PROPOSITION XCV.-THEOREM.

All the plane angles which form any solid angle, are together less than four right angles.

E

B

Let there be a solid at the point A contained by the plane 4s BAC, CAD, DAE, EAB, these Ls, taken together, are four Ls. For, through any point B in AB, let a plane be extended to meet the sides of the solid in the lines of common section BC, CD, DE, EB, thereby forming the pyramid BCDE-A; and from any point O within the rectilineal figure BCDE, which is the base of the pyramid, let the Is OB, OC, OD, OE, be drawn to the angular points of the figure, which will thereby be divided into as many As as the pyramid has sides. Then each of the solid Ls at the base of the pyramid is contained by three plane Ls, any two of them are together the third, (Prop. 94.) Thus, ABE, ABC, are together EBC. .. the Ls at the bases of the As which have their vertex at A, are together the s at the bases of the As which have their vertex at 0. But all the Ls of the former are together all the Ls of the latter. .. the remaining Ls at A are together the remaining Ls at O, that is, four right angles, (Prop. 1, cor. 3.)

PROPOSITION XCVI.-THEorem.

If two solid angles be each of them contained by three plane angles, and have these angles equal, each to each, and alike situated, the two solid angles are equal.

Let there be a solid angle at A contained by the three plane Ls BAC, CAD, DAB, and a solid angle at E, contained by the three plane Ls FEG, GEH, HEF, = the former, each to each, and alike situated, these solid angles are equal.

AA

For let the s AB, AC, AD, EF, EG, EH, be all equal, and let their extremities be joined by the lines BC, CD, DB, FG, GH, HF; and thus there are formed two isosceles pyramids, BCD-A, and

FGH-E. Upon the bases BCD, FGH, let the 8 AK,

EL, fall from the vertices A, E. Then the Ld As AKB, AKC, AKD, have equal hypotenuses, and the side AK common, their other sides, KB, KC, KD, (Prop.39, cor. 2), are also equal; ... K is the centre of the circle that circumscribes the ▲ BDC. In like manner, L is the centre of the circle that circumscribes the AFHG. But these As are equal in every respect. For the sides BC, FG, are equal, they are the bases of equal and similar As BAC, EFG; and for the same reason, CD=GH, and BD=HF. If, .. the pyramid BCD-A be applied to the pyramid FGH-E, their bases BCD and FGH will coincide. Also the point K will fall on L, in the plane of the circle about FGH, no other point than the centre is equally distant from the three points in the circumference, the perpendicular KA will coincide with LE, (Prop. 86, cor. 2), and the point A will coincide with E, the rLd As AKB, ELF, have equal hypotenuses, (Prop. 39, cor. 2), and BK=FL. But the points B, D, C, coincide with F, H, G, each with each; .. the Is AB, AC, AD, coincide with EF, EG, EH, and the plane Ls BAC, CAD, DAB, with FEG, GEH, HEF, each with each. Consequently the solid Ls themselves coincide and are equal.

Cor. 1. If two solid angles, each contained by three plane angles, have their linear sides, or the planes that bound them, parallel each to each, the solid angles are equal, (Props. 84 and 89.)

Cor. 2. If two solid angles, each contained by the same number of plane angles, have their linear or plane sides parallel, each to each, the solid angles are equal. For each of the solid angles may be divided into solid Ls, each contained by three plane Ls, and the parts being equal, and alike situated, the wholes are equal.

PROPOSITION XCVII.—THEOREM.

If two triangular prisms, ABC-DEF, and GHK-LMN, have the plane angles BAC, CAD, DAB, and HGK, KGL, LGH, and also the linear sides AB, AC, AD, and GH, GK, GL, about two of their solid angles at A and G equal, each to each, and alike situated, the prisms are equal and similar.

For, since the solid Ls A, G, are each contained by three plane Ls, which are

E

B

M

F

B

equal each to each, these solid Ls being applied to each other, coincide, (Prop. 96.) ... the Is AB, AC, AD, coincide with the s GH, GK, GL, each with each. But AB coinciding with GH, and the point D with L, the DE must fall upon LM, (Ax. 16), and BE on HM; .. the point E coincides with M, and the point F with N. Consequently the rectilineal figures which bound the one prism, coincide with, and are equal and similar to the rectilineal figures which bound the other, each to each; the solid s of the one coincide with and are equal to the solid Ls of the other, each to each; and the solids themselves are equal and similar.

Cor. 1. If two triangular pyramids have the plane angles and linear sides about two of their solid angles equal, each to each, and alike situated, the pyramids are equal and similar.

Cor. 2. If two triangular prisms have the linear sides about two of their solid angles both equal and parallel, each to each, the prisms are equal and similar.

Cor. 3. If two pyramids have the linear sides about their vertical angles both equal and parallel, each to each, the pyramids are equal and similar.

PROPOSITION XCVIII.-THEOREM.

The opposite sides, as ABCD, EFGH, of a parallelopiped ABCD-EFGH, are similar and equal parallelograms, and the diagonal plane divides it into two equal and similar prisms.

For, since the opposite planes, AF, DG, are parallel, and cut by a third plane BD, the common sections AB, DČ, are в parallel, (Prop. 89); for a similar reason AD and BC are parallel; hence the figure ABCD is a

.

F

=

E

G

H

In like manner, all the figures which bound the solid are s. Hence AD =EH, and DC=HG, and the LADC is EHG, (Prop. 85); the As ADC, EHG, (Prop. 5), and consequently the s ABCD, EFGH, are equal and similar. Again, AE, CG, are each and || DH; they are and || one another; ... the figure ACGE is a □, (Prop. 34, cor. 1), and divides the whole parallelopiped AG into two triangular prisms ABC-EFG, and ACD-EGH, and these prisms are equal and similar, (Prop. 97); the plane Ls and linear sides about the solid Zs at F and D

are equal each to each. For the plane Дs EFG, ADC, are = one another, each being = the EHG, and the linear sides FG, DA, are = one another, each being =EH, and so of the others.

Cor.

1. The opposite solid angles of a parallelopiped

are equal.

Cor. 2. Every rectangular parallelopiped is bounded by rectangles.

Cor. 3. The ends of a prism are similar and equal figures, and their planes parallel.

Cor. 4. Every parallelopiped is a quadrangular prism, of which the ends are parallelograms, and conversely.

Cor. 5. If two parallelopipeds have the plane angles and linear sides about two of their solid angles equal, each to each, and alike situated, or the linear sides, about two of their solid angles, both equal and parallel, each to each, the solids are equal and similar.

Cor. 6. If from the angular points of any rectilineal figure, there be drawn straight lines above its plane, all equal and parallel, and their extremities be joined, the figure so formed is a prism. If the rectilinear figure be a parallelogram, the prism is a parallelopiped. If the rectilineal figure be a rectangle, and the straight lines be perpendicular to its plane, the prism is a rectangular parallelopiped.

Cor. 7. Every triangular prism is equal to another triangular prism, having its base equal to the half of one of the sides of the prism, and its altitude the perpendicular distance of the opposite edge. The prism FEG-BAC= ABF-DCG, (formed by the diagonal plane AFGD), which has for its base half the side ABFE, and for its altitude the perpendicular distance of the edge CG; for they are each half of the parallelopiped FD.

PROPOSITION XCIX.

Parallelopipeds ABCD-EFGH, and ABCD-KLMN upon the same base ABCD, and between the same parallel planes AC,

EM, are equal.

Because the s EF, GH, KL, MN, " are || AB, CD, they are one another, and being all in the same plane, NK, ML, being produced, will .. meet both EF and HG; let them meet the former in O, P, and the lat

N

M

R, Q, and let AO, BP, CQ, DR, be joined. The ABCD-OPQR, is a parallelopiped. For, by hypothe plane EQ is || AC, the plane of the parallels IQ is the plane of the parallels AB-EP, and ane AD-NO parallel to BC-MP. Hence AEOand BFP-CGQ are two triangular prisms, which the linear sides AE, AD, AO, about the solid A, equal and the linear sides BF, BC, BP, about olid LB, (Prop. 98), each to each. Consequently prisms are equal, (Prop. 97), and each of them being away from the whole solid ABCD-EPQH, the reers, the parallelopipeds AG, AQ, are equal. In the manner, the parallelopipeds AM and AQ may be d equal.. the parallelopipeds AG, AM, are equal

e another.

r. Triangular prisms, upon the same base, and bethe same parallel planes, are equal.

r, if two planes be made to pass, the one through EG, and the other through AC, KM, they will bisect arallelopipeds AG, AM, (Prop. 98); .. the prisms -EHG, ADC-KNM, will be equal.

PROPOSITION C.-THEOREM.

rallelopipeds ABCD-EFGH, and ABKL-OPQR, equal bases, and between the same parallel planes, EQ, are equal.

SE I. When the bases have side AB common, and lie ben the same parallel lines AB, these parallelopipeds AG, are equal.

D

or let the planes EAL, FBK,
= the plane HC of the former,
he lines of common section A
KN, and the plane HF in

R

P

N

K

lines EM, FN. Then since EA, AL, are || FB, BK, planes are parallel, and the plane EB is || the plane HK; e figure AN is a parallelopiped. But ADL-EHM, BCK-FGN, are two triangular prisms, which have linear sides, AD, AE, AL, about the solid ▲A, both = || the linear sides, BC, BF, BK, about the solid [B, to each; . these prisms are equal, (Prop. 97, cor. 2); each of them being taken away from the whole solid, KD-EFNH, the remainders, the parallelopipeds AG, I, are equal. But AN is =AQ, (Prop. 99); AG is 1Q.