* 4. 6.

3. 2.

*

* 32. 1. segment; the triangles ABD, AEC are equiangular* to one another: therefore as BA to AD, so is* EA to AC, * 16. 6. and consequently the rectangle BA, AC is equal* to the rectangle EA, AD, that is, to the rectangle ED, DA, together with the square of AD: but the rec*35. 3. tangle ED, DA is equal to the rectangle BD, DC: therefore the rectangle BA, AC is equal to the rectangle BD, DC, together with the square of AD. Wherefore, if an angle, &c. Q. E. D.

5. 4.

If from an angle of a triangle a straight line be drawn perpendicular to the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the circle described about the triangle.

Let ABC be a triangle, and AD the perpendicular from the angle A to the base BC; the rectangle BA, AC shall be equal to the rectangle contained by AD and the diameter of the circle described about the triangle.

Describe the circle ACB about the triangle, and draw the diameter AE, and join EC. Because the 31. 3. right angle BDA is equal to the angle ECA in a semicircle, and 21. 3. the angle ABD equal to the angle AEC in the same segment; the

4. 6.

triangles ABD, AEC are equiangular: therefore as BA to AD, so is EA to AC: and consequently the rec* 16. 6. tangle BA, AC is equal to the rectangle EA, AD. If, therefore, from an angle, &c. Q. E. D.

*

PROP. D.

THEOR.

The rectangle contained by the diagonals of a quadri

lateral inscribed in a circle, is equal to both the rectangles contained by its opposite sides.

Let ABCD be any quadrilateral inscribed in a circle, and join AC, BD; the rectangle contained by AC, BD shall be equal to the two rectangles contained by AB, CD, and by AD, BC.a

Make the angle ABE equal to the angle DBC; add * 23. 1. to each of these equals the common angle EBD, then

the angle ABD is equal to the angle EBC: and the

*

B

A

* 4. 6.

/D

16. 6.

angle BDA is equal to the angle BCE, because they * 21. 3. are in the same segment; therefore the triangle ABD is equiangular to the triangle BCE: wherefore* as BC is to CE, so is BD to DA; and consequently the rectangle BC, AD is equal to the rectangle BD, CE. Again, because the angle ABE is equal to the angle DBC, and the angle* BAE * 21. 3. to the angle BDC, the triangle ABE is equiangular to the triangle BCD: therefore as 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 shown equal to the rectangle BD, CE; therefore b the whole rectangle AC, BD* is equal to the rectangle * 1. 2. AB, DC, together with the rectangle AD, BC. Therefore, the rectangle, &c. Q. E. D.

This is a Lemma of Cl. Ptolomæus, in page 9 of his Meya2 Συνταξις.

The rectangles BC, AD and BA, DC are together equal to the rectangles BD, CE and BD, AE; that is to the whole rectangle BD, AC.

2 Ax.

1. 2.

214

THE

ELEMENTS OF EUCLID.

BOOK XI.

DEFINITIONS.

J.

A SOLID is that which hath length, breadth, and thick

ness.

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

X

'The tenth definition is omitted for reasons given in the notes.' See the Octavo Edition.

XI.

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.

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 obtuseangled; and if greater, an acute-angled cone.