The first who addressed me filled me with a great ambition—to do a feat I had never yet heard of as accomplished by man, namely, to convince a 'Circle-Squarer' of his error! The value my friend had selected for '' was not an original one—being 3·2: but the enormous error, beginning as early as the first decimal place, tempted one with the idea that it could be easily demonstrated to be an error. I should think more than a score of letters were interchanged before I became sadly convinced that I had no chance. What man could still hope on, after receiving such a rebuff as the following? "You persuade yourself," so my friend wrote, "that you have made your circumscribed polygon equal to the circle, which you know cannot be, and have thereby pushed the quadrant beyond 90°, valuing the circumference at 360°." I meekly begged to be referred to the actual words in which I had advanced this startling assertion: but I never succeeded in getting the quotation verified. My second 'Circle-Squarer' went to work in quite another fashion. His object was not so much to obtain an arithmetical value for ',' as to construct a geometrical straight Line which, given the radius, should exhibit to the eye the actual length of the circumference. His diagram was a most imposing one—Triangles and Parallels were interlaced in bewildering profusion—and it used up no less than 23 letters of the alphabet. Some of the Lines had arithmetical values assigned to them: and there was one value, '1·8879020478639098461 &c.', which for a long time baffled all my endeavours to guess how in the world he had invented it. Of course one might have taken exception[Pg xviii] to such a construction at the very outset, and have said "I will admit the possibility of constructing a Line, which shall bear to the unit-line any arithmetical ratio you like, so long as you express it as an exact decimal: but what can I do with your '&c.'?" But his was not the kind of mind to which the geometrical construction of an '&c.' presents any difficulty. At length, after many failures, I chanced on the discovery that this portentous number was of the decimal part of '.' After this it was no wonder, considering that, in the course of construction, he had taken of this Line, and afterwards divided by 10, that the resulting Line, added to 3 times the unit-Line, was triumphantly proved to represent ''! I ventured to ask if this was the way he had obtained the long decimal quoted above, namely, by multiplying the decimal part of '' by , and received the courteous reply "your suggestion is perfectly correct"! Another ignis fatuus—though not numbering so many victims as the 'Quadrature of the Circle'—is 'the Trisection of an Angle' (that is, its trisection by Euclid's machinery). And yet another ignis fatuus—the one with which the following treatise is concerned—is the attempt to dispense with Euclid's celebrated 12th Axiom.