The Problem of squaring the circle solved; or, the circumference and
area of the circle discovered. By James Smith.[200] London, 1859, 8vo.
On the relations of a square inscribed in a circle. Read at the British
Association, Sept. 1859, published in the Liverpool Courier, Oct. 8,
1859, and reprinted in broadsheet.
The question: Are there any commensurable relations between a circle
and other Geometrical figures? Answered by a member of the British
Association ... London, 1860, 8vo.--[This has been translated into
French by M. Armand Grange, Bordeaux, 1863, 8vo.]
The Quadrature of the Circle. Correspondence between an eminent
mathematician and James Smith, Esq. (Member of the Mersey Docks and
Harbour Board), London, 1861, 8vo. (pp. 200).
Letter to the ... British Association ... by James Smith, Esq.
Liverpool, 1861, 8vo.
Letter to the ... British Association ... by James Smith, Esq.
Liverpool, 1862, 8vo.--[These letters the author promised to continue.]
A Nut to crack for the readers of Professor De Morgan's 'Budget of
Paradoxes.' By James Smith, Esq. Liverpool, 1863, 8vo.
Paper read at the Liverpool Literary and Philosophical Society,
reported in the Liverpool Daily Courier, Jan. 26, 1864. Reprinted as a
pamphlet.
The Quadrature of the circle, or the true ratio between the diameter
and circumference geometrically and mathematically demonstrated. By
James Smith, Esq. Liverpool, 1865, 8vo.
{104}
[On the relations between the dimensions and distances of the Sun,
Moon, and Earth; a paper read before the Literary and Philosophical
Society of Liverpool, Jan. 25, 1864. By James Smith, Esq.
The British Association in Jeopardy, and Dr. Whewell, the Master of
Trinity, in the stocks without hope of escape. Printed for the authors
(J. S. confessed, and also hidden under _Nauticus_). (No date, 1865).
The British Association in Jeopardy, and Professor De Morgan in the
Pillory without hope of escape. London, 1866, 8vo.]
When my work appeared in numbers, I had not anything like an adequate idea
of Mr. James Smith's superiority to the rest of the world in the points in
which he is superior. He is beyond a doubt the ablest head at unreasoning,
and the greatest hand at writing it, of all who have tried in our day to
attach their names to an error. Common cyclometers sink into puny orthodoxy
by his side.
The behavior of this singular character induces me to pay him the
compliment which Achilles paid Hector, to drag him round the walls again
and again. He was treated with unusual notice and in the most gentle
manner. The unnamed mathematician, E. M. bestowed a volume of mild
correspondence upon him; Rowan Hamilton[201] quietly proved him wrong in a
way accessible to an ordinary schoolboy; Whewell,[202] as we shall see,
gave him the means of seeing himself wrong, even more easily than by
Hamilton's method. Nothing would do; it was small kick and silly fling at
all; and he exposed his conceit by alleging that he, James Smith, had
placed Whewell in the stocks. He will therefore be universally pronounced a
proper object of the severest literary punishment: but the opinion of all
who can put two propositions together will be that of the many strokes I
have given, the hardest and most telling are my republications of his own
attempts to reason.
He will come out of my hands in the position he ought {105} to hold, the
Supreme Pontiff of cyclometers, the vicegerent of St. Vitus upon earth, the
Mamamouchi of burlesque on inference. I begin with a review of him which
appeared in the _Athenaeum_ of May 11, 1861. Mr. Smith says I wrote it: this
I neither affirm nor deny; to do either would be a sin against the
editorial system elsewhere described. Many persons tell me they know me by
my style; let them form a guess: I can only say that many have declared as
above while fastening on me something which I had never seen nor heard of.
The Quadrature of the Circle: Correspondence between an Eminent
Mathematician and James Smith, Esq. (Edinburgh, Oliver & Boyd; London,
Simpkin, Marshall & Co.)
"A few weeks ago we were in perpetual motion. We did not then suppose that
anything would tempt us on a circle-squaring expedition: but the
circumstances of the book above named have a peculiarity which induces us
to give it a few words.
"Mr. James Smith, a gentleman residing near Liverpool, was some years ago
seized with the _morbus cyclometricus_.[203] The symptoms soon took a
defined form: his circumference shrank into exactly 3-1/8 times his
diameter, instead of close to 3-16/113, which the mathematician knows to be
so near to truth that the error is hardly at the rate of a foot in 2,000
miles. This shrinking of the circumference remained until it became
absolutely necessary that it should be examined by the British Association.
This body, which as Mr. James Smith found to his sorrow, has some interest
in 'jealously guarding the mysteries of their profession,' refused at first
to entertain the question. On this Mr. Smith changed his 'tactics' and the
name of his paper, and smuggled in the subject under the form of 'The
Relations of a Circle inscribed in a Square'! The paper was thus forced
upon the Association, for Mr. Smith informs us that he {106} 'gave the
Section to understand that he was not the man that would permit even the
British Association to trifle with him.' In other words, the Association
bore with and were bored with the paper, as the shortest way out of the
matter. Mr. Smith also circulated a pamphlet. Some kind-hearted man, who
did not know the disorder as well as we do, and who appears in Mr. Smith's
handsome octavo as E. M.--the initials of 'eminent mathematician'--wrote to
him and offered to show him in a page that he was all wrong. Mr. Smith
thereupon opened a correspondence, which is the bulk of the volume. When
the correspondence was far advanced, Mr. Smith announced his intention to
publish. His benevolent instructor--we mean in intention--protested against
the publication, saying 'I do not wish to be gibbeted to the world as
having been foolish enough to enter upon what I feel now to have been a
ridiculous enterprise.'
"For this Mr. Smith cared nothing: he persisted in the publication, and the
book is before us. Mr. Smith has had so much grace as to conceal his kind
adviser's name under E. M., that is to say, he has divided the wrong among
all who may be suspected of having attempted so hopeless a task as that of
putting a little sense into his head. He has violated the decencies of
private life. Against the will of the kind-hearted man who undertook his
case, he has published letters which were intended for no other purpose
than to clear his poor head of a hopeless delusion. He deserves the
severest castigation; and he will get it: his abuse of confidence will
stick by him all his days. Not that he has done his benefactor--in
intention, again--any harm. The patience with which E. M. put the blunders
into intelligible form, and the perseverance with which he tried to find a
cranny-hole for common reasoning to get in at, are more than respectable:
they are admirable. It is, we can assure E. M., a good thing that the
nature of the circle-squarer should be so completely exposed as in this
volume. The benefit which he intended Mr. James Smith may be {107}
conferred upon others. And we should very much like to know his name, and
if agreeable to him, to publish it. As to Mr. James Smith, we can only say
this: he is not mad. Madmen reason rightly upon wrong premises: Mr. Smith
reasons wrongly upon no premises at all.
"E. M. very soon found out that, to all appearance, Mr. Smith got a circle
of 3-1/8 times the diameter by making it the supposition to set out with
that there was such a circle; and then finding certain consequences which,
so it happened, were not inconsistent with the supposition on which they
were made. Error is sometimes self-consistent. However, E. M., to be quite
sure of his ground, wrote a short letter, stating what he took to be Mr.
Smith's hypothesis, containing the following: 'On AC as diameter, describe
the circle D, which by hypothesis shall be equal to three and one-eighth
times the length of AC.... I beg, before proceeding further, to ask whether
I have rightly stated your argument.' To which Mr. Smith replied: 'You have
stated my argument with perfect accuracy.' Still E. M. went on, and we
could not help, after the above, taking these letters as the initials of
Everlasting Mercy. At last, however, when Mr. Smith flatly denied that the
area of the circle lies between those of the inscribed and circumscribed
polygons, E. M. was fairly beaten, and gave up the task. Mr. Smith was left
to write his preface, to talk about the certain victory of truth--which,
oddly enough, is the consolation of all hopelessly mistaken men; to compare
himself with Galileo; and to expose to the world the perverse behavior of
the Astronomer Royal, on whom he wanted to fasten a conversation, and who
replied, 'It would be a waste of time, Sir, to listen to anything you could
have to say on such a subject.'
"Having thus disposed of Mr. James Smith, we proceed to a few remarks on
the subject: it is one which a journal would never originate, but which is
rendered necessary from time to time by the attempts of the autopseustic to
become {108} heteropseustic. To the mathematician we have nothing to say:
the question is, what kind of assurance can be given to the world at large
that the wicked mathematicians are not acting in concert to keep down their
superior, Mr. James Smith, the current Galileo of the quadrature of the
circle.
"Let us first observe that this question does not stand alone:
independently of the millions of similar problems which exist in higher
mathematics, the finding of the diagonal of a square has just the same
difficulty, namely, the entrance of a pair of lines of which one cannot be
definitely expressed by means of the other. We will show the reader who is
up to the multiplication-table how he may go on, on, on, ever nearer, never
there, in finding the diagonal of a square from the side.
"Write down the following rows of figures, and more, if you like, in the
way described:
1 2 5 12 29 70 169 408 985
1 3 7 17 41 99 239 577 1393
After the second, each number is made up of double the last increased by
the last but one: thus, 5 is 1 more than twice 2, 12 is 2 more than twice
5, 239 is 41 more than twice 99. Now, take out two adjacent numbers from
the upper line, and the one below the first from the lower: as
70 169
99.
Multiply together 99 and 169, giving 16,731. If, then, you will say that 70
diagonals are exactly equal to 99 sides, you are in error about the
diagonal, but an error the amount of which is not so great as the 16,731st
part of the diagonal. Similarly, to say that five diagonals make exactly
seven sides does not involve an error of the 84th part of the diagonal.
"Now, why has not the question of _crossing the square_ been as celebrated
as that of _squaring the circle_? Merely because Euclid demonstrated the
impossibility of the first {109} question, while that of the second was not
demonstrated, completely, until the last century.
"The mathematicians have many methods, totally different from each other,
of arriving at one and the same result, their celebrated approximation to
the circumference of the circle. An intrepid calculator has, in our own
time, carried his approximation to what they call 607 decimal places: this
has been done by Mr. Shanks,[204] of Houghton-le-Spring, and Dr.
Rutherford[205] has verified 441 of these places. But though 607 looks
large, the general public will form but a hazy notion of the extent of
accuracy acquired. We have seen, in Charles Knight's[206] _English
Cyclopaedia_, an account of the matter which may illustrate the
unimaginable, though rationally conceivable, extent of accuracy obtained.
"Say that the blood-globule of one of our animalcules is a millionth of an
inch in diameter. Fashion in thought a globe like our own, but so much
larger that our globe is but a blood-globule in one of its animalcules:
never mind the microscope which shows the creature being rather a bulky
instrument. Call this the first globe _above_ us. Let the first globe above
us be but a blood-globule, as to size, in the animalcule of a still larger
globe, which call the second globe above us. Go on in this way to the
twentieth globe above us. Now go down just as far on the other side. Let
the blood-globule with which we started be a globe peopled with animals
like ours, but rather smaller: {110} and call this the first globe below
us. Take a blood-globule out of this globe, people it, and call it the
second globe below us: and so on to the twentieth globe below us. This is a
fine stretch of progression both ways. Now give the giant of the twentieth
globe _above_ us the 607 decimal places, and, when he has measured the
diameter of his globe with accuracy worthy of his size, let him calculate
the circumference of his equator from the 607 places. Bring the little
philosopher from the twentieth globe _below_ us with his very best
microscope, and set him to see the small error which the giant must make.
He will not succeed, unless his microscopes be much better for his size
than ours are for ours.
"Now it must be remembered by any one who would laugh at the closeness of
the approximation, that the mathematician generally goes _nearer_; in fact
his theorems have usually no error at all. The very person who is
bewildered by the preceding description may easily forget that if there
were _no error at all_, the Lilliputian of the millionth globe below us
could not find a flaw in the Brobdingnagian of the millionth globe above.
The three angles of a triangle, of perfect accuracy of form, are
_absolutely_ equal to two right angles; no stretch of progression will
detect _any_ error.
"Now think of Mr. Lacomme's mathematical adviser (_ante_, Vol. I, p. 46)
making a difficulty of advising a stonemason about the quantity of pavement
in a circular floor!
"We will now, for our non-calculating reader, put the matter in another
way. We see that a circle-squarer can advance, with the utmost confidence,
the assertion that when the diameter is 1,000, the circumference is
accurately 3,125: the mathematician declaring that it is a trifle more than
3,141-1/2. If the squarer be right, the mathematician has erred by about a
200th part of the whole: or has not kept his accounts right by about 10s.
in every 100l. Of course, if he set out with such an error he will
accumulate blunder upon blunder. Now, if there be a process in which {111}
close knowledge of the circle is requisite, it is in the prediction of the
moon's place--say, as to the time of passing the meridian at Greenwich--on
a given day. We cannot give the least idea of the complication of details:
but common sense will tell us that if a mathematician cannot find his way
round the circle without a relative error four times as big as a
stockbroker's commission, he must needs be dreadfully out in his attempt to
predict the time of passage of the moon. Now, what is the fact? His error
is less than a second of time, and the moon takes 27 days odd to revolve.
That is to say, setting out with 10s. in 100l. of error in his
circumference, he gets within the fifth part of a farthing in 100l. in
predicting the moon's transit. Now we cannot think that the respect in
which mathematical science is held is great enough--though we find it not
small--to make this go down. That respect is founded upon a notion that
right ends are got by right means: it will hardly be credited that the
truth can be got to farthings out of data which are wrong by shillings.
Even the celebrated Hamilton[207] of Edinburgh, who held that in
mathematics there was no way of going wrong, was fully impressed with the
belief that this was because error was avoided from the beginning. He never
went so far as to say that a mathematician who begins wrong must end right
somehow.
"There is always a difficulty about the mode in which the thinking man of
common life is to deal with subjects he has not studied to a professional
extent. He must form opinions on matters theological, political, legal,
medical, and social. If he can make up his mind to choose a guide, there
is, of course, no perplexity: but on all the subjects mentioned the
direction-posts point different ways. Now why should he not form his
opinion upon an abstract mathematical question? Why not conclude that, as
to the circle, it is possible Mr. James Smith may be the man, just {112} as
Adam Smith[208] was the man of things then to come, or Luther, or Galileo?
It is true that there is an unanimity among mathematicians which prevails
in no other class: but this makes the chance of their all being wrong only
different in degree. And more than this, is it not generally thought among
us that priests and physicians were never so much wrong as when there was
most appearance of unanimity among them? To the preceding questions we see
no answer except this, that the individual inquirer may as rationally
decide a mathematical question for himself as a theological or a medical
question, so soon as he can put himself into a position in mathematics,
level with that in which he stands in theology or medicine. The every-day
thought and reading of common life have a certain resemblance to the
thought and reading demanded by the learned faculties. The research, the
balance of evidence, the estimation of probabilities, which are used in a
question of medicine, are closely akin in character, however different the
matter of application, to those which serve a merchant to draw his
conclusions about the markets. But the mathematicians have methods of their
own, to which nothing in common life bears close analogy, as to the nature
of the results or the character of the conclusions. The logic of
mathematics is certainly that of common life: but the data are of a
different species; they do not admit of doubt. An expert arithmetician,
such as is Mr. J. Smith, may fancy that calculation, merely as such, is
mathematics: but the value of his book, and in this point of view it is not
small, is the full manner in which it shows that a practised arithmetician,
venturing into the field of mathematical demonstration, may show himself
utterly destitute of all that distinguishes the reasoning geometrical
investigator from the calculator.
{113}
"And further, it should be remembered that in mathematics the power of
verifying results far exceeds that which is found in anything else: and
also the variety of distinct methods by which they can be attained. It
follows from all this that a person who desires to be as near the truth as
he can will not judge the results of mathematical demonstration to be open
to his criticism, in the same degree as results of other kinds. Should he
feel compelled to decide, there is no harm done: his circle may be 3-1/8
times its diameter, if it please him. But we must warn him that, in order
to get this circle, he must, as Mr. James Smith has done, _make it at
home_: the laws of space and thought beg leave respectfully to decline the
order."
I will insert now at length, from the _Athenaeum_ of June 8, 1861, the easy
refutation given by my deceased friend, with the remarks which precede.
"Mr. James Smith, of whose performance in the way of squaring the circle we
spoke some weeks ago in terms short of entire acquiescence, has advertised
himself in our columns, as our readers will have seen. He has also
forwarded his letter to the Liverpool _Albion_, with an additional
statement, which he did not make in _our_ journal. He denies that he has
violated the decencies of private life, since his correspondent revised the
proofs of his own letters, and his 'protest had respect only to making his
name public.' This statement Mr. James Smith precedes by saying that we
have treated as true what we well knew to be false: and he follows by
saying that we have not read his work, or we should have known the above
facts to be true. Mr. Smith's pretext is as follows. His correspondent
E. M. says, 'My letters were not intended for publication, and I protest
against their being published,' and he subjoins 'Therefore I must desire
that my name may not be used.' The obvious meaning is that E. M. protested
against the publication altogether, but, judging that Mr. Smith was {114}
determined to publish, desired that his name should not be used. That he
afterwards corrected the proofs merely means that he thought it wiser to
let them pass under his own eyes than to leave them entirely to Mr. Smith.
"We have received from Sir W. Rowan Hamilton[209] a proof that the
circumference is more than 3-1/8 diameters, requiring nothing but a
knowledge of four books of Euclid. We give it in brief as an exercise for
our juvenile readers to fill up. It reminds us of the old days when real
geometers used to think it worth while seriously to demolish pretenders.
Mr. Smith's fame is now assured: Sir W. R. Hamilton's brief and easy
exposure will procure him notice in connection with this celebrated
problem.
"It is to be shown that the perimeter of a regular polygon of 20 sides is
greater than 3-1/8 diameters of the circle, and still more, of course, is
the circumference of the circle greater than 3-1/8 diameters.
"1. It follows from the 4th Book of Euclid, that the rectangle under the
side of a regular decagon inscribed in a circle, and that side increased by
the radius, is equal to the square of the radius. But the product 791 (791
+ 1280) is less than 1280 x 1280; if then the radius be 1280 the side of
the decagon is greater than 791.
"2. When a diameter bisects a chord, the square of the chord is equal to
the rectangle under the doubles of the segments of the diameter. But the
product 125 (4 x 1280 - 125) is less than 791 x 791. If then the bisected
chord be a side of the decagon, and if the radius be still 1280, the double
of the lesser segment exceeds 125.
"3. The rectangle under this doubled segment and the radius is equal to the
square of the side of an inscribed regular polygon of 20 sides. But the
product 125 x 1280 is equal to 400 x 400; therefore, the side of the
last-mentioned polygon is greater than 400, if the radius be still 1280. In
other words, if the radius be represented by the new {115} member 16, and
therefore the diameter by 32, this side is greater than 5, and the
perimeter exceeds 100. So that, finally, if the diameter be 8, the
perimeter of the inscribed regular polygon of 20 sides, and still more the
circumference of the circle, is greater than 25: that is, the circumference
is more than 3-1/8 diameters."
The last work in the list was thus noticed in the _Athenaeum_, May 27, 1865.
"Mr. James Smith appears to be tired of waiting for his place in the Budget
of Paradoxes, and accordingly publishes a long letter to Professor De
Morgan, with various prefaces and postscripts. The letter opens by a hint
that the Budget appears at very long intervals, and 'apparently without any
sufficient reason for it.' As Mr. Smith hints that he should like to see
Mr. De Morgan, whom he calls an 'elephant of mathematics,' 'pumping his
brains' 'behind the scenes'--an odd thing for an elephant to do, and an odd
place to do it in--to get an answer, we think he may mean to hint that the
Budget is delayed until the pump has worked successfully. Mr. Smith is
informed that we have had the whole manuscript of the Budget, excepting
only a final summing-up, in our hands since October, 1863. [This does not
refer to the Supplement.] There has been no delay: we knew from the
beginning that a series of historical articles would be frequently
interrupted by the things of the day. Mr. James Smith lets out that he has
never been able to get a private line from Mr. De Morgan in answer to his
communications: we should have guessed it. He says, 'The Professor is an
old bird and not to be easily caught, and by no efforts of mine have I been
able, up to the present moment, either to induce or twit him into a
discussion....' Mr. Smith curtails the proverb: old birds are not to be
caught with _chaff_, nor with _twit_, which seems to be Mr. Smith's word
for his own chaff, and, so long as the first letter is sounded, a very
proper word. Why does he not try a little grain of sense? Mr. Smith
evidently {116} thinks that, in his character as an elephant, the Professor
has not pumped up brain enough to furnish forth a bird. In serious earnest,
Mr. Smith needs no answer. In one thing he excites our curiosity: what is
meant by demonstrating 'geometrically _and_ mathematically?'"