It is well-known that a line can be easily divided into any given number of equal parts, using ruler and compass. However, if we try to do the same with an arbitrary angle, the situation is quite different. For example it has been proven by P. L. Wantzel (1836)[1] using group theory developed by Galois [2], that it is impossible to trisect every angle using only ruler and compass. However, the search for approximate solutions to this problem turned to be very popular.

In this paper will be shown how we can approximately divide a given angle by a constructible number using a ruler and a compass. It is a trivial task to divide by , , and for any other we have

where .

Let us begin with , and let the given angle be shown in Fig.1.

In Fig.1 the following relations hold

and

Next we will show that . Let . From Fig.1 follows

and finally

If we expand the right-hand side of (1) in Taylor series in terms of , we get

From (2) we can conclude that .

In Fig.2 is shown the relative error for and .

Let us now discuss the general case, Fig.3.

Suppose that , and , where , so we have

Then with obvious notation we may represent by the line point on the line segment in Fig.3 . So we have

Let , then

From (3) we see that if , then , and likewise if , then as expected. For such that , it is , from (3) follows

so . In this case bisects and . Hence for , and the construction is exact.

It is convenient to expand (3) in Taylor series in terms of for , aided by the fact that it must be exact for . We find

where the first few are

It is interesting that have all their roots in the interval .

If we write error , and relative error

then for from (4) and (5) we get

Then for and , numerical search using (3) and (5) gives maximum occurring at approximately, with (slightly over 3.5%), and (6) gives , close to the correct value, suggesting that we have convergence in (4) for and , see Fig.4 and Fig.5.