C 4. 6.

Book VI. angle DBE be added to each of them, the angle ABD is equal

to EBC.

Because the angle ABD is equal B
to EBC, and the angle BDA to the
angle BCE, for they are in the same

С b 21. 3. segment BCDA6; the triangles ABD,

BCE are equiangular: Wherefore",
as BD to DA, fo is BC to CE ; and

confequently the rectangle BD, CE d:6.6. is equal to the rectangle AD, BC. Again, because the angle ABE is

А. equal to the angle DBC, and the angle BAE to the angle b BDC; the triangle ABE is equiangular to BCD: as, therefore, BA to AE, lo cis BD to 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

B e 1.2. whole rectangle AC, BD is equal

to the rectangle AB, DC, together
with the rectangle AD, BC. Where-

С fore, &c. Q. E. D.

Cor. If AD be equal to DC, or the angle ABD equal to DBC, the

E rectangle AC, BD is equal to the A rectangles AD, BC and AD, AB; that is, to the rectangle contained by AD and the sum of AB and BC. Consequently AB, BC together are to BD, as AC to AD,



F two points be taken in the semidiameter of a

circle, such that the rectangle contained by the segments between them and the centre is equal to the square of the semidiameter : straight lines drawn from these points to any point of the circumference Thall have the same ratio that the segments of the diameter between them and the circumference have to one another.

Let ABC be a circle, of which AC is the diameter, and D the centre; and let E, F be two points in AC, on the same side of the centre, so that the rectangle ED, DF is equal to the {quare of AD; and draw EB, FB to any point B of the circumference : EB is to FB, as EA to AF.

€ 32. I.

Join AB, BD; and because the rectangle ED, DF is equal Book. VI. to the square of AD or DB, FD is to DB, as - DB to DE; that is, the sides of the tri

a 17.6. angles FBD, EBD, about

G their common angle D, are proportionals; therefore the triangles are equiangular

b 6.6.

С and have the angle FBDF equal to BED: But BED is equal o to the two EAB, ABE; therefore the two EAB, ABE are equal to FBD: of which EAB is equal d to ABD, because BD is equal d

5. 2. to DA; therefore the remaining angle ABE is equal to the remaining angle ABF; that is, the angle FBE is bifected by BA: Wherefore, as FB to BE, fo eis FA to AE. Therefore, e 3.6. &c. Q. E. D.

Cor. Hence, AB bisects the angle FBE. And if BC be joined, and FB produced to G: because the angle ABC in a semicircle is a right angle f, it is half the sum of the angles [ 31. 3. FBE and EBG 8; of which the angle ABE is the half of g 13. 1. FBE; therefore the remaining angle EBC is the half of the remaining angle EBG: Therefore BC bisects the exterior angle EBG.

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Book XI. Solid is that which hath length, breadth, and thickness.

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.

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

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 firit line above the plane, meets the same plane.

VI. See N. The inclination of a plane to a plane is the angle contained by two straight lines drain from any the same point of their


common section at right angles to it, one upon one plane, and Book XI. the other upon the other plane.

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 as do not meet one another though produced.

Similar solid figures are such as are contained by the same num. See N.

ber of fimilar planes having the same inclination to one an-
X. Omitted.

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

See N. two plane angles, which are not in the same plane, in one point, the inclinations of all the planes being inwards.

A parallelopiped is a solid figure contained by fix quadrilateral
figures, whereof every opposite two are parallel.

A pyramid is a solid figure contained by planes that are consti-

tuted betwixt one plane and one point above it in which they


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 sphere is a solid figure described by the revolution of a semicircle about its diameter, which remains unmoved.

The axis of a sphere is the fixed straight line about which the
semicircle revolves.

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

The diameter of a sphere is any straight line which passes

through the centre, and is terminated both ways by the super-
ficies of the sphere.

A cone is a solid figure described by the revolution of a right

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


Y 2

Book XI.

XIX. The axis of a cone is the fixed straight line about which the triangle revolves.

-The base of a cone is the circle described by that fide containing
the right angle, which revolves.

A cylinder is a solid figure described by the revolution of a right

angled parallelogram about one of its fides, which remains

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

The bases of a cylinder are the circles described by the two re-
volving opposite sides of the parallelogram.

Similar cones and cylinders are those which have their axes and

the diameters of their bases proportionals.


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VE part of a straight line cannot be in a plane,

and another part above it. If it be pofible, let AB, part of the straight line ABC, be in the plane, and the part BC above it : And since the straight line AB is in the plane, it can Le produced in that plane : Let it be produced to D: And let any plane pass through the straight line AD, and be turned

A about it until it pass through

the point C; and because the points B, C, are in this plane, a 7. Def. 1. the straight line BC is in it ?: Therefore there are two straight

lines ABC, ABD in the same plane that have a common segb10Ax.I. ment AB; which is impoffible b. Therefore one part, &c.

Q. E. D.



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TWO firaight lines which cut one another are in

one plane, and three straight lines which meet one another are in one plane.


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