14.6.

* 11. 5.

AE is to E K. fo is † the Parallelogram A G to the Parallelogram GK; but as G E is to EL, fo is GK to K L ; and as HE is + to EM, fo is PE to K M ; Therefore, as the Parallelogram A G is to the Parallelogram GK, fo is GK to KL, and PE to K M. 132 of this. But as AG is to G K, fo is the Solid A B to the Solid E X; and as G K is to K L, fo is the Solid E X to the Solid PL; and as PE is to KM, fo is the Solid PL to the Solid KO: Therefore, as the Solid A B is to the Solid E X, fo is * E X to PL, and PL to KO: But if four Magnitudes be continually proportional, the first to the fourth hath + a triplicate Proportion of that which it has to the fecand. Therefore, alfo, the Solid A B to the Solid K O, bath a triplicate Proportion of that which A B has to EX: But as A B is to E X, fo is the Parallelogram A G tỏ the Parallelogram, G K; and fo is the Right Line AE to the Right Line EK: Wherefore the Solid AB to the Solid K O, hath a Proportion triplicate of that which AE has to EK. But the Solid K O is equal to the Solid CD, and the Right Line EK equal to the Right Line C F Therefore, the Solid AB, to the Solid CD, has a Proportion triplicate of that which the homologous Side AE bas to the homologous Side CF; which was to be demonftrated.

+ Def. 11.5.

Coroll. From hence it is manifeft, if four Right Lines be continually proportional, as the firft is to the fourth, fo is a folid Parallelop pedon described upon the firft, to a fimilar folid Parallelopipedon, alike tuate, defcribed upon the fecond; because the first to the fourth, has a Proportion triplicate of that which it has to the second.

PROPOSITION XXXIV.

THEORE M.

The Bafes and Altitudes of equal folid Parallelopi pedons are reciprocally proportional; and thofe folid Parallelopipedons, whofe Bafes and Altitudes are reciprocally proportional, are equal.

LETAB, CD, be equal folid Parallelopipedons. I fay, their Bales and Altitudes are reciprocally proportional; that is, as the Bafe E H is to the Bafe NP, fo is the Altitude of the Solid CD to the Alti tude of the Solid A B.

First, let the infiftent Lines A G, EF, LB, HK, CM, NX, OD, PR, be at Right Angles to their Bafes: I fay, as the Bale E H is to the Bafe NP, fo is CM to A G. For, if the Bafe EH be equal to the Base N P, and the Solid A B is equal to the Solid CD; the Altitude C M hall also be equal to the Altitude AG: For if, when the Bafes EH, NP, are equal, the Altitudes AG, CM, are not fo; then the Solid A B will not be equal to the folid CD, but it is put equal to it: Therefore the Altitude C M is not unequal to the Altitude A G, and fo they are neceffarily equal to one another; and, confequently, as the Bafe E H is to the Bafe N P, fo fhall C M be to A G. But now let the Base EH be unequal to the Bafe NP, and let E H be the greater; then, fince the Solid A B is equal to the Solid CD, CM is greater than A G; for otherwife, it would follow, that the Solids A B, C D, are not equal, which are put fuch: Therefore, make CT equal to A G, and compleat the folid Parallelopipedon VC upon the Bafe NP, having the Altitude CT: Then, because the Solid A B is equal to the Solid CD, and VC is fome other Solid; and fince equal Magnitudes have the fame Proportion to the fame Magui-* 7. 5. tudes; it shall be, as the Solid A B is to the Solid C V, fo is the Solid CD to the Solid CV: But as the Sond AB is to the Solid C V, fo is + the Bafe EH to the † 32 of ili Base N P ; for AB, C V, are Solids having equal Altitudes: And as the Solid CD is to the Solid CV, fo

Q2

+25 of this is the Bafe MP to the Báfe PT, and fo is MC to CT: Therefore, as the Bafe EH is to the Bafe NP, fo is MC to CT. But CT is equal to AG; wherefore, as the Bafe EH is to the Bafe NP, fo is MC to AG: Therefore the Bafes and Altitudes of the equal folid Parallelopipedons AB, CD, are reciprocally proportional.

Now, let the Bafes and Altitudes of the folid Parallelopipedons AB, CD, be reciprocally proportional; that is, let the Bafe EH be to the Bafe N P, as the Antitude of the Solid CD is to the Altitude of the Solid AB. I fay, the Solid A B is equal to the Solid CD.

For, let again the infiftent Lines be at Right Angles to the Bafes; then, if the Bafe EH be equal to the Base N P, and EH is to NP as the Altitude of the Solid CD is to the Altitude of the Solid AB; the Altitude of the Solid C D fhall be equal to the Altitude of the Solid A B. But Solid Parallelopipedons, that ftand upon equal Bafes, and have the fame Alti13 of bis. tude are equal to each other; therefore the Solid A B is equal to the Solid CD.

.

*

But now let the Base E H not be equal to the Bale N P, and let EH be the greater; then the Altitude of the Solid CD is greater than the Altitude of the Solid AB; that is, CM is greater than AG: Again, put CT equal to A G, and compleat the Solid C V, as before; and then, because the Bafe E H is to the Base NP, as MC is to A G, and AG is equal to CT; it fhall be as the Bafe E H is to the Bafe N P, fo is M C to CT: But as the Bafe E H is to the Bafe N P, fo is the Solid AB the Solid CV; for the Solids AB, CV, have equal Altitudes; and as MC is to CT, fo is the Bafe M P to the Bafe PT, and fo the Solid CD to the Solid CV: Therefore as the Solid A B is to the Solid CV, fo is the Solid CD to the Solid C V: But fince each of the Solids A B, CD, has the fame Proportion to CV; the Solid A B fhall be equal to the Solid CD; whence, the two folid Parallelopipedons AB, CD, whofe Pafes and Altitudes are reciprocally proportional, are equal, which was to be demonftrated.

Now, let the infiftent Lines F E, BL, GA, KH, XN, DO, MC, R P, not be at Right Angles to the Bafes; and from the Points F, G, B, K, X, M, D, R, let there be drawn Perpendiculars to the Planes

of

of the Bafes EH, NP, meeting the fame in the Points S, T, Y, V, Q, Z, N, P, and compleat the Solids FV. ΧΩ. Then, I fay, if the Solids AB, CD, be equal, their Bafes and Altitudes are reciprocally proportional: viz. as the Bafe E H is to the Bafe NP, fo is the Altitude of the Solid C D to the Altitude of the Solid A B.

*

For, because the Solid AB is equal to the Solid CD; and the Solid AB is equal to the Solid BT; for they 30 of this. stand upon the fame Bafe FK, and have the fame Altitude; and the Solid DC is equal to the Solid DZ, fince they ftand upon the fame Bafe XR, and have the fame Altitude; therefore the Solid BT fhall be equal to the Solid D Z. But the Bafes and Altitudes of thofe equal Solids, whofe Altitudes are at Right Angles to their Bafes, are + reciprocally proportional; therefore as + From the Bafe FK is to the Bafe X R, fo is the Altitude of what bes the Solid DZ to the Altitude of the Solid B T. But been before the Bafe FK is equal to the Bafe EH, and the Bafe provid XR to the Bafe NP; wherefore, as the Bafe EH is to the Bafe N P, fo is the Altitude of the Solid D Z to the Altitude of the Solid BT. But the Solids D Z, DC, have the fame Altitude, and fo have the Solids BT, BA; therefore the Bafe EH is to the Base N P, as the Altitude of the Solid DC is to the Altude of the Solid A B; and fo, the Bafes and Altitudes of equal folid Parallelopipedons are reciprocally propertional.

Again, let the Bafes and Altitudes of the fold Parallelopipedons AB, C D, be reciprocally proportional; viz. as the Bafe E H is to the Bafe N P, to let the Aititude of the Solid C D be to the Altitude of the Solid AB: I fay, the Solid A B is equal to the Solid CD..

For, the fame Conftruction remaining, because the Bafe EH is to the Bafe NP, as the Altitude of the Solid CD is to the Altitude of the Solid AB; and fince the Bafe EH is equal to the Bafe FK and NP to XR; it fhall be as the Base FK is to the Base XR, fo is the Altitude of the Solid CD to the Altitude of the Solid A B. But the Altitudes of the folids AB, BT, are the fame; as alfo of the Solids C D, DZ; therefore the Bafe F K is to the Bafe XR, as the Altitude of the Solid DZ is to the Altitude of the Solid BT; wherefore the Bafes and Altitudes of the folid Parallelopipe

Q.3

dons

+ From what bas been before

proved.

dons BT, DZ, are reciprocally proportional; but thofe folid Parallelopipedons, whofe Altitudes are at Right Angles to their Bafes, and the Bafes and Altitudes are reciprocally proportional, are equal to + each other. But the Solid B T is equal to the Solid B A, for they ftand upon the fame Bafe F K, and have the fame Altitude; and the Solid DZ is alfo equal to the Solid DC, fince they ftand upon the fame Base XR, and have the fame Altitude: Therefore the Solid A B is equal to the Solid CD; whence folid Parallelopipedons whofe Bafes and Altitudes are reciprocally propertional, are equal; which was to be demonftrated.

PROPOSITION XXXV.

THEOREM.

If there be two plane Angles equal, and from the Vertices of thofe Angles two Right Lines be elevated above the Planes, in which the Angles are, containing equal Angles with the Lines firft given, each to its correspondent one; and if in thofe elevated Lines any Points be taken, from which Lines be drawn perpendicular to the Planes in which the Angles firft given are, and Right Lines be drawn to the Angles first given from the Points made by the Perpendiculars in the Planes; thofe Right Lines will contain equal Angles to the elevated Lines.

LET BAC, EDF, be two equal Right-lined plane

Angles, and from A and D, the Vertices of thofe Angles, let two Right Lines, A G and D M be elevated above the Planes of the said Angles, making equal Angles with the Lines firft given, each to its correfpondent one; viz. the Angle MDE equal to the Angle GA B, and the Angle M DF to the Angle GAC and take any Points G and M in the Right Lines AG, DM; from which let G L and M N be drawn perpendicular to the Planes paffing thro' BAC, EDF, meeting the fame in the Points L and N; and join LA and ND. I fay, the Angle G A L is equal to the Angle M D N.

Make