plane (I. Def. 1). But the straight lines GH, GK, in that plane meet it; therefore each of the angles BGH, BGK, is a right angle. And because BA is parallel (I. 8) to GH (for each of them is parallel to DE), the angles GBA, BGH, are together equal (Pl. Ge. I. 29) to two right angles. And BGH is a right angle; therefore also GBA is a right angle, and GB perpendicular to BA. For the same reason, GB is perpendicular to BC. Since therefore the straight line GB stands at right angles to the two straight lines BA, BC, that cut one another in B, GB is perpendicular (I. 4) to the plane through BA, BC; and it is perpendicular to the plane through DE, EF; therefore BG is perpendicular to each of the planes through AB, BC, and DE, EF. But planes to which the same straight line is perpendicular, are parallel (I. 12) to one another. Therefore the plane through AB, BC, is parallel to the plane through DE, EF.

PROPOSITION XIV. THEOREM.

If two parallel planes be cut by another plane, their common sections with it are parallels.

Let the parallel planes AB, CD, be cut by the plane EFHG, and let their common sections. with it be EF, GH; EF is parallel to GH.

F

B

H

D

For the straight lines EF and GH are in the same plane, namely, EFHG, which cuts the planes AB and CD; and they do not meet though produced ; for the planes in which they are, do not meet; therefore EF and A GH are parallel.

PROPOSITION XV. THEOREM.

E

If two parallel planes be cut by a third plane, they have the same inclination to that plane.

Let AB and CD be two parallel planes, and EH a third plane cutting them. The planes AB and CD are equally inclined to EH.

Let the straight lines EF and GH be the common sections of the plane EH with the two planes AB and CD; and from K any point in EF, draw in the plane EII the straight line KM at right angles to EF, and let it meet GH in

E

G

A

C

K

M

N

O

D

H

L; draw also KN at right angles to EF in the plane AB; and through the straight lines KM, KN, let a plane be made to pass cutting the plane CD in the line LO. And because EF and GH are the com- B mon sections of the plane EH with the two parallel planes AB and CD, EF is parallel to GH (I. 14). But EF is at right angles to the plane that passes through KN and KM (I. 4), because it is at right angles to the lines KM and KN; therefore GH is also at right angles to the same plane (I. 7), and it is therefore at right angles to the lines LM, LO, which it meets in that plane. Therefore, since LM and LO are at right angles to LG, the common section of the two planes CD and EH, the angle OLM is the inclination of the plane CD to the plane EH (I. Def. 4). For the same reason the angle MKN is the inclination of the plane AB to the plane EH. But because KN and LO are parallel, being the common sections of the parallel planes AB and CD with a third plane, the interior angle NKM is equal to the exterior angle OLM; that is, the inclination of the plane AB to the plane EH, is equal to the inclination of the plane CD to the same plane EH.

PROPOSITION XVI. THEOREM.

If two straight lines be cut by parallel planes, they shall be cut in the same ratio.

Let the straight lines AB, CD, be cut by the parallel planes GH, KL, MN, in the points A, E, B; C, F, D. As AE is to EB, so is CF to FD.

K

A

F

H

Join AC, BD, AD, and let AD meet the plane KL in the point X; and join EX, XF. Because the two parallel planes KL, MN, are cut by the plane EBDX, G the common sections EX, BD, are parallel (I. 3). For the same reason, because the two parallel planes GH, KL, are cut by the plane AXFC, the common sections AČ, XF, are parallel. And because EX is parallel to BD, M/ a side of the triangle ABD, as AE to EB, so is (Pl. Ge. VI. 2) AX to XD. Again, because XF is parallel to AC, a side of the triangle ADC, as AX to XD, so

E X

B

D

N

is CF to FD. And it was proved that AX is to XD, as Therefore (Pl. Ge. V. 11), as AE to EB, so

AE to EB.

is CF to FD.

PROPOSITION XVII. THEOREM.

If a straight line be at right angles to a plane, every plane which passes through it shall be at right angles to that plane. Let the straight line AB be at right angles to a plane CK; every plane which passes through AB shall be at right angles to the plane CK.

D G A H

K

C F B E

Let any plane DE pass through AB, and let CE be the common section of the planes DE, CK; take any point F in CE, from which draw FG in the plane DE at right angles to CE. And because AB is perpendicular to the plane CK, therefore it is also perpendicular to every straight line meeting it in that plane (I. Def. 1); and consequently it is perpendicular to CE. Wherefore ABF is a right angle; but GFB is likewise a right angle; therefore AB is parallel to FG. And AB is at right angles to the plane CK; therefore FG is also at right angles to the same plane (I. 7). But one plane is at right angles to another plane when the straight lines drawn in one of the planes, at right angles to their common section, are also at right angles to the other plane (I. Def. 2); and any straight line FG in the plane DE, which is at right angles to CE, the common section of the planes, has been proved to be perpendicular to the other plane CK; therefore the plane DE is at right angles to the plane CK. In like manner, it may be proved that all the planes which pass through AB are at right angles to the plane CK.

PROPOSITION XVIII. THEOREM.

If two planes cutting one another be each of them perpendicular to a third plane, their common section shall be perpendicular to the same plane.

Let the two planes AB, BC, be each of them perpendicular to a third plane, and let BD be the common section of the first two; BD is perpendicular to the plane ADC. From D in the plane ADC, draw DE perpendicular to

B

AD, and DF to DC. Because DE is perpendicular to AD, the common section of the planes AB and ADC; and because the plane AB is at right angles to ADC, DE is at right angles to the plane AB (I. Def. 2), and therefore also to the straight line BD in that plane (I. Def. 1). For the same reason, DF is at right angles to DB. Since A F E C BD is therefore at right angles to both the lines DE and DF, it is at right angles to the plane in which DE and DF are, that is, to the plane ADC (I. 4).

SECOND BOOK.

DEFINITIONS.

1. A solid is a figure that has length, breadth, and thick

ness.

2. A solid angle is formed by the meeting, in one point, of more than two plane angles not in the same plane.

When the solid angle is contained by three plane angles, it is called a trihedral angle; and if it be contained by more than three, it is said to be polyhedral. The plane angles are called the sides or faces; and the common sections of their planes, the edges of the angle.

3. A pyramid is a solid having a rectilineal figure for its base, and, for its sides, it has triangles that meet in a point without the base, and having for their bases the sides of the base of the pyramid.

The common vertex of the sides is called the vertex of the pyramid; and the altitude of a pyramid is the perpendicular from its vertex to the plane of its base.

4. A prism is a solid contained by plane figures, of which two are opposite, equal, similar, and parallel to one another, and the others are parallelograms.

The parallelograms are called the sides; and the other two plane figures the ends, one of which is called the base. The altitude is the perpendicular distance of its two ends. It is said to be a right prism when the edges are perpendi

B

cular to the base; in other cases it is said to be oblique. The surface of the sides of a pyramid or prism is called the lateral or convex surface. A pyramid or prism is named according to the figure of its base. According as the base is a triangle, a rectangle, a square, or a polygon, it is said to be triangular, rectangular, square, or polygonal.

5. A parallelopiped is a solid figure contained by six quadrilateral figures, of which every opposite two are parallel.

Any side of a parallelopiped may be called its base; and its base, and the side opposite, are called its ends. A parallelopiped is just a prism, with a parallelogram for its base; and the definitions of the terms right, oblique, altitude, lateral and convex surface, are the same as in the case of the prism. A right parallelopiped is said to be contained by any three of its edges that belong to one of its trihedral angles, or by any three lines equal to them. If A, B, C, be three of these edges, or any three lines, the parallelopiped contained by them is expressed thus, A· B ̊C. 6. A cube is a solid figure contained by six equal squares. The cube described upon any line, as a line M, is expressed thus, M3.

7. A polyhedron is any solid contained by more than three planes. If it has four sides, it is called a tetrahedron ; if six, a hexahedron; if eight, an octahedron; if twelve, a dodecahedron; and if twenty, an icosahedron.

8. Two polyhedrons are said to be similar, when they are contained by the same number of similar faces, similarly situated, and containing equal dihedral angles.

9. A polyhedron is said to be regular, when its sides are equal regular polygons of the same kind, and its solid angles equal.

There are only five regular polyhedrons, of 4, 6, 8, 12, and 20 sides respectively, which are named as in the seventh definition. The first is contained by equilateral triangles; the second by squares; the third by equilateral triangles; the fourth by pentagons; and the fifth by equilateral triangles.

10. A sphere is a solid described by the revolution of a semicircle about its diameter, which remains fixed.