8 Proceedings of Royal Society of Edinburgh. [sess. 
words, the straight lines B C and A C are equal to each other. 
Consequently A C B is also an isosceles triangle. 
The arm I F may be extended indefinitely beyond F. The 
piece B H 0 admits also of indefinite extension beyond 0. 
The drawing exhibits the instrument as in the position in which 
it trisects the angle B AO, which may he any angle less than 135°, 
that is, than a right angle and half of a right angle. For, as A B C 
is an isosceles triangle, so the angles B A C and A B C at its base 
A B are equal to each other. And, as A C 0 is also an isosceles 
triangle, so the angles A C 0 and C A 0, at its base A C, are also 
equal to each other. But the angle A C 0 is exterior to the 
isosceles triangle A C B, and therefore it, as also the angle C A O, 
is equal to twice the angle B A C. The angle B A C is therefore 
equal to a third of the angle B A O. 
When the instrument is being used to trisect any angle under 
135°, as for instance the angle B A O, a sharp cylindrical pin or a 
needle is inserted through the hole at A into the vertex of the 
angle. The point B is then moved along a side, as A B, of the 
angle until the borders E O and I O meet in a point (which may 
he termed O) in the other side of the angle. The angle is thereby 
trisected, or may he so, by bisecting the angle C A O. 
An angle of 135° or larger may he trisected by first dividing it 
into two or four equal angles. Two or four thirds respectively of 
one of those divisions of the angle will be equal to a third of the 
entire angle. 
If the instrument be made and used with precision, any angle 
may be trisected with it. 
The reasons why an angle of 135° or upwards cannot be directly 
trisected by the instrument are — that, if the angle to be trisected 
were 135°, then the angles ABC and BAC would each be half 
of a right angle, and the angle OCA would be a right angle, and 
consequently the borders I 0 and E 0, inasmuch as they would be 
parallels, would never meet ; nor would they ever meet if the angle 
B A O were upwards of 135°. 
