Sidebilder
PDF
ePub

See N.

Book VI.

PRO P. B. THE OR.
F an angle of a triangle be bisected by a straight line,

which likewise curs 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 bisected by the straight line AD; the rectangle BA, AC is equal to the rectangle

BD, DC together with the square of AD. a. $. 4.

Describe the circle * ACB about the triangle, and produce AD to the circumference in E, and join

A
EC. then because the angle BAD

is equal to the angle CAE, and the
b. 21. 3. angle ABD to the angle b AEC, for
they are in the same segment; the B

D triangles ABD, AEC are equiangu

lar to one another. therefore as 6. 4. 6. BA to AD, so is EA to AC, and

confequently the rectangle BA, AC d. 16. 6. is equal to the rectangle EA, AD,

E that ise to the rectangle ED, DA

together with the square of AD. but the rectangle ED, DA is £. 35. 3. 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.

[ocr errors]
[ocr errors]

See N.

IF from an angle of a triangle a straight line be drawn

PROP. C. THEOR.
F

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 base BC; the rectangle BA, AC is equal to the rect. angle contained by AD and the diameter of the circle described about the triangle.

Describe

A

a. s. 4

b. 31. 3.

Describe a the circle ACB about

Book VI. the triangle, and draw its diameter AE, and join EC. because the right angle BDA is equal to the angle ECA in a semicircle, and the angle B

с

D ABD to the angle AEC in the same segment"; the triangles ABD,

c. 21. 3. AEC are equiangular, therefore as

d. 4. 6. BA to AD, so is EA to AC, and consequently the rectangle BA, AC

E is equal to the rectangle EA, AD. If therefore from an angle, &c. c. 16. 6. 0. E. D.

PRO P. D. THE OR.

TH

HE rectangle contained by the diagonals of a quadri

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

Let ABCD be any quadrilateral inscribed in a circle, and join AC, BD; the rectangle contained by AC, 5D 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, be- a. 21. 3. cause they are in the same segment; therefore the triangle ABD is equiangular to the triangle BCE. wherefore b as BC is to CE, so is BD

В.

b. 4. 6. 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

E ABE is equiangular to the triangle BCD. as therefore BA to AE, fo is BD to DC; wherefore the rectangle BA, DC is equal to the rectangle BD, AE, but the rectangle BC, AD has been Mewn 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 payaan Burtažis.

THE

c. 16. 6.

[blocks in formation]

A

I.
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 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 per-
pendicular 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 same point of their com-
mon section at right angles to it, one upon one plane, and the
other upon the other plane.

VII. Two

See N.

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' pro-
duced:

IX.
A solid angle is that which is made by the meeting of more than two See N.
plane angles, which are not in the same plane, in one point.

X.
' The tenth Definition is omitted for reasons given in the Notes.'

XI.
Similar solid figures are such as have all their folid angles equal, each Sce N.

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 remainis unmoved.

XV.
The axis of a sphere is the fixed straight line about which the semi-
circle revolves.

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

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

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.
If the fixed side be equal to the other side containing the right angle,

thc

1

Book XI. 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 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 remains fixed.

XXI. 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 equila.
teral 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 equi-
lateral triangles.

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

gures whereof every opposite two are parallel.

PROP.

« ForrigeFortsett »