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mid deh, the angle A B C [by 9. def. 11] will be equal
to the angle der, the angle GBC to the angle her, and
the angle ABG to
the angle deh, and K L
A B is to DE, as BC

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is to EF, and as BG

P is to EH. Therefore because it is as A B is to D'E, fo is BC to EF, and the sides a.

G bout the equal angles

H are proportional: the

B

I E parallelogram в м will be similar to the parallelogram EP. By the same reason the parallelogram en is similar to the parallelogram ER, and the parallelogram Bk to the parallelogram ex: Therefore the three parallelograms BM, KB, BN are similar to the three parallelograms EP, EX, E R. But [by 24. 11.) the three parallelograms MB, BK, BN are similar and equal to the three opposite parallelograms, and the three parallelograms EP, EX, ER fimilar and equal to the three opposite ones; wherefore the solids BGML, EH PO are contained under equal numbers of similar planes; and so (by 9. def. 11.) the solid BGML is similar to the solid EHPO. But [by 33. 11.) similar solid parallelepipedons are in the triplicate ratio of their homologous fides: Therefore the solid BGMs to the solid Eh po is in the triplicate ratio of the homologous fide BC to the homologous fide Ef. But [by 15. 5.] as the solid BGML is to the solid E H PO, so is the pyramid ABCG to the pyramid DE FH; for the pyramid is a sixth part of that solid, since the prism, which is one half of the solid parallelepipedon is thrice the pyramid: Therefore the pyramid ABCG to the pyramid DEFH is in the triplicate ratio of BC to E F. Which was to be demonstrated.

Corollary. From hence it is manifeft, that similar pyramids having polygonous bafes, are to one another in the triplicate ratio of the homologous fides: For they being divided into pyramids having triangular bases, because [by 20.6.] fimilar polygonous bases are divided into equal numbers of similar triangles homologous to the wholes ; it will be as one of the pyramids having a triangular base in one pyramid, is to one pyramid having a triangular base in

the

the other, so are all the pyramids having triangular bases in the one pyramid, to all those having triangular bases in the other; that is, so is one pyramid having the polygo. nous base to the other pyramid having the polygonous bale. But one pyramid with a triangular base is to another pyramid with a triangular base in the triplicate ratio of their homologous fides: And therefore one pyramid having a polygonous base to a similar pyramid having such a bale, is in the triplicate ratio of one of its homologous sides to the other:

PROP. IX. THEOR. The bases of equal pyramids, having triangular bases;

are reciprocally proportional to their altitudes, and those triangular pyramids whose bases are reciprocally proportional to their altitudes, are equal to one another.

For let there be equal pyramids having the triangular bases ABC, DEF, and vertexes the points G, H: I say the bases of the pyramids ABCG, DE FH are reciprocally proportional to their altitudes; that is, as the base ABC is to the base DEF, fo is the altitude of the pyramid DEFH to the altitude of the pyramid A BCG.

For complete the folid parallelepipedons BGML, EHPO. Then because the pyramid aecg is equal to the pyramid DEFH, and the pyramid A BCG is the fixth part of the solid BGML, and the pyramid DEFH the fixth part of the solid EHPO: The solid B GML [by 15. 5.) will be equal to the solid E H PO. But [by 34. 11.) the bases of equal solid parallelepipedons are reciprocally proportional to their altitudes : Therefore as the

bare øm is to the base H.

EP, so is the altitude of the solid E HPO to the altitude of the folid BGML. But as the base

BM is to the base E P, AM

fo is the triangle 'A BC to the triangle DEF: Therefore as the trian.

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the pyramid DEF H, and the altitude of the solid BGML Book Xit. Euclid's Elements. 385 gle A B C is to the triangle der, fo is the altitude of the solid EH PO to the altitude of the solid BGM L. But the altitude of the solid EhPo is the same as the altitude of

the same as the altitude of the pyramid ABCG: Therefore as the base a BC is to the base D E F; fo is the altitude of the pyramid De FH to the altitude of the pyramid, ABCG: Wherefore the bases of the pyramids A B CG; D'EF H are reciprocally proportional to their altitudes.

Now let the bases of the pyramids ABCG, DEFH be reciprocally proportional to their altitudes, and let the base A B C be to the base D E F, as the altitude of the pyramid D'EFH is to the altitude of the pyramid ABCG: I say the pyramid ABCG is equal to the pyramid DEFH,

For the same construction remaining, because as the base A B C is to the base D E F, so is the altitude of the pytamid DEFH to the altitude of the pyramid ABCG And as the base ABC is to the base D E F, so is the parallelos: gram B M to the parallelogram EP; it will be as the pare allelogram B M is to the parallelogram EP, so is the altitude : of the pyramid D.EFH to the altitude of the pyramid. ABC G. But the altitude of the pyramid DEFH is fame as the altitude of the solid parallelepipedon EH PO; and the altitude of the pyramid, ABCG the same as the altitude of the solid parallelepipedon BGML. Therefore.. as the base B M is to the base EP, so is the altitude of the folid parallelepipedon eh po to the altitude of the solid parallelepipedon BGML. But those folid parallelepipedons. whose bases are reciprocally proportional to their altitudes . [by 34. 11.) are equal to one another. Therefore the solid parallelepipedon BGML is equal to the solid parallelepipedon EHPO. And the pyramid, ABCG is a, fqxth. part of the folid BGML, and the pyramid D E F H is also a fixth part of the solid EHPO; therefore the pyramid A B C G is equal to the pyramid D E F H.

Wherefore the bases of equal pyramids having triangu-, lar bases, are reciprocally proportional to their altitudes ; : and those triangular pyramids, whose bases are reciprocally proportional to their altiiudes, are equal to one another. Which was to be demonstrated.

"9. the

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PROP. X. THEOR.

Every cone is a third part of a cylinder which has

the same base and an equal altituded. For let a cone have the same base as a cylinder, viz. the circle A B C D, and an equal altitude: I say the cone is a third part of the cylinder ; that is, the cylinder is

thrice the cone. For if the cylinder H

be not thrice the cone; it will be either greater than thrice the cone, or less. First let it be greater than thrice the conę; and describe a square ABCD in the circle ABCD: Then the square A B C D is greater than one half the

circle A B C D. And upon the square A B C D erect a prism of the fame altitude as the cylinder, which prism will be greater than one half the cylinder ; because if a square be described about the circle A B C D, the inscribed square will be one half the circumscribed square; and there are erected upon those bases folid parallelepipedons of the fame altitude, viz. the prisms themselves; and fo the prisms are to one another as their bases; therefore the prism erected upon the square A BCD is one half the prism erected upon the square described about the circle ABCD; and the cylinder is less than the prism erected

upon the square described about the circle ABCD: Therefore the prism described upon the square A B C D of the same altitude as the cylinder, is greater than one half that cylinder. Bisect the parts A B; BC, CD, DÅ of the circumference in the points, E, F, G, H, and join A EE B, B F, FC, CG, GD, DH, HA. Then each of the triangles A E B, BFC, CGD, DHA is greater than one half the lege ment of the circle A BCD wherein it stands, as has been already proved [fee 2. 12.) Upori each of the triangles A E B, B FC, CGD, DHA erect a prifm of the same altitude as the cylinder: Then will each of these prisms be greater than one half the respective segment of the cylin. der; because if thro’ the points E, F, G, H parallels be drawn to A B, BC, CD, DA, and parallelograms at them

be

.

last [by 1.

be completed, upon which folid parallelepipedons of the fame altitude as the cylinder are erected; each of these erected parallelepipeduns will be double the prisms which are in the triangles A EB, BFC, CGD, DHA; and the segments of the cylinder are less than the erected folid parallelepipedons; therefore the prisms in the triangles A E B, B FC, CGD, DH Å are greater than one half the respective fegments of the cylinder; and to let the remaining parts of the circumference be bisected, right lines be joined, and upon each of the triangles erect prisms of the fame altitude as the cylinder, and do this always, till at

10.) some segments of the cylinder remain being less than the excess of the cylinder above thrice the cône. Let such fegments be left, and let them be À E, EB, BF, FC, CG, G1, OH, HA. Then the remaining prism, whose base is the polygon A E B FCGDH, and altitude the same as that of the cylinder, is greater than thrice the cone. But [by cor. 7: 12.) the prisın whose base is the polygon A E B FCGDH, and altitude the same as that of the cylinder, is tbrice the pyramid whose base is the polygon A È BFCGDH, and vertex the same as thas of the cone; and therefore the pyramid whose base is the polygon A E B FCGDH, and vertex the same as that of the cone, is greater than the cone whose base is the circle A B CD : but it is less too, because it is contained in it, which is impossible: Therefore the cylinder is not greater than thrice the cone. I lay moreover, that the cylinder is not less than ihrice the cone. For if possible let the cylinder be less than thrice the cone; then inversely the cone will be greater than a third part of the cylinder. Describe ihe square ABCD in the circle Å BCD; shis' Square will be greater than one half the circle A B C D, and upon the square ABCD erect a pyramid, having the same vërtex as that of the cone; then the erected pyramid will be greater than one half the cone; because, as we have already demonstrated, if a square be described about the cisele, the square A B C D will be one half of it. And if folid parallelepipedons be erected upon those squares, having the same altitude with that of the cone, which are also called prisms; that erected upon the square À BCD will be one half of that ere sed upon the square described about the

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