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Book XI.

XIX. m The axis of a cone is the ixed straight line about which the triangle revolves.

XX.
The base of a cone is the circle described by that fide contain,
ing 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 fides which re-
mains 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 re, volving 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 fix equal squares,

XXVI.
A tetrahedron is a solid figure contained by four equal and e-
quilateral triangles.

XXVII. An octahedron is a folid 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 fix quadrilateral

figures whereof every opposite two are parallel.

PROP.

Book XI,

PRO P. I.

THEO R.

ON

NE part of a straight line cannot be in a plane and Sec N.

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

с produced in that plane: Let it be produced to D: And let any plane pass thro'the straight line AD, and be turned about it until it pass

A B D thro' the point C; and because the points B, C are in this plane, the straight line BC is in ita : a 7. def. 1, Therefore there are two straight lines ABC, ABD in the same plane that have a common segment AB, which is impollible b. Cor. II. 1. Therefore one part, &c. Q. E. D.

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WO straight lines which cut one another are in one See N.

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 met one another, are in one plane.
Let any plane pass through the straight

A

D line EB, and let the plane be turned about EB, produced, if necefiary, until it pass through the point C: Then because the points E, C are in this plane, the Ε E straight line EC is in ita : For the same

a 7. def. I. reason, the straight line BC is in the same ; and, by the hypothelis, 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 fame are b

B 11,11. CD, AB: Therefore AB, CD are in one plane. Wherefore two straight lines, &c. Q. E. D.

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•Book XI.

PRO P. III.

THE O R.

Se N.

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 is a
straight line : If it be not, from the point

B
D to B, draw, in the plane AB, the
straight line DEB, and in the plane BC
the straight line DFB : Then two straight E
lines DEB, DFB have the same extre-

mities, and therefore include a space be C
a 10. Ax. 1. twixt them; which is impossible: There-
fore BDthe common section of the planes

A

D
AB, BC cannot but be a straight line.
Wherefore, if two planes, &c. O, E. D.

PRO P. IV. THE O R.

See N.

IF a straight line stand at right angles to each of two

straight lines in the point of their interfection, it shall also be at right angles to the plane which pafles through them, that is, to the plane in which they are.

a is. I.

Let the straight line EF stand at right angles to each of the traight 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 cqual to one another; and thro' E draw, in the plane in which are AB, CD, any straight line GFH; 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 a AED, BEC, the base AD is equal o to the base BC, and the angle DAE to the angle EBC: And the angle AEG is equal to the angle BEH"; there fore the triangles AEG, BEH have two angles of one equal to two angles of the other, each to each, and the Gides AE, EB, adjacent to the equal angles, equal to one another; wherefore they thall have their other fides equal : GE is therefore

equal

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b 4. 1.

d 8.1.

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: And because AD is equal to BC, and AF to FB, the two fides FA, AD are equal to the two FB,BC, each to each; and the bafe 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

A to FB ; FA then and AG, are equal

с to FB and BH, and the angle FAG

G has been proved equal to the angle FBH; therefore the base GF is equal to the base FH: Again, because it

E H was proved, that GE is equal to EH, and EF is common; GE, EF are e. D

B qual to HE, EF; and the base GF is equal to the base FH; therefore the angle GEF is equal d to the angle HEF; and consequently each of these angles is a right angle. Therefore FE makes right angles with GH, 10. def. I. that is, with any straight line drawn through E in the plane paling 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 planer Therefore EF is at right angles to the plane f 3. def. 11. in which are AB, CD. Wherefore, if a straight line, &c. Q E. D.

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If three straight lines meet all in one point, and a soe N. straight line

stands at right angles to each of them in that point ; these three straight lines are in one and the

fame 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,

A 3. II.

Book XI. are, shall be a straight line; let this be BF: Therefore the three W straight lines AB, BC, BP 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 b 4. 11.

b to the plane passing through them; and therefore makes 63. def. 15. right angles with every straight A,

it
BF which is 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

-D
both in the same plane, which is B
impossible : Therefore the straight

E
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 thre
straight lines, &c. Q. E. D.

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def. 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 fame 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 BE, AE, AD. Then, because

C
AB is perpendicular to the plane, it
shall make right a 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, B

D
ABE is a right angle: For the same rea-
son, each of the angles CDB, CDE is
a right angle: And because AB is equal
to DE, and BD common, the two

E
fides 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

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