its .base, and the other a triangle that is half of the parallelogram, Book XII.
these prisms are equal to one ano her ; therefore the prisim having my
the parallelogram EBFG for its base, and the straight line KH 1. 49. 11.
opposite to it, is equal to the prifin having the triangle GFC for its
bale, and the triangle HKL opposite to it; for they are of the
fame altitude, because they are between the parallel m planes ABC, m. 15. 15.
HKL. and it is manifest that each of these prisms is greater than
either of the pyramids of which the triangles AEG, HKL are the
bases, and the vertices the points H, D, because if EF be joined,
the prism having the parallelogram EBFG for its base, and KH the
straight line oppofite to it, is greater than the pyramid of which the
bafe is the triangle EBF, and vertex the point K; but this pyramid
is equal to the pyramid the base of which is the triangle AEG, f. C. 17.
and vertex the point H; because they are contained by equal and
similar planes. wherefore the prism having the parallelogram EBFG
for its base, and opposite side KH, is greater than the pyramid of
which the base is the triangle AEG, and vertex the point H. and
the prism of which the base is the parallelogram EBFG, and op-
posite side KH is equal to the prism having the triangle GFC for
its base, and HKL the triangle opposite to it; and the pyramid
of which the base is the triangle AEG, and vertex H, is equal
to the pyramid of which the bafe is the triangle HKL, and vertex
D. therefore the two prisms before-mentioned are greater than the
two pyramids of which the bases are the triangles AEG, HKL,
and vertices the points H, D. Therefore the whole pyramid of
which the base is the triangle ABC, and vertex the point D, is
divided into two equal pyramids similar to one another, and to the
whole pyramid ; and into two equal prisms; and the two prisms
are together greater than half of the whole pyramid. Q. E. D.

Book XII.

PROP. IV. THEO R.

Sec N.

IF

there be two pyramids of the fame altitude, upon

triangular bases, and each of them be divided into two equal pyramids similar to the whole pyranid, and also into two equal prisms; and if each of these pyramids be divided in the same manner as the first two, and lo on. as the base of one of the first two pyramids is to the base of the other, so shall all the praisins in one of them be to all the prisms in the other, that are produced by the same number of divisions.

Let there be two pyramids of the same altitude upon the triangular bases ABC, DEF, and having their vertices in the points G, H; and let each of them be divided into two equal pyramids fimilar to the whole, and into two equal prisms; and let each of the pyramids thus made be conceived to be divided in the like manner, and so on. as the base ABC is to the base DEF, so are all the prisms in the pyramid ABCG to all the prisins in the pyramid DEFH made by the fame number of divifions.

Make the same construction as in the foregoing proposition, and

because BX is equal to XC, and AL to LC, therefore XL is paral. 2. 6.

lel to AB, and the triangle ABC fimilar to the triangle LXC. for the same reason, the triangle DEF is similar to RVF. and because BC is double of CX, and EF double of FV, therefore BC is to CX, as EF to FV. and upon BC, CX are described the finilar and fimilarly situated rectilineal figures ABC, LYC; and upon EF, FV,

in like manner, are described the similar figures DEF, RVF. thereb. 42. 6. fore as the triangle ABC is to the triangle LXC, fo b is the triangle

DEF to the triangle RVF, and, by permutation, as the triangle
ABC to the triangle DEF, so is the triangle LXC to the triangle

RVF. and because the planes ABC, OMN, as also the planes 2.15.11. DEF, STY are parallel “, the perpendiculars drawn from the points

G, H to the bafes ABC, DEF, which, by the Hypothesis, are equal d. :7.11. to one another, shall be cut each into two equal & parts by the

planes OMN, STY, because the straight lines GC, HF are cut into two equal parts in the points N, Y by the same planes. therefore the prisis LXCOMN, RVFSTY are of the same altitude; and

1

therefore as the base LXC to the base RVF; that is, as the triangle Book XII.
ABC to the triangle DEF, so is the priíın having the triangle LXC
for its base, and OMN the triangle opposite to it, to the prism of c. Cor. 32.
which the base is the triangle RVF, and the opposite triangle STY.
and because the two prisms in the pyramid ABCG are equal to one
arother, and also the two prisms in the pyramid DEFH equal to
one another, as the prism of which the base is the parallelogram
K3XL and opposite side MO, to the prism having the triangle LXC
for its base, and OMN the triangle opposite to it; so f is the prism f. 9. s.
of which the base is the parallelogram PEVR, and opposite fide
TS, to the prism of which the base is the triangle RVF, and oppo-
fute triangleSTY. therefore, componendo, as the prisms KBXLMO,

B х с Е V F
LXCOMN together are unto the prism LXCOMN; so are the
prisins PEVRTS, RVFSTY to the prism RVFSTY. and, per-
mutando, as the prisms KBXLMO, LXCOMN are to the prisms
PEVRTS, RVFSTY ; fo is the prism LXCOMN to the prism
RVFSTY. but as the prism LXCOMN to the prism RVFSTY,
fo is, as has been proved, the base ABC to the bale DEF. therefore
as the base ABC to the base DEF, fo are the two prisms in the
pyramid ABCG to the two prisms in the pyrainid DEFH. and like-
wise if the pyramids now made, for example the two OMNG,
STYH be divided in the same manner; as the base OMN is to the
balc STY, so shall the two prisms in the pyramid OMNG be to
the two priims in the pyramid STYH. but the base OMN is to the
bale STY, as the bale ABC to the base DEF; therefore as the base

Book XII. ABC to the base DEF, so are the two prisins in the pyramid ABCO

to the two prisins 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 fame thing may be shewn of the prisms made by dividing the pyramids AKLO and DPRS, and of all made by the same number of divisions. Q. E. D.

PRO P. V. THE O R.

Sec N.

PYRAMIDS of the

fame altitude which have triangular bases, are to one another as their bases.

2. 3. 12.

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 fame altitude. as the base ABC to the base DEF, so is the pyramid ABCG to the pyramid DEFH.

For, if it be not fo, 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 folid Q. and divide the pyramid DETII into two equal pyrainids, similar to the whole, and into two equal prisms. therefore these two prisms are greater a 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 folid Q. let thefe, for example, be the pyramids DPRS, STYH. therefore the prisms, which make the rest of the pyramid DEFH, are greater than the folid Q. divide likewise the pyramid ABCG in the fame manner, and into as many parts, as the pyramid DEFH. therefore as the base ABC to the base DEF, fo b are the prisms in the pyramid ABCG to the prisms in the pyramid DEFH. but as the base ABC to the bafe DEF, fo, by hypothesis, is the pyramid ABCG to the solid Q; and therefore, as the pyramid ABCG to the folid Q , fo are the prisms in the pyramid ABCG to the prisms in the pyramid DEFH. but the pyramid APCG is greater than the prisms contained in it; wherefore o also the solid Q is greater than the prisms in the pyramid DEFH. but it is also less, which is impoffible.

• This may be explained the same way as at the note in Proposition 2. ist the like case.

. 4. 12.

6. 14. 5.

therefore the base ABC is not to the base DEF, as the pyramid Book XII. ABCG to any folid which is less than the pyramid DEFH. in the same manner it may be demonstrated, that the base DEF is not to the base ABC, as the pyramid DEFH to any folid which is less than the pyramid ABCG. Nor can the bafe ABC be to the base DEF, as the pyramid ABCG to any solid which is greater than the pyramid DEFH. for, if it be poffible, let it be fo to a greater, viz. the folid Z. and because the base ABC is to the base DEF, as the pyramid ABCG to the folid Z; by inversion, as the base DEF to the base ABC, so is the solid Z to the pyramid ABCG. but as the folid Z is to the pyramid ABCG, so is the pyramid

DEFH to some solid t, which must be less than the pyramid d. 14. 5. ABCG, because the folid Z 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 folid which is less than the pyramid DEFH. Therefore as the base ABC is to the base DEF, fo is the pyramid ABCG to the pyramid DEFH. Wherefore pyramids, &c. Q. E. D.

† This may be cxplained the same way as the like at the mark + in Prop. 2.