If from any 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 A B C be a triangle, and A D the perpendicular drawn from the angle A to the base B Č. The rectangle B A.A C is equal to the rectangle contained by AD and the diameter of the circle described about the triangle.

Describe the circle ACB about the triangle (IV. 5); draw the diameter A E, and join E C.

Because the right angle BDA is equal to the angle E CA in a semicircle (III. 31); and the angle ABD to the angle A EC in the same segment (III. 21). Therefore the triangles ABD and AEC are equiangular, and BA is to AD, as EA is to AC

D

(VI. 4). Wherefore the rectangle B A.A C is equal to the rectangle È A.AD (VI. 16). Therefore, if from any angle, &c. Q. E. D.

PROP. D. THEOREM.

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

Let A B CD be any quadrilateral figure inscribed in a circle, and AC and B D its diagonals. The rectangle A C.B D is equal to the two rectangles A B.CD, and A D.B C.

B

Make the angle AB E equal to the angle DBC (I. 23). Because the angle ABE is equal to the angle DBC. To each of these equals add the common angle EBD. Therefore the angle ABD is equal to the angle E B C. But the angle B D A is equal to the angle BCE, because they are in the same segment (III. 21). Therefore the triangle ABD is equiangular to the triangle BCE; and BC is to CE, as BD is to DA (VI. 4). Wherefore the rectangle B C.AD is equal to the rectangle B D.CE (VI. 16).

D

Again, because the angle ABE is equal to the angle D B C, and the angle B A E to the angle B D C (III. 21). Therefore the triangle ABE is equiangular to the triangle B CD; and BA is to AE, as BD is to DC. Wherefore the rectangle B A.DC is equal to the rectangle B D.A E. but the rectangle B C.AD has been shown to be equal to the rectangle BD.CE. Therefore the rectangles B C.AD and B A.DC are together equal (I. Ax. 2) to the rectangles BD.CE and BD.A E, that is, to the whole rectangle B D.A C. Therefore the whole rectangle AC.BD is equal to the two rectangles A B.D C, and A D.BC (II. 1). Therefore the rectangle, &c. Q. E. D.

Corollary.-The sum of the chords drawn from the extremities of any arc of a circle to any point in the remaining part of the circumference, is to the chord drawn from the middle of the arc to the same point, as the chord of the whole arc is to the chord of half the arc.

This Proposition D is a Lemma of Cl. Ptolemæus, in page 9 of his Mɛyákr Euvažis, or "Great Construction."

BOOK XI.

DEFINITIONS.

I.

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

A solid is extension in any three directions, uniform or variable; and strictly speaking, signifies a definite portion of space.

II.

That which bounds a solid is a superficies or surface.

This definition simply signifies that the boundaries of solids are surfaces.

III.

A straight line is perpendicular, or at right angles, to a plane, wheL 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.

The common section of two planes is the line in which the one outs the other, when they intersect or cross each other.

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 it 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.

The meaning of this definition will be more easily understood, by conceiving a plane to pass through the straight line, cutting the plane at right angles. The angle between the straight line and the common section of these planes is the inclination of the straight line to the plane.

VI.

The inclination of one plane to another is the acute angle contained by two straight lines drawn from any point in their common section at right angles to it, one upon each plane.

The meaning of this definition will be best understood by conceiving a plane to cut both planes at right angles to their common section. The angle between the common sections of this third plane with the other two is their inclination.

VII.

Two planes are said to have the same inclination to one another which two other planes have, when their angles of inclination are equal.

VIII.

Parallel planes are such as do not meet one another though produced ever so far in all directions.

The meaning of this definition is that that the space between the planes is always of the same width.

IX.

A solid angle is that which is made by the meeting of more than two plane angles in one point, but which are not in the same plane. The term solid, here applied to an angle, merely indicates that the angle described belongs to a solid figure, or one that has length, breadth, and thickness. A solid angle does not enclose space. The vertex of a solid angle is the point where all its plane angles meet.

X.

Equal and similar solid figures are such as are contained by similar planes equal in number, magnitude, and inclination to one another.

Dr. Simson has in his edition omitted this definition, on the ground that it is a theorem and not a definition.

XI.

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.

The planes containing the solid angles of any solid figure are similar and similarly situated to the planes containing the corresponding solid angle in another solid figure, only when the vertices of these solid angles being made to coincide, and a plane of the one applied to the corresponding plane of the other, the remaining planes of the one coincide with the remaining planes of the other, each to each.

XII.

A pyramid is a solid figure contained by planes that are constituted between one plane and a point above it in which they meet.

A pyramid may be defined as the solid figure formed by a solid angle and a plane intersecting all its plane angles at any distance from its vertex. This plane is called the base of the pyramid.

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 are parallelograms.

The opposite ends or faces of a prism are generally called its bases; although the term base is sometimes applied to any parallelogram on which it is supposed to to stand. The parallelograms are generally called the sides of the prism. Pyramids and prisms are called triangular, quadrangular, pentagonal, polygonal, &c., according as their bases are triangles, quadrangles, pentagons, polygons, &c. A prism is called right, when its sides are rectangles; oblique, when otherwise.

XIV.

A sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains unmoved.

A sphere may be defined as a solid figure bounded by one surface of such a kind that all straight lines, drawn from a certain point within the solid to its superficies, are equal to one another.

XV.

The axis of a sphere is the fixed straight line about which the semicircle revolves.

Any diameter of a sphere may be made, or supposed to be, an axis of revolution.

XVI.

The centre of a sphere is the same with that of the generating semieircle.

The certain point within the sphere, from which all the equal straight lines are drawn to the superficies, is called the centre.

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. The straight line drawn from the centre to the superficies of a sphere is called its radius

XVIII.

A cone is a solid figure described by the revolution of a rightangled triangle about one of the sides containing the right angle, that side remaining 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 cone; and if greater, an acute-angled cone.

A cone may be defined as a solid figure bounded by a circle and a superficies terminating in a point, of such a kind, that all straight lines drawn in it from that point to the circumference of the circle are equal to one another. This point ia called the vertex of the cone.

XIX.

The axis of a cone is the fixed straight line about which the triangle revolves.

The axis of a cone is the straight line drawn from its vertex to the centre of its base. XX.

The base of a cone is the circle described by the revolving leg of the right angle.

The base of a cone is the circle which forms one of its boundaries.

XXI.

A cylinder is a solid figure described by the revolution of a rectangle about one of its sides which remains fixed.

A cylinder may be defined as a solid figure bounded by two opposite equal and parallel circles, and a superficies of such a kind that all straight lines drawn in it between their circumferences parallel to the straight lines joining their centres, are equal to one another.

XXII.

The axis of a cylinder is the fixed straight line about which the rectangle revolves.

The axis of a cylinder is the straight line which joins the centres of its bases.

XXIII.

The bases of a cylinder are the circles described by the two revolving opposite sides of the rectangle.

The ends or bases of a cylinder are the two opposite, equal and parallel ciroles which form two of its boundaries.

XXIV.

Similar cones and cylinders are those which have their axes and the diameters of their bases proportionals.

A.

A parallelopiped is a solid figure contained by six quadrilateral figures, of which every opposite two are parallel.

A parallelopiped is a prism of which the bases are parallelograms. If the bases and sides of a parallelopiped be rectangles, it is called right, if otherwise oblique.

B.

A polyhedron is any solid figure bounded by plane figures. If these plane figures are all equal and similar, the polyhedron is called regular.

XXV.

A hexahedron, or cube is a solid figure contained by six equal squares. A cube is a right parallelopiped of which the sides are 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.

XXIX.

An icosahedron is a solid figure contained by twenty equal and equilateral triangles.

The five preceding definitions relate only to the five regular polyhedrons, or five regular bodies, as they are called; because no greater number than these five can exist. The irregular polyhedrons are innumerable.

PROP. I. THEOREM.

One part of a straight line cannot be in a plane and another part out of it.

If it be possible, let A B, part of the straight line A B C, be in a plane, and the part B C be out of it.

Because the straight line A B is in the plane, it can be produced in the plane (I. Post. 1). Let AB be produced to D; and let any plane be made to pass through the straight line AD, and turn about A D until it pass through the point C.

Because the points B and C are in this plane, the

straight line (I Def. 7) BC is in it. Therefore the two straight lines A B C and ABD in the same plane, have a common segment A B ( 11 Cor.); which is impossible. Therefore one part of a straight line, &c. Q. E. D.

PROP. 11. THEOREM.

Two straight lines which cut one another are in one plane, and three straight lines which meet one another in three points, are in one plane. Let two straight lines A B and CD cut one another in E; A B and