*

whose Base is the Parallelogram ACBL, being oppofite to XPRO, for they ftand upon the fame Base ACBL; and the infiftent Lines AF, AX, LM, LO, CD, CP, BH, BR, are in the fame Right Lines, FO, DR; but the Solid CO, whofe Bafe is the Parallelogram ACBL, being oppofite to XPRO, is equal to the Solid CN, whofe Bafe is the Paral-* 29 of this, lelogram ACBL, being oppofite to GEKN; for they ftand upon the fame Base ACBL, and their infiftent Lines AG, AX, CE, CP, LN, LO, BK, BR, are in the fame Right Lines GP, NR; wherefore the Solid CM fhall be equal to the Solid CN. Therefore, folid Parallelepipedons, being conftituted upon the fame Bafe, and having the fame Altitude, whofe infiftent Lines are not placed in the fame Right Lines, are equal to one another; which was to be de monftrated.

PROPOSITION XXXI.

THEOREM.

Solid Parallelepipedons, being conftituted upon equal Bafes, and having the fame Altitude, are equal to one another.

ET AE, CF, be folid Parallelepipedons conftituted upon the equal Bafes AB, CD, and having the fame Altitude. I fay, the Solid AE is equal to the Solid CF.

First, let HK, BE, AG, LM, OP, DF, C2, RS; be at Right Angles to the Bafes AB, CD; let the Angle ALB not be equal to the Angle CRD, and produce CR to T, so that RT be equal to AL: Then make the Angle TRY, at the Point R, in the Right Line RT, equal to the Angle AL B; make * 23. I. RY equal to LB; draw XY thro' the Point Y parallel to RT, and complete the Parallelogram RX, and the Solid Y. Therefore because the two Sides TR, RY, are equal to the two Sides AL, LB, and they contain equal Angles, the Parallelogram RX fhall be equal and fimilar to the Parallelogram HL. And again, becaufe AL is equal to RT, and LM to RS, and they contain equal Angles, the Paral4

lelogram

lelogram R fhall be equal and fimilar to the Parallelogram AM. For the fame Reason, the Parallelogram LE is equal and fimilar to the Parallelogram SY. Therefore three Parallelograms of the Solid AE, are equal and fimilar to three Parallelograms of the SolidY; and fo the three oppofite ones of one t24 of this. Solid, are + alfo equal and fimilar to the three oppofite ones of the other. Therefore the whole folid Parallelepipedon AE is equal to the whole folid Parallelepipedon Y. Let DR, XY, be produced, and meet each other in the Point 9, and let TQ be drawn thro' T parallel to D, and produce TQ, OD, till they meet in V, and complete the Solid

RI: Then the Solid, whofe Bafe is the Pa29 of this. rallelogram R, and or is that oppofite to it, is equal to the Solid + Y, whofe Bafe is the Parallelogram R, and Y is that oppofite to it; for they ftand upon the fame Base R, have the fame Altitude, and their infiftent Lines R, RY, TQ, TX, SZ, SN, г, , are in the fame Right Lines

X, Z: But the Solid Y is equal to the Solid AE; and fo AE is equal to the Solid . Again, because the Parallelogram RY XT is equal to the Parallelogram T, for it ftands on the fame Base RT, and between the fame Parallels RT, X; and the Parallelogram RYXT is equal to the Parallelogram CD, because it is alfo equal to AB; the Parallelogram T is equal to the Parallelogram CD, and DT is fome other Parallelogram. Therefore as the Base CD is to the Base DT, fo is T to TD; and because the folid Parallelepipedon CI is cut by the Plane RF, being parallel to two oppofite Planes, it *25 of thi• fhall be as the Bafe CD is to the Base DT, fo is the Solid CF to the Solid RI. For the fame Reafon, because the folid Parallelepipedon I is cut by the Plane R✶ parallel to two oppofite Planes; as the Bafe T is to the Bafe DT, fo fhall* the Solid Q be to the Solid RI; but as the Bafe CD is to the Bafe DT, fo is the Bafe T to TD. Therefore as the Solid CF is to the Solid RI, fo is the Solid to the Solid RI; and fince each Q, has the fame Proportion to Solid CF is equal to the Solid

has been proved equal to the

of the Solids CF, the Solid RI, the

; but the Solid Solid AE; there

for the Solid AE fhall be + equal to the Solid CF.

But now let the infiftent Lines AG, HK, BE, LM, CN, OP, DF, RS, not be at Right Angles to the Bases AB, CD. I fay, again, that the Solid AE is equal to the Solid CF. Let there be drawn from the Points K, E, G, M, P, F, N, S, to the Plane wherein are the Bases AB, CD, the Perpendiculars KE, ET, GY, MO, SI, F, N 2, PX, meeting the Plane in the Points F, T, Y, 4, I, 4, 0, X, and join ET, YO, EY, To, X+, Xo, I, I; then the Solid K is equal to the Solid PI, for they stand on equal Bafes KM, PS, have the fame Altitude, and the infiftent Lines are at Right Angles to the Bafes; but the Solid K is equal to the Solid AE, and the Solid PI to the Solid CF, fince they stand up- ‡ 29 of this on the fame Bafe, have the fame Altitude, and their infiftent Lines are in the fame Right Lines. Therefore the Solid AE fhall be equal to the Solid CF. Wherefore, folid Parallelepipedons, being constituted upon equal Bafes, and having the fame Altitude, are equal to one another; which was to be demonftrated.

PROPOSITION XXXII.

THEOREM.

Solid Parallelepipedons, that have the fame Altitude, are to each other as their Bases.

ET AB, CD, be folid Parallelepipedons that

another as their Bases; that is, as the Base AE is to the Bafe CF, fo is the Solid AB to the Solid CD.

*

For apply a Parallelogram FH to the Right Line FG, equal to the Parallelogram AE, and complete the folid Parallelepipedon GK upon the Base FH, having the fame Altitude as CD has. Then the Solid AB is equal to the Solid GK; for they stand * 31 of this. upon equal Bafes AE, FH, and have the fame Altitude; and fo because the folid Parallelepipedon CK is cut by the Plane DG, parallel to two oppofite Planes, it fhall be + as the Bafe HF, is to the Bafet 25 of this FC, fo is the Solid HD to the Solid DC; but the

Bafe

Bafe F H is equal to the Bafe AE, and the Solid AB to the Solid FK. Therefore as the Bafe AE is to the Bafe CF, fo is the Solid AB to the Solid CD. Wherefore, folid Parallelepipedons, that have the fame Altitude, are to each other as their Bafes; which was to be demonstrated.

PROPOSITION XXXIII.

THEOR E M.,

Similar folid Parallelepipedons, are to one another in the triplicate Proportion of their homologous Sides.

ET AB, CD, be folid Parallelepipedons, and let the Side AE be homologous to the Side CF. I fay, the Solid AB, to the Solid CD, hath a Proportion triplicate of that which the Side AE has to the Side CF.

For produce AE, GE, HE, to EK, EL, EM; and make EK equal to CF, and EL to FN, and EM to FR; and let the Parallelogram KL, and likewise the Solid KO be compleated. Then because the two Sides KE, EL, are equal to the two Sides CF, FN, and the Angle KEL equal to the Angle CFN; fince the Angle AEG is alfo equal to the Angle CFN, because of the Similarity of the Solids AB, CD) the Parallelogram KL fhall be fimilar and equal to the Parallelogram CN. For the fame Reafon, the Parallelogram KM is equal and fimilar to the Parallelogram CR, and the Parallelogram OE to DF. Therefore three Parallelograms of the Solid KO, are equal and fimilar to three Parallelograms of 24 of this, the Solid CD: But thofe three Parallelograms are equal and fimilar to the three oppofite Parallelograms. Therefore the whole Solid KO is equal and fimilar to the whole Solid CD. Let the Parallelogram GK be compleated, as alfo the Solids EX, LP, upon the Bases GK, KL, having the fame Altitude as AB. And fince, because of the Similarity of the Solids. AB and CD, it is as AE is to CF, fo is EG to FN; and fo EH to FR, and F C is equal to E K, and FN to EL, and FR to EM. It shall be as AE is to

*

EK,

EK, fo is the Parallelogram A G to the Parallelo- † 1. 6. gram GK; but as GE is to EL, fo is GK to KL; and as HE is to EM, fo is PE to KM. Therefore as the Parallelogram AG is to the Parallelogram GK, fo is GK to KL, and PE to KM. But as AG is to GK, fo is the Solid A B to the Solid EX; and ‡ 32 of this. as GK is to KL, fo is the Solid EX to the Solid PL; and as PE is to KM, fo is the Solid PL to the Solid KO. Therefore as the Solid AB is to the Solid E X, fo is* EX to PL, and PL to KO. But * 11. 5. if four Magnitudes be continually proportional, the firft to the fourth hath † a triplicate Proportion of † Def. 11. g

that which it has to the fecond. Therefore alfo the Solid AB to the Solid KO, hath a triplicate Proportion of that which AB has to EX: But as AB is to EX, fo is the Parallelogram A G to the Parallelogram GK; and fo is the Right Line AE to the Right Line EK. Wherefore the Solid AB, to the Solid KO, hath a Proportion triplicate of that which AE has to EK; but the Solid KO is equal to the Solid CD, and the Right Line EK equal to the Right Line CF. Therefore the Solid AB to the Solid CD, has a Proportion triplicate of that which the homologous Side AE has to the homologous Side CF; which was to be demonftrated.

Coroll. From hence it is manifeft, if four Right Lines be proportional, as the firft is to the fourth, fo is a folid Parallelepipedon described upon the firft, to a fimilar folid Parallelepipedon alike, fituate, - defcribed upon the fecond; because the firft to the fourth, has a Proportion triplicate of that which it has to the fecond.

PRO