DEMONSTRATION. Describe the circle ACB about the triangle (a), and draw its diameter AE, and join EC: because the right angle BDA is equal to the angle ECA in a semicircle (6), and the angle ABD equal to the angle AEC in the same segment (c); the triangles ABD, AEC are equiangular: therefore, as BA is to AD, so is EA to AC (d); and consequently the rectangle BA, AO is equal to the rectangle EA, AD (e).

PROPOSITION D.

THEOREM.-The rectangle under the diagonals of a quadrilateral figure inscribed in a circle, is equal to both the rectangles contained by its opposite sides.

DEMONSTRATION. Let ABCD be any quadrilateral figure 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.

Make the angle ABE equal to the angle DBC (a); add to each of these the common angle EBD, then the angle ABD is equal to the angle EBC: and the angle BDA is equal to the angle BCE, because they are in the same segment (6); therefore the triangle ABD is equiangular to the triangle BCE: wherefore, as BC is to CE, so is BD to DÁ (c); and consequently the rectangle BC, AD is equal to the rectangle BD, CE (d): again, because the angle ABE is equal to the angle DBC, and the angle BAE to the angle BDC (6), the triangle ABE is equiangular to the triangle BCD; therefore as BA is to AE, so is BD

B

I. 23.
(b) III. 21.
(c) VI. 4.
(d) VI. 16.

to DC (c); wherefore the rectangle BA, DC is equal to the rectangle BD, AE (đ): but the rectangle BC, AD has been shown equal to the rectangle BD, CE; therefore 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 (e); therefore the whole rectangle AC, BD is equal to the rectangle AB, DC, together with the rectangle AD, BC.

SCHOLIUM. This proposition is a Lemma of Cl. Ptolomæus, in page 9 of the Meydan Zuvragis, or " Great Construction."

THE

ELEMENTS OF EUCLID.

BOOK XI.

DEFINITIONS.

1. A SOLID is a magnitude, having length, breadth, and thickness. COROLLARY. All solids are bounded by superficies, or surfaces.

2. A straight line AB is said to be perpendicular to a plane, when it makes right angles with all straight lines which meet it in that place.

A

3. A plane is said to be perpendicular to a plane, when any straight line AB, drawn in one of the planes perpendicular to the common section of the two planes, is perpendicular to the other plane.

SCHOLIUM. The common section of two planes is the line in which they mutually cut or intersect each other.

4. The inclination of a straight line AC to a plane is the acute angle Č formed by that straight line, and another CB drawn from the point C, in which the first line meets the plane, to the point B in which a perpendicular AB to the plane drawn from any point A of the first line above the plane, meets the same plane.

A

5. The inclination of one plane to another is the acute angle ABC, formed by two straight lines drawn from any the same point B of their common section at right angles to it, one AB upon one plane, and the other BC upon the other plane.

C

6. PARALLEL PLANES are such as do not meet one another, though produced ever so far in every direction.

7. 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.

9. SIMILAR SOLID FIGURES are such as have all their solid angles equal, each to each, and are contained by the same number of planes similarly situated.

10. A PYRAMID is a solid figure contained by planes that are constituted between one plane figure and a point above it.

SCHOLIUM. The last-named plane figure is called the base, and the point above it the vertex of the pyramid; and all the planes meeting together in the vertex are triangles. The altitude of a pyramid is the perpendicular drawn from its vertex to its base.

11. 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 are parallelograms.

SCHOLIA. 1. The opposite ends are termed the bases of the prism, and the parallelograms its sides; but the term base is sometimes applied to any side upon which it is supposed to

stand. The altitude of a prism is a perpendicular from one of its ends or bases to the other.

2. A prism, the ends or bases of which are perpendicular to its sides, is said to be a right prism; any other is an oblique prism.

3. Pyramids and prisms are said to be triangular, quadrangular, pentagonal, or polygonal, according as their bases are triangles, quadrangles, pentagons, or polygons.

12. A SPHERE is a solid figure described by the revolution of a semicircle (ABC) about its diameter (AC), which remains unmoved.

13. The AXIS OF A SPHERE is the fixed straight line (AC) about which the semicircle revolves.

14. The CENTER OF A SPHERE is the same with that of the generating semicircle.

15. The DIAMETER OF A SPHERE is any straight line which passes through its center, and is terminated both ways by the superficies of the sphere."

16. 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 (AB) be equal to the other side containing the right angle (CB), the cone is said to be right-angled; if it (DF) be less than the other side (EF), obtuse-angled; and if greater (as GH and HI) acute-angled.

17. The AXIS OF A CONE is the fixed straight line about which the triangle revolves.

18. The BASE OF A CONE is the circle described by that side containing the right angle, which revolves.

19. A CYLINDER is a solid figure described by the revolution of a right-angled parallelogram (ABC) about one of its sides (AB), which remains fixed.

20. The AXIS OF A CYLINDER is the fixed straight line (AB) about which the parallelogram revolves.

21. The BASES OF A CYLINDER are the circles described by the two revolving opposite sides of the parallelogram.