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Book VI.

PRO P. B. THE O R.

Soe N.

a. 5.4.

IF

an angle of a triangle be bifected 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 bisecting the angle.

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

Describe the circle a ACB about the triangle, and produce AD to the circumference in E, and join EC. then because the angle BAD is equal to the angle CAE, and the

А. bo 21. 3. angle ABD to the angle b AEC, for

they are in the same segment; the
triangles ABD, AEC are equian-
gular to one another. therefore as

B €. 4. 6. BA to AD, so is © EA to AC, and

consequently the rectangle BA, AC
is equal 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 ! 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.

d. 16. 6.

c. 3. 2.

f. 35. 3.

Sec N.

I

PROP. C. THEO R.
F from an 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 ABC be a triangle, and AD the perpendicular from the angle A to the bafc BCthe rectangle BA, AC is equal to the rectangle contained by AD and the diameter of the circle defcribed about the triangle.

b. 31. 3•

Describe the circle ACB about

Book VI.

А. the triangle, and draw its diameter

a. 5. 4. AE, and join EC. because the right angle BDA is equal o to the angle

B

C ECA ¡a a semicircle, and the angle

D ABD to the angle AEC in the same fegmento; tbe triangles ABD, AEC

d. 4. 6. are equiangular. therefore as - BA to AD, fo is EA to AC, and conse E quently the rectangle BA, AC is equal to the rectangle EA, AD. If therefore from an angle, &c. c. 16. 6. O. E. D.

C. 21. 3•

E

THE

PROP. D. THEOR.
HE rectangle contained by the diagonals of a qua-

drilateral inscribed in a circle, is equal to both the rectangles contained by its opposite sides.

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 *.

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 angle BCE, because a. 21. 3. they are in the same segment; therefore the triangle ABD is equiangular to the triangle BCE. whereforebas BCis to CE, fo is BD to DA,

B

b. 4. 6. and consequently the rectangle BC,

с 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

D

A BD to DC; wherefore 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 is equal to the rectangle AB, DC together with the rectangle AD, BC. Therefore the rectangle, &c. Q.E.D.

This is a Lemma of Cl. Ptolomacus in page 9. of his siyáan ouvražis.

C. 16. 6.

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A sou

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

II.
That which bounds a solid is a superficies.

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 con-

tained 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.

Book XI. 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 tho' produced.

IX. A solid angle is that which is made by the meeting of more than Se N.

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.' See N.

XI. Similar solid figures are such as have all their solid angles equal, See N.

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 se-
micircle revolves.

XVI.
The center of a sphere is the fame with that of the semicircle.

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 solid figure described by the revolution of a right

angled triangle about one of the sides containing the right
angle, which fide remains fixed.

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Book XI. If the fixed fide 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 fide 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 fides which remains fix-
ed.

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 re-
volving 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 fix equal squares.

XXVI.
A Tetrahedron is a solid figure contained by four equal and equi-
lateral 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 pen-
tagons which are equilateral and equiangular.

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

DEF: A.
A Parallelepiped is a solid figure' contained by six quadrilateral

figures whereof every opposite two are parallel.

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