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 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_:
"The uncultivated reason proceeds by a process almost entirely material.
Though the necessary law of thought {121} must determine the conclusion of
the ploughboy as much as that of Aristotle himself, the ploughboy's
conclusion will only be tolerably sure when the matter of it is such as
comes within his usual cognizance. He knows that geese being all birds does
not make all birds geese, but mainly because there are ducks, chickens,
partridges, etc. A beginner in geometry, when asked what follows from
'Every A is B,' answers 'Every B is A.' That is, the necessary laws of
thought, except in minds which have examined their tools, are not very sure
to work correct conclusions except upon familiar matter.... As the
cultivation of the individual increases, the laws of thought which are of
most usual application are applied to familiar matter with tolerable
safety. But difficulty and risk of error make a new appearance with a new
subject; and this, in most cases, until new subjects are familiar things,
unusual matter common, untried nomenclature habitual; that is, until it is
a habit to be occupied upon a novelty. It is observed that many persons
reason well in some things and badly in others; and this is attributed to
the consequence of employing the mind too much upon one or another subject.
But those who know the truth of the preceding remarks will not have far to
seek for what is often, perhaps most often, the true reason.... I maintain
that logic tends to make the power of reason over the unusual and
unfamiliar more nearly equal to the power over the usual and familiar than
it would otherwise be. The second is increased; but the first is almost
created."
Mr. James Smith, by bringing ignorance, folly, dishonesty into contact with
my name, in the way of conditional insinuation, has done me a good turn: he
has given me right to a freedom of personal remark which I might have
declined to take in the case of a person who is useful and respected in
matters which he understands.
Tit for tat is logic all the world over. By the way, what has become of the
rest of the maxim: we never hear it {122} now. When I was a boy, in some
parts of the country at least, it ran thus:
"Tit for tat;
Butter for fat:
If you kill my dog,
I'll kill your cat."
He is a glaring instance of the truth of the observations quoted above. I
will answer for it that, at the Mersey Dock Board, he never dreams of
proving that the balance at the banker's is larger than that in the book by
assuming that the larger sum is there, and then proving that the other
supposition--the smaller balance--is upon that assumption, an absurdity. He
never says to another director, How can you dare to refuse me a right to
assume the larger balance, when you yourself, the other day,
said,--Suppose, for argument's sake, we had 80,000l. at the banker's,
though you knew the book only showed 30,000l.? This is the way in which he
has supported his geometrical paradox by Euclid's example: and this is not
the way he reasons at the board; I know it by the character of him as a man
of business which has reached my ears from several quarters. But in
geometry and rational arithmetic he is a smatterer, though expert at
computation; at the board he is a trained man of business. The language of
geometry is so new to him that he does not know what is meant by "two mean
proportionals:" but all the phrases of commerce are rooted in his mind. He
is most unerasably booked in the history of the squaring of the circle, as
the speculator who took a right to assume a proposition for the destruction
of other propositions, on the express ground that Euclid assumes a
proposition to show that it destroys itself: which is as if the curate
should demand permission to throttle the squire because St. Patrick drove
the vermin to suicide to save themselves from slaughter. He is conspicuous
as a speculator who, more visibly than almost any other known to history,
reasoned in a circle by way of reasoning on a circle. But {123} what I have
chiefly to do with is the force of instance which he has lent to my
assertion that men who have not had real training in pure logic are unsafe
reasoners in matter which is not familiar. It is hard to get first-rate
examples of this, because there are few who find the way to the printer
until practice and reflection have given security against the grossest
slips. I cannot but think that his case will lead many to take what I have
said into consideration, among those who are competent to think of the
great mental disciplines. To this end I should desire him to continue his
efforts, to amplify and develop his great principle, that of proving a
proposition by assuming it and taking as confirmation every consequence
that does not contradict the assumption.
Since my Budget commenced, Mr. Smith has written me notes: the portion
which I have preserved--I suppose several have been mislaid--makes a
hundred and seven pages of note-paper, closely written. To all this I have
not answered one word: but I think I cannot have read fewer than forty
pages. In the last letter the writer informs me that he will not write at
greater length until I have given him an answer, according to the "rules of
good society." Did I not know that for every inch I wrote back he would
return an ell? Surely in vain the net is spread in the eyes of anything
that hath a wing. There were several good excuses for not writing to Mr. J.
Smith: I will mention five. First, I distinctly announced at the beginning
of this Budget that I would not communicate with squarers of the circle.
Secondly, any answer I might choose to give might with perfect propriety be
reserved for this article; had the imputation of incivility been made after
the first note, I should immediately have replied to this effect: but I
presumed it was quite understood. Thirdly, Mr. Smith, by his publication of
E. M.'s letters against the wish of the writer, had put himself out of the
pale of correspondence. Fourthly, he had also gone beyond the rules of good
society in sending {124} letter after letter to a person who had shown by
his silence an intention to avoid correspondence. Fifthly, these same rules
of good society are contrived to be flexible or frangible in extreme cases:
otherwise there would be no living under them; and good society would be
bad. Father Aldrovand has laid down the necessary distinction--"I tell
thee, thou foolish Fleming, the text speaketh but of promises made unto
Christians, and there is in the rubric a special exemption of such as are
made to Welchmen." There is also a rubric to the rules of good society; and
squarers of the circle are among those whom there is special permission not
to answer: they are the wild Welchmen of geometry, who are always
assailing, but never taking, the Garde Douloureuse[219] of the circle. "At
this commentary," proceeds the story, "the Fleming grinned so broadly as to
show his whole case of broad strong white teeth." I know not whether the
Welchman would have done the like, but I hope Mr. James Smith will: and I
hope he has as good a case to show as Wilkin Flammock. For I wish him long
life and long health, and should be very glad to see so much energy
employed in a productive way. I hope he wishes me the same: if not, I will
give him what all his judicious friends will think a good reason for doing
so. His pamphlets and letters are all tied up together, and will form a
curious lot when death or cessation of power to forage among book-shelves
shall bring my little library to the hammer. And this time may not be far
off: for I was X years old in A.D. X^2; not 4 in A.D. 16, nor 5 in A.D. 25,
but still in one case under that law. And now I have made my own age a
problem of quadrature, and Mr. J. Smith may solve it. But I protest against
his method of assuming a result, and making itself prove itself: he might
in this way, as sure as eggs is eggs (a corruption of X is X), make me
1,864 years old, which is a great deal too much.
{125}
_April 5, 1864._--Mr. Smith continues to write me long letters, to which he
hints that I am to answer. In his last, of 31 closely written sides of
note-paper, he informs me, with reference to my obstinate silence, that
though I think myself and am thought by others to be a mathematical
Goliath, I have resolved to play the mathematical snail, and keep within my
shell. A mathematical _snail_! This cannot be the thing so called which
regulates the striking of a clock; for it would mean that I am to make Mr.
Smith sound the true time of day, which I would by no means undertake upon
a clock that gains 19 seconds odd in every hour by false quadrature. But he
ventures to tell me that pebbles from the sling of simple truth and common
sense will ultimately crack my shell, and put me _hors de combat_.[220] The
confusion of images is amusing: Goliath turning himself into a snail to
avoid [pi] = 3-1/8, and James Smith, Esq., of the Mersey Dock Board: and
put _hors de combat_--which should have been _cache_[221]--by pebbles from
a sling. If Goliath had crept into a snail-shell, David would have cracked
the Philistine with his foot. There is something like modesty in the
implication that the crack-shell pebble has not yet taken effect; it might
have been thought that the slinger would by this time have been singing--
"And thrice [and one-eighth] I routed all my foes,
And thrice [and one-eighth] I slew the slain."
But he promises to give the public his nut-cracker if I do not, before the
Budget is concluded, "unravel" the paradox, which is the
mathematico-geometrical nut he has given me to crack. Mr. Smith is a crack
man: he will crack his own nut; he will crack my shell; in the mean time he
cracks himself up. Heaven send he do not crack himself into lateral
contiguity with himself.
On June 27 I received a letter, in the handwriting of Mr. James Smith,
signed Nauticus. I have ascertained {126} that one of the letters to the
_Athenaeum_ signed Nauticus is in the same handwriting. I make a few
extracts:
"... The important question at issue has been treated by a brace of
mathematical birds with too much levity. It may be said, however, that
sarcasm and ridicule sometimes succeed, where reason fails.... Such a
course is not well suited to a discussion.... For this reason I shall for
the future [this implies there has been a past, so that Nauticus is not
before me for the first time] endeavor to confine myself to dry reasoning
from incontrovertible premises.
... It appears to me that so far as his theory is concerned he comes off
unscathed. You might have found "a hole in Smith's circle" (have you seen a
pamphlet bearing this title? [I never heard of it until now]), but after
all it is quite possible the hole may have been left by design, for the
purpose of entrapping the unwary."
[On the publication of the above, the author of the pamphlet obligingly
forwarded a copy to me of _A Hole in Smith's Circle_--by a Cantab: Longman
and Co., 1859, (pp. 15). "It is pity to lose any fun we can get out of the
affair," says my almamaternal brother: to which I add that in such a case
warning without joke is worse than none at all, as giving a false idea of
the nature of the danger. The Cantab takes some absurdities on which I have
not dwelt: but there are enough to afford a Cantab from every college his
own separate hunting ground.]
Does this hint that his mode of proof, namely, assuming the thing to be
proved, was a design to entrap the unwary? if so, it bangs Banagher. Was
his confounding two mean proportionals with one mean proportional found
twice over a trick of the same intent? if so, it beats cockfighting. That
Nauticus is Mr. Smith appears from other internal evidence. In 1819, Mr.
J. C. Hobhouse[222] was sent to Newgate for a {127} libel on the House of
Commons which was only intended for a libel on Lord Erskine.[223] The
ex-Chancellor had taken Mr. Hobhouse to be thinking of him in a certain
sentence; this Mr. Hobhouse denied, adding, "There is but one man in the
country who is always thinking of Lord Erskine." I say that there is but
one man of our day who would couple me and Mr. James Smith as a "brace of
mathematical birds."
Mr. Smith's "theory" is unscathed by me. Not a doubt about it: but how does
he himself come off? I should never think of refuting a theory proved by
assumption of itself. I left Mr. Smith's [pi] untouched: or, if I put in my
thumb and pulled out a plum, it was to give a notion of the cook, not of
the dish. The "important question at issue" was not the circle: it was,
wholly and solely, whether the abbreviation of _James_ might be spelled
_Jimm_.[224] This is personal to the verge of scurrility: but in literary
controversy the challenger names the weapons, and Mr. Smith begins with
charge of ignorance, folly, and dishonesty, by conditional implication. So
that the question is, not the personality of a word, but its applicability
to the person designated: it is enough if, as the Latin grammar has it,
_Verbum personale concordat cum nominativo_.[225]
I may plead precedent for taking a liberty with the orthography of _Jem_.
An instructor of youth was scandalized at the abrupt and irregular--but
very effective--opening of Wordsworth's little piece:
{128}
"A simple child
That lightly draws its breath,
And feels its life in every limb,
What should it know of death?"
So he mended the matter by instructing his pupils to read the first line
thus:
"A simple child, dear brother ----."
The brother, we infer from sound, was to be James, and the blank must
therefore be filled up with _Jimb_.
I will notice one point of the letter, to make a little more distinction
between the two birds. Nauticus lays down--quite correctly--that the sine
of an angle is less than its circular measure. He then takes 3.1416 for
180 deg., and finds that 36' is .010472. But this is exactly what he finds
for the sine of 36' in tables: he concludes that either 3.1416 or the
tables must be wrong. He does not know that sines, as well as [pi], are
interminable decimals, of which the tables, to save printing, only take in
a finite number. He is a six-figure man: let us go thrice again to make up
nine, and we have as follows:
Circular measure of 36' .010471975...
Sine of 36' .010471784...
Excess of measure over sine .000000191...
Mr. Smith invites me to say which is wrong, the quadrature, or the tables:
I leave him to guess. He says his assertions "arise naturally and
necessarily out of the arguments of a circle-squarer:" he might just as
well lay down that all the pigs went to market because it is recorded that
"_This_ pig went to market." I must say for circle-squarers that very few
bring their pigs to so poor a market. I answer the above argument because
it is, of all which Mr. James Smith has produced, the only one which rises
to the level of a schoolboy: to meet him halfway I descend to that level.
Mr. Smith asks me to solve a problem in the _Athenaeum_: {129} and I will do
it, because the question will illustrate what is _below_ schoolboy level.
"Let x represent the circular measure of an angle of 15 deg., and y half
the sine of an angle of 30 deg. = area of the square on the radius of a
circle of diameter unity = .25. If x - y = xy, firstly, what is the
arithmetical value of xy? secondly, what is the angle of which xy
represents the circular measure?"
If x represent 15 deg. and y be 1/4, xy represents 3 deg. 45', whether
x - y be xy or no. But, y being 1/4, x - y is _not_ xy unless x be 1/3,
that is, unless 12x or [pi] be 4, which Mr. Smith would not admit. How
could a person who had just received such a lesson as I had given
immediately pray for further exposure, furnishing the stuff so liberally
as this? Is it possible that Mr. Smith, because he signs himself
Nauticus, means to deny his own very regular, legible, and peculiar
hand? It is enough to make the other members of the Liverpool Dock Board
cry, Mersey on the man!
Mr. Smith says that for the future he will give up what he calls sarcasm,
and confine himself, "as far as possible," to what he calls dry reasoning
from incontrovertible premises. If I have fairly taught him that _his_
sarcasm will not succeed, I hope he will find that his wit's end is his
logic's beginning.
I now reply to a question I have been asked again and again since my last
Budget appeared: Why do you take so much trouble to expose such a reasoner
as Mr. Smith? I answer as a deceased friend of mine used to answer on like
occasions--A man's capacity is no measure of his power to do mischief. Mr.
Smith has untiring energy, which does something; self-evident honesty of
conviction, which does more; and a long purse, which does most of all. He
has made at least ten publications, full of figures which few readers can
criticize. A great many people are staggered to this extent, that they
imagine there must be {130} the indefinite _something_ in the mysterious
_all this_. They are brought to the point of suspicion that the
mathematicians ought not to treat "all this" with such undisguised
contempt, at least. Now I have no fear for [pi]: but I do think it possible
that general opinion might in time demand that the crowd of
circle-squarers, etc. should be admitted to the honors of opposition; and
this would be a time-tax of five per cent., one man with another, upon
those who are better employed. Mr. James Smith may be made useful, in hands
which understand how to do it, towards preventing such opinion from
growing. A speculator who expressly assumes what he wants to prove, and
argues that all which contradicts it is absurd, _because_ it cannot stand
side by side with his assumption, is a case which can be exposed to all.
And the best person to expose it is one who has lived in the past as well
as the present, who takes misthinking from points of view which none but a
student of history can occupy, and who has something of a turn for the
business.
Whether I have any motive but public good must be referred to those who can
decide whether a missionary chooses his pursuit solely to convert the
heathen. I shall certainly be thought to have a little of the spirit of
Col. Quagg, who delighted in strapping the Grace-walking Brethren. I must
quote this myself: if I do not, some one else will, and then where am I?
The Colonel's principle is described as follows:
"I licks ye because I kin, and because I like, and because ye'se critters
that licks is good for. Skins ye have on, and skins I'll have off; hard or
soft, wet or dry, spring or fall. Walk in grace if ye like till pumpkins is
peaches; but licked ye must be till your toe-nails drop off and your noses
bleed blue ink. And--licked--they--were--accordingly."
I am reminded of this by the excessive confidence with which Mr. James
Smith predicted that he would treat me as Zephaniah Stockdolloger (Sam
Slick calls it _slockdollager_) treated Goliah Quagg. He has announced his
{131} intention of bringing me, with a contrite heart, and clean
shaved,--4159265... razored down to 25,--to a camp-meeting of
circle-squarers. But there is this difference: Zephaniah only wanted to
pass the Colonel's smithy in peace; Mr. James Smith sought a fight with me.
As soon as this Budget began to appear, he oiled his own strap, and
attempted to treat me as the terrible Colonel would have treated the
inoffensive brother.
He is at liberty to try again.