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Book VI

PROP, B.

THEOR.

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F an angle of a triangle be bisected by a straight 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 bisected by the ftraight line AD; the rectangle BA, AC is equal to the rectangle BD, DC, together with the fquare of AD.

Defcribe the circle ACB about the triangle, and produce

B

A

C

D

AD to the circumference in E,
and join EC: Then, because the
angle BAD is equal to the angle
CÃE, and the angle ABD to the
angle AEC, for they are in the
fame fegment; the triangles ABD,
AEC are equiangular to one an-
other: Therefore as BA to AD,
fo is EA to AC, and confe-
quently 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. P.

PROP. C.

E

THEOR.

[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 circle defcribed about the triangle.

Let ABC be a triangle, and AD the perpendicular from the angle A to the base BC; the rectangle BA, AC is equal to the rectangle contained by AD and the diameter of the circle defcribed about the triangle.

Defcribe

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THE

PROP. D.

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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.

Let ABCD be any quadrilateral infcribed 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

b 4 6.

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 bas 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

c 16.5.

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 is equal to dr. 2. 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 Cl. Ptolomaeus, in page 9, of his eyxan GUVTZĝis,

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A

DEFINITION S.

I.

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

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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 straight 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 straight line to a plane is the acute angle contained by that ftraight 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 fame plane.

VI.

The inclination of a plane to a plane is the acute angle contain ed 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 ate 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.

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

XI.

Similar folid figures are fuch as have all their folid angles equal, See 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.

Á prism 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 defcribed by the revolution of a femicircle about its diameter, which remains unmoved.

XV.

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

XVI.

The centre 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 centre, and is terminated both ways by the fuperficies of the 1phere.

XVIII.

A cone is a folid figure defcribed 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 lefs than the other fide, an obtufe angled, and if greater, an acute angled cone.

N 3

XIX. The

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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 defcribed by that fide containing the right angle, which revolves.

XXI.

A cylinder is a folid figure defcribed by the revolution of a right angled parallelogram about one of its fides which remains fixed.

XXII.

The axis of a cylinder is the fixed ftraight line about which the parallelogram revolves.

XXIII.

The bafes of a cylinder are the circles described by the two revolving oppofite fides of the parallelogram.

XXIV.

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

XXV.

A cube is a folid figure contained by fix equal fquares.

XXVI.

A tetrahedron is a folid figure contained by four equal and equilateral triangles.

XXVII.

An octahedron is a folid figure contained by eight equal and equilateral triangles.

XXVIII.

A dodecachedron is a folid figure contained by twelve equal pentagons which are equilateral and equiangular.

XXIX.

An icofahedron is a folid figure contained by twenty equal and equilateral triangles.

DE F. A.

A parallelopiped is a folid figure contained by fix quadrilateral figures whereof every oppofite two are parallel.

PROP.

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