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THE

ELEMENTS OF EUCLID.

BOOK XI.

DEFINITIONS,

I. A solid is that which hath length, breadth, and thickness.

II. That which bounds a solid is a superficies.

III. A straight line is perpendicular, or at right angles, to a

plane, when it makes right angles with every straight line in that plane which meets it.

IV. A plane is perpendicular to a plane, when the straight

lines drawn in one of the planes perpendicular to the common section of the two planes, are perpendicular to the other plane.

v. The inclination of a straight line to a plane, is the acute

angle contained by that straight line, and another drawn from the point in which the first line meets the plane, to the point in which a perpendicular to the plane drawn from any point of the first line above the plane, meets the same plane.

VI. The inclination of a plane to a plane, is the acute angle contained by two straight lines drawn from any

the same point of their common section at right angles to it, one upon one plane, and the other upon the other plane.

VII. Two planes are said to have the same or a like inclina

tion to one another which two other planes have, when the said angles of inclination are equal to one another.

VIII. Parallel planes are such as do not meet one another

though produced.

See N.

IX. A solid angle is that which is made by the meeting of

more than two plane angles, which are not in the same plane, in one point.

See N.

X. • The tenth definition is omitted for reasons given in the

notes.'

See N.

XI. Similar solid figures are such as have all their solid angles

equal, cach to each, and are contained by the same number of similar planes.

XII. A pyramid is a solid figure contained by planes that are

constituted betwixt one plane and one point above it in which they meet.

XIII. A prism is a solid figure contained by plane figures, of

which two that are opposite are equal, similar, and parallel to one another; and the others parallelograms.

XIV. A sphere is a solid figure described by the revolution of

a semicircle about its diameter, which remains unmoved.

XV. The axis of a sphere is the fixed straight line about

which the semicircle revolves.

XVI. The centre of a sphere is the same with that of the

semicircle.

XVII. The diameter of a sphere is any straight line which passes through the centre, and is terminated both ways by the superficies of the sphere.

XVIII. A cone is a solid figure described by the revolution of

a right-angled triangle about one of the sides containing

the right angle, which side remains fixed. If the fixed side be equal to the other side containing the

right angle, the cone is called a right-angled cone ; if it be less than the other side, an obtuse-angled ; and if greater, an acute-angled cone.

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.

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.

PROPOSITION I.

See N. THEOREM.-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 * 7 Def. I. because the points B, C are in this plane, the straight line

BC is in it: therefore there are two straight lines ABC, ABD

in the same plane that have a common segment AB; which * Cor. 11.1. is * impossible. Therefore, one part, &c.

B

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PROPOSITION 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 in one plane; and three straight lines EC, CB, BE, which meet one another, shall be 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 *

А,
EC is in it; for the same reason, the straight
line BC is in the same; and by the hypothesis, EX
EB is in it; therefore the three straight lines
EC, CB, BE are in one plane: but in the planec

B
in which EC, EB are, in the same are CD,
AB: therefore AB, CD are in one plane. Wherefore, two
straight lines, &c.

D

* 7 Def. 1.

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* 1. 11.

Q. E. D.

PROPOSITION III.

THEOR.-If two planes cut one another, their common section See N.

is a straight line.

1 Post.

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 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.

• 10 Ax. 1.

Q. E. D.

PROPOSITION IV.

THEOR.--If a straight line stand at right angles to each of See N.

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 also 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

С
straight lines AE, ED are equal to the two BE,
EC, each to each, and that they contain equal

H angles * AED, BEC, the base AD is equal * to

• 15. ). the base BC, and the angle DAE to the angle

B

• 15. 1. 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 they have their other sides equal *; therefore GE is * 26. 1.

• 4.1.

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