PROP. XIN. THEOR.

In every triangle, the square of the side subtending either of the acute angles, is less than the squares of the sides containing that angle, by twice the rectangle contained by either of these sides, and the straight line intercepted between the perpendicular let fall upon it from the opposite angle, and the acute angle.

*12. 1.

Let ABC be any triangle, and the angle at B one of its acute angles, and upon BC, one of the sides containing it, let fall the perpendicular* AD from the opposite angle: the square of AC, opposite the angle B, shall be less than the squares of CB, BA, by twice the rectangle CB BD.

*7. 2.

First, let AD fall within the triangle ABC: and because the straight line CB is divided into two parts in the point D, the squares of CB, BD are equal to twice the rectangle contained by CB, BD, and the square of DC; to each of these equals add the square of AD; therefore the squares of CB, BD, DA, are equalf to twice the rectangle CB BD, and the squares of AD, DC: but the square of AB is equal to the squares of

12 Ax.

#47. 1.

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

#16. 1.

Secondly, let AD fall without the triangle ABC: then, because the angle at D is a right angle, the angle ACB is greater than a right angle; and therefore the square of AB is equal to the squares of AC, CB, and twice the rectangle BC CD: to each of these equals

#12. 2.

B

12 AX.

#3. 2.

add the square of BC; therefore the squares of AB, BC are equal to the square of AC, and twice the square of 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 rectangle BC-CD and the square 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 less than the squares of AB, BC, by twice the rectangle DB BC.

Lastly, let the side AC be perpendicular to BC: then BC is the straight line between the perpendicular and the acute angle at B: and it is manifest that the squares of AB, BC, are equal to the square of AC, and twice the square of BC: therefore in every triangle, &c. Q. E. D.

#47. 1.

and
2 Ax.

PROP. XIV. PROB.

B

To describe a square that shall be equal to a given
rectilineal figure.

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

#45. 1.

Describe the rectangular parallelogram BCDE equal to the rectilineal figure A. Then if the sides of it, BE, ED, are

equal to one another,

+26 Def. it ist a square, and what was required is

13. 1.

+10. 1.

now done: but if they

are not equal, pro-
duce one of them

BE to F, and maket

EF equal to ED, and bisect† BF in G; and from the centre G, at the distance GB, or GF, describe the semicircle BHF, and produce DE to H. The square described upon EH shall be equal to the given rectilineal figure A.

H

#5. 2.
*12 Def.

#47. 1.

+1 Ax.

Join G, H: and 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 square of GF: but GF is equalt to GH: therefore the rectangle BE EF, together with the square of EG, is equal to the square of GH: but the squares of HE, EG are equal to the square of GH: therefore the rectangle BE-EF, together with the square of EG, is equal to the squares of HE, EG: take away the square of EG, which is common to both; and the remaining rectangle BE EF is equal to the square of EH: but the 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 +Const. BD is equal† to the rectilineal figure A; therefore the rectilineal figure A is equal to the square 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.

+3 Ax.

1 Ax.

A straight line is said to touch a circle, when it meets the circle, and being produced does not cut it.

II.

Circles are said to touch one another, which meet, but do not cut one another.

III.

Straight lines are said to be equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal.

IV.

And the straight line on which the greater perpendicular falls, is said to be farther from the centre.

V.

An arc of a circle is any part of the circumference.

VI.

A segment of a circle is the figure contained by an arc of a circle and the straight line joining its extremities; which straight line is called the chord of the arc, or the base of the segment.

VII.

An angle in a segment is the angle contained by two straight lines drawn from any point in the arc of the segment to the extremities of the chord of the arc, or base of the segment.

VIII.

And an angle is said to insist or stand upon the arc intercepted between the straight lines that contain the angle.

IX.

A sector of a circle is the figure contained by two straight lines drawn from the centre, and the arc between them.

X.

Similar segments of circles are those in which the angles are equal, or which contain equal angles.