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and they are upon equal straight lines BC, BK: But similar seg- Book VI. ments of circles upon equal straight lines, are equal to one another: Therefore the fegment BXC is equal to the segment COK: 8 24 3 And the triangle BGC is equal to the triangle CGK; therefore the whole, the sector BGC, is equal to the whole, the fector CGK: For the fame reason, the sector KGL is equal to each of the fectors BGC, CGK: In the fame manner, the sectors 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 reason, whatever multiple the circumference EN is of EF, the fame multiple is the sector EHN of the fector EHF: And if the circumference BL be equal to EN, the

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fector BGL is equal to the sector EHN; and if the circumfe-
rence BL be greater than EN, the sector BGL is greater than
the sector EHN; and if less, less: Since then, there are four
magnitudes, the two circumferences BC, EF, and the two sec-
tors BGC, EHF, and of the circumference BC, and sector
BGC, the circumference BL and sector BGL are any equi-
multiples whatever; and of the circumference EF and sector
EHF, the circumference EN and sector EHN are any equi-
multiples whatever; and that it has been proved, if the circum-
ference BL be greater than EN, the sector BGL is greater than
the sector EHN; and if equal, equal; and if less, less. There-
foreb, as the circumference BC is to the circumference EF, so bs.
is the sector BGC to the sector EHF. Wherefore, in equal cir-
cles, &c. Q. E. D.

def. S.

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Book VF.

Sec N.

PROP. В. THEOR.

IF an angle of a triangle be bisected by a straight line, which likewife cuts the base; the rectangle contained by the fides of the triangle is equal to the rectangle con tained by the fegments of the base, together with the square of the straight line bisecting the angle.

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

a 5.4.

Describe the circle ACB about the triangle, and produce

AD to the circumference in E,

A

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angle AEC, for they are in the
same segment; the triangles ABD, B

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с 4 б.

d 16.6,

€ 3.2.

f 35 3.

See N.

E

AEC are equiangular to one an-
other: Therefore as BA to AD,
so is EA to AC, and confe-
quently the rectangle BA, AC is
equal d to the rectangle EA, AD,
that is, to the rectangle ED, DA
together with the square 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
square of AD. Wherefore, if an angle, &c. Q. E. D.

PROP. C. THEOR.

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

Let ABC be a triangle, and AD the perpendicular from the angle As the bate BC; the rectangle BA, AC is equal to the rectang. co. ained by AD and the diameter of the circle defcrined about the triangle.

Describe

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TH

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

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 same segment; therefore

the triangle ABD is equiangular to

B

: the triangle BCE: Wherefore & as BC is to CE, so is BD to DA; and consequently 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

b 4. 6.

C

с 16.5.

E

D

AE, so is BD to DC; wherefore

triangle BCD: As therefore BA to A

the rectangle BA, DC is equal to the rectangle BD, AE: But the rectangle BC, AD has been shewn 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.

N2

THE

† This is a Lemma of Cl Ptolomapus, in page 9. of his μεγαλη συνταξις.

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DEFINITIONS.

I.

is that which hath length, breadth, and thicknefs.

II.

That which bounds a solid is a fuperficies.

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 is the acute angle contained by two straight lines drawn from any the fame point of their common section 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 such 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 such as have all their folid angles equal, Sce 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 constituted betwixt one plane and one point above it in which they meet.

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A prism is a solid figure contained by plane figures of which two that are opposite 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 femicircle about its diameter, which remains unmoved.

XV.

The axis of a sphere is the fixed straight 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 straight line which passes thro' the center, and is terminated both ways by the superficies 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 side be equal to the other fide containing the right angle, the cone is called a right angled cone; if it be less than the other fide, an obtuse angled, and if greater, an acute angled cone.

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