A Budget of Paradoxes

Introductory

PREFACE TO THE FIRST EDITION. (1872)

It is not without hesitation that I have taken upon myself the editorship of a work left avowedly imperfect by the author, and, from its miscellaneous and discursive character, difficult of completion with due regard to editorial limitations by a less able hand.

Had the author lived to carry out his purpose he would have looked through his Budget again, amplifying and probably rearranging some of its contents. He had collected materials for further illustration of Paradox of the kind treated of in this book; and he meant to write a second part, in which the contradictions and inconsistencies of orthodox learning would have been subjected to the same scrutiny and castigation as heterodox ignorance had already received.

It will be seen that the present volume contains more than the Athenæum Budget. Some of the additions formed a Supplement to the original articles. These supplementary paragraphs were, by the author, placed after those to which they respectively referred, being distinguished from the rest of the text by brackets. I have omitted these brackets as useless, except where they were needed to indicate subsequent writing.

Another and a larger portion of the work consists of discussion of matters of contemporary interest, for the Budget was in some degree a receptacle for the author's thoughts on any literary, scientific, or social question. Having grown thus gradually to its present size, the book as it was left was not quite in a fit condition for publication, but the alterations which have been made are slight and few, being in most cases verbal, and such as the sense absolutely required, or transpositions of sentences to secure coherence with the rest, in places where the author, in his more recent insertion of them, had overlooked the connection in which they stood. In no case has the meaning been in any degree modified or interfered with.

One rather large omission must be mentioned here. It is an account of the quarrel between Sir James South and Mr. Troughton on the mounting, etc. of the equatorial telescope at Campden Hill. At some future time when the affair has passed entirely out of the memory of living Astronomers, the appreciative sketch, which is omitted in this edition of the Budget, will be an interesting piece of history and study of character.

A very small portion of Mr. James Smith's circle-squaring has been left out, with a still smaller portion of Mr. De Morgan's answers to that Cyclometrical Paradoxer.

In more than one place repetitions, which would have disappeared under the author's revision, have been allowed to remain, because they could not have been taken away without leaving a hiatus, not easy to fill up without damage to the author's meaning.

I give these explanations in obedience to the rules laid down for the guidance of editors at page 15. If any apology for the fragmentary character of the book be thought necessary, it may be found in the author's own words at page 281 of the second volume.

The publication of the Budget could not have been delayed without lessening the interest attaching to the writer's thoughts upon questions of our own day. I trust that, incomplete as the work is compared with what it might have been, I shall not be held mistaken in giving it to the world. Rather let me hope that it will be welcomed as an old friend returning under great disadvantages, but bringing a pleasant remembrance of the amusement which its weekly appearance in the Athenæum gave to both writer and reader.

The Paradoxes are dealt with in chronological order. This will be a guide to the reader, and with the alphabetical Index of Names, etc., will, I trust, obviate all difficulty of reference.

Sophia De Morgan.

6 Merton Road, Primrose Hill.

PREFACE TO THE SECOND EDITION.

If Mrs. De Morgan felt called upon to confess her hesitation at taking upon herself the labor of editing these Paradoxes, much more should one who was born two generations later, who lives in another land and who was reared amid different influences, confess to the same feeling when undertaking to revise this curious medley. But when we consider the nature of the work, the fact that its present rarity deprives so many readers of the enjoyment of its delicious satire, and the further fact that allusions that were commonplace a half century ago are now forgotten, it is evident that some one should take up the work and perform it con amore.

Having long been an admirer of De Morgan, having continued his work in the bibliography of early arithmetics, and having worked in his library among the books of which he was so fond, it is possible that the present editor, whatever may be his other shortcomings, may undertake the labor with as much of sympathy as any one who is in a position to perform it. With this thought in mind, two definite rules were laid down at the beginning of the task: (1) That no alteration in the text should be made, save in slightly modernizing spelling and punctuation and in the case of manifest typographical errors; (2) That whenever a note appeared it should show at once its authorship, to the end that the material of the original edition might appear intact.

In considering, however, the unbroken sequence of items that form the Budget, it seems clear that readers would be greatly aided if the various leading topics were separated in some convenient manner. After considerable thought it was decided to insert brief captions from time to time that might aid the eye in selecting the larger subjects of the text. In some parts of the work these could easily be taken from the original folio heads, but usually they had to be written anew. While, therefore, the present editor accepts the responsibility for the captions of the various subdivisions, he has endeavored to insert them in harmony with the original text.

As to the footnotes, the first edition had only a few, some due to De Morgan himself and others to Mrs. De Morgan. In the present edition those due to the former are signed A. De M., and those due to Mrs. De Morgan appear with her initials, S. E. De M. For all other footnotes the present editor is responsible. In preparing them the effort has been made to elucidate the text by supplying such information as the casual reader might wish as he passes over the pages. Hundreds of names are referred to in the text that were more or less known in England half a century ago, but are now forgotten there and were never familiar elsewhere. Many books that were then current have now passed out of memory, and much that agitated England in De Morgan's prime seems now like ancient history. Even with respect to well-known names, a little information as to dates and publications will often be welcome, although the editor recognizes that it will quite as often be superfluous. In order, therefore, to derive the pleasure that should come from reading the Budget, the reader should have easy access to the information that the notes are intended to supply. That they furnish too much here and too little there is to be expected. They are a human product, and if they fail to serve their purpose in all respects it is hoped that this failure will not seriously interfere with the reader's pleasure.

In general the present editor has refrained from expressing any opinions that would strike a discordant note in the reading of the text as De Morgan left it. The temptation is great to add to the discussion at various points, but it is a temptation to be resisted. To furnish such information as shall make the reading more pleasant, rather than to attempt to improve upon one of the most delicious bits of satire of the nineteenth century, has been the editor's wish. It would have been an agreeable task to review the history of circle squaring, of the trisection problem, and of the duplication of the cube. This, however, would be to go too far afield. For the benefit of those who wish to investigate the subject the editor can only refer to such works and articles as the following: F. Rudio, Archimedes, Huygens, Lambert, Legendre,—mit einer Uebersicht über die Geschichte des Problemes von der Quadratur des Zirkels, Leipsic, 1892; Thomas Muir, "Circle," in the eleventh edition of the Encyclopædia Britannica; the various histories of mathematics; and to his own article on "The Incommensurability of π" in Prof. J. W. A. Young's Monographs on Topics of Modern Mathematics, New York, 1911.

The editor wishes to express his appreciation and thanks to Dr. Paul Carus, editor of The Monist and The Open Court for the opportunity of undertaking this work; to James Earl Russell, LL.D., Dean of Teachers College, Columbia University, for his encouragement in its prosecution; to Miss Caroline Eustis Seely for her intelligent and painstaking assistance in securing material for the notes; and to Miss Lydia G. Robinson and Miss Anna A. Kugler for their aid and helpful suggestions in connection with the proof-sheets. Without the generous help of all five this work would have been impossible.

David Eugene Smith.

Teachers College, Columbia University.

A BUDGET OF PARADOXES

INTRODUCTORY.

If I had before me a fly and an elephant, having never seen more than one such magnitude of either kind; and if the fly were to endeavor to persuade me that he was larger than the elephant, I might by possibility be placed in a difficulty. The apparently little creature might use such arguments about the effect of distance, and might appeal to such laws of sight and hearing as I, if unlearned in those things, might be unable wholly to reject. But if there were a thousand flies, all buzzing, to appearance, about the great creature; and, to a fly, declaring, each one for himself, that he was bigger than the quadruped; and all giving different and frequently contradictory reasons; and each one despising and opposing the reasons of the others—I should feel quite at my ease. I should certainly say, My little friends, the case of each one of you is destroyed by the rest. I intend to show flies in the swarm, with a few larger animals, for reasons to be given.

In every age of the world there has been an established system, which has been opposed from time to time by isolated and dissentient reformers. The established system has sometimes fallen, slowly and gradually: it has either been upset by the rising influence of some one man, or it has been sapped by gradual change of opinion in the many.

I have insisted on the isolated character of the dissentients, as an element of the a priori probabilities of the case. Show me a schism, especially a growing schism, and it is another thing. The homeopathists, for instance, shall be, if any one so think, as wrong as St. John Long; but an organized opposition, supported by the efforts of many acting in concert, appealing to common arguments and experience, with perpetual succession and a common seal, as the Queen says in the charter, is, be the merit of the schism what it may, a thing wholly different from the case of the isolated opponent in the mode of opposition to it which reason points out.

During the last two centuries and a half, physical knowledge has been gradually made to rest upon a basis which it had not before. It has become mathematical. The question now is, not whether this or that hypothesis is better or worse to the pure thought, but whether it accords with observed phenomena in those consequences which can be shown necessarily to follow from it, if it be true. Even in those sciences which are not yet under the dominion of mathematics, and perhaps never will be, a working copy of the mathematical process has been made. This is not known to the followers of those sciences who are not themselves mathematicians and who very often exalt their horns against the mathematics in consequence. They might as well be squaring the circle, for any sense they show in this particular.

A great many individuals, ever since the rise of the mathematical method, have, each for himself, attacked its direct and indirect consequences. I shall not here stop to point out how the very accuracy of exact science gives better aim than the preceding state of things could give. I shall call each of these persons a paradoxer, and his system a paradox. I use the word in the old sense: a paradox is something which is apart from general opinion, either in subject-matter, method, or conclusion.

Many of the things brought forward would now be called crotchets, which is the nearest word we have to old paradox. But there is this difference, that by calling a thing a crotchet we mean to speak lightly of it; which was not the necessary sense of paradox. Thus in the sixteenth century many spoke of the earth's motion as the paradox of Copernicus, who held the ingenuity of that theory in very high esteem, and some, I think, who even inclined towards it. In the seventeenth century, the depravation of meaning took place, in England at least. Phillips says paradox is "a thing which seemeth strange"—here is the old meaning: after a colon he proceeds—"and absurd, and is contrary to common opinion," which is an addition due to his own time.

Some of my readers are hardly inclined to think that the word paradox could once have had no disparagement in its meaning; still less that persons could have applied it to themselves. I chance to have met with a case in point against them. It is Spinoza's Philosophia Scripturæ Interpres, Exercitatio Paradoxa, printed anonymously at Eleutheropolis, in 1666. This place was one of several cities in the clouds, to which the cuckoos resorted who were driven away by the other birds; that is, a feigned place of printing, adopted by those who would have caught it if orthodoxy could have caught them. Thus, in 1656, the works of Socinus could only be printed at Irenopolis. The author deserves his self-imposed title, as in the following:

"Quanto sane satius fuisset illam [Trinitatem] pro mysterio non habuisse, et Philosophiæ ope, antequam quod esset statuerent, secundum veræ logices præcepta quid esset cum Cl. Kleckermanno investigasse; tanto fervore ac labore in profundissimas speluncas et obscurissimos metaphysicarum speculationum atque fictionum recessus se recipere ut ab adversariorum telis sententiam suam in tuto collocarent. Profecto magnus ille vir ... dogma illud, quamvis apud theologos eo nomine non multum gratiæ iniverit, ita ex immotis Philosophiæ fundamentis explicat ac demonstrat, ut paucis tantum immutatis, atque additis, nihil amplius animus veritate sincere deditus desiderare possit."

This is properly paradox, though also heterodox. It supposes, contrary to all opinion, orthodox and heterodox, that philosophy can, with slight changes, explain the Athanasian doctrine so as to be at least compatible with orthodoxy. The author would stand almost alone, if not quite; and this is what he meant. I have met with the counter-paradox. I have heard it maintained that the doctrine as it stands, in all its mystery is a priori more likely than any other to have been Revelation, if such a thing were to be; and that it might almost have been predicted.

After looking into books of paradoxes for more than thirty years, and holding conversation with many persons who have written them, and many who might have done so, there is one point on which my mind is fully made up. The manner in which a paradoxer will show himself, as to sense or nonsense, will not depend upon what he maintains, but upon whether he has or has not made a sufficient knowledge of what has been done by others, especially as to the mode of doing it, a preliminary to inventing knowledge for himself. That a little knowledge is a dangerous thing is one of the most fallacious of proverbs. A person of small knowledge is in danger of trying to make his little do the work of more; but a person without any is in more danger of making his no knowledge do the work of some. Take the speculations on the tides as an instance. Persons with nothing but a little geometry have certainly exposed themselves in their modes of objecting to results which require the higher mathematics to be known before an independent opinion can be formed on sufficient grounds. But persons with no geometry at all have done the same thing much more completely.

There is a line to be drawn which is constantly put aside in the arguments held by paradoxers in favor of their right to instruct the world. Most persons must, or at least will, like the lady in Cadogan Place, form and express an immense variety of opinions on an immense variety of subjects; and all persons must be their own guides in many things. So far all is well. But there are many who, in carrying the expression of their own opinions beyond the usual tone of private conversation, whether they go no further than attempts at oral proselytism, or whether they commit themselves to the press, do not reflect that they have ceased to stand upon the ground on which their process is defensible. Aspiring to lead others, they have never given themselves the fair chance of being first led by other others into something better than they can start for themselves; and that they should first do this is what both those classes of others have a fair right to expect. New knowledge, when to any purpose, must come by contemplation of old knowledge in every matter which concerns thought; mechanical contrivance sometimes, not very often, escapes this rule. All the men who are now called discoverers, in every matter ruled by thought, have been men versed in the minds of their predecessors, and learned in what had been before them. There is not one exception. I do not say that every man has made direct acquaintance with the whole of his mental ancestry; many have, as I may say, only known their grandfathers by the report of their fathers. But even on this point it is remarkable how many of the greatest names in all departments of knowledge have been real antiquaries in their several subjects.

I may cite, among those who have wrought strongly upon opinion or practice in science, Aristotle, Plato, Ptolemy, Euclid, Archimedes, Roger Bacon, Copernicus, Francis Bacon, Ramus, Tycho Brahé, Galileo, Napier, Descartes, Leibnitz, Newton, Locke. I take none but names known out of their fields of work; and all were learned as well as sagacious. I have chosen my instances: if any one will undertake to show a person of little or no knowledge who has established himself in a great matter of pure thought, let him bring forward his man, and we shall see.

This is the true way of putting off those who plague others with their great discoveries. The first demand made should be—Mr. Moses, before I allow you to lead me over the Red Sea, I must have you show that you are learned in all the wisdom of the Egyptians upon your own subject. The plea that it is unlikely that this or that unknown person should succeed where Newton, etc. have failed, or should show Newton, etc. to be wrong, is utterly null and void. It was worthily versified by Sylvanus Morgan (the great herald who in his Sphere of Gentry gave coat armor to "Gentleman Jesus," as he said), who sang of Copernicus as follows (1652):

"If Tellus winged be,

The earth a motion round;

Then much deceived are they

Who nere before it found.

Solomon was the wisest,

His wit nere this attained;

Cease, then, Copernicus,

Thy hypothesis is vain."

Newton, etc. were once unknown; but they made themselves known by what they knew, and then brought forward what they could do; which I see is as good verse as that of Herald Sylvanus. The demand for previous knowledge disposes of twenty-nine cases out of thirty, and the thirtieth is worth listening to.

I have not set down Copernicus, Galileo, etc. among the paradoxers, merely because everybody knows them; if my list were quite complete, they would have been in it. But the reader will find Gilbert, the great precursor of sound magnetical theory; and several others on whom no censure can be cast, though some of their paradoxes are inadmissible, some unprovoked, and some capital jokes, true or false: the author of Vestiges of Creation is an instance. I expect that my old correspondent, General Perronet Thompson, will admit that his geometry is part and parcel of my plan; and also that, if that plan embraced politics, he would claim a place for his Catechism on the Corn Laws, a work at one time paradoxical, but which had more to do with the abolition of the bread-tax than Sir Robert Peel.

My intention in publishing this Budget in the Athenæum is to enable those who have been puzzled by one or two discoverers to see how they look in a lump. The only question is, has the selection been fairly made? To this my answer is, that no selection at all has been made. The books are, without exception, those which I have in my own library; and I have taken all—I mean all of the kind: Heaven forbid that I should be supposed to have no other books! But I may have been a collector, influenced in choice by bias? I answer that I never have collected books of this sort—that is, I have never searched for them, never made up my mind to look out for this book or that. I have bought what happened to come in my way at show or auction; I have retained what came in as part of the undescribed portion of miscellaneous auction lots; I have received a few from friends who found them among what they called their rubbish; and I have preserved books sent to me for review. In not a few instances the books have been bound up with others, unmentioned at the back; and for years I knew no more I had them than I knew I had Lord Macclesfield's speech on moving the change of Style, which, after I had searched shops, etc. for it in vain, I found had been reposing on my own shelves for many years, at the end of a summary of Leibnitz's philosophy. Consequently, I may positively affirm that the following list is formed by accident and circumstance alone, and that it truly represents the casualties of about a third of a century. For instance, the large proportion of works on the quadrature of the circle is not my doing: it is the natural share of this subject in the actual run of events.

[I keep to my plan of inserting only such books as I possessed in 1863, except by casual notice in aid of my remarks. I have found several books on my shelves which ought to have been inserted. These have their titles set out at the commencement of their articles, in leading paragraphs; the casuals are without this formality.]

Before proceeding to open the Budget, I say something on my personal knowledge of the class of discoverers who square the circle, upset Newton, etc. I suspect I know more of the English class than any man in Britain. I never kept any reckoning; but I know that one year with another—and less of late years than in earlier time—I have talked to more than five in each year, giving more than a hundred and fifty specimens. Of this I am sure, that it is my own fault if they have not been a thousand. Nobody knows how they swarm, except those to whom they naturally resort. They are in all ranks and occupations, of all ages and characters. They are very earnest people, and their purpose is bona fide the dissemination of their paradoxes. A great many—the mass, indeed—are illiterate, and a great many waste their means, and are in or approaching penury. But I must say that never, in any one instance, has the quadrature of the circle, or the like, been made a pretext for begging; even to be asked to purchase a book is of the very rarest occurrence—it has happened, and that is all.

These discoverers despise one another: if there were the concert among them which there is among foreign mendicants, a man who admitted one to a conference would be plagued to death. I once gave something to a very genteel French applicant, who overtook me in the street, at my own door, saying he had picked up my handkerchief: whether he picked it up in my pocket for an introduction, I know not. But that day week came another Frenchman to my house, and that day fortnight a French lady; both failed, and I had no more trouble. The same thing happened with Poles. It is not so with circle-squarers, etc.: they know nothing of each other. Some will read this list, and will say I am right enough, generally speaking, but that there is an exception, if I could but see it.

I do not mean, by my confession of the manner in which I have sinned against the twenty-four hours, to hold myself out as accessible to personal explanation of new plans. Quite the contrary: I consider myself as having made my report, and being discharged from further attendance on the subject. I will not, from henceforward, talk to any squarer of the circle, trisector of the angle, duplicator of the cube, constructor of perpetual motion, subverter of gravitation, stagnator of the earth, builder of the universe, etc. I will receive any writings or books which require no answer, and read them when I please: I will certainly preserve them—this list may be enlarged at some future time.

There are three subjects which I have hardly anything upon; astrology, mechanism, and the infallible way of winning at play. I have never cared to preserve astrology. The mechanists make models, and not books. The infallible winners—though I have seen a few—think their secret too valuable, and prefer mutare quadrata rotundis—to turn dice into coin—at the gaming-house: verily they have their reward.

I shall now select, to the mystic number seven, instances of my personal knowledge of those who think they have discovered, in illustration of as many misconceptions.

1. Attempt by help of the old philosophy, the discoverer not being in possession of modern knowledge. A poor schoolmaster, in rags, introduced himself to a scientific friend with whom I was talking, and announced that he had found out the composition of the sun. "How was that done?"—"By consideration of the four elements."—"What are they?"—"Of course, fire, air, earth, and water."—"Did you not know that air, earth, and water, have long been known to be no elements at all, but compounds?"—"What do you mean, sir? Who ever heard of such a thing?"

2. The notion that difficulties are enigmas, to be overcome in a moment by a lucky thought. A nobleman of very high rank, now long dead, read an article by me on the quadrature, in an early number of the Penny Magazine. He had, I suppose, school recollections of geometry. He put pencil to paper, drew a circle, and constructed what seemed likely to answer, and, indeed, was—as he said—certain, if only this bit were equal to that; which of course it was not. He forwarded his diagram to the Secretary of the Diffusion Society, to be handed to the author of the article, in case the difficulty should happen to be therein overcome.

3. Discovery at all hazards, to get on in the world. Thirty years ago, an officer of rank, just come from foreign service, and trying for a decoration from the Crown, found that his claims were of doubtful amount, and was told by a friend that so and so, who had got the order, had the additional claim of scientific distinction. Now this officer, while abroad, had bethought himself one day, that there really could be no difficulty in finding the circumference of a circle: if a circle were rolled upon a straight line until the undermost point came undermost again, there would be the straight line equal to the circle. He came to me, saying that he did not feel equal to the statement of his claim in this respect, but that if some clever fellow would put the thing in a proper light, he thought his affair might be managed. I was clever enough to put the thing in a proper light to himself, to this extent at least, that, though perhaps they were wrong, the advisers of the Crown would never put the letters K.C.B. to such a circle as his.

4. The notion that mathematicians cannot find the circle for common purposes. A working man measured the altitude of a cylinder accurately, and—I think the process of Archimedes was one of his proceedings—found its bulk. He then calculated the ratio of the circumference to the diameter, and found it answered very well on other modes of trial. His result was about 3.14. He came to London, and somebody sent him to me. Like many others of his pursuit, he seemed to have turned the whole force of his mind upon one of his points, on which alone he would be open to refutation. He had read some of Kater's experiments, and had got the Act of 1825 on weights and measures. Say what I would, he had for a long time but one answer—"Sir! I go upon Captain Kater and the Act of Parliament." But I fixed him at last. I happened to have on the table a proof-sheet of the Astronomical Memoirs, in which were a large number of observed places of the planets compared with prediction, and asked him whether it could be possible that persons who did not know the circle better than he had found it could make the calculations, of which I gave him a notion, so accurately? He was perfectly astonished, and took the titles of some books which he said he would read.