nals, the first to the fourth has a ratio [by 11 def. 5.] the triplicate of the ratio that the first has to the fecond; and therefore the folid A B to the folid Ko will be in the triplicate ratio of AB to E X. But [by 32 11.] as A B is to Ex, fo is the parallelogram A G, to the parallelogram G K, and [by 16] the right line A E to EK: Wherefore the folid A B will be to the folid & o in the triplicate ratio of AE to E K. But the folid Ko is equal to the folid CD, and the right line E K to the right line CF: Wherefore the folid AB has a ratio to the solid c D, triplicate of the ratio of the homologous fide AE to the homologous fide CF. Which was to be demonftrated.

Corollary. From hence it is manifeft, if four right lines be* proportional, as the first is to the fourth, fo is the folid parallelepipedon described upon the firft, to a folid parallelepipedon fimilar and alike fituate described upon the fourth; because the first line to the fourth, is in the triplicate ratio of the first to the second.

*Continual it fhould be.

PROP. XXXIV.

THEOR.

The bafes of equal folid parallelepipedons are recipro cally proportional to their altitudes. And those folid parallelepipedons whofe bafes are reciprocally proportional to their altitudes, are equal to one av other".

Let A B, CD be equal folid parallepipedons: I fay their bafes are reciprocally proportional to their altitudes; that is, as the base EH is to the bafe N P, fo is the altitude of the folid C D to the altitude of the folid A B.

For firft, let the fides ftanding upon the bases AG,

Book XI. Cм be equal to AG; for if when the bafes EH, NP are equal, the altitudes AG, CM are not equal: the folid A B [by 31. 11.] will not be equal to the folid c D. But it is fuppofed to be equal to it. Therefore the altitude C M is not unequal to the altitude AG: Wherefore they are equal: and fo the bafe E H will be to the bafe NP, as CM is to a G. Therefore it is manifeft that the bafes and altitudes of the folid parallelepipedons AB, CD are reciprocally proportional.

But now let the bafe EH be unequal to the base N P, viz. greater than it, and the folid A B is equal to the folid CD: Therefore CM is greater than A G. For if it be not,

the folids A B,CD [by. 31.11.] would again X be unequal: But they are fuppofed to be equal: now make CT equal to A G and upon the base NP altitude CT, complete

and

the folid parallelepipedon QC. Then be

cause the folid A B is equal to the folid c D, and QC is another folid, and [by 7.5.] equal magnitudes have the fame ratio to the fame magnitude; the fol AB. will be to the folid co, as the folid cn is to the folid co. But, [by 32.41.} as the folid A B is to the folid co fo is the bafe EH to the base NP, for the altitudes of the folids A B, CQ are equal. But as the folid CD is to co, fo [by 25.11.] is the base MP to the base PT, and [by 1. 6.] fo is MC to CT: Therefore as the bafe E H is to the bafe NP, fo is the bafe MC to CT. But CT is equal to AG: Therefore as the bafe E H is to the bafe NP, fo is M C to A G. Wherefore the bafes and altitudes of the folid parallelepipedons A.B, CD are reciprocally proportional.

AGAIN. Let the bafes and altitudes of the folid parallelepipedons A B, C D be reciprocally proportional; and let the base EH be to the bafe N P, as the altitude of the fo Jid C D is to the altitude of the folid A B: I fay the folid A B is equal to the folid c D.

For again let the fides ftanding upon the bafes be at right angles to the bafes; and if the bafe EH be equal to

the

the bafe NP; and it is as the bafe E H is to the base P fo is the altitude of the folid C D to the altitude of the folid AB; the altitude of the folid CD will be equal to the altitude of the folid AB. But folid parallelepipedons which ftand upon the same base, and have the fame altitude, are [by 31 11.] equal to one another. Therefore the folid A B is equal to the folid CD.

But now let the base E H not be equal to the base NP, viz. let it be greater; then the altitude of the folid CD is greater than the altitude of the folid A B; that is, CM greater than A G. Again, make CT equal to A G, and complete the folid cq in like manner as before. Then because as the base H is to the bafe N P, fo is CM to AG; and AG is equal to CT; it will be as the bafé E H is to the base NP, fo is MC to CT: But [by 32. 11.] as the bafe EH is to the base N P, fo is the folid A B to the folid CQ; for the folids A B, C Q have equal altitudes. And as MC is to CT, fo (by 1. 6.] is the bafe M P to the base PT, and by [25. 11.] fois the folid CD to the folid' co.. Therefore as the folid A B is to the folid co, fo is the folid CD to the folid c Q. Therefore each of the folids A B, CD have the fame ratio to the folid c Q: Wherefore [by 9. 5.] the folid A B is equal to the folid c D. was to be demonstrated.

Which

But now let the fides standing upon the bafes F E, BL GA, KH, X N, DO, MC, RP not be at right angles to the bafes, and [by 11. 11.] from the points F, G, B, K, X, M, D, R, draw perpendiculars to the planes of the bases E H, NP meeting the planes in the points S, T, Y, V, Q, Z, I, W, and complete the folids F v, xw: I fay, and thus alfo when the folids A B, C D are equal, the bafes are reciprocally proportional to their altitudes: and as the bafe EH is to the bafe NP, fo is the altitude of the folid C D to the altitude of the folid A B. For because the folid A B is equal. to the folid CD, [and by 30. 11.]the folid B T is equal to the folid A B, for they both ftand upon the fame base F K, have the fame altitude, and their fides ftanding upon the' bafes do not terminate in the fame right lines; and the folid DC is equal to the folid Dz; for they both ftand upon the fame base x R, have the fame altitude, and their fides flanding upon the bafe do not terminate in the fame right lines; the folid ET will be equal to the folid Dz. But

Book XI. [by the part aforegoing of this] the bafes of equal folid parallelepipedons at right angles to their bafes, are reciprocally proportional to their altitudes: Therefore as the base

fo is the altitude of the folid Dz, to the altitude of the solid BT. But the folids DZ, BT have the fame altitude as the folids DC, BA. Therefore as the base EH is to the bafe N P, fo is the altitude of the folid D C to the altitude of the folid B A. Wherefore the bafes and altitudes of the folid parallelepipedons A B, C D are reciprocally proportional.

AGAIN. Let the bafes and altitudes of the folid parallelepipedons A B, CD be reciprocally proportional; and let it be as the base E H to the bafe N P, fo is the altitude of the folid CD to the altitude of the folid A B: I say the solid A B is equal to the folid c D.

For the fame conftruction remaining. Becaufe as the base E H is to the base NP, fo is the altitude of the folid CD to the altitude of the folid AB; and the base EH is equal to the base F K, and NP to X R; the base FK will be to the base x R, as the altitude of the folid CD, is to the altitude of the folid AB: for the altitudes of the folids A B, CD are the fame as those of the folids BT, D Z. Therefore as the base F K is to the bafe XR, fo is the altitude of the folid D Z, to the altitude of the folid BT: Wherefore the bafes of the folid parallepipedons BT, Dz are reciprocally proportional to their altitudes. But thofe folid parallelepipedons, whofe fides are at right angles to their bafes, and the bafes are reciprocally proportional to their altitudes, are [by the former part of this] equal to one another. Therefore the folid BT is equal to the folid Dz. But [by 30. 11.] the folid BA is equal to the folid BT; for they stand upon the fame base FK, have the fame alti

tude,

tude, and their fides standing upon the base do not terminate in the fame right lines: Also the folid D z is equal to the folid DC; for they stand upon the fame base x R, have the fame altitude, and their fides ftanding upon the bafe do not terminate in the fame right lines. Therefore the folid A B is equal to the folid c D. Which was to be demonftrated.

Such a theorem as this holds good in triangles, viz. equa triangles have their bases and altitudes reciprocally proportional; and thofe triangles whofe bafes and altitudes are reciprocally proportional, are equal,

PROP. XXXV. THEOR.

If there be two equal plane angles, and right lines be elevated from their vertexes, containing equal angles with the fides of the given plane angles, each to each; and if any points be taken in thofe elevated right lines, and from them be drawn perpendiculars to the planes wherein are the first mentioned angles; and from the points wherein thefe perpendiculars meet the planes, right lines be drawn to the first mentioned angles; these right lines will make equal angles with the elevated lines.

Let there be two equal right lined angles BAC, EDF, and let the right lines A G, DM be elevated from the points A,D, making equal angles with the fides of the first mentioned angles each to each, the angle MDE equal to the angle GAB, and the angle MDF equal to the angle GAC; and take any points G, M in AG, DM, from which draw the perpendiculars GL, MN, to the planes paffing thro' BAC, EDF, meeting

D