and they are upon equal ftraight lines BC, BK: But fimilar feg- Book VI." ments of circles upon equal ftraight lines, are equal to one another: Therefore the fegment BXC is equal to the fegment COK: 8 24 3. And the triangle BGC is equal to the triangle CGK; therefore the whole, the fector BGC, is equal to the whole, the fector CGK: For the fame reason, the fector KGL is equal to each of the, fectors BGC, CGK: In the fame manner, the fectors EHF, FHM, MHN may be proved equal to one another: Therefore, what multiple foever the circumference BL is of the circumference BC, the fame multiple is the sector BGL of the sector BGC: For the fame reafon, whatever multiple the circumference EN is of EF, the fame multiple is the fector EHN of the fector EHF: And if the circumference BL be equal to EN, the

fector BGL is equal to the fector EHN; and if the circumference BL be greater than EN, the fector BGL is greater than the fector EHN; and if lefs, lefs: Since then, there are four magnitudes, the two circumferences BC, EF, and the two fectors BGC, EHF, and of the circumference BC, and sector BGC, the circumference BL and fector BGL are any equimultiples whatever; and of the circumference EF and fector EHF, the circumference EN and fector EHN are any equimultiples whatever; and that it has been proved, if the circumference BL be greater than EN, the fector BGL is greater than the fector EHN; and if equal, equal; and if lefs, lefs. Therefore, as the circumference BC is to the circumference EF, fobs. def. s. is the fector BGC to the fector EHF. Wherefore, in equal circles, &c. Q. E. D.

Book VI.

See N.

a 5.4.

b 21. 3.

C 4 6.

d 16. 6,

€ 3.2.

£ 35. 3.

See N.

PROP. B. THEOR.

IF an angle of a triangle be bifected by a ftraight line, which likewife cuts the bafe; the rectangle contained by the fides of the triangle is equal to the rectangle contained by the fegments of the bafe, together with the fquare of the ftraight line bifecting the angle.

Let ABC be a triangle, and let the angle BAC be bifected by the ftraight line AD; the rectangle BA, AC is equal to the rectangle BD, DC, together with the fquare of AD.

B

A

C

D

Defcribe the circle ACB about the triangle, and produce AD to the circumference in E, and join EC: Then, becaufe the angle BAD is equal to the angle CAE, and the angle ABD to the angle AEC, for they are in the fame fegment; the triangles ABD, AEC are equiangular to one another: Therefore as BA to AD, fo is EA to AC, and confequently the rectangle BA, AC is equal to the rectangle EA, AD, that is, to the rectangle ED, DA together with the fquare of AD: But the rectangle ED, DA is equal to the rectangle f BD, DC. Therefore the rectangle BA, AC is equal to the rectangle BD, DC, together with the fquare of AD. Wherefore, if an angle, &c. Q. E. D.

IF

E

PROP. C. THEO R.

F from any angle of a triangle a ftraight line be drawn perpendicular to the bafe; the rectangle contained by the fides of the triangle is equal to the rectangle contained by the perpendicular and the diameter of the cirde defcribed about the triangle.

Let ABC be a triangle, and AD the perpendicular from the angle Athene BC; the rectangle BA, AC is equal to the reelang. co. *ained by AD and the diameter of the circle deforised about the triangic.

Describe

THE

PROP. D.

E

e 16. 6.

THEOR.

HE rectangle contained by the diagonals of a quadrilateral infcribed in a circle, is equal to both the rectangles contained by its oppofite fides.

b

b 4.6.

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 †. 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 to the a 21. 3. angle BCE, because they are in the fame fegment; therefore the triangle ABD is equiangular to the triangle BCE: Wherefore as BC is to CE, fo is BD to DA; and confequently 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 to the angle BDC, the triangle ABE is equiangular to the triangle BCD: As therefore BA to AE, fo is BD to DC; wherefore

a

E

the rectangle BA, DC is equal to the rectangle BD, AE: But the rectangle BC, AD has been fhewn equal to the rectangle BD, CE; therefore the whole rectangle AC, BD d is equal to the rectangle AB, DC, together with the rectangle AD, BC. Therefore the rectangle, &c. Q. E D.

N 2

THE

This is a Lemma of C Ptolomagus, in page 9. of his ueyaλn ouytaŽIS.

c 16. 5.

A

DEFINITIONS.

I.

Solid is that which hath length, breadth, and thickness.

A ftraight 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 ftraight lines drawn in one of the planes perpendicularly to the common fection of the two planes, are perpendicular to the other plane.

V.

The inclination of a ftraight line to a plane is the acute angle contained by that ftraight line, and another drawn from the point in which the firft 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 fame plane.

VI.

The inclination of a plane to a plane is the acute angle contained by two ftraight lines drawn from any the fame point of their common fection at right angles to it, one upon one plane, and the other upon the other plane.

VII. Two

VII.

Two planes are faid to have the fame, or a like inclination to one another, which two other planes have, when the faid angles of inclination are equal to one another.

VIII.

Parallel planes are fuch which do not meet one another though produced.

IX.

Book XI.

A folid angle is that which is made by the meeting of more See N. than two plane angles, which are not in the fame plane, in

one point.

X.

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

XI.

Similar folid figures are fuch as have all their folid angles equal, Se N. each to each, and which are contained by the fame number

of fimilar planes.

XII.

A pyramid is a folid figure contained by planes that are conftituted betwixt one plane and one point above it in which they meet.

XIII.

A prifm is a folid figure contained by plane figures of which two that are oppofite are equal, fimilar, and parallel to one another; and the others parallelograms.

XIV.

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

XV.

The axis of a sphere is the fixed ftraight line about which the femicircle revolves.

XVI.

The center of a sphere is the fame with that of the femicircle.

XVII.

The diameter of a sphere is any ftraight line which paffes thro' the center, and is terminated both ways by the fuperficies of the sphere.

XVIII.

A cone is a folid figure described by the revolution of a right angled triangle about one of the fides containing the right angle, which fide remains fixed.

If the fixed fide be equal to the other fide containing the right angle, the cone is called a right angled cone; if it be iefs than the other fide, an obtufe angled, and if greater, an acute angled cone.