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contained by two straight lines drawn from any the
tion to one another which two other planes have,
more than two plane angles, which are not in the
angles equal, each to each, and are contained by the
constituted betwixt one plane and one point above it
which two that are opposite are equal, similar, and
of a semicircle about its diameter, which remains
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.
their axes and the diameters of their bases propor-
A cube is a solid figure contained by six equal squares.
A tetrahedron is a solid figure contained by four equal and equilateral triangles.
equal pentagons which are equilateral and equian-
An icosahedron is a solid figure contained by twenty
equal and equilateral triangles.
A parallelopiped is a solid figure contained by six qua
drilateral figures, whereof every opposite two are
PROP. I. THEOR.
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 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.
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
* 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.
PROP. III. THEOR. If two planes cut one another, their common section is a See N.
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 *
* 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.
* 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 * A ED, BEC,
GEF is equal * to the angle HEF; and consequently * 10 Def. 1. each of these angles is a right * angle. Therefore FE
* 8. 1.
* 4. 1.
* 8. 1.