Mr. James Smith will, I have no doubt, be the most uneclipsed
circle-squarer of our day. He will not owe this distinction to his being an
influential and respected member of the commercial world of Liverpool, even
though the power of publishing which his means give him should induce him
to issue a whole library upon one paradox. Neither will he owe it to the
pains taken with him by a mathematician who corresponded with him until the
joint letters filled an octavo volume. Neither will he owe it to the notice
taken of him by Sir William Hamilton, of Dublin, who refuted him in a
manner intelligible to an ordinary student of Euclid, which refutation he
calls a remarkable paradox easily explainable, but without explaining it.
What he will owe it to I proceed to show.
Until the publication of the _Nut to Crack_ Mr. James Smith stood among
circle-squarers in general. I might have treated him with ridicule, as I
have done others: and he says that he does not doubt he shall come in for
his share at the tail end of my Budget. But I can make a better job of him
than so, as Locke would have phrased it: he is such a very striking example
of something I have said on the use of logic that I prefer to make an
example of his writings. On one point indeed he well deserves the
_scutica_,[210] if not the _horribile flagellum_.[211] He tells me that he
will bring his solution to me in such a form as shall compel me to admit it
as _un fait accompli_ [_une faute accomplie?_][212] {117} or leave myself
open to the humiliating charge of mathematical ignorance and folly. He has
also honored me with some private letters. In the first of these he gives
me a "piece of information," after which he cannot imagine that I, "as an
honest mathematician," can possibly have the slightest hesitation in
admitting his solution. There is a tolerable reservoir of modest assurance
in a man who writes to a perfect stranger with what he takes for an
argument, and gives an oblique threat of imputation of dishonesty in case
the argument be not admitted without hesitation; not to speak of the minor
charges of ignorance and folly. All this is blind self-confidence, without
mixture of malicious meaning; and I rather like it: it makes me understand
how Sam Johnson came to say of his old friend Mrs. Cobb,[213]--"I love Moll
Cobb for her impudence." I have now done with my friend's _suaviter in
modo_,[214] and proceed to his _fortiter in re_[215]: I shall show that he
_has_ convicted himself of ignorance and folly, with an honesty and candor
worthy of a better value of [pi].
Mr. Smith's method of proving that every circle is 3-1/8 diameters is to
assume that it is so,--"if you dislike the term datum, then, by hypothesis,
let 8 circumferences be exactly equal to 25 diameters,"--and then to show
that every other supposition is thereby made absurd. The right to this
assumption is enforced in the "Nut" by the following analogy:
"I think you (!) will not dare (!) to dispute my right to this hypothesis,
when I can prove by means of it that every other value of [pi] will lead to
the grossest absurdities; unless indeed, you are prepared to dispute the
right of Euclid to adopt a false line hypothetically for the purpose {118}
of a '_reductio ad absurdum_'[216] demonstration, in pure geometry."
Euclid assumes what he wants to _disprove_, and shows that his _assumption_
leads to absurdity, and so _upsets itself_. Mr. Smith assumes what he wants
to _prove_, and shows that _his_ assumption makes _other propositions_ lead
to absurdity. This is enough for all who can reason. Mr. James Smith cannot
be argued with; he has the whip-hand of all the thinkers in the world.
Montucla would have said of Mr. Smith what he said of the gentleman who
squared his circle by giving 50 and 49 the same square root, _Il a perdu le
droit d'etre frappe de l'evidence_.[217]
It is Mr. Smith's habit, when he finds a conclusion agreeing with its own
assumption, to regard that agreement as proof of the assumption. The
following is the "piece of information" which will settle me, if I be
honest. Assuming [pi] to be 3-1/8, he finds out by working instance after
instance that the mean proportional between one-fifth of the area and
one-fifth of eight is the radius. That is,
if [pi] = 25/8, sqrt(([pi]r^2)/5 . 8/5) = r.
This "remarkable general principle" may fail to establish Mr. Smith's
quadrature, even in an honest mind, if that mind should happen to know
that, a and b being any two numbers whatever, we need only assume--
[pi] = a^2/b, to get at sqrt(([pi]r^2)/a . b/a) = r.
We naturally ask what sort of glimmer can Mr. Smith have of the subject
which he professes to treat? On this point he has given satisfactory
information. I had mentioned the old problem of finding two mean
proportionals, {119} as a preliminary to the duplication of the cube. On
this mention Mr. Smith writes as follows. I put a few words in capitals;
and I write rq[218] for the sign of the square root, 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