BOOK III.

SOLID GEOMETRY.

SEC. XIII.-PLANES AND THEIR INCLINATIONS.

DEFINITIONS.

1. A straight line is said to be perpendicular to a plane when it is perpendicular to every straight line it can meet in that plane.

2. The line in which two planes cut one another is called their INTERSECTION.

Schol. The intersection of two planes is a straight line. For, if A and D are any two points in the intersection of the planes FC and EB, the straight line joining these points must be in both planes (Def. 8, Sec. I); it is, therefore, their intersection.

3. The angle of inclination of two planes is that contained by any two straight lines perpendicular to their intersection, one in each plane.

If the line in each plane is perpendicular to the other plane, the two planes are said to be perpendicular to each other, the angle of inclination being a right angle.

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F

C

I

Thus, if AB and CB are both perpendicular to the intersection EH, then ABC is the angle of inclination of the planes DH and HF. If AB is also perpendicular to the plane HF, and CB to DH, the two D planes are perpendicular to each other.

E

G

B

H

4. Two planes are said to be parallel with one another when their intersections by any third plane are parallel.

Cor. Two parallel planes can never meet; for, if they were to meet in any point, their intersections by a third plane passing through that point would also meet; which is contrary to the definition.

5. The divergence of three or more planes from a single point constitutes a SOLID ANGLE. Thus, A is a solid angle contained by the three A planes BAC, CAD, DAB.

D

THEOREM I.

Two straight lines perpendicular to the same plane are parallel to each other.

Let AB and CD be two perpendiculars to the plane

A.

C

EF. It is to be shown that they are parallel.

E

B

D

F

Join BD. Now, AB and

CD, being perpendicular to

the plane, are also perpendicular to the line BD

which they meet in it (Def. 1). Also, a plane perpendicular to EF and passing through BD will contain both AB and CD (Def. 3). But straight lines in the same plane and perpendicular to the same straight line are parallel (Cor. 2, Theo. IV, B. I). Therefore, AB is parallel to CD.

That is, two straight lines, etc.

THEOREM II.

Parallel lines intercepted between parallel planes are equal.

Let FH and EG be any A two parallel lines intercepted between two parallel planes, AB and CD. It is to be shown that FH is equal to

EG.

Through the parallels FH

and EG let the plane FG pass. Its intersections with the parallel planes AB and CD will be parallel lines (Def. 4); that is, FE is parallel to HG. Hence, FEGH is a parallelogram, and FH is equal to EG (Theo. XIII, B. I).

Therefore, parallel lines, etc.

Cor. 1. Any straight line, as FH, perpendicular to one of two parallel planes, as CD, will be perpendicular to the other. For FHG being a right angle (Def. 1), HFE will also be a right angle (Cor. 1, Theo. IV, B. I), whatever may be the position of the parallels FE and HG in their planes.

Cor. 2. The perpendicular distance between two parallel planes is everywhere the same.

SEC. XIV. OF PARALLELOPIPEDS.

F

E

H

1. A PARALLELOPIPED is a solid bounded by six plane faces, of which each one is parallel to its opposite. The intersections of the faces are called edges.

Cor. Any face of a parallelo

piped is a parallelogram. For

A

the lines GH and BC, being the intersections of two parallel planes, FH and AC, with a third plane GC, are parallel (Def. 4, Sec. XIII); and in the same way it may be shown that BG and CH are parallel; hence, BCHG is a parallelogram (Def. 2, Sec. VI, B. I).

2. A right PARALLELOPIPED is one whose edges are perpendicular to the faces. Any other parallelopiped is called oblique.

3. A CUBE is a right parallelopiped whose length, breadth, and hight are equal.

4. Two parallelopipeds, or other solids, are called equivalent when they contain the same amount of solid space.

THEOREM III.

Any two opposite faces of a parallelopiped are equal.

Let AH be a parallelopiped. It is to be proved that any two opposite faces, as DEFA, CHGB,, are equal. Join DF and CG. Now, since EFGH is

E

D

A

F

H

G

B

a parallelogram (Cor. Def. 1), EF is equal to HG (Theo. XIII, B. I); and in the same manner it may be shown that DE is equal to CH. But FG and DC,

being both equal and parallel to EH (Theo. XIII, B. I), must be equal and parallel to each other. Hence, DCGF is a parallelogram (Theo. XIV, B. I), and DF is equal CG. Wherefore, the triangles DEF, CHG, are mutually equilateral, and consequently equal. But the former triangle is half the parallelogram DEFA (Cor. 1, Theo. XIII, B. I), and the latter triangle is half the parallelogram CHGB: hence, these two parallelograms are equal.

Therefore, any two opposite faces, etc.

Schol. Any face of a parallelopiped may be taken as the base. We may designate DABC and EFGH as the lower and upper bases. The perpendicular distance between the bases is called the hight or altitude.

Cor. 1. If EB be a right parallelopiped, the edges AF and BG, since, by definition, they are both perpendicular to the face DB, must be perpendicular also to the straight line AB, which they meet in that plane (Def. 1, Sec. XIII). Hence, ABGF, or any face of a right parallelopiped, is a rectangle.

Cor. 2. The faces of a cube are all equal squares.

THEOREM IV.

The solidity of a right parallelopiped is equal to the area of its base multiplied by its hight.

Let ABC be a right parallelopiped. It is to be proved that its solidity is equal to the area of its base AB, multiplied by its hight DC.

Let planes pass through the solid parallel to the three faces

A

D

B