25 : 16 :: rq 20 : rq 8.192: and (rq 8.192)^2 x [pi]/4 = 8.192 x .78125 = 6.4 will be the second number; therefore rq(15.625 x 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. The curious phenomena of thought here exhibited illustrate, as above said, a remark I have long ago made on the effect of proper study of logic. Most persons reason well enough on matter to which they are accustomed, and in terms with which they are familiar. But in unaccustomed matter, and with use of strange terms, few except those who are practised in the abstractions of pure logic can be tolerably sure to keep their feet. And one of the reasons is easily stated: terms which are not quite familiar partake of the vagueness of the X and Y on which the student of logic learns to see the formal force of a proposition independently of its material elements. I make the following quotation from my fourth paper on logic in the _Cambridge Transactions_: