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*7.2.

+2 Ax.

* 47.1.

16. 1.

*12.2.

† 2 Ax.

* 3.2.

A

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 equal+ to twice the
rectangle CB, BD, and the squares of
AD, DC: but the square of AB is equal*
to the squares of BD, DA, because the
angle BDA is a right angle; and the
square of AC is equal to the squares of AD, DC; there-
fore 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.

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

B D

B

rectangle BC, CD: to each of these equals 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, *47.1. and BC, are equal* to the square of AC and twice the square of BC: therefore, in every triangle, &c. Q. E. D.

2 Ax.

See N.

PROP. XIV. PROB.

B

A

To describe a square that shall be equal to a given recti

lineal figure.

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

H

+ 30 Def.

Describe the rectangular parallelogram BCDE * 45. 1. equal to the rectilineal figure A. 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 they are not equal, produce one of them BE to F, and make EF equal to ED, and

B

G

D

† 3. 1.

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.

*

† 10. 1.

15 Def.

*47.1.

Join GH: and because the straight line BF is divided into two equal parts in the point G, and into two unequal at F, the rectangle BE, EF, together with the square of EG, is equal to the square of GF: but GF 5. 2. is equal to GH: therefore the rectangle BE, EF, to- † gether 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 +1 Ax. 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 con- † 3 Ax. tained 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; † Constr. therefore the rectilineal figure A is equal to the square + 1 Ax. 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

ELEMENTS OF EUCLID.

BOOK III.

DEFINITIONS.

I.

EQUAL circles are those of which the diameters are equal, or from the centres 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 centres coincide, the circles must likewise coincide, since the straight lines from the tres are equal."

II.

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

III.

cen

&

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

IV.

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.

V.

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

VI.

A segment of a circle is the figure contained by a straight line and the circumference 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 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.

IX.

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

X.

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

XI.

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

See N.

10. 1. * 11. 1.

+ Constr.

8.1.

PROP. I. PROB.

To find the centre of a circle.

Let ABC be the given circle; it is required to find its centre.

Draw within it any straight line AB, and bisect * it in D; from the point D draw* DC at right angles to AB, and produce it to E, and bisect CE in F: the point F shall be the centre of the circle ABC.

FG

E

D B

For if it be not, let, if possible, G be the centre, and join GA, GD, GB: then, because DA is equal to DB, and DG common to the two triangles ADG, BDG, the two sides AD, DG, are equal to the two BD, DG, each to each; and the base GA is +15 Def. 1. equal to the base GB, because they are drawn from the centre G‡: therefore the angle ADG is equal✶ to the angle GDB: but when a straight line standing upon another straight line makes the adjacent angles equal to one another, *10 Def. 1. each of the angles is a right angle; therefore the angle GDB is a right angle: but FDB is likewise at right angle; wherefore the angle FDB is equal to the angle GDB, the greater to the less, which is impossible therefore G is not the centre of the circle ABC. In the same manner it can be shewn that no other point but F is the centre; that is, F is the centre of the circle ABC. Which was to be found.

+ Constr. + 1 Ax.

*

COR. From this it is manifest, that if in a circle a straight line bisect another at right angles, the centre of the circle is in the line which bisects the other.

[blocks in formation]

If any two points be taken in the circumference of a circle, the straight line which joins them shall fall within the circle.

Let ABC be a circle, and A, B any two points in the circumference; the straight line drawn from A to B shall fall within the circle.

N. B.-Whenever the expression "straight lines from the cen tre," or drawn "from the centre" occurs, it is to be understood that they are drawn to the circumference.

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