"This establishes the following _infallible_ rule, for finding two mean proportionals OF EQUAL VALUE, and is more than a preliminary, to the famous old problem of 'Squaring the circle.' Let any finite number, say 20, and its fourth part = ¼(20) = 5, be given numbers. Then rq(20 × 5) = rq 100 = 10, is their mean proportional. Let this be a given mean proportional TO FIND ANOTHER MEAN PROPORTIONAL OF EQUAL VALUE. Then
20 × [pi]/4 = 20 × 3.125/4 = 20 × .78125 = 15.625
25 : 16 :: rq 20 : rq 8.192: and (rq 8.192)^2 × [pi]/4 = 8.192 × .78125 = 6.4
will be the second number; therefore rq(15.625 × 6.4) = rq 100 = 10, is the required mean proportional.... Now, my good Sir, however competent you may be to prove every man a fool [not _every_ man, Mr. Smith! only _some_; pray learn logical quantification] who now thinks, or in times gone by has thought, the 'Squaring of the Circle' _a possibility_; I doubt, and, on the evidence afforded by your Budget, I cannot help doubting, whether you were ever before competent to find two mean proportionals _by my unique method_."--(_Nut_, pp. 47, 48.) [That I never was, I solemnly declare!]
All readers can be made to see the following exposure. When 5 and 20 are given, x is a mean proportional when in 5, x, 20, 5 is to x as x to 20. And x must be 10. But x and y are two mean proportionals when in 5, x, y, 20, x { 120} is a mean proportional between 5 and y, and y is a mean proportional between x and 20. And these means are x = 5 [cuberoot]4, y = 5 [cuberoot]16. But Mr. Smith finds _one_ mean, finds it _again_ in a roundabout way, and produces 10 and 10 as the two (equal!) means, in solution of the "famous old problem." This is enough: if more were wanted, there is more where this came from. Let it not be forgotten that Mr. Smith has found a translator abroad, two, perhaps three, followers at home, and--most surprising of all--a real mathematician to try to set him right. And this mathematician did not discover the character of the subsoil of the land he was trying to cultivate until a goodly octavo volume of letters had passed and repassed. I have noticed, in more quarters than one, an apparent want of perception of the _full_ amount of Mr. Smith's ignorance: persons who have not been in contact with the non-geometrical circle-squarers have a kind of doubt as to whether anybody can carry things so far. But I am an "old bird" as Mr. Smith himself calls me; a Simorg, an "all-knowing Bird of Ages" in matters of cyclometry.