RVFSTY; so is the prism LXCOMN to the prism Book XII. RVFSTY: But as the prism LXCOMN to the prism RVFSTY, so is, as has been proved, the base ABC to the

base DEF: Therefore, as the base ABC to the base DEF, so are the two prisms in the pyramid ABCG to the two prisms in the pyramid DEFH: And likewise, if the pyramids now made, for example, the two OMNG, STYH be divided in the same manner; as the base OMN is to the base STY, so are the two prisms in the pyramid OMNG to the two prisms in the pyramid STYH: But the base OMN is to the base STY, as the base ABC to the base DEF: therefore, as the base ABC to the base DEF, so are the two prisms in the pyramid ABCG to the two prisms in the pyramid DEFH; and so are the two prisms in the pyramid OMNG to the two prisms in the pyramid STYH; and so are all four to all four : And the same thing may be shown of the prisms made by dividing the pyramids AKLO, and DPRS, and of all made by the same number of divisions. Q. E. D.

PROP. V. THEOR.

Pyramids of the same altitude which have trian- See N. gular bases, are to one another as their bases.

Let the pyramids of which the triangles ABC, DEF are the bases, and of which the vertices are the points G, H, be of the same altitude: As the base ABC to the base DEF, so is the pyramid ABCG to the pyramid DEFH.

Book XII,

For, if it be not so, the base ABC must be to the base DEF, as the pyramid ABCG to a solid either less than the pyramid DEFH, or greater than it *. First, let it be to a solid less than it, viz. to the solid Q: And divide the pyramid DEFH into two equal pyramids, similar to the whole, and into two equal

a 3. 12. prisms: Therefore these two prisms are greater than the half of the whole pyramid. And again, let the pyramids made by this division be in like manner divided, and so on, until the pyramids which remain undivided in the pyramid DEFH be, all of them together, less than the excess of the pyramid DEFH above the solid Q: Let these, for example, be the pyramids DPRS, STYH: Therefore the prisms which make the rest of the pyramid DEFH, are greater than the solid Q: Divide likewise the pyramid ABCG in the same manner, and into as many parts as the pyramid DEFH: Therefore as the base

b 4. 12. ABC to the base DEF, sob are the prisms in the pyramid ABCG to the prisms in the pyramid DEFH: But as the base ABC to the base DEF, so, by hypothesis, is the pyramid ABCG to the solid Q; and therefore, as the pyramid ABCG to the solid Q, so are the prisms in the pyramid ABCG to the prisms in the pyramid DEFH: But the pyramid ABCG is greater than the prisms contained in it; wherefore also the solid Q is greater than the prisms in the pyramid DEFH. But it is also less, which is impossible. Therefore the base ABC is not to the base DEF, as the pyramid ABCG to any solid which is

с 14. 5.

less than the pyramid DEFH. In the same manner, it may Book XII. be demonstrated, that the base DEF is not to the base ABC, as the pyramid DEFH to any solid which is less than the pyramid ABCG. Nor can the base ABC be to the base DEF, as the pyramid ABCG to any solid which is greater than the pyramid DEFH. For, if it be possible, let it be so to a greater, viz. the solid Z. And because the base ABC is to the base DEF as the pyramid ABCG to the solid Z: by inversion, as the base DEF to the base ABC, so is the solid Z to the pyramid ABCG. But as the solid Z is to the pyramid ABCG, so is the pyramid DEFH to some solid *, which must be less than the pyramid ABCG, because the solid Z a 14 5. is greater than the pyramid DEFH. And therefore, as the base DEF to the base ABC, so is the pyramid DEFH to a solid less than the pyramid ABCG; the contrary to which has been proved. Therefore, the base ABC is not to the base DEF, as the pyramid ABCG to any solid, which is greater than the pyramid DEFH. And it has been proved, that neither is the base ABC to the base DEF, as the pyramid ABCG to any solid, which is less than the pyramid DEFH. Therefore, as the base ABC is to the base DEF, so is the pyramid ABCG to the pyramid DEFH. Wherefore, " pyramids," &c. Q. E. D.

PROP. VI. THEOR.

Pyramids of the same altitude which have polygons See N. for their bases, are to one another as their bases.

Let the pyramids which have the polygons ABCDE, FGHKL for their bases, and their vertices in the points M, N, be of the same altitude: As the base ABCDE to the base FGHKL, so is the pyramid ABCDEM to the pyramid FGHKLN.

Divide the base ABCDE into the triangles ABC, ACD, ADE; and the base FGHKL into the triangles FGH, FHк, FKL: And upon the bases ABC, ACD, ADE let there be as many pyramids, of which the common vertex is the point M, and upon the remaining bases as many pyramids having their common vertex in the point N: Therefore, since the triangle

• This may be explained the same way as the like in the note at the mark † in Prop. 2.

Book XII. ABC is the triangle FGH, as the pyramid ABCM to the pyramid FGHN; and the triangle ACD to the triangle FGH, as the pyramid ACDM to the pyramid FGHN; and also the

a 5. 12.

24. 5.

triangle ADE to the triangle FGH, as the pyramid ADEM to the pyramid FGHN; as all the first antecedents to their b 2. Cor. common consequent, sob are all the other antecedents to their common consequent; that is, as the base ABCDE to the base FGH, so is the pyramid ABCDEM to the pyramid FGHN. And, for the same reason, as the base FGHKL to the base FGH, so is the pyramid FGHKLN to the pyramid FGHN: And, by inversion, as the base FGH to the base FGHKL, so is the pyramid FGHN to the pyramid FGHKLN: Then, because as the base ABCDE to the base FGH, so is the pyramid ABCDEM to the pyramid FGHN; and as the base FGH to the base FGHKL, so is the pyramid FGHN to the pyramid FGHKLN: c 22. 5. therefore ex æquali, as the base ABCDE to the base FGHLK, so is the pyramid ABCDEM to the pyramid FGHKLN. Therefore, " pyramids," &c. Q. E. D.

PROP. VII. THEOR.

Every prism having a triangular base may be divided into three pyramids that have triangular bases, and are equal to one another.

Let there be a prism of which the base is the triangle ABC, and let DEF be the triangle opposite to it: The prism ABCDEF may be divided into three equal pyramids, having triangular bases.

Join BD, EC, CD; and because ABED is a parallelogram a 34. 1. of which BD is the diameter, the triangle ABD is equal a to the triangle EBD; therefore the pyramid of which the base is the triangle ABD, and vertex the point C, is equal to the Book XII. pyramid of which the base is the triangle EBD, and vertex

the point C: But this pyramid is the same with the pyramid b 5. 12. the base of which is the triangle EBC, and vertex the point D; for they are contained by the same planes: Therefore the pyramid of which the base is the triangle ABD, and vertex the point C, is equal to the pyramid, the base of which is the triangle EBC, and vertex the point D: Again, because FCBE is

a parallelogram of which the diameter

is CE, the triangle ECF is equal a to

F

the triangle ECB: therefore the pyra

mid of which the base is the triangle D

E

ECB, and vertex the point D, is equal

to the pyramid, the base of which is
the triangle ECF, and vertex the point
D: But the pyramid of which the base
is the triangle ECB, and vertex the A

C

B

point D has been proved to be equal to
the pyramid of which the base is the triangle ABD, and vertex
the point C. Therefore the prism ABCDEF is divided into
three equal pyramids having triangular bases, viz. into the
pyramids ABDC, EBDC, ECFD: And because the pyra-
mid of which the base is the triangle ABD, and vertex the
point C, is the same with the pyramid of which the base is
the triangle ABC, and vertex the point D, for they are con-
tained by the same planes; and because the pyramid of which
the base is the triangle ABD, and vertex the point C, has
been demonstrated to be a third part of the prism, the base of
which is the triangle ABC, and to which DEF is the oppo-
site triangle; therefore the pyramid of which the base is the
triangle ABC, and vertex the point D, is the third part of
the prism which has the same base, viz. the triangle ABC,
and DEF is the opposite triangle. Q. E. D.

COR. 1. From this it is manifest, that every pyramid is the third part of a prism which has the same base, and is of an equal altitude with it; for if the base of the prism be any other figure than a triangle, it may be divided into prisms having triangular bases.

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.

с 6. 12.