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Book VI.

PROB. B. THEOR.

See Note.

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

b 21. 3.

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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.

Describe the circlea ACB about the triangle, and produce
AD to the circumference in E,

Α.
and join EC: then because the
angle BAD is equal to the angle
CĂE, and the angle ABD to the
angleb AEC, for they are in the B

segment; the triangles
ABD, AEC are equiangular to
one another; therefore as BA to
AD, so ise EA to AC, and con-
sequently the rectangle BA, AC
is equald to the rectangle EA,

E AD, that is °, to the rectangle ED, DA, together with the square of AD : but the rectangle ED, DA is equal to the rectanglef 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.

C 4.6.

d 16. 6. e 3. 2.

f 35. 3.

PROP. C. THEOR.

See Note.

IF from any angle of a triangle a straight line be drawn perpendicular to the base; the rectangle con. tained 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.

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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 described about the triangle.

Book VI.

2 5. 4.
b 31. 3.

raigh

с

Describe a the circle ACB about
the triangle, and draw its diameter
AE, and join EC:because the right
angle BDA is equal b to the angie
ECA in a semicircle, and the angle B

ABD to the angle AEC in the saine
i segmento; the triangles ABD, AEC
are equiangular: therefore as a BA
to AD, so is EA to AC; and conse-
quently the rectangle BA, AC is
equal e to the rectangle EA, AD.
If, therefore, from an angle, &c.
Q. E. D.

c'21.3

CO recta

d 4. 6.

ther

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THE rectangle contained by the diagonals of a See Note: quadilateral inscribed in a circle, is equal to both the rectangles contained by its opposite sides.

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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

a 21. 3.
to the angle EBC: and the angle BDA is equal a to the an-
gle BCE, because they are in the same segment; therefore
the triangle ABD is equiangular

b 4. 6.'
to the triangle BCE: wherefore b

с
as BC is to CE, so is BD to DA;
and consequently the rectangle

c 16.6.
BC, AD is equal to the rectangle
BD,CE: again, because the angle
ABE is equal to the angle DBC,
and the angle a BAE to the angle
BDC, the triangle ABE is equi-

D angular to the triangle BCD: as therefore BA to AE, so is BD to A DC; wherefore the rectangle BA, DC is equal to the rectangle BD, AE: but the rectangle BC, AD has been shown equal to the rectangle BD, CE: therefore the whole rectangle AC, BD d is equal to the rect-d42 angle AB, DC, together with the rectangle AD, BC. Therefore the rectangle, &c. Q. E. D."

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• This is a Lemma of Cl. Ptolomæus, in page 9 of his peyaan pyro Tuis.

THE

ELEMENTS OF EUCLID.

BOOK XI.

DEFINITIONS.

I.
Book X1. A SOLID is that which hath length, breadth, and thickness.

II.
That which bounds a solid is a superficies.

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 con-

tained by two straight lines drawn from any the same point
of their common section at right angles to it, one upon one
plane, and the other upon the other plane.

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VII.
Two planes are said to have the same, or a like inclination to Book XI.
Х

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 though
produced.

IX. A solid angle is that which is made by the meeting of more See Note. than 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 Note.
XI.
Similar solid figures are such as have all their solid angles See Notes

equal, 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 consti-

tuted 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 onc
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
semicircle revolves.

XVI.
The centre of a sphere is the same with that of a semicircle.

XVII.
The diameter of a sphere is any straight line which passes

through the centre, and is terminated both ways log the super-
ficies 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 side remains fixed. If the fixed side be equal to the other side containing the

right angle, the cone is called a right angled.cone; if it be i less than the other side, an obtuse angled, and if greater, an acute angled cone,

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1

Book XI.

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 side 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 sides, which re.
mains fixed.

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 revolving 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 six equal squares.

XXVI.
A tetrahedron is a solid figure contained by four equal and
equilateral 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 pentagons which are equilateral and equiangular.

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

DEF. A.
A parallelopiped is a solid figure contained by six quadrifa-

teral figures, whereof every opposite two are parallel.

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