therefore the pyramid ABCG is * 2 A. 5. equal to the pyramid DEFH. Therefore the bases, &c. Q. E. D.

*

PROP. X. THEOR.

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

Let a cone have the same base with a cylinder, viz. the circle ABCD, and the same altitude: the cone shall be the third part of the cylinder; that is, the cylinder shall be triple of the cone.

If the cylinder be not triple of the cone, it must either be greater than the triple, or less than it. First, if possible, let it be greater than the triple; and inscribe the square ABCD in the circle; this square is greater than the half of the circle ABCD. Upon the

a As was shown in Prop. ii. of this Book,

square ABCD erect a prism of the same altitude with the cylinder; this prism shall be greater than half of the cylinder; because if a square is described about the circle, and a prism erected upon the square, of the same altitude with the cylinder, then the inscribed square is half of that circumscribed; and upon these square bases are erected parallelopipeds, viz. the prisms of the same altitude; therefore the prism upon the square ABCD is the half of the prism upon the square described about the circle: because they are to one another as their bases: but the cylinder is less than * 32. 11. the prism upon the square described about the circle ABCD: therefore the prism upon the square ABCD of the same altitude with the cylinder, is greater than half of the cylinder. Bisect the circumferences AB, BC, CD, DA in the points E, F, G, H; and join AE, EB, BF, FC, CG, GD, DH, HA: then, each of the triangles AEB, BFC, CGD, DHA is greater than the half of the segment of the circle in which it stands, as

was proved in Prop. ii. of this Book. Erect prisms upon each of these triangles of the same altitude with the cylinder; each of these prisms shall be greater than half of the segment of the cylinder in which it is; because, if, through the points E, F, G, H, parallels be drawn to AB, BC, CD, DA, and parallelograms be completed upon the same AB, BC, CD, DA, and parallelopipeds be erected upon the parallelograms; the prisms upon the triangles AEB, BFC, CGD, DHA are the halves of the parallelopipeds: * and the seg- 2 Cor.

DD

7. 12.