Chapter 5

1 2 5 12 29 70 169 408 9851 3 7 17 41 99 239 577 1393

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 16999.

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 ofcrossing the squarebeen as celebrated as that ofsquaring the circle? Merely because Euclid demonstrated the impossibility of the firstquestion, 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 Cyclopædia, 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 globeaboveus. 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: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 globeaboveus 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 globebelowus 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 goesnearer; 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 wereno 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, areabsolutelyequal to two right angles; no stretch of progression will detectanyerror.

"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 3,141½. 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 whichclose 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, justas 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.

"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 3⅛ 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 theAthenæumof 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 LiverpoolAlbion, with an additional statement, which he did not make inourjournal. 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 wasdetermined 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 3⅛ 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 3⅛ diameters of the circle, and still more, of course, is the circumference of the circle greater than 3⅛ 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 newmember 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 3⅛ diameters."

The last work in the list was thus noticed in theAthenæum, 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 withchaff, nor withtwit, 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 evidentlythinks 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 'geometricallyandmathematically?'"

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 theNut to CrackMr. 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 thescutica,[210]if not thehorribile flagellum.[211]He tells me that he will bring his solution to me in such a form as shall compel me to admit it asun fait accompli[une faute accomplie?][212]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'ssuaviter in modo,[214]and proceed to hisfortiter in re[215]: I shall show that hehasconvicted himself of ignorance and folly, with an honesty and candor worthy of a better value ofπ.

Mr. Smith's method of proving that every circle is 3⅛ 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 ofπwill lead to the grossest absurdities; unless indeed, you are prepared to dispute the right of Euclid to adopt a false line hypothetically for the purposeof a 'reductio ad absurdum'[216]demonstration, in pure geometry."

Euclid assumes what he wants todisprove, and shows that hisassumptionleads to absurdity, and soupsets itself. Mr. Smith assumes what he wants toprove, and shows thathisassumption makesother propositionslead 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'être frappé de l'évidence.[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. Assumingπto be 3⅛, 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,

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,aandbbeing any two numbers whatever, we need only assume—

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,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 writerq[218]for the sign of the square root, which embarrasses small type:

"This establishes the followinginfalliblerule, for finding two mean proportionalsOF 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 = ¼(20) = 5, be given numbers. Thenrq(20 × 5) =rq100 = 10, is their mean proportional. Let this be a given mean proportionalTO FIND ANOTHER MEAN PROPORTIONAL OF EQUAL VALUE. Then

will be the first number; as

will be the second number; thereforerq(15.625 × 6.4) =rq100 = 10, is the required mean proportional.... Now, my good Sir, however competent you may be to prove every man a fool [noteveryman, Mr. Smith! onlysome; 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 proportionalsby 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,xis a mean proportional when in 5,x, 20, 5 is toxasxto 20. Andxmust be 10. Butxandyaretwomean proportionals when in 5,x,y, 20,xis a mean proportional between 5 andy, andyis a mean proportional betweenxand 20. And these means arex= 5 ³√4,y= 5 ³√16. But Mr. Smith findsonemean, finds itagainin 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 thefullamount 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 theCambridge Transactions:

"The uncultivated reason proceeds by a process almost entirely material. Though the necessary law of thoughtmust 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 itnow. 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."

"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. Butwhat 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 sendingletter 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 inA.D.X2; not 4 inA.D.16, nor 5 inA.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.

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 mathematicalsnail! 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 mehors de combat.[220]The confusion of images is amusing: Goliath turning himself into a snail to avoidπ= 3⅛, and James Smith, Esq., of the Mersey Dock Board: and puthors de combat—which should have beencaché[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."

"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 ascertainedthat one of the letters to theAthenæumsigned 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 ofA 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 alibel 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'sπuntouched: or, if I put in my thumbandpulled 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 ofJamesmight be spelledJimm.[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 ofJem. An instructor of youth was scandalized at the abrupt and irregular—but very effective—opening of Wordsworth's little piece:

"A simple childThat lightly draws its breath,And feels its life in every limb,What should it know of death?"

"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 withJimb.

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°, 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 asπ, 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:

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 "Thispig 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 theAthenæum:and I will do it, because the question will illustrate what isbelowschoolboy level.

"Letxrepresent the circular measure of an angle of 15°, andyhalf the sine of an angle of 30° = area of the square on the radius of a circle of diameter unity = .25. Ifx-y=xy, firstly, what is the arithmetical value ofxy? secondly, what is the angle of whichxyrepresents the circular measure?"

Ifxrepresent 15° andybe ¼,xyrepresents 3° 45', whetherx-ybexyor no. But,ybeing ¼,x-yisnotxyunlessxbe ⅓, that is, unless 12xorπbe 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 thathissarcasm 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 bethe indefinitesomethingin the mysteriousall 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 forπ: 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,becauseit 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 itslockdollager) treated Goliah Quagg. He has announced hisintention 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.

THE MOON HOAX.

The Moon-hoax; or the discovery that the moon has a vast population of human beings. By Richard Adams Locke.[226]New York, 1859, 8vo.

This is a reprint of the hoax already mentioned. I suppose R. A. Locke is the name assumed by M. Nicollet.[227]The publisher informs us that when the hoax first appeared day by day in a morning paper, the circulation increased fivefold, and the paper obtained a permanent footing. Besides this, an edition of 60,000 was sold off in less than one month.

The discovery was also published under the name of A. R. Grant.[228]Sohncke's[229]Bibliotheca Mathematicaconfounds this Grant with Prof. R. Grant[230]of Glasgow, the author of theHistory of Physical Astronomy, who is accordingly made to guarantee the discoveries in the moon. I hope Adams Locke will not merge in J. C. Adams,[231]the co-discoverer of Neptune. Sohncke gives the titles ofthree French translations of the Moon hoax at Paris, of one at Bordeaux, and of Italian translations at Parma, Palermo, and Milan.

A Correspondent, who is evidently fully master of details, which he has given at length, informs me that the Moon hoax appeared first in theNew York Sun, of which R. A. Locke was editor. It so much resembled a story then recently published by Edgar A. Poe, in a Southern paper, "Adventures of Hans Pfaal," that some New York journals published the two side by side. Mr. Locke, when he left theNew York Sun, started another paper, and discovered the manuscript of Mungo Park;[232]but this did not deceive. TheSun, however, continued its career, and had a great success in an account of a balloon voyage from England to America, in seventy-five hours, by Mr. Monck Mason,[233]Mr. Harrison Ainsworth,[234]and others. I have no doubt that M. Nicollet was the author of the Moon hoax,[235]written in a way which marks the practised observatory astronomer beyond all doubt, and by evidence seen in the most minute details. Nicollet had an eye to Europe. I suspect that he took Poe's story, and made it a basis for his own. Mr. Locke, it would seem, when he attempted a fabrication for himself, did not succeed.

The Earth we inhabit, its past, present, and future. By Capt. Drayson.[236]London, 1859, 8vo.

The earth is growing; absolutely growing larger: its diameter increases three-quarters of an inch per mile every year. The foundations of our buildings will give way intime: the telegraph cables break, and no cause ever assigned except ships' anchors, and such things. The book is for those whose common sense is unwarped, who can judge evidence as well as the ablest philosopher. The prospect is not a bad one, for population increases so fast that a larger earth will be wanted in time, unless emigration to the Moon can be managed, a proposal of which it much surprises me that Bishop Wilkins has a monopoly.

IMPALEMENT BY REQUEST.

Athenæum, August, 19, 1865.Notice to Correspondents.

"R. W.—If you will consult the opening chapter of the Budget of Paradoxes, you will see that the author presents only works in his own library at a given date; and this for a purpose explained. For ourselves we have carefully avoided allowing any writers to present themselves in our columns on the ground that the Budget has passed them over. We gather that Mr. De Morgan contemplates additions at a future time, perhaps in a separate and augmented work; if so, those who complain that others of no greater claims than themselves have been ridiculed may find themselves where they wish to be. We have done what we can for you by forwarding your letter to Mr. De Morgan."

The author of "An Essay on the Constitution of the Earth," published in 1844, demanded of theAthenæum, as anact of fairness, that a letter from him should be published, proving that he had as much right to be "impaled" as Capt. Drayson. He holds, on speculative grounds, what the other claims to have proved by measurement, namely, that the earth is growing; and he believes that in time—a good long time, notourtime—the earth and other planets may grow into suns, with systems of their own.

This gentleman sent me a copy of his work, after the commencement of my Budget; but I have no recollection of having received it, and I cannot find it on the (nursery?quarantine?) shelves on which I keep my unestablished discoveries. Had I known of this work in time, (see the Introduction) I should of course, have impaled it (heraldically) with the other work; but the two are very different. Capt. Drayson professes to prove his point by results of observation; and I think he does not succeed. The author before me only speculates; and a speculator can get any conclusion into his premises, if he will only build or hire them of shape and size to suit. It reminds me of a statement I heard years ago, that a score of persons, or near it, were to dine inside the skull of one of the aboriginal animals, dear little creatures! Whereat I wondered vastly, nothing doubting; facts being stubborn and not easy drove, as Mrs. Gamp said. But I soon learned that the skull was not a real one, but artificially constructed by the methods—methods which have had striking verifications, too—which enable zoologists to go the whole hog by help of a toe or a bit of tail. This took off the edge of the wonder: a hundred people can dine inside an inference, if you draw it large enough. The method might happen to fail for once: for instance, the toe-bone might have been abnormalized by therian or saurian malady; and the possibility of such failure, even when of small probability, is of great alleviation. The author before me is, apparently, the sole fabricator of his own premises. With vital force in the earth and continual creation on the part of the original Creator, he expands our bit of a residence as desired. But, as the Newtoness of Cookery observed, First catch your hare. When this is done, when youhavea growing earth, you shall dress it with all manner of proximate causes, and serve it up with a growing Moon for sauce, a growing Sun, if it please you, at the other end, and growing planets for side-dishes. Hoping this amount of impalement will be satisfactory, I go on to something else.

THE HAILESEAN SYSTEM OF ASTRONOMY.

The Hailesean System of Astronomy.By John Davey Hailes[237](two pages duodecimo, 1860).

He offers totake100,000l.to 1,000l.that he shows the sun to be less than seven millions of miles from the earth. The earth in the center, revolving eastward, the sun revolving westward, so that they "meet at half the circle distance in the 24 hours." The diameter of the circle being 9839458303, the circumference is 30911569920.

The following written challenge was forwarded to the Council of the Astronomical Society: it will show the "general reader"—and help him towards earning his name—what sort of things come every now and then to our scientific bodies. I have added punctuation:

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