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C 12.

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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 b; and

make AK equal to DG; at the point K, in the plain BAL, h 23. I. raise a perpendicular HK: which make equal to GF; and join

HA; then the solid angle at A, which is contained by the plain angles BAL, BAH, HAL, is equal to the solid 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, because GF is perpendicular d def. 3.

to the plain EDC 4, the angles FGD, FGE, FGC, are right angles; for the same reason, HKA, HKB, HKL, are right angles; and, because the two sides KA, AB, are equal to the two sides GD, DE, and contain equal angles, the bases BK, EG, are equal ; and, because BK, KH, are equal to EG, GF, each to each, and contain equal angles, the bales HB, FE, are equal. Again, because AK, KH, are equal to DG, GF, each to each, and contain equal angles, the bale 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; but the angle BAL is equal to EDC, and a part BAK equal to EDG; therefore the remainders KAL, GDC, are equal, and the base KL to GC'; and, because HK, KL, are equal to FG, GC, cach 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 solid angle at D, each to each ; therefore the solid angle at def. 10. A is made equal to the folid angle at D 8; which was to be

done.

4. 1.

1.

PRO P. XXVII. PROB.

То o describe a parallelopipedon from a given right line, fimilar

and alike ftuate to a solid parallelopipedon given.

It is required to describe, from the right line AB, a solid parallelopipedon, fimilar and alike situate to the given folid parallelopipedon CD.

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

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AK as EC is to CG b; and KA to AH as GC is to CF ; then“, Book XI• by equality, as BA is to AH, so is CE to CF. Compleat the parallelogram BH, and folid AL: Then, because the three plain b 12. 6angles, containing the folid angle at A, are equal to the three plain C 22. 5. angles containing the folid angle at C, and the fides about the equal angles proportional, the parallelogram KB is similar to the parallelogram GE. For the same reason, KH is similar to GF, and HB to FE ; therefore the three parallelograms of the solid AL are fimilar to the three parallelograms of the folid CD; but these three parallelograms are equal and similar to the three opposite ones d; therefore the solid AL is similar to the solid CD': Which was to be done.

d 24.

e def. 9.

PRO P. XXVIII. T H E O R.

IF a solid parallelopipedon be cut by a plain paling through the di

agonals of two opposite plains, that solid will be bisečted by the plain.

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'If the folid parallelopipedon AB be cut by the plain GAEF, passing through the diagonals GF, AE, of two opposite plains, then the solid AB is bisected by the plain GAEF. For, because the triangles CGF, GBF, are equal, and likewise the triangles ADE, AEH", and the parallelograms AC, BE 6, for they are: 34. Ja opposite, and likewise GH equal to CE ; the prism contained by the two triangles CGF, ADE, and the three parallelograms GE, AC, CE, is equal to the prism contained by the triangles GFB, AEH, and the three parallelograms GÉ, BE, AB “. Wherefore, &c.

b 24.

c def. 10.

PRO P. XXIX. THE O R.

OLID parallelopipedons, constitute upon the same base, ba

ving the fame altitude, and whose insistent right lines are in the same right line, are equal to one another.

Let'the folid parallelopipedons CM, BF, be constitute upon the same base AB, having the same altitude, and whose insistent right lines AF, AG, LM, LN, CD, CE, BH, BK, are in the same right lines FN, DK, then the solid CM is equal to the folid CŇ. For, because CH, CK, are parallelograms, DH, EK, are each equal to CB“; therefore DH is equal to a 34. 1. EK. Take EH from, or add to both, then there will remain

HK

b 8. I. c 24

Book XI. HK equal to DE, and the triangle DEC equal to HKB b, and

the parallelogram DG to HN; but the parallelogram CF is equal to BM, and CG to BN, for they are opposite; therefore the prism contained by the two triangles AFG, DEC, and the three parallelograms CF, DG, CG, is equal to the prism con

tained by the two triangles LMN, HKB, and the three paralled def. Io. lograms BM, HN, BN ; add, or take away the solid whose base

is the parallelogram AB, opposite to the parallelogram GEHM; then the folid ČM is equal to the solid CN. Wherefore, &c.

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OLID parallelopipedons, constitute upon the same base, hain the saine right line, are equal to one another,

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Let there be folid parallelopipedons CM, CN, having equal altitudes, standing on the same base AB, and whose insistent right lines AF, AG, LM, LN, CD, CE, BH, BK, are not in the same 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; likewise produce GE, FM, to the points o, P; join AX, LO, CP, BR; then the folid CM, whose base is the parallelogram ACBL, opposite to the equal parallelogram FDHM“, is equal to the folid CO b, whose base is the same parallelogram AB, opposite 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 same right lines FO, DR; but the folid CO is equal to the folid CN, for they have the same base AB, opposite 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 same right lines GP, NR; therefore the folid CM is equal to the solid CN. Wherefore, &c.

PROP. XXXI. THEOR.

S

OLID parallelopipedons, constitute upon equal bases, and having the same altitudes, are equal.

Let AE, CF, be solid parallelopipedons, constitute upon the equal bases AB, CD; and having the same altitude, the solid AE is

equal

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equal to the folid CF. First, let the solids AE, CF, have the Book XI. infiftent lines AG, HK, LM, BE, OP, DF, CG, RS, at right angles to the bases AB, CD, and let the angle ALB be equal to the angle CRD. Produce CR to T; and make RT equal LB; compleat the parallelogram Df, equiangular to AB or CD, and the folid ki, having its infiftent right lines at right angles to DT, and cf the same altitude with AE or CF. Then, because the right lines DF, RS, are at right angles to the plain OT, they are parallel á and equal ; therefore the a 6.

b conft. 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 bale DT°; c 25. but OR is equal to DT ; therefore the folid CF is equal to the folid RI. But the solid RI is equal to the solid AE ; therefore the solid 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 solid 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, because RX is equiangular to AB, and the infift-d 35. F. ent lines at right angles to the base RX, and of the same altitude with the folid AE, the plains in the folid AE are equal and similar to these in the folid YW ; therefore the folid YW is e

d def. Io qual to the solid AEd. For the same reason, the folid aW, whose base is the parallelogram RW, and ae, that opposite to it, is equal to the folid YW, whose base is the parallelogram RW, and Yf, that opposite to it; for they stand upon the fame base RW, have the fame altitude, and their infiftent lines Ra, RY, TX, TQ, SZ, SN, We, Wf, are in the same right lines aX, Zf; but the folid YW is equal to the solid CF; therefore the solid aW is equal to the solid CF.

Now, let the insistent lines ML, EB, GA, KH, NO, SD, PC, FR, not be at right angles to the bases AB, CD, the solid AE will be equal to the solid 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, ÇD®, meeting them in the points f, Y, g, T, X, W, I, a ;e II. and join fY, Yg, gT, Tf, Xa, XW, WI, Ia ; then, because GY, Kg, are at right angles to the same plain, they are paralJelf. For the same reason, Mf is parallel to ET.' But MG is f 6. 6. parallel to EK; therefore the plains MY, KT, of which the one passes through GY, Yf, and the other through Kg, gT, which are parallel to GY, YF, and not in the same plain with them, are parallel to one anothers, and equal and parallel to their 8 15 opposite plains ; therefore fE is a parallelopipedon. It may be proved in the same manner, that aF is a parallelopipedon ; but

the

Book XI. the folid GT is equal to the solid PI; for they are upon equal

bases, and of the same altitude, from what has been demonstrah 29. or 30. ted; 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 solid
CF. Wherefore, &c.

PRO P. XXXII. THEO R.

SOLID parallelopipedons that have the same 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 solid AB to the solid CD.

For, to the right line FG, apply the parallelogram FH, equal to the parallelogram AE; upon the base FH, compleat the solid GK, of the same altitude as CD; then the solid AB is equal to the solid GK a; but the folid CK is cut by the plain DG, parallel to the opposite plain ; therefore the folid CD is to the solid GK, as CF is to FHb; that is, AB is to CD as AE is to CF. Wherefore, &c.

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PRO P. XXXIII. THE O R.

SIMILAR solid parallelopipedons are to one another in the

triplicate ratio of their homologous fides.

Let AB, CD be similar solid parallelopipedons, and let the fide AE be homologous to the side CF; then the folid AB has to the solid CD a triplicate ratio of that which the side 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 similar and equal to the parallelogram CN. For the same reason, the parallelogram KM is, equal and similar to the parallelogram CR, and OE to FD; therefore the whole folid KO is equal and fimilar to the solid CD. Likewise, compleat the parallelogram HL, and solids EX, LP, upon the bafes GK, KL, having the same altitude as AB, for EH is an insistent line to both; but the folid OK is proved similar to CD; and AB is

a 24.

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