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21. 3.

§ 1. 2.

Again, because the angle ABE is equal to the angle DBC, and the angle || BAE to the angle BDC, the triangle ABE is equiangular to the triangle BCD; as therefore BA to AE, so is BD to DC; wherefore the rectangle BA, DC is equal to the rectangle BD, AE:

But the rectangle BC, AD has been shewn equal to the rectangle BD, CE; therefore the whole rectangle AC, BDS is equal to the rectangle AB, DC, together with the rectangle AD, BC. Therefore, the rectangle, &c. Q. E. D.

BOOK XI.

DEFINITIONS.

I.

A SOLID is that which hath length, breadth, and thickness.

[blocks in formation]

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.

Thus the plane in which the right line

AB is drawn is perpendicular to the plane in which right line BC is drawn, for AB is at right angles to BC.

ས.

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.

B

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, the other upon the other plane.

VII.

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

XIa.

Similar solid figures are such as have all their solid angles equal, each to each, and which are contained by the same number of similar planes.

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a The 10th Definition, as it usually stands, is suspected to be an interpolation, and is therefore omitted.

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.

Thus the side AC, revolving round AB, one of the sides containing the right angle, and remains fixed, gene

rates a cone.

B

XIX.

B

C

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. Thus the revolution of the parallelogram AC about its side AB, which remains fixed, generates a cylinder.

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.

[graphic]
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