d

Book XII. fquare of FH, as the circle ABCD is to the space S: Therefore as the circle ABCD is to the space S, fo is the polygon AXBOCPDR to the polygon EKFLGMHN: But the circle ABCD is greater than the polygon contained in it; wherefore 14. 5. the space S is greater & than the polygon EKFLGMHN: But it is likewife lefs, as has been demonftrated; which is impoffible. Therefore the fquare of BD is not to the fquare of FH, as the circle ABCD is to any space less than the circle EFGH. In the fame manner, it may be demonstrated, that neither is the fquare of FH to the fquare of BD, as the circle EFGH is to any space less than the circle ABCD. Nor is the square of BD to the fquare of FH, as the circle ABCD is to any space greater than the circle EFGH: For, if poffible, let it be so to T, a fpace greater than the circle EFGH: Therefore inverfely as the fquare of FH to the fquare of BD, so is the space T to

the circle ABCD. But as the space + T is to the circle ABCD, fo is the circle EFGH to fome space, which must be less than the circle ABCD, because the space T is greater, by hypothefis, than the circle EFGH. Therefore as the fquare of FH is to the

For as in the foregoing note, at *, it was explained how it was poffible there could be a fourth proportional to the fquares of BD, FH, and the circle ABCD, which was named S. So in like

manner there can be a fourth proportional to this other space, named T, and the circles ABCD, EFGH. And the like is to be understood in fome of the following propositions.

the fquare of BD, fo is the circle EFGH to a space less than Book XII. the circle ABCD, which has been demonftrated to be impoffible: Therefore the square of BD is not to the square of FH as the circle ABCD is to any space greater than the circle EFGH And it has been demonftrated, that neither is the fquare of BD to the fquare of FH, as the circle ABCD to any fpace less than the circle EFGH: Wherefore, as the fquare of BD to the fquare of FH, fo is the circle ABCD to the circle EFGH+. Circles therefore are, &c. Q. E. D.

E

PROP. III. THEOR.

VERY pyramid having a triangular base, may be See N. divided into two equal and fimilar pyramids having triangular bafes, and which are fimilar to the whole pyramid; and into two equal prisms which together are greater than half of the whole pyramid.

D

Let there be a pyramid of which the base is the triangle ABC and its vertex the point D: The pyramid ABCD may be divided into two equal and fimilar pyramids having triangular bases, and fimilar to the whole; and into two equal prifms which together are greater than half of the whole pyramid.

Divide AB, BC, CA,AD, DB, DC, each into two equal parts in the points E, F, G, H, K, L, and join EH, EG, GH, HK, KL, LH, EK, KF, FG. Because AE is equal to EB, and AH to HD, HE is parallel a to DB: For the fame reason, HK is parallel to AB: Therefore HEBK is a parallelogram, and HK equal to EB: But EB is equal to AE; therefore alfo AE is equal to HK: And AH is equal to HD; wherefore EA, AH are equal to KH, HD,

B

K

E

H

L

a 2. 6.

G

b 34.

F

each to each; and the angle EAH is equal to the angle KHD'; © 29. 1. therefore the bafe EH is equal to the base KD, and the triangle

Because as a fourth proportional to the fquares of BD, FH and the circle ABCD is poffible, and that it can neither be lefs nor greater than the circle EFGH, it must be equal to it,

d 4. 1.

e

Book XII. AEH equal d and fimilar to the triangle HKD: For the fame reason, the triangle AGH is equal and fimilar to the triangle HLD: And because the two straight lines EH, HG which meet one another are parallel to KD, DL that meet one another, and are not in the fame plane with them, they contain e 10. 11. equal angles; therefore the angle EHG is equal to the angle KDL. Again, because EH, HG are equal to KD, DL, each to each, and the angle EHG equal to the angle KDL; therefore the base EG is equal to the base KL: And the triangle EHG equal d and fimilar to the triangle KDL: For the fame reason, the triangle AEG is also equal and fimilar to the triangle HKL. Therefore the pyramid of which the base is the triangle AEG, and of which the vertex is the point H, is efc. 11. qualf and fimilar to the pyramid the

g 4.6.

B

D

K

L

E

F C

base of which is the triangle KHL, and vertex the point D: And because HK is parallel to AB a fide of the triangle ADB, the triangle ADB is equiangular to the triangle HDK, and their fides are proportionals g: Therefore the triangle ADB is fimilar to the triangle HDK: And for the fame reason, the triangle DBC is fimilar to the triangle DKL; and the triangle ADC to the triangle HDL; and alfo the triangle ABC to the triangle AEG: But the triangle AEG is fimilar to the triangle HKL, as before was proved; therefore h 21.6. the triangle ABC is fimilar to the triangle HKL. And the pyramid of which the base is the triangle ABC, and vertex the point D, is therefore fimilar i to the pyramid of which the base is the triangle HKL, and vertex the fame point D: But the pyramid of Def. II. which the base is the triangle HKL, and vertex the point D, is fimilar, as has been proved, to the pyramid the base of which is the triangle AEG, and vertex the point H: Wherefore the pyramid the base of which is the triangle ABC, and vertex the point D, is fimilar to the pyramid of which the base is the triangle AEG and vertex H: Therefore each of the pyramids AEGH, HKLD is fimilar to the whole pyramid ABCD: And 41. 1. because BF is equal to FC, the parallelogram EBFG is double k of the triangle GFC: But when there are two prifms of the

i B. II. & 11.

fame

с

b 15. 11.

fame altitude, of which one has a parallelogram for its bafe, Book XII.
and the other a triangle that is half of the parallelogram, these
prifms are equal a to one another; therefore the prifm having a 40. 11.
the parallelogram EBFG for its bafe, and the ftraight line KH
oppofite to it, is equal to the prism having the triangle GFC
for its bafe, and the triangle HKL oppofite to it; for they are
of the fame altitude, because they are between the parallel
planes ABC, HKL: And it is manifeft 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 ftraight line oppofite to it, is
greater than the pyramid of which the base is the triangle EBF,
and vertex the point K; but this pyramid is equal to the py- c C.11.
ramid the base of which is the triangle AEG, and vertex the
point H; because they are contained by equal and fimilar planes:
Wherefore the prism having the parallelogram EBFG for its
bafe, and oppofite fide KH, is greater than the pramid 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 oppofite fide KH is equal to the prism 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 base is the triangle HKL,
and vertex D: Therefore the two prifms before mentioned are
greater than the two pyramids of which the bases are the tri-
angles 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 fimilar
to one another, and to the whole pyramid; and into two equal
prifms; and the two prisms are together greater than half of
the whole pyramid, Q. E. D.

1

Book XII.

See N.

a 2. 6.

It

F 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 pyramid, and alfo into two equal prifms; and if each of these pyramids be divided in the fame manner as the first two, and fo on: As the base of one of the first two pyramids is to the base of the other, fo fhall all the prisms in one of them be to all the prisms in the other that are produced by the fame number of divifions.

Let there be two pyramids of the fame 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 prifms; 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, fo are all the prifms in the pyramid ABCG to all the prifms 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 a to AB, and the triangle ABC fimilar to the triangle LXC: For the fame reafon, 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 fimilar and fimilarly fituated rectilineal figures ABC, LXC; and upon EF, FV, in like manner, are defcribed the fimilar figures DEF, RVF: Therefore, as the trib22. 6. angle 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 because the planes ABC, OMN, as alfo the planes 5. II. DEF, STY are parallel c, the perpendiculars drawn from the points G, H to the bases ABC, DEF, which, by the hypothefis, are equal to one another, fhall be cut each into two equal d 17.11d 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 prifms LXCOMN, RVFSTÝ are of the fame altitude; and therefore, as the bafe LXC to

the