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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 AF 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).

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SECOND BOOK.

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

i 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

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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.

The axis of the sphere is the fixed diameter about which the semicircle revolves; and its centre is the same as that of the generating semicircle.

11. The diameter of a sphere is a straight line passing through the centre of the sphere, and terminated at each extremity by the surface.

12. A right cone is a solid described by the revolution of a right-angled triangle about one of the sides containing the right angle, which remains fixed.

The axis of the cone is the fixed line about which the generating triangle revolves; and its base is the circle de scribed by the revolving side containing the right angle.

13. A right cylinder is a solid described by the revolution of a rectangle about one of its sides, which remains fixed.

The axis of the cylinder is the fixed line about which the rectangle revolves; and its bases or ends are the circles described by the opposite revolving sides of the rectangle.

14. Similar cones and cylinders are those that have their axes and the diameters of their bases proportional.

It is evident (Def. 12) that the axis of a cone is the straight line joining its vertex and the centre of its base; and (Def. 13) that the axis of a cylinder is the straight line joining the centres of its two ends.

PROPOSITION I. THEOREM. Any two of the plane angles that form a trihedral angle, are together greater than the third.

Let the solid angle at A be contained by the three plane angles BAC, CAD, DAB. Any two of them are greater than the third.

If the angles BAC, CAD, DAB, be all equal, it is evident that any two of them are greater than the third. But if they are not, let BAC be that angle which is not less than either of the other two, and is greater than one of them DAB; and at the point A in the straight line AB, make, in the plane which passes through BA, AC, the angle BAE equal (Pl. Ge. I. 23) to the angle DAB; and make AE equal to AD, and through

D

B

E draw BEC cutting AB, AC, in the points B, C, and join DB, DC. And because DA is equal to AE, and AB is common to the two triangles ABD, ABE, and also the angle DAB equal to the angle EAB; therefore the base DB is equal to the base BE. And because BD, DC, are greater (Pl. Ge. I. 20) than CB, and one of them BD has been proved equal to BE a part of CB, therefore the other DC is greater than the remaining part EC. And because DA is equal to AE, and AC common, but the base DC greater than the base EC ; therefore the angle DAC is greater (Pl. Ge. I. 25) than the angle EAC; and, by the construction, the angle DAB is equal to the angle BAE; wherefore the angles DAB, DAC, are together greater than BAE, EAC, that is, than the angle BĂC. But BAC is not less than either of the angles DAB, DAC; therefore BAC, with either of them, is greater than the other.

PROPOSITION II. THEOREM. Every solid angle is contained by plane angles which together are less than four right angles.

First, let the solid angle at A be contained by three plane angles BAC, CAD, DAB. These three together are less than four right angles.

Take in each of the straight lines AB, AC, AD, any points B, C, D, and join BC, CD, DB; then, because the solid angle at B is contained by the three plane angles CBA, ABD, DBC, any two of them are greater (II. 1) than the third ; therefore the angles CBA, ABD, are greater than the angle DBC. For the B same reason, the angles BCA, ACD, are greater than the angle DCB; and the angles CDA, ADB, greater than BDC; wherefore the six angles CBA, ABD, BCA, ACD, CDA, ADB, are greater than the three angles DBC, BCD, CDB; but the three angles DBC, BCD, CDB, are equal to two right angles (Pl. Ge. I. 32); therefore the six angles CBA, ABD, BCA, ACD, CDA, ADB, are greater than two right angles; and because the three angles of each of the triangles ABC, ACD, ADB, are equal to two right angles, therefore the nine angles of these three triangles,

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A

D

namely, the angles CBA, BAC, ACB, ACD, CDA, DAC, ADB, DBA, BAD, are equal to six right angles. Of these, the six angles CBA, ACB, ACD, CDA, ADB, DBA, are greater than two right angles; therefore the remaining three angles BAC, DAC, BAD, which contain the solid angle at A, are less than four right angles.

Next, let the solid angle at A be contained by any number of plane angles BAČ, CAD, DAE, EAF, FAB; these together are less than four right angles,

Let the planes in which the angles are, be cut by a plane, and let the common sections of it with those planes be BC, CD, DE, EF, FB; and because the solid angle at B is contained by three plane angles CBA, ABF, FBC, of which any two are greater (II. 1) than the third, the angles B CBA, ABF, are greater than the angle FBC. For the same reason, the two plane angles at each of the points C, D, E, F, namely, the angles which are at the bases of the triangles having the common vertex A, are greater than the third angle at the same point, which is one of the angles of the polygon BCDEF; therefore all the angles at the bases of the triangles are together greater than all the angles of the polygon; and because all the angles of the triangles are together equal to twice as many right angles as there are triangles (Pl. Ge. I. 32); that is, as there are sides in the polygon BCDEF; and because all the angles of the polygon, together with four right angles, are likewise equal to twice as many right angles as there are sides in the polygon (Pl. Ge. I. 32, Cor. 1); therefore all the angles of the triangles are equal to all the angles of the polygon, together with four right angles. But all the angles at the bases of the triangles are greater than all the angles of the polygon, as has been proved. Wherefore, the remaining angles of the triangles, namely, those at the vertex, which contain the solid angle at A, are less than four right angles.

PROPOSITION III. THEOREM. If two solids be contained by the same number of equal and similar planes, similarly situated, and if the inclination

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