angles at their centres, and BAC and EDF angles at their circumferences. The arc BC is to the arc EF, as the angle BGC is to the angle EHF; and as the angle BAC is to the angle EDF. Also, the arc BC is to the arc EF, as the sector BGC is to the sector EHF. Take any number of arcs CK and K L, each equal to BC; and any number F M and MN, each equal to E F. Join GK, A GL, H M and HN.

B

H

K

M

E

Because the arcs BC, CK and KL are all equal, the angles BGC, CGK and K GL are also all equal (III.27). Therefore what multiple soever the arc BL is of the arc BC, the same multiple is the angle BGL of the angle BGC.__For the same reason, whatever multiple the arc EN is of the arc EF, the same multiple is the angle EHN of the angle EHF. If the arc BL be equal to the arc EN, the angle BGL is equal to the angle EHN (III. 27); if the arc BL be greater than the arc EN, the angle BGL is greater than the angle EHN; and if less, less. But there are four magnitudes, the two ares BC and EF, and the two angles BGC and EHF; and of the arc BC, and the angle BGC, have been taken any equimultiples whatever, viz., the arc BL, and the angle BGL; and of the arc EF, and of the angle EHF, any equimultiples whatever, viz., the arc EN, and the angle EHN. And it has been proved, that if the arc BL be greater than the arc EN; the angle BGL is greater than the angle EHN. if equal, equal; and if less, less. Therefore the arc BC is to the arc EF, as the angle BGC is to the angle EHF (V. Def. 5). But the angle BG.C is to the angle EHF, as the angle BAC is to the angle EDF (V. 15); each being double of each (III. 20). Therefore the arc BC is to the arc EF, as the angle BGC is to the angle EHF; and as the angle BAC is to the angle EDF.

Next, the arc B C is to the arc E F, as the sector B G C is to the sector EHF.

Join B C and CK; and in the arcs BC and CK take any points X and O. Join B X, X C, CO and OK.

Because in the triangles G B C and GCK, the two sides B G and G C are equal to the two sides CG and G K, each to each, and they contain equal angles (Const.). Therefore the base BC is equal to the base CK (I. 4), and the triangle G B C to the triangle GCK. Because the arc B C is equal

D

N

M

E

to the arc CK, the remaining part of the whole circumference of the circle A B C, is equal to the remaining part of the whole circumference of the same circle (I. Ax. 3). Therefore the angle B XC is equal to the angle COK (III. 27). Wherefore the segment BXC is similar to the segment COK (III. Def. 11). But they are upon equal straight lines, BC and CK; and similar segments of circles upon equal straight lines are equal to one another (III. 24). Therefore the segment BXC is equal to the segment COK. But the triangle BGC was proved to be

equal to the triangle C G K. Therefore the whole, the sector B G C, is equal to the whole, the sector CGK. For the same reason, the sector KGL is equal to each of the sectors BGC and CGK. In the same manner, the sectors EHF, FHM and MHN may be proved equal to one another. Therefore, what multiple soever the arc BL is of the arc B C, the same multiple is the sector B G L of the sector B GC. For the same reason, whatever multiple the arc E N is of the arc EF, the same multiple is the sector EHN of the sector EHF. If the arc BL be equal to the arc EN, the sector BGL is equal to the sector E HN; if the arc BL be greater than the arc EN, the sector BGL is greater than the sector EHÑ; and if less, less. But there are four magnitudes, the two arcs B C and E F, and the two sectors B G C and EHF; and of the arc B C and the sector B G C, the arc B L and the sector BGL are any equimultiples whatever; and of the arc EF and the sector EHF, the arc EN and the sector EHN are any equimultiples whatever. And it has been proved, that if the arc BL be greater than the arc EN, the sector B GL is greater than the sector EH N; if equal, equal; and if less, less. Therefore, the arc BC is to the arc EF, as the sector B G C is to the sector EHF (V. Def. 5). Wherefore, in equal circles, &c. Q. E. D.

Corollary 1.-In the same circle, the angles, either at the centres or circumferences, are to one another, as the arcs on which they stand; so also are the sectors. Corollary 2.-In the same circle, any angle is to a right angle, as the arc on which it stands is to a quadrant; and any angle is to four right angles, as the arc on which it stands is to the whole circumference.

The following propositions, marked B, C, and D, were added to this Book by Dr. Simson, because they are frequently made use of by geometers.

PROP. B. THEOREM

If any angle of a triangle be bisected by a straight line which likewise cuts the base; the rectangle contained by the sides of the triangle is equal to the rectangle contained by the segments of the base, together with the square of the straight line which bisects the angle.

Let A B C be a triangle, and let the angle BA C be bisected by the straight line AD. The rectangle B A.AČ is equal to the rectangle B D.D C, together with the square of A D.

Describe the circle A CB about the triangle (IV. 5). Produce AD to meet the circumference in E, and join E C.

B

Because the angle B AD is equal to the angle CAE (Hyp.); and the angle ABD to the angle AEC (III. 21), for they are in the same segment. Therefore the triangles ABD and AEC are equiangular to one another (I. 32); and B A is to AD, as EA is to AC (VI. 4). But the rectangle B A.A C is equal to the rectangle EA.AD (VI. 16); that is, to the rectangle ED.DA, together with the square of AD (II. 3). Because the rectangle ED.DA is equal to the rectangle B D.DC (III. 35). Therefore the rectangle BA.A C is equal to the rectangle B D.D C, together with the square of AD (I. Ac. 1). Wherefore, if an angle, &c. Q. E. D.

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 AD 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 ECA in a semicircle (III. 31); and the angle ABD to the angle AEC in the same segment (III. 21). Therefore the triangles ABD and AEC are equiangular, and BA is to AD, as EA is to AC

B

D

(VI. 4). Wherefore the rectangle B A.AC 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 ABCD be any quadrilateral figure inscribed in a circle, and AC and B D its diagonals. The rectangle A C.BD is equal to the two rectangles A B.CD, and A D.B C.

E

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 DB C, and the angle B A E to the angle B D C (III. 21). Therefore the triangle A B E is equiangular to the triangle BCD; and BA is to AE, as BĎ is to D C. Wherefore the rectangle B A.DC is equal to the rectangle BD.AE. b t the rectangle B C.AD has been shown to be equal to the rectangle BD.CE. Therefore the rectangles B C.AD and BA.D C are together equal (I. Ax. 2) to the rectangles BD.CE and BD.AE, 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 AD.B C (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 Mɛyákn Lúvratis, 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, 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.

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

V.

The inclination of a straight line to a plane, is the acute angle con tained 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,