Sidebilder
PDF
ePub

e. Cor. 32.

II.

therefore as the bafe LXC to the bafe RVF; that is, as the triangle Book XII. ABC to the triangle DEF, fo is the prism having the triangle LXC for its base, and OMN the triangle oppofite to it, to the prism of 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 another, and also the two prisms in the pyramid DEFH equal to one another, as the prifm of which the base is the parallelogram KBXL and oppofite fide MO, to the prism having the triangle LXC for its base, and OMN the triangle oppofite to it; fo f is the prifm f. 7. 5. of which the base is the parallelogram PEVR, and opposite fide TS, to the prism of which the base is the triangle RVF, and oppofite triangleSTY. therefore, componendo, as the prisms KBXLMO,

[blocks in formation]

LXCOMN together are unto the prism LXCÓMN; so are the prifms PEVRTS, RVFSTY to the prifm RVFSTY. and, permutando, as the prisms KBXLMO, LXCOMN are to the prisms PEVRTS, RVFSTY; fo is the prism LXCOMN to the prism RVFSTY. but as the prifm LXCOMN to the prifm RVFSTY, fo is, as has been proved, the bafe ABC to the base DEF. therefore as the base ABC to the base DEF, fo are the two prifms in the pyramid ABCG to the two prifms in the pyramid DEFH. and likewife if the pyramids now made, for example the two OMNG, STYH be divided in the fame manner; as the base OMN is to the bafe STY, fo fhall the two prisms in the pyramid OMNG be to the two prisms in the pyramid STYH. but the bafe OMN is to the bafe STY, as the bafe ABC to the base DEF; therefore as the base

[ocr errors]

Book XII. ABC to the bafe DEF, fo are the two prifms in the pyramid ABCG to the two prifms in the pyramid DEFH; and fo are the two prisms 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. Q. E. D.

See N.

a. 3. 12.

b. 4. 12.

C. 14. 5.

PYR

PROP. V. THEOR.

of

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

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 base ABC to the base DEF, fo is the ругаmid ABCG to the pyramid DEFH.

a

For, if it be not fo, the base ABC must be to the base DEF, as the pyramid ABCG to a folid either less 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 prisms. therefore these two prifms 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 rest 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 base ABC to the bafe DEF, fob are the prisms in the pyramid ABCG to the prifms in the pyramid DEFH. but as the base 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 ABCG 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. in the like cafe.

1

therefore the base ABC is not to the base DEF, as the pyramid Book XII. ABCG to any solid which is less than the pyramid DEFH. in the fame 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 folid which is greater than the pyramid DEFH. for, if it be poffible, let it be so 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 inverfion, as the base DEF to the base ABC, fo is the folid Z to the pyramid ABCG. but as the folid Z is to the pyramid ABCG, fo is the pyramid

AAB

E V F

d

DEFH to fome folid +, which must be less than the pyramid ABCG, because the solid Z is greater than the pyramid DEFH. and therefore, as the base DEF to the base ABC, fo 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 folid which is greater than the pyramid DEFH. and it has been proved that neither is the base ABC to the bafe DEF, as the pyramid ABCG to any folid which is lefs than the pyramid DEFH. Therefore as the base 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 † in Prop. z.

d. 14. 5

Book XII.

See N.

2. 5. 12.

P

PROP. VI. THEOR.

YRAMIDS of the same altitude which have polygons for their bafes, are to one another as their bases.

Let the pyramids which have the polygons ABCDE, FGHKL for their bafes, and their vertices in the points M, N, be of the fame altitude. as the base ABCDE to the base FGHKL, fo 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, FHK, 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 bafes as many pyramids having their common vertex in the point N. therefore, fince the triangle ABC is to 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 triangle ADE to the triangle FGH, as the pyramid

[blocks in formation]

b. 2. Cor. 24. 5.

ADEM to the pyramid FGHN; as all the first antecedents to their common confequent, fob are all the other antecedents to their common confequent; that is, as the base ABCDE to the base FGH, fo is the pyramid ABCDEM to the pyramid FGHN. and for the fame reafon, as the base FGHKL to the base FGH, fo is the pyramid FGHKLN to the pyramid FGHN. and, by inverfion, as the bafe FGH to the bafe FGHKL, fo is the pyramid FGHN to the pyramid FGHKLN. then because as the base ABCDE to the bafe FGH, fo is the pyramid ABCDEM to the pyramid FGHN; and as the base FGH to the base FGHKL, fo is the pyramid FGHN to the £. 22.5. pyramid FGHKLN; therefore, ex aequali, as the bafe ABCDE to

the base FGHKL, fo the pyramid ABCDEM to the pyramid Book XII. FGHKLN. Therefore pyramids, &c. Q. E. D.

E

PROP. VII. THEOR.

VERY prifm having a triangular base may be divided into three pyramids that have triangular báfes, and are equal to one another.

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

a

Join BD, EC, CD; and because ABED is a parallelogram of which BD is the diameter, the triangle ABD is equal to the tri- a. 34. I. angle EBD; therefore the pyramid of which the base is the triangle ABD, and vertex the point C, is equal to the pyramid of which b. 5. 17. the base is the triangle EBD, and vertex the point C. but this pyramid is the fame with the pyramid the base of which is the triangle EBC, and vertex the point D; for they are contained by the fame 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

A

F

E

B

FCBE is a parallelogram of which the dia-
meter is CE, the triangle ECF is equal a to
the triangle ECB; therefore the pyramid
of which the bafe is the triangle ECB, and
vertex the point D, is equal to the pyra-
mid 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 point D has been proved equal
to the pyramid of which the bafe is the triangle ABD, and vertex
the point C. Therefore the prifm ABCDEF is divided into three
equal pyramids having triangular bafes, viz. into the pyramids
ABDC, EBDC, ECFD. and because the pyramid of which the
bafe is the triangle ABD, and vertex the point C, is the fame with
the pyramid of which the base is the triangle ABC, and vertex the
point D, for they are contained by the fame planes; and that the
pyramid of which the bafe is the triangle ABD, and vertex the
point C, has been demonstrated to be a third part of the prism the

R

« ForrigeFortsett »