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its bafe, and the other a triangle that is half of the parallelogram, Book XII.
there prifms are equal to one ano her; therefore the prifm having

the parallelogram EBFG for its base, and the straight line KH 1. 49. 11.
oppofite to it, is equal to the prifin having the triangle GFC for its
bafe, 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. 11.
HKL. and it is manifeft that each of thefe prifms is greater than
either of the pyramids of which the triangles AEG, HKL are the
bafes, and the vertices the points H, D; becaufe if EF be joined,
the prifm having the parallelogram EBFG for its bafe, and KH the
ftraight 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 bafe of which is the triangle AEG, f. C. II.
and vertex the point H; because they are contained by equal and
fimilar planes. wherefore the prifm having the parallelogram EBFG
for its bafe, and oppofite fide KH, is greater than the pyramid of
which the bafe is the triangle AEG, and vertex the point H. and
the prifm of which the bafe is the parallelogram EBFG, and op-
pofite fide KH is equal to the prifm having the triangle GFC for
its bafe, and HKL the triangle oppofite to it; and the pyramid
of which the bafe 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 prifms before-mentioned are greater than the
two pyramids of which the bafes are the triangles AEG, HKL,
and vertices the points H, D. Therefore the whole pyramid of
which the bafe is the triangle ABC, and vertex the point D, is
divided into two equal pyramids fimilar to one another, and to the
whole pyramid; and into two equal prifms; and the two prifms
are together greater than half of the whole pyramid. Q. E. D.

Book XII.

See N.

4.2. 6.

b. 21. 6.

IF

PROP. IV. THEOR.

there be two pyramids of the fame altitude, upon triangular bafes, and each of them be divided into two equal pyramids fimilar to the whole pyranid, and alfo into two equal prifms; and if each of these pyramids be divide 1 in the fame manner as the first two, and to on. as the bafe of one of the first two pyramids is to the bafe of the other, fo fhall all the pifins in one of them be to all the prifms in the other, that are produced by the fame number of divifions.

Let there be two pyramids of the fame altitude upon the triangular bafes 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 prifms; and let each of the pyramids thus made be conceived to be divided in the like manner, and fo on. as the bafe ABC is to the base DEF, fo are all the prifms in the pyramid ABCG to all the prifins in the pyramid DEFH made by the fame number of divifions.

Make the fame conftruction as in the foregoing propofition. and because BX is equal to XC, and AL to LC, therefore XL is parallel to AB, and the triangle ABC fimilar to the triangle LXC. for the fame reason, the triangle DEF is fimilar 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 defcribed the fimilar and fimilarly fituated rectilineal figures ABC, LXC; and upon EF, FV, in like manner, are described the fimilar figures DEF, RVF. therefore as the triangle ABC is to the triangle LXC, fob is the triangle DEF to the triangle RVF, and, by permutation, as the triangle ABC to the triangle DEF, fo is the triangle LXC to the triangle RVF. and becaufe the planes ABC, OMN, as alfo the planes c. 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 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 fame planes. therefore the prifins LXCOMN, RVFSTY are of the fame altitude; and

d. 17. 11.

!

J

therefore as the bafe LXC to the bafe RVF; that is, as the triangle Book XII. ABC to the triangle DEF, fo is the prifin having the triangle LXC

for its bafe, and OMN the triangle oppofite to it, to the prifm of e. Cor. 32. which the bafe is the triangle RVF, and the oppofite triangle STY.

II.

and because the two prifms in the pyramid ABCG are equal to one
another, and alfo the two prifms in the pyramid DEFH equal to
one another, as the prifm of which the bafe is the parallelogram
K3XL and oppofite fide MO, to the prifmm having the triangle LXC
for its bafe, and OMN the triangle oppofite to it; fo f is the prifm f. 7. 5.

of which the bafe is the parallelogram PEVR, and opposite fide

TS, to the prifm of which the bafe is the triangle RVF, and oppo-
fite triangle STY. therefore, componendo, as the prisms KBXLMO,

[blocks in formation]

LXCOMN together are unto the prifm LXCOMN; fo are the
prifins PEVRTS, RVFSTY to the prifm RVFSTY. and, per-
mutando, as the prifms KBXLMO, LXCOMN are to the prifms
PEVRTS, RVFSTY; fo is the priẩm LXCOMN to the prifm
RVFSTY. but as the prifm LXCOMN to the prifm RVFSTY,
fo is, as has been proved, the base ABC to the bale DEF. therefore
as the bafe ABC to the bafe DEF, fo are the two prifms in the
pyramid ABCG to the two prifins in the pyramid DEFH. and like-
wife if the pyramids now made, for example the two OMNG,
STYH be divided in the fame manner; as the bafe OMN is to the
bale STY, fo fhall the two prifms in the pyramid OMNG be to
the two prifms in the pyramid STYH. but the bafe OMN is to the
base STY, as the base ABC to the bafe DEF; therefore as the bafe

Book XII. ABC to the bafe DEF, fo are the two prifins in the pyramid ABCO to the two prifins in the pyramid DEFH; and fo are the two prifms in the pyramid OMNG to the two prifms in the pyramid STYH; and fo are all four to all four. and the fame thing may be fhewn of the prisms made by dividing the pyramids AKLO and DPRS, and of all made by the fame number of divifions.

Sce N.

2. 3. 12.

b. 4. 12.

C. 34. 5.

PYR

PROP. V. THEO R.

Q. E. D.

YRAMIDS of the fame altitude which have triangular bafes, are to one another as their bafes.

Let the pyramids of which the triangles ABC, DEF are the bafes, and of which the vertices are the points G, H, be of the fame altitude. as the bafe ABC to the base DEF, fo is the pyramid ABCG to the pyramid DEFH.

a

For, if it be not fo, the bafe ABC must be to the bafe DEF, as the pyramid ABCG to a folid either lefs than the pyramid DEFH, or greater than it *. First, let it be to a folid less than it, viz. to the folid Q. and divide the pyramid DEFH into two equal pyramids, fimilar to the whole, and into two equal prifms. therefore thefe two prisms are greater than the half of the whole pyramid. and, again, let the pyramids made by this divifion be in like manner divided, and fo on, until the pyramids which remain undivided in the pyramid DEFH be, all of them together, lefs than the excefs of the pyramid DEFH above the folid Q. let thefe, for example, be the pyramids DPRS, STYH. therefore the prifms, which make the reft of the pyramid DEFH, are greater than the folid Q divide likewife the pyramid ABCG in the fame manner, and into as many parts, as the pyramid DEFH. therefore as the bafe ABC to the bafe DEF, fob are the prifms in the pyramid ABCG to the prifms in the pyramid DEFH. but as the bafe ABC to the bafe DEF, fo, by hypothefis, is the pyramid ABCG to the folid Q; and therefore, as the pyramid ABCG to the folid Q, fo are the prifms in the pyramid ABCG to the prifms in the pyramid DEFH. but the pyramid APCG is greater than the prifms contained in it; wherefore alfo the folid Q is greater than the prifms in the pyramid DEFH. but it is alfo lefs, which is impoffible.

* This may be explained the fame way as at the note in Propofition 2. im the like cafe.

I

therefore the bafe ABC is not to the bafe DEF, as the pyramid Book XII.
ABCG to any folid which is lefs than the pyramid DEFH. in the
fame manner it may be demonftrated, that the bafe DEF is not to
the bafe ABC, as the pyramid DEFH to any folid which is lefs
than the pyramid ABCG. Nor can the bafe ABC be to the base
DEF, as the pyramid ABCG to any folid 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 bafe ABC is to the bafe DEF, as
the pyramid ABCG to the folid Z; by inverfion, as the bafe
DEF to the bafe ABC, fo is the folid Z to the pyramid ABCG.
but as the folid Z is to the pyramid ABCG, fo is the pyramid

[blocks in formation]

DEFH to fome folid t, which must be lefs than the pyramid d. 14. 5.
ABCG, because the folid Z is greater than the pyramid DEFH.
and therefore, as the bafe DEF to the bafe ABC, fo is the pyra-
mid DEFH to a folid lefs than the pyramid ABCG; the contrary
to which has been proved. therefore the base ABC is not to the
bafe DEF, as the pyramid ABCG to any folid which is greater
than the pyramid DEFH. and it has been proved that neither is
the bafe ABC to the bafe DEF, as the pyramid ABCG to any fo-
lid which is lefs than the pyramid DEFH. Therefore as the bafe
ABC is to the bafe DEF, fo is the pyramid ABCG to the pyra-
mid DEFH. Wherefore pyramids, &c. Q. E. D.

This may be explained the fame way as the like at the mark f in Prop. 2.

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