O B, which is also moveable on the point O, until A D becomes equal to A B; this can be easily effected, (because A D & A B have each a
scale of equal parts (of degrees) attached to them, commencing from
A.) The given angle aocis now trisected, and A O B is the third
part of it.
Proof. The two triangles a o b, ab d, are equiangular, for the angle a b d is common to both triangles, as well as equal, both to the angle b a o, (because rad. b o~ rad. a o,) and to the angle b d a, (because ab —a d, by construction or application of the instrument.) The two triangles, having 2 angles in the one, equal to two angles in the other, have therefore their third angles equal, namely, the angle a o b —angle bad. But the angle b a d, at the circumference of the circle a b c, being measured by half the arc it stands upon, b c, is half of the angle b o c, at the centre of the same circle, measured by the whole arc be; therefore, the angle a o b, its equal, is half of the same angle b o c, or one-third of the whole angle a o c, which was to be trisected.
The legs of the instrument may be reduced, in number, to three, thus — The piece A B is absolutely necessary to the first construction, in order to affix a scale of equal parts to A D, similar to that upon A B, (one of degrees is the best, because the angle can then be read off, not forgetting, in dividing it, that A D is to be considered the chord of an arc ; but as soon as the parallels have been cut upon both legs, A B becomes superfluous.
To show this, I will state how A D is divided ; — 1st, set off upon A B a scale of equal parts, making their productions pass through the cen- tre O, at whatever position of the instrument, but governed by O B. 2dly, when thesa have been cut upon the leg A B, set off an equal scale upon the other leg A D, and make their productions pass through the corresponding lines upon A B, conveying to the centre O like- wise.
This being done, it must be evident, that the subsequent use of A B is superfluous, because the leg O B, after it has been placed parallel to, that is, immediately over and upon, any line that has been cut upon A D, and that was made by construction, to pass through corresponding parallels on the leg A B, no longer requires, for this very reason, the aid of the latter piece; so that the trisector, in this state, will, as before mentioned, consist of only three pieces, (as shewn in fig. 3.) But, in this case, having applied A O and A D, as before, to the side and chord of the angle to be trisected, O B must be moved towards A, until O B become parallel to, or coincide with one of these
