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Secondly. Let AD fall without the triangle ABC.

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D."

Then, because the angle at D is a right angle,

1. The angle ACB is greater than a right angle (I. 16) ; and therefore

2. The square on AB is equal to the squares on AC, CB, and

twice the rectangle BC, CD (II. 12); to each of these equals add the square on BC; therefore

3. The squares on AB, BC are equal to the square on AC,

twice the square on BC, and twice the rectangle BC, CD; but because BD is divided into two parts in C, therefore

4. The rectangle DB, BC is equal to the rectangle BC, CD, and

the square on BC (II. 3) ; and the doubles of these are equal ; that is, twice the rectangle DB, BC is equal to twice the rectangle BC, CD, and twice the square on BC; therefore

5. The squares on AB, BC are equal to the square on AC, and

twice the rectangle DB, BC. wherefore the square on AC alone is less than the squares on AB, BC; by twice the rectangle DB, BC.

Lastly. Let the side AC be perpendicular to BC.

B

Then BC is the straight line between the perpendicular and the acute angle at B; and it is manifest, that

The squares on AB, BC, are equal to the square on AC, and twice the square on BC (I. 47).

Therefore in any triangle, &c. Q.E.D.

PROPOSITION 14.- Problem.

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.

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Construction. Describe the rectangular parallelogram BCDE equal to the rectilineal figure A (I. 45). Then, if the sides of it, BE, ED, are equal to one another, it is a square, and what was required is now done. But if BE, ED are not equal, produce one of them BE to F, and ake F equal to ED; bisect BF in G (I. 10); from the centre G, at the distance GB or GF, describe the semicircle BHF, and produce DE to meet the circumference in H.

Then the square described upon EH shall be equal to the given rectilineal figure A.

Join GH.

Proof. Then, because the straight line BF is divided into two equal parts in the point G, and into two unequal parts in the point E, therefore

1. The rectangle BE, EF, together with the square on EG, is

equal to the square on GF (II. 5); but GF is equal to GH (Def. 15); therefore

2. The rectangle BE, EF, together with the square on EG, is

equal to the square on GH; but the squares on HE, EG are equal to the square on GH (I. 47); therefore

3. The rectangle BE, EF, together with the square on EG, is

equal to the squares on HE, EG; take away the square on EG, which is common to both; therefore

4. The rectangle BE, EF is equal to the square on HE. But the rectangle contained by BE, EF is the parallelogram BD, because EF is equal to ED; therefore

5. BD is equal to the square on EH; but BD is equal to the rectilineal figure A (constr.); therefore

6. The square on EH is equal to the rectilineal figure A. Wherefore a square has been made equal to the given rectilineal figure A, namely, the square

Q.E.F.

described

upon EH.

BOOK III.

DEFINITIONS.

1. EQUAL circles are those of which the diameters are equal, or from the centres of which the straight lines to the circumferences are equal.

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

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

4. 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.

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

6. A segment of a circle is the figure contained by a straight line, and the circumference which it cuts off.

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

8. An angle in a segment is the angle contained by two straight lines drawn from any point in the circumference of the segment, to the extremities of the straight line which is the base of the segment.

9. An angle is said to insist or stand upon the circumference intercepted between the straight lines that contain the angle.

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

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