 |
Throughout this book I have called people by the names which denote them in their books, or by our vernacular names. This is the intelligible way of proceeding. I might, for instance (Vol. I, p. 44), have spoken of Charles de Bovelles,[604] of Lefevre d'Etaples,[605] of Pelerin,[606] and of Etienne.[607] But I prefer the old plan. Those who like another plan better, are welcome to substitute with a pen, when they know what to write; when they do not, it is clear that they would not have understood me if I had given modern names.
The principal advisers of King Custom are as follows. First, there is Etymology, the chiffonnier, or general rag-merchant, who has made such a fortune of late years in his own business that he begins to be considered highly respectable. He gives advice which is more thought of than followed, partly on account of the fearful extremes into which he runs. He lately asked some boys of sixteen, at a matriculation examination in English, to what branch of {325} the Indo-Germanic family they felt inclined to refer the Pushto language, and what changes in the force of the letters took place in passing from Greek into Moeso-Gothic. Because all syllables were once words, he is a little inclined to insist that they shall be so still. He would gladly rule English with a Saxon rod, which might be permitted with a certain discretion which he has never attained: and when opposed, he defends himself with analogies of the Aryan family until those who hear him long for the discovery of an Athanasyus. He will transport a word beyond seas—he is recorder of Rhematopolis—on circumstantial evidence which looks like mystery gone mad; but, strange to say, something very often comes to light after sentence is passed which proves the soundness of the conviction.
The next adviser is Logic, a swearing old justice of peace, quorum, and rotulorum, whose excesses brought on such a fit of the gout that for many years he was unable to move. He is now mending, and his friends say he has sown his wild oats. He has some influence with the educated subjects of Custom, and will have more, if he can learn the line at which interference ought to stop: with them he has succeeded in making an affirmative of two negatives; but the vulgar won't never have nothing to say to him. He has always railed at Milton for writing that Eve was the fairest of her daughters; but has never satisfactorily shown what Milton ought to have said instead.
The third adviser has more influence with the mass of the subjects of King Custom than the other two put together; his name is Fiddlefaddle, the toy-shop keeper; and the other two put him forward to do their worst work. In return, he often uses their names without authority. He took Etymology to witness that means to an end must be plural: and he would have any one method to be a mean. But Etymology proved him wrong, King Custom referred him to his Catechism, in which is "a means whereby we receive the same," and Analogy—a subordinate of {326} Etymology—asked whether he thought it a great new to hear that he was wrong. It was either this Fiddlefaddle, or Lindley Murray[608] his traveler, who persuaded the Miss Slipslops, of the Ladies Seminary, to put "The Misses Slipslop" over the gate. Sixty years ago, this bagman called at all the girls' schools, and got many of the teachers to insist on the pupils saying "Is it not" and "Can I not" for "Isn't it" and "Can't I": of which it came that the poor girls were dreadfully laughed at by their irreverent brothers when they went home for the holidays. Had this bad adviser not been severely checked, he might by this time have proposed our saying "The Queen's of England son," declaring, in the name of Logic, that the prince was the Queen's son, not England's.
Lastly, there is Typography the metallurgist, an executive officer who is always at work in secret, and whose lawless mode of advising is often done by carrying his notions into effect without leave given. He it is who never ceases suggesting that the same word is not to occur in a second place within sight of the first. When the Authorized Version was first printed, he began this trick at the passage, "Let there be light, and there was light;" he drew a line on the proof under the second light, and wrote "luminosity?" opposite. He is strongest in the punctuations and other signs; he has a pepper-box full of commas always by his side. He puts everything under marks of quotation which he has ever heard before. An earnest preacher, in a very moving sermon, used the phrase Alas! and alack a day! Typography stuck up the inverted commas because he had read the old Anglo-Indian toast, "A lass and a lac a day!" If any one should have the sense to leave out of his Greek {327} the unmeaning scratches which they call accents, he goes to a lexicon and puts them in. He is powerful in routine; but when two routines interlace or overlap, he frequently takes the wrong one.
Subject to bad advice, and sometimes misled for a season, King Custom goes on his quiet way and is sure to be right at last.
"Treason does never prosper: what's the reason? Why, when it prospers, none dare call it treason."
Language is in constant fermentation, and all that is thrown in, so far as it is not fit to assimilate, is thrown off; and this without any obvious struggle. In the meanwhile every one who has read good authors, from Shakspeare downward, knows what is and what is not English; and knows, also, that our language is not one and indivisible. Two very different turns of phrase may both be equally good, and as good as can be: we may be relieved of the consequences of contempt of one court by habeas corpus issuing out of another.
TEST OF LANGUAGE.
Hallam remarks that the Authorized Version of the Bible is not in the language of the time of James the First: that it is not the English of Raleigh or of Bacon. Here arises the question whether Raleigh and Bacon are the true expositors of the language of their time; and whether they were not rather the incipient promoters of a change which was successfully resisted by—among other things—the Authorized Version of the Testaments. I am not prepared to concede that I should have given to the English which would have been fashioned upon that of Bacon by imitators, such as they usually are, the admiration which is forced from me by Bacon's English from Bacon's pen. On this point we have a notable parallel. Samuel Johnson {328} commands our admiration, at least in his matured style: but we nauseate his followers. It is an opinion of mine that the works of the leading writers of an age are seldom the proper specimens of the language of their day, when that language is in its state of progression. I judge of a language by the colloquial idiom of educated men: that is, I take this to be the best medium between the extreme cases of one who is ignorant of grammar and one who is perched upon a style. Dialogue is what I want to judge by, and plain dialogue: so I choose Robert Recorde[609] and his pupil in the Castle of Knowledge, written before 1556. When Dr. Robert gets into his altitudes of instruction, he differs from his own common phraseology as much as probably did Bacon when he wrote morals and philosophy. But every now and then I come to a little plain talk about a common thing, of which I propose to show a specimen. Anything can be made to look old by such changes as makes into maketh, with a little old spelling. I shall invert these changes, using the newer form of inflexion, and the modern spelling: with no other variation whatever.
"Scholar. Yet the reason of that is easy enough to be conceived, for when the day is at the longest the Sun must needs shine the more time, and so must it needs shine the less time when the day is at the shortest: this reason I have heard many men declare.
Master. That may be called a crabbed reason, for it {329} goes backward like a crab. The day makes not the Sun to shine, but the Sun shining makes the day. And so the length of the day makes not the Sun to shine long, neither the shortness of the day causes not [sic] the Sun to shine the lesser time, but contrariwise the long shining of the Sun makes the long day, and the short shining of the Sun makes the lesser day: else answer me what makes the days long or short?
Scholar. I have heard wise men say that Summer makes the long days, and Winter makes the long nights.
Master. They might have said more wisely, that long days make summer and short days make winter.
Scholar. Why, all that seems one thing to me.
Master. Is it all one to say, God made the earth, and the earth made God? Covetousness overcomes all men, and all men overcome covetousness?
Scholar. No, not so; for here the effect is turned to be the cause, and the agent is made the patient.
Master. So is it to say Summer makes long days, when you should say: Long days make summer.
Scholar. I perceive it now: but I was so blinded with the vulgar error, that if you had demanded of me further what did make the summer, I had been like to have answered that green leaves do make summer; and the sooner by remembrance of an old saying that a year should come in which the summer should not be known but by the green leaves.
Master. Yet this saying does not import that green leaves do make summer, but that they betoken summer; so are they the sign and not the cause of summer."
I have taken a whole page of our author, without omission, that the reader may see that I do not pick out sentences convenient for my purpose. I have done nothing but alter the third person of the verb and the spelling: but great is the effect thereof. We say "the Sun shining makes the day"; Recorde, "the Sonne shynynge maketh the daye." {330} These points apart, we see a resemblance between our English and that of three hundred years ago, in the common talk of educated persons, which will allow us to affirm that the language of the authorized Bible must have been very close to that of its time. For I cannot admit that much change can have taken place in fifty years: and the language of the version represents both our common English and that of Recorde with very close approximation. Take sentences from Bacon and Raleigh, and it will be apparent that these writers will be held to differ from all three, Recorde, the version, and ourselves, by differences of the same character. But we speak of Recorde's conversation, and of our own. We conclude that it is the plain and almost colloquial character of the Authorized Version which distinguishes it from the English of Bacon and Raleigh, by approximating it to the common idiom of the time. If any one will cast an eye upon the letters of instruction written by Cecil[610] and the Bishop of London to the translators themselves, or to the general directions sent to them in the King's name, he will find that these plain business compositions differ from the English of Bacon and Raleigh by the same sort of differences which distinguish the version itself.
PRONUNCIATION.
The foreign word, or the word of a district, or class of people, passes into the general vernacular; but it is long before the specially learned will acknowledge the right of those with whom they come in contact to follow general usage. The rule is simple: so long as a word is technical or local, those who know its technical or local pronunciation may reasonably employ it. But when the word has become general, the specialist is not very wise if he refuse to follow {331} the mass, and perfectly foolish if he insist on others following him. There have been a few who demanded that Euler should be pronounced in the German fashion:[611] Euler has long been the property of the world at large; what does it matter how his own countrymen pronounce the letters? Shall we insist on the French pronouncing Newton without that final tong which they never fail to give him? They would be wise enough to laugh at us if we did. We remember that a pedant who was insisting on all the pronunciations being retained, was met by a maxim in contradiction, invented at the moment, and fathered upon Kaen-foo-tzee,[612] an authority which he was challenged to dispute. Whom did you speak of? said the bewildered man of accuracy. Learn your own system, was the answer, before you impose it on others; Confucius says that too.[613]
The old English has fote, fode, loke, coke, roke, etc., for foot, etc. And above rhymes in Chaucer to remove. Suspecting that the broader sounds are the older, we may surmise that remove and food have retained their old sounds, and that cook, once coke, would have rhymed to our Luke, the vowel being brought a little nearer, perhaps, to the o in our present coke, the fuel, probably so called as used by cooks. If this be so, the Chief Justice Cook[614] of our lawyers, and the Coke (pronounced like the fuel) of the greater part of the world, are equally wrong. The lawyer has no right whatever to fasten his pronunciation upon us: even leaving aside the general custom, he cannot prove himself right, and is probably wrong. Those who {332} know the village of Rokeby (pronounced Rookby) despise the world for not knowing how to name Walter Scott's poem: that same world never asked a question about the matter, and the reception of the parody of Jokeby, which soon appeared, was a sufficient indication of their notion. Those who would fasten the hodiernal sound upon us may be reminded that the question is, not what they call it now, but what it was called in Cromwell's time. Throw away general usage as a lawgiver, and this is the point which emerges. Probably Rūke-by would be right, with a little turning of the Italian ū towards ō of modern English.
[Some of the above is from an old review. I do not always notice such insertions: I take nothing but my own writings. A friend once said to me, "Ah! you got that out of the Athenaeum!" "Excuse me," said I. "the Athenaeum got that out of me!"]
APOLOGIES TO CLUVIER.
It is part of my function to do justice to any cyclometers whose methods have been wrongly described by any orthodox sneerers (myself included). In this character I must notice Dethlevus Cluverius,[615] as the Leipzig Acts call him (probably Dethleu Cluvier), grandson of the celebrated geographer, Philip Cluvier. The grandson was a Fellow of the Royal Society, elected on the same day as Halley,[616] November 30, 1678: I suppose he lived in England. This {333} man is quizzed in the Leipzig Acts for 1686; and, if Montucla insinuate rightly, by Leibnitz, who is further suspected of wanting to embroil Cluvier with his own opponent Nieuwentiit,[617] on the matter of infinitesimals. So far good: I have nothing against Leibnitz, who though he was ironical, told us what he laughed at. But Montucla has behaved very unfairly: he represents Cluvier as placing the essence of his method in the solution of the problem construere mundum divinae menti analogum, to construct a world corresponding to the divine mind. Nothing to begin with: no way of proceeding. Now, it ought to have been ex data linea construere,[618] etc.: there is a given line, which is something to go on. Further, there is a way of proceeding: it is to find the product of 1, 2, 3, 4, etc. for ever. Moreover, Montucla charges Cluvier with unsquaring the parabola, which Archimedes had squared as tight as a glove. But he never mentions how very nearly Cluvier agrees with the Greek: they only differ by 1 divided by 3n^2, where n is the infinite number of parts of which a parabola is composed. This must have been the conceit that tickled Leibnitz, and made him wish that Cluvier and Nieuwentiit should fight it out. Cluvier, was admitted, on terms of irony, into the Leipzig Acts: he appeared on a more serious footing in London. It is very rare for one cyclometer to refute another: les corsaires ne se battent pas.[619] The only instance I recall is that of M. Cluvier, who (Phil. Trans., 1686, No. 185) refuted M. Mallemont de Messange,[620] who {334} published at Paris in 1686. He does it in a very serious style, and shows himself a mathematician. And yet in the year in which, in the Phil. Trans., he was a geometer, and one who rebukes his squarer for quoting Matthew xi. 25, in that very year he was the visionary who, in the Leipzig Acts, professed to build a world resembling the divine mind by multiplying together 1, 2, 3, 4, etc. up to infinity.
THE RAINBOW PARADOX.
There is a very pretty opening for a paradox which has never found its paradoxer in print. The philosophers teach that the rainbow is not material: it comes from rain-drops, but those rain-drops do not take color. They only give it, as lenses and mirrors; and each one drop gives all the colors, but throws them in different directions. Accordingly, the same drop which furnishes red light to one spectator will furnish violet to another, properly placed. Enter the paradoxer whom I have to invent. The philosopher has gulled you nicely. Look into the water, and you will see the reflected rainbow: take a looking-glass held sideways, and you see another reflection. How could this be, if there were nothing colored to reflect? The paradoxer's facts are true: and what are called the reflected rainbows are other rainbows, caused by those other drops which are placed so as to give the colors to the eye after reflection, at the water or the looking-glass. A few years ago an artist exhibited a picture with a rainbow and its apparent reflection: he simply copied what he had seen. When his picture was examined, some started the idea that there could be no reflection of a rainbow; they were right: they inferred that the artist had made a mistake; they were wrong. When it was explained, some agreed and some dissented. Wanted, {335} immediately, an able paradoxer: testimonials to be forwarded to either end of the rainbow, No. 1. No circle-squarer need apply, His Variegatedness having been pleased to adopt 3.14159... from Noah downwards.
TYCHO BRAHE REVIVED.
The system of Tycho Brahe,[621] with some alteration and addition, has been revived and contended for in our own day by a Dane, W. Zytphen,[622] who has published The Motion of the Sun in the Universe, (second edition) Copenhagen, 1865, 8vo, and Le Mouvement Sideral, 1865, 8vo. I make an extract.
"How can one explain Copernically that the velocity of the Moon must be added to the velocity of the Earth on the one place in the Earth's orbit, to learn how far the Moon has advanced from one fixed star to another; but in another place in the orbit these velocities must be subtracted (the movements taking place in opposite directions) to attain the same result? In the Copernican and other systems, it is well known that the Moon, abstracting from the insignificant excentricity of the orbit, always in twenty-four hours performs an equally long distance. Why has Copernicus never been denominated Fundamentus or Fundator? Because he has never convinced anybody so thoroughly that this otherwise so natural epithet has occurred to the mind."
Really the second question is more effective against Newton than against Copernicus; for it upsets gravity: the first is of great depth.
{336}
JAMES SMITH WILL NOT DOWN.
The Correspondent journal makes a little episode in the history of my Budget (born May, 1865, died April, 1866). It consisted entirely of letters written by correspondents. In August, a correspondent who signed "Fair Play"—and who I was afterwards told was a lady—thought it would be a good joke to bring in the Cyclometers. Accordingly a letter was written, complaining that though Mr. Sylvester's[623] demonstration of Newton's theorem—then attracting public attention—was duly lauded, the possibly greater discovery of the quadrature seemed to be blushing unseen, and wasting etc. It went on as follows:
"Prof. De Morgan, who, from his position in the scientific world, might fairly afford to look favourably on less practised efforts than his own, seems to delight in ridiculing the discoverer. Science is, of course, a very respectable person when he comes out and makes himself useful in the world [it must have been a lady; each sex gives science to the other]: but when, like a monk of the Middle Ages, he shuts himself up [it must have been a lady; they always snub the bachelors] in his cloistered cell, repeating his mumpsimus from day to day, and despising the labourers on the outside, we begin to think of Galileo,[624] Jenner,[625] Harvey,[626] and other glorious trios, who have been contemned ..."
The writer then called upon Mr. James Smith[627] to come {337} forward. The irony was not seen; and that day fortnight appeared the first of more than thirty letters from his pen. Mr. Smith was followed by Mr. Reddie,[628] Zadkiel,[629] and others, on their several subjects. To some of the letters I have referred; to others I shall come. The Correspondent was to become a first-class scientific journal; the time had arrived at which truth had an organ: and I received formal notice that I could not stifle it by silence, nor convert it into falsehood by ridicule. When my reader sees my extracts, he will readily believe my declaration that I should have been the last to stifle a publication which was every week what James Mill[630] would call a dose of capital for my Budget. A few anti-paradoxers brought in common sense: but to the mass of the readers of the journal it all seemed to be the difference between Tweedledum and Tweedledee. Some said that the influx of scientific paradoxes killed the journal: but my belief is that they made it last longer than it otherwise would have done. Twenty years ago I recommended the paradoxers to combine and publish their views in a common journal: with a catholic editor, who had no pet theory, but a stern determination not to exclude anything merely for absurdity. I suspect it would answer very well. A strong title, or motto, would be wanted: not so coarse as was roared out in a Cambridge mob when I was an undergraduate—"No King! No Church! No House of Lords! No nothing, blast me!"—but something on that principle.
At the end of 1867 I addressed the following letter to the Athenaeum:
PSEUDOMATH, PHILOMATH, AND GRAPHOMATH.
December 31, 1867
Many thanks for the present of Mr. James Smith's letters {338} of Sept. 28 and of Oct. 10 and 12. He asks where you will be if you read and digest his letters: you probably will be somewhere first. He afterwards asks what the WE of the Athenaeum will be if, finding it impossible to controvert, it should refuse to print. I answer for you, that We-We of the Athenaeum, not being Wa-Wa the wild goose, so conspicuous in "Hiawatha," will leave what controverts itself to print itself, if it please.
Philomath is a good old word, easier to write and speak than mathematician. It wants the words between which I have placed it. They are not well formed: pseudomathete and graphomathete would be better: but they will do. I give an instance of each.
The pseudomath is a person who handles mathematics as the monkey handled the razor. The creature tried to shave himself as he had seen his master do; but, not having any notion of the angle at which the razor was to be held, he cut his own throat. He never tried a second time, poor animal! but the pseudomath keeps on at his work, proclaims himself clean-shaved, and all the rest of the world hairy. So great is the difference between moral and physical phenomena! Mr. James Smith is, beyond doubt, the great pseudomath of our time. His 3-1/8 is the least of a wonderful chain of discoveries. His books, like Whitbread's barrels, will one day reach from Simpkin & Marshall's to Kew, placed upright, or to Windsor laid length-ways. The Queen will run away on their near approach, as Bishop Hatto did from the rats: but Mr. James Smith will follow her were it to John o' Groats.
The philomath, for my present purpose, must be exhibited as giving a lesson to presumption. The following anecdote is found in Thiebault's[631] Souvenirs de vingt ans de sejours a Berlin, published in 1804. The book itself got a high character for truth. In 1807 Marshal Mollendorff[632] {339} answered an inquiry of the Duc de Bassano,[633] by saying that it was the most veracious of books, written by the most honest of men. Thiebault does not claim personal knowledge of the anecdote, but he vouches for its being received as true all over the north of Europe.[634]
Diderot[635] paid a visit to Russia at the invitation of Catherine the Second. At that time he was an atheist, or at least talked atheism: it would be easy to prove him either one thing or the other from his writings. His lively sallies on this subject much amused the Empress, and all the younger part of her Court. But some of the older courtiers suggested that it was hardly prudent to allow such unreserved exhibitions. The Empress thought so too, but did not like to muzzle her guest by an express prohibition: so a plot was contrived. The scorner was informed that an eminent mathematician had an algebraical proof of the existence of God, which he would communicate before the whole Court, if agreeable. Diderot gladly consented. The mathematician, who is not named, was Euler.[636] He came to Diderot with the gravest air, and in a tone of perfect conviction said, "Monsieur!
(a + b^n)/n = x
donc Dieu existe; repondez!"[637] Diderot, to whom algebra was Hebrew, though this is expressed in a very roundabout way by Thiebault—and whom we may suppose to have expected some verbal argument of alleged algebraical closeness, was disconcerted; while peals of laughter sounded on all sides. Next day he asked permission to return to France, which was granted. An algebraist would have {340} turned the tables completely, by saying, "Monsieur! vous savez bien que votre raisonnement demande le developpement de x suivant les puissances entieres de n".[638] Goldsmith could not have seen the anecdote, or he might have been supposed to have drawn from it a hint as to the way in which the Squire demolished poor Moses.
The graphomath is a person who, having no mathematics, attempts to describe a mathematician. Novelists perform in this way: even Walter Scott now and then burns his fingers. His dreaming calculator, Davy Ramsay, swears "by the bones of the immortal Napier." Scott thought that the the philomaths worshiped relics: so they do, in one sense. Look into Hutton's[639] Dictionary for Napier's Bones, and you shall learn all about the little knick-knacks by which he did multiplication and division. But never a bone of his own did he contribute; he preferred elephants' tusks. The author of Headlong Hall[640] makes a grand error, which is quite high science: he says that Laplace proved the precession of the equinoxes to be a periodical inequality. He should have said the variation of the obliquity. But the finest instance is the following: Mr. Warren,[641] in his well-wrought tale of the martyr-philosopher, was incautious enough to invent the symbols by which his savant satisfied himself Laplace[642] was right on a doubtful point. And this is what he put together—
[sqrt]-3a^2, [rectangle]y^2 / z^2 + 9 - n = 9, n x log e.
Now, to Diderot and the mass of mankind this might be Laplace all over: and, in a forged note of Pascal, would {341} prove him quite up to gravitation. But I know of nothing like it, except in the lately received story of the American orator, who was called on for some Latin, and perorated thus: "Committing the destiny of the country to your hands, Gentlemen, I may without fear declare, in the language of the noble Roman poet,
E pluribus unum, Multum in parvo, Ultima Thule, Sine qua non."[643]
But the American got nearer to Horace than the martyr-philosopher to Laplace. For all the words are in Horace, except Thule, which might have been there. But [rectangle] is not a symbol wanted by Laplace; nor can we see how it could have been; in fact, it is not recognized in algebra. As to the junctions, etc., Laplace and Horace are about equally well imitated.
Further thanks for Mr. Smith's letters to you of Oct. 15, 18, 19, 28, and Nov. 4, 15. The last of these letters has two curious discoveries. First, Mr. Smith declares that he has seen the editor of the Athenaeum: in several previous letters he mentions a name. If he knew a little of journalism he would be aware that editors are a peculiar race, obtained by natural selection. They are never seen, even by their officials; only heard down a pipe. Secondly, "an ellipse or oval" is composed of four arcs of circles. Mr. Smith has got hold of the construction I was taught, when a boy, for a pretty four-arc oval. But my teachers knew better than to call it an ellipse: Mr. Smith does not; but he produces from it such confirmation of 3-1/8 as would convince any honest editor.
Surely the cyclometer is a Darwinite development of a spider, who is always at circles, and always begins again when his web is brushed away. He informs you that he {342} has been privileged to discover truths unknown to the scientific world. This we know; but he proceeds to show that he is equally fortunate in art. He goes on to say that he will make use of you to bring those truths to light, "just as an artist makes use of a dummy for the purpose of arranging his drapery." The painter's lay-figure is for flowing robes; the hairdresser's dummy is for curly locks. Mr. James Smith should read Sam Weller's pathetic story of the "four wax dummies." As to his use of a dummy, it is quite correct. When I was at University College, I walked one day into a room in which my Latin colleague was examining. One of the questions was, "Give the lives and fates of Sp. Maelius,[644] and Sp. Cassius."[645] Umph! said I, surely all know that Spurius Maelius was whipped for adulterating flour, and that Spurius Cassius was hanged for passing bad money. Now, a robe arranged on a dummy would look just like the toga of Cassius on the gallows. Accordingly, Mr. Smith is right in the drapery-hanger which he has chosen: he has been detected in the attempt to pass bad circles. He complains bitterly that his geometry, instead of being read and understood by you, is handed over to me to be treated after my scurrilous fashion. It is clear enough that he would rather be handled in this way than not handled at all, or why does he go on writing? He must know by this time that it is a part of the institution that his "untruthful and absurd trash" shall be distilled into mine at the rate of about 3-1/8 pages of the first to one column of the second. Your readers will never know how much they gain by the process, until Mr. James Smith publishes it all in a big book, or until they get hold of what he has already published. I have six pounds avoirdupois of pamphlets and letters; and there is more than half a pound of letters {343} written to you in the last two months. Your compositor must feel aggrieved by the rejection of these clearly written documents, without erasures, and on one side only. Your correspondent has all the makings of a good contributor, except the knowledge of his subject and the sense to get it. He is, in fact, only a mask: of whom the fox
"O quanta species, inquit, cerebrum non habet."[646]
I do not despair of Mr. Smith on any question which does not involve that unfortunate two-stick wicket at which he persists in bowling. He has published many papers; he has forwarded them to mathematicians: and he cannot get answers; perhaps not even readers. Does he think that he would get more notice if you were to print him in your journal? Who would study his columns? Not the mathematician, we know; and he knows. Would others? His balls are aimed too wide to be blocked by any one who is near the wicket. He has long ceased to be worth the answer which a new invader may get. Rowan Hamilton,[647] years ago, completely knocked him over; and he has never attempted to point out any error in the short and easy method by which that powerful investigator condescended to show that, be right who may, he must be wrong. There are some persons who feel inclined to think that Mr. Smith should be argued with: let those persons understand that he has been argued with, refuted, and has never attempted to stick a pen into the refutation. He stated that it was a remarkable paradox, easily explicable; and that is all. After this evasion, Mr. James Smith is below the necessity of being told that he is unworthy of answer. His friends complain that I do nothing but chaff him. Absurd! I winnow him; and if nothing but chaff results, whose fault is that? I am usefully employed: for he is the type of a class which ought to be known, and which I have done much to make known.
{344}
Nothing came of this until July 1869, when I received a reprint of the above letter, with a comment, described as Appendix D of a work in course of publication on the geometry of the circle. The Athenaeum journal received the same: but the Editor, in his private capacity, received the whole work, being The Geometry of the Circle and Mathematics as applied to Geometry by Mathematicians, shown to be a mockery, delusion, and a snare, Liverpool, 8vo, 1869. Mr. J. S. here appears in deep fight with Professor Whitworth,[648] and Mr. Wilson,[649] the author of the alleged amendment of Euclid. How these accomplished mathematicians could be inveigled into continued discussion is inexplicable. Mr. Whitworth began by complaining of Mr. Smith's attacks upon mathematicians, continued to correspond after he was convinced that J. S. proved an arc and its chord to be equal, and only retreated when J. S. charged him with believing in 3-1/8, and refusing acknowledgment. Mr. Wilson was introduced to J. S. by a volunteer defense of his geometry from the assaults of the Athenaeum. This the editor would not publish; so J. S. sent a copy to Mr. Wilson himself. Some correspondence ensued, but Mr. Wilson soon found out his man, and withdrew.
There is a little derision of the Athenaeum and a merited punishment for "that unscrupulous critic and contemptible mathematical twaddler, De Morgan."
MR. REDDIE'S ASTRONOMY.
At p. 183 I mentioned Mr. Reddie,[650] the author of Vis Inertiae Victa and of Victoria toto coelo,[651] which last is not {345} an address to the whole heaven, either from a Roman Goddess or a British Queen, whatever a scholar may suppose. Between these Mr. Reddie has published The Mechanics of the Heavens, 8vo, 1862: this I never saw until he sent it to me, with an invitation to notice it, he very well knowing that it would catch. His speculations do battle with common notions of mathematics and of mechanics, which, to use a feminine idiom, he blasphemes so you can't think! and I suspect that if you do not blaspheme them too, you can't think. He appeals to the "truly scientific," and would be glad to have readers who have read what he controverts, i.e., Newton's Principia: I wish he may get them; I mean I hope he may obtain them. To none but these would an account of his speculations be intelligible: I accordingly disposed of him in a very short paragraph of description. Now many paradoxers desire notice, even though it be disparaging. I have letters from more than one—besides what have been sent to the Editor of the Athenaeum—complaining that they are not laughed at; although they deserve it, they tell me, as much as some whom I have inserted. Mr. Reddie informs me that I have not said a single word against his books, though I have given nearly a column to sixteen-string arithmetic, and as much to animalcule universes. What need to say anything to readers of Newton against a book from which I quoted that revolution by gravitation is demonstrably impossible? It would be as useless as evidence against a man who has pleaded guilty. Mr. Reddie derisively thanks me for "small mercies"; he wrote me private letters; he published them, and more, in the Correspondent. He gave me, pro viribus suis,[652] such a dressing you can't think, both for my Budget non-notice, and for reviews which he assumed me to have written. He outlawed himself by declaring (Correspondent, Nov. 11, 1856) that I—in a review—had made a quotation which was "garbled, evidently on purpose {346} to make it appear that" he "was advocating solely a geocentric hypothesis, which is not true." In fact, he did his best to get larger "mercy." And he shall have it; and at a length which shall content him, unless his mecometer be an insatiable apparatus. But I fear that in other respects I shall no more satisfy him than the Irish drummer satisfied the poor culprit when, after several times changing the direction of the stroke at earnest entreaty, he was at last provoked to call out, "Bad cess to ye, ye spalpeen! strike where one will, there's no plasing ye!"
Mr. Reddie attaches much force to Berkeley's[653] old arguments against the doctrine of fluxions, and advances objections to Newton's second section, which he takes to be new. To me they appear "such as have been often made," to copy a description given in a review: though I have no doubt Mr. Reddie got them out of himself. But the whole matter comes to this: Mr. Reddie challenged answer, especially from the British Association, and got none. He presumes that this is because he is right, and cannot be answered: the Association is willing to risk itself upon the counter-notion that he is wrong, and need not be answered; because so wrong that none who could understand an answer would be likely to want one.
Mr. Reddie demands my attention to a point which had already particularly struck me, as giving the means of showing to all readers the kind of confusion into which paradoxers are apt to fall, in spite of the clearest instruction. It is a very honest blunder, and requires notice: it may otherwise mislead some, who may suppose that no one able to read could be mistaken about so simple a matter, {347} let him be ever so wrong about Newton. According to his own mis-statement, in less than five months he made the Astronomer Royal abandon the theory of the solar motion in space. The announcement is made in August, 1865, as follows: the italics are not mine:
"The third (Victoria ...), although only published in September, 1863, has already had its triumph. It is the book that forced the Astronomer Royal of England, after publicly teaching the contrary for years, to come to the conclusion, "strange as it may appear," that "the whole question of solar motion in space is at the present time in doubt and abeyance." This admission is made in the Annual Report of the Council of the Royal Astronomical Society, published in the Society's Monthly Notices for February, 1864."
It is added that solar motion is "full of self-contradiction, which "the astronomers" simply overlooked, but which they dare not now deny after being once pointed out."
The following is another of his accounts of the matter, given in the Correspondent, No. 18, 1865:
"... You ought, when you came to put me in the 'Budget,' to have been aware of the Report of the Council of the Royal Astronomical Society, where it appears that Professor Airy,[654] with a better appreciation of my demonstrations, had admitted—'strange,' say the Council, 'as it may appear,'—that 'the whole question of solar motion in space [and here Mr. Reddie omits some words] is now in doubt and abeyance.' You were culpable as a public teacher of no little pretensions, if you were 'unaware' of this. If aware of it, you ought not to have suppressed such an important testimony to my really having been 'very successful' in drawing the teeth of the pegtops, though you thought them so firmly fixed. And if you still suppress {348} it, in your Appendix, or when you reprint your 'Budget,' you will then be guilty of a suppressio veri, also of further injury to me, who have never injured you...."
Mr. Reddie must have been very well satisfied in his own mind before he ventured such a challenge, with an answer from me looming in the distance. The following is the passage of the Report of the Council, etc., from which he quotes:
"And yet, strange to say, notwithstanding the near coincidence of all the results of the before-mentioned independent methods of investigation, the inevitable logical inference deduced by Mr. Airy is, that the whole question of solar motion in space, so far at least as accounting for the proper motion of the stars is concerned, [I have put in italics the words omitted by Mr. Reddie] appears to remain at this moment in doubt and abeyance."
Mr. Reddie has forked me, as he thinks, on a dilemma: if unaware, culpable ignorance; if aware, suppressive intention. But the thing is a trilemma, and the third horn, on which I elect to be placed, is surmounted by a doubly-stuffed seat. First, Mr. Airy has not changed his opinion about the fact of solar motion in space, but only suspends it as to the sufficiency of present means to give the amount and direction of the motion. Secondly, all that is alluded to in the Astronomical Report was said and printed before the Victoria proclamation appeared. So that the author, instead of drawing the tooth of the Astronomer Royal's pegtop, has burnt his own doll's nose.
William Herschel,[655] and after him about six other astronomers, had aimed at determining, by the proper motions of the stars, the point of the heavens towards which the solar system is moving: their results were tolerably accordant. Mr. Airy, in 1859, proposed an improved method, and, applying it to stars of large proper motion, produced {349} much the same result as Herschel. Mr. E. Dunkin,[656] one of Mr. Airy's staff at Greenwich, applied Mr. Airy's method to a very large number of stars, and produced, again, nearly the same result as before. This paper was read to the Astronomical Society in March, 1863, was printed in abstract in the Notice of that month, was printed in full in the volume then current, and was referred to in the Annual Report of the Council in February, 1864, under the name of "the Astronomer Royal's elaborate investigation, as exhibited by Mr. Dunkin." Both Mr. Airy and Mr. Dunkin express grave doubts as to the sufficiency of the data: and, regarding the coincidence of all the results as highly curious, feel it necessary to wait for calculations made on better data. The report of the Council states these doubts. Mr. Reddie, who only published in September, 1863, happened to see the Report of February, 1864, assumes that the doubts were then first expressed, and declares that his book of September had the triumph of forcing the Astronomer Royal to abandon the fact of motion of the solar system by the February following. Had Mr. Reddie, when he saw that the Council were avowedly describing a memoir presented some time before, taken the precaution to find out when that memoir was presented, he would perhaps have seen that doubts of the results obtained, expressed by one astronomer in March, 1863, and by another in 1859, could not have been due to his publication of September, 1863. And any one else would have learnt that neither astronomer doubts the solar motion, though both doubt the sufficiency of present means to determine its amount and direction. This is implied in the omitted words, which Mr. Reddie—whose omission would have been dishonest if he had seen their meaning—no doubt took for pleonasm, superfluity, overmuchness. The rashness which pushed him headlong {350} into the quillet that his thunderbolt had stopped the chariot of the Sun and knocked the Greenwich Phaeton off the box, is the same which betrayed him into yet grander error—which deserves the full word, quidlibet—about the Principia of Newton. There has been no change of opinion at all. When a person undertakes a long investigation, his opinion is that, at a certain date, there is prima facie ground for thinking a sound result may be obtained. Should it happen that the investigation ends in doubt upon the sufficiency of the grounds, the investigator is not put in the wrong. He knew beforehand that there was an alternative: and he takes the horn of the alternative indicated by his calculations. The two sides of this case present an instructive contrast. Eight astronomers produce nearly the same result, and yet the last two doubt the sufficiency of their means: compare them with the what's-his-name who rushes in where thing-em-bobs fear to tread.
I was not aware, until I had written what precedes, that Mr. Airy had given a sufficient answer on the point. Mr. Reddie says (Correspondent, Jan. 20, 1866):
"I claim to have forced Professor Airy to give up the notion of 'solar motion in space' altogether, for he admits it to be 'at present in doubt and abeyance.' I first made that claim in a letter addressed to the Astronomer Royal himself in June, 1864, and in replying, very courteously, to other portions of my letter, he did not gainsay that part of it."
Mr. Reddie is not ready at reading satire, or he never would have so missed the meaning of the courteous reply on one point, and the total silence upon another. Mr. Airy must be one of those peculiar persons who, when they do not think an assertion worth notice, let it alone, without noticing it by a notification of non-notice. He would never commit the bull of "Sir! I will not say a word on that subject." He would put it thus, "Sir! I will only say ten words on that subject,"—and, having thus said them, would {351} proceed to something else. He assumed, as a matter of form, that Mr. Reddie would draw the proper inference from his silence: and this because he did not care whether or no the assumption was correct.
The Mechanics of the Heavens, which Mr. Reddie sends to be noticed, shall be noticed, so far as an extract goes:
"My connection with this subject is, indeed, very simply explained. In endeavoring to understand the laws of physical astronomy as generally taught, I happened to entertain some doubt whether gravitating bodies could revolve, and having afterwards imbibed some vague idea that the laws of the universe were chemical and physical rather than mechanical, and somehow connected with electricity and magnetism as opposing correlative forces—most probably suggested to my mind, as to many others, by the transcendent discoveries made in electro-magnetism by Professor Faraday[657]—my former doubts about gravitation were revived, and I was led very naturally to try and discover whether a gravitating body really could revolve; and I became convinced it could not, before I had ever presumed to look into the demonstrations of the Principia."
This is enough against the book, without a word from me: I insert it only to show those who know the subject what manner of writer Mr. Reddie is. It is clear that "presumed" is a slip of the pen; it should have been condescended.
Mr. Reddie represents me as dreaming over paltry paradoxes. He is right; many of my paradoxes are paltry: he is wrong; I am wide awake to them. A single moth, beetle, or butterfly, may be a paltry thing; but when a cabinet is arranged by genus and species, we then begin to admire the {352} infinite variety of a system constructed on a wonderful sameness of leading characteristics. And why should paradoxes be denied that collective importance, paltry as many of them may individually be, which is accorded to moths, beetles, or butterflies? Mr. Reddie himself sees that "there is a method in" my "mode of dealing with paradoxes." I hope I have atoned for the scantiness of my former article, and put the demonstrated impossibility of gravitation on that level with Hubongramillposanfy arithmetic and inhabited atoms which the demonstrator—not quite without reason—claims for it.
In the Introduction to a collected edition of the three works, Mr. Reddie describes his Mechanism of the Heavens, from which I have just quoted, as—
"a public challenge offered to the British Association and the mathematicians at Cambridge, in August, 1862, calling upon them to point to a single demonstration in the Principia or elsewhere, which even attempts to prove that Universal Gravitation is possible, or to show that a gravitating body could possibly revolve about a center of attraction. The challenge was not accepted, and never will be. No such demonstration exists. And the public must judge for themselves as to the character of a so-called "certain science," which thus shrinks from rigid examination, and dares not defend itself when publicly attacked: also of the character of its teachers, who can be content to remain dumb under such circumstances."
ON PARADOXERS IN GENERAL.
The above is the commonplace talk of the class, of which I proceed to speak without more application to this paradoxer than to that. It reminds one of the funny young rascals who used, in times not yet quite forgotten, to abuse the passengers, as long as they could keep up with the {353} stage coach; dropping off at last with "Why don't you get down and thrash us? You're afraid, you're afraid!" They will allow the public to judge for themselves, but with somewhat of the feeling of the worthy uncle in Tom Jones, who, though he would let young people choose for themselves, would have them choose wisely. They try to be so awfully moral and so ghastly satirical that they must be answered: and they are best answered in their own division. We have all heard of the way in which sailors cat's-pawed the monkeys: they taunted the dwellers in the trees with stones, and the monkeys taunted them with cocoa-nuts in return. But these were silly dendrobats: had they belonged to the British Association they would have said—No! No! dear friends; it is not in the itinerary: if you want nuts, you must climb, as we do. The public has referred the question to Time: the procedure of this great king I venture to describe, from precedents, by an adaptation of some smart anapaestic tetrameters—your anapaest is the foot for satire to halt on, both in Greek and English—which I read about twenty years ago, and with the point of which I was much tickled. Poetasters were laughed at; but Mr. Slum, whom I employed—Mr. Charles Dickens obliged me with his address—converted the idea into that of a hit at mathematicasters, as easily as he turned the Warren acrostic into Jarley. As he observed, when I settled his little account, it is cheaper than any prose, though the broom was not stolen quite ready made:
Forty stripes save one for the smaller Paradoxers.
Hark to the wisdom the sages preach Who never have learnt what they try to teach. We are the lights of the age, they say! We are the men, and the thinkers we! So we build up guess-work the livelong day, In a topsy-turvy sort of way, Some with and some wanting a plus b. Let the British Association fuss; What are theirs to the feats to be wrought by us? {354} Shall the earth stand still? Will the round come square? Must Isaac's book be the nest of a mare? Ought the moon to be taught by the laws of space To turn half round without right-about-face? Our whimsey crotchets will manage it all; Deep! Deep! posterity will them call! Though the world, for the present, lets them fall Down! Down! to the twopenny box of the stall!
Thus they—But the marplot Time stands by, With a knowing wink in his funny old eye. He grasps by the top an immense fool's cap, Which he calls a philosophaster-trap: And rightly enough, for while these little men Croak loud as a concert of frogs in a fen, He first singles out one, and then another, Down goes the cap—lo! a moment's pother, A spirit like that which a rushlight utters As just at the last it kicks and gutters: When the cruel smotherer is raised again Only snuff, and but little of that, will remain.
But though uno avulso thus comes every day Non deficit alter is also in play: For the vacant parts are, one and all, Soon taken by puppets just as small; Who chirp, chirp, chirp, with a grasshopper's glee, We're the lamps of the Universe, We! We! We! But Time, whose speech is never long,— He hasn't time for it—stops the song And says—Lilliput lamps! leave the twopenny boxes, And shine in the Budget of Paradoxes!
When a paradoxer parades capital letters and diagrams which are as good as Newton's to all who know nothing about it, some persons wonder why science does not rise and triturate the whole thing. This is why: all who are fit to read the refutation are satisfied already, and can, if they please, detect the paradoxer for themselves. Those who are not fit to do this would not know the difference between the true answer and the new capitals and diagrams on which the delighted paradoxer would declare {355} that he had crumbled the philosophers, and not they him. Trust him for having the last word: and what matters it whether he crow the unanswerable sooner or later? There are but two courses to take. One is to wait until he has committed himself in something which all can understand, as Mr. Reddie has done in his fancy about the Astronomer Royal's change of opinion: he can then be put in his true place. The other is to construct a Budget of Paradoxes, that the world may see how the thing is always going on, and that the picture I have concocted by cribbing and spoiling a bit of poetry is drawn from life. He who wonders at there being no answer has seen one or two: he does not know that there are always fifty with equal claims, each of whom regards his being ranked with the rest as forty-nine distinct and several slanders upon himself, the great Mully Ully Gue. And the fifty would soon be five hundred if any notice were taken of them. They call mankind to witness that science will not defend itself, though publicly attacked in terms which might sting a pickpocket into standing up for his character: science, in return, allows mankind to witness or not, at pleasure, that it does not defend itself, and yet receives no injury from centuries of assault. Demonstrative reason never raises the cry of Church in Danger! and it cannot have any Dictionary of Heresies except a Budget of Paradoxes. Mistaken claimants are left to Time and his extinguisher, with the approbation of all thinking non-claimants: there is no need of a succession of exposures. Time gets through the job in his own workmanlike manner as already described.
On looking back more than twenty years, I find among my cuttings the following passage, relating to a person who had signalized himself by an effort to teach comets to the conductor of the Nautical Almanac:
"Our brethren of the literary class have not the least idea of the small amount of appearance of knowledge {356} which sets up the scientific charlatan. Their world is large, and there are many who have that moderate knowledge, and perception of what is knowledge, before which extreme ignorance is detected in its first prank. There is a public of moderate cultivation, for the most part sound in its judgment, always ready in its decisions. Accordingly, all their successful pretenders have some pretension. It is not so in science. Those who have a right to judge are fewer and farther between. The consequence is, that many scientific pretenders have nothing but pretension."
This is nearly as applicable now as then. It is impossible to make those who have not studied for themselves fully aware of the truth of what I have quoted. The best chance is collection of cases; in fact, a Budget of Paradoxes. Those who have no knowledge of the subject can thus argue from the seen to the unseen. All can feel the impracticability of the Hubongramillposanfy numeration, and the absurdity of the equality of contour of a regular pentagon and hexagon in one and the same circle. Many may accordingly be satisfied, on the assurance of those who have studied, that there is as much of impracticability, or as much of absurdity, in things which are hidden under
"Sines, tangents, secants, radius, cosines Subtangents, segments and all those signs; Enough to prove that he who read 'em Was just as mad as he who made 'em."
Not that I mean to be disrespectful to mathematical terms: they are short and easily explained, and compete favorably with those of most other subjects: for instance, with
"Horse-pleas, traverses, demurrers, Jeofails, imparlances, and errors, Averments, bars, and protestandos, And puis d'arreign continuandos."
{357}
From which it appears that, taking the selections made by satirists for our samples, there are, one with another, four letters more in a law term than in one of mathematics. But pleading has been simplified of late years.
All paradoxers can publish; and any one who likes may read. But this is not enough; they find that they cannot publish, or those who can find they are not read, and they lay their plans athwart the noses of those who, they think, ought to read. To recommend them to be content with publication, like other authors, is an affront: of this I will give the reader an amusing instance. My good nature, of which I keep a stock, though I do not use it all up in this Budget, prompts me to conceal the name.
I received the following letter, accompanied by a prospectus of a work on metaphysics, physics, astronomy, etc. The author is evidently one whom I should delight to honor:
"Sir,—A friend of mine has mentioned your name in terms of panigeric [sic], as being of high standing in mathematics, and of greatly original thought. I send you the enclosed without comment; and, assuming that the bent of your mind is in free inquiry, shall feel a pleasure in showing you my portfolio, which, as a mathematician, you will acknowledge to be deeply interesting, even in an educational point of view. The work is complete, and the system so far perfected as to place it above criticism; and, so far as regards astronomy, as will Ptolemy beyond rivalry [sic: no doubt some words omitted]. Believe me to be, Sir, with the profoundest respect, etc. The work is the result of thirty-five years' travel and observation, labor, expense, and self-abnegation."
I replied to the effect that my time was fully occupied, and that I was obliged to decline discussion with many persons who have views of their own; that the proper way is to publish, so that those who choose may read when they can find leisure. I added that I should advise a precursor in the shape of a small pamphlet, as two octavo volumes {358} would be too much for most persons. This was sound advice; but it is not the first, second, or third time that it has proved very unpalatable. I received the following answer, to which I take the liberty of prefixing a bit of leonine wisdom:
"Si doceas stultum, laetum non dat tibi vultum; Odit te multum; vellet te scire sepultum.[658]"
"Sir,—I pray you pardon the error I unintentionally have fallen into; deceived by the F.R.S. [I am not F.R.S.] I took you to be a man of science [omnis homo est animal, Sortes est homo, ergo Sortes est animal][659] instead of the mere mathematician, or human calculating-machine. Believe me, Sir, you also have mistaken your mission, as I have mine. I wrote to you as I would to any other man well up in mathematics, with the intent to call your attention to a singular fact of omission by Euclid, and other great mathematicians: and, in selecting you, I did you an honor which, from what I have just now heard, was entirely out of place. I think, considering the nature of the work set forth in the prospectus, you are guilty of both folly and presumption, in assuming the character of a patron; for your own sense ought to have assured you that was such my object I should not have sought him in a De Morgan, who exists only by patronage of others. On the other hand, I deem it to be an unpardonable piece of presumption in offering your advice upon a subject the magnitude, importance, and real utility of which you know nothing about: by doing so you have offered me a direct insult. The system is a manual of Philosophy, a one inseparable whole of metaphysics and physic; embracing points the most interesting, laws the most important, {359} doctrines the most essential to advance man in accordance with the spirit of the times. I may not live to see it in print; for, at ——, life at best is uncertain: but, live or die, be assured Sir, it is not my intention to debase the work by seeking patronage, or pandering to the public taste. Your advice was the less needed, seeing I am an old-established ——. I remain, etc.—P.S. You will oblige me by returning the prospectus of my work."
My reader will, I am sure, not take this transition from the "profoundest respect" to the loftiest insolence for an apocraphical correspondence, to use a word I find in the Prospectus: on my honor it is genuine. He will be better employed in discovering whether I exist by patronizing others, or by being patronized by them. I make any one who can find it out a fair offer: I will give him my patronage if I turn out to be Bufo, on condition he gives me his, if I turn out to be Bavius.[660] I need hardly say that I considered the last letter to be one of those to which no answer is so good as no answer.
These letters remind me in one respect of the correspondents of the newspapers. My other party wrote because a friend had pointed me out: but he would not have written if he had known what another friend told him just in time for the second letter. The man who sends his complaint to the newspaper very often says, in effect, "Don't imagine, Sir, that I read your columns; but a friend who sometimes does has told me ..." It is worded thus: "My attention {360} has been directed to an article in your paper of ..." Many thanks to my friend's friends for not mentioning the Budget: had my friend's attention been directed to it I might have lost a striking example of the paradoxer in search of a patron. That my Friend was on this scent in the first letter is revealed in the second. Language was given to man to conceal his thoughts; but it is not every one who can do it.
Among the most valuable information which my readers will get from me is comparison of the reactions of paradoxers, when not admitted to argument, or when laughed at. Of course, they are misrepresented; and at this they are angry, or which is the same thing, take great pains to assure the reader that they are not. So far natural, and so far good; anything short of concession of a case which must be seriously met by counter-reasons is sure to be misrepresentation. My friend Mr. James Smith and my friend Mr. Reddie are both terribly misrepresented: they resent it by some insinuations in which it is not easy to detect whether I am a conscious smotherer of truth, or only muddle-headed and ignorant. [This was written before I received my last communication from Mr. James Smith. He tells me that I am wrong in saying that his work in which I stand in the pillory is all reprint: I have no doubt I confounded some of it with some of the manuscript or slips which I had received from my much not-agreed-with correspondent. He adds that my mistake was intentional, and that my reason is obvious to the reader. This is information, as the sea-serpent said when he read in the newspaper that he had a mane and tusks.]
THE DOUBLE VAHU PROCESS.
My friend Dr. Thorn[661] sees deeper into my mystery. By the way, he still sends an occasional touch at the old {361} subject; and he wants me particularly to tell my readers that the Latin numeral letters, if M be left out, give 666. And so they do: witness DCLXVI. A person who thinks of the origin of symbols will soon see that 666 is our number because we have five fingers on each hand: had we had but four, our mystic number would have been expressed by 555, and would have stood for our present 365. Had n been the number on each hand, the great number would have been
(n + 1) (4n^2 + 2n + 1)
With no finger on each hand, the number would have been 1: with one finger less than none at all on each hand, it would have been 0. But what does this mean? Here is a question for an algebraical paradoxer! So soon as we have found out how many fingers the inhabitants of any one planet have on each hand, we have the means of knowing their number of the Beast, and thence all about them. Very much struck with this hint of discovery, I turned my attention to the means of developing it. The first point was to clear my vision of all the old cataracts. I propose the following experiment, subject of course to the consent of parties. Let Dr. Thorn Double-Vahu Mr. James Smith, and Thau Mr. Reddie: if either be deparadoxed by the treatment, I will consent to undergo it myself. Provided always that the temperature required be not so high as the Doctor hints at: if the Turkish Baths will do for this world, I am content.
The three paradoxers last named and myself have a pentasyllable convention, under which, though we go far beyond civility, we keep within civilization. Though Mr. James Smith pronounced that I must be dishonest if I did not see his argument, which he knew I should not do [to say nothing of recent accusation]; though Dr. Thorn declared me a competitor for fire and brimstone—and my wife, too, which doubles the joke: though Mr. Reddie {362} was certain I had garbled him, evidently on purpose to make falsehood appear truth; yet all three profess respect for me as to everything but power to see truth, or candor to admit it. And on the other hand, though these were the modes of opening communication with me, and though I have no doubt that all three are proper persons of whom to inquire whether I should go up-stairs or down-stairs, etc., yet I am satisfied they are thoroughly respectable men, as to everything but reasoning. And I dare say our several professions are far more true in extent than in many which are made under more parliamentary form. We find excuses for each other: they make allowances for my being hoodwinked by Aristotle, by Newton, by the Devil; and I permit them to feel, for I know they cannot get on without it, that their reasons are such as none but a knave or a sinner can resist. But they are content with cutting a slice each out of my character: neither of them is more than an uncle, a Bone-a-part; I now come to a dreadful nephew, Bone-the-whole.
I will not give the name of the poor fellow who has fallen so far below both the honestum and the utile, to say nothing of the decorum or the dulce.[662] He is the fourth who has taken elaborate notice of me; and my advice to him would be, Nec quarta loqui persona laboret.[663] According to him, I scorn humanity, scandalize learning, and disgrace the press; it admits of no manner of doubt that my object is to mislead the public and silence truth, at the expense of the interests of science, the wealth of the nation, and the lives of my fellow men. The only thing left to be settled is, whether this is due to ignorance, natural distaste for truth, personal malice, a wish to curry favor with the Astronomer Royal, or mere toadyism. The only accusation which has truth in it is, that I have made myself a "public scavenger of science": the assertion, which is the {363} most false of all is, that the results of my broom and spade are "shot right in between the columns of" the Athenaeum. I declare I never in my life inserted a word between the columns of the Athenaeum: I feel huffed and miffed at the very supposition. I have made myself a public scavenger; and why not? Is the mud never to be collected into a heap? I look down upon the other scavengers, of whom there have been a few—mere historical drudges; Montucla, Hutton, etc.—as not fit to compete with me. I say of them what one crossing-sweeper said of the rest: "They are well enough for the common thing; but put them to a bit of fancy-work, such as sweeping round a post, and see what a mess they make of it!" Who can touch me at sweeping round a paradoxer? If I complete my design of publishing a separate work, an old copy will be fished up from a stall two hundred years hence by the coming man, and will be described in an article which will end by his comparing our century with his own, and sighing out in the best New Zealand pronunciation—
"Dans ces tems-la C'etait deja comme ca!"[664]
ORTHODOX PARADOXERS.
And pray, Sir! I have been asked by more than one—do your orthodox never fall into mistake, nor rise into absurdity? They not only do both, but they admit it of each other very freely; individually, they are convinced of sin, but not of any particular sin. There is not a syndoxer among them all but draws his line in such a way as to include among paradoxers a great many whom I should exclude altogether from this work. My worst specimens are but exaggerations of what may be found, occasionally, in the thoughts of sagacious investigators. At the end of the {364} glorious dream, we learn that there is a way to Hell from the gates of Heaven, as well as from the City of Destruction: and that this is true of other things besides Christian pilgrimage is affirmed at the end of the Budget of Paradoxes. If D'Alembert[665] had produced enough of a quality to match his celebrated mistake on the chance of throwing head in two throws, he would have been in my list. If Newton had produced enough to match his reception of the story that Nausicaa, Homer's Phaeacian princess, invented the celestial sphere, followed by his serious surmise that she got it from the Argonauts,—then Newton himself would have had an appearance entered for him, in spite of the Principia. In illustration, I may cite a few words from Tristram Shandy:
"'A soldier,' cried my uncle Toby, interrupting the Corporal, 'is no more exempt from saying a foolish thing, Trim, than a man of letters.'—'But not so often, an' please your honor,' replied the Corporal. My uncle Toby gave a nod."
I now proceed to die out. Some prefatory remarks will follow in time.[666] I shall have occasion to insist that all is not barren: I think I shall find, on casting up, that two out of five of my paradoxers are not to be utterly condemned. Among the better lot will be found all gradations of merit; at the same time, as was remarked on quite a different subject, there may be little to choose between the last of the saved and the first of the lost. The higher and better class is worthy of blame; the lower and worse class is worthy of praise. The higher men are to be reproved for not taking up things in which they could do some good: the lower men are to be commended for taking up things in which they can do no great harm. The circle problem is like Peter Peebles's lawsuit:
{365}
"'But, Sir, I should really spoil any cause thrust on me so hastily.'—'Ye cannot spoil it, Alan,' said my father, 'that is the very cream of the business, man,—... the case is come to that pass that Stair or Arniston could not mend it, and I don't think even you, Alan, can do it much harm.'"
I am strongly reminded of the monks in the darker part of the Middle Ages. To a certain proportion of them, perhaps two out of five, we are indebted for the preservation of literature, and their contemporaries for good teaching and mitigation of socials evils. But the remaining three were the fleas and flies and thistles and briars with whom the satirist lumps them, about a century before the Reformation:
"Flen, flyys, and freris, populum domini male caedunt; Thystlis and breris crescentia gramina laedunt. Christe nolens guerras qui cuncta pace tueris, Destrue per terras breris, flen, flyys, and freris. Flen, flyys, and freris, foul falle hem thys fyften yeris, For non that her is lovit flen, flyys ne freris."[667]
I should not be quite so savage with my second class. Taken together, they may be made to give useful warning to those who are engaged in learning under better auspices: aye, even useful hints; for bad things are very often only good things spoiled or misused. My plan is that of a predecessor in the time of Edward the Second:
"Meum est propositum genti imperitae Artes frugi reddere melioris vitae."[668]
To this end I have spoken with freedom of books as books, of opinions as opinions, of ignorance as ignorance, of {366} presumption as presumption; and of writers as I judge may be fairly inferred from what they have written. Some—to whom I am therefore under great obligation—have permitted me to enlarge my plan by assaults to which I have alluded; assaults which allow a privilege of retort, of which I have often availed myself; assaults which give my readers a right of partnership in the amusement which I myself have received.
For the present I cut and run: a Catiline, pursued by a chorus of Ciceros, with Quousque tandem? Quamdiu nos? Nihil ne te?[669] ending with, In te conferri pestem istam jam pridem oportebat, quam tu in nos omnes jamdiu machinaris! I carry with me the reflection that I have furnished to those who need it such a magazine of warnings as they will not find elsewhere; a signatis cavetote:[670] and I throw back at my pursuers—Valete, doctores sine doctrina; facite ut proxima congressu vos salvos corporibus et sanos mentibus videamus.[671] Here ends the Budget of Paradoxes.
{367}
* * * * *
APPENDIX.
I think it right to give the proof that the ratio of the circumference to the diameter is incommensurable. This method of proof was given by Lambert,[672] in the Berlin Memoirs for 1761, and has been also given in the notes to Legendre's[673] Geometry, and to the English translation of the same. Though not elementary algebra, it is within the reach of a student of ordinary books.[674]
Let a continued fraction, such as
a ——- b + c ——- d + e - f + etc.,
be abbreviated into a/b+ c/d+ e/f+ etc.: each fraction being understood as falling down to the side of the preceding sign +. In every such fraction we may suppose b, d, f, etc. {368} positive; a, c, e, &c. being as required: and all are supposed integers. If this succession be continued ad infinitum, and if a/b, c/d, e/f, etc. all lie between -1 and +1, exclusive, the limit of the fraction must be incommensurable with unity; that is, cannot be A/B, where A and B are integers.
First, whatever this limit may be, it lies between -1 and +1. This is obviously the case with any fraction p/(q + [omega]), where [omega] is between +-1: for, p/q, being < 1, and p and q integer, cannot be brought up to 1, by the value of [omega]. Hence, if we take any of the fractions
a/b, a/b+ c/d, a/b+ c/d+ e/f, etc.
say a/b+ c/d+ e/f+ g/h we have, g/h being between +-1, so is e/f+ g/h, so therefore is c/d+ e/f+ g/h; and so therefore is a/b+ c/d+ e/f+ g/h.
Now, if possible, let a/b+ c/d+ etc. be A/B at the limit; A and B being integers. Let
P = A c/d+ e/f+ etc., Q = P e/f+ g/h+ etc., R = Q g/h + i/k + etc.
P, Q, R, etc. being integer or fractional, as may be. It is easily shown that all must be integer: for
{369}
A/B = a/b+ P/A, or, P = aB - bA
P/A = c/d+ Q/P, or, Q = cA - dP
Q/P = e/f+ R/Q, or, R = eP - fQ
etc., etc. Now, since a, B, b, A, are integers, so also is P; and thence Q; and thence R, etc. But since A/B, P/A, Q/P, R/Q, etc. are all between -1 and +1, it follows that the unlimited succession of integers P, Q, R, are each less in numerical value than the preceding. Now there can be no such unlimited succession of descending integers: consequently, it is impossible that a/b+ c/d+, etc. can have a commensurable limit.
It easily follows that the continued fraction is incommensurable if a/b, c/d, etc., being at first greater than unity, become and continue less than unity after some one point. Say that i/k, l/m,... are all less than unity. Then the fraction i/k+ l/m+ ... is incommensurable, as proved: let it be [kappa]. Then g/(h + [kappa]) is incommensurable, say [lambda]; e/(f + [lambda]) is the same, say [mu]; also c/(d + [mu]), say [nu], and a/(b + [nu]), say [rho]. But [rho] is the fraction a/b+ c/d+ ... itself; which is therefore incommensurable.
Let [phi]z represent
a a^2 a^3 1 + - + ———- + ——————— + .... z 2z(z+1) 2.3.z(z+1)(z+2)
{370} Let z be positive: this series is convergent for all values of a, and approaches without limit to unity as z increases without limit. Change z into z + 1, and form [phi]z - [phi](z+1): the following equation will result—
a [phi]z-[phi](z+1) = ———([phi](z+2)) z(z+1)
a [phi](z+1) a [phi](z+1) a [phi](z+2) or a = - ————— . z + - ————— . —- ————— z [phi]z z [phi]z z+1 [phi](z+1)
a = [psi]z(z+[psi](z+1))
[psi]z being (a/z)([phi](z+1)/[phi]z); of which observe that it diminishes without limit as z increases without limit. Accordingly, we have
[psi]z = a/z+ [psi](z+1) = a/z+ a/(z+1)+ [psi](z+2) = a/z+ a/(z+1)+ a/(z+2)+ [psi](z+3), etc.
And, [psi](z + n) diminishing without limit, we have
a/z . [phi](z+1)/[phi]z = (a/z+) (a/(z+1)+) (a/(z+2)+) (a/((z+3)+ ...))
Let z = 1/2; and let 4a = -x^2. Then (a/z)[phi](z+1) is -(x^2/2) ( 1 - x^2/(2.3) + x^4/(2.3.4.5...)) or -(x/2) sin x. Again [phi]z is 1 - x^2/2 + x^4/(2.3.4) or cos x: and the continued fraction is
(1/4)x^2/(1/2)+ (1/4)x^2/(3/2)+ (1/4)x^2/(5/2)+ ... or -x/2 x/1+ -x^2/3+ -x^2/5+ ...
{371} whence tan x = x/1+ -x^2/3+ -x^2/5+ -x^2/7+ ...
Or, as written in the usual way,
tan x = x ———- 1 - x^2 ———- 3 - x^2 ———- 5 - x^2 ———- 7 - ...
This result may be proved in various ways: it may also be verified by calculation. To do this, remember that if
a1/b1+ a2/b2+ a3/b3+ ... an/bn = Pn/Qn; then
P_1=a_1, P_2=b_2 P_1, P_3=b_3 P_2+a_3 P_1, P_4=b_4 P_3+a_4 P_2, etc. Q_1=b_1, Q_2=b_2 Q_1+a_2, Q_3=b_3 Q_2+a_3 Q_1, Q_4=b_4 Q_3+a_4 Q_2, etc.
in the case before us we have
a1=x, a2=-x^2, a3=-x^2, a4=-x^2, a5=-x^2, etc. b1=1, b2=3, b3=5, b4=7, b5=9, etc.
P1=x Q1=1 P2=3x Q2=3-x^2 P3=15x-x^3 Q3=15-6x^2 P4=105x-10x^3 Q4=105-45x^2+x^4 P5=945x-105x^3+x^5 Q5=945-420x^2+15x^4 P6=10395x-1260x^3+21x^5 Q6=10395-4725x^2+210x^4-x^6
We can use this algebraically, or arithmetically. If we divide Pn by Qn, we shall find a series agreeing with the known series for tan x, as far as n terms. That series is
x + x^3/3 + 2x^5/15 + 17x^7/315 + 62x^9/2835 + ...
{372} Take P5, and divide it by Q5 in the common way, and the first five terms will be as here written. Now take x = .1, which means that the angle is to be one tenth of the actual unit, or, in degrees 5 deg..729578. We find that when x = .1, P6 = 1038.24021, Q6 = 10347.770999; whence P6 divided by Q6 gives .1003346711. Now 5 deg..729578 is 5 deg.43'46-1/2"; and from the old tables of Rheticus[675]—no modern tables carry the tangents so far—the tangent of this angle is .1003347670.
Now let x = (1/4)[pi]; in which case tan x = 1. If (1/4)[pi] be commensurable with the unit, let it be (m/n), m and n being integers: we know that (1/4)[pi] < 1. We have then
1=(m/n)/1- (m^2/n^2)/3- (m^2/n^2)/5- ... = m/n- m^2/3n- m^2/5n- m^2/7n- ...
Now it is clear that m^2/3n, m^2/5n, m^2/7n, etc. must at last become and continue severally less than unity. The continued fraction is therefore incommensurable, and cannot be unity. Consequently [pi]^2 cannot be commensurable: that is, [pi] is an incommensurable quantity, and so also is [pi]^2.
I thought I should end with a grave bit of appendix, deeply mathematical: but paradox follows me wherever I go. The foregoing is—in my own language—from Dr. (now Sir David) Brewster's[676] English edition of Legendre's Geometry, (Edinburgh, 1824, 8vo.) translated by some one who is not named. I picked up a notion, which others had at Cambridge in 1825, that the translator was the late Mr. Galbraith,[677] then known at Edinburgh as a writer and teacher.
{373} But it turns out that it was by a very different person, and one destined to shine in quite another walk; it was a young man named Thomas Carlyle.[678] He prefixed, from his own pen, a thoughtful and ingenious essay on Proportion, as good a substitute for the fifth Book of Euclid as could have been given in the space; and quite enough to show that he would have been a distinguished teacher and thinker on first principles. But he left the field immediately.
* * * * *
(The following is the passage referred to at Vol. II, page 54.)
Michael Stifelius[679] edited, in 1554, a second edition of the Algebra (Die Coss.), of Christopher Rudolff.[680] This is one of the earliest works in which + and - are used.
Stifelius was a queer man. He has introduced into this very work of Rudolff his own interpretation of the number of the Beast. He determined to fix the character of Pope Leo: so he picked the numeral letters from LEODECIMVS, and by taking in X from LEO X. and striking out M as standing for mysterium, he hit the number exactly. This discovery completed his conversion to Luther, and his determination to throw off his monastic vows. Luther dealt with him as straight-forwardly as with Melanchthon about his astrology: he accepted the conclusions, but told him to clear his mind of all the premises about the Beast. Stifelius {374} did not take the advice, and proceeded to settle the end of the world out of the prophet Daniel: he fixed on October, 1533. The parishioners of some cure which he held, having full faith, began to spend their savings in all kinds of good eating and drinking; we may charitably hope this was not the way of preparing for the event which their pastor pointed out. They succeeded in making themselves as fit for Heaven as Lazarus, so far as beggary went: but when the time came, and the world lasted on, they wanted to kill their deceiver, and would have done so but for the interference of Luther. {375}
* * * * *
INDEX.
Pages denoted by numerals of this kind (287) refer to biographical notes, chiefly by the editor. Numerals like 426 refer to books discussed by De Morgan, or to leading topics in the text. Numerals like 126 indicate minor references.
Abbott, Justice, I, 181. Abernethy, J., II, 219. Aboriginal Britons, a poem, II, 270. Academy of Sciences, French, I, 163. Adair, J., I, 223. Adam, M., I, 65. Adams, J. C., I, 43, 82, 385, 388; II, 131, 135, 140, 303. Ady, Joseph, II, 42, 42. Agnew, H. C., I, 328. Agricola, J., I, 394. Agricultural Laborer's letter, II, 16. Agrippa, H. C., I, 48, 48. Ainsworth, W. H., II, 132. Airy, I, 85, 88, 152, 242; II, 85, 140, 150, 303, 347. Alchemy, I, 125. Alfonso X (El Sabio), II, 269. Alford, H., II, 221. Alfred, King, Ballad of, II, 22. Algebra, I, 121. Algebraic symbols, I, 121. Almanac, I, 300; II, 147, 148, 207. (See Easter.) Aloysius Lilius, I, 362. Alsted, J. H., II, 282. Ameen Bey, II, 15. Amicable Society, I, 347. Ampere, I, 86. Amphisbaena serpent, I, 31. Anagrams, De Morgan, I, 138. Anaxagoras, II, 59. Anghera, II, 60, 60, 61, 279. Annuities, Fallacies of, I, 157. Antichrist, I, 130. Antimony, I, 125. Antinewtonism, I, 162. Antinomians, I, 394. Antiphon, II, 59. Antonie, F., I, 126, 126. Apollonius, I, 41, 107. Apparitions, II, 47. Arago, I, 243, 390. Aratus, II, 167. Arbuthnot, I, 133, 134. Archer, H., II, 90. Archimedes, I, 5, 11, 42, 83, 107. Archytas, I, 53. Argoli, I, 104. Aristocrat, as a scientist, I, 131. Aristotle, I, 5, 331. Arnobius, II, 73. Arson, P. J., II, 207. Ashton, R., II, 99. Astrology, I, 118, 127, 128, 350; II, 43. Astronomer's Drinking Song, I, 380. Astronomical Aphorisms, I, 398. Paradox, I, 394. Police Report, I, 390. Society. (See Royal Astronomical Society.) Astronomy, Bailly's exaggerated view of, I, 166. Astunica, Didacas, I, 90. Athanasian Creed, I, 371. Atheists, Philosophical, I, 1. Atoms, II, 191. {376} Attraction, I, 136, 151, 155. Augustine, St., II, 23. Aurora borealis, I, 134. Austen, Jane, I, 191. Auzout, A., II, 300. Aviation, Early ideas of, II, 8.
Babbage, C., I, 207, 290, 291; II, 181. Bachet, de Meziriac, I, 161. Bacon, F., I, 5, 75, 75, 76, 79, 89, 145, 331. Bacon, R., I, 5, 126, 126, 360; II, 94. Baconian controversy, I, 141. Baden Powell, II, 267. Bailly, J. S., I, 166, 166, 308. Baily, F., I, 308, 309; II, 16, 143, 188. Baily, R., II, 16. Baker, T., II, 302. Bakewell, F. C., II, 156, 156. Banks, J., I, 28. Barberini, M., II, 267. Barker, C., II, 262. Baronius, I, 33, 35; II, 62. Barreme, I, 42. Barrett, G., II, 188. Barrow, I., I, 160; II, 302. Baruel, de, I, 165. Bassano, Duc de, II, 3, 339. Baxter, T., I, 146. Bayle, P., II, 73. Beaufort, F., II, 267. Beaugrand, I, 119, 121. Beaulieu, I, 119, 119, 121. Beaune, de, II, 59. Becourt, R., II, 277. Bedford, Duke of, (6th), I, 182. Behmen, I, 168, 254; II, 317. Bellenden, W., I, 175. Bentley, I, 110. Berkeley, G., II, 346. Bernard, E., II, 297, 300. Bernardus Trevisanus, I, 126, 126. Bernoullis, I, 130, 150, 335, 336. Bertius, P., II, 300. Bese, I, 66. Bessel, I, 384; II, 2. Bethune, I, 99, 279, 291. Bettesworth, I, 19. Beza. (See Bese.) Bickersteth, E. H., I, 238. Bidder, I, 86. Biden, J., II, 158, 160. Bidle, (Biddle), I, 239. Biot, I, 85. Birch, T., I, 108; II, 304, 313. Birks, T. R., II, 158, 158. Bishop, G., I, 386. Bishops as Paradoxers, I, 226. Boccaccio, I, 38. Boethius, I, 42, 45. Boehme. (See Behmen.) Boncompagni, I, 298. Boniface, St., I, 32. Bonnycastle, J., II, 16. Booker, I, 115. Boole, G., I, 261, 332; II, 75, 79. —A tribute to, II, 79. Borelli, G. A., II, 300. Borello, I, 69. Boreman, I, 113. Borron, Mrs., II, 7. Boscovich, I, 156, 164. Bouguer, II, 301. Bouillaud, I, 87; II, 295. Bouvard, A., I, 327. Bovillus, I, 44; II, 324. —Epitome of, I, 44. Bowdler, H. M., I, 194. Bowring, J., I, 352; II, 256. Boyle, R., I, 24, 125; II, 300. Bradley, I, 24. Bradwardine, I, 227, 228, 229. Brahe. (See Tycho B.) Brancker, I, 107; II, 300. Brenan, J., I, 330, 330. Brewster, D., I, 39, 137, 140; II, 214, 288, 372. Briggs, I, 69; II, 299, 302. Bright, J., II, 235. Brinkley, J., I, 311. Britannicus, D., II, 8. British Museum library, I, 151. Brothers, R., I, 315; II, 97. Brougham, Henry, Lord, I, 191. Brouncker (Brounker), I, 132; II, 302. Brown, W., II, 168. Browne, T., I, 31. Brucker, I, 61. Brunet, I, 402. Bruennow, I, 386. Bruno, I, 59, 59. Bryson, II, 59. Buergi, I, 52. Buffon, I, 282. Bulstrode, II, 84. Bungus, I, 55, 55, 57. Buoncompagno, U., I, 362. {377} Burgon, J. W., II, 30. Buridan, I, 37. —Questiones morales, I, 37. Buridan's Ass, I, 37. Burke, E., I, 173. Burlesque, Frend's, I, 208. Burnet, G., I, 107. Burney, Frances, I, 190. Burton, Frances B., I, 374. Busby, R., II, 313. Buteo, I, 51. Butler, G., I, 199. Butler, S., II, 218. Buxton, J., I, 86. Byrgius. (See Buergi.) Byrne, O., I, 329; II, 186, 190. Byron, I, 186; II, 270, 273. |
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