Page images
PDF
EPUB

VI.

See N.

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.

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.

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

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

angles equal, each to each, and are contained by the
same number of similar planés.

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.

See N.

See N.

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 con

taining 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 propor-
tionals.

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 equian-
gular.

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 qua

drilateral 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.

See N.

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

с

B

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. Therefore, one part, &c. Q. E. D. •Cor. 11.1.

[blocks in formation]

A

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

D about EB, produced if necessary, until it pass through the point C: then because E the points E, C are in this plane the straight line * EC is in it: for the same

C4

* 7 Def. 1. 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 * CD, AB: therefore AB, CD, are in one plane. • 1. 11. Wherefore two straight lines, &c. Q. E. D.

B

PROP. III. THEOR. If two planes cut one another, their common section is a See N.

straight line.

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 t, draw, in the + 1 Post. plane AB, the straight line DEB, and in the plane BC, the straight line DFB;

B then two straight lines DEB, DFB bave the same extremities and therefore include a space betwixt them: which is *

impos

* 10 Ax. 1. sible: 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.

See N.

* 4. 1.

* 15. 1.

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 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 straight

lines AE, ED are equal to the two BE, EC, each to 15. 1.

each, 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 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 *: there-
fore GE is 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 * 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 FC; there-
fore the angle FAD is equal * to the angle FBC: again,
it was proved that GA is equal to BH, and also AF to
FB; therefore FA and AG, are equal to FB and BH,
each to each ; and the angle FAG has
been proved equal to the angle FBH;
therefore the base GF is equal

to the
base FH: again, because it was proved A
that GE is equal to EH, and EF is com-
mon; therefore GE, EF are equal to HE,

H
EF, each to each ; and the base GF is

В equal to the base FH; therefore the angle

GEF is equal * to the angle HEF; and consequently * 10 Def, 1. each of these angles is a right * angle. Therefore FE

26. 1.

4. 1.

8. 1.

F

* 4. 1.

G

D

* 8. 1.

« PreviousContinue »