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I.
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 meeting it
in that plane.

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

in one of the planes perpendicularly 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 con-

tained 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 lection at right angles to it, one upon one plane, and
the other upon the other plane.

VII.

Book XI. Two planes are said to have the same, or a like inclination to one

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

VIII. Parallel planes are such which do not meet one another tho' produced.

IX. A solid angle is that which is made by the meeting of more than See N.

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

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, See N.

each to each, and which are contained by the same number of fimilar 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 se-
micircle revolves.

XVI.
The center of a sphere is the fame with that of the femicircle.

XVII.
The diameter of a sphere is any straight line which passes thro'

the center, and is terminated both ways by the superficies of
the fphere.

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 fide remains fixed.

Book XI. If the fixed fide be equal to the other fide containing the right

angle, the Cone is called a right angled Cone; if it be less
than the other fide, 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.

The base of a Cone is the circle described by that fide 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 fides 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 axis 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 equi-
lateral 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 Parallelepiped is a solid figure contained by fix quadrilateral

figures whereof every opposite two are parallel.

Book XI.

PROP. 1. THEOR.

ON

NE part of a straight line cannot be in a plane and See 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 produced in that

C plane. let it be produced to D. and let any plane, pass thro' the straight line AD, and be turned about it until A B it pass thro' the point C; and because the points B, C are in this plane, the straight line BC is in it a. therefore there are two straight lines a. 7. Def. I. ABC, ABD in the same plane that have a common segment AB, which is impoffible b. Therefore one part, &c. Q. E. D.

b.Cor.11.1.

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

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

TWO

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 thro' the straight line A D EB, and let the plane be turned about EB, produced if necessary, until it pass thro' the point C. then because the points E, C are

E in this plane, the straight line EC is in it a.

a. 7.

.Def. 86 for the same reason, the straight line BC is in the fame; and, by the Hypothesis, EB is

in it. therefore the three straight lines EC, : CB, BE are in one pláne . but in the plane C

B in which EC, EB are, in the same are 5 CD,

b. 1. II. AB. therefore AB, CD are in one plane. Wherefore two straight lines, &c. Q. E. D.

Book XI.

PROP. III.' THEOR.

See N.

IF
(F 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

B
draw in the plane AB the straight line
DEB, and in the plane BC the straight
line DFB. then two straight lines DEB,

E

F
DFB have the same extremities, and there-

fore include a space betwixt them; which
a. 10. Ax.l. is impossible a. therefore BD the common D

section of the planes AB, BC cannot but
be a straight line. Wherefore if two planes, &c. Q. E. D.

A

PROP. IV.. THEOR.

See N.

[F 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 paffing thro' AB, CD.

Take the straight lines AE, EB, CE, ED all equal to one another; and thro' 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 a. 15. I. equal angles & AED, BEC, the base AD is equal to the base BC, b. 4. I.

and the angle DAE to the angļe EBC. and the angle AEG is equal to the angle BEH ~; therefore the triangles AEG, BEH have two angles of one equal to two angles of the other, each to each, and the

fides AE, EB, adjacent to the equal angles, equal to one another; c. 26. I.

wherefore they shall have their other sides equal c. GE is therefore

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