Describe (5. 4.) the circle ACB about the triangle, and draw its diameter AE, and join EC; because the right angle BDA is equal (31. 3.) to the angle ECA in a semicircle, and the angle ABD to B the angle AEC in the same segment (21. 3.); the triangles ABD, AEC are equiangular: therefore as (4. 6.) BA to AD, so is EA to AC; and consequently the rectangle BA, AC is equal (16. 6.) 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 quadrilateral 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 is equal to the two rectangles contained by AB, CD, and by AD, BC.†

B

C

Make the angle ABE equal to the angle DBC; 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 (21. 3.) to the angle BCE, because they are in the same segment; therefore the triangle ABD is equiangular to the triangle BCE; wherefore (4. 6.) as BC is to CE, so is BD to DA; and consequently the rectangle BC, AD is equal (16. 6.) to the rectangle BD, CE: again, because the angle ABE is equal to the angle DBC, and the angle (21. 3.) 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 shown equal to the rectangle BD, CE; therefore the whole rectangle AC, BD (1. 2.) is equal to the rectangle AB, DC, together with the rectangle AD, BC. Therefore the rectangle, &c. Q. E. D.

* See Note.

E

D

A

+ This is a Lemma of Cl. Ptolomæus, in page 9 of his usyan ourTaĢIS.

THE

ELEMENTS OF EUCLID.

BOOK XI.

DEFINITIONS.

I.

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

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 perpendicularly 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 as 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 which 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.*

X.

'The tenth definition is omitted for reasons given in the notes.*

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 a 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.

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.

* See Note.

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.

XXIX.

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

DEF. A.

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

PROP. I. THEOR.

ONE part of a straight line cannot be in a plane, and 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:

A

B

D

C

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 (Cor. 11. 1.). Therefore, one part, &c. Q. E. D.

PROP. II. THEOR.

Two straight lines which cut one another are in one plane,

* See Note.

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 are in one plane: and three straight lines EC, CB, BE which meet one another, are in one plane.

A

D

E

X

Let any plane pass through the straight line EB, and let the plane be turned about EB, produced, if necessary, until it pass through the point C: then because the points E, C are in this plane, the straight line EC is in it (7. def. 1.) for the same 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 (1. 11.) CD, AB: C

:

B

therefore AB, CD are in one plane. Wherefore two straight lines, &c. Q. E. D.

PROP. III. THEOR.

Ir two planes cut one another, their common section is a straight line.*

B

Let two planes AB, BC cut one another, and let the line DB be their common section: DB is a 'straight line : if it be not, from the point D to B, draw, in the plane AB, the straight line DEB, and in the plane BC the straight line DFB; then two straight lines DEB, DFB have the same extremities, and therefore include a space betwixt them; which is impossible (10. Ax. 1.): 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.

C

F

D

Α

Ir 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 is also 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

* See Note.