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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.

ΧΧΙ.

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 equi

lateral triangles.

XXVII.

An octahedron is a solid figure contained by eight equal and equi

lateral triangles.

XXVIII.

A dodecahedron is a solid figure contained by twelve equal penta

gons which are equilateral and equiangular.

ΧΧΙΧ.

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

DEF. A.

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

Book XI.

PROP. I. THEOR.

ONE part of a straight line cannot be in a plane See N. 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 pro

duced in that plane: let it be pro

duced to D: and let any plane pass through the straight line AD, and

A

C

B

D

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 ita: therefore there are two a7.def.1. straight lines ABC, ABD in the same plane that have a common segment AB, which is impossible b. Therefore, one part, &c. b Cor.11. Q. E. D.

1.

PROP. II. THEOR.

TWO straight lines which cut one another are in See N. 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 are 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 ita: 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 b CD, AB: therefore, AB, CD are in one plane. Wherefore, two straight lines, &c. Q. E. D.

A

D

E

a 7. def.1.

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B

b 1. 11.

Book XI.

See N.

1.

PROP. III. THEOR.

IF two planes cut one another, their common sec. tion is a straight line.

B

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 there

fore include a space betwixt them; which a 10. Ax. 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.

C

E

F

A

D

See N.

a 15.1.

b 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 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 two straight lines AE, ED are equal to the two BE, EC, and that they contain equal angles & AED, BEC, the base AD is equal to the base BC, and the angle DAE to the angle EBC: and the angle AEG is equal to the angle BEHa; 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; where

c 26. 1. fore they shall have their other sides equal: GE is therefore

equal to EH, and AG to BH: and because AE is equal to EB, Book XI.
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: b 4.1.
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 equal d 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
equal to the angle FBH; therefore the G

F

d 8. 1.

C

A

base GF is equal to the base FH: again, because it was proved, that GE is equal to EH, and EF is common; CE, EF are

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equal to HE, EF; and the base GF is D equal to the base FH; therefore the an

B

gle GEF is equal d to the angle HEF; and consequently each of these angles is a right angle. Therefore FE makes right an-e 10. def. gles with GH, that is, with any straight line drawn through E in 1 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 planef: therefore EF is at right an- f 3. def. gles to the plane in which are AB, CD. Wherefore, if a straight 11. line, &c. Q. E. D.

PROP. V. THEOR.

IF three straight lines meet all in one point, and a See N. 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,

Book XI. shall be a straight a line; let this be BF: therefore the three straight lines AB, BC, BF are all in one plane, viz. that which a 3. 11. 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 b to the plane passing through them; and therefore makes right c3. def. angles with every straight line meet

b 4. 11.

11.

ing it in that plane; but BF which is A

in that plane meets it: therefore 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

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straight lines BC, BD, BE are in one and the same plane. Therefore, if three straight lines, &c. Q. E. D.

PROP. VI. THEOR.

11.

IF 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 BD, to which draw DE at right angles, in the same plane; and make DE equal to AB, and join A

B

C

D

BE, AE, AD. Then, because AB is pera 3. def. pendicular to the plane, it shall make righta 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 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 to the base BE: again, because AB is equal to DE, and BE to AD; AB, BE are equal

b 4. 1.

E

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