ot, which embarrasses
small type:

"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 = (1/4)(20) = 5, be given numbers. Then rq(20 x 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 x [pi]/4 = 20 x 3.125/4 = 20 x .78125 = 15.625

will be the first number; as

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