Book II.

See N.

PROP. XIII. THEOR.

N every triangle the square of the fide fubtending any of the acute angles, is lefs than the fquares of the fides containing that angle, by twice the rectangle contained by either of thefe fides, and the ftraight line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle.

a

Let ABC be any triangle, and the angle at B one of its acute angles, and upon BC one of the fides containing it let fall the perpen. 3. 12. 1. dicular AD from the oppofite angle. the fquare of AC oppofite to the angle B, is lefs than the fquares of CB, BA by twice the rectangle CB, BD.

First, Let AD fall within the triangle ABC; and because the ftraight line CB is divided into two

parts in the point D, the fquares of

C

A

47. 1. equal to the fquares of BD, DA, because the angle BDA is a right angle; and the square of AC is equal to the fquares of AD, DC. therefore the fquares of CB, BA are equal to the fquare of AC, and twice the rectangle CB, BD; that is, the square of AC alone is lefs than the fquares of CB, BA by twice the rectangle CB, BD.

Secondly, Let AD fall without the triangle ABC. then because the angle at D is a right and. 16. 1. gle, the angle ACB is greater than a right angle; and therefore the e. 12. 2. fquare of AB is equal to the squares of AC, CB and twice the rectangle BC, CD. to thefe equals add the square of BC, and the fquares of B

A

C D

AB,

AB, BC are equal to the fquare of AC, and twice the fquare of Book II. BC, and twice the rectangle BC, CD. but because BD is divided into two parts in C, the rectangle DB, BC is equal to the rectan- f. 3. 2. gle BC, CD and the fquare of BC. and the doubles of these are equal. therefore the squares of AB, BC are equal to the square of AC, and twice the rectangle DB, BC. therefore the square of AC alone, is lefs than the fquares of AB, BC, by twice the rectangle DB, BC.

Laftly, Let the fide AC be perpendicular to BC; then is BC the straight line between the perpendicular and the acute angle atB. and it is ma nifeft that the fquares of AB, BC are equal to the fquare of AC, and twice the fquare of BC, Therefore in every triangle, &c. Q. E. D.

A

C. 47. E

T

PROP. XIV. PROB.

O defcribe a fquare that fhall be equal to a given See N. rectilineal figure.

Let A be the given rectilineal figure; it is required to describe a fquare that shall be equal to A.

Describe the rectangular parallelogram BCDE equal to the rectilineal figure A. If then the fides of it BE, ED are equal to one ano

BF in G; and from the center G, at the distance GB or GF describe the femicircle BHF, and produce DE to H, and join GH. therefore because the straight line BF is divided into two equal parts in the point G, and into two unequal at E, the rectangle BE, EF, together with the square of EG, is equal to the fquare of GF. but b. s. a. GF is equal to GH; therefore the rectangle BE, EF, together with

the fquare of EG, is equal to the fquare of GH. but the fquares

of

Book II. of HE, EG are equal to the fquare of GH. therefore the rectangle BE, EF together with the fquare of EG is equal to the fquares of

rectangle contained by BE, EF is the parallelogram BD, because EF is equal to ED; therefore BD is equal to the square of EH, but BD is equal to the rectilineal figure A; therefore the rectilineal figure A is equal to the fquare of EH. wherefore a square has been made equal to the given rectilineal figure A, viz. the square described upon EH. Which was to be done.

THE

E

BOOK III.

DEFINITION S.

I.

QUAL circles are thofe of which the diameters are equal, or from the centers of which the straight lines to the circumferences are equal.

'This is not a Definition but a Theorem, the truth of which is 'evident; for if the circles be applied to one another, so that their 'centers coincide, the circles muft likewife coincide, fince the 'ftraight lines from their centers are equal.'

Book III.

VI.

A fegment of a circle is the figure con-
tained by a straight line and the cir-
cumference it cuts off.

VII.

"The angle of a segment is that which is contained by the straight "line and the circumference."

VIII.

An angle in a fegment is the angle con-
tained by two straight lines drawn
from any point in the circumference
of the fegment, to the extremities of
the straight line which is the base of
the fegment,

IX.

And an angle is faid to infift or stand
upon the circumference intercepted be-
tween the straight lines that contain
the angle.

X.

The sector of a circle is the figure con-
tained by two straight lines drawn from
the center, and the circumference be-
tween them.

XI.

Similar fegments of a circle,
are thofe in which the an-
gles are equal, or which
contain equal angles.

PROP, I. PROB.

2. 10. I.

Let ABC be the given circle; it is required to find its center. Draw within it any straight line AB, and bifect it in D; from b. 11. 1. the point D draw b DC at right angles to AB, and produce it to E, and bisect CE in F. the point F is the center of the circle ABC.

For