137. Definition. In two symmetrical figures the corresponding symmetrical lines are called homologous.

Thus, in the symmetrical figures ABCDE, A'B'C'D'E', the homologous lines are AB and A'B', BC and B'C', etc.

In all cases, two figures, symmetrical with respect to an axis, can be brought into coincidence by the revolution of either about the axis.

D

E

с

B

M

N

B

b. Symmetry with respect to a centre.

ED

D'

138. Definition. Two points are symmetrical with respect to a fixed point, called the centre of symmetry, when this centre bisects the straight line joining the two points.

Thus, A and A' are symmetrical with respect to the centre O, if the line AA' passes through O and is bisected at 0.

A

The distance of a point from the centre is called its radius of symmetry. A point A is brought into coincidence with its symmetrical point A', by revolving its radius OA through two right angles in its own plane (16).

139. Definition. Any two figures are symmetrical with respect to a centre, when every point of one figure has its symmetrical point on the other.

Thus, A'B' is the symmetrical figure of the straight line AB with respect to the centre O.

BI

Since the triangles AOB, A' OB',

A

C

B

are equal (76), the angle B is equal

to the angle B'; therefore, AB and A'B' are parallel. In general,

the homologous lines of two figures, symmetrical with respect to a centre,

are parallel. Thus, in the symmetri

cal figures ABCD, A'B'C'D', the homologous lines AB and A'B' are D' parallel, BC and B'C' are parallel,

etc.

A

A

B

Two figures symmetrical with respect to a centre can be brought

into coincidence by revolving one of them, in its own plane, about the centre; every radius of symmetry revolving through two right angles at the same time.

140. Definition. Any single figure is called a symmetrical figure, either when it can be divided by an axis into two figures symmetrical with respect to that axis, or when it has a centre such that every straight line drawn through it cuts the figure in two points which are symmetrical with respect to this centre.

Thus, ABCDC'B' is a symmetrical figure with respect to the axis MN, being divided by MN into two figures, ABCD and AB'C'D, which are symmetrical with respect to MN.

B

N

B

E

D

L

Also, the figure ABCDEF is symmetrical with respect to the centre O, its vertices, taken two and two, being symmetrical with respect to 0. In this case, any straight line KL drawn through the centre and terminated by the perimeter, is called a diameter.

K

B

PROPOSITION XLIV.-THEOREM.

141. If a figure is symmetrical with respect to two AXES perpendicular to each other, it is also symmetrical with respect to the intersection of these axes as a CENTRE of symmetry.

Let the figure ABCDEFGH be symmetrical with respect to the two perpendicular axes MN, PQ, which intersect in O; then, the point is also the centre of symmetry of the figure.

For, let T be any point in the perimeter of the figure; draw TRT'

perpendicular to MN, and TSt per

M

pendicular to PQ; join T'O, Ot and RS.

T

H

P

N

B

E

Since the figure is symmetrical with respect to MN, we have RT" =RT; and since RT OS, it follows that RT' OS; therefore,

=

=

RT' OS is a parallelogram (108), and RS is equal and parallel to OT'.

=

Again, since the figure is symmetrical with respect to PQ, we have St STOR; therefore, SROt is a parallelogram, and RS is equal and parallel to Ot. Hence, T', O and t, are in the same straight line, since there can be but one parallel to RS drawn through the same point O.

=

Now we have OT" =RS and Ot= RS, and consequently OT' Ot; therefore, any straight line T'Ot, drawn through O, is bisected at C; that is, O is the centre of symmetry of the figure.

BOOK II.

THE CIRCLE.

1. DEFINITIONS. A circle is a portion of a plane bounded by a curve, all the points of which are equally distant from a point within it called the centre.

The curve which bounds the circle is called its circumference.

Any straight line drawn from the centre to the circumference is called a radius.

Any straight line drawn through the centre and terminated each way by the circumference is called a diameter.

с

B

In the figure, O is the centre, and the curve ABCEA is the circumference of the circle; the circle is the space included within the circumference; OA, OB, OC, are radii; AOC is a diameter.

By the definition of a circle, all its radii are equal; also all its diameters are equal, each being double the radius.

If one extremity, O, of a line OA is fixed, while the line revolves in a plane, the other extremity, A, will describe a circumference, whose radii are all equal to OA.

2. Definitions. An arc of a circle is any portion of its circumference; as DEF.

A chord is any straight line joining two points of the circumference; as DF. The arc DEF is said to be subtended by its chord DF.

Every chord subtends two arcs, which together make up the whole circumference. Thus DF subtends both the arc DEF and the arc DCBAF. Wher an arc and its chord are spoken of, the arc less than

a semi-circumference, as DEF, is always understood, unless otherwise stated.

A segment is a portion of the circle included between an arc and its chord; thus, by the segment DEF is meant the space included between the arc DF and its chord.

A sector is the space included between an arc and the two radii drawn to its extremities; as AOB.

3. From the definition of a circle it follows that every point within the circle is at a distance from the centre which is less than the radius; and every point without the circle is at a distance from the centre which is greater than the radius. Hence (I. 40), the locus of all the points in a plane which are at a given distance from a given point is the circumference of a circle described with the given point as a centre and with the given distance as a radius.

4. It is also a consequence of the definition of a circle, that two circles are equal when the radius of one is equal to the radius of the other, or when (as we usually say) they have the same radius. For if one circle be superposed upon the other so that their centres coincide, their circumferences will coincide, since all the points of both are at the same distance from the centre.

If when superposed the second circle is made to turn upon its centre as upon a pivot, it must continue to coincide with the first. 5. Postulate. A circumference may be described with any point as a centre and any distance as a radius.

ARCS AND CHORDS.

PROPOSITION I.-THEOREM.

6. A straight line cannot intersect a circumference in more than two points.

For, if it could intersect it in three points, the three radii drawn to these three points would be three equal straight lines drawn from the same point to the same straight line, which is impossible (I. 36).

5*