500 Description of an Instrument for trisecting Angles, [Nov. 



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 pa- 

 rallel 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 to- 

 wards A, until O B become parallel to, or coincide with one of these 



