THE

ELEMENTS OF EUCLID.

BOOK III.

DEFINITIONS.

I.

IF the radii of two circles are equal, the circles are equal, and their circumferences are equal; and if two circles are equal, their radii are equal and their circumferences are equal.

OBS. This is neither a defn nor an axiom, but consists of two props which may be thus proved.

(1) Let ABC, DEF be two circles, of which the centres are G, H, and which have equal radii. Then the circle ABC shall be equal to the circle DEF, and the circumference ABC to the circumference DEF.

For if the circle ABC be applied to the circle DEF, so that the centre G may coincide with the centre H, then every point in the circumference ABC will coincide with a point in the circumference DEF, since the radii of the two circles are equal; that is, the circumference ABC will coincide with the circumference DEF and the circle ABC with the circle DEF. But magnitudes which coincide are equal (Ax. 8): therefore the circumference ABC is equal to the circumference DEF, and the circle ABC to the circle DEF. Which was to be proved.

(2) Let ABC, DEF be two equal circles, of which the centres are G, H: then the radii of ABC shall be equal to the radii of DEF, and the circumference ABC to the circumference DEF.

For if the radii of the two circles are unequal, let DEF be that

B

A

с

D

E

H

F

circle, which, if possible, has the larger radii. Apply ABC to DEF as before, then the extremity of each radius of ABC will be nearer to the coincident centres than the extremity of the radius of DEF on which it falls, and therefore the circle ABC will fall entirely within, and be a part of the circle DEF. Therefore the circle ABC is less (Ax. 9) than the circle DEF: which is by hyps impossible. Therefore the radii of the two circles are not unequal: that is, they are equal, and hence by (1), the circumference ABC is equal to the circumference DEF. Which was to be proved.

The distance of a straight line from the centre of a circle is the length of the perpendicular drawn to it from the centre.

V.

Straight lines are accordingly equally distant from the centre of a circle, when the perpendiculars drawn to them from the centre are equal; and one straight line is said to be farther from or nearer to the centre than another, according as the greater or less perpendicular falls on it.

VI.

A segment of a circle is the figure contained by a part of the circumference of a circle, called an arc of a circle, and the straight line drawn joining the two extremities of the arc.

VII.

D

An angle of a segment is either of the two curvilinear angles contained by the straight line and the arc of the circle.

VIII.

An angle in a segment is any angle contained by two straight lines drawn from any point in the arc of the segment to the extremities of the straight line (which is the base of the segment).

IX.

An angle is said to stand on a part of the circumference, or arc of the circle, when such arc is intercepted by the straight lines that include the angle.

X.

A sector of a circle is the figure contained by two radii, and the arc intercepted between them.

Segments of cir

cles are defined to be similar, when the angles in them are equal.

XI.

PROPOSITIONS.

PROP. I. THEOR.

To find the centre of a given circle.

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

Draw any straight line AB, so as to fall entirely within the circle, and have its extremities A, B on the circumference. Bisect (i. 10) AB in D; from D draw (i. 11) a straight line perpendicular to AB; and produce it both ways. Then because AB was taken within the circle, this straight line must meet the circumference in two points; let them be c and E;

C

G+

H

F

D

B

E

bisect CE in F. Then F shall be the centre of the circle

ABC.

For if F be not the centre, some other point must be the centre, either a point in the straight line CE, or a point without it.

I. Let, if possible, some point &, in the straight line CE, be the centre.

Then because CE is bisected in F by const", any other point in CE besides F divides CE into two unequal parts ; and therefore GE, GC are unequal. But because & is

the centre of the circle ABC, GC is equal to GE by def1 : which is impossible. Therefore & is not the centre of the circle ABC; and it can be shewn in like manner that no other point in CE but F can be the centre.

II. Let, if possible, some point H, without the straight line CE, be the centre.

Join HA, HD, HB.

Because DA is equal to DB by const", HA to Hв by the def of a circle, and HD common to the two triangles HAD, HDB; therefore these two triangles have the three sides HA, AD, DH respectively equal to the three sides HB, BD, DH. Therefore they are equal in every respect (i. 8); and hence the angle ADH is equal to the angle BDH. That is, the straight line HD standing on AB makes with it the adjacent angles HDA, HDB equal to one another: therefore by defn each of these angles is a right angle. But FDB is likewise a right angle by const; and all right angles are equal (Ax. 11); therefore the angle HDB is equal to the angle FDB, that is, the part.equal to the whole: which is impossible (Ax. 9). Therefore H is not the centre of the circle ABC; and it can be shewn in like manner that no other point out of CE is the centre.

Hence the point F must be the centre of the circle. Which was to be proved.

COR.-Hence if a straight line be drawn bisecting at right angles a straight line in a circle (i. e. within the circle, and having its extremities on the circumference), and be produced, the centre of the circle shall be in this straight line produced.

PROP. II. THEOR.

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

For if it do not fall within, it must either fall without the circle, or partly within and partly without the circle, or else fall on the circumference.

I. Let AB fall, if possible, without the circle (Fig. 1), or partly within and partly without the circle (Fig. 2).