The Elements of Euclid

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.

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 in this plane, the straight line BC is in it (7. def. 1.): therefore there are two straight lines ABC, ABD in the same plane that have a common segment AB, which is impossible (Cor. 11. 1.). Therefore, one part, &c. Q. E. D.

PROP. II. THEOR.

Two straight lines which cut one another are in one plane,

* See Note.

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 are in one plane: and three straight lines EC, CB, BE which meet one another, are in one plane.

Let any plane pass through the straight line EB, and let the plane be turned about EB, 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 (7. def. 1.) for the same reason, the 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 (1. 11.) CD, AB: C'

therefore AB, CD are in one plane. Wherefore two straight lines, &c. Q. E. D.

PROP. III. THEOR.

Ir 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 is 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 space betwixt them; which is impossible (10. Ax. 1.): therefore BD the common section of the planes AB, BC cannot but be a straight line. Wherefore, if two planes, &c. Q. E. D.

PROP. IV. THEOR.

Ir 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 is also 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

* See Note.

two straight lines, AE, ED are equal to the two BE, EC, and that they contain equal angles (15. 1.) AED, BEC, the base AD is equal (4. 1.) to the base BC, and the angle DAE to the angle EBC: and the angle AEG is equal to the angle BEH (15. 1.); therefore the triangles AEG, BEH have two angles of 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 they shall have their other sides equal (26. 1.): GE is therefore 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 (4. 1.) 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 EC; therefore the angle FAD is equal (8. 1.) to the angle FBC: again, it was proved that GA is equal to BH, and also AF to FB; FA, then, and AG are equal to FB and BH, and the angle FAG has been proved A equal to the angle FBH; therefore the base GF (4. 1.) to the base FH: again, because it was proved, that GE is equal to EH, and EF is common; GE, EF are equal to HE, EF; and the base GF is equal to the base FH; therefore the angle GEF is equal (8. 1.) to the angle HEF; and consequently each of these angles is a right (10. def. 1.) 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. 11.): 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 stands 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 are 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 the plane, in which BD and BE are, shall be a straight

• See Note.

(3. 11.) line; let this be BF: therefore the three straight lines AB, BC, BF are all in one plane, viz. that which passes through AB, BC: and because AB stands at right angles to each of the straight lines BD, BE, it is also at right angles (4. 11.) to the plane passing through them; and therefore makes right angles (3. def. 11.) with every straight line meeting it in that plane; but

BF, which is in that plane, meets it: there- A fore the angle ABF is a right angle; but the angle ABC, by the hypothesis, is also a right angle; therefore the angle ABF is equal to the angle ABC, and they are both in the same plane which is impossible: therefore the straight line BC is not above the plane in which are BD and BE: wherefore the three straight lines BC, BD, BE are in one and the same plane. Therefore, if three straight lines, &c. Q. E. D.

PROP. VI. THEOR.

-E

Ir two straight lines be at right angles to the same plane, they shall be parallel to one another.

Let the straight lines AB, CD be at right angles to the same plane; AB is parallel to CD.

Let them meet the plane in the points B, D, and draw the straight line B, D, to which draw DE at right angles, in the same plane; and make DE equal to AB, and join BE, AE, AD. A Then because AB is perpendicular to the plane, it shall make right (3. def. 11.) angles with every straight line which meets it, and is in that plane: but BD, BE, which are in that plane, do each of them meet AB. Therefore, each of the angles ABD, ABE is a right angle: for the same B reason, each of the angles CDB, CDE is a right angle: and because AB is equal to DE, and BD common, the two sides AB, BD are equal to the two ED, DB: and they contain right angles; therefore the base AD is equal (4. 1.) to the base BE: again, because AB is equal to DE, and BE to AD; AB, BE are equal to ED, DA; and, in the triangles ABE, EDA, the base AE is common; therefore the angle ABE is equal (8. 1.) to the angle EDA: but ABE is a right angle; therefore EDA is also a right angle, and ED perpendicular to DA: but it is also perpendicular to each of the two BD, DC: wherefore ED is at right angles to each of the three straight lines BD, DA, DC, in the point in which they meet: therefore these three straight lines are all in the same plane (5. 11.): but AB is in the plane in which are BD, DA, because any three straight lines which meet one another are in one plane (2. 11.): therefore AB, BD, DC are in one plane:

and each of the angles ABD, BDC is a right angle; therefore AB is parallel (28. 1.) to CD. Wherefore, if two straight lines, &c. Q. E. D.

PROP. VII. THEOR.

Ir two straight lines be parallel, the straight line drawn from any point in the one to any point in the other is in the same plane with the parallels.*

Let AB, CD be parallel straight lines, and take any point E in the one, and the point F in the other: the straight line which joins E and F is in the same plane with the parallels.

If not, let it be, if possible, above the plane, as EGF; and in the

plane ABCD in which the parallels A

are, draw the straight line EHF from

E to F; and since EGF also is a straight line, the two straight lines EHF, EGF includes a space between them, which is impossible (10. Ax. 1.). Therefore the straight line joining the points E, F is not above the plane in which the parallels AB, CD are, and is therefore in that plane. Wherefore, if two straight lines, &c. Q. E. D.

PROP. VIII. THEOR.

Ir two straight lines be parallel, and one of them be at right angles to a plane, the other also shall be at right angles to the same plane.*

Let AB, CD be two parallel straight lines, and let one of them AB be at right angles to a plane: the other CD is at right angles to the same plane.

Let AB, CD meet the plane in the points B, D, and join BD: therefore (7. 11.) AB, CD, BD are in one plane. In the plane to which AB is at right angles, draw DE at right angles to BD, and make DE equal to AB, and join BE, AE, AD. And because AB is perpendicular to the plane, it is perpendicular to every straight line which meets it, and is in that plane (3. def. 11.); therefore each of the angles ABD, ABE is a right angle; and because the straight line BD meets the parallel straight lines AB, CD, the angles ABD, CDB are together equal (29. 1.) to two right angles and ABD is a right angle; therefore also CDB is a right angle, and CD perpendicular to BD: and because AB is equal to DE, and BD common, the two AB, BD are equal to the two ED, DB, and the angle ADB is equal to the angle EDB, because each of * See Note.

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