{155}

I had written as far as damnatum when in came the letter of Nauticus as a printed slip, with a request that I would consider the slip as a 'revised copy.' Not a word of alteration in the part I have quoted! And in the evening came a letter desiring that I would alter a gross error; but not the one above: this is revising without revision! If there were cyclometers enough of this stamp, they would, as cultivation progresses—and really, with John Stuart Mill in for Westminster, it seems on the move, even though, as I learn while correcting the proof, Gladstone be out from Oxford; for Oxford is no worse than in 1829, while Westminster is far above what she ever has been: election time excuses even such a parenthesis as this—be engaged to amuse those who can afford it with paralogism at their meals, after the manner of the other jokers who wore the caps and bells. The rich would then order their dinners with panem et Circenses,—up with the victuals and the circle-games—as the poor did in the days of old.

Mr. Smith is determined that half a column shall not do. Not a day without something from him: letter, printed proof, pamphlet. In what is the last at this moment of writing he tells me that part of the title of a work of his will be "Professor De Morgan in the pillory without hope of escape." And where will he be himself? This I detected by an effort of reasoning which I never could have made except by following in his steps. In all matters connected with [pi] the letters l and g are closely related: this appears in the well-known formula for the time of oscillation [pi] [sqrt](l : g). Hence g may be written for l, but only once: do it twice, and you require the time to be [pi] [sqrt](l^2 : g^2). This may be reinforced by observing that if as a datum, or if you dislike that word, by hypothesis, the first l be a g, it is absurd that it should be an l. Write g for the first l, and we have un fait accompli. I shall be in pillory; and overhead, in a cloud, will sit Mr. James Smith on one stick laid across two others, under a nimbus of 3-1/8 diameters to {156} the circumference—in [pi]-glory. Oh for a drawing of this scene! Mr. De Morgan presents his compliments to Mr. James Smith, and requests the honor of an exchange of photographs.

July 26.—Another printed letter.—Mr. James Smith begs for a distinct answer to the following plain question: "Have I not in this communication brought under your notice truths that were never before dreamed of in your geometrical and mathematical philosophy?" To which, he having taken the precaution to print the word truths in italics, I can conscientiously answer, Yes, you have. And now I shall take no more notice of these truths, until I receive something which surpasses all that has yet been done.



A FEW SMALL PARADOXERS.

The Circle secerned from the Square; and its area gauged in terms of a triangle common to both. By Wm. Houlston,[274] Esq. London and Jersey, 1862, 4to.

Mr. Houlston squares at about four poetical quotations in a page, and brings out [pi] = 3.14213.... His frontispiece is a variegated diagram, having parts designated Inigo and Outigo. All which relieves the subject, but does not remove the error.



Considerations respecting the figure of the Earth.... By C. F. Bakewell.[275] London, 1862, 8vo.

Newton and others think that in a revolving sphere the {157} loose surface matter will tend to the equator: Mr. Bakewell thinks it will tend to the poles.



On eccentric and centric force: a new theory of projection. By H. F. A. Pratt, M.D.[276] London, 1862, 8vo.

Dr. Pratt not only upsets Newton, but cuts away the very ground he stands on: for he destroys the first law of motion, and will not have the natural tendency of matter in motion to be rectilinear. This, as we have seen, was John Walsh's[277] notion. In a more recent work "On Orbital Motion," London, 1863, 8vo., Dr. Pratt insists on another of Walsh's notions, namely, that the precession of the equinoxes is caused by the motion of the solar system round a distant central sun. In this last work the author refers to a few notes, which completely destroy the theory of gravitation in terms "perfectly intelligible as well to the unlearned as to the learned": to me they are quite unintelligible, which rather tends to confirm a notion I have long had, that I am neither one thing nor the other. There is an ambiguity of phrase which delights a writer on logic, always on the look-out for specimens of homonymia or aequivocatio. The author, as a physician, is accustomed to "appeal from mere formulae": accordingly, he sets at nought the whole of the mathematics, which he does not understand. This equivocation between the formula of the physician and that of the mathematician is as good, though not so perceptible to the world at large, as that made by Mr. Briggs's friend in Punch's picture, which I cut out to paste into my Logic. Mr. Briggs wrote for a couple of bruisers, meaning to prepare oats for his horses: his friend sent him the Whitechapel Chicken and the Bayswater Slasher, with the gloves, all ready.

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On matter and ether, and the secret laws of physical change. By T. R. Birks, M.A.[278] Cambridge, 1862, 8vo.

Bold efforts are made at molecular theories, and the one before me is ably aimed. When the Newton of this subject shall be seated in his place, books like the present will be sharply looked into, to see what amount of anticipation they have made.



DR. THORN AND MR. BIDEN.

The history of the 'thorn tree and bush' from the earliest to the present time: in which is clearly and plainly shown the descent of her most gracious Majesty and her Anglo-Saxon people from the half tribe of Ephraim, and possibly from the half tribe of Manasseh; and consequently her right and title to possess, at the present moment, for herself and for them, a share or shares of the desolate cities and places in the land of their forefathers! By Theta, M.D.[279] (Private circulation.) London, 1862, 8vo.

This is much about Thorn, and its connected words, Thor, Thoth, Theta, etc. It is a very mysterious vagary. The author of it is the person whom I have described elsewhere as having for his device the round man in the three-cornered hole, the writer of the little heap of satirical anonymous letters about the Beast and 666. By accident I discovered the writer: so that if there be any more thorns to crackle under the pot, they need not be anonymous.

Nor will they be anonymous. Since I wrote the above, I have received onymous letters, as ominous as the rest. The writer, William Thorn, M.D., is obliged to reveal {159} himself, since it is his object to prove that he himself is one 666. By using W for double Vau (or 12) he cooks the number out of his own name. But he says it is the number not of a beast but of a man, and adds, "Thereby hangs a tale!" which sounds like contradiction. He informs me that he will talk the matter over with me: but I shall certainly have nothing to say to a gentleman of his number; it is best to keep on the safe side.

In one letter I am informed that not a line should I have had, but for my "sneer at 666," which, therefore, I am well pleased to have given. I am also told that my name means the "'garden of death,' that place in which the tree of knowledge was plucked, and so you are like your name 'dead' to the fact that you are an Israelite, like those in Ezekiel 37 ch." Some hints are given that I shall not fare well in the next world, which any one who reads the chapter in Ezekiel will see is quite against his comparison. The reader must not imagine that my prognosticator means Morgan to be a corruption of Mortjardin; he proves his point by Hebrew: but any philologist would tell him the true derivation of the name, and how Glamorgan came to get it. It will be of much comfort to those young men who have not got through to know that the tree of knowledge itself was once in the same case. And so good bye to 666 for the present, and the assumption that the enigma is to be solved by the united numeral forces of the letters of a word.

It is worthy of note that, as soon as my Budget commenced, two guardian spirits started up, fellow men as to the flesh, both totally unknown to me: they have stuck to me from first to last. James Smith, Esq., finally Nauticus, watches over my character in this world, and would fain preserve me from ignorance, folly, and dishonesty, by inclosing me in a magic circle of 3-1/8 diameters in circumference. The round man in the three-cornered hole, finally William Thorn, M.D., takes charge of my future destiny, {160} and tries to bring me to the truth by unfolding a score of meanings—all right—of 666. He hints that I, and my wife, are servants of Satan: at least he desires us both to remember that we cannot serve God and Satan; and he can hardly mean that we are serving the first, and that he would have us serve the second. As becomes an interpreter of the Apocalypse, he uses seven different seals; but not more than one to one letter. If his seals be all signet-rings, he must be what Aristophanes calls a sphragidonychargocometical fellow. But—and many thanks to him for the same—though an M.D., he has not sent me a single vial. And so much for my tree of secular knowledge and my tree of spiritual life: I dismiss them with thanks from myself and thanks from my reader. The dual of the Pythagorean system was Isis and Diana; of the Jewish law, Moses and Aaron; and of the City of London, Gog and Magog; of the Paradoxiad, James Smith, Esq., and William Thorn, M.D.

September, 1866. Mr. James Biden[280] has favored me with some of his publications. He is a rival of Dr. Thorn; a prophet by name-right and crest-right. He is of royal descent through the De Biduns. He is the watchman of Ezekiel: God has told him so. He is the author of The True Church, a phrase which seems to have a book-meaning and a mission-meaning. He shall speak for himself:

"A crest of the Bidens has significance. It is a lion rampant between wings—wings in Scripture denote the flight of time. Thus the beasts or living creatures of the Revelations have each six wings, intimating a condition of mankind up to and towards the close of six thousand years of Bible teaching. The two wings of the crest would thus intimate power towards the expiration of 2000 years, as time is marked in the history of Great Britain.

{161}

"In a recent publication, The Pestilence, Why Inflicted, are given many reasons why the writer thinks himself to be the appointed watchman foretold by Ezekiel, chapters iii. and xxxiii. Among the reasons are many prophecies fulfilled in him. Of these it is now needful to note two as bearing especially on the subject of the reign of Darius.

"1.—In Daniel it is said, 'Darius the Median took the kingdom, being about threescore and two years old.'—Daniel v. 31.

"When 'Belshazzar' the king of the Chaldeans is found wanting, Darius takes the kingdom. It is not given him by the popular voice; he asserts his right, and this is not denied. He takes it when about sixty-two years of age. The language of Daniel is prophetic, and Darius has in another an antitype. The writer was born July 18th, 1803; and the claim was asserted at the close of 1865, when he was about sixty-two years of age.

"The claims which have been asserted demand a settled faith, and which could only be reached through a long course of divine teaching."

When I was a little boy at school, one of my school-fellows took it into his head to set up a lottery of marbles: the thing took, and he made a stony profit. Soon, one after another, every boy had his lottery, and it was, "I won't put into yours unless you put into mine." This knocked up the scheme. It will be the same with the prophets. Dr. Thorn, Mr. Biden, Mrs. Cottle,[281] etc. will grow imitators, until we are all pointed out in the Bible: but A will not admit B's claim unless B admits his. For myself, as elsewhere shown, I am the first Beast in the Revelations.

Every contraband prophet gets a few followers: it is a great point to make these sequacious people into Buridan's asses, which they will become when prophets are so numerous that there is no choosing.

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SIR G. C. LEWIS.

An historical survey of the Astronomy of the Ancients. By the Rt. Hon. Sir G. C. Lewis.[282] 8vo. 1862.

There are few men of our day whom I admire more than the late Sir G. Lewis: he was honest, earnest, sagacious, learned, and industrious. He probably sacrificed his life to his conjunction of literature and politics: and he stood high as a minister of state in addition to his character as a man of letters. The work above named is of great value, and will be read for its intrinsic merit, consulted for its crowd of valuable references, quoted for its aid to one side of many a discussion, and opposed for its force against the other. Its author was also a wit and a satirist. I know of three classical satires of our day which are inimitable imitations: Mr. Malden's[283] Pragmatized Legends, Mr. Mansel's[284] Phrontisterion, and Sir G. Cornewall Lewis's Inscriptio Antiqua. In this last, HEYDIDDLEDIDDLETHECATANDTHEFIDDLE etc. is treated as an Oscan inscription, and rendered into Latin by approved methods. As few readers have seen it, I give the result:

"Hejus dedit libenter, dedit libenter. Deus propitius [est], deus [donatori] libenter favet. Deus in viarum {163} junctura ovorum dape [colitur], deus mundi. Deus in litatione voluit, benigno animo, haedum, taurum intra fines [loci sacri] portandos. Deus, bis lustratus, beat fossam sacrae libationis."[285]

How then comes the history of astronomy among the paradoxes? Simply because the author, so admirably when writing about what he knew, did not know what he did not know, and blundered like a circle-squarer. And why should the faults of so good a writer be recorded in such a list as the present? For three reasons: First, and foremost, because if the exposure be not made by some one, the errors will gradually ooze out, and the work will get the character of inaccurate. Nothing hurts a book of which few can fathom the depths so much as a plain blunder or two on the surface. Secondly, because the reviews either passed over these errors or treated them too gently, rather implying their existence than exposing them. Thirdly, because they strongly illustrate the melancholy truth, that no one knows enough to write about what he does not know. The distinctness of the errors is a merit; it proceeds from the clear-headedness of the author. The suppression in the journals may be due partly to admiration of the talent and energy which lived two difficult lives at once, partly to respect for high position in public affairs, partly to some of the critics being themselves men of learning only, unable to detect the errors. But we know that action and reaction are equal and contrary. If our generation take no notice of defects, and allow them to go down undetected among merits, the next generation will discover them, will perhaps believe us incapable of detecting them, at least will pronounce our judgment good for nothing, and will form an {164} opinion in which the merits will be underrated: so it has been, is, and will be. The best thing that can be done for the memory of the author is to remove the unsound part that the remainder may thrive. The errors do not affect the work; they occur in passages which might very well have been omitted: and I consider that, in making them conspicuous, I am but cutting away a deleterious fungus from a noble tree.

(P. 154). The periodic times of the five planets were stated by Eudoxus,[286] as we learn from Simplicius;[287] the following is his statement, to which the true times are subjoined, for the sake of comparison:

STATEMENT OF EUDOXUS TRUE TIME Mercury 1 year — 87d. 23h. Venus 1 " — 224d. 16h. Mars 2 " 1y. 321d. 23h. Jupiter 12 " 11y. 315d. 14h. Saturn 30 " 29y. 174d. 1h.

Upon this determination two remarks may be made. First, the error with respect to Mercury and Venus is considerable; with respect to Mercury, it is, in round numbers, 365 instead of 88 days, more than four times too much. Aristotle remarks that Eudoxus distinguishes Mercury and Venus from the other three planets by giving them one sphere each, with the poles in common. The proximity of Mercury to the sun would render its course difficult to observe and to measure, but the cause of the large error with respect to Venus (130 days) is not apparent.

{165}

Sir G. Lewis takes Eudoxus as making the planets move round the sun; he has accordingly compared the geocentric periods of Eudoxus with our heliocentric periods. What greater blunder can be made by a writer on ancient astronomy than giving Eudoxus the Copernican system? If Mercury were a black spot in the middle of the sun it would of course move round the earth in a year, or appear to do so: let it swing a little on one side and the other of the sun, and the average period is still a year, with slight departures both ways. The same for Venus, with larger departures. Say that a person not much accustomed to the distinction might for once write down the mistake; how are we to explain its remaining in the mind in a permanent form, and being made a ground for such speculation as that of the difficulty of observing Mercury leading to a period four times what it ought to be, corrected in proof and published by an industrious and thoughtful person? Only in one way: the writer was quite out of his depth. This one case is conclusive; be it said with all respect for the real staple of the work and of the author. He knew well the difference of the systems, but not the effect of the difference: he is another instance of what I have had to illustrate by help of a very different person, that it is difficult to reason well upon matter which is not familiar.



(P. 254). Copernicus, in fact, supposed the axis of the earth to be always turned towards the Sun.^{(169)} [(169). See Delambre, Hist. Astr. Mod., Vol. I, p. 96]. It was reserved to Kepler to propound the hypothesis of the constant parallelism of the earth's axis to itself.



If there be one thing more prominent than another in the work of Copernicus himself, in the popular explanations of it, and in the page of Delambre[288] cited, it is that the parallelism of the earth's axis is a glaring part of the {166} theory of Copernicus. What Kepler[289] did was to throw away, as unnecessary, the method by which Copernicus, per fas et nefas,[290] secured it. Copernicus, thinking of the earth's orbital revolution as those would think who were accustomed to the solid orbs—and much as the stoppers of the moon's rotation do now: why do they not strengthen themselves with Copernicus?—thought that the earth's axis would always incline the same end towards the sun, unless measures were taken to prevent it. He did take measures: he invented a compensating conical motion of the axis to preserve the parallelism; and, which is one of the most remarkable points of his system, he obtained the precession of the equinoxes by giving the necessary trifle more than compensation. What stares us in the face at the beginning of the paragraph to which the author refers?

"C'est donc pour arriver a ce parallelisme, ou pour le conserver, que Copernic a cru devoir recourir a ce mouvement egal et oppose qui detruit l'effet qu'il attribue si gratuitement au premier, de deranger le parallelisme."[291]

Parallelism at any price, is the motto of Copernicus: you need not pay so dear, is the remark of Kepler.

The opinions given by Sir G. Lewis about the effects of modern astronomy, which he does not understand and singularly undervalues, will now be seen to be of no authority. He fancies that—to give an instance—for the determination of a ship's place, the invention of chronometers has been far more important than any improvement in astronomical theory (p. 254). Not to speak of latitude,—though the omission is not without importance,—he ought to have known that longitude is found by the difference between what o'clock it is at Greenwich and at the ship's place, at {167} one absolute moment of time. Now if a chronometer were quite perfect—which no chronometer is, be it said—and would truly tell Greenwich mean time all over the world, it ought to have been clear that just as good a watch is wanted for the time at the place of observation, before the longitude of that place with respect to Greenwich can be found. There is no such watch, except the starry heaven itself: and that watch can only be read by astronomical observation, aided by the best knowledge of the heavenly motions.

I think I have done Sir G. Lewis's very excellent book more good than all the reviewers put together.

I will give an old instance in which literature got into confusion about astronomy. Theophrastus,[292] who is either the culprit or his historian, attributes to Meton,[293] the contriver of the lunar calendar of nineteen years, which lasts to this day, that his solstices were determined for him by a certain Phaeinus of Elis on Mount Lycabettus. Nobody else mentions this astronomer: though it is pretty certain that Meton himself made more than one appointment with him for the purpose of observing solstices; and we may be sure that if either were behind his time, it was Meton. For Phaeinus Helius is the shining sun himself; and in the astronomical poet Aratus[294] we read about the nineteen years of the shining sun:

[Greek: Enneakaideka kukla phaeinou eelioio].[295]

Some man of letters must have turned Apollo into Phaeinus of Elis; and there he is in the histories of astronomy to {168} this day. Salmasius[296] will have Aratus to have meant him, and proposes to read [Greek: eleioio]: he did not observe that Phaeinus is a very common adjective of Aratus, and that, if his conjecture were right, this Phaeinus would be the only non-mythical man in the poems of Aratus.

[When I read Sir George Lewis's book, the points which I have criticized struck me as not to be wondered at, but I did not remember why at the time. A Chancellor of the Exchequer and a writer on ancient astronomy are birds of such different trees that the second did not recall the first. In 1855 I was one of a deputation of about twenty persons who waited on Sir G. Lewis, as Chancellor of the Exchequer, on the subject of a decimal coinage. The deputation was one of much force: Mr. Airy, with myself and others, represented mathematics; William Brown,[297] whose dealings with the United States were reckoned by yearly millions, counted duodecimally in England and decimally in America, was the best, but not the only, representative of commerce. There were bullionists, accountants, retailers, etc. Sir G. L. walked into the room, took his seat, and without waiting one moment, began to read the deputation a smart lecture on the evils of a decimal coinage; it would require alteration of all the tables, it would impede calculation, etc. etc. Of those arguments against it which weighed with many of better knowledge than his, he obviously knew nothing. The members of the deputation began to make their statements, and met with curious denials. He interrupted me with "Surely there is no doubt that the calculations of our books of arithmetic are easier {169} than those in the French books." He was not aware that the universally admitted superiority of decimal calculation made many of those who prefer our system for the market and the counter cast a longing and lingering look towards decimals. My answer and the smiles which he saw around, made him give a queer puzzled look, which seemed to say, "I may be out of my depth here!" His manner changed, and he listened. I saw both the slap-dash mode in which he dealt with subjects on which he had not thought, and the temperament which admitted suspicion when the means of knowledge came in his way. Having seen his two phases, I wonder neither at his more than usual exhibition of shallowness when shallow, nor at the intensity of the contrast when he had greater depth.]



DECIMAL COINAGE.

Among the paradoxers are the political paradoxers who care not how far they go in debate, their only object being to carry the House with them for the current evening. What I have said of editors I repeat of them. The preservation of a very marked instance, the association of political recklessness with cyclometrical and Apocalyptic absurdity, may have a tendency to warn, not indeed any hardened public-man and sinner, but some young minds which have yearnings towards politics, and are in formation of habits.

In the debate on decimal coinage of July 12, 1855, Mr. Lowe,[298] then member for Kidderminster, an effective speaker and a smart man, exhibited himself in a speech on which I wrote a comment for the Decimal Association. I have seldom seen a more wretched attempt to distort the points of a public question than the whole of this speech. Looking at the intelligence shown by the speaker on other occasions, {170} it is clear that if charity, instead of believing all things, believed only all things but one, he might tremble for his political character; for the honesty of his intention on this occasion might be the incredible exception. I give a few paragraphs with comments:

"In commenting on the humorous, but still argumentative speech of Mr. Lowe, the member for Kidderminster, we may observe, in general, that it consists of points which have been several times set forth, and several times answered. Mr. Lowe has seen these answers, but does not allude to them, far less attempt to meet them. There are, no doubt, individuals, who show in their public speaking the outward and visible signs of a greater degree of acuteness than they can summon to guide their private thinking. If Mr. Lowe be not one of these, if the power of his mind in the closet be at all comparable to the power of his tongue in the House, it may be suspected that his reserve with respect to what has been put forward by the very parties against whom he was contending, arises from one or both of two things—a high opinion of the arguments which he ignored—a low opinion of the generality of the persons whom he addressed. [Both, I doubt not].

"Did they calculate in florins In the name of common sense, ?" how can it be objected to a system that people do not use it before it is introduced? Let the decimal system be completed, and calculation shall be made in florins; that is, florins shall take their proper place. If florins were introduced now, there must be a column for the odd shilling. "He was glad that some hon. If the hon. gentleman make gentleman had derived benefit this assertion of himself, it from the issue of florins. His is not for us to gainsay it. only experience of their It only proves that he is one convenience was, that when he of that class of {171} men who ought to have received are described in the old song, half-a-crown, he had generally of which one couplet runs received a florin, and when he thus: ought to have paid a florin, he had generally paid I sold my cow to buy me a half-a-crown." (Hear, hear, calf; and laughter.) I never make a bargain but I lose half, With a etc. etc. etc.

But he cannot mean that Englishmen in general are so easily managed. And as to Jonathan, who is but John lengthened out a little, he would see creation whittled into chips before he would even split what may henceforth be called the Kidderminster difference. The House, not unmoved—for it laughed—with sly humor decided that the introduction of the florin had been "eminently successful and satisfactory."

The truth is that Mr. Lowe here attacks nothing except the coexistence of the florin and half-crown. We are endeavoring to abolish the half-crown. Let Mr. Lowe join us; and he will, if we succeed, be relieved from the pressure on his pocket which must arise from having the turn of the market always against him.

"From a florin they get to 2 Note the sophism of expressing 2-5ths of a penny, but who our coin in terms of the ever bought anything, who ever penny, which we abandon, reckoned or wished to reckon instead of the florin, which in such a coin as that?" we retain. Remember that this (Hear, hear.) 2 2-5ths is the hundredth part of the pound, which is called, as yet, a cent. Nobody buys anything at a cent, because the cent is not yet introduced. Nobody reckons in cents for the same reason. Everybody wishes to reckon in cents, who wishes to combine the advantage of decimal reckoning with the preservation of the pound as {172} the highest unit of account; amongst others, a majority of the House of Commons, the Bank of England, the majority of London bankers, the Chambers of Commerce in various places, etc. etc. etc. "Such a coin could never come Does 2-1/2d. never pass from into general circulation hand to hand? And is 2-1/2d. because it represents nothing so precisely the modulus of which corresponds with any of popular wants, that an the wants of the people." alteration of 4 per cent. would make it useless? Of all the values which 2-1/2d. measures, from three pounds of potatoes down to certain arguments used in the House of Commons, there is not one for which a cent would not do just as well. Mr. Lowe has fallen into the misconception of the person who admired the dispensation of Providence by which large rivers are made to run through cities so great and towns so many. If the cent were to be introduced to-morrow, straightway the buns and cakes, the soda-water bottles, the short omnibus fares, the bunches of radishes, etc. etc. etc., would adapt themselves to the coin. "If the proposed system were The confusion of ideas here adopted, they would all be exhibited is most instructive. compelled to live in decimals The speaker is under the for ever; if a man dined at a impression that we are public house he would have to introducing fractions: the pay for his dinner in decimal truth is, that we only want to fractions. (Hear, hear.) He abandon the more difficult objected to that, for he fractions which we have got, thought that a man ought to be and to introduce easier able to pay for his dinner in fractions. Does he deny this? integers." (Hear, hear, and a Let us trace his denial to its laugh.) legitimate consequences. A man ought to pay for his dinner in integers.

{173}

Now, if Mr. Lowe insists on it that our integer is the pound, he is bound to admit that the present integer is the pound, of which a shilling, etc., are fractions. The next time he has a chop and a pint of stout in the city, the waiter should say—"A pound, sir, to you," and should add, "Please to remember the waiter in integers." Mr. Lowe fancies that when he pays one and sixpence, he pays in integers, and so he does, if his integer be a penny or a sixpence. Let him bring his mind to contemplate a mil as the integer, the lowest integer, and the seven cents five mils which he would pay under the new system would be payment in integers also. But, as it happens with some others, he looks up the present system, with Cocker,[299] and Walkingame,[300] and always looks down the proposed system. The word decimal is obstinately associated with fractions, for which there is no need. Hence it becomes so much of a bugbear, that, to parody the lines of Pope, which probably suggested one of Mr. Lowe's phrases—

"Dinner he finds too painful an endeavor, Condemned to pay in decimals for ever."

"The present system, however, A pleasant sum even for an had not yet been changed into accomplished mathematician. decimal system. That change What does divided by the might appear very easy to decimal of a pound mean? accomplished mathematicians Perhaps it means reduced to and men of science, but it was the decimal of a pound! Mr. one which it would be very Lowe supposes, as many others difficult to carry out. (Hear, do, that, after the change, hear). What would have to be all calculations will be done? Every sum would have to proposed in old money, and be reduced into a vulgar then converted into new. He fraction of a pound, and then cannot hit the {174} idea that divided by the decimal of a the new coins will take the pound—a pleasant sum for an place of the old. This lack of old applewoman to work out!" apprehension will presently (Hear, hear, and laughter.) appear further. "It would not be an agreeable Let the members be assured task, even for some members of that nine half-pence will be, that House, to reduce 4-1/2d., for every practical purpose, or nine half-pence, to mils." 18 mils. But now to the fact (Hear, hear.) asserted. Davies Gilbert[301] used to maintain that during the long period he sat in the House, he never knew more than three men in it, at one time, who had a tolerable notion of fractions. [I heard him give the names of three at the time when he spoke: they were Warburton,[302] Pollock,[303] and Hume.[304] He himself was then out of Parliament.] Joseph Hume affirmed that he had never met with more than ten members who were arithmeticians. But both these gentlemen had a high standard. Mr. Lowe has given a much more damaging opinion. He evidently means that the general run of members could not do his question. It is done as follows: Since farthings gain on mils, at the rate of a whole mil in 24 farthings (24 farthings being 25 mils), it is clear that 18 farthings being three-quarters of 24 farthings, will gain three-quarters of a mil; that is, 18 farthings are eighteen {175} mils and three-quarters of a mil. Any number of farthings is as many mils and as many twenty-fourths of a mil. To a certain extent, we feel able to protest against the manner in which Kidderminster has treated the other constituencies. We do not hold it impossible to give the Members of the House in general a sufficient knowledge of the meaning and consequences of the decimal succession of units, tens, hundreds, thousands, etc.; and we believe that there are in the House itself competent men, in number enough to teach all the rest. All that is wanted is the power of starting from the known to arrive at the unknown. Now there is one kind of decimals with which every member is acquainted—the Chiltern Hundreds. If public opinion would enable the competent minority to start from this in their teaching, not as a basis, but as an alternative, in three weeks the fundamentals would be acquired, and members in general would be as fit to turn 4-1/2d. into mils, as any boys on the lower forms of a commercial school.

For a long period of years, allusion to the general ignorance of arithmetic, has been a standing mode of argument, and has always been well received: whenever one member describes others as knownothings, those others cry Hear to the country in a transport of delight. In the meanwhile the country is gradually arriving at the conclusion that a true joke is no joke. "The main objection was, if Fine words, wrongly used. The they went below 6d., that the new coins are commensurable new scale of coins would not with, and in a finite ratio be commensurate in any finite to, the old ones. The farthing ratio with anything in this is to the mil as 25 to 24. The new currency of mils." speaker has something here in the bud, which we shall presently meet with in the flower; and fallacies are more easily nipped in flower than in bud. {176} "No less than five of our This dreadful change of value present coins must be called consists in sixpence farthing in, or else—which would be going to the half-shilling worse—new values must be instead of sixpence. Whether given to them." the new farthings be called mils or not is of no consequence. "If a poor man put a penny in Mr. Lowe, who cannot pass a his pocket, it would come out half-crown for more than a a coin of different value, florin, or get in a florin at which he would not understand. less than half-a-crown, has Suppose he owed another man a such a high faith in the penny, how was he to pay him ? sterner stuff of his fellow Was he to pay him in mils? countrymen, that he believes Four mils would be too little, any two of them would go to and five mils would be too fisty cuffs for the 25th part much. The hon. gentlemen said of a farthing. He reasons there would be only a mil thus: He has often heard in between them. That was exactly the streets, "I'd fight you it. He believed there would be for the fiftieth part of a a 'mill' between them." (Much farden:" and having (that is, laughter.) for a Member) a notion both of fractions and logic, he infers that those who would fight for the 50th of a farthing would, a fortiori, fight for a 25th. His mistake arises from his not knowing that when a person offers to fight another for 1/200d., he really means to fight for love; and that the stake is merely a matter of form, a feigned issue, a pro forma report of progress. Do the Members of the House think they have all the forms to themselves? "What would be the present We should hardly believe all expression for four-pence? this to be uttered in earnest, Why, 0.166 (a laugh); for if we had not known {177} that threepence? .0125; for a several persons who have not penny? .004166, and so on ad Mr. Lowe's humor, nevertheless infinitum (a laugh); for a have his impressions on this half-penny? .002083 ad point. It must therefore be infinitum. (A laugh). What answered; but how is this to would be the present be done seriously? expression for a farthing? Why, .0010416 ad infinitum. Dialogue between a member of (A laugh). And this was the Parliament and an orange-boy, system which was to cause such three days after the a saving in figures, and these introduction of the complete were the quantities into which decimal system. The member, the poor would have to reduce going down to the House, wants the current coin of the realm. oranges to sustain his voice (Cheers). With every respect in a two hours' speech on for decimal fractions, of moving that 100000l. be placed which he boasted no profound at the disposal of Her knowledge, he doubted whether Majesty, to supply the poor the poor were equal to mental with ready-reckoners. arithmetic of this kind, (hear, hear) and he hoped the Boy. Fine oranges! two a adoption of the system would penny! two a penny! {178} be deferred until there were some proof that they would be Member. Here boy, two! Now, able to understand it; for, how am I to pay you? after all, this was the question of the poor, and the Boy. Give you change, your whole weight of the change honor. would fall upon them. Let the rich by all means have Member. Ah! but how? Where's permission to perplex your ready-reckoner? themselves by any division of a pound they pleased; but do Boy. I sells a better sort not let them, by any nor them. Mine's real Cheyny. experiment like this, impose difficulties upon the poor and Member. But you see a compel men to carry farthing is now .0014166666 ready-reckoners in their ad infinitum, and if we pocket to give them all these multiply this by 4—— fractional quantities." (Hear, hear.) Boy. Hold hard, Guv'ner; I sees what you're arter. Now what'll you stand if I puts you up to it? which Bill Smith he put me up in two minutes, cause he goes to the Ragged School.

Member. You don't mean that you do without a book!

Boy. Book be blowed. Come now, old un, here's summut for both on us. I got a florin, you gives me a half-a-crown for it, and I larns you the new money, gives you your oranges, and calls you a brick into the bargain.

Member (to himself). Never had such a chance of getting off half-a-crown for value since that —— fellow Bowring carried his crochet. (Aloud.) Well, boy, it's a bargain. Now!

Boy. Why, look 'e here, my trump, its a farden more to the tizzy—that's what it is.

Member. What's that?

Boy. Why, you knows a sixpence when you sees it. (Aside). Blest if I think he does! Well, its six browns and a farden now. A lady buys two oranges, and forks {179} out a sixpence; well in coorse, I hands over fippence farden astead of fippence. I always gives a farden more change, and takes according.

Member (in utter surprise, lets his oranges tumble into the gutter). Never mind! They won't be wanted now. (Walks off one way. Boy makes a pass of naso-digital mesmerism, and walks off the other way).

To the poor, who keep no books, the whole secret is "Sixpence farthing to the half shilling, twelve pence halfpenny to the shilling." The new twopence halfpenny, or cent, will be at once five to the shilling.

In conclusion, we remark that three very common misconceptions run through the hon. Member's argument; and, combined in different proportions, give variety to his patterns.

First, he will have it that we design to bring the uneducated into contact with decimal fractions. If it be so, it will only be as M. Jourdain was brought into contact with prose. In fact, Quoi! quand je dis, Nicole, apportez-moi mes pantoufles, c'est de la prose?[305] may be rendered: "What! do you mean that ten to the florin is a cent a piece must be called decimal reckoning?" If we had to comfort a poor man, horror-struck by the threat of decimals, we should tell him what manner of fractions had been inflicted upon him hitherto; nothing less awful than quarto-duodecimo-vicesimals, we should assure him.

Secondly, he assumes that the penny, such as it now is, will remain, as a coin of estimation, after it has ceased to be a coin of exchange; and that the mass of the people will continue to think of prices in old pence, and to calculate them in new ones, or else in new mils. No answer is required to this, beyond the mere statement of the nature of the assumption and denial.

{180}

Thirdly, he attributes to the uneducated community a want of perception and of operative power which really does not belong to them. The evidence offered to the Committee of the House shows that no fear is entertained on this point by those who come most in contact with farthing purchasers. And this would seem to be a rule,—that is, fear of the intelligence of the lower orders in the minds of those who are not in daily communication with them, no fear at all in the minds of those who are.

A remarkable instance of this distinction happened five-and-twenty years ago. The Admiralty requested the Astronomical Society to report on the alterations which should be made in the Nautical Almanac, the seaman's guide-book over the ocean. The greatest alteration proposed was the description of celestial phenomena in mean (or clock time), instead of apparent (or sundial) time, till then always employed. This change would require that in a great many operations the seaman should let alone what he formerly altered by addition or subtraction, and alter by addition or subtraction what he formerly let alone; provided always that what he formerly altered by addition he should, when he altered at all, alter by subtraction, and vice versa. This was a tolerably difficult change for uneducated skippers, working by rules they had only learned by rote. The Astronomical Society appointed a Committee of forty, of whom nine were naval officers or merchant seamen [I was on this Committee]. Some men of science were much afraid of the change. They could not trust an ignorant skipper or mate to make those alterations in their routine, on the correctness of which the ship might depend. Had the Committee consisted of men of science only, the change might never have been ventured on. But the naval men laughed, and said there was nothing to fear; and on their authority the alteration was made. The upshot was, that, after the new almanacs appeared, not a word of complaint was ever heard on the matter. Had the House of Commons had to {181} decide this question, with Mr. Lowe to quote the description given by Basil Hall[306] (who, by the way, was one of the Committee) of an observation on which the safety of the ship depended, worked out by the light of a lantern in a gale of wind off a lee shore, this simple and useful change might at this moment have been in the hands of its tenth Government Commission.



[Aug. 14, 1866. The Committee was appointed in the spring of 1830: it consisted of forty members. Death, of course, has been busy; there are now left Lord Shaftesbury,[307] Mr. Babbage,[308] Sir John Herschel,[309] Sir Thomas Maclear[310] (Astronomer Royal at the Cape of Good Hope), Dr. Robinson[311] (of Armagh), Sir James South,[312] Lord Wrottesley,[313] and myself].

{182}



THE TONAL SYSTEM.

Project of a new system of arithmetic, weight, measure, and coins, proposed to be called the tonal system, with sixteen to the base. By J. W. Mystrom.[314] Philadelphia, 1862, 8vo.

That is to say, sixteen is to take the place of ten, and to be written 10. The whole language is to be changed; every man of us is to be sixteen-stringed Jack and every woman sixteen-stringed Jill. Our old one, two, three, up to sixteen, are to be (Noll going for nothing, which will please those who dislike the memory of Old Noll) replaced by An, De, Ti, Go, Su, By, Ra, Me, Ni, Ko, Hu, Vy, La, Po, Fy, Ton; and then Ton-an, Ton-de, etc. for 17, 18, etc. The number which in the system has the symbol

28(13)5(11)7(14)0(15)

(using our present compounds instead of new types) is to be pronounced

Detam-memill-lasan-suton-hubong-ramill-posanfy.

The year is to have sixteen months, and here they are:

Anuary, Debrian, Timander, Gostus, Suvenary, Bylian, Ratamber, Mesudius, Nictoary, Kolumbian, Husamber, Vyctorius, Lamboary, Polian, Fylander, Tonborius.

Surely An-month, De-month, etc. would do as well. Probably the wants of poetry were considered. But what are we to do with our old poets? For example—

"It was a night of lovely June, High rose in cloudless blue the moon."

Let us translate—

"It was a night of lovely Nictoary, High rose in cloudless blue the (what, in the name of all that is absurd?)."

And again, Fylander thrown into our December! What is {183} to become of those lines of Praed, which I remember coming out when I was at Cambridge,—

"Oh! now's the time of all the year for flowers and fun, the Maydays; To trim your whiskers, curl your hair, and sinivate the ladies."

If I were asked which I preferred, this system or that of Baron Ferrari[315] already mentioned, proceeding by twelves, I should reply, with Candide, when he had the option given of running the gauntlet or being shot: Les volontes sont libres, et je ne veux ni l'un ni l'autre.[316] We can imagine a speculator providing such a system for Utopia as it would be in the mind of a Laputan: but to explain how an engineer who has surveyed mankind from Philadelphia to Rostof on the Don should for a moment entertain the idea of such a system being actually adopted, would beat a jury of solar-system-makers, though they were shut up from the beginning of Anuary to the end of Tonborius. When I see such a scheme as this imagined to be practicable, I admire the wisdom of Providence in providing the quadrature of the circle, etc., to open a harmless sphere of action to the possessors of the kind of ingenuity which it displays. Those who cultivate mathematics have a right to speak strongly on such efforts of arithmetic as this: for, to my knowledge, persons who have no knowledge are frequently disposed to imagine that their makers are true brothers of the craft, a little more intelligible than the rest.



SOME SMALL PARADOXERS.

Vis inertiae victa,[317] or Fallacies affecting science. By James Reddie.[318] London, 1862, 8vo.

{184}

An attack on the Newtonian mechanics; revolution by gravitation demonstrably impossible; much to be said for the earth being the immovable center. A good analysis of contents at the beginning, a thing seldom found. The author has followed up his attack in a paper submitted to the British Association, but which it appears the Association declined to consider. It is entitled—



Victoria Toto Coelo; or, Modern Astronomy recast. London, 1863, 8vo.

At the end is a criticism of Sir G. Lewis's History of Ancient Astronomy.



On the definition and nature of the Science of Political Economy. By H. Dunning Macleod,[319] Esq. Cambridge, 1862, 8vo.

A paper read—but, according to the report, not understood—at the British Association. There is a notion that political economy is entirely mathematical; and its negative quantity is strongly recommended for study: it contains "the whole of the Funds, Credit, 32 parts out of 33 of the value of Land...." The mathematics are described as consisting of—first, number, or Arithmetic; secondly, the theory of dependent quantities, subdivided into dependence by cause and effect, and dependence by simultaneous variations; thirdly, "independent quantities or unconnected events, which is the theory of probabilities." I am not ashamed, having the British Association as a co-non-intelligent, to say I do not understand this: there is a paradox in it, and the author should give further explanation, especially of his negative quantity. Mr. Macleod has gained {185} praise from great names for his political economy; but this, I suspect, must have been for other parts of his system.



On the principles and practice of just Intonation, with a view to the abolition of temperament.... By General Perronet Thompson.[320] Sixth Edition. London, 1862, 8vo.

Here is General Thompson again, with another paradox: but always master of the subject, always well up in what his predecessors have done, and always aiming at a useful end. He desires to abolish temperament by additional keys, and has constructed an enharmonic organ with forty sounds in the octave. If this can be introduced, I, for one, shall delight to hear it: but there are very great difficulties in the way, greater than stood even in the way of the repeal of the bread-tax.

In a paper on the beats of organ-pipes and on temperament published some years ago, I said that equal temperament appeared to me insipid, and not so agreeable as the effect of the instrument when in progress towards being what is called out of tune, before it becomes offensively wrong. There is throughout that period unequal temperament, determined by accident. General Thompson, taking me one way, says I have launched a declaration which is likely to make an epoch in musical practice; a public musical critic, taking me another way, quizzes me for preferring music out of tune. I do not think I deserve either one remark or the other. My opponent critic, I suspect, takes equally tempered and in tune to be phrases of one meaning. But by equal temperament is meant equal distribution among all the keys of the error which an instrument must have, which, with twelve sounds only in the octave, professes to be fit for all the keys. I am reminded of the equal temperament which was once applied to the postmen's jackets. The coats were all made for the average man: the {186} consequence was that all the tall men had their tails too short; all the short men had them too long. Some one innocently asked why the tall men did not change coats with the short ones.



A diagram illustrating a discovery in the relation of circles to right-lined geometrical figures. London, 1863, 12mo.

The circle is divided into equal sectors, which are joined head and tail: but a property is supposed which is not true.



An attempt to assign the square roots of negative powers; or what is [sqrt] -1? By F.H. Laing.[321] London, 1863, 8vo.

If I understand the author, -a and +a are the square roots of -a^2, as proved by multiplying them together. The author seems quite unaware of what has been done in the last fifty years.



BYRNE'S DUAL ARITHMETIC.

Dual Arithmetic. A new art. By Oliver Byrne.[322] London, 1863, 8vo.

The plan is to throw numbers into the form a(1.1)^{b} (1.01)^{c} (1.001)^{d}... and to operate with this form. This is an ingenious and elaborate speculation; and I have no doubt the author has practised his method until he could surprise any one else by his use of it. But I doubt if he will persuade others to use it. As asked of Wilkins's universal language, Where is the second man to come from?

An effective predecessor in the same line of invention {187} was the late Mr. Thomas Weddle,[323] in his "New, simple, and general method of solving numeric equations of all orders," 4to, 1842. The Royal Society, to which this paper was offered, declined to print it: they ought to have printed an organized method, which, without subsidiary tables, showed them, in six quarto pages, the solution (x=8.367975431) of the equation

1379.664 x^{622} + 2686034 x 10^{432} x^{152} - 17290224 x 10^{518} x^{60} + 2524156 x 10^{574} = 0.

The method proceeds by successive factors of the form, a being the first approximation, a x 1.b x 1.0c x 1.00d.... In my copy I find a few corrections made by me at the time in Mr. Weddle's announcement. "It was read before that learned body [the R. S.] and they were pleased [but] to transmit their thanks to the author. The en[dis]couragement which he received induces [obliges] him to lay the result of his enquiries in this important branch of mathematics before the public [, at his own expense; he being an usher in a school at Newcastle]." Which is most satirical, Mr. Weddle or myself? The Society, in the account which it gave of this paper, described it as a "new and remarkably simple method" possessing "several important advantages." Mr. Rutherford's[324] extended value of [pi] was read at the very next meeting, and was printed in the Transactions; and very properly: Mr. Weddle's paper was excluded, and very very improperly.



HORNER'S METHOD.

I think it may be admited that the indisposition to look at and encourage improvements of calculation which once {188} marked the Royal Society is no longer in existence. But not without severe lessons. They had the luck to accept Horner's[325] now celebrated paper, containing the method which is far on the way to become universal: but they refused the paper in which Horner developed his views of this and other subjects: it was printed by T. S. Davies[326] after Horner's death. I make myself responsible for the statement that the Society could not reject this paper, yet felt unwilling to print it, and suggested that it should be withdrawn; which was done.

But the severest lesson was the loss of Barrett's Method,[327] now the universal instrument of the actuary in his highest calculations. It was presented to the Royal Society, and refused admission into the Transactions: Francis Baily[328] printed it. The Society is now better informed: "live and learn," meaning "must live, so better learn," ought to be the especial motto of a corporation, and is generally acted on, more or less.

Horner's method begins to be introduced at Cambridge: it was published in 1820. I remember that when I first went to Cambridge (in 1823) I heard my tutor say, in conversation, there is no doubt that the true method of solving equations is the one which was published a few years ago in the Philosophical Transactions. I wondered it was not taught, but presumed that it belonged to the higher mathematics. This Horner himself had in his head: and in a sense it is true; for all lower branches belong to the higher: but he would have stared to have been told that he, Horner, {189} was without a European predecessor, and in the distinctive part of his discovery was heir-at-law to the nameless Brahmin—Tartar—Antenoachian—what you please—who concocted the extraction of the square root.

It was somewhat more than twenty years after I had thus heard a Cambridge tutor show sense of the true place of Horner's method, that a pupil of mine who had passed on to Cambridge was desired by his college tutor to solve a certain cubic equation—one of an integer root of two figures. In a minute the work and answer were presented, by Horner's method. "How!" said the tutor, "this can't be, you know." "There is the answer, Sir!" said my pupil, greatly amused, for my pupils learnt, not only Horner's method, but the estimation it held at Cambridge. "Yes!" said the tutor, "there is the answer certainly; but it stands to reason that a cubic equation cannot be solved in this space." He then sat down, went through a process about ten times as long, and then said with triumph: "There! that is the way to solve a cubic equation!"

I think the tutor in this case was never matched, except by the country organist. A master of the instrument went into the organ-loft during service, and asked the organist to let him play the congregation out; consent was given. The stranger, when the time came, began a voluntary which made the people open their ears, and wonder who had got into the loft: they kept their places to enjoy the treat. When the organist saw this, he pushed the interloper off the stool, with "You'll never play 'em out this side Christmas." He then began his own drone, and the congregation began to move quietly away. "There," said he, "that's the way to play 'em out!"

I have not scrupled to bear hard on my own university, on the Royal Society, and on other respectable existences: being very much the friend of all. I will now clear the Royal Society from a very small and obscure slander, simply because I know how. This dissertation began with {190} the work of Mr. Oliver Byrne, the dual arithmetician, etc. This writer published, in 1849, a method of calculating logarithms.[329] First, a long list of instances in which, as he alleges, foreign discoverers have been pillaged by Englishmen, or turned into Englishmen: for example, O'Neill,[330] so called by Mr. Byrne, the rectifier of the semi-cubical parabola claimed by the Saxons under the name of Neal: the grandfather of this mathematician was conspicuous enough as Neal; he was archbishop of York. This list, says the writer, might be continued without end; but he has mercy, and finishes with his own case, as follows:—"About twenty years ago, I discovered this method of directly calculating logarithms. I could generally find the logarithm of any number in a minute or two without the use of books or tables. The importance of the discovery subjected me to all sorts of prying. Some asserted that I committed a table of logarithms to memory; others attributed it to a peculiar mental property; and when Societies and individuals failed to extract my secret, they never failed to traduce the inventor and the invention. Among the learned Societies, the Royal Society of London played a very base part. When I have more space and time at my disposal, I will revert to this subject again."

Such a trumpery story as this remains unnoticed at the time; but when all are gone, a stray copy from a stall falls into hands which, not knowing what to make of it, make history of it. It is a very curious distortion. The reader may take it on my authority, that the Royal Society played no part, good or bad, nor had the option of playing a part. {191} But I myself pars magna fui:[331] and when the author has "space and time" at his disposal, he must not take all of them; I shall want a little of both.



ARE ATOMS WORLDS?

The mystery of being; or are ultimate atoms inhabited worlds? By Nicholas Odgers.[332] Redruth and London, 1863, 8vo.

This book, as a paradox, beats quadrature, duplication, trisection, philosopher's stone, perpetual motion, magic, astrology, mesmerism, clairvoyance, spiritualism, homoeopathy, hydropathy, kinesipathy, Essays and Reviews, and Bishop Colenso,[333] all put together. Of all the suppositions I have given as actually argued, this is the one which is hardest to deny, and hardest to admit. Reserving the question—as beyond human discussion—whether our particles of carbon, etc. are clusters of worlds, the author produces his reasons for thinking that they are at least single worlds. Of course—though not mentioned—the possibility is to be added of the same thing being true of the particles which make up our particles, and so down, for ever: and, on the other hand, of our planets and stars as being particles in some larger universe, and so up, for ever.

"Great fleas have little fleas upon their backs to bite 'em, And little fleas have lesser fleas, and so ad infinitum. And the great fleas themselves, in turn, have greater fleas to go on; While these again have greater still, and greater still, and so on."[334]

I have often had the notion that all the nebulae we see, including our own, which we call the Milky Way, may be particles of snuff in the box of a giant of a proportionately {192} larger universe. Of course the minim of time—a million of years or whatever the geologists make it[335]—which our little affair has lasted, is but a very small fraction of a second to the great creature in whose nose we shall all be in a few tens of thousands of millions of millions of millions of years.

All this is quite possible, and the probabilities for and against are quite out of reach. Perhaps also all the worlds, both above and below us, are fac-similes of our own. If so, away goes free will for good and all; unless, indeed, we underpin our system with the hypothesis that all the fac-simile bodies of different sizes are actuated by a common soul. These acute supplementary notions of mine go far to get rid of the difficulty which some have found in the common theory that the soul inhabits the body: it has been stated that there is, somewhere or another, a world of souls which communicate with their bodies by wondrous filaments of a nature neither mental nor material, but of a tertium quid fit to be a go-between; as it were a corporispiritual copper encased in a spiritucorporeal gutta-percha. My theory is that every soul is everywhere in posse, as the schoolmen said, but not anywhere in actu, except where it finds one of its bodies. These a priori difficulties being thus removed, the system of particle-worlds is reduced to a dry question of fact, and remitted to the decision of the microscope. And a grand field may thus be opened, as optical science progresses! For the worlds are not fac-similes of ours in time: there is not a moment of our past, and not a moment of our future, but is the present of one or more of the particles. A will write the death of Caesar, and B the building of the Pyramids, by actual observation of the processes with a power of a thousand millions; C will discover the commencement of the Millennium, and D the {193} termination of Ersch and Gruber's Lexicon,[336] as mere physical phenomena. Against this glorious future there is a sad omen: the initials of the forerunner of this discovery are—NO!



THE SUPERNATURAL.

The History of the Supernatural in all ages and nations, and in all Churches, Christian and Pagan: demonstrating a universal faith. By Wm. Howitt.[337] London, 2 vols. 8vo. 1863.

Mr. Howitt is a preacher of spiritualism. He cements an enormous collection of alleged facts with a vivid outpouring of exhortation, and an unsparing flow of sarcasm against the scorners of all classes. He and the Rev. J. Smith[338] (ante, 1854) are the most thoroughgoing universalists of all the writers I know on spiritualism. If either can insert the small end of the wedge, he will not let you off one fraction of the conclusion that all countries, in all ages, have been the theaters of one vast spiritual display. And I suspect that this consequence cannot be avoided, if any part of the system be of truly spiritual origin. Mr. Howitt treats the philosophers either as ignorant babies, or as conscious spirit-fearers: and seems much inclined to accuse the world at large of dreading, lest by the actual presence of the other world their Christianity should imbibe a spiritual element which would unfit it for the purposes of their lives.

{194}



FROM MATTER TO SPIRIT.

From Matter to Spirit. By C. D. With a preface by A. B.[339] London, 1863, 8vo.

This is a work on Spiritual Manifestations. The author upholds the facts for spiritual phenomena: the prefator suspends his opinion as to the cause, though he upholds the facts. The work begins systematically with the lower class of phenomena, proceeds to the higher class, and offers a theory, suggested by the facts, of the connection of the present and future life. I agree in the main with A. B.; but can, of course, make none but horrescent reference to his treatment of the smaller philosophers. This is always the way with your paradoxers: they behave towards orthodoxy as the thresher fish behaves towards the whale. But if true, as is said, that the drubbing clears the great fish of parasites which he could not otherwise get rid of, he ought to bear no malice. This preface retorts a little of that contempt which the "philosophical world" has bestowed with heaped measure upon those who have believed their senses, and have drawn natural, even if hasty, inferences. There is philosophercraft as well as priestcraft, both from one source, both of one spirit. In English cities and towns, the minister of religion has been tamed: so many weapons are bared against him when he obtrudes his office in a dictatory manner, that, as a rule, there is no more quiet and modest member of society than the urban clergyman. Domination over religious belief is reserved for the exclusive use of those who admit the right: the rare exception to this mode of behavior is laughed at as a bigot, or shunned as a nuisance. But the overbearing minister of nature, who snaps you with unphilosophical as the clergyman once frightened you with infidel, is still a recognized member of society, wants taming, and will get it. He wears the priest's cast-off {195} clothes, dyed to escape detection: the better sort of philosophers would gladly set him to square the circle.

The book just named appeared about the same time as this Budget began in the Athenaeum. It was commonly attributed, the book to my wife, the preface to myself. Some time after, our names were actually announced by the publisher, who ought to know. It will be held to confirm this statement that I announce our having in our possession some twenty reviews of different lengths, and of all characters: who ever collects a number of reviews of a book, except the author?

A great many of these reviews settle the matter a priori. If there had been spirits in the matter, they would have done this, and they would not have done that. Jean Meslier[340] said there could be no God over all, for, if there had been one, He would have established a universal religion. If J. M. knew that, J. M. was right: but if J. M. did not know that, then J. M. was on the "high priori road," and may be left to his course. The same to all who know what spirits would do and would not do.

A. B. very distinctly said that he knew some of the asserted facts, believed others on testimony, but did not pretend to know whether they were caused by spirits, or had some unknown and unimagined origin. This he said as clearly as I could have said it myself. But a great many persons cannot understand such a frame of mind: their own apparatus is a kind of spirit-level, and their conclusion on any subject is the little bubble, which is always at one end or the other. Many of the reviewers declare that A. B. is a secret believer in the spirit-hypothesis: and one of them wishes that he had "endorsed his opinion more boldly." According to this reviewer, any one who writes "I boldly {196} say I am unable to choose," contradicts himself. In truth, a person who does say it has a good deal of courage, for each side believes that he secretly favors the other; and both look upon him as a coward. In spite of all this, A. B. boldly repeats that he feels assured of many of the facts of spiritualism, and that he cannot pretend to affirm or deny anything about their cause.

The great bulk of the illogical part of the educated community—whether majority or minority I know not; perhaps six of one and half-a-dozen of the other—have not power to make a distinction, cannot be made to take a distinction, and of course, never attempt to shake a distinction. With them all such things are evasions, subterfuges, come-offs, loopholes, etc. They would hang a man for horse-stealing under a statute against sheep-stealing; and would laugh at you if you quibbled about the distinction between a horse and a sheep. I divide the illogical—I mean people who have not that amount of natural use of sound inference which is really not uncommon—into three classes:—First class, three varieties: the Niddy, the Noddy, and the Noodle. Second class, three varieties: the Niddy-Noddy, the Niddy-Noodle, and the Noddy-Noodle. Third class, undivided: the Niddy-Noddy-Noodle. No person has a right to be angry with me for more than one of these subdivisions.

The want of distinction was illustrated to me, when a boy, about 1820, by the report of a trial which I shall never forget: boys read newspapers more keenly than men. Every now and then a bench of country magistrates rather astonishes the town populations, accustomed to rub their brains[341] against one another. Such a story as the following would, {197} in our day, bring down grave remarks from above: but I write of the olden (or Eldon[342]) time, when nothing but conviction in a court of record would displace a magistrate. In that day the third-class amalgamator of distinct things was often on the bench of quarter-sessions.

An attorney was charged with having been out at night poaching. A clear alibi was established; and perjury had certainly been committed. The whole gave reason to suspect that some ill-willers thought the bench disliked the attorney so much that any conviction was certain on any evidence. The bench did dislike the attorney: but not to the extent of thinking he could snare any partridges in the fields while he was asleep in bed, except the dream-partridges which are not always protected by the dream-laws. So the chairman said, "Mr. ——, you are discharged; but you should consider this one of the most fortunate days of your life." The attorney indignantly remonstrated, but the magistrate was right; for he said, "Mr. ——, you have frequently been employed to defend poachers: have you been careful to impress upon them the enormity of their practices?" It appeared in a wrangling conversation that the magistrates saw little moral difference between poaching and being a poacher's professional defender without lecturing him on his wickedness: but they admitted with reluctance, that there was a legal distinction; and the brain of N^3 could no further go. This is nearly fifty years ago; and Westernism was not quite extinct. If the present lords of the hills and the valleys want to shine, let them publish a true history of their own order. I am just old enough to remember some of the last of the squires and parsons who protested against teaching the poor to read and write. They now write books for the working classes, give them lectures, and the like. There is now no class, as a class, more highly educated, broadly educated, and deeply educated, {198} than those who were, in old times, best described as partridge-popping squireens. I have myself, when a boy, heard Old Booby speaking with pride of Young Booby as having too high a spirit to be confined to books: and I suspected that his dislike to teaching the poor arose in fact from a feeling that they would, if taught a little, pass his heir.

A. B. recommended the spirit-theory as an hypothesis on which to ground inquiry; that is, as the means of suggestion for the direction of inquiry. Every person who knows anything of the progress of physics understands what is meant; but not the reviewers I speak of. Many of them consider A. B. as adopting the spirit-hypothesis. The whole book was written, as both the authors point out, to suggest inquiry to those who are curious; C. D. firmly believing, A. B. as above. Neither C. D. nor A. B. make any other pretence. Both dwell upon the absence of authentications and the suppression of names as utterly preventive of anything like proof. And A. B. says that his reader "will give him credit, if not himself a goose, for seeing that the tender of an anonymous cheque would be of equal effect, whether drawn on the Bank of England or on Aldgate Pump." By this test a number of the reviewers are found to be geese: for they take the authors as offering proof, and insist, against the authors, on the very point on which the authors had themselves insisted beforehand.

Leaving aside imperceptions of this kind, I proceed to notice a clerical and medical review. I have lived much in the middle ages, especially since the invention of printing; and from thence I have brought away a high respect for and grateful recollection of—the priest in everything but theology, and the physician in everything but medicine. The professional harness was unfavorable to all progress, except on a beaten road; the professional blinkers prevented all but the beaten road from being seen: the professional reins were pulled at the slightest attempt to quicken pace, even on the permitted path; and the {199} professional whip was heavily laid on at the slightest attempt to diverge. But when the intelligent man of either class turned his attention out of his ordinary work, he had, in most cases, the freshness and vigor of a boy at play, and like the boy, he felt his freedom all the more from the contrast of school-restraint.

In the case of medicine, and physics generally, the learned were, in some essential points, more rational than many of their present impugners. They pass for having put a priori obstacles in the way of progress: they might rather be reproved for too much belief in progress obtained by a priori means. They would have shouted with laughter at a dunce who—in a review I read, but without making a note—declared that he would not believe his senses except when what they showed him was capable of explanation upon some known principle. I have seen such stuff as this attributed to the schoolmen; but only by those who knew nothing about them. The following, which I wrote some years ago, will give a notion of a distinction worth remembering. It is addressed to the authorities of the College of Physicians.

"The ignominy of the word empiric dates from the ages in which scholastic philosophy deduced physical consequences a priori;—the ages in which, because a lion is strong, rubbing with lion's fat would have been held an infallible tonic. In those happy days, if a physician had given decoction of a certain bark, only because in numberless instances that decoction had been found to strengthen the patient, he would have been a miserable empiric. Not that the colleges would have passed over his returns because they were empirical: they knew better. They were as skilful in finding causes for facts, as facts for causes. The president and the elects of that day would have walked out into the forest with a rope, and would have pulled heartily at the tree which yielded the bark: nor would they ever have left it until they had pulled out a legitimate {200} reason. If the tree had resisted all their efforts, they would have said, 'Ah! no wonder now; the bark of a strong tree makes a strong man.' But if they had managed to serve the tree as you would like to serve homoeopathy, then it would have been 'We might have guessed it; all the virtus roborativa has settled in the bark.' They admitted, as we know from Moliere, the virtus dormitiva[343] of opium, for no other reason than that opium facit dormire.[344] Had the medicine not been previously known, they would, strange as it may seem to modern pharmacopoeists, have accorded a virtus dormitiva to the new facit dormire. On this point they have been misapprehended. They were prone to infer facit from a virtus imagined a priori; and they were ready in supplying facit in favor of an orthodox virtus. They might have gone so far, for example, under pre-notional impressions, as the alliterative allopath, who, when maintenance of truth was busy opposing the progress of science called vaccination, declared that some of its patients coughed like cows, and bellowed like bulls; but they never refused to find virtus when facit came upon them, no matter whence. They would rather have accepted Tenterden steeple than have rejected the Goodwin Sands. They would have laughed their modern imitators to scorn: but as they are not here, we do it for them.

"The man of our day—the a priori philosopher—tries the question whether opium can cause sleep by finding out in the recesses of his own noddle whether the drug can have a dormitive power: Well! but did not the schoolman do the same? He did; but mark the distinction. The schoolman had recourse to first principles, when there was no opium to try it by: our man settles the point in the same way with a lump of opium before him. The schoolman shifted his principles with his facts: the man of our drawing-rooms will fight facts with his principles, just as an old {201} physician would have done in actual practice, with the rod of his Church at his back.

"The story about Galileo—which seems to have been either a joke made against him, or by him—illustrates this. Nature abhors a vacuum was the explanation of the water rising in a pump: but they found that the water would not rise more than 32 feet. They asked for explanation: what does the satirist make the schoolmen say? That the stoppage is not a fact, because nature abhors a vacuum? No! but that the principle should be that nature abhors a vacuum as far as 32 feet. And this is what would have been done.

"There are still among us both priests and physicians who would have belonged, had they lived three or four centuries ago, to the glorious band of whom I have spoken, the majority of the intelligent, working well for mankind out of the professional pursuit. But we have a great many who have helped to abase their classes. Go where we may, we find specimens of the lower orders of the ministry of religion and the ministry of health showing themselves smaller than the small of other pursuits. And how is this? First, because each profession is entered upon a mere working smack of its knowledge, without any depth of education, general or professional. Not that this is the whole explanation, nor in itself objectionable: the great mass of the world must be tended, soul and body, by those who are neither Hookers[345] nor Harveys[346]: let such persons not venture ultra crepidam, and they are useful and respectable. But, secondly, there is a vast upheaving of thought from the depths of commonplace learning. I am a clergyman! Sir! I am a medical man! Sir! and forthwith the nature of things is picked to pieces, and there is a race, with the last the winner, between Philosophy mounted on Folly's donkey, and Folly mounted on Philosophy's donkey. How fortunate {202} it is for Law that her battles are fought by politicians in the Houses of Parliament. Not that it is better done: but then politics bears the blame."

I now come to the medical review. After a quantity of remark which has been already disposed of, the writer shows Greek learning, a field in which the old physician would have had a little knowledge. A. B., for the joke's sake, had left untranslated, as being too deep, a remarkably easy sentence of Aristotle, to the effect that what has happened was possible, for if impossible it would not have happened. The reviewer, in "simple astonishment,"—it was simple—at the pretended incapacity—I was told by A. B. that the joke was intended to draw out a reviewer—translates:—He says that this sentence is A. B.'s summing up of the evidence of Spiritualism. Now, being a sort of alter ego[347] of A. B., I do declare that he is not such a fool as to rest the evidence of Spiritualism—the spirit explanation—upon the occurrence of certain facts proving the possibility of those very facts. In truth, A. B. refuses to receive spiritualism, while he receives the facts: this is the gist of his whole preface, which simply admits spiritualism among the qualified candidates, and does not know what others there may be.

The reviewer speaks of Aristotle as "that clear thinker and concise writer." I strongly suspect that his knowledge of Aristotle was limited to the single sentence which he had translated or got translated. Aristotle is concise in phrase, not in book, and is powerful and profound in thought: but no one who knows that his writing, all we have of him, is the very opposite of clear, will pretend to decide that he thought clearly. As his writing, so probably was his thought; and his books are, if not anything but clear, at least anything good but clear. Nobody thinks them clear except a person who always clears difficulties: which I have no doubt was the reviewer's habit; that is, if he ever took the field {203} at all. The gentleman who read Euclid, all except the As and Bs and the pictures of scratches and scrawls, is the type of a numerous class.

The reviewer finds that the word amosgepotically, used by A. B., is utterly mysterious and incomprehensible. He hopes his translation of the bit of Greek will shield him from imputation of ignorance: and thinks the word may be referred to the "obscure dialect" out of which sprung aneroid, kalos geusis sauce, and Anaxyridian trousers. To lump the first two phrases with the third smacks of ignorance in a Greek critic; for [Greek: anaxuridia], breeches, would have turned up in the lexicon; and kalos geusis, though absurd, is not obscure. And [Greek: amosgepos], somehow or other, is as easily found as [Greek: anaxuridia]. The word aneroid, I admit, has puzzled better scholars than the critic: but never one who knows the unscholarlike way in which words ending in [Greek: eides] have been rendered. The aneroid barometer does not use a column of air in the same way as the old instrument. Now [Greek: aeroeides]—properly like the atmosphere—is by scientific non-scholarship rendered having to do with the atmosphere; and [Greek: anaeroeides]—say anaeroid—denies having to do with the atmosphere; a nice thing to say of an instrument which is to measure the weight of the atmosphere. One more absurdity, and we have aneroid, and there you are. The critic ends with a declaration that nothing in the book shakes his faith in a Quarterly reviewer who said that suspension of opinion, until further evidence arrives, is justifiable: a strange summing up for an article which insists upon utter rejection being unavoidable.[348] The expressed aim of both A. B. and C. D. was to excite inquiry, and get further evidence: until this is done, neither asks for a verdict.

Oh where! and oh where! is old Medicine's learning gone! There was some in the days of yore, when Popery {204} was on! And it's oh! for some Greek, just to find a word upon! The reviewer who, lexicon in hand, can neither make out anaxyridical, amosgepotical, kalos geusis, nor distinguish them from aneroid, cannot be trusted when he says he has translated a sentence of Aristotle. He may have done it; but, as he says of spiritualism, we must suspend our opinion until further evidence shall arrive.

We now come to the theological review. I have before alluded to the faults of logic which are Protestant necessities: but I never said that Protestant argument had nothing but paralogism. The writer before me attains this completeness: from beginning to end he is of that confusion and perversion which, as applied to interpretation of the New Testament, is so common as to pass unnoticed by sermon-hearers; but which, when applied out of church, is exposed with laughter in all subjects except theology. I shall take one instance, putting some words in italics.

A. B. Theological Critic.

My state of mind, which refers ... he proceeds to argue that the whole either to unseen he himself is outside its intelligence, or something sacred pale because he refers which man has never had any all these strange phenomena to conception of, proves me to unseen spiritual be out of the pale of the intelligence. Royal Society.

The possibility of a yet unimagined cause is insisted on in several places. On this ground it is argued by A. B. that spiritualists are "incautious" for giving in at once to the spirit doctrine. But, it is said, they may be justified by the philosophers, who make the flint axes, as they call them, to be the works of men, because no one can see what else they can be. This kind of adoption, condemned as a conclusion, is approved as a provisional theory, suggestive of direction of inquiry: experience having shown that {205} inquiry directed by a wrong theory has led to more good than inquiry without any theory at all. All this A. B. has fully set forth, in several pages. On it the reviewer remarks that "with infinite satisfaction he tries to justify his view of the case by urging that there is no other way of accounting for it; after the fashion of the philosophers of our own day, who conclude that certain flints found in the drift are the work of men, because the geologist does not see what else they can be." After this twist of meaning, the reviewer proceeds to say, and A. B. would certainly join him, "There is no need to combat any such mode of reasoning as this, because it would apply with equal force and justice to any theory whatever, however fantastic, profane, or silly." And so, having shown how the reviewer has hung himself, I leave him funipendulous.

One instance more, and I have done. A reviewer, not theological, speaking of the common argument that things which are derided are not therefore to be rejected, writes as follows:—"It might as well be said that they who laughed at Jenner[349] and vaccination were, in a certain but very unsatisfactory way, witnesses to the possible excellence of the system of St. John Long."[350] Of course it might: and of course it is said by all people of common sense. In introducing the word "possible," the reviewer has hit the point: I suspect that this word was introduced during revision, to put the sentence into fighting order; hurry preventing it being seen that the sentence was thus made to fight on the wrong side. Jenner, who was laughed at, was right; therefore, it is not impossible—that is, it is possible—that a derided system may be right. Mark the three gradations: in medio tutissimus ibis.[351]

{206}

Reviewer.—If a system be derided, it is no ground of suspense that derided systems have turned out true: if it were, you would suspend your opinion about St. John Long on account of Jenner.—Ans. You ought to do so, as to possibility; and before examination; not with the notion that J. proves St. J. probable; only possible.

Common Sense.—The past emergence of truths out of derided systems proves that there is a practical certainty of like occurrence to come. But, inasmuch as a hundred speculative fooleries are started for one truth, the mind is permitted to approach the examination of any one given novelty with a bias against it of a hundred to one: and this permission is given because so it will be, leave or no leave. Every one has licence not to jump over the moon.

Paradoxer.—Great men have been derided, and I am derided: which proves that my system ought to be adopted. This is a summary of all the degrees in which paradoxers contend for the former derision of truths now established, giving their systems probability. I annex a paragraph which D [e &c.] inserted in the Athenaeum of October 23, 1847.

"Discoverers and Discoveries.

"Aristotle once sent his servant to the cellar to fetch wine:—and the fellow brought him back small beer. The Stagirite (who knew the difference) called him a blockhead. 'Sir,' said the man, 'all I can say is, that I found it in the cellar.' The philosopher muttered to himself that an affirmative conclusion could not be proved in the second figure,—and Mrs. Aristotle, who was by, was not less effective in her remark, that small beer was not wine because it was in the same cellar. Both were right enough: and our philosophers might take a lesson from either—for they insinuate an affirmative conclusion in the second figure. Great discoverers have been little valued by established {207} schools,—and they are little valued. The results of true science are strange at first,—and so are their's. Many great men have opposed existing notions,—and so do they. All great men were obscure at first,—and they are obscure. Thinking men doubt,—and they doubt. Their small beer, I grant, has come out of the same cellar as the wine; but this is not enough. If they had let it stand awhile in the old wine-casks, it might have imbibed a little of the flavor."



There are better reviews than I have noticed; which, though entirely dissenting, are unassailable on their own principles. What I have given represents five-sixths of the whole. But it must be confessed that the fraction of fairness and moderation and suspended opinion which the doctrine of Spirit Manifestations has met with—even in the lower reviews—is strikingly large compared to what would have been the case fifty years ago. It is to be hoped that our popular and periodical literatures are giving us one thinker created for twenty geese double-feathered: if this hope be realized, we shall do! Seeing all that I see, I am not prepared to go the length of a friend of mine who, after reading a good specimen of the lower reviewing, exclaimed—Oh! if all the fools in the world could be rolled up into one fool, what a reviewer he would make!



Calendrier Universel et Perpetuel; par le Commandeur P. J. Arson.[352] Publie par ses Enfans (Oeuvre posthume). Nice, 1863, 4to.

I shall not give any account of this curious calendar, with all its changes and symbols. But there is one proposal, which, could we alter the general notions of time—a thing of very dubious possibility—would be convenient. The week is made to wax and wane, culminating on the Sunday, {208} which comes in the middle. Thursday, Friday, Saturday, are ascending or waxing days; Monday, Tuesday, Wednesday, are descending or waning days. Our six days, lumped together after the great distinguishing day, Sunday, are too many to be distinctly thought of together: a division of three preceding and three following the day of most note would be much more easily used. But all this comes too late. It may be, nevertheless, that some individuals may be able to adjust their affairs with advantage by referring Thursday, Friday, Saturday, to the following Sunday, and Monday, Tuesday, Wednesday, to the preceding Sunday. But M. Arson's proposal to alter the names of the days is no more necessary than it is practicable.



CYCLOMETRY.

I am not to enter anything I do not possess. The reader therefore will not learn from me the feats of many a man-at-arms in these subjects. He must be content, unless he will bestir himself for himself, not to know how Mr. Patrick Cody trisects the angle at Mullinavat, or Professor Recalcati squares the circle at Milan. But this last is to be done by subscription, at five francs a head: a banker is named who guarantees restitution if the solution be not perfectly rigorous; the banker himself, I suppose, is the judge. I have heard of a man of business who settled the circle in this way: if it can be reduced to a debtor and creditor account, it can certainly be done; if not, it is not worth doing. Montucla will give the accounts of the lawsuits which wagers on the problem have produced in France.

Neither will I enter at length upon the success of the new squarer who advertises (Nov. 1863) in a country paper that, having read that the circular ratio was undetermined, "I thought it very strange that so many great scholars in all ages should have failed in finding the true ratio, and have been determined to try myself.... I am about to secure the {209} benefit of the discovery, so until then the public cannot know my new and true ratio." I have been informed that this trial makes the diameter to the circumference as 64 to 201, giving [pi] = 3.140625 exactly. The result was obtained by the discoverer in three weeks after he first heard of the existence of the difficulty. This quadrator has since published a little slip, and entered it at Stationers' Hall. He says he has done it by actual measurement; and I hear from a private source that he uses a disk of 12 inches diameter, which he rolls upon a straight rail. Mr. James Smith did the same at one time; as did also his partisan at Bordeaux. We have, then, both 3.125 and 3.140625, by actual measurement. The second result is more than the first by about one part in 200. The second rolling is a very creditable one; it is about as much below the mark as Archimedes was above it. Its performer is a joiner, who evidently knows well what he is about when he measures; he is not wrong by 1 in 3,000.

The reader will smile at the quiet self-sufficiency with which "I have been determined to try myself" follows the information that "so many great scholars in all ages" have failed. It is an admirable spirit, when accompanied by common sense and uncommon self-knowledge. When I was an undergraduate there was a little attendant in the library who gave me the following,—"As to cleaning this library, Sir, if I have spoken to the Master once about it, I have spoken fifty times: but it is of no use; he will not employ littery men; and so I am obliged to look after it myself."

I do not think I have mentioned the bright form of quadrature in which a square is made equal to a circle by making each side equal to a quarter of the circumference. The last squarer of this kind whom I have seen figures in the last number of the Athenaeum for 1855: he says the thing is no longer a problem, but an axiom. He does not know that the area of the circle is greater than that of any other figure of the same circuit. This any one might see without {210} mathematics. How is it possible that the figure of greatest area should have any one length in its circuit unlike in form to any other part of the same length?