Budget of Paradoxes/P

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A Budget of Paradoxes by Augustus De Morgan


The system of Tycho Brahé,[621] with some alteration and addition, has been revived and contended for in our own day by a Dane, W. Zytphen,[622] who has published The Motion of the Sun in the Universe, (second edition) Copenhagen, 1865, 8vo, and Le Mouvement Sidéral, 1865, 8vo. I make an extract.

"How can one explain Copernically that the velocity of the Moon must be added to the velocity of the Earth on the one place in the Earth's orbit, to learn how far the Moon has advanced from one fixed star to another; but in another place in the orbit these velocities must be subtracted (the movements taking place in opposite directions) to attain the same result? In the Copernican and other systems, it is well known that the Moon, abstracting from the insignificant excentricity of the orbit, always in twenty-four hours performs an equally long distance. Why has Copernicus never been denominated Fundamentus or Fundator? Because he has never convinced anybody so thoroughly that this otherwise so natural epithet has occurred to the mind."

Really the second question is more effective against Newton than against Copernicus; for it upsets gravity: the first is of great depth.

[ 336 ]



The Correspondent journal makes a little episode in the history of my Budget (born May, 1865, died April, 1866). It consisted entirely of letters written by correspondents. In August, a correspondent who signed "Fair Play"—and who I was afterwards told was a lady—thought it would be a good joke to bring in the Cyclometers. Accordingly a letter was written, complaining that though Mr. Sylvester's[623] demonstration of Newton's theorem—then attracting public attention—was duly lauded, the possibly greater discovery of the quadrature seemed to be blushing unseen, and wasting etc. It went on as follows:

"Prof. De Morgan, who, from his position in the scientific world, might fairly afford to look favourably on less practised efforts than his own, seems to delight in ridiculing the discoverer. Science is, of course, a very respectable person when he comes out and makes himself useful in the world [it must have been a lady; each sex gives science to the other]: but when, like a monk of the Middle Ages, he shuts himself up [it must have been a lady; they always snub the bachelors] in his cloistered cell, repeating his mumpsimus from day to day, and despising the labourers on the outside, we begin to think of Galileo,[624] Jenner,[625] Harvey,[626] and other glorious trios, who have been contemned ..."

The writer then called upon Mr. James Smith[627] to come [ 337 ] forward. The irony was not seen; and that day fortnight appeared the first of more than thirty letters from his pen. Mr. Smith was followed by Mr. Reddie,[628] Zadkiel,[629] and others, on their several subjects. To some of the letters I have referred; to others I shall come. The Correspondent was to become a first-class scientific journal; the time had arrived at which truth had an organ: and I received formal notice that I could not stifle it by silence, nor convert it into falsehood by ridicule. When my reader sees my extracts, he will readily believe my declaration that I should have been the last to stifle a publication which was every week what James Mill[630] would call a dose of capital for my Budget. A few anti-paradoxers brought in common sense: but to the mass of the readers of the journal it all seemed to be the difference between Tweedledum and Tweedledee. Some said that the influx of scientific paradoxes killed the journal: but my belief is that they made it last longer than it otherwise would have done. Twenty years ago I recommended the paradoxers to combine and publish their views in a common journal: with a catholic editor, who had no pet theory, but a stern determination not to exclude anything merely for absurdity. I suspect it would answer very well. A strong title, or motto, would be wanted: not so coarse as was roared out in a Cambridge mob when I was an undergraduate—"No King! No Church! No House of Lords! No nothing, blast me!"—but something on that principle.

At the end of 1867 I addressed the following letter to the Athenæum:

Pseudomath, Philomath, and Graphomath.

December 31, 1867

Many thanks for the present of Mr. James Smith's letters [ 338 ] of Sept. 28 and of Oct. 10 and 12. He asks where you will be if you read and digest his letters: you probably will be somewhere first. He afterwards asks what the WE of the Athenæum will be if, finding it impossible to controvert, it should refuse to print. I answer for you, that We-We of the Athenæum, not being Wa-Wa the wild goose, so conspicuous in "Hiawatha," will leave what controverts itself to print itself, if it please.

Philomath is a good old word, easier to write and speak than mathematician. It wants the words between which I have placed it. They are not well formed: pseudomathete and graphomathete would be better: but they will do. I give an instance of each.

The pseudomath is a person who handles mathematics as the monkey handled the razor. The creature tried to shave himself as he had seen his master do; but, not having any notion of the angle at which the razor was to be held, he cut his own throat. He never tried a second time, poor animal! but the pseudomath keeps on at his work, proclaims himself clean-shaved, and all the rest of the world hairy. So great is the difference between moral and physical phenomena! Mr. James Smith is, beyond doubt, the great pseudomath of our time. His 3⅛ is the least of a wonderful chain of discoveries. His books, like Whitbread's barrels, will one day reach from Simpkin & Marshall's to Kew, placed upright, or to Windsor laid length-ways. The Queen will run away on their near approach, as Bishop Hatto did from the rats: but Mr. James Smith will follow her were it to John o' Groats.

The philomath, for my present purpose, must be exhibited as giving a lesson to presumption. The following anecdote is found in Thiébault's[631] Souvenirs de vingt ans de séjours à Berlin, published in 1804. The book itself got a high character for truth. In 1807 Marshal Mollendorff[632] [ 339 ] answered an inquiry of the Duc de Bassano,[633] by saying that it was the most veracious of books, written by the most honest of men. Thiébault does not claim personal knowledge of the anecdote, but he vouches for its being received as true all over the north of Europe.[634]

Diderot[635] paid a visit to Russia at the invitation of Catherine the Second. At that time he was an atheist, or at least talked atheism: it would be easy to prove him either one thing or the other from his writings. His lively sallies on this subject much amused the Empress, and all the younger part of her Court. But some of the older courtiers suggested that it was hardly prudent to allow such unreserved exhibitions. The Empress thought so too, but did not like to muzzle her guest by an express prohibition: so a plot was contrived. The scorner was informed that an eminent mathematician had an algebraical proof of the existence of God, which he would communicate before the whole Court, if agreeable. Diderot gladly consented. The mathematician, who is not named, was Euler.[636] He came to Diderot with the gravest air, and in a tone of perfect conviction said, "Monsieur!


donc Dieu existe; répondez!"[637] Diderot, to whom algebra was Hebrew, though this is expressed in a very roundabout way by Thiébault—and whom we may suppose to have expected some verbal argument of alleged algebraical closeness, was disconcerted; while peals of laughter sounded on all sides. Next day he asked permission to return to France, which was granted. An algebraist would have [ 340 ] turned the tables completely, by saying, "Monsieur! vous savez bien que votre raisonnement demande le développement de x suivant les puissances entières de n".[638] Goldsmith could not have seen the anecdote, or he might have been supposed to have drawn from it a hint as to the way in which the Squire demolished poor Moses.

The graphomath is a person who, having no mathematics, attempts to describe a mathematician. Novelists perform in this way: even Walter Scott now and then burns his fingers. His dreaming calculator, Davy Ramsay, swears "by the bones of the immortal Napier." Scott thought that the the philomaths worshiped relics: so they do, in one sense. Look into Hutton's[639] Dictionary for Napier's Bones, and you shall learn all about the little knick-knacks by which he did multiplication and division. But never a bone of his own did he contribute; he preferred elephants' tusks. The author of Headlong Hall[640] makes a grand error, which is quite high science: he says that Laplace proved the precession of the equinoxes to be a periodical inequality. He should have said the variation of the obliquity. But the finest instance is the following: Mr. Warren,[641] in his well-wrought tale of the martyr-philosopher, was incautious enough to invent the symbols by which his savant satisfied himself Laplace[642] was right on a doubtful point. And this is what he put together—

√-3a2, Budget of Paradoxes rectangle.png y2 / z2 + 9 - n = 9, n × log e.

Now, to Diderot and the mass of mankind this might be Laplace all over: and, in a forged note of Pascal, would [ 341 ] prove him quite up to gravitation. But I know of nothing like it, except in the lately received story of the American orator, who was called on for some Latin, and perorated thus: "Committing the destiny of the country to your hands, Gentlemen, I may without fear declare, in the language of the noble Roman poet,

E pluribus unum,
Multum in parvo,
Ultima Thule,
Sine qua non."[643]

But the American got nearer to Horace than the martyr-philosopher to Laplace. For all the words are in Horace, except Thule, which might have been there. But Budget of Paradoxes rectangle.png is not a symbol wanted by Laplace; nor can we see how it could have been; in fact, it is not recognized in algebra. As to the junctions, etc., Laplace and Horace are about equally well imitated.

Further thanks for Mr. Smith's letters to you of Oct. 15, 18, 19, 28, and Nov. 4, 15. The last of these letters has two curious discoveries. First, Mr. Smith declares that he has seen the editor of the Athenæum: in several previous letters he mentions a name. If he knew a little of journalism he would be aware that editors are a peculiar race, obtained by natural selection. They are never seen, even by their officials; only heard down a pipe. Secondly, "an ellipse or oval" is composed of four arcs of circles. Mr. Smith has got hold of the construction I was taught, when a boy, for a pretty four-arc oval. But my teachers knew better than to call it an ellipse: Mr. Smith does not; but he produces from it such confirmation of 3⅛ as would convince any honest editor.

Surely the cyclometer is a Darwinite development of a spider, who is always at circles, and always begins again when his web is brushed away. He informs you that he [ 342 ] has been privileged to discover truths unknown to the scientific world. This we know; but he proceeds to show that he is equally fortunate in art. He goes on to say that he will make use of you to bring those truths to light, "just as an artist makes use of a dummy for the purpose of arranging his drapery." The painter's lay-figure is for flowing robes; the hairdresser's dummy is for curly locks. Mr. James Smith should read Sam Weller's pathetic story of the "four wax dummies." As to his use of a dummy, it is quite correct. When I was at University College, I walked one day into a room in which my Latin colleague was examining. One of the questions was, "Give the lives and fates of Sp. Mælius,[644] and Sp. Cassius."[645] Umph! said I, surely all know that Spurius Mælius was whipped for adulterating flour, and that Spurius Cassius was hanged for passing bad money. Now, a robe arranged on a dummy would look just like the toga of Cassius on the gallows. Accordingly, Mr. Smith is right in the drapery-hanger which he has chosen: he has been detected in the attempt to pass bad circles. He complains bitterly that his geometry, instead of being read and understood by you, is handed over to me to be treated after my scurrilous fashion. It is clear enough that he would rather be handled in this way than not handled at all, or why does he go on writing? He must know by this time that it is a part of the institution that his "untruthful and absurd trash" shall be distilled into mine at the rate of about 3⅛ pages of the first to one column of the second. Your readers will never know how much they gain by the process, until Mr. James Smith publishes it all in a big book, or until they get hold of what he has already published. I have six pounds avoirdupois of pamphlets and letters; and there is more than half a pound of letters [ 343 ] written to you in the last two months. Your compositor must feel aggrieved by the rejection of these clearly written documents, without erasures, and on one side only. Your correspondent has all the makings of a good contributor, except the knowledge of his subject and the sense to get it. He is, in fact, only a mask: of whom the fox

"O quanta species, inquit, cerebrum non habet."[646]

I do not despair of Mr. Smith on any question which does not involve that unfortunate two-stick wicket at which he persists in bowling. He has published many papers; he has forwarded them to mathematicians: and he cannot get answers; perhaps not even readers. Does he think that he would get more notice if you were to print him in your journal? Who would study his columns? Not the mathematician, we know; and he knows. Would others? His balls are aimed too wide to be blocked by any one who is near the wicket. He has long ceased to be worth the answer which a new invader may get. Rowan Hamilton,[647] years ago, completely knocked him over; and he has never attempted to point out any error in the short and easy method by which that powerful investigator condescended to show that, be right who may, he must be wrong. There are some persons who feel inclined to think that Mr. Smith should be argued with: let those persons understand that he has been argued with, refuted, and has never attempted to stick a pen into the refutation. He stated that it was a remarkable paradox, easily explicable; and that is all. After this evasion, Mr. James Smith is below the necessity of being told that he is unworthy of answer. His friends complain that I do nothing but chaff him. Absurd! I winnow him; and if nothing but chaff results, whose fault is that? I am usefully employed: for he is the type of a class which ought to be known, and which I have done much to make known.

[ 344 ]

Nothing came of this until July 1869, when I received a reprint of the above letter, with a comment, described as Appendix D of a work in course of publication on the geometry of the circle. The Athenæum journal received the same: but the Editor, in his private capacity, received the whole work, being The Geometry of the Circle and Mathematics as applied to Geometry by Mathematicians, shown to be a mockery, delusion, and a snare, Liverpool, 8vo, 1869. Mr. J. S. here appears in deep fight with Professor Whitworth,[648] and Mr. Wilson,[649] the author of the alleged amendment of Euclid. How these accomplished mathematicians could be inveigled into continued discussion is inexplicable. Mr. Whitworth began by complaining of Mr. Smith's attacks upon mathematicians, continued to correspond after he was convinced that J. S. proved an arc and its chord to be equal, and only retreated when J. S. charged him with believing in 3⅛, and refusing acknowledgment. Mr. Wilson was introduced to J. S. by a volunteer defense of his geometry from the assaults of the Athenæum. This the editor would not publish; so J. S. sent a copy to Mr. Wilson himself. Some correspondence ensued, but Mr. Wilson soon found out his man, and withdrew.

There is a little derision of the Athenæum and a merited punishment for "that unscrupulous critic and contemptible mathematical twaddler, De Morgan."



At p. 183 I mentioned Mr. Reddie,[650] the author of Vis Inertiæ Victa and of Victoria toto cœlo,[651] which last is not [ 345 ] an address to the whole heaven, either from a Roman Goddess or a British Queen, whatever a scholar may suppose. Between these Mr. Reddie has published The Mechanics of the Heavens, 8vo, 1862: this I never saw until he sent it to me, with an invitation to notice it, he very well knowing that it would catch. His speculations do battle with common notions of mathematics and of mechanics, which, to use a feminine idiom, he blasphemes so you can't think! and I suspect that if you do not blaspheme them too, you can't think. He appeals to the "truly scientific," and would be glad to have readers who have read what he controverts, i.e., Newton's Principia: I wish he may get them; I mean I hope he may obtain them. To none but these would an account of his speculations be intelligible: I accordingly disposed of him in a very short paragraph of description. Now many paradoxers desire notice, even though it be disparaging. I have letters from more than one—besides what have been sent to the Editor of the Athenæum—complaining that they are not laughed at; although they deserve it, they tell me, as much as some whom I have inserted. Mr. Reddie informs me that I have not said a single word against his books, though I have given nearly a column to sixteen-string arithmetic, and as much to animalcule universes. What need to say anything to readers of Newton against a book from which I quoted that revolution by gravitation is demonstrably impossible? It would be as useless as evidence against a man who has pleaded guilty. Mr. Reddie derisively thanks me for "small mercies"; he wrote me private letters; he published them, and more, in the Correspondent. He gave me, pro viribus suis,[652] such a dressing you can't think, both for my Budget non-notice, and for reviews which he assumed me to have written. He outlawed himself by declaring (Correspondent, Nov. 11, 1856) that I—in a review—had made a quotation which was "garbled, evidently on purpose [ 346 ] to make it appear that" he "was advocating solely a geocentric hypothesis, which is not true." In fact, he did his best to get larger "mercy." And he shall have it; and at a length which shall content him, unless his mecometer be an insatiable apparatus. But I fear that in other respects I shall no more satisfy him than the Irish drummer satisfied the poor culprit when, after several times changing the direction of the stroke at earnest entreaty, he was at last provoked to call out, "Bad cess to ye, ye spalpeen! strike where one will, there's no plasing ye!"

Mr. Reddie attaches much force to Berkeley's[653] old arguments against the doctrine of fluxions, and advances objections to Newton's second section, which he takes to be new. To me they appear "such as have been often made," to copy a description given in a review: though I have no doubt Mr. Reddie got them out of himself. But the whole matter comes to this: Mr. Reddie challenged answer, especially from the British Association, and got none. He presumes that this is because he is right, and cannot be answered: the Association is willing to risk itself upon the counter-notion that he is wrong, and need not be answered; because so wrong that none who could understand an answer would be likely to want one.

Mr. Reddie demands my attention to a point which had already particularly struck me, as giving the means of showing to all readers the kind of confusion into which paradoxers are apt to fall, in spite of the clearest instruction. It is a very honest blunder, and requires notice: it may otherwise mislead some, who may suppose that no one able to read could be mistaken about so simple a matter, [ 347 ] let him be ever so wrong about Newton. According to his own mis-statement, in less than five months he made the Astronomer Royal abandon the theory of the solar motion in space. The announcement is made in August, 1865, as follows: the italics are not mine:

"The third (Victoria ...), although only published in September, 1863, has already had its triumph. It is the book that forced the Astronomer Royal of England, after publicly teaching the contrary for years, to come to the conclusion, "strange as it may appear," that "the whole question of solar motion in space is at the present time in doubt and abeyance." This admission is made in the Annual Report of the Council of the Royal Astronomical Society, published in the Society's Monthly Notices for February, 1864."

It is added that solar motion is "full of self-contradiction, which "the astronomers" simply overlooked, but which they dare not now deny after being once pointed out."

The following is another of his accounts of the matter, given in the Correspondent, No. 18, 1865:

"... You ought, when you came to put me in the 'Budget,' to have been aware of the Report of the Council of the Royal Astronomical Society, where it appears that Professor Airy,[654] with a better appreciation of my demonstrations, had admitted—'strange,' say the Council, 'as it may appear,'—that 'the whole question of solar motion in space [and here Mr. Reddie omits some words] is now in doubt and abeyance.' You were culpable as a public teacher of no little pretensions, if you were 'unaware' of this. If aware of it, you ought not to have suppressed such an important testimony to my really having been 'very successful' in drawing the teeth of the pegtops, though you thought them so firmly fixed. And if you still suppress [ 348 ] it, in your Appendix, or when you reprint your 'Budget,' you will then be guilty of a suppressio veri, also of further injury to me, who have never injured you...."

Mr. Reddie must have been very well satisfied in his own mind before he ventured such a challenge, with an answer from me looming in the distance. The following is the passage of the Report of the Council, etc., from which he quotes:

"And yet, strange to say, notwithstanding the near coincidence of all the results of the before-mentioned independent methods of investigation, the inevitable logical inference deduced by Mr. Airy is, that the whole question of solar motion in space, so far at least as accounting for the proper motion of the stars is concerned, [I have put in italics the words omitted by Mr. Reddie] appears to remain at this moment in doubt and abeyance."

Mr. Reddie has forked me, as he thinks, on a dilemma: if unaware, culpable ignorance; if aware, suppressive intention. But the thing is a trilemma, and the third horn, on which I elect to be placed, is surmounted by a doubly-stuffed seat. First, Mr. Airy has not changed his opinion about the fact of solar motion in space, but only suspends it as to the sufficiency of present means to give the amount and direction of the motion. Secondly, all that is alluded to in the Astronomical Report was said and printed before the Victoria proclamation appeared. So that the author, instead of drawing the tooth of the Astronomer Royal's pegtop, has burnt his own doll's nose.

William Herschel,[655] and after him about six other astronomers, had aimed at determining, by the proper motions of the stars, the point of the heavens towards which the solar system is moving: their results were tolerably accordant. Mr. Airy, in 1859, proposed an improved method, and, applying it to stars of large proper motion, produced [ 349 ] much the same result as Herschel. Mr. E. Dunkin,[656] one of Mr. Airy's staff at Greenwich, applied Mr. Airy's method to a very large number of stars, and produced, again, nearly the same result as before. This paper was read to the Astronomical Society in March, 1863, was printed in abstract in the Notice of that month, was printed in full in the volume then current, and was referred to in the Annual Report of the Council in February, 1864, under the name of "the Astronomer Royal's elaborate investigation, as exhibited by Mr. Dunkin." Both Mr. Airy and Mr. Dunkin express grave doubts as to the sufficiency of the data: and, regarding the coincidence of all the results as highly curious, feel it necessary to wait for calculations made on better data. The report of the Council states these doubts. Mr. Reddie, who only published in September, 1863, happened to see the Report of February, 1864, assumes that the doubts were then first expressed, and declares that his book of September had the triumph of forcing the Astronomer Royal to abandon the fact of motion of the solar system by the February following. Had Mr. Reddie, when he saw that the Council were avowedly describing a memoir presented some time before, taken the precaution to find out when that memoir was presented, he would perhaps have seen that doubts of the results obtained, expressed by one astronomer in March, 1863, and by another in 1859, could not have been due to his publication of September, 1863. And any one else would have learnt that neither astronomer doubts the solar motion, though both doubt the sufficiency of present means to determine its amount and direction. This is implied in the omitted words, which Mr. Reddie—whose omission would have been dishonest if he had seen their meaning—no doubt took for pleonasm, superfluity, overmuchness. The rashness which pushed him headlong [ 350 ] into the quillet that his thunderbolt had stopped the chariot of the Sun and knocked the Greenwich Phaeton off the box, is the same which betrayed him into yet grander error—which deserves the full word, quidlibet—about the Principia of Newton. There has been no change of opinion at all. When a person undertakes a long investigation, his opinion is that, at a certain date, there is prima facie ground for thinking a sound result may be obtained. Should it happen that the investigation ends in doubt upon the sufficiency of the grounds, the investigator is not put in the wrong. He knew beforehand that there was an alternative: and he takes the horn of the alternative indicated by his calculations. The two sides of this case present an instructive contrast. Eight astronomers produce nearly the same result, and yet the last two doubt the sufficiency of their means: compare them with the what's-his-name who rushes in where thing-em-bobs fear to tread.

I was not aware, until I had written what precedes, that Mr. Airy had given a sufficient answer on the point. Mr. Reddie says (Correspondent, Jan. 20, 1866):

"I claim to have forced Professor Airy to give up the notion of 'solar motion in space' altogether, for he admits it to be 'at present in doubt and abeyance.' I first made that claim in a letter addressed to the Astronomer Royal himself in June, 1864, and in replying, very courteously, to other portions of my letter, he did not gainsay that part of it."

Mr. Reddie is not ready at reading satire, or he never would have so missed the meaning of the courteous reply on one point, and the total silence upon another. Mr. Airy must be one of those peculiar persons who, when they do not think an assertion worth notice, let it alone, without noticing it by a notification of non-notice. He would never commit the bull of "Sir! I will not say a word on that subject." He would put it thus, "Sir! I will only say ten words on that subject,"—and, having thus said them, would [ 351 ] proceed to something else. He assumed, as a matter of form, that Mr. Reddie would draw the proper inference from his silence: and this because he did not care whether or no the assumption was correct.

The Mechanics of the Heavens, which Mr. Reddie sends to be noticed, shall be noticed, so far as an extract goes:


"My connection with this subject is, indeed, very simply explained. In endeavoring to understand the laws of physical astronomy as generally taught, I happened to entertain some doubt whether gravitating bodies could revolve, and having afterwards imbibed some vague idea that the laws of the universe were chemical and physical rather than mechanical, and somehow connected with electricity and magnetism as opposing correlative forces—most probably suggested to my mind, as to many others, by the transcendent discoveries made in electro-magnetism by Professor Faraday[657]—my former doubts about gravitation were revived, and I was led very naturally to try and discover whether a gravitating body really could revolve; and I became convinced it could not, before I had ever presumed to look into the demonstrations of the Principia."


This is enough against the book, without a word from me: I insert it only to show those who know the subject what manner of writer Mr. Reddie is. It is clear that "presumed" is a slip of the pen; it should have been condescended.

Mr. Reddie represents me as dreaming over paltry paradoxes. He is right; many of my paradoxes are paltry: he is wrong; I am wide awake to them. A single moth, beetle, or butterfly, may be a paltry thing; but when a cabinet is arranged by genus and species, we then begin to admire the [ 352 ] infinite variety of a system constructed on a wonderful sameness of leading characteristics. And why should paradoxes be denied that collective importance, paltry as many of them may individually be, which is accorded to moths, beetles, or butterflies? Mr. Reddie himself sees that "there is a method in" my "mode of dealing with paradoxes." I hope I have atoned for the scantiness of my former article, and put the demonstrated impossibility of gravitation on that level with Hubongramillposanfy arithmetic and inhabited atoms which the demonstrator—not quite without reason—claims for it.

In the Introduction to a collected edition of the three works, Mr. Reddie describes his Mechanism of the Heavens, from which I have just quoted, as—


"a public challenge offered to the British Association and the mathematicians at Cambridge, in August, 1862, calling upon them to point to a single demonstration in the Principia or elsewhere, which even attempts to prove that Universal Gravitation is possible, or to show that a gravitating body could possibly revolve about a center of attraction. The challenge was not accepted, and never will be. No such demonstration exists. And the public must judge for themselves as to the character of a so-called "certain science," which thus shrinks from rigid examination, and dares not defend itself when publicly attacked: also of the character of its teachers, who can be content to remain dumb under such circumstances."



The above is the commonplace talk of the class, of which I proceed to speak without more application to this paradoxer than to that. It reminds one of the funny young rascals who used, in times not yet quite forgotten, to abuse the passengers, as long as they could keep up with the [ 353 ] stage coach; dropping off at last with "Why don't you get down and thrash us? You're afraid, you're afraid!" They will allow the public to judge for themselves, but with somewhat of the feeling of the worthy uncle in Tom Jones, who, though he would let young people choose for themselves, would have them choose wisely. They try to be so awfully moral and so ghastly satirical that they must be answered: and they are best answered in their own division. We have all heard of the way in which sailors cat's-pawed the monkeys: they taunted the dwellers in the trees with stones, and the monkeys taunted them with cocoa-nuts in return. But these were silly dendrobats: had they belonged to the British Association they would have said—No! No! dear friends; it is not in the itinerary: if you want nuts, you must climb, as we do. The public has referred the question to Time: the procedure of this great king I venture to describe, from precedents, by an adaptation of some smart anapæstic tetrameters—your anapæst is the foot for satire to halt on, both in Greek and English—which I read about twenty years ago, and with the point of which I was much tickled. Poetasters were laughed at; but Mr. Slum, whom I employed—Mr. Charles Dickens obliged me with his address—converted the idea into that of a hit at mathematicasters, as easily as he turned the Warren acrostic into Jarley. As he observed, when I settled his little account, it is cheaper than any prose, though the broom was not stolen quite ready made:

Forty stripes save one for the smaller Paradoxers.
Hark to the wisdom the sages preach
Who never have learnt what they try to teach.
We are the lights of the age, they say!
We are the men, and the thinkers we!
So we build up guess-work the livelong day,
In a topsy-turvy sort of way,
Some with and some wanting a plus b.
Let the British Association fuss;
What are theirs to the feats to be wrought by us?
[ 354 ]
Shall the earth stand still? Will the round come square?
Must Isaac's book be the nest of a mare?
Ought the moon to be taught by the laws of space
To turn half round without right-about-face?
Our whimsey crotchets will manage it all;
Deep! Deep! posterity will them call!
Though the world, for the present, lets them fall
Down! Down! to the twopenny box of the stall!
Thus they—But the marplot Time stands by,
With a knowing wink in his funny old eye.
He grasps by the top an immense fool's cap,
Which he calls a philosophaster-trap:
And rightly enough, for while these little men
Croak loud as a concert of frogs in a fen,
He first singles out one, and then another,
Down goes the cap—lo! a moment's pother,
A spirit like that which a rushlight utters
As just at the last it kicks and gutters:
When the cruel smotherer is raised again
Only snuff, and but little of that, will remain.
But though uno avulso thus comes every day
Non deficit alter is also in play:
For the vacant parts are, one and all,
Soon taken by puppets just as small;
Who chirp, chirp, chirp, with a grasshopper's glee,
We're the lamps of the Universe, We! We! We!
But Time, whose speech is never long,—
He hasn't time for it—stops the song
And says—Lilliput lamps! leave the twopenny boxes,
And shine in the Budget of Paradoxes!

When a paradoxer parades capital letters and diagrams which are as good as Newton's to all who know nothing about it, some persons wonder why science does not rise and triturate the whole thing. This is why: all who are fit to read the refutation are satisfied already, and can, if they please, detect the paradoxer for themselves. Those who are not fit to do this would not know the difference between the true answer and the new capitals and diagrams on which the delighted paradoxer would declare [ 355 ] that he had crumbled the philosophers, and not they him. Trust him for having the last word: and what matters it whether he crow the unanswerable sooner or later? There are but two courses to take. One is to wait until he has committed himself in something which all can understand, as Mr. Reddie has done in his fancy about the Astronomer Royal's change of opinion: he can then be put in his true place. The other is to construct a Budget of Paradoxes, that the world may see how the thing is always going on, and that the picture I have concocted by cribbing and spoiling a bit of poetry is drawn from life. He who wonders at there being no answer has seen one or two: he does not know that there are always fifty with equal claims, each of whom regards his being ranked with the rest as forty-nine distinct and several slanders upon himself, the great Mully Ully Gue. And the fifty would soon be five hundred if any notice were taken of them. They call mankind to witness that science will not defend itself, though publicly attacked in terms which might sting a pickpocket into standing up for his character: science, in return, allows mankind to witness or not, at pleasure, that it does not defend itself, and yet receives no injury from centuries of assault. Demonstrative reason never raises the cry of Church in Danger! and it cannot have any Dictionary of Heresies except a Budget of Paradoxes. Mistaken claimants are left to Time and his extinguisher, with the approbation of all thinking non-claimants: there is no need of a succession of exposures. Time gets through the job in his own workmanlike manner as already described.

On looking back more than twenty years, I find among my cuttings the following passage, relating to a person who had signalized himself by an effort to teach comets to the conductor of the Nautical Almanac:


"Our brethren of the literary class have not the least idea of the small amount of appearance of knowledge [ 356 ] which sets up the scientific charlatan. Their world is large, and there are many who have that moderate knowledge, and perception of what is knowledge, before which extreme ignorance is detected in its first prank. There is a public of moderate cultivation, for the most part sound in its judgment, always ready in its decisions. Accordingly, all their successful pretenders have some pretension. It is not so in science. Those who have a right to judge are fewer and farther between. The consequence is, that many scientific pretenders have nothing but pretension."


This is nearly as applicable now as then. It is impossible to make those who have not studied for themselves fully aware of the truth of what I have quoted. The best chance is collection of cases; in fact, a Budget of Paradoxes. Those who have no knowledge of the subject can thus argue from the seen to the unseen. All can feel the impracticability of the Hubongramillposanfy numeration, and the absurdity of the equality of contour of a regular pentagon and hexagon in one and the same circle. Many may accordingly be satisfied, on the assurance of those who have studied, that there is as much of impracticability, or as much of absurdity, in things which are hidden under

"Sines, tangents, secants, radius, cosines
Subtangents, segments and all those signs;
Enough to prove that he who read 'em
Was just as mad as he who made 'em."

Not that I mean to be disrespectful to mathematical terms: they are short and easily explained, and compete favorably with those of most other subjects: for instance, with

"Horse-pleas, traverses, demurrers,
Jeofails, imparlances, and errors,
Averments, bars, and protestandos,
And puis d'arreign continuandos."

[ 357 ]

From which it appears that, taking the selections made by satirists for our samples, there are, one with another, four letters more in a law term than in one of mathematics. But pleading has been simplified of late years.

All paradoxers can publish; and any one who likes may read. But this is not enough; they find that they cannot publish, or those who can find they are not read, and they lay their plans athwart the noses of those who, they think, ought to read. To recommend them to be content with publication, like other authors, is an affront: of this I will give the reader an amusing instance. My good nature, of which I keep a stock, though I do not use it all up in this Budget, prompts me to conceal the name.

I received the following letter, accompanied by a prospectus of a work on metaphysics, physics, astronomy, etc. The author is evidently one whom I should delight to honor:

"Sir,—A friend of mine has mentioned your name in terms of panigeric [sic], as being of high standing in mathematics, and of greatly original thought. I send you the enclosed without comment; and, assuming that the bent of your mind is in free inquiry, shall feel a pleasure in showing you my portfolio, which, as a mathematician, you will acknowledge to be deeply interesting, even in an educational point of view. The work is complete, and the system so far perfected as to place it above criticism; and, so far as regards astronomy, as will Ptolemy beyond rivalry [sic: no doubt some words omitted]. Believe me to be, Sir, with the profoundest respect, etc. The work is the result of thirty-five years' travel and observation, labor, expense, and self-abnegation."

I replied to the effect that my time was fully occupied, and that I was obliged to decline discussion with many persons who have views of their own; that the proper way is to publish, so that those who choose may read when they can find leisure. I added that I should advise a precursor in the shape of a small pamphlet, as two octavo volumes [ 358 ] would be too much for most persons. This was sound advice; but it is not the first, second, or third time that it has proved very unpalatable. I received the following answer, to which I take the liberty of prefixing a bit of leonine wisdom:

"Si doceas stultum, lætum non dat tibi vultum;
Odit te multum; vellet te scire sepultum.[658]"

"Sir,—I pray you pardon the error I unintentionally have fallen into; deceived by the F.R.S. [I am not F.R.S.] I took you to be a man of science [omnis homo est animal, Sortes est homo, ergo Sortes est animal][659] instead of the mere mathematician, or human calculating-machine. Believe me, Sir, you also have mistaken your mission, as I have mine. I wrote to you as I would to any other man well up in mathematics, with the intent to call your attention to a singular fact of omission by Euclid, and other great mathematicians: and, in selecting you, I did you an honor which, from what I have just now heard, was entirely out of place. I think, considering the nature of the work set forth in the prospectus, you are guilty of both folly and presumption, in assuming the character of a patron; for your own sense ought to have assured you that was such my object I should not have sought him in a De Morgan, who exists only by patronage of others. On the other hand, I deem it to be an unpardonable piece of presumption in offering your advice upon a subject the magnitude, importance, and real utility of which you know nothing about: by doing so you have offered me a direct insult. The system is a manual of Philosophy, a one inseparable whole of metaphysics and physic; embracing points the most interesting, laws the most important, [ 359 ] doctrines the most essential to advance man in accordance with the spirit of the times. I may not live to see it in print; for, at ——, life at best is uncertain: but, live or die, be assured Sir, it is not my intention to debase the work by seeking patronage, or pandering to the public taste. Your advice was the less needed, seeing I am an old-established ——. I remain, etc.—P.S. You will oblige me by returning the prospectus of my work."


My reader will, I am sure, not take this transition from the "profoundest respect" to the loftiest insolence for an apocraphical correspondence, to use a word I find in the Prospectus: on my honor it is genuine. He will be better employed in discovering whether I exist by patronizing others, or by being patronized by them. I make any one who can find it out a fair offer: I will give him my patronage if I turn out to be Bufo, on condition he gives me his, if I turn out to be Bavius.[660] I need hardly say that I considered the last letter to be one of those to which no answer is so good as no answer.

These letters remind me in one respect of the correspondents of the newspapers. My other party wrote because a friend had pointed me out: but he would not have written if he had known what another friend told him just in time for the second letter. The man who sends his complaint to the newspaper very often says, in effect, "Don't imagine, Sir, that I read your columns; but a friend who sometimes does has told me ..." It is worded thus: "My attention [ 360 ] has been directed to an article in your paper of ..." Many thanks to my friend's friends for not mentioning the Budget: had my friend's attention been directed to it I might have lost a striking example of the paradoxer in search of a patron. That my Friend was on this scent in the first letter is revealed in the second. Language was given to man to conceal his thoughts; but it is not every one who can do it.

Among the most valuable information which my readers will get from me is comparison of the reactions of paradoxers, when not admitted to argument, or when laughed at. Of course, they are misrepresented; and at this they are angry, or which is the same thing, take great pains to assure the reader that they are not. So far natural, and so far good; anything short of concession of a case which must be seriously met by counter-reasons is sure to be misrepresentation. My friend Mr. James Smith and my friend Mr. Reddie are both terribly misrepresented: they resent it by some insinuations in which it is not easy to detect whether I am a conscious smotherer of truth, or only muddle-headed and ignorant. [This was written before I received my last communication from Mr. James Smith. He tells me that I am wrong in saying that his work in which I stand in the pillory is all reprint: I have no doubt I confounded some of it with some of the manuscript or slips which I had received from my much not-agreed-with correspondent. He adds that my mistake was intentional, and that my reason is obvious to the reader. This is information, as the sea-serpent said when he read in the newspaper that he had a mane and tusks.]



My friend Dr. Thorn[661] sees deeper into my mystery. By the way, he still sends an occasional touch at the old [ 361 ] subject; and he wants me particularly to tell my readers that the Latin numeral letters, if M be left out, give 666. And so they do: witness DCLXVI. A person who thinks of the origin of symbols will soon see that 666 is our number because we have five fingers on each hand: had we had but four, our mystic number would have been expressed by 555, and would have stood for our present 365. Had n been the number on each hand, the great number would have been

(n + 1) (4n2 + 2n + 1)

With no finger on each hand, the number would have been 1: with one finger less than none at all on each hand, it would have been 0. But what does this mean? Here is a question for an algebraical paradoxer! So soon as we have found out how many fingers the inhabitants of any one planet have on each hand, we have the means of knowing their number of the Beast, and thence all about them. Very much struck with this hint of discovery, I turned my attention to the means of developing it. The first point was to clear my vision of all the old cataracts. I propose the following experiment, subject of course to the consent of parties. Let Dr. Thorn Double-Vahu Mr. James Smith, and Thau Mr. Reddie: if either be deparadoxed by the treatment, I will consent to undergo it myself. Provided always that the temperature required be not so high as the Doctor hints at: if the Turkish Baths will do for this world, I am content.

The three paradoxers last named and myself have a pentasyllable convention, under which, though we go far beyond civility, we keep within civilization. Though Mr. James Smith pronounced that I must be dishonest if I did not see his argument, which he knew I should not do [to say nothing of recent accusation]; though Dr. Thorn declared me a competitor for fire and brimstone—and my wife, too, which doubles the joke: though Mr. Reddie [ 362 ] was certain I had garbled him, evidently on purpose to make falsehood appear truth; yet all three profess respect for me as to everything but power to see truth, or candor to admit it. And on the other hand, though these were the modes of opening communication with me, and though I have no doubt that all three are proper persons of whom to inquire whether I should go up-stairs or down-stairs, etc., yet I am satisfied they are thoroughly respectable men, as to everything but reasoning. And I dare say our several professions are far more true in extent than in many which are made under more parliamentary form. We find excuses for each other: they make allowances for my being hoodwinked by Aristotle, by Newton, by the Devil; and I permit them to feel, for I know they cannot get on without it, that their reasons are such as none but a knave or a sinner can resist. But they are content with cutting a slice each out of my character: neither of them is more than an uncle, a Bone-a-part; I now come to a dreadful nephew, Bone-the-whole.

I will not give the name of the poor fellow who has fallen so far below both the honestum and the utile, to say nothing of the decorum or the dulce.[662] He is the fourth who has taken elaborate notice of me; and my advice to him would be, Nec quarta loqui persona laboret.[663] According to him, I scorn humanity, scandalize learning, and disgrace the press; it admits of no manner of doubt that my object is to mislead the public and silence truth, at the expense of the interests of science, the wealth of the nation, and the lives of my fellow men. The only thing left to be settled is, whether this is due to ignorance, natural distaste for truth, personal malice, a wish to curry favor with the Astronomer Royal, or mere toadyism. The only accusation which has truth in it is, that I have made myself a "public scavenger of science": the assertion, which is the [ 363 ] most false of all is, that the results of my broom and spade are "shot right in between the columns of" the Athenæum. I declare I never in my life inserted a word between the columns of the Athenæum: I feel huffed and miffed at the very supposition. I have made myself a public scavenger; and why not? Is the mud never to be collected into a heap? I look down upon the other scavengers, of whom there have been a few—mere historical drudges; Montucla, Hutton, etc.—as not fit to compete with me. I say of them what one crossing-sweeper said of the rest: "They are well enough for the common thing; but put them to a bit of fancy-work, such as sweeping round a post, and see what a mess they make of it!" Who can touch me at sweeping round a paradoxer? If I complete my design of publishing a separate work, an old copy will be fished up from a stall two hundred years hence by the coming man, and will be described in an article which will end by his comparing our century with his own, and sighing out in the best New Zealand pronunciation—

"Dans ces tems-là
C'était déjà comme ça!"[664]



And pray, Sir! I have been asked by more than one—do your orthodox never fall into mistake, nor rise into absurdity? They not only do both, but they admit it of each other very freely; individually, they are convinced of sin, but not of any particular sin. There is not a syndoxer among them all but draws his line in such a way as to include among paradoxers a great many whom I should exclude altogether from this work. My worst specimens are but exaggerations of what may be found, occasionally, in the thoughts of sagacious investigators. At the end of the [ 364 ] glorious dream, we learn that there is a way to Hell from the gates of Heaven, as well as from the City of Destruction: and that this is true of other things besides Christian pilgrimage is affirmed at the end of the Budget of Paradoxes. If D'Alembert[665] had produced enough of a quality to match his celebrated mistake on the chance of throwing head in two throws, he would have been in my list. If Newton had produced enough to match his reception of the story that Nausicaa, Homer's Phæacian princess, invented the celestial sphere, followed by his serious surmise that she got it from the Argonauts,—then Newton himself would have had an appearance entered for him, in spite of the Principia. In illustration, I may cite a few words from Tristram Shandy:


"'A soldier,' cried my uncle Toby, interrupting the Corporal, 'is no more exempt from saying a foolish thing, Trim, than a man of letters.'—'But not so often, an' please your honor,' replied the Corporal. My uncle Toby gave a nod."


I now proceed to die out. Some prefatory remarks will follow in time.[666] I shall have occasion to insist that all is not barren: I think I shall find, on casting up, that two out of five of my paradoxers are not to be utterly condemned. Among the better lot will be found all gradations of merit; at the same time, as was remarked on quite a different subject, there may be little to choose between the last of the saved and the first of the lost. The higher and better class is worthy of blame; the lower and worse class is worthy of praise. The higher men are to be reproved for not taking up things in which they could do some good: the lower men are to be commended for taking up things in which they can do no great harm. The circle problem is like Peter Peebles's lawsuit:

[ 365 ]


"'But, Sir, I should really spoil any cause thrust on me so hastily.'—'Ye cannot spoil it, Alan,' said my father, 'that is the very cream of the business, man,—... the case is come to that pass that Stair or Arniston could not mend it, and I don't think even you, Alan, can do it much harm.'"


I am strongly reminded of the monks in the darker part of the Middle Ages. To a certain proportion of them, perhaps two out of five, we are indebted for the preservation of literature, and their contemporaries for good teaching and mitigation of socials evils. But the remaining three were the fleas and flies and thistles and briars with whom the satirist lumps them, about a century before the Reformation:

"Flen, flyys, and freris, populum domini male cædunt;
Thystlis and breris crescentia gramina lædunt.
Christe nolens guerras qui cuncta pace tueris,
Destrue per terras breris, flen, flyys, and freris.
Flen, flyys, and freris, foul falle hem thys fyften yeris,
For non that her is lovit flen, flyys ne freris."[667]

I should not be quite so savage with my second class. Taken together, they may be made to give useful warning to those who are engaged in learning under better auspices: aye, even useful hints; for bad things are very often only good things spoiled or misused. My plan is that of a predecessor in the time of Edward the Second:

"Meum est propositum genti imperitæ
Artes frugi reddere melioris vitæ.[668]

To this end I have spoken with freedom of books as books, of opinions as opinions, of ignorance as ignorance, of [ 366 ] presumption as presumption; and of writers as I judge may be fairly inferred from what they have written. Some—to whom I am therefore under great obligation—have permitted me to enlarge my plan by assaults to which I have alluded; assaults which allow a privilege of retort, of which I have often availed myself; assaults which give my readers a right of partnership in the amusement which I myself have received.

For the present I cut and run: a Catiline, pursued by a chorus of Ciceros, with Quousque tandem? Quamdiu nos? Nihil ne te?[669] ending with, In te conferri pestem istam jam pridem oportebat, quam tu in nos omnes jamdiu machinaris! I carry with me the reflection that I have furnished to those who need it such a magazine of warnings as they will not find elsewhere; a signatis cavetote:[670] and I throw back at my pursuers—Valete, doctores sine doctrina; facite ut proxima congressu vos salvos corporibus et sanos mentibus videamus.[671] Here ends the Budget of Paradoxes.


621 ^  On Tycho Brahe see Vol. I, note 112.

622 ^  Wilhelm Frederik von Zytphen also published the Tidens Ström, a chronological table, in 1840. The work to which De Morgan refers, the Solens Bevægelse i Verdensrummet, appeared first in 1861. De Morgan seems to have missed his Nogl Ord om Cirkelens Quadratur which appeared in 1865, at Copenhagen.

623 ^  James Joseph Sylvester (1814-1897), professor of natural philosophy at University College, London (1837-1841), professor of mathematics at the University of Virginia (1841-1845), actuary in London (1845-1855), professor of mathematics at Woolwich (1877-1884) and at Johns Hopkins University, Baltimore (1877-1884), and Savilian professor of geometry at Oxford (1884-1894).

624 ^  See Vol. I, note 112.

625 ^  See Vol. II, note 349.

626 ^  See Vol. I, note 112.

627 ^  See Vol. I, note 42.

628 ^  See Vol. II, note 318.

629 ^  See Vol. I, note 691.

630 ^  James Mill, born 1773, died 1836.

631 ^  See Vol. II, note 11.

632 ^  See Vol. II, note 13.

633 ^  See Vol. II, note 14.

634 ^  This anecdote is printed at page 4 (Vol. II); but as it is used in illustration here, and is given more in detail, I have not omitted it.—S.E. De M.

635 ^  See Vol. II, note 15.

636 ^  See Vol. I, note 786.

637 ^  "Monsieur, \scriptstyle\frac{(a+b^n)}{n}=x,, whence God exists; answer that!"

638 ^  "Monsieur, you know very well that your argument requires the development of x according to integral powers of n."

639 ^  See Vol. I, note 337.

640 ^  Thomas Love Peacock (1785-1866) an English novelist and poet.

641 ^  Perhaps Dr. Samuel Warren (1807-1877), the author of Ten Thousand a Year (serially in Blackwood's in 1839; London, 1841).

642 ^  See Vol. I, note 584.

643 ^  "From many, one; much in little; Ultima Thule (the most remote region); without which not."

644 ^  Spurius Mælius (fl. 440 B. C.), who distributed corn freely among the poor in the famine of 440 B. C. and was assassinated by the patricians.

645 ^  Spurius Cassius Viscellinus, Roman consul in 502, 493, and 486 B. C. Put to death in 485.

646 ^  "O what a fine bearing, he said, that has no brain."

647 ^  Sir William Rowan Hamilton. See Vol. I, note 709.

648 ^  William Allen Whitworth, the author of the well-known Choice and Chance (Cambridge, 1867), and other works.

649 ^  James Maurice Wilson, whose Elementary Geometry appeared in 1868 and went through several editions.

650 ^  See Vol. II, note 315.

651 ^  "Force of inertia conquered," and "Victory in the whole heavens."

652 ^  "With all his might."

653 ^  George Berkeley (1685-1753), Bishop of Cloyne, the idealistic philosopher and author of the Principles of Human Knowledge (1710), The Analyst, or a Discourse addressed to an Infidel Mathematician (1734), and A Defense of Freethinking in Mathematics (1735). He asserted that space involves the idea of movement without the sensation of resistance. Space sensation less than the "minima sensibilia" is, therefore, impossible. From this he argues that infinitesimals are impossible concepts.

654 ^  See Vol. I, note 129.

655 ^  See Vol. I, note 120.

656 ^  Edwin Dunkin revised Lardner's Handbook of Astronomy (1869) and Milner's The Heavens and the Earth (1873) and wrote The Midnight Sky (1869).

657 ^  Michael Faraday (1791-1867) the celebrated physicist and chemist. He was an assistant to Sir Humphrey Davy (1813) and became professor of chemistry at the Royal Institution, London, in 1827.

658 ^  "If you teach a fool he shows no joyous countenance; he cordially hates you; he wishes you buried."

659 ^  "Every man is an animal, Sortes is a man, therefore Sortes is an animal."

660 ^ 

"May some choice patron bless each grey goose quill;
May every Bavius have his Bufo still."—Pope, Prologue to the Satires.

Bavius has become proverbial as a bad poet from the lines in Vergil's Eclogues (III, 90):

"Qui Bavium non odit, amet tua carmina, Maevi,
Atque idem jungat vulpes, et mulgeat hircos."

"He who does not hate Bavius shall love thy verses, O Maevius; and the same shall yoke foxes and shall milk he-goats."

Bavius and Maevius were the worst of Latin poets, condemned by Horace as well as Vergil.

661 ^  See Vol. II, note 279.

662 ^  "Honest," "useful," "handsome," "sweet."

663 ^  "Let not the fourth man attempt to speak."

664 ^ 

"In those old times,—ah
'Twas just like this, ah!"

665 ^  See Vol. I, note 785.

666 ^  These remarks were never written.—S. E. De M.

667 ^ 

"Fleas, flies, and friars, are masters who sadly the people abuse,
And thistles and briars are sure growing grains to abuse.
O Christ, who hatest strife and slayst all things in peace,
Destroy where'er are rife, briars, friars, flies and fleas.
Fleas, flies, and friars foul fall them these fifteen years
For none that there is loveth fleas, flies, nor freres."

668 ^  "It is my plan to restore to an unskilled race the worthy arts of a better life."

669 ^  The first sentences of the first oration of Cicero against Catiline: "Quo usque tandem abutere, Catilina, patientia nostra?" (How long, O Catiline, will you abuse our patience?) "Quamdiu etiam furor iste tuus nos eludet?" (How long will this your madness baffle us?) "Nihilne te nocturnum praesidium Palati, ... nihil horum ora voltusque moverunt?" (Does the night watch of the Palatium, ... do the faces and expressions of all these men fail to move you?) "In te conferri ..." (This plague should have been inflicted upon you long ago, which you have plotted against us so long.)

670 ^  "Beware of the things that are marked."

671 ^  "Farewell, ye teachers without learning! See to it that at our next meeting we may find you strong in body and sound in mind."