XIX.

The axis of a cone is the fixed straight line about which the triangle revolves.

XX.

The base of a cone is the circle described by that side containing the right angle, which revolves.

XXI.

A cylinder is a solid figure described by the revolution of a right-angled parallelogram about one of its sides which remains fixed.

XXII.

The axis of a cylinder is the fixed straight line about which the parallelogram revolves.

XXIII.

The bases of a cylinder are the circles described by the two revolving opposite sides of the parallelogram.

XXIV.

Similar cones and cylinders are those which have their axes and the diameters of their bases proportionals.

XXV.

A cube is a solid figure contained by six equal squares.

XXVI.

A tetrahedron is a solid figure contained by four equal and equilateral triangles.

XXVII.

An octahedron is a solid figure contained by eight equal and equilateral triangles.

U

XXVIII.

A dodecahedron is a solid figure contained by twelve equal pentagons which are equilateral and equiangular.

XXIX.

An icosahedron is a solid figure contained by twenty equal and equilateral triangles.

Def. A.

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

PROP I. THEOR.

One part of a straight line cannot be in a plane, and another part above it.

If it be possible, let AB, part of the straight line ABC, be in the plane, and the part BC above it: and since the straight line AB is in the plane, it can be produced in that plane. Let it be produced to D: and let any plane pass through the

straight line AD, and be turned about it until it pass through the point C; and because the points B, C are *7 Def. in this plane, the straight line* BC is in it: therefore there are two straight lines ABC, ABD in the same *Cor. 11. plane that have a common segment AB, which is impossible. Therefore, one part, &c. Q. E. D.

1.

1.

PROP. II. THEOR.

Two straight lines which cut one another are in one plane, and three straight lines which meet one another are in one plane.

Let two straight lines AB, CD, cut one another in E; AB, CD shall be one plane: and three straight lines EC, CB, BE which meet one another, shall be in one plane.

Let any plane pass through A the straight line EB, and let

the plane be turned about EB,

C

D

1.

produced, if necessary, until it pass through the point C: then because the points E, C are in this plane, the straight line* EC is in it: for the same reason, the * 7 Def. straight line BC is in the same; and, by the hypothesis, EB is in it: therefore the three straight lines EC, CB, BE are in one plane: but in the plane in which EC, EB are, in the same are* CD, AB; therefore AB, CD * 1. 11. are in one plane. Wherefore two straight lines, &c. Q. E. D.

PROP. III. THEOR.

If two planes cut one another, their common section is a straight line.

Let two planes AB, BC, cut one another, and let the line DB be their common section: DB shall be a straight line.

If it be not, from the point D to B draw, in the plane AB, the straight line DEB, and in the plane BC the straight line DFB: then two straight lines DEB, DFB have the same extremities, and therefore include a

1.

10 Ax. space betwixt them; which is impossible: therefore BD the common section of the planes AB, BC cannot but be a straight line. Wherefore, if two planes, &c. Q. E. D.

* 4. 1.

PROP. IV. THEOR.

If a straight line stand at right angles to each of two straight lines in the point of their intersection, it shall also be at right angles to the plane which passes through them, that is, to the plane in which they are.

Let the straight line EF stand at right angles to each of the straight lines AB, CD in E, the point of their intersection: EF shall be at right angles to the plane passing through AB, CD.

*

Take the straight lines AE, EB, CE, ED all equal to one another; and through E draw, in the plane in which are AB, CD, any straight line GEH; and join AD, CB; then, from any point F in EF, draw FA, FG, FD, FC, FH, FB. And because the two straight lines AE, ED are equal to the two BE, EC, and that 15. 1. they contain equal angles* AED, BEC, the base AD is equal to the base BC, and the angle DAE to the * 15. 1. angle EBC: and the angle AEG is equal to the angle BEH; therefore the triangles AEG, BEH have two angles of the one equal to two angles of the other, each to each, and the sides AE, EB, adjacent to the equal angles, equal to one another: wherefore their other * 26. 1. sides are equal:* therefore GE is equal to EH, and AG to BH. And because AE is equal to EB, and FE common and at right angles to them, the base AF is equal to the base FB; for the same reason, CF is equal to FD: and because AD is equal to BC, and AF to FB, the two sides FA, AD are equal to the two FB, BC, each to each; and the base DF was proved equal to the base FC; therefore the angle FAD is

* 4. 1.

*

equal to the angle FBC. Again, it was proved that * 8. 1. GA is equal to BH, and also AF to FB; therefore FA

*

H

* 4.1.

8. 1.

and AG, are equal to FB and BH, each to each; and the angle FAG has been proved equal to the angle FBH; therefore the base GF is equal* to the base FH: again, because it was proved, that GE is equal to EH, and EF is common; therefore GE, EF are equal to HE, EF; and the base GF is equal to the base FH; therefore the angle GEF is equal to the angle HEF; and consequently each of these angles is a right angle: therefore FE makes right angles with GH, that is, with any straight line drawn through E in the plane passing through AB, CD. In like manner, it may be proved, that FE makes right angles with every straight line which meets it in that plane: but a straight line is at right angles to a plane when it makes right angles with every straight line which meets it in that plane; * * 3 Def. therefore EF is at right angles to the plane in which are AB, CD. Wherefore, if a straight line, &c. Q. E. D.

PROP. V. THEOR.

If three straight lines meet all in one point, and a straight line stand at right angles to each of them in that point; these three straight lines are in one and the same plane.

Let the straight line AB stand at right angles to each of the straight lines BC, BD, BE, in B the point where they meet; BC, BD, BE shall be in one and the same plane.

If not, let, if it be possible, BD and BE be in one plane, and BC be above it; and let a plane pass through AB, BC the common section of which with

1

11.

10 Def.