a 11.

h 23. I. C 12.

d def. 3.

Book XI. G; join DG; at the point A, with the right line AB, make the angles BAL, BAK, equal to the angles EDC, EDG; and make AK equal to DG; at the point K, in the plain BAL, raife a perpendicular HK: which make equal to GF; and join HA; then the folid angle at A, which is contained by the plain angles BAL, BAH, HAL, is equal to the folid angle at D, contained by the plain angles EDC, EDF, FDC. For, take the right line AB, equal to DE; AL to DC; and join HB, KB, FE, GC, FC; then, becaufe GF is perpendicular to the plain EDC, the angles FGD, FGE, FGC, are right angles; for the fame reafon, HKA, HKB, HKL, are right angles; and, because the two fides KA, AB, are equal to the two fides GD, DE, and contain equal angles, the bafes BK, EG, are equal; and, because BK, `KH, are equal to EG, GF, each to each, and contain equal angles, the bafes HB, FE, are equal. Again, because AK, KH, are equal to DG, GF, each to each, and contain equal angles, the bafe AH is equal to DF; but AB, AH, are equal to DE, DF, and the base BH equal to EF; therefore the angle BAH is equal to EDF f; but the angle BAL is equal to EDC, and a part BAK equal to EDG; therefore the remainders KAL, GDC, are equal, and the bafe KL to GC; and, because HK, KL, are equal to FG, GC, each to each, and the angle HKL equal to FGC, the base HL is equal to FC; but HA, AL, are equal to FD, DC, and the base HL equal to FC; the angle HAL is equal to FDC; therefore the plain angles BAL, BAH, HAL, containing the folid angle A. are equal to the plain angles EDC, EDF, FDC, containing the folid angle at D, each to each; therefore the folid angle at def. 10. A is made equal to the folid angle at D 8; which was to be done.

4. I.

8, 1.

26.

PROP. XXVII. PRO B.

To defcribe a parallelopipedon from a given right line, fimilar and alike fituate to a folid parallelopipedon given.

It is required to defcribe, from the right line AB, a folid parallelopipedon, fimilar and alike fituate to the given folid parallelopipedon CD.

At the point A, in the given right line AB, make a folid angle A, contained by the plain angles BAH, HAK, KAB, equal to the folid angle at C, fo that the angle BAH be equal to ECF; BAK to ECG; and HAK to FCG; and make BA to

с

AK as EC is to CG b; and KA to AH as GC is to CF; then, Book XIby equality, as BA is to AH, fo is CE to CF. Compleat the parallelogram BH, and folid AL: Then, because the three plain b 12. 6 angles, containing the folid angle at A, are equal to the three plain 22. 5. angles containing the folid angle at C, and the fides about the equal angles proportional, the parallelogram KB is fimilar to the parallelogram GE. For the fame reafon, KH is fimilar to GF, and HB to FE; therefore the three parallelograms of the folid AL are fimilar to the three parallelograms of the folid CD; but these three parallelograms are equal and fimilar to the three oppofite ones 4; therefore the folid AL is fimilar to the folid CD: Which was to be done.

d 24.
e def. 9.

IF

PRO P. XXVIII. T HEOR.

a folid parallelopipedon be cut by a plain passing through the diagonals of two oppofite plains, that folid will be bifected by the plain.

b 24.

If the folid parallelopipedon AB be cut by the plain GAEF, palling through the diagonals GF, AE, of two oppofite plains, then the folid AB is bifected by the plain GAEF. For, because the triangles CGF, GBF, are equal, and likewife the triangles ADE, AEH, and the parallelograms AC, BE, for they are 2 34. 1. oppofite, and likewife GH equal to CE; the prifm contained by the two triangles CGF, ADE, and the three parallelograms GE, AC, CE, is equal to the prifm contained by the triangles GFB, AEH, and the three parallelograms GE, BE, AB cc def. 10. Wherefore, &c.

PRO P. XXIX. THE OR.

OLID parallelopipedons, conftitute upon the fame base, having the fame altitude, and whose infiftent right lines are in the fame right line, are equal to one another.

Let the folid parallelopipedons CM, BF, be conftitute upon the fame base AB, having the fame altitude, and whofe infiftent right lines AF, AG, LM, LN, CD, CE, BH, BK, are in the fame right lines FN, DK; then the folid CM is equal to the folid CN. For, becaufe CH, CK, are parallelograms, DH, EK, are each equal to CB; therefore DH is equal to 34. I. EK. Take EH from, or add to both, then there will remain

HK

b 8. I. € 24.

BOOK XI. HK equal to DE, and the triangle DEC equal to HKB, and the parallelogram DG to HN; but the parallelogram CF is equal to BM, and CG to BN, for they are oppofite; are oppofite; therefore the prifm contained by the two triangles AFG, DEC, and the three parallelograms CF, DG, CG, is equal to the prifm contained by the two triangles LMN, HKB, and the three paralled def. 1o. lograms BM, HN, BN d; add, or take away the folid whose base is the parallelogram AB, oppofite to the parallelogram GEHM; then the folid CM is equal to the folid CN. Wherefore, &c.

a 24.

b 29.

OLID parallelopipedons, conflitute upon the fame base, having the fame altitude, and whofe infiftent right lines are not in the fame right line, are equal to one another.

Let there be folid parallelopipedons CM, CN, having equal altitudes, ftanding on the fame base AB, and whose infiftent right lines AF, AG, LM, LN, CD, CE, BH, BK, are not in the fame right lines; then the folid CM will be equal to the folid CN. For, produce NK, DH, till they meet in R; and draw GE, FM, meeting in X; likewife produce GE, FM, to the points O, P; join AX, LO, CP, BR; then the folid CM, whofe bafe is the parallelogram ACBL, oppofite to the equal parallelogram FDHM, is equal to the folid CO, whose base is the fame parallelogram AB, oppofite to the equal parallelogram XR; for they stand upon the fame base AB, and the infiftent lines AF, AX, LM, LO, CD, CP, BH, BR, are in the fame right lines FO, DR; but the folid CO is equal to the folid CN, for they have the fame bafe AB, oppofite to the parallelograms XR, GK, each equal to AB, and their infiftent right lines AG, AX, CE, CP, LN, LO, BK, BR, are in the fame right lines GP, NR; therefore the folid CM is equal to the folid CN. Wherefore, &c.

S

PROP. XXXI. THEOR.

OLID parallelopipedons, conftitute upon equal bafes, and having the fame altitudes, are equal.

Let AE, CF, be folid parallelopipedons, conftitute upon the equal bases AB, CD; and having the fame altitude, the folid AE is

a

b conft

equal to the folid CF. Firft, let the folids AE, CF, have the Book XI. infiftent lines AG, HK, LM, BE, OP, DF, CG, RS, at right angles to the bafes AB, CD; and let the angle ALB be equal to the angle CRD. Produce CR to T; and make RT equal to LB; compleat the parallelogram DF, equiangular to AB or CD; and the folid I, having its infiftent right lines at right angles to DT, and cf the fame altitude with AE or CF. Then, because the right lines DF, RS, are at right angles to the plain OT, they are parallel and equal; therefore the a 6. parallelogram DS is equal and parallel to CP, TI: Therefore the folid CF is to the folid RI as the base OR is to the bafe DT°; c 25. but OR is equal to DT; therefore the folid CF is equal to the folid RI. But the folid RI is equal to the folid AE; therefore the folid AE is equal to the folid CF: But, if the angle ALB is not equal to CRD, at the point R, with the right line RF, make the angle TRY equal to the angle ALB; and make RY equal to AL; and compleat the parallelogram RX, and folid YW. Produce DR, VT, XY, to the points Q and a ; and compleat the folid ae; then the parallelograms RX, RQ, are equal d; and, becaufe RX is equiangular to AB, and the infift- d 35. F ent lines at right angles to the bafe RX, and of the fame altitude with the folid AE, the plains in the folid AE are equal and fimilar to thefe in the folid YW; therefore the folid YW is equal to the folid AEd. For the fame reason, the folid aW, whofe bafe is the parallelogram RW, and ae, that oppofite to it, is equal to the folid YW, whofe bafe is the parallelogram RW, and Yf, that oppofite to it; for they ftand upon the fame base RW, have the fame altitude, and their infiftent lines Ra, RY, TX, TQ, SZ, SN, We, Wf, are in the fame right lines aX, Zf; but the folid YW is equal to the folid CF; therefore the folid aW is equal to the folid CF.

d def. Io

e II.

Now, let the infiftent lines ML, EB, GA, KH, NO, SD, PC, FR, not be at right angles to the bafes AB, CD, the folid AE will be equal to the folid CF. For, from the points G, K, E, M, P, F, S, N, let fall the right lines Mf, ET, GY, Kg, PX, FW, Na, SI, perpendicular to the plain of the bases AB, CD, meeting them in the points ƒ, Y, g, T, X, W, I, a;e and join ƒY, Yg, gT, Tƒ, Xa, XW, WI, la; then, because GY, Kg, are at right angles to the fame plain, they are parallel f. For the fame reafon, Mf is parallel to ET. But MG isf 6. 6. parallel to EK; therefore the plains MY, KT, of which the one paffes through GY, Yf, and the other through Kg, gT, which are parallel to GY, YF, and not in the fame plain with them, are parallel to one anothers, and equal and parallel to their 15. oppofite plains; therefore ƒE is a parallelopipedon. It may be proved in the fame manner, that aF is a parallelopipedon; but

the

Book XI. the folid GT is equal to the folid PI; for they are upon equal bafes, and of the fame altitude, from what has been demonftrated; h 29. or 30. and the folid GT is equal to the folid AE; and the solid XF to the folid aS; therefore the folid AE is equal to the folid CF. Wherefore, &c.

a 31.

b 25.

SOL

PRO P. XXXII. THE OR.

LID parallelopipedons that have the fame altitude are to one another as their bafes.

Let AB, CD, be folid parallelopipedons, having the fame altitude; as the base AE is to the base CF, fo is the folid AB to the folid CD.

For, to the right line FG, apply the parallelogram FH, equal to the parallelogram AE; upon the bafe FH, compleat the folid GK, of the fame altitude as CD; then the folid AB is equal to the folid GK ; but the folid CK is cut by the plain DG, parallel to the oppofite plain; therefore the folid CD is to the folid GK, as CF is to FH"; that is, AB is to CD as AE is to CF. Wherefore, &c.

a 24,

S

PRO P. XXXIII. THE O R.

IMILAR folid parallelopipedons are to one another in the triplicate ratio of their homologous fides.

Let AB, CD be fimilar folid parallelopipedons, and let the fide AE be homologous to the fide CF; then the folid AB has to the folid CD a triplicate ratio of that which the fide AE has to CF.

For, produce AE, GE, HE, to K, L, M; make EK equal to CF, EL to FN, and EM to FR; and the angle KEL is e qual to CFN; for AEG is equal to CFN; and compleat the parallelogram KL, and the folid KO; then the parallelogram KL is fimilar and equal to the parallelogram CN. For the fame reason, the parallelogram KM is equal and fimilar to the parallelogram CR, and OE to FD; therefore the whole folid KO is equal and fimilar to the folid CD. Likewise, compleat the parallelogram HL, and folids EX, LP, upon the bases GK, KL, having the fame altitude as AB, for EH is an infiftent line to both; but the folid OK is proved fimilar to CD; and AB is