1

twice the rectangle under AD, AC; but the square of AB Воок II. is equal to the squares of BD, DA; therefore the square of ✓ BC is equal to the squares of BA, AC, and twice the rectangle c 4. under DA, AC; therefore the square of BC is greater than the squares of BA, AC, by twice the rectangle under DA, AC. Wherefore, &c.

b 47. 1.

PROP. XIII. THEOR.

Nevery acute angled triangle, the square of the fide fubtending the acute angle, is less than the squares of the fide containing the acute angle, by twice a rectangle contained under one of the fides about the acute angle, and that part of the fide lying between the acute angle and the perpendicular let fall from the opposite angle.

Let B be an acute angle in the triangle ABC; from the angle A let fall the perpendicular AD, cutting BC in D; 2 12. 1. the square of AC is less than the squares of AB, BC, by twice the rectangle under CB, BD.

For the square of AC is equal to the squares of AD, DCb; b 47. 1. and the square of AB is equal to the squares of AD, DB b; but the squares of BC, BD, are equal to twice the rectangle under BC, BD, together with the square of DC ; therefore the c 7. squares of AB, BC, are equal to the squares of AD, DC, and twice the rectangle under CB, BD; but the square of AC is equal to the squares of AD, DC; therefore the square of AC is less than the squares of AB, BC, by twice the rectangle under CB, BD. Therefore, &c.

PROP. XIV. PROB.

T

O make a square equal to a given right lined figure.

Make the rectangle BCDE equal to a given right lined figure A*; If BE be equal to ED, then BCDE is a square; 2 45. 1. and what was required is done. If not, produce EE to F; make EF equal to ED, and bisect BF in Gb; with the center b 10. 1. G, and distance GB, describe a femicircle BHF; produce DE to H, and join GH.

E

Then

{

& Def. 15 d Def. 15

BOOK II. Then the rectangle under BE, EF, together with the square of GE, are equal to the square of GF, or GH d; but the square of GH is equal to the squares of GE, EH °; therefore the rectangle under BE, EF, together with the square of GE, are equal to the squares of HE, EG. Take the square of GE from both, and the rectangle under BE, EF, that is, BD, is equal to the square of EH. Wherefore, &c,

I.

C 47. 1,

THE

A right line is faid to touch a circle, when drawn to the fame, and being produced, does not cut the circle.

III.

Circles are said to touch each other, which, meeting, do not cut one another.

IV.

Right lines in a circle are said to be equally distant from the center, when perpendiculars drawn from the center to each of them are equal, and that line upon which the greatest perpendicular falls is the least line.

Definition 19th, 1.

V.

VI.

An angle of a segment is the angle contained by the right line and circumference of the circle.

VII.

Book III.

An angle is faid to be in a segment, when right lines are drawn from fome point in the circumference to the ends of that line which is the base of the segment, which lines contain the

angle.

VIII. But,