Abbildungen der Seite
PDF
EPUB

Bonk XII.

PRO P. IX.

THEOR.

THE bases and, altitudes of equal pyramids having

triangular bases are reciprocally proportional : And triangular pyramids of which the bases and altitudes are reciprocally proportional, are equal to one another.

Let the pyramids of which the triangles ABC, DEF are the bases, and which have their vertices in the points G, H, be equal to one another: The bases and altitudes of the pyramids ABOG, DEFH are reciprocally proportional, viz. the base ABC is to the base DEF, as the altitude of the pyramid DEFH to the alirude of the pyramid ABCG.

Complete the parallelograns AC, AG, GC, DF, DH, HF; and the folid parallelepipeds BGML, EHPO contained by

[ocr errors]
[merged small][ocr errors][merged small][merged small][ocr errors][merged small][merged small][merged small][merged small][merged small][merged small]

these planes and those opposite to them: And because the py: ramid ABCG is equal to the pyramid DEFH, and that the folid BGML is fextuple of the pyramid ABCG, and the folid

EHPO sextuple of the pyramid DEFH ; therefore the folid a 1. Ax. s. BGML is equal a to the solid EHPO: But the bases and alti

tudes of equal solid pas allelepipeds are reciprocally proporb 34. 11.

tional b; therefore as the base BM to the bafe EP, so is the ala tirude of the folid EHPO to the altitude of the solid BGML: But as the bale BM to the base EP, lo is c the triangle ABC to the triangle DEF ; therefore as the triangle ABC to the trio an le DEF, fo is the alrirude of the folid EHPO to the altitvde of the solid BGML: But she altitude of the folid EHPO is the same with the aititude of the pyramid DEFH ; and the altitude of the folid BGML is the faine with the altitude of the

piramid

GIS. 5.

pyramid ABCG : Therefore, as the base ABC to the base DEF, Book XII. fo is the altitude of the pyramid DEFH to the altitude of the pyramid ABCG : Wherefore the bases and altitudes of the pyra. mids ABCG, DEFH are reciprocally proportional.

Again, Let the bases and altitudes of the pyramids ABCG, DEFH be reciprocally proportional, viz, the base ABC to the bate DEF, as the altitude of the pyramid DEFH to the altitude of the pyramid ABCG: The pyramid ABCG is equal to the pyramid DEFH.

The same construction being made, because as the base ABC to the base DEF, so is the aititude of the pyramid DEFA 10 the altitude of the pyramid ABCG: and as the bale ABC 10 the bafe DEF, fo is the parallelogram BM to the parallelogram EP; therefore the parallelogram BM is to EP, as the altitude of the pyramid DEFH to the altitude of the pyramid ABCG : But the altitude of the pyramid DEFH is the same with the altitude of the folid parallelepiped EAPO; and the al. titude of the pyramid ABCG is the same with the alricude of the solid parallelepiped BGML : As, therefore, the base BM 10 the base EP, fo is the altitude of the solid parallelepiped EHPO to the altitude of the folid parallelepiped BGML. But folid parallelepipeds having their bases and altitudes reciprocally proportional, are equal b

to one another. Therefore the solid pa-b 34. II. rallelepiped BGML is equal to the solid parallelepiped EHPO. And the pyramid ABCG is the fixth part of the folid BGML, and the pyramid DEFH the fixth part of the folid EHPO. Therefore the pyramid ABCG is equal to the pyramid DEFH. Therefore the bases, &c.

Q. E. D.

PRO P. X. THEOR.

EVERY cone is the third part of a cylinder which

has the fame base, and is of an equal altitude with it.

Let a cone have the same base with a cylinder, viz. the circle ABCD, and the fame altitude. The cone is the third part of the cylinder ; that is, the cylinder is triple of the cone.

If the cylinder be not triple of the cone, it must either be greater than the triple, or less than it. First, Let it be greater than the triple; and describe the square ABCD in the circle ; this fquare is greater than the half of the circle ABCD.

As was Dicwn in prop. 2. of this book.

Upoa

[ocr errors]

Book XII. Upon the square ABCD erec? a prism of the same altitude with

the cylinder ; this priiin is greater than half of the cylinder ; because if a square be described about the circle, and a prilaz erected upon the square, of the same altitude with the cylinder, the inscribed square is half of that circumscribed ; and upon these square bases are erected solid parallelepipeus, viz. the prisms, of the same altitude ; therefore the prism upon the {quare ABCD is the half of the prism upon the square descri

bed about the circle ; because they are to one another 'as their 4 32. 11. bases a : And the cylinder is less than

the prism upon the square
defcribed about the circle ABCD: Therefore the prism upon
the square ABCD of the same altitude with the cylinder, is
greater than half of the cylinder. Bifect the circumferences
AB, BC, CD. DA in the points E, F, G, H; and join AE,
EB, BF, FC, CG, GD, DH, HA : Then, each of the triangles
AEB, BFC, CGD, DHA is greater than the half of the leg,
ment of the circle in which it Aands,
as was shewn in prop. 2. of this

A
book. Erect prisms upon cach of
these triangles of the fame altitude E

H
with the cylinder ; each of thefe
prisins is greater than half of the seg.
ment of ihe cylinder in which it is; B

D
because if, through the points E, F,
GH, parallels be drawn to AB, BC,
CD, DA, and parallelograms be F
completed upon the same AB, BC,

С CD, DA, and solid parallelepipeds, be erected upon the parallelograms; the prisms upon the

triangles AEB, BFC, CGD, DHA are the balves of the fob 2. Cor. lid parallelepipeds b. And the segments of the cylinder which 7. 12. are upon the segments of the circle cut off by AB, BC, CD,

DA, are less than the folid parallelepipeds which contain them. Therefore the prisms upon the triangles AEB, BFC, CGD, DHA, are greater than half of the segments of the cylinder in which they are ; therefore if each of the circumferences be divided into two equal parts, and ftraight lines be drawn from the points of division to the extremities of the circumferences, and upon the triangles thus made, prisus be erected of the fame

altitude with the cylinder, and fo on, there must at length re. c Lenma. main some fegments of the cylir.der which together are lefs c

than the excess of the cylinder above the triple of the cone. Let them be those upon the segments of the circle AE, EB, BF,

FC,

[ocr errors]

.

7. 12.

FC, CG, GD, DH, HA. Therefore the rest of the cylin.Book XII. der, that is, the prism of which the base is the polygon AEBFCGDH, and of which the altitude is the same with that of the cylinder, is greater than the triple of the cone : But this prism is triple d of the pyramid upon the fame base, of which à 1. Cor.. the vertex is the same with the vertex of the cone ; therefore the pyramid upon the base AEBFCGDH, having the fame vertex with the cone, is greater than the cone, of which the base is the circle ABCD : But it is also less, for the pyramid is contained within the cone; which is impossible. Nor can the cyiinder be less than the triple of the cone. Le ic he less, if pofiible : Therefore, inversely, the cone is greater than the third part of the cylinder. In the circle ABCD dcfcribe a square ; This square is greater than the half of the circle : And upon the square ABCD erect a pyramid having the same vertex with the cone ; this pyramid is greater than the half of the cone; because, as was before demonstrated, if a square be described about the circle, the square ABCD is the half of it; and if, upon these squares

H there be erected solid parallelepi- A

D peds of the fame altitude with the cone, which are also prisms, the prism upon the square ABCD

E

G shall be the half of that which is upon the square described about the circle; for they are to one another D as their bafes a ; as are also the third parts of them : Therefore the py

F ramid, the base of which is the square ABCD, is half of the pyramid upon the square defcribed about the circle : But this last pyramid is greater than the cone which it contains ; therefore the pyramid upon the square ABCD having the same vertex with the cone, is greater than the half of the cone. Bisect the circumferences AB, BC, CD, DA in the points E, F, G, H, and join AE, EB, BF, FC, CG, GD, DH, HA: Therefore each of the triangles AEB, BFC, CGD, DHA is greater than half of the segment of the circle in which it is : Upon each of these triangles erect pyramids having the same vertex with the cone. Therefore each of these pyramids is greater than the half of the segment of the cone in which it is, as before was demonstrated of the prisms and segments of the cylinder ; and thus dividing each of the circumferences into two equal parts, and joining the

points

a 32. 11.

Book XII. points of division and their extremities by straight lines, and

upon the triangies erecting pyramids having their vertices the
same with that of the cone, and so on, there must at length
remain fome fegments of the cone, which together shall be less
than she excess of the cone above the third part of the cylin-
der. Let these be the fegnienis upcn AE, EB, BF, FC, CG, GD,
DH, HA. Therefore the set of
the cone, that is, the pyramid, of

H
which the base is the polygon

A

D
ALBFCGDH, and of which the
vertex is the lame with that of the
cone, is greater that the third part

E

G of the cylinder. But this pyramid is the third part of the prisni upon the same bale AEBFCCDH. and

B of the same altiiude with the cylin. der. Therefore this prism is greato er than the cylinder of which the base is the circle ABCD. But it is also less, for it is contained within the cylinder ; which is impoflibie. Therefore the cylinder is no less than the triple of the cone. And it has been demonstrated that neither is it greater than the triple. Therefore the cylinder is tip'e of the cone, or, the cone is the part of the cylinder. "Wherefore every cone, &c. Q. E. D.

Scc N.

P R 0 P. XI. T H E O R.
CONES
COnes and cylinders of the fame altitude, are to one

Es
another as their bases.

"Let he cones and cylinders, of which the bases are the circles ABCD, EFGH, and the axes KL, MN, and AC, EG the diameters of their basis, be of the fame altitude. As the circle ABCD to the circle EFGH, so is ihe cone AL to the cone EN.

If it be not so, let the circle ABCD be to the circle EFGH, as the cone AL to some folid either less than the cone EN, or greater than it.

First, let it be to a solid less than EN, viz. (Q the folid X ; and ler Z be the folid which is equal to the excess of the cone EN above the solid X ; therefore the cone EN is equal to the solids X. Z together. In the circle EFGH de. feribe the square EFGH, therefore this fquare is greater than the half of the circle : Upon the square EFGH erect a pyramid of the fame alitude with the cone; this pyramid is greater than half of the cone. For, if a squarę be described a out the circle, and a pyramid be elected upon it, h2 1

ving

« ZurückWeiter »