Here we show how the use of neusis can be used to trisect arbitrary angles. In fact, we shall give four different constructions: two ancient, one medieval, and one modern.
Archimedes of Syracuse 287 BC — c. 212 BC) is not only unquestionably the greatest mathematician of antiquity, but one of the greatest of all time. His writings reveal a mind of unsurpassed inventiveness, as well as extraordinary subtlety and suppleness of thought. His trisection is perhaps the best known, and maybe the simplest. Suppose the acute angle to be trisected is placed in a circle of radius one and diameter . Extend the diameter of the circle, and from construct a line which crosses the circle and the diameter extension at and , and for which . (This is the neusis construction.) Here is what it looks like:
Then the angle .
This is easily proved by some angle chasing. Construct the line . Since the triangles and are isosceles. Suppose angle , then angle and so angles . Then angle and by adding all the angles at : , or .
Pappus of Alexandria (c. 290 — c. 350 AD) is still considered an “ancient” mathematician, even though he succeeded Archimedes by nearly half a millennium. He wrote at a time when mathematics (and science generally) was at a low ebb; his achievements are all the more remarkable.
Suppose is the angle to be trisected, and let . By using appropriate perpendiculars, construct a rectangle for which is a diagonal. Extend , and from construct a line crossing at and at and for which . Then angles .
The proof is again easy angle chasing: let be the midpoint of and construct . All these new lengths are the same, so that triangles , and are isosceles. The rest is straightforward.
Jordanus (1225 — 1260) is a shadowy figure about whom nothing is known aside from some books which bear this name. One theory is that “Nemorarius” is simply a copying error by a later scribe; at any rate there is no baptismal or any other record of “Nemorarius”. Although he has been described in Carl Boyer’s history as an “able, but less gifted contemporary” of Leonardo of Pisa (Fibonacci), his works on geometry, statics and mechanics embody the best thought of his time. His trisection is a neat and elegant variant of Archimedes, and again starts with a semicircle with the angle to be trisected. Construct the radius perpendicular to , and from draw a line cutting this radius and the circle at and respectively, and for which . Then angle .
The proof is pretty much the same as for Archimedes’ method: since triangle is isosceles (by construction) if angle then angles . The rest of the proof is as above.
David Allan Brooks published a delightful trisection method in The College Mathematics Journal in 2007. Brooks is not a professional mathematician, and so his work gives added credence (should any be needed) to excellent work produced by amateurs. This method is in fact not a neusis method, as it uses a right angled ruler (a “carpenter’s square“) instead of two marks, but it very much in the spirit of neusis. Let be the angle to be trisected, and let be the midpoint of . Construct a circle with centre and radius and from construct the perpendicular to . Now lay the carpenter’s square so that one side goes through , the right angle lies on at and the other edge is tangent to . Then the angle
Here is a picture:
To prove it, suppose the tangent crosses at , and construct the radius to the tangent. Let angles and be and respectively, so that . By similar triangles. angle . We shall assume that .
Since we have and so , . Then and so . But considering triangle we also have , and so . Expanding both sides produces
and this can be written as
so that or . Given that it follows that . This is a slightly different proof than that given in Brook’s article, and maybe slightly simpler, in requiring less extra points and lines.