Let AP be any chord through A. Bisect AP in Q and join CQ.

Therefore the locus of Q is the circle of which AC is the diameter (Bk. II. Prop. 22, Cor.).

EXAMPLES.

1. A straight line of constant length is always parallel to a given line, and one of its extremities describes a given circumference. Prove that the locus of the other extremity is a circle whose centre is in the given line.

2. Prove that the locus of the middle points of all the straight lines drawn from a given point to a given circle is another circle.

3. Prove that the locus of the middle points of all chords to a circle drawn through a given point is another circle.

4. A ladder is raised gradually against a wall; prove that the locus of the middle point is a circle; find its centre and radius.

5. A circle rolls within another fixed circle whose radius is equal to its own diameter. Prove that the locus of a point on the circumference of the rolling circle is a diameter of the fixed circle.

6. Prove that the locus of the middle point of a straight line, which moves in such a way that the sum of the perpendiculars upon it from two fixed points is constant, is a circle, and find the centre of this circle.

7. Through one extremity of the diameter of a semicircle lines are drawn and produced beyond the semicircle, so that the part produced is equal to the chord joining the point of intersection of the line and semicircle with the other extremity of the diameter. Find the locus of the extremities

of the parts so produced.

Note. When we speak of one straight line as being equal to another, or, more accurately, as being equal in length to another, we have a very exact and precise idea of what is meant, viz. that the one line can be applied to the other so that the extremities and every intermediate point of the one may coincide with the extremities and some intermediate point of the other. So, again, when we speak of one circular arc as being equal to another circular arc of equal radius, we mean that the one arc can be applied to the other, so that the extremities and every intermediate point of the one may coincide with the extremities and some intermediate point of the other. But our ideas of equality of length are not confined to lines which may be superposed upon one another in this manner, and in practice we speak familiarly of the lengths of curves or of circular arcs, meaning thereby the lengths of straight lines equal in length to these curves or circular arcs, although a straight line and the arc of a circle cannot of course be superposed one upon the other. It is well therefore to have an accurate definition of the length of a curved line, which we will now proceed to investigate.

If a circle be made to roll along a straight line, as the wheel of a coach rolls along a road, every point of the circle in each revolution will coincide in succession with every point of the straight line lying between the first and last points of contact, and the length of the straight line between these points is said to be the length of the circle, or of the circumference of the circle. In the same way, if the length of the line be measured before the circle has made a complete revolution—that is, before the point of the circle first of all in contact with the line is brought into coincidence

with a point of the line again—the length of this smaller line is called the length of the arc of the circle between the points of such first and last contact with the line.

In this motion it is supposed that no point of the circle slides along any portion of the line, but that successive points of the circle are brought into contact successively with fresh points of the line, and that fresh points of the line are brought into contact successively with fresh points of the circle.

It is clear that the straight line may be supposed to roll on the circle as well as the circle on the straight line.

Hence we have this general definition, not only applicable to a circle or circular arc, but to any curved line, viz. :

DEFINITION.

43.-If a straight line be made to roll upon a curved line, the length of the straight line between its first and last points of contact with the curved line, is defined to be the length of the curved line between its first and last points of contact with the straight line.

Again, suppose AB

to be a curved line, and C, D, E, &c., successive points upon it, intermediate between A and B, and let the chords AC, CD, DE, &c., be drawn.

Let AH be any indefinite straight line, and let one of its extremities be made to

coincide with the point A, the line itself coinciding with

AC produced.

Let the straight line AH revolve about the point C until it coincides with CD, and let d be the point of AH which coincides with the point D in this position.

Again, let AH revolve about D or d until it coincides with DE, and let e be the point upon it which coincides with E in this position; and so on.

It is clear that if H be the point with which B coincides in the final position of the line, AH will be equal to the sum of the lengths of the chords AC, CD, DE, &c.

Now let the number of intermediate points C, D, E, &c., be indefinitely increased, and let the distance between each successive pair of points be indefinitely diminished, it is clear that the points c, d, e, &c., will ultimately, when their number is very greatly increased, coincide successively with every point in the curved line AB, in which case, if K be the point upon it which ultimately coincides with B, AK will be the length of the curve AB by previous definition.

Therefore the length of the curved line AB will be the same as the sum of the lengths of the chords AC, CD, &c., when the number of these chords is indefinitely increased and the length of each is indefinitely diminished; whence we have this additional and very important definition of the length of a curved line.

DEFINITION.

44.-If a number of points be taken between the first and last points on any finite curved line, and the chords between each pair of points in succession be drawn, and if the number of such points be indefinitely increased and the length of each chord be indefinitely diminished, the ultimate sum of the lengths of these chords is the length of the curved line.

BOOK III.

PROBLEMS OF CONSTRUCTION CONNECTED WITH THE STRAIGHT LINE AND CIRCLE.

OUR investigations hitherto have been purely theoretical, and have been confined to the demonstration of certain properties of figures, assumed to exist, satisfying certain proposed conditions. Such investigations are called theorems. We now pass to the more practical portion of the subject, that is to say, the determination of methods by means of which such figures are to be drawn, or rather approximately drawn, for, owing to the imperfection of our instruments, the ideal state contemplated in theoretical geometry cannot be attained by them.

Many instruments have been invented by means of which certain constructions can be effected, such as squares, parallel rulers, elliptic compasses, graduated sectors of circles, and so forth; but in elementary geometry we suppose ourselves confined to the use of a ruler, by means of which straight lines may be drawn from one point to another, or given lines may be produced if necessary, and compasses, by means of which distances may be measured off from given larger lines equal to given smaller lines, and circles may be described round given points as centres, and having lines of given length for their radii.

The determination of the method of constructing a given figure with given instruments is called a problem, and the solution of a problem requires us to show how the required construction can be effected by the use of these instruments, and to prove that the construction so effected is

correct.