A Budget of Paradoxes Volume II Part 10

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 circ.u.mference 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 ill.u.s.trate 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 circ.u.mference 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 quant.i.ty 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 a.s.sertion that when the diameter is 1,000, the circ.u.mference is accurately 3,125: the mathematician declaring that it is a trifle more than 3,141. 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 acc.u.mulate 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 pa.s.sing 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 pa.s.sage 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 circ.u.mference, 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 cla.s.s: 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 a.n.a.logy, 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 dest.i.tute 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 s.p.a.ce 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 pa.s.s 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 circ.u.mference 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 a.s.sured: 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 circ.u.mference 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 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 1280 - 125) is less than 791 791. 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 1280 is equal to 400 400; 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 circ.u.mference of the circle, is greater than 25: that is, the circ.u.mference 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 ma.n.u.script 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?'"

I now proceed to my original treatment of the case.

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 a.s.surance 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 _fort.i.ter 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 a.s.sume that it is so,--"if you dislike the term datum, then, by hypothesis, let 8 circ.u.mferences be exactly equal to 25 diameters,"--and then to show that every other supposition is thereby made absurd. The right to this a.s.sumption is enforced in the "Nut" by the following a.n.a.logy:

"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 a.s.sumes what he wants to _disprove_, and shows that his _a.s.sumption_ leads to absurdity, and so _upsets itself_. Mr. Smith a.s.sumes what he wants to _prove_, and shows that _his_ a.s.sumption 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 a.s.sumption, to regard that agreement as proof of the a.s.sumption. The following is the "piece of information" which will settle me, if I be honest. a.s.suming [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 a.s.sume--

[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 embarra.s.ses 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 = (20) = 5, be given numbers. Then rq(20 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 [pi]/4 = 20 3.125/4 = 20 .78125 = 15.625

will be the first number; as

25 : 16 :: rq 20 : rq 8.192: and (rq 8.192)^2 [pi]/4 = 8.192 .78125 = 6.4

will be the second number; therefore rq(15.625 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 pa.s.sed and repa.s.sed. 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 ill.u.s.trate, 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.