What to do when the trisector comes

Abstract

A trisector is a pe r son w h o has, he thinks, succeeded in d iv id ing any angle into three equal par ts us ing s t ra ightedge and compass alone. He comes w h e n he sends y o u his t r isect ion in the mail and asks y o u r op in ion , or (worse) calls y o u to discuss his work , or (worse still) shows up in person . You m a y th ink that the p r o b l e m of h o w to deal wi th tr isectors is not an i m p o r t a n t one; I in tend to show that it is. Tr isectors form a subse t of ma themat i ca l cranks , wh ich in turn are a subse t of all cranks. By " c r a n k " I m e a n wha t Augus tus D e M o r g a n mean t b y " p a r a d o x e r" in his wonder fu l book A Budget of Paradoxers: A great m a n y indiv idua ls , ever since the rise of the mathemat ica l me thod , have , each for himself , attacked its direct and indi rec t consequences . I shall call each of these pe r sons a paradoxer, and his syst em a paradox. I use the word in the old sense: a pa radox is some th ing w h i c h is apar t f rom general op in ion , e i ther in subjec t -mat ter , me thod , or conclusion. I pul led that book off a l ibrary shelf nearly th i r ty years ago not k n o w i n g wha t was in it, and it s tar ted m y interes t in cranks in general and mathemat ica l cranks in part icular . You m i g h t th ink that it would be eas ier for m a t h e m a t i c i a n s to h a n d l e c r a n k s t h a n , say , economis ts . An advocate of H e n r y George ' s single tax could argue endlessly and inconclusively, bu t the only w a y to see if the tax wou ld work would be to get a na t ion or collection of na t ions to give the idea a try. In ma themat i c s , there ough t to be less endless a rgument : g iven a " p r o o f " of Fe rmat ' s Conjecture, s o m e t h i n g like " the re is an error on page 4, line 12" should settle the mat ter , and g iven a tr isect ion, there is no need even to read as far as page 4 since there is no chance w h a t s o e v e r that it is correct. But unfor tuna te ly it is not that way . Trisectors can and do argue just as endless ly as any other cranks. By the way, if you have any tr isect ions or any works of ma themat ica l cranks I wou ld be mos t grateful to have copies. I have sough t such things out for years , and m y collection of ana lyzed t r i s ec t ions s t anda rd ized d iagrams wi th nea t letters, and compu te r pr int outs of e r ro r s i s in its second hundred . But I have the collector 's lust: I w a n t t h e m all. Quadra tu re s of the circle (the mos t popu la r target of cranks in the last century bu t now not common) , dupl ica t ions of the cube, proofs of Fermat ' s Conjecture: all are welcome.