(1. def. 6.) to BC as DE to EF; that is, the sides about the equal 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 (24. 11.) 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 equal (B. 11.); and therefore the solid BGML, is similar (11. def. 11.) to the solid EHPO: but similar solid parallelopipeds have the triplicate (33. 11.) ratio of that which their homologous sides 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 (15. 5.) the pyramid ABCG to the pyramid DEFH; because the pyramids are the sixth part of the solids; since the prism, which is the half (28. 11.) of the solid parallelopiped is triple (7. 12.) of the 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 polygons; therefore as one of the triangular pyramid in the first multangular pyramids 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. 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 are reciprocally proportional, viz. the base ABC is 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 solid parallelopipeds BGML, EHPO contained by these planes and those opposite to them: and because the pyramid ABCG is equal to the pyramid DEFH, and that the solid BGML is sectuple of the pyramid ABCG, and the solid EHPO sectuple of the pyramid DEFH; therefore the solid BGML is equal (1. Ax. 5.) to the solid EHPO: but the bases and altitudes of equal solid parallelopipeds are reciprocally proportional (34. 11.); therefore as the base BM to the base EP, so is the altitude of the solid EHPO to the altitude of the solid R

BGML: but as the base BM to the base EP, so is (15. 5.) the triangle ABC to the triangle DEF; therefore 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: wherefore 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 is 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 solid parallelopiped EHPO: and the altitude of the pyramid ABCG is the same with the altitude of of the solid parallelopiped BGML: as, therefore, the base BM to the base EP, so is the altitude of the solid parallelopiped EHPO to the altitude of the solid parallelopiped BGML. But solid parallelopipeds having their bases and altitudes reciprocally proportional, are equal (34. 11.) to one another. Therefore the solid parallelopiped BGML is equal to the solid 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 EPHO. Therefore the pyramid ABCD is equal to the pyramid DEFH. Therefore the bases, &c. Q. È. D.,

PROP. X. THEOR.

EVERY Cone is a 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 is the third part of the cylinder; that is, the cylinder is 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, let it be greater than the triple: and describe the square ABCD in the circle; this square is greater than the half of the circle ABCD:* Upon the square ABCD erect a prism of the same altitude with the cylinder; this prism is greater than half of the cylinder: because if a square be described about the circle, and a prism erected upon the square, of the same altitude with the cylinder, the inscribed square is half of that circumscribed; and upon these square bases are erected solid parallelopipeds, viz. the prisms of the same altitude; therefore the prism upon the square ABCD is half of the prism upon the square described about the circle: because they are to one another as their bases (32. 11.); and the cylinder is less than 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 half of the segment of the circle in which it stands, as was shown in prop. 2. of this book. Erect prisms upon each of these triangles, of the same altitude with the cylinder; each of these prisms is 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 solid parallelopipeds be erected upon the parallelograms; the prisms upon the triangles AEB, BFC, CGD, DHA are the halves of the solid parallelopipeds (2.

B

E

A

C

H

G

D

Cor. 7. 12.). And the segments of the cylinder which are upon the segments of the circle cut off by AB, BC, CD, DA, are less than the solid parallelopipeds which contain them. Therefore the prisms upon the triangles AEB, BFC, CGD, DHA, are greater than half of the segments of the cylinder in which they are; there

* As was shown in prop. 2. of this book.

:

H

fore if each of the circumferences be divided into two equal parts,
and straight lines be drawn from the points of division to the ex-
tremities of the circumferences, and upon the triangles thus made,
prisms be erected of the same altitude with the cylinder, and so on,
there must at length remain some segments of the cylinder which
together are less (Lem.) than the excess of the cylinder above
the triple of the cone. Let them be those upon the segments of the
circle AE, EB, BF, FC, CG, GD, DH, HA. Therefore the rest of the
cylinder, that is, the prism of which the base is the polygon
AEBFCGDH, and of which the altitude is the same with that of the
cylinder, is greater than the triple of the cone: but this prism is triple
(1. Cor. 7. 12.) of the pyramid upon the same base, of which the ver-
tex is the same with the vertex of the cone; therefore the pyramid
upon the base AEBFCGDH, having the same vertex with the cone,
is greater than the cone, of which the base is the circle ABCD: but
it is also less, for the pyramid is contained within the cone; which
is impossible. Nor can the cylinder be less than the triple of the
cone. Let it be less, if possible: therefore, inversely, the cone is
greater than the third part of the cylinder. In the circle ABCD de-
scribe a square; this square is greater than the half of the circle: and
upon the square ABCD erect a pyramid having the same vertex with
the cone: this pyramid is greater than the half of the cone; because, as
was before demonstrated, if a square be described about the circle,
the square ABCD is the half of it; and if,
upon these squares there be erected solid
parallelopipeds of the same altitudes with
the cone, which are also prisms, the prism
upon the square ABCD shall be the half
of that which is upon the square described E
about the circle; for they are to one another
as their bases (32. 11.); as are also the
third parts of them; therefore the pyramid,
the base of which is the square ABCD, is
half of the pyramid upon the square de-
scribed about the circle: but this last pyra-
mid is greater than the cone which it contains; therefore the pyra-
mid upon the square ABCD, having the same vertex with the cone,
is greater than the half of the cone. Bisect the circumferences AB,
BC, CD, DA in the points E, F, G, H, and join AE, EB, BF, FC, CG,
GD, DH, HA: therefore each of the triangles AEB, BFC, CGD, DHA
is greater than half of the segment of the circle in which it is: upon
each of these triangles erect pyramids having the same vertex with
the cone. Therefore each of these pyramids is greater than the half
of the segment of the cone in which it is, as before was demonstrated
of the prisms and segments of the cylinder: and thus dividing each
of the circumferences into two equal parts, and joining the points of
division and their extremities by straight lines, and upon the triangles
erecting pyramids having their vertices the same with that of the
cone, and so on, there must at length remain some segments of the

A

D

B

C

F

G

A

D

G

cone, which together shall be less than the excess of the cone above the third part of the cylinder. Let these be the segments upon AE, EB, BF, FC, CG, GD, DH, HA. Therefore the rest of the cone, that is, the pyramid, of which the base is the H polygon AEBFCGDH, and of which the vertex is the same with that of the cone, is greater than the third part of the cylinder. But this pyramid is the third part of the prism upon the same base AEBFCGDH, E and of the same altitude with the cylinder. Therefore this prism is greater than the cylinder of which the base is the circle ABCD. But it is also less; for it is contained within the cylinder; which is impossible. Therefore the cylinder is not less than the triple of the And it has been demonstrated that neither is it greater than the triple. Therefore the cylinder is triple of the cone, or the cone is the third part of the cylinder. Wherefore every cone, &c. Q. E. D.

cone.

PROP. XI. THEOR.

B

F

C

CONES and cylinders of the same altitude, are to one another as their bases.*

Let the cones and cylinders, of which the bases are the circles ABCD, EFGH, and the axes KL, MN, and AC, EG the diameters of their bases, be of the same altitude. As the circle ABCD to the circle EFGH, so is the cone AL to the cone EN.

If it be not so, let the circle ABCD be to the circle EFGH, as the cone AL to some solid either less than the cone EN, or greater than it. First, let it be to a solid less than EN, viz. to the solid X; and let Z be the solid which is equal to the excess of the cone EN above the solid X; therefore the cone EN is equal to the solids X, Z together. In the circle EFGH describe the square EFGH, therefore this square is greater than the half of the circle: upon the square EFGH erect a pyramid of the same altitude with the cone; this pyramid is greater than half of the cone. For, if a square be described about the circle, and a "pyramid be erected upon it, having the same vertex with the cone,t the pyramid inscribed in the cone is half of the pyramid circumscribed about it, because they are to one another as their bases (6. 12.): but the cone is less than the circumscribed pyramid; therefore the pyramid of which the base is the square EFGH, and its vertex the same with that of the cone, is greater than half of the cone: divide the circumferences EF, FG, GH, HE, each into two

* See Note.

+ Vertex is put in the place of altitude, which is in the Greck, because the pyramid, in what follows, is supposed to be circumscribed about the cone, and so must have the same vertex. And the same change is made in some places following.