Budget of Paradoxes/K

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A Budget of Paradoxes by Augustus De Morgan

1850. A letter in the handwriting of an educated man, dated from a street in which it must be taken that educated persons live, is addressed to the Secretary of the [ 45 ] Astronomical Society about a matter on which the writer says "his professional pursuit will enable him to give a satisfactory reply." In a question before a court of law it is sworn on one side that the moon was shining at a certain hour of a certain night on a certain spot in London; on the other side it is affirmed that she was clouded. The Secretary is requested to decide. This is curious, as the question is not astrological. Persons still send to Greenwich, now and then, to have their fortunes told. In one case, not very many years ago, a young gentleman begged to know who his wife was to be, and what fee he was to remit.

Sometimes the astronomer turns conjurer for fun, and his prophesies are fulfilled. It is related of Flamsteed[87] that an old woman came to know the whereabouts of a bundle of linen which had strayed. Flamsteed drew a circle, put a square into it, and gravely pointed out a ditch, near her cottage, in which he said it would be found. He meant to have given the woman a little good advice when she came back: but she came back in great delight, with the bundle in her hand, found in the very place. The late Baron Zach[88] received a letter from Pons,[89] a successful finder of comets, complaining that for a certain period he had found no comets, though he had searched diligently. Zach, a man of much sly humor, told him that no spots had been seen on the sun for about the same time—which was true,—and assured him that when the spots came back, the comets would come with them. Some time after he got a letter [ 46 ] from Pons, who informed him with great satisfaction that he was quite right, that very large spots had appeared on the sun, and that he had found a fine comet shortly after. I do not vouch for the first story, but I have the second in Zach's handwriting. It would mend the joke exceedingly if some day a real relation should be established between comets and solar spots: of late years good reason has been shown for advancing a connection between these spots and the earth's magnetism.[90] If the two things had been put to Zach, he would probably have chosen the comets. Here is a hint for a paradox: the solar spots are the dead comets, which have parted with their light and heat to feed the sun, as was once suggested. I should not wonder if I were too late, and the thing had been actually maintained. My list does not contain the twentieth part of the possible whole.

The mention of coincidences suggests an everlasting source of explanations, applicable to all that is extraordinary. The great paradox of coincidence is that of Leibnitz, known as the pre-established harmony, or law of coincidences, by which, separately and independently, the body receives impressions, and the mind proceeds as if it had perceived them from without. Every sensation, and the consequent state of the soul, are independent things coincident in time by the pre-established law. The philosopher could not otherwise account for the connection of mind and matter; and he never goes by so vulgar a rule as Whatever is, is; to him that which is not clear as to how, is not at all. Philosophers in general, who tolerate each other's theories much better than Christians do each other's failings, seldom revive Leibnitz's fantasy: they seem to act upon the maxim quoted by Father Eustace[91] from the [ 47 ] Decretals, Facinora ostendi dum puniuntur, flagitia autem abscondi debent.[92]

The great ghost-paradox, and its theory of coincidences, will rise to the surface in the mind of every one. But the use of the word coincidence is here at variance with its common meaning. When A is constantly happening, and also B, the occurrence of A and B at the same moment is the mere coincidence which may be casualty. But the case before us is that A is constantly happening, while B, when it does happen, almost always happens with A, and very rarely without it. That is to say, such is the phenomenon asserted: and all who rationally refer it to casualty, affirm that B is happening very often as well as A, but that it is not thought worthy of being recorded except when A is simultaneous. Of course A is here a death, and B the spectral appearance of the person who dies. In talking of this subject it is necessary to put out of the question all who play fast and loose with their secret convictions: these had better give us a reason, when they feel internal pressure for explanation, that there is no weathercock at Kilve; this would do for all cases. But persons of real inquiry will see that first, experience does not bear out the asserted frequency of the spectre, without the alleged coincidence of death: and secondly, that if the crowd of purely casual spectres were so great that it is no wonder that, now and then the person should have died at or near the moment, we ought to expect a much larger proportion of cases in which the spectre should come at the moment of the death of one or another of all the cluster who are closely connected with the original of the spectre. But this, we know, is almost without example. It remains then, for all, who speculate at all, to look upon the asserted phenomenon, think what they may of it, the thing which is to be explained, as a connection in time of the death, and the [ 48 ] simultaneous appearance of the dead. Any person the least used to the theory of probabilities will see that purely casual coincidence, the wrong spectre being comparatively so rare that it may be said never to occur, is not within the rational field of possibility.

The purely casual coincidence, from which there is no escape except the actual doctrine of special providences, carried down to a very low point of special intention, requires a junction of the things the like of each of which is always happening. I will give three instances which have occurred to myself within the last few years: I solemnly vouch for the literal truth of every part of all three:

In August 1861, M. Senarmont,[93] of the French Institute, wrote to me to the effect that Fresnel[94] had sent to England, in or shortly after 1824, a paper for translation and insertion in the European Review, which shortly afterwards expired. The question was what had become of that paper. I examined the Review at the Museum, found no trace of the paper, and wrote back to that effect at the Museum, adding that everything now depended on ascertaining the name of the editor, and tracing his papers: of this I thought there was no chance. I posted this letter on my way home, at a Post Office in the Hampstead Road at the junction with Edward Street, on the opposite side of which is a bookstall. Lounging for a moment over the exposed books, sicut meus est mos,[95] I saw, within a few minutes of the posting of the letter, a little catch-penny book of anecdotes of Macaulay, which I bought, and ran over for a minute. My eye was soon caught by this sentence: "One of the young fellows immediately wrote to the editor (Mr. Walker) [ 49 ] of the European Review." I thus got the clue by which I ascertained that there was no chance of recovering Fresnel's paper. Of the mention of current reviews, not one in a thousand names the editor.

In the summer of 1865 I made my first acquaintance with the tales of Nathaniel Hawthorne, and the first I read was about the siege of Boston in the War of Independence. I could not make it out: everybody seemed to have got into somebody else's place. I was beginning the second tale, when a parcel arrived: it was a lot of old pamphlets and other rubbish, as he called it, sent by a friend who had lately sold his books, had not thought it worth while to send these things for sale, but thought I might like to look at them and possibly keep some. The first thing I looked at was a sheet which, being opened, displayed "A plan of Boston and its environs, shewing the true situation of his Majesty's army and also that of the rebels, drawn by an engineer, at Boston Oct. 1775." Such detailed plans of current sieges being then uncommon, it is explained that "The principal part of this plan was surveyed by Richard Williams, Lieutenant at Boston; and sent over by the son of a nobleman to his father in town, by whose permission it was published." I immediately saw that my confusion arose from my supposing that the king's troops were besieging the rebels, when it was just the other way.

April 1, 1853, while engaged in making some notes on a logical point, an idea occurred which was perfectly new to me, on the mode of conciliating the notions omnipresence and indivisibility into parts. What it was is no matter here: suffice it that, since it was published elsewhere (in a paper on Infinity, Camb. Phil. Trans. vol. xi. p. 1) I have not had it produced to me. I had just finished a paragraph on the subject, when a parcel came in from a bookseller containing Heywood's[96] Analysis of Kant's Critick, 1844.

[ 50 ] On turning over the leaves I found (p. 109) the identical thought which up to this day, I only know as in my own paper, or in Kant. I feel sure I had not seen it before, for it is in Kant's first edition, which was never translated to my knowledge; and it does not appear in the later editions. Mr. Heywood gives some account of the first edition.

In the broadsheet which gave account of the dying scene of Charles II, it is said that the Roman Catholic priest was introduced by P. M. A. C. F. The chain was this: the Duchess of Portsmouth[97] applied to the Duke of York, who may have consulted his Cordelier confessor, Mansuete, about procuring a priest, and the priest was smuggled into the king's room by the Duchess and Chiffinch.[98] Now the letters are a verbal acrostic of Père Mansuete a Cordelier Friar, and a syllabic acrostic of PortsMouth and ChifFinch. This is a singular coincidence. Macaulay adopted the first interpretation, preferring it to the second, which I brought before him as the conjecture of a near relative of my own. But Mansuete is not mentioned in his narrative: it may well be doubted whether the writer of a broadside for English readers would use Père instead of Father. And the person who really "reminded" the Duke of "the duty he owed to his brother," was the Duchess and not Mansuete. But my affair is only with the coincidence.

But there are coincidences which are really connected without the connection being known to those who find in them matter of astonishment. Presentiments furnish marked cases: sometimes there is no mystery to those who have the clue. In the Gentleman's Magazine (vol. 80, part 2, p. 33) we read, the subject being presentiment of death, as follows: "In 1778, to come nearer the recollection of [ 51 ] survivors, at the taking of Pondicherry, Captain John Fletcher, Captain De Morgan, and Lieutenant Bosanquet, each distinctly foretold his own death on the morning of his fate." I have no doubt of all three; and I knew it of my grandfather long before I read the above passage. He saw that the battery he commanded was unduly exposed: I think by the sap running through the fort when produced. He represented this to the engineer officers, and to the commander-in-chief; the engineers denied the truth of the statement, the commander believed them, my grandfather quietly observed that he must make his will, and the French fulfilled his prediction. His will bore date the day of his death; and I always thought it more remarkable than the fulfilment of the prophecy that a soldier should not consider any danger short of one like the above, sufficient reason to make his will. I suppose the other officers were similarly posted. I am told that military men very often defer making their wills until just before an action: but to face the ordinary risks intestate, and to wait until speedy death must be the all but certain consequence of a stupid mistake, is carrying the principle very far. In the matter of coincidences there are, as in other cases, two wonderful extremes with every intermediate degree. At one end we have the confident people who can attribute anything to casual coincidence; who allow Zadok Imposture and Nathan Coincidence to anoint Solomon Selfconceit king. At the other end we have those who see something very curious in any coincidence you please, and whose minds yearn for a deep reason. A speculator of this class happened to find that Matthew viii. 28-33 and Luke viii. 26-33 contain the same account, that of the demons entering into the swine. Very odd! chapters tallying, and verses so nearly: is the versification rightly managed? Examination is sure to show that there are monstrous inconsistencies in the mode of division, which being corrected, the verses tally as well as the chapters. And then how comes it? I cannot go on, [ 52 ] for I have no gift at torturing a coincidence, but I would lay twopence, if I could make a bet—which I never did in all my life—that some one or more of my readers will try it. Some people say that the study of chances tends to awaken a spirit of gambling: I suspect the contrary. At any rate, I myself, the writer of a mathematical book and a comparatively popular book, have never laid a bet nor played for a stake, however small: not one single time.

It is useful to record such instances as I have given, with precision and on the solemn word of the recorder. When such a story as that of Flamsteed is told, a priori assures us that it could not have been: the story may have been a ben trovato,[99] but not the bundle. It is also useful to establish some of the good jokes which all take for inventions. My friend Mr. J. Bellingham Inglis,[100] before 1800, saw the tobacconist's carriage with a sample of tobacco in a shield, and the motto Quid rides[101] (N. & Q., 3d S. i. 245). His father was able to tell him all about it. The tobacconist was Jacob Brandon, well known to the elder Mr. Inglis, and the person who started the motto, the instant he was asked for such a thing, was Harry Calender of Lloyd's, a scholar and a wit. My friend Mr. H. Crabb Robinson[102] remembers the King's Counsel (Samuel Marryat) who took the motto Causes produce effects, when his success enabled him to start a carriage.

The coincidences of errata are sometimes very remarkable: it may be that the misprint has a sting. The death of Sir W. Hamilton[103] of Edinburgh was known in London on a Thursday, and the editor of the Athenæum wrote to [ 53 ] me in the afternoon for a short obituary notice to appear on Saturday. I dashed off the few lines which appeared without a moment to think: and those of my readers who might perhaps think me capable of contriving errata with meaning will, I am sure, allow the hurry, the occasion, and my own peculiar relation to the departed, as sufficient reasons for believing in my entire innocence. Of course I could not see a proof: and two errata occurred. The words "addition to Stewart"[104] require "for addition to read edition of." This represents what had been insisted on by the Edinburgh publisher, who, frightened by the edition of Reid,[105] had stipulated for a simple reprint without notes. Again "principles of logic and mathematics" required "for mathematics read metaphysics." No four words could be put together which would have so good a title to be Hamilton's motto.


April 1850, found in the letter-box, three loose leaves, well printed and over punctuated, being

Chapter VI. Brethren, lo I come, holding forth the word of life, for so I am commanded.... Chapter VII. Hear my prayer, O generations! and walk by the way, to drink the waters of the river.... Chapter VIII. Hearken o earth, earth, earth, and the kings of the earth, and their armies....

A very large collection might be made of such apostolic writings. They go on well enough in a misty—meant for mystical—imitation of St. Paul or the prophets, until at last some prodigious want of keeping shows the education of the writer. For example, after half a page which might [ 54 ] pass for Irving's[106] preaching—though a person to whom it was presented as such would say that most likely the head and tail would make something more like head and tail of it—we are astounded by a declaration from the Holy Spirit, speaking of himself, that he is "not ashamed of the Gospel of Christ." It would be long before we should find in educated rhapsody—of which there are specimens enough—such a thing as a person of the Trinity taking merit for moral courage enough to stand where St. Peter fell. The following declaration comes next—"I will judge between cattle and cattle, that use their tongues."



The figure of the earth. By J. L. Murphy,[107] of Birmingham. (London and Birmingham, 4 pages, 12mo.) (1850?)

Mr. Murphy invites attention and objection to some assertions, as that the earth is prolate, not oblate. "If the philosopher's conclusion be right, then the pole is the center of a valley (!) thirteen miles deep." Hence it would be very warm. It is answer enough to ask—Who knows that it is not?


 *** A paragraph in the MS. appears to have been inserted in this place by mistake. It will be found in the Appendix at the end of this volume.—S. E. De M.



1851. The following letter was written by one of a class of persons whom, after much experience of them, I [ 55 ] do not pronounce insane. But in this case the second sentence gives a suspicion of actual delusion of the senses; the third looks like that eye for the main chance which passes for sanity on the Stock Exchange and elsewhere:

15th Sept. 1851.

"Gentlemen,—I pray you take steps to make known that yesterday I completed my invention which will give motion to every country on the Earth;—to move Machinery!—the long sought in vain 'Perpetual Motion'!!—I was supported at the time by the Queen and H.R.H. Prince Albert. If, Gentlemen, you can advise me how to proceed to claim the reward, if any is offered by the Government, or how to secure the Patent for the machine, or in any way assist me by advice in this great work, I shall most graciously acknowledge your consideration.

These are my convictions that my SEVERAL discoveries will be realized: and this great one can be at once acted upon: although at this moment it only exists in my mind, from my knowledge of certain fixed principles in nature:—the Machine I have not made, as I only completed the discovery YESTERDAY, Sunday!

I have, etc. —— ——"
To the Directors of the
London University, Gower Street.



The Divine Drama of History and Civilisation. By the Rev. James Smith, M.A.[108] London, 1854, 8vo.

I have several books on that great paradox of our day, Spiritualism, but I shall exclude all but three. The bibliography of this subject is now very large. The question is one both of evidence and speculation;—Are the facts [ 56 ] true? Are they caused by spirits? These I shall not enter upon: I shall merely recommend this work as that of a spiritualist who does not enter on the subject, which he takes for granted, but applies his derived views to the history of mankind with learning and thought. Mr. Smith was a man of a very peculiar turn of thinking. He was, when alive, the editor, or an editor, of the Family Herald: I say when alive, to speak according to knowledge; for, if his own views be true, he may have a hand in it still. The answers to correspondents, in his time, were piquant and original above any I ever saw. I think a very readable book might be made out of them, resembling "Guesses at Truth:" the turn given to an inquiry about morals, religion, or socials, is often of the highest degree of unexpectedness; the poor querist would find himself right in a most unpalatable way.

Answers to correspondents, in newspapers, are very often the fag ends of literature. I shall never forget the following. A person was invited to name a rule without exception, if he could: he answered "A man must be present when he is shaved." A lady—what right have ladies to decide questions about shaving?—said this was not properly a rule; and the oracle was consulted. The editor agreed with the lady; he said that "a man must be present when he is shaved" is not a rule, but a fact.


[Among my anonymous communicants is one who states that I have done injustice to the Rev. James Smith in "referring to him as a spiritualist," and placing his "Divine Drama" among paradoxes: "it is no paradox, nor do spiritualistic views mar or weaken the execution of the design." Quite true: for the design is to produce and enforce "spiritualistic views"; and leather does not mar nor weaken a shoemaker's plan. I knew Mr. Smith well, and have often talked to him on the subject: but more testimony from me is unnecessary; his book will speak for itself. [ 57 ] His peculiar style will justify a little more quotation than is just necessary to prove the point. Looking at the "battle of opinion" now in progress, we see that Mr. Smith was a prescient:

(P. 588.) "From the general review of parties in England, it is evident that no country in the world is better prepared for the great Battle of Opinion. Where else can the battle be fought but where the armies are arrayed? And here they all are, Greek, Roman, Anglican, Scotch, Lutheran, Calvinist, Established and Territorial, with Baronial Bishops, and Nonestablished of every grade—churches with living prophets and apostles, and churches with dead prophets and apostles, and apostolical churches without apostles, and philosophies without either prophets or apostles, and only wanting one more, 'the Christian Church,' like Aaron's rod, to swallow up and digest them all, and then bud and flourish. As if to prepare our minds for this desirable and inevitable consummation, different parties have been favored with a revival of that very spirit of revelation by which the Church itself was originally founded. There is a complete series of spiritual revelations in England and the United States, besides mesmeric phenomena that bear a resemblance to revelation, and thus gradually open the mind of the philosophical and infidel classes, as well as the professed believers of that old revelation which they never witnessed in living action, to a better understanding of that Law of Nature (for it is a Law of Nature) in which all revelation originates and by which its spiritual communications are regulated."

Mr. Smith proceeds to say that there are only thirty-five incorporated churches in England, all formed from the New Testament except five, to each of which five he concedes a revelation of its own. The five are the Quakers, the Swedenborgians, the Southcottians, the Irvingites, and the Mormonites. Of Joanna Southcott he speaks as follows: [ 58 ]

(P. 592.) "Joanna Southcott[109] is not very gallantly treated by the gentlemen of the Press, who, we believe, without knowing anything about her, merely pick up their idea of her character from the rabble. We once entertained the same rabble idea of her; but having read her works—for we really have read them—we now regard her with great respect. However, there is a great abundance of chaff and straw to her grain; but the grain is good, and as we do not eat either the chaff or straw if we can avoid it, nor even the raw grain, but thrash it and winnow it, and grind it and bake it, we find it, after undergoing this process, not only very palatable, but a special dainty of its kind. But the husk is an insurmountable obstacle to those learned and educated gentlemen who judge of books entirely by the style and the grammar, or those who eat grain as it grows, like the cattle. Such men would reject all prological revelation; for there never was and probably never will be a revelation by voice and vision communicated in classical manner. It would be an invasion of the rights and prerogatives of Humanity, and as contrary to the Divine and Established order of mundane government, as a field of quartern loaves or hot French rolls."


Mr. Smith's book is spiritualism from beginning to end; and my anonymous gainsayer, honest of course, is either ignorant of the work he thinks he has read, or has a most remarkable development of the organ of imperception.]



I cut the following from a Sunday paper in 1849:

"X. Y.—The Chaldeans began the mathematics, in which the Egyptians excelled. Then crossing the sea, by means [ 59 ] of Thales,[110] the Milesian, they came into Greece, where they were improved very much by Pythagoras,[111] Anaxagoras,[112] and Anopides[113] of Chios. These were followed by Briso,[114] Antipho, [two circle-squarers; where is Euclid?] and Hippocrates,[115] but the excellence of the algebraic art was begun by Geber,[116] an Arabian astronomer, and was carried on by Cardanus,[117] Tartaglia,[118] Clavius,[119], Stevinus,[120] Ghetaldus,[121] Herigenius,[122] Fran. Van Schooten [meaning Francis Van Schooten[123]], Florida de Beaume,[124] etc."

Bryso was a mistaken man. Antipho had the disadvantage of being in advance of his age. He had the notion of which the modern geometry has made so much, that of [ 60 ] a circle being the polygon of an infinitely great number of sides. He could make no use of it, but the notion itself made him a sophist in the eyes of Aristotle, Eutocius,[125] etc. Geber, an Arab astronomer, and a reputed conjurer in Europe, seems to have given his name to unintelligible language in the word gibberish. At one time algebra was traced to him; but very absurdly, though I have heard it suggested that algebra and gibberish must have had one inventor.

Any person who meddles with the circle may find himself the crane who was netted among the geese: as Antipho for one, and Olivier de Serres[126] for another. This last gentleman ascertained, by weighing, that the area of the circle is very nearly that of the square on the side of the inscribed equilateral triangle: which it is, as near as 3.162 ... to 3.141.... He did not pretend to more than approximation; but Montucla and others misunderstood him, and, still worse, misunderstood their own misunderstanding, and made him say the circle was exactly double of the equilateral triangle. He was let out of limbo by Lacroix, in a note to his edition of Montucla's History of Quadrature.



Quadratura del cerchio, trisezione dell' angulo, et duplicazione del cubo, problemi geometricamente risolute e dimostrate dal Reverendo Arciprete di San Vito D. Domenico Angherà,[127] Malta, 1854, 8vo.
[ 61 ]
Equazioni geometriche, estratte dalla lettera del Rev. Arciprete ... al Professore Pullicino[128] sulla quadratura del cerchio. Milan, 1855 or 1856, 8vo.
Il Mediterraneo gazetta di Malta, 26 Decembre 1855, No. 909: also 911, 912, 913, 914, 936, 939.
The Malta Times, Tuesday, 9th June 1857.
Misura esatta del cerchio, dal Rev. D. Angherà. Malta, 1857, 12mo.
Quadrature of the circle ... by the Rev. D. Angherà, Archpriest of St. Vito. Malta, 1858, 12mo.

I have looked for St. Vitus in catalogues of saints, but never found his legend, though he figures as a day-mark in the oldest almanacs. He must be properly accredited, since he was an archpriest. And I pronounce and ordain, by right accruing from the trouble I have taken in this subject, that he, St. Vitus, who leads his votaries a never-ending and unmeaning dance, shall henceforth be held and taken to be the patron saint of the circle-squarer. His day is the 15th of June, which is also that of St. Modestus,[129] with whom the said circle-squarer often has nothing to do. And he must not put himself under the first saint with a slantendicular reference to the other, as is much to be feared was done by the Cardinal who came to govern England with a title containing St. Pudentiana,[130] who shares a day with St. Dunstan. The Archpriest of St. Vitus will have it that the square inscribed in a semicircle is half of the semicircle, or the circumference 3-1/5 diameters. He is active and able, with [ 62 ] nothing wrong about him except his paradoxes. In the second tract named he has given the testimonials of crowned heads and ministers, etc. as follows. Louis-Napoleon gives thanks. The minister at Turin refers it to the Academy of Sciences, and hopes so much labor will be judged degna di pregio.[131] The Vice-Chancellor of Oxford—a blunt Englishman—begs to say that the University has never proposed the problem, as some affirm. The Prince Regent of Baden has received the work with lively interest. The Academy of Vienna is not in a position to enter into the question. The Academy of Turin offers the most distinct thanks. The Academy della Crusca attends only to literature, but gives thanks. The Queen of Spain has received the work with the highest appreciation. The University of Salamanca gives infinite thanks, and feels true satisfaction in having the book. Lord Palmerston gives thanks, by the hand of "William San." The Viceroy of Egypt, not being yet up in Italian, will spend his first moments of leisure in studying the book, when it shall have been translated into French: in the mean time he congratulates the author upon his victory over a problem so long held insoluble. All this is seriously published as a rate in aid of demonstration. If these royal compliments cannot make the circumference of a circle about 2 per cent. larger than geometry will have it —which is all that is wanted—no wonder that thrones are shaky.

I am informed that the legend of St. Vitus is given by Ribadeneira[132] in his lives of Saints, and that Baronius,[133] in [ 63 ] his Martyrologium Romanum, refers to several authors who have written concerning him. There is an account in Mrs. Jameson's[134] History of Sacred and Legendary Art (ed. of 1863, p. 544). But it seems that St. Vitus is the patron saint of all dances; so that I was not so far wrong in making him the protector of the cyclometers. Why he is represented with a cock is a disputed point, which is now made clear: next after gallus gallinaceus[135] himself, there is no crower like the circle-squarer.



The following is an extract from the English Cyclopædia, Art. Tables:

"1853. William Shanks,[136] Contributions to Mathematics, comprising chiefly the Rectification of the Circle to 607 Places of Tables, London, 1853. (Quadrature of the Circle.) Here is a table, because it tabulates the results of the subordinate steps of this enormous calculation as far as 527 decimals: the remainder being added as results only during the printing. For instance, one step is the calculation of the reciprocal of 601.5601; and the result is given. The number of pages required to describe these results is 87. Mr. Shanks has also thrown off, as chips or splinters, the values of the base of Napier's logarithms, and of its logarithms of 2, 3, 5, 10, to 137 decimals; and the value of the modulus .4342 ... to 136 decimals: with the 13th, 25th, 37th ... up to the 721st powers of 2. These tremendous stretches of calculation—at least we so call them in our day—are useful in several respects; they prove more than [ 64 ] the capacity of this or that computer for labor and accuracy; they show that there is in the community an increase of skill and courage. We say in the community: we fully believe that the unequalled turnip which every now and then appears in the newspapers is a sufficient presumption that the average turnip is growing bigger, and the whole crop heavier. All who know the history of the quadrature are aware that the several increases of numbers of decimals to which π has been carried have been indications of a general increase in the power to calculate, and in courage to face the labor. Here is a comparison of two different times. In the day of Cocker,[137] the pupil was directed to perform a common subtraction with a voice-accompaniment of this kind: '7 from 4 I cannot, but add 10, 7 from 14 remains 7, set down 7 and carry 1; 8 and 1 which I carry is 9, 9 from 2 I cannot, etc.' We have before us the announcement of the following table, undated, as open to inspection at the Crystal Palace, Sydenham, in two diagrams of 7 ft. 2 in, by 6 ft. 6 in.: 'The figure 9 involved into the 912th power, and antecedent powers or involutions, containing upwards of 73,000 figures. Also, the proofs of the above, containing upwards of 146,000 figures. By Samuel Fancourt, of Mincing Lane, London, and completed by him in the year 1837, at the age of sixteen. N.B. The whole operation performed by simple arithmetic.' The young operator calculated by successive squaring the 2d, 4th, 8th, etc., powers up to the 512th, with proof by division. But 511 multiplications by 9, in the short (or 10-1) way, would have been much easier. The 2d, 32d, 64th, 128th, 256th, and 512th powers are given at the back of the announcement. The powers of 2 have been calculated for many purposes. In Vol. II of his Magia Universalis Naturæ et Artis, Herbipoli, 1658, 4to, the Jesuit Gaspar Schott[138] having discovered, on some grounds of theological [ 65 ] magic, that the degrees of grace of the Virgin Mary were in number the 256th power of 2, calculated that number. Whether or no his number correctly represented the result he announced, he certainly calculated it rightly, as we find by comparison with Mr. Shanks."


There is a point about Mr. Shanks's 608 figures of the value of π which attracts attention, perhaps without deserving it. It might be expected that, in so many figures, the nine digits and the cipher would occur each about the same number of times; that is, each about 61 times. But the fact stands thus: 3 occurs 68 times; 9 and 2 occur 67 times each; 4 occurs 64 times; 1 and 6 occur 62 times each; 0 occurs 60 times; 8 occurs 58 times; 5 occurs 56 times; and 7 occurs only 44 times. Now, if all the digits were equally likely, and 608 drawings were made, it is 45 to 1 against the number of sevens being as distant from the probable average (say 61) as 44 on one side or 78 on the other. There must be some reason why the number 7 is thus deprived of its fair share in the structure. Here is a field of speculation in which two branches of inquirers might unite. There is but one number which is treated with an unfairness which is incredible as an accident; and that number is the mystic number seven! If the cyclometers and the apocalyptics would lay their heads together until they come to a unanimous verdict on this phenomenon, and would publish nothing until they are of one mind, they would earn the gratitude of their race.—I was wrong: it is the Pyramid-speculator who should have been appealed to. A correspondent of my friend Prof. Piazzi Smyth[139] notices that 3 is the number of most frequency, and that 3-1/7 is the nearest approximation to it in simple digits. Professor Smyth himself, whose word on Egypt is paradox of a very high order, backed by a great quantity of useful labor, the results which will be made available by those who do not receive [ 66 ] the paradoxes, is inclined to see confirmation for some of his theory in these phenomena.



These paradoxes of calculation sometimes appear as illustrations of the value of a new method. In 1863, Mr. G. Suffield,[140] M.A., and Mr. J. R. Lunn,[141] M.A., of Clare College and of St. John's College, Cambridge, published the whole quotient of 10000 ... divided by 7699, throughout the whole of one of the recurring periods, having 7698 digits. This was done in illustration of Mr. Suffield's method of Synthetic division.

Another instance of computation carried to paradoxical length, in order to illustrate a method, is the solution of x3 - 2x = 5, the example given of Newton's method, on which all improvements have been tested. In 1831, Fourier's[142] posthumous work on equations showed 33 figures of solution, got with enormous labor. Thinking this a good opportunity to illustrate the superiority of the method of W. G. Horner,[143] not yet known in France, and not much known in [ 67 ] England, I proposed to one of my classes, in 1841, to beat Fourier on this point, as a Christmas exercise. I received several answers, agreeing with each other, to 50 places of decimals. In 1848, I repeated the proposal, requesting that 50 places might be exceeded: I obtained answers of 75, 65, 63, 58, 57, and 52 places. But one answer, by Mr. W. Harris Johnston,[144] of Dundalk, and of the Excise Office, went to 101 decimal places. To test the accuracy of this, I requested Mr. Johnston to undertake another equation, connected with the former one in a way which I did not explain. His solution verified the former one, but he was unable to see the connection, even when his result was obtained. My reader may be as much at a loss: the two solutions are:


The results are published in the Mathematician, Vol. III, p. 290. In 1851, another pupil of mine, Mr. J. Power Hicks,[145] carried the result to 152 decimal places, without knowing what Mr. Johnston had done. The result is in the English Cyclopædia, article Involution and Evolution.

I remark that when I write the initial of a Christian name, the most usual name of that initial is understood. I never saw the name of W. G. Horner written at length, until I applied to a relative of his, who told me that he was, as I supposed, Wm. George, but that he was named after a relative of that surname.

The square root of 2, to 110 decimal places, was given [ 68 ] me in 1852 by my pupil, Mr. William Henry Colvill, now (1867) Civil Surgeon at Baghdad. It was


Mr. James Steel[146] of Birkenhead verified this by actual multiplication, and produced

2 - \frac{2580413}{10^{117}}

as the square.


Calcolo decidozzinale del Barone Silvio Ferrari. Turin, 1854, 4to.

This is a serious proposal to alter our numeral system and to count by twelves. Thus 10 would be twelve, 11 thirteen, etc., two new symbols being invented for ten and eleven. The names of numbers must of course be changed. There are persons who think such changes practicable. I thought this proposal absurd when I first saw it, and I think so still:[147] but the one I shall presently describe beats it so completely in that point, that I have not a smile left for this one.


ON COMETS.[edit]

The successful and therefore probably true theory of Comets. London, 1854. (4pp. duodecimo.)

The author is the late Mr. Peter Legh,[148] of Norbury Booths Hall, Knutsford, who published for eight or ten [ 69 ] years the Ombrological Almanac, a work of asserted discovery in meteorology. The theory of comets is that the joint attraction of the new moon and several planets in the direction of the sun, draws off the gases from the earth, and forms these cometic meteors. But how these meteors come to describe orbits round the sun, and to become capable of having their returns predicted, is not explained.



The Mormon, New York, Saturday, Oct. 27, 1855.

A newspaper headed by a grand picture of starred and striped banners, beehive, and eagle surmounting it. A scroll on each side: on the left, "Mormon creed. Mind your own business. Brigham Young;"[149] on the right, "Given by inspiration of God. Joseph Smith."[150] A leading article on the discoveries of Prof. Orson Pratt[151] says, "Mormonism has long taken the lead in religion: it will soon be in the van both in science and politics." At the beginning of the paper is Professor Pratt's "Law of Planetary Rotation." The cube roots of the densities of the planets are as the square roots of their periods of rotation. The squares of the cube roots of the masses divided by the squares of the diameters are as the periods of rotation. Arithmetical verification attempted, and the whole very modestly stated [ 70 ] and commented on. Dated G. S. L. City, Utah Ter., Aug. 1, 1855. If the creed, as above, be correctly given, no wonder the Mormonites are in such bad odor.



The two estates; or both worlds mathematically considered. London, 1855, small (pp. 16).

The author has published mathematical works with his name. The present tract is intended to illustrate mathematically a point which may be guessed from the title. But the symbols do very little in the way of illustration: thus, x being the present value of the future estate (eternal happiness), and a of all that this world can give, the author impresses it on the mathematician that, x being infinitely greater than a, x + a = x, so that a need not be considered. This will not act much more powerfully on a mathematician by virtue of the symbols than if those same symbols had been dispensed with: even though, as the author adds, "It was this method of neglecting infinitely small quantities that Sir Isaac Newton was indebted to for his greatest discoveries."

There has been a moderate quantity of well-meant attempt to enforce, sometimes motive, sometimes doctrine, by arguments drawn from mathematics, the proponents being persons unskilled in that science for the most part. The ground is very dangerous: for the illustration often turns the other way with greater power, in a manner which requires only a little more knowledge to see. I have, in my life, heard from the pulpit or read, at least a dozen times, that all sin is infinitely great, proved as follows. The greater the being, the greater the sin of any offence against him: therefore the offence committed against an infinite being is infinitely great. Now the mathematician, of which the proposers of this argument are not aware, is perfectly familiar with quantities which increase together, and never cease increasing, but so that one of them remains finite when [ 71 ] the other becomes infinite. In fact, the argument is a perfect non sequitur.[152] Those who propose it have in their minds, though in a cloudy and indefinite form, the idea of the increase of guilt being proportionate to the increase of greatness in the being offended. But this it would never do to state: for by such statement not only would the argument lose all that it has of the picturesque, but the asserted premise would have no strong air of exact truth. How could any one undertake to appeal to conscience to declare that an offence against a being 4-7/10 times as great as another is exactly, no more and no less, 4-7/10 times as great an offence against the other?

The infinite character of the offence against an infinite being is laid down in Dryden's Religio Laici,[153] and is, no doubt, an old argument:

"For, granting we have sinned, and that th' offence
Of man is made against Omnipotence,
Some price that bears proportion must be paid,
And infinite with infinite be weighed.
See then the Deist lost; remorse for vice
Not paid; or, paid, inadequate in price."

Dryden, in the words "bears proportion" is in verse more accurate than most of the recent repeaters in prose. And this is not the only case of the kind in his argumentative poetry.

My old friend, the late Dr. Olinthus Gregory,[154] who was a sound and learned mathematician, adopted this dangerous kind of illustration in his Letters on the Christian Religion. [ 72 ] He argued, by parallel, from what he supposed to be the necessarily mysterious nature of the impossible quantity of algebra to the necessarily mysterious nature of certain doctrines of his system of Christianity. But all the difficulty and mystery of the impossible quantity is now cleared away by the advance of algebraical thought: and yet Dr. Gregory's book continues to be sold, and no doubt the illustration is still accepted as appropriate.

The mode of argument used by the author of the tract above named has a striking defect. He talks of reducing this world and the next to "present value," as an actuary does with successive lives or next presentations. Does value make interest? and if not, why? And if it do, then the present value of an eternity is not infinitely great. Who is ignorant that a perpetual annuity at five per cent is worth only twenty years' purchase? This point ought to be discussed by a person who treats heaven as a deferred perpetual annuity. I do not ask him to do so, and would rather he did not; but if he will do it, he must either deal with the question of discount, or be asked the reason why.

When a very young man, I was frequently exhorted to one or another view of religion by pastors and others who thought that a mathematical argument would be irresistible. And I heard the following more than once, and have since seen it in print, I forget where. Since eternal happiness belonged to the particular views in question, a benefit infinitely great, then, even if the probability of their arguments were small, or even infinitely small, yet the product of the chance and benefit, according to the usual rule, might give a result which no one ought in prudence to pass over. They did not see that this applied to all systems as well as their own. I take this argument to be the most perverse of all the perversions I have heard or read on the subject: there is some high authority for it, whom I forget.

The moral of all this is, that such things as the preceding should be kept out of the way of those who are not [ 73 ] mathematicians, because they do not understand the argument; and of those who are, because they do.

[The high authority referred to above is Pascal, an early cultivator of mathematical probability, and obviously too much enamoured of his new pursuit. But he conceives himself bound to wager on one side or the other. To the argument (Pensées, ch. 7)[155] that "le juste est de ne point parier," he answers, "Oui: mais il faut parier: vous êtes embarqué; et ne parier point que Dieu est, c'est parier qu'il n'est pas."[156] Leaving Pascal's argument to make its way with a person who, being a sceptic, is yet positive that the issue is salvation or perdition, if a God there be,—for the case as put by Pascal requires this,—I shall merely observe that a person who elects to believe in God, as the best chance of gain, is not one who, according to Pascal's creed, or any other worth naming, will really secure that gain. I wonder whether Pascal's curious imagination ever presented to him in sleep his convert, in the future state, shaken out of a red-hot dice-box upon a red-hot hazard-table, as perhaps he might have been, if Dante had been the later of the two. The original idea is due to the elder Arnobius,[157] who, as cited by Bayle,[158] speaks thus:

"Sed et ipse [Christus] quæ pollicetur, non probat. Ita est. Nulla enim, ut dixi, futurorum potest existere comprobatio. Cum ergo hæc sit conditio futurorum, ut teneri et comprehendi nullius possint anticipationis attactu; nonne [ 74 ] purior ratio est, ex duobus incertis, et in ambigua expectatione pendentibus, id potius credere, quod aliquas spes ferat, quam omnino quod nullas? In illo enim periculi nihil est, si quod dicitur imminere, cassum fiat et vacuum: in hoc damnum est maximum, id est salutis amissio, si cum tempus advenerit aperiatur non fuisse mendacium."[159]

Really Arnobius seems to have got as much out of the notion, in the third century, as if he had been fourteen centuries later, with the arithmetic of chances to help him.]



The Sentinel, vol. ix. no. 27. London, Saturday, May 26, 1855.

This is the first London number of an Irish paper, Protestant in politics. It opens with "Suggestions on the subject of a Novum Organum Moralium," which is the application of algebra and the differential calculus to morals, socials, and politics. There is also a leading article on the subject, and some applications in notes to other articles. A separate publication was afterwards made, with the addition of a long Preface; the author being a clergyman who I presume must have been the editor of the Sentinel.

Suggestions as to the employment of a Novum Organum Moralium. Or, thoughts on the nature of the Differential Calculus, and on the application of its principles to metaphysics, with a view to the attainment of demonstration and certainty in moral, [ 75 ] political and ecclesiastical affairs. By Tresham Dames Gregg,[160] Chaplain of St. Mary's, within the church of St. Nicholas intra muros, Dublin. London, 1859, 8vo. (pp. xl + 32).

I have a personal interest in this system, as will appear from the following extract from the newspaper:

"We were subsequently referred to De Morgan's Formal Logic and Boole's Laws of Thought[161] both very elaborate works, and greatly in the direction taken by ourselves. That the writers amazingly surpass us in learning we most willingly admit, but we venture to pronounce of both their learned treatises, that they deal with the subject in a mode that is scholastic to an excess.... That their works have been for a considerable space of time before the world and effected nothing, would argue that they have overlooked the vital nature of the theme.... On the whole, the writings of De Morgan and Boole go to the full justification of our principle without in any wise so trenching upon our ground as to render us open to reproach in claiming our Calculus as a great discovery.... But we renounce any paltry jealousy as to a matter so vast. If De Morgan and Boole have had a priority in the case, to them we cheerfully shall resign the glory and honor. If such be the truth, they have neither done justice to the discovery, nor to themselves [quite true]. They have, under the circumstances, acted like 'the foolish man, who roasteth not that which he taketh [ 76 ] in hunting.... It will be sufficient for us, however, to be the Columbus of these great Americi, and popularize what they found, if they found it. We, as from the mountain top, will then become their trumpeters, and cry glory to De Morgan and glory to Boole, under Him who is the source of all glory, the only good and wise, to Whom be glory for ever! If they be our predecessors in this matter, they have, under Him, taken moral questions out of the category of probabilities, and rendered them perfectly certain. In that case, let their books be read by those who may doubt the principles this day laid before the world as a great discovery, by our newspaper. Our cry shall be ευρηκασι![162] Let us hope that they will join us, and henceforth keep their light [sic] from under their bushel."

For myself, and for my old friend Mr. Boole, who I am sure would join me, I disclaim both priority, simultaneity, and posteriority, and request that nothing may be trumpeted from the mountain top except our abjuration of all community of thought or operation with this Novum Organum.

To such community we can make no more claim than Americus could make to being the forerunner of Columbus who popularized his discoveries. We do not wish for any ευρηκασι and not even for εὑρηκασι. For self and Boole, I point out what would have convinced either of us that this house is divided against itself.

Α being an apostolic element, δ the doctrinal element, and Χ the body of the faithful, the church is Α δ Χ, we are told. Also, that if Α become negative, or the Apostolicity become Diabolicity [my words]; or if δ become negative, and doctrine become heresy; or if Χ become negative, that is, if the faithful become unfaithful; the church becomes negative, "the very opposite to what it ought to be." For self and Boole, I admit this. But—which is not noticed—if Α and δ should both become negative, diabolical origin [ 77 ] and heretical doctrine, then the church, Α δ Χ, is still positive, what it ought to be, unless Χ be also negative, or the people unfaithful to it, in which case it is a bad church. Now, self and Boole—though I admit I have not asked my partner—are of opinion that a diabolical church with false doctrine does harm when the people are faithful, and can do good only when the people are unfaithful. We may be wrong, but this is what we do think. Accordingly, we have caught nothing, and can therefore roast nothing of our own: I content myself with roasting a joint of Mr. Gregg's larder.

These mathematical vagaries have uses which will justify a large amount of quotation: and in a score of years this may perhaps be the only attainable record. I therefore proceed.

After observing that by this calculus juries (heaven help them! say I) can calculate damages "almost to a nicety," and further that it is made abundantly evident that c e x is "the general expression for an individual," it is noted that the number of the Beast is not given in the Revelation in words at length, but as χξϜ'.[163] On this the following remark is made:

"Can it be possible that we have in this case a specimen given to us of the arithmetic of heaven, and an expression revealed, which indicates by its function of addibility, the name of the church in question, and of each member of it; and by its function of multiplicability the doctrine, the mission, and the members of the great Synagogue of Apostacy? We merely propound these questions;—we do not pretend to solve them."

After a translation in blank verse—a very pretty one—of the 18th Psalm, the author proceeds as follows, to render it into differential calculus:

[ 78 ]

"And the whole tells us just this, that David did what he could. He augmented those elements of his constitution which were (exceptis excipiendis)[164] subject to himself, and the Almighty then augmented his personal qualities, and his vocational status. Otherwise, to throw the matter into the expression of our notation, the variable e was augmented, and c x rose proportionally. The law of the variation, according to our theory, would be thus expressed. The resultant was David the king c e x [c = r?] (who had been David the shepherd boy), and from the conditions of the theorem we have

\frac{du}{de} = ce\frac{dx}{de} + ex\frac{dc}{de}x + cx

which, in the terms of ordinary language, just means, the increase of David's educational excellence or qualities—his piety, his prayerfulness, his humility, obedience, etc.—was so great, that when multiplied by his original talent and position, it produced a product so great as to be equal in its amount to royalty, honor, wealth, and power, etc.: in short, to all the attributes of majesty."[165]

The "solution of the family problem" is of high interest. It is to determine the effect on the family in general from a change [of conduct] in one of them. The person chosen is one of the maid-servants.

"Let c e x be the father; c1e1x1 the mother, etc. The family then consists of the maid's master, her mistress, her young master, her young mistress, and fellow servant. Now the master's calling (or c) is to exercise his share of control over this servant, and mind the rest of his business: call this remainder a, and let his calling generally, or all his affairs, be to his maid-servant as m : y, i.e., y = (mz/c); ... [ 79 ] and this expression will represent his relation to the servant. Consequently,

c e x = \bigl(a + \frac{mz}{c}\bigr) e x; otherwise \bigl(a + \frac{mz}{c}\bigr) e x

is the expression for the father when viewed as the girl's master."

I have no objection to repeat so far; but I will not give the formula for the maid's relation to her young master; for I am not quite sure that all young masters are to be trusted with it. Suffice it that the son will be affected directly as his influence over her, and inversely as his vocational power: if then he should have some influence and no vocational power, the effect on him would be infinite. This is dismal to think of. Further, the formula brings out that if one servant improve, the other must deteriorate, and vice versa. This is not the experience of most families: and the author remarks as follows:

"That is, we should venture to say, a very beautiful result, and we may say it yielded us no little astonishment. What our calculation might lead to we never dreamt of; that it should educe a conclusion so recondite that our unassisted power never could have attained to, and which, if we could have conjectured it, would have been at best the most distant probability, that conclusion being itself, as it would appear, the quintessence of truth, afforded us a measure of satisfaction that was not slight."

That the writings of Mr. Boole and myself "go to the full justification of" this "principle," is only true in the sense in which the Scotch use, or did use, the word justification.



[The last number of this Budget had stood in type for months, waiting until there should be a little cessation of correspondence more connected with the things of the day. [ 80 ] I had quite forgotten what it was to contain; and little thought, when I read the proof, that my allusions to my friend Mr. Boole, then in life and health, would not be printed till many weeks after his death. Had I remembered what my last number contained, I should have added my expression of regret and admiration to the numerous obituary testimonials, which this great loss to science has called forth.

The system of logic alluded to in the last number of this series is but one of many proofs of genius and patience combined. I might legitimately have entered it among my paradoxes, or things counter to general opinion: but it is a paradox which, like that of Copernicus, excited admiration from its first appearance. That the symbolic processes of algebra, invented as tools of numerical calculation, should be competent to express every act of thought, and to furnish the grammar and dictionary of an all-containing system of logic, would not have been believed until it was proved. When Hobbes,[166] in the time of the Commonwealth, published his Computation or Logique, he had a remote glimpse of some of the points which are placed in the light of day by Mr. Boole. The unity of the forms of thought in all the applications of reason, however remotely separated, will one day be matter of notoriety and common wonder: and Boole's name will be remembered in connection with one of the most important steps towards the attainment of this knowledge.]



The Decimal System as a whole. By Dover Statter.[167] London and Liverpool, 1856, 8vo.

[ 81 ]

The proposition is to make everything decimal. The day, now 24 hours, is to be made 10 hours. The year is to have ten months, Unusber, Duober, etc. Fortunately there are ten commandments, so there will be neither addition to, nor deduction from, the moral law. But the twelve apostles! Even rejecting Judas, there is a whole apostle of difficulty. These points the author does not touch.



The first book of Phonetic Reading. London, Fred. Pitman,[168] Phonetic Depot, 20, Paternoster Row, 1856, 12mo.
The Phonetic Journal. Devoted to the propagation of phonetic reading, phonetic longhand, phonetic shorthand, and phonetic printing. No. 46. Saturday, 15 November 1856. Vol. 15.

I write the titles of a couple out of several tracts which I have by me. But the number of publications issued by the promoters of this spirited attempt is very large indeed.[169] The attempt itself has had no success with the mass of the public. This I do not regret. Had the world found that the change was useful, I should have gone contentedly with the stream; but not without regretting our old language. I admit the difficulties which our unpronounceable spelling puts in the way of learning to read: and I have no doubt that, as affirmed, it is easier to teach children phonetically, and afterwards to introduce them to our common system, than to proceed in the usual way. But by the usual way I mean proceeding by letters from the very beginning. If, which I am sure is a better plan, children be taught at the commencement very much by complete words, as if they were learning Chinese, and be gradually accustomed to [ 82 ] resolve the known words into letters, a fraction, perhaps a considerable one, of the advantage of the phonetic system is destroyed. It must be remembered that a phonetic system can only be an approximation. The differences of pronunciation existing among educated persons are so great, that, on the phonetic system, different persons ought to spell differently.

But the phonetic party have produced something which will immortalize their plan: I mean their shorthand, which has had a fraction of the success it deserves. All who know anything of shorthand must see that nothing but a phonetic system can be worthy of the name: and the system promulgated is skilfully done. Were I a young man I should apply myself to it systematically. I believe this is the only system in which books were ever published. I wish some one would contribute to a public journal a brief account of the dates and circumstances of the phonetic movement, not forgetting a list of the books published in shorthand.

A child beginning to read by himself may owe terrible dreams and waking images of horror to our spelling, as I did when six years old. In one of the common poetry-books there is an admonition against confining little birds in cages, and the child is asked what if a great giant, amazingly strong, were to take you away, shut you up,

And feed you with vic-tu-als you ne-ver could bear.

The book was hyphened for the beginner's use; and I had not the least idea that vic-tu-als were vittles: by the sound of the word I judged they must be of iron; and it entered into my soul.

The worst of the phonetic shorthand book is that they nowhere, so far as I have seen, give all the symbols, in every stage of advancement, together, in one or following pages. It is symbols and talk, more symbols and more talk, etc. A universal view of the signs ought to begin the works. [ 83 ]



Ombrological Almanac. Seventeenth year. An essay on Anemology and Ombrology. By Peter Legh,[170] Esq. London, 1856, 12mo.

Mr. Legh, already mentioned, was an intelligent country gentleman, and a legitimate speculator. But the clue was not reserved for him.

The proof that the three angles of a triangle are equal to two right angles looked for in the inflation of the circle. By Gen. Perronet Thompson. London, 1856, 8vo. (pp. 4.)

Another attempt, the third, at this old difficulty, which cannot be put into few words of explanation.[171]

Comets considered as volcanoes, and the cause of their velocity and other phenomena thereby explained. London (circa 1856), 8vo.

The title explains the book better than the book explains the title.


1856. A stranger applied to me to know what the ideas of a friend of his were worth upon the magnitude of the earth. The matter being one involving points of antiquity, I mentioned various persons whose speculations he seemed to have ignored; among others, Thales. The reply was, "I am instructed by the author to inform you that he is perfectly acquainted with the works of Thales, Euclid, Archimedes, ..." I had some thought of asking whether he had used the Elzevir edition of Thales,[172] which is known to be very incomplete, or that of Professor Niemand with the lections, Nirgend, 1824, 2 vols. folio; just to see whether the [ 84 ] last would not have been the very edition he had read. But I refrained, in mercy.


The moon is the image of the Earth, and is not a solid body. By The Longitude.[173] (Private Circulation.) In five parts. London, 1856, 1857, 1857; Calcutta, 1858, 1858, 8vo.

The earth is "brought to a focus"; it describes a "looped orbit round the sun." The eclipse of the sun is thus explained: "At the time of eclipses, the image is more or less so directly before or behind the earth that, in the case of new moon, bright rays of the sun fall and bear upon the spot where the figure of the earth is brought to a focus, that is, bear upon the image of the earth, when a darkness beyond is produced reaching to the earth, and the sun becomes more or less eclipsed." How the earth is "brought to a focus" we do not find stated. Writers of this kind always have the argument that some things which have been ridiculed at first have been finally established. Those who put into the lottery had the same kind of argument; but were always answered by being reminded how many blanks there were to one prize. I am loath to pronounce against anything: but it does force itself upon me that the author of these tracts has drawn a blank.



Times, April 6 or 7, 1856. The moon has no rotary motion.

A letter from Mr. Jellinger Symons,[174] inspector of schools, which commenced a controversy of many letters and pamphlets. This dispute comes on at intervals, and will continue to do so. It sometimes arises from inability to understand the character of simple rotation, geometrically; sometimes from not understanding the mechanical doctrine of rotation.

[ 85 ]


Lunar Motion. The whole argument stated, and illustrated by diagrams; with letters from the Astronomer Royal. By Jellinger C. Symons. London, 1856, 8vo.

The Astronomer Royal endeavored to disentangle Mr. J. C. Symons, but failed. Mr. Airy[175] can correct the error of a ship's compasses, because he can put her head which way he pleases: but this he cannot do with a speculator.

Mr. Symons, in this tract, insinuated that the rotation of the moon is one of the silver shrines of the craftsmen. To see a thing so clearly as to be satisfied that all who say they do not see it are telling wilful falsehood, is the nature of man. Many of all sects find much comfort in it, when they think of the others; many unbelievers solace themselves with it against believers; priests of old time founded the right of persecution upon it, and of our time, in some cases, the right of slander: many of the paradoxers make it an argument against students of science. But I must say for men of science, for the whole body, that they are fully persuaded of the honesty of the paradoxers. The simple truth is, that all those I have mentioned, believers, unbelievers, priests, paradoxers, are not so sure they are right in their points of difference that they can safely allow themselves to be persuaded of the honesty of opponents. Those who know demonstration are differently situated. I suspect a train might be laid for the formation of a better habit in this way. We know that Suvaroff[176] taught his Russians at Ismail not to fear the Turks by accustoming them to charge bundles of faggots dressed in turbans, etc.

At which your wise men sneered in phrases witty,
He made no answer—but he took the city!

Would it not be a good thing to exercise boys, in pairs, in the following dialogue:—Sir, you are quite wrong!—Sir, [ 86 ] I am sure you honestly think so! This was suggested by what used to take place at Cambridge in my day. By statute, every B.A. was obliged to perform a certain number of disputations, and the father of the college had to affirm that it had been done. Some were performed in earnest: the rest were huddled over as follows. Two candidates occupied the places of the respondent and the opponent: Recte statuit Newtonus, said the respondent: Recte non statuit Newtonus,[177] said the opponent. This was repeated the requisite number of times, and counted for as many acts and opponencies. The parties then changed places, and each unsaid what he had said on the other side of the house: I remember thinking that it was capital drill for the House of Commons, if any of us should ever get there. The process was repeated with every pair of candidates.

The real disputations were very severe exercises. I was badgered for two hours with arguments given and answered in Latin,—or what we called Latin—against Newton's first section, Lagrange's[178] derived functions, and Locke[179] on innate principles. And though I took off everything, and was pronounced by the moderator to have disputed magno honore,[180] I never had such a strain of thought in my life. For the inferior opponents were made as sharp as their betters by their tutors, who kept lists of queer objections, drawn from all quarters. The opponents used to meet the day before to compare their arguments, that the same might not come twice over. But, after I left Cambridge, it became the fashion to invite the respondent to be present, who therefore learnt all that was to be brought against him. This made the whole thing a farce: and the disputations were abolished.

[ 87 ]


The Doctrine of the Moon's Rotation, considered in a letter to the Astronomical Censor of the Athenæum. By Jones L. MacElshender.[181] Edinburgh, 1856, 8vo.

This is an appeal to those cultivated persons who will read it "to overrule the dicta of judges who would sacrifice truth and justice to professional rule, or personal pique, pride, or prejudice"; meaning, the great mass of those who have studied the subject. But how? Suppose the "cultivated persons" were to side with the author, would those who have conclusions to draw and applications to make consent to be wrong because the "general body of intelligent men," who make no special study of the subject, are against them? They would do no such thing: they would request the general body of intelligent men to find their own astronomy, and welcome. But the truth is, that this intelligent body knows better: and no persons know better that they know better than the speculators themselves.

But suppose the general body were to combine, in opposition to those who have studied. Of course all my list must be admitted to their trial; and then arises the question whether both sides are to be heard. If so, the general body of the intelligent must hear all the established side have to say: that is, they must become just as much of students as the inculpated orthodox themselves. And will they not then get into professional rule, pique, pride, and prejudice, as the others did? But if, which I suspect, they are intended to judge as they are, they will be in a rare difficulty. All the paradoxers are of like pretensions: they cannot, as a class, be right, for each one contradicts a great many of the rest. There will be the puzzle which silenced the crew of the cutter in Marryat's novel of the Dog Fiend.[182] "A tog is a tog," said Jansen.—"Yes," replied another, "we all know a dog is a dog; but the question is—Is this dog [ 88 ] a dog?" And this question would arise upon every dog of them all.


87 ^  See Vol. I, note 133.

88 ^  Franz Xaver, Freiherr von Zach (1754-1832) was director of the observatory at Seeberge near Gotha. He wrote the Tabulae speciales aberrationis et mutationis (1806-7), Novae et correctae tabulae solis (1792), and L'attraction des montagnes et ses effets sur le fil à plomb (1814).

89 ^  Jean Louis Pons (1761-1831) was connected with the observatory at Marseilles for thirty years (1789-1819). He later became director of the observatory at Marlia, near Lucca, and subsequently filled the same office at Florence. He was an indefatigable searcher for comets, discovering 37 between 1801 and 1827, among them being the one that bears Encke's name.

90 ^  This hypothesis has now become an established fact.

91 ^  John Chetwode Eustace (c. 1762-1815) was born in Ireland. Although a Roman Catholic priest he lived for a time at Cambridge where he did some tutoring. His Classical Tour appeared in 1813 and went through several editions.

92 ^  "Crimes should be exposed when they are punished, but disgraceful acts should be hidden."

93 ^  Henri Hureau de Sénarmont (1808-1862) was professor of mineralogy at the Ecole des mines and examiner at the Ecole polytechnique at Paris.

94 ^  Augustin Jean Fresnel (1788-1827), "Ingenieur des ponts et chaussées," gave the first experimental proofs of the wave theory of light. He studied the questions of interference and polarization, and determined the approximate velocity of light.

95 ^  "As is my custom."

96 ^  Francis Heywood (1796-1858) made the first English translation of Kant's Critick of Pure Reason (1838, reprinted in 1848). The Analysis came out, as here stated, in 1844.

97 ^  Louise Renée de Keroualle, Duchess of Portsmouth and Aubigny (1649-1734), was a favorite of Charles II. She used her influence to keep him under the control of Louis XIV.

98 ^  William Chiffinch (c. 1602-1688) was page of the king's bed-chamber and keeper of the private closet to Charles II. He was one of the king's intimates and was an unscrupulous henchman.

99 ^  "Well devised."

100 ^  "John Bellingham Inglis. His Philobiblion "translated from the first edition (of Ricardus d'Aungervile, Bishop of Durham), 1473," appeared at London in 1832. It was republished in America (Albany, N. Y.) in 1864.

101 ^  "What are you laughing at?"

102 ^  See Vol. I, note 681.

103 ^  See Vol. I, note 211.

104 ^  Referring to Hamilton's edition of the Collected Works of Dugald Stewart, 10 volumes, Edinburgh, 1854-58. It is not commonly remembered that Stewart (1753-1828) taught mathematics at the University of Edinburgh before he took up philosophy.

105 ^  This was Hamilton's edition of the Works of Thomas Reid (2 vols., Edinburgh, 1846-1863). Reid (1710-1796) included mathematics in his work in philosophy at Aberdeen. In 1764 he succeeded Adam Smith at Glasgow.

106 ^  Edward Irving (1792-1834), the famous preacher. At first he assisted Dr. Chalmers at Glasgow, but in 1822 he went to London where he met with great success. A few years later he became mentally unbalanced and was finally expelled from his church (1832) for heresy. He was a great friend of Carlyle.

107 ^  He also wrote a number of other paradoxes, including An Essay towards a Science of Consciousness (1838), Instinctive Natural Religion (1858), Popular Treatise on the structure, diseases, and treatment of the human teeth (1837), and On Headache (1859).

108 ^  James Smith (1801-1857), known as Shepherd Smith, was a socialist and a mystic, with a philosophy that was wittily described as "Oriental pantheism translated into Scotch." He was editor of several journals.

109 ^  Joanna Southcott (1750-1814) was known for her rhyming prophecies in which she announced herself as the woman spoken of in Revelations xii. She had at one time as many as 100,000 disciples, and she established a sect that long survived her.

110 ^  Thales, c. 640-548 B. C.

111 ^  Pythagoras, 580-501 B. C.

112 ^  Anaxagoras, 499-428 B. C., the last of the Ionian school, teacher of Euripides and Pericles. Plutarch speaks of him as having squared the circle.

113 ^  Oinopides of Chios, contemporary of Anaxagoras. Proclus tells us that Oinopides was the first to show how to let fall a perpendicular to a line from an external point.

114 ^  Bryson and Antiphon, contemporaries of Socrates, invented the so-called method of exhaustions, one of the forerunners of the calculus.

115 ^  He wrote, c. 440 B. C., the first elementary textbook on mathematics in the Greek language. The "lunes of Hippocrates" are well known in geometry.

116 ^  Jabir ben Aflah. He lived c. 1085, at Seville, and wrote on astronomy and spherical trigonometry. The Gebri filii Affla Hispalensis de astronomia libri IX was published at Nuremberg in 1533.

117 ^  Hieronymus Cardanus, or Girolamo Cardano (1501-1576), the great algebraist. His Artis magnae sive de regulis Algebrae was published at Nuremberg in 1545.

118 ^  Nicolo Tartaglia (c. 1500-1557), the great rival of Cardan.

119 ^  See note 98, Vol. I.

120 ^  See note 124, Vol. I.

121 ^  See note 123, Vol. I.

122 ^  Pierre Hérigone lived in Paris the first half of the 17th century. His Cours mathématique (6 vols., 1634-1644) had some standing but was not at all original.

123 ^  Franciscus van Schooten (died in 1661) was professor of mathematics at Leyden. He edited Descartes's La Géométrie.

124 ^  Florimond de Beaune (1601-1652) was the first Frenchman to write a commentary on Descartes's La Géométrie. He did some noteworthy work in the theory of curves.

125 ^  See note 23, Vol. I.

126 ^  Olivier de Serres (b. in 1539) was a writer on agriculture. Montucla speaks of him in his Quadrature du cercle (page 227) as having asserted that the circle is twice the inscribed equilateral triangle, although, as De Morgan points out, this did not fairly interpret his position.

127 ^  Angherà wrote not only the three works here mentioned, but also the Problemi del più alto interesse scientifico, geometricamente risoluti e dimostrati, Naples, 1861. His quadrature was defended by Giovanni Motti in a work entitled Matematica Vera. Falsità del sistema ciclometrico d'Archimede, quadratura del cerchio d'Angherà, ricerca algebraica dei lati di qualunque poligono regolare inscritto in un circolo, Voghera, 1877. The Problemi of 1861 contains Angherà's portrait, and states that he lived at Malta from 1849 to 1861. It further states that the Malta publications are in part reproduced in this work.

128 ^  This was his friend Paolo Pullicino whose Elogio was pronounced by L. Farrugia at Malta in 1890. He wrote a work La Santa Effegie della Blata Vergine Maria, published at Valetta in 1868.

129 ^  St. Vitus, St. Modestus, and St. Crescentia were all martyred the same day, being torn limb from limb after lions and molten lead had proved of no avail. At least so the story runs.

130 ^  The reference is to Cardinal Wiseman. See Vol. II, note 56.

131 ^  "Worthy of esteem."

132 ^  Pedro de Ribadeneira (Ribadeneyra, Rivadeneira), was born at Toledo in 1526 and died in 1611. He held high position in the Jesuit order. The work referred to is the Flos Sanctorum o libro de las vidas de los santos, of which there was an edition at Barcelona in 1643. His life of Loyola (1572) and Historia ecclesiástica del Cisma del reino de Inglaterra were well known.

133 ^  Cæsar Baronius (1538-1607) was made a cardinal in 1595 and became librarian at the Vatican in 1597. The work referred to appeared at Rome in 1589.

134 ^  Mrs. Jameson's (1794-1860) works were very popular half a century ago, and still have some circulation among art lovers. The first edition of the work mentioned appeared in 1848.

135 ^  The barnyard cock.

136 ^  Shanks did nothing but computing. The title should, of course, read "to 607 Places of Decimals." He later carried the computation to 707 decimal places. (Proc. Roy. Society, XXI, p. 319.) He also prepared a table of prime numbers up to 60,000. (Proc. Roy. Society, XXII, p. 200.)

137 ^  See Vol. I, note 24.

138 ^  See Vol. I, note 78.

139 ^  See Vol. I, note 704.

140 ^  George Suffield published Synthetic Division in Arithmetic, to which reference is made, in 1863.

141 ^  John Robert Lunn wrote chiefly on Church matters, although he published a work on motion in 1859.

142 ^  Jean Baptiste Joseph, Baron Fourier (1768-1830), sometime professor in the Military School at Paris, and later at the Ecole polytechnique. He is best known by his Théorie analytique de la chaleur (Paris, 1822), in which the Fourier series is used. The work here referred to is the Analyse des équations déterminées (Paris, 1831).

143 ^  William George Horner (1786-1837) acquired a name for himself in mathematics in a curious manner. He was not a university man nor was he a mathematician of any standing. He taught school near Bristol and at Bath, and seems to have stumbled upon his ingenious method for finding the approximate roots of numerical higher equations, including as a special case the extracting of the various roots of numbers. Davies Gilbert presented the method to the Royal Society in 1819, and it was reprinted in the Ladies' Diary for 1838, and in the Mathematician in 1843. The method was original as far as Horner was concerned, but it is practically identical with the one used by the Chinese algebraist Ch'in Chiu-shang, in his Su-shu Chiu-chang of 1247. But even Ch'in Chiu-shang can hardly be called the discoverer of the method since it is merely the extension of a process for root extracting that appeared in the Chiu-chang Suan-shu of the second century B. C.

144 ^  He afterwards edited Loftus's Inland Revenue Officers' Manual (London, 1865). The two equations mentioned were x3 - 2x = 5 and y3 - 90y2 + 2500y - 16,000 = 0, in which y = 30 - 10x. Hence each place of y is the complement of the following place of x with respect to 9.

145 ^  Probably the John Power Hicks who wrote a memoir on T. H. Key, London, 1893.

146 ^  Possibly the one who wrote on the quadrature of the circle in 1881.

147 ^  As it is. But what a pity that we have not 12 fingers, with duodecimal fractions instead of decimals! We should then have 0.6 for ½, 0.4 for ⅓, 0.8 for ⅔, 0.3 for ¼, 0.9 for ¾, and 0.16 for ⅛, instead of 0.5, 0.333+, 0.666+, 0.25, 0.75, and 0.125 as we now have with our decimal system. In other words, the most frequently used fractions in business would be much more easily represented on the duodecimal scale than on the decimal scale that we now use.

148 ^  He wrote Hints for an Essay on Anemology and Ombrology (London, 1839-40) and The Music of the Eye (London, 1831).

149 ^  Brigham Young (1801-1877) was born at Whitingham, Vermont, and entered the Mormon church in 1832. In 1840 he was sent as a missionary to England. After the death of Joseph Smith he became president of the Mormons (1847), leading the church to Salt Lake City (1848).

150 ^  Joseph Smith (1805-1844) was also born in Vermont, and was four years the junior of Brigham Young. The Book of Mormon appeared in 1827, and the church was founded in 1830. He was murdered in 1844.

151 ^  Orson Pratt (1811-1881) was one of the twelve apostles of the Mormon Church (1835), and made several missionary journeys to England. He was professor of mathematics in the University of Deseret (the Mormon name for Utah). Besides the paper mentioned Pratt wrote the Divine Authenticity of the Book of Mormon (1849), Cubic and Biquadratic Equations (1866), and a Key to the Universe (1866).

152 ^  "It does not follow."

153 ^  Dryden (1631-1700) published his Religio Laici in 1682. The use of the word "proportion" in the sense of ratio was common before his time, but he uses it in the sense of having four terms; that is, that price is to price as offence is to offence.

154 ^  Olinthus Gilbert Gregory (1774-1841) succeeded Hutton as professor of mathematics at Woolwich. He was, with De Morgan, much interested in founding the University of London. He wrote on astronomy (1793), mechanics (1806), practical mathematics (1825), and Christian evidences (1811).

155 ^  See Vol. I, note 482. The Pensées appeared posthumously in 1670.

156 ^  "The right thing to do is not to wager at all." "Yes, but you ought to wager; you have started out; and not to wager at all that God exists is to wager that he does not exist."

157 ^  He lived about 300 A.D., in Africa, and wrote Libri septem adversus Gentes. This was printed at Rome in 1542-3.

158 ^  Pierre Bayle (1647-1706) was professor of philosophy at the Prostestant University at Sedan from 1675 until its dissolution in 1681. He then became professor at Rotterdam (1681-1693). In 1684 he began the publication of his journal of literary criticism Nouvelles de la République des Lettres. He is best known for his erudite Dictionnaire historique et critique (1697).

159 ^  "But Christ himself does not prove what he promises. It is true. For, as I have said, there cannot be any absolute proof of future events. Therefore since it is a condition of future events that they cannot be grasped or comprehended by any efforts of anticipation, is it not more reasonable, out of two alternatives that are uncertain and that are hanging in doubtful expectation, to give credence to the one that gives some hope rather than to the one that offers none at all? For in the former case there is no danger if, as is said to threaten, it becomes empty and void; while in the latter case the danger is greatest, that is, the loss of salvation, if when the time comes it is found that it was not a falsehood."

160 ^  Gregg wrote several other paradoxes, including the following: The Authentic Report of the extraordinary case of Tresham Dames Gregg ... his committal to Bridewell for refusing to give his recognizance (Dublin, 1841), An Appeal to Public Opinion upon a Case of Injury and Wrong ... in the case of a question of prerogative that arose between [R. Whately] ... Archbishop of Dublin and the author (London, 1861), The Cosmology of Sir Isaac Newton proved to be in accordance with the Bible (London, 1871), The Steam Locomotive as revealed in the Bible (London 1863) and On the Sacred Law of 1866, conferring perpetual life with immunity from decay and disease. A cento of decisive scriptural oracles strangely discovered (London and Dublin, 1875). These titles will help the reader to understand the man whom De Morgan so pleasantly satirizes.

161 ^  See Vol. I, note 592.

162 ^  "They have found it."

163 ^  The late Greeks used the letters of their alphabet as numerals, adding three early alphabetic characters. The letter χ}} represented 600, ξ}} represented 60, and Ϝ}} stood for 6. This gives 666, the number of the Beast given in the Revelation.

164 ^  "Allowing for necessary exceptions."

165 ^  Mr. Gregg is not alone in his efforts to use the calculus in original lines, as any one who has read Herbart's application of the subject to psychology will recall.

166 ^  See Vol. I, note 188; note 197.

167 ^  The full title shows the plan,—The Decimal System as a whole, in its relation to time, measure, weight, capacity, and money, in unison with each other. But why is this so much worse than the French plan of which we have only the metric system and the decimal division of the angle left?

168 ^  One of the brothers of Sir Isaac Pitman (1813-1897), the inventor of modern stenography. Of these brothers, Benjamin taught the art in America, Jacob in Australia, and Joseph, Henry, and Frederick in England.

169 ^  For example, The Phonographic Lecturer (London, 1871 etc.), The Phonographic Student (1867, etc.), and The Shorthand Magazine (1866, etc.).

170 ^  See Vol. II, note 148.

171 ^  It involves the theory of non-Euclidean geometry, Euclid's postulate of parallels being used in proving this theorem.

172 ^  Referring to the fact that none of the works of Thales is extant.

173 ^  The author was one B. Bulstrode. Parts 4 and 5 were printed at Calcutta.

174 ^  See Vol. II, note 18.

175 ^  See Vol. I, note 129.

176 ^  Alexander Vasilievich Suvaroff (1729-1800), a Russian general who fought against the Turks, in the Polish wars, and in the early Napoleonic campaigns. When he took Ismail in 1790 he sent this couplet to Empress Catherine.

177 ^  "Newton hath determined rightly," "Newton hath not determined rightly."

178 ^  See Vol. I, note 621.

179 ^  See Vol. I, note 700.

180 ^  "With great honor."

181 ^  Apparently unknown to biographers. He seems to have written nothing else.

182 ^  Captain Marryat (1792-1848) published Snarley-yow, or the Dog Fiend in 1837.