It should be noticed that in the usual solution two arcs intersect, and the point thus determined is connected with the vertex. Now these two arcs intersect twice, and since one of the points of intersection may be the vertex itself, the other point of intersection must be taken. It is not, however, worth while to make much of this matter with pupils. Proclus calls attention to the possible suggestion that the point of intersection may be imagined to lie outside the angle, and he proceeds to show the absurdity; but here, again, the subject is not one of value to beginners. He also contributes to the history of the trisection of an angle. Any angle is easily trisected by means of certain higher curves, such as the conchoid of Nicomedes (_ca._ 180 B.C.), the quadratrix of Hippias of Elis (_ca._ 420 B.C.), or the spiral of Archimedes (_ca._ 250 B.C.). But since this problem, stated algebraically, requires the solution of a cubic equation, and this involves, geometrically, finding three points, we cannot solve the problem by means of straight lines and circles alone. In other words, the trisection of _any_ angle, by the use of the straightedge and compasses alone, is impossible. Special angles may however be trisected. Thus, to trisect an angle of 90 deg. we need only to construct an angle of 60 deg., and this can be done by constructing an equilateral triangle. But while we cannot trisect the angle, we may easily approximate trisection. For since, in the infinite geometric series 1/2 + 1/8 + 1/32 + 1/128 + ..., _s_ = _a_ / (1 - _r_), we have _s_ = 1/2 / 3/4 = 2/3. In other words, if we add 1/2 of the angle, 1/8 of the angle, 1/32 of the angle, and so on, we approach as a limit 2/3 of the angle; but all of these fractions can be obtained by repeated bisections, and hence by bisections we may approximate the trisection. The approximate bisection (or any other division) of an angle may of course be effected by the help of the protractor and a straightedge. The geometric method is, however, usually more accurate, and it is advantageous to have the pupils try both plans, say for bisecting an angle of about 49 1/2 deg.. [Illustration] Applications of this problem are numerous. It may be desired, for example, to set a lamp-post on a line bisecting the angle formed by two streets that come together a little unsymmetrically, as here shown, in which case the bisecting line can easily be run by the use of a measuring tape, or even of a stout cord. A more interesting illustration is, however, the following: [Illustration]