William Wallace Proof of the Butterfly Theorem
In a letter to William Wallace, dated 7 April 1805, Sir William Herschel - the discoverer of Uranus and its two major moons, Titania and Oberon - wrote:
I have kept a little problem for you which a friend of mine has sent me who says he cannot find a solution of it. I mentioned to him that I had a friend who would probably help him to one. The problem is this. Given AB the diameter of a circle. CD a chord cutting it at right angles in K. EF, and HG two other chords drawn any how through the point K; and HF, EG chords joining the extremes of EF, HG. Required to prove that MK is equal to LK.
Until very recently, this is the first recorded mention of the Butterfly theorem. The previous record, according to [Coxeter and Greitzer] was the 1815 solution by W. G. Horner of Horner's method fame. Wallace solution appears to predate that of Horner by 10 years. Sir Herschel's letter may has been thought to be the original source of the problem. However, the latest discovery of a 1803 publication with W. Wallace's posting of a generalization of the problem shows convincingly that Wallace has been aware of the problem that is known now as the "Butterfly Theorem" as early as 1802.
What follows is the original solution by William Wallace as was recently published in the Journal of the British Society for the History of Mathematics:
- H. S. M. Coxeter, S. L. Greitzer, Geometry Revisited, MAA, 1967
- Alex D D Craik and John J O'Connor, Some unknown documents associated with William Wallace (1768-1843), BSHM Bulletin: Journal of the British Society for the History of Mathematics, 26:1, 17-28