A Budget of Paradoxes Volume I Part 24

ON CURIOSITIES OF [pi].

1830. The celebrated interminable fraction 3.14159..., which the mathematician calls [pi], is the ratio of the circ.u.mference to the diameter. But it is thousands of things besides. It is constantly turning up in mathematics: and if arithmetic and algebra had been studied without geometry, [pi] must have come in somehow, though at what stage or under what name must have depended upon the casualties of algebraical invention.

This will readily be seen when it is stated that [pi] is nothing but four times the series

1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + ...

_ad infinitum_.[616] It would be wonderful if so simple a series {285} had but one kind of occurrence. As it is, our trigonometry being founded on the circle, [pi] first appears as the ratio stated. If, for instance, a deep study of probable fluctuation from average had preceded, [pi] might have emerged as a number perfectly indispensable in such problems as: What is the chance of the number of aces lying between a million + x and a million - x, when six million of throws are made with a die? I have not gone into any detail of all those cases in which the paradoxer finds out, by his una.s.sisted ac.u.men, that results of mathematical investigation _cannot be_: in fact, this discovery is only an accompaniment, though a necessary one, of his paradoxical statement of that which _must be_. Logicians are beginning to see that the notion of _horse_ is inseparably connected with that of _non-horse_: that the first without the second would be no notion at all. And it is clear that the positive affirmation of that which contradicts mathematical demonstration cannot but be accompanied by a declaration, mostly overtly made, that demonstration is false. If the mathematician were interested in punishing this indiscretion, he could make his denier ridiculous by inventing a.s.serted results which would completely take him in.

More than thirty years ago I had a friend, now long gone, who was a mathematician, but not of the higher branches: he was, _inter alia_, thoroughly up in all that relates to mortality, life a.s.surance, &c. One day, explaining to him how it should be ascertained what the chance is of the survivors of a large number of persons now alive lying between given limits of number at the end of a certain time, I came, of course upon the introduction of [pi], which I could only describe as the ratio of the circ.u.mference of a circle to its diameter. "Oh, my dear friend! that must be a delusion; what can the circle have to do with the numbers alive at the end of a given time?"--"I cannot demonstrate it to you; but it is demonstrated."--"Oh! stuff! I think you can prove anything with your differential calculus: figment, {286} depend upon it." I said no more; but, a few days afterwards, I went to him and very gravely told him that I had discovered the law of human mortality in the Carlisle Table, of which he thought very highly. I told him that the law was involved in this circ.u.mstance. Take the table of expectation of life, choose any age, take its expectation and make the nearest integer a new age, do the same with that, and so on; begin at what age you like, you are sure to end at the place where the age past is equal, or most nearly equal, to the expectation to come. "You don't mean that this always happens?"--"Try it." He did try, again and again; and found it as I said. "This is, indeed, a curious thing; this _is_ a discovery." I might have sent him about trumpeting the law of life: but I contented myself with informing him that the same thing would happen with any table whatsoever in which the first column goes up and the second goes down; and that if a proficient in the higher mathematics chose to palm a figment upon him, he could do without the circle: _a corsaire, corsaire et demi_,[617] the French proverb says. "Oh!" it was remarked, "I see, this was Milne!"[618] It was _not_ Milne: I remember well showing the formula to him some time afterwards. He raised no difficulty about [pi]; he knew the forms of Laplace's results, and he was much interested. Besides, Milne never said stuff! and figment! And he would not have been taken in: he would have quietly tried it with the Northampton and all the other tables, and would have got at the truth.

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EUCLID WITHOUT AXIOMS.

The first book of Euclid's Elements. With alterations and familiar notes. Being an attempt to get rid of axioms altogether; and to establish the theory of parallel lines, without the introduction of any principle not common to other parts of the elements. By a member of the University of Cambridge. Third edition. In usum serenissimae filiolae.

London, 1830.

The author was Lieut. Col. (now General) Perronet Thompson,[619] the author of the "Catechism on the Corn Laws." I reviewed the fourth edition--which had the name of "Geometry without Axioms," 1833--in the quarterly _Journal of Education_ for January, 1834. Col. Thompson, who then was a contributor to--if not editor of--the _Westminster Review_, replied in an article the authorship of which could not be mistaken.

Some more attempts upon the problem, by the same author, will be found in the sequel. They are all of acute and legitimate speculation; but they do not conquer the difficulty in the manner demanded by the conditions of the problem. The paradox of parallels does not contribute much to my pages: its cases are to be found for the most part in geometrical systems, or in notes to them. Most of them consist in the proposal of additional postulates; some are attempts to do without any new postulate. Gen. Perronet Thompson, whose paradoxes are always constructed on much study of previous writers, has collected in the work above named, a budget of attempts, the heads of which are in the _Penny_ and _English Cyclopaedias_, at "Parallels." He has given thirty instances, selected from what he had found.[620]

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Lagrange,[621] in one of the later years of his life, imagined that he had overcome the difficulty. He went so far as to write a paper, which he took with him to the Inst.i.tute, and began to read it. But in the first paragraph something struck him which he had not observed: he muttered _Il faut que j'y songe encore_,[622] and put the paper in his pocket.

THE LUNAR CAUSTIC JOKE.

The following paragraph appeared in the _Morning Post_, May 4, 1831:

"We understand that although, owing to circ.u.mstances with which the public are not concerned, Mr. Goulburn[623] declined becoming a candidate for University honors, that his scientific attainments are far from inconsiderable. He is well known to be the author of an essay in the Philosophical Transactions on the accurate rectification of a circular arc, and of an investigation of the equation of a lunar caustic--a problem likely to become of great use in nautical astronomy."

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This hoax--which would probably have succeeded with any journal--was palmed upon the _Morning Post_, which supported Mr. Goulburn, by some Cambridge wags who supported Mr. Lubbock, the other candidate for the University of Cambridge. Putting on the usual concealment, I may say that I always suspected Dr-nkw-t-r B-th-n-[624] of having a share in the matter. The skill of the hoax lies in avoiding the words "quadrature of the circle,"

which all know, and speaking of "the accurate rectification of a circular arc," which all do not know for its synonyme. The _Morning Post_ next day gave a reproof to hoaxers in general, without referring to any particular case. It must be added, that although there are _caustics_ in mathematics, there is no _lunar_ caustic.

So far as Mr. Goulburn was concerned, the above was poetic justice. He was the minister who, in old time, told a deputation from the Astronomical Society that the Government "did not care twopence for all the science in the country." There may be some still alive who remember this: I heard it from more than one of those who were present, and are now gone. Matters are much changed. I was thirty years in office at the Astronomical Society; and, to my certain knowledge, every Government of that period, Whig and Tory, showed itself ready to help with influence when wanted, and with money whenever there was an answer for the House of Commons. The following correction subsequently appeared. Referring to the hoax about Mr. Goulburn, Messrs. C. H. and Thompson Cooper[625] have corrected an error, by stating that the election which gave rise to the hoax was that in which Messrs.

Goulburn {290} and Yates Peel[626] defeated Lord Palmerston[627] and Mr.

Cavendish.[628] They add that Mr. Gunning, the well-known Esquire Bedell of the University, attributed the hoax to the late Rev. R. Sheepshanks, to whom, they state, are also attributed certain clever fict.i.tious biographies--of public men, as I understand it--which were palmed upon the editor of the _Cambridge Chronicle_, who never suspected their genuineness to the day of his death. Being in most confidential intercourse with Mr.

Sheepshanks,[629] both at the time and all the rest of his life (twenty-five years), and never heard him allude to any such things--which were not in his line, though he had satirical power of quite another {291} kind--I feel satisfied he had nothing to do with them. I may add that others, his nearest friends, and also members of his family, never heard him allude to these hoaxes as their author, and disbelieve his authorship as much as I do myself. I say this not as imputing any blame to the true author, such hoaxes being fair election jokes in all time, but merely to put the saddle off the wrong horse, and to give one more instance of the insecurity of imputed authorship. Had Mr. Sheepshanks ever told me that he had perpetrated the hoax, I should have had no hesitation in giving it to him. I consider all clever election squibs, free from bitterness and personal imputation, as giving the mult.i.tude good channels for the vent of feelings which but for them would certainly find bad ones.

[But I now suspect that Mr. Babbage[630] had some hand in the hoax. He gives it in his "Pa.s.sages, &c." and is evidently writing from memory, for he gives the wrong year. But he has given the paragraph, though not accurately, yet with such a recollection of the points as brings suspicion of the authorship upon him, perhaps in conjunction with D. B.[631] Both were on Cavendish's committee. Mr. Babbage adds, that "late one evening a cab drove up in hot haste to the office of the _Morning Post_, delivered the copy as coming from Mr. Goulburn's committee, and at the same time ordered fifty extra copies of the _Post_ to be sent next morning to their committee-room." I think the man--the only one I ever heard of--who knew all about the cab and the extra copies must have known more.]

ON M. DEMONVILLE.

_Demonville._--A Frenchman's Christian name is his own secret, unless there be two of the surname. M. Demonville is a very good instance of the difference between a {292} French and English discoverer. In England there is a public to listen to discoveries in mathematical subjects made without mathematics: a public which will hear, and wonder, and think it possible that the pretensions of the discoverer have some foundation. The unnoticed man may possibly be right: and the old country-town reputation which I once heard of, attaching to a man who "had written a book about the signs of the zodiac which all the philosophers in London could not answer," is fame as far as it goes. Accordingly, we have plenty of discoverers who, even in astronomy, p.r.o.nounce the learned in error because of mathematics. In France, beyond the sphere of influence of the Academy of Sciences, there is no one to cast a thought upon the matter: all who take the least interest repose entire faith in the Inst.i.tute. Hence the French discoverer turns all his thoughts to the Inst.i.tute, and looks for his only hearing in that quarter. He therefore throws no slur upon the means of knowledge, but would say, with M. Demonville: "A l'egard de M. Poisson,[632] j'envie loyalement la millieme partie de ses connaissances mathematiques, pour prouver mon systeme d'astronomie aux plus incredules."[633] This system is that the only bodies of our system are the earth, the sun, and the moon; all the others being illusions, caused by reflection of the sun and moon from the ice of the polar regions. In mathematics, addition and subtraction are for men; multiplication and division, which are in truth creation and destruction, are prerogatives of deity. But _nothing_ multiplied by _nothing_ is _one_. M. Demonville obtained an introduction to William the Fourth, who desired the opinion of the Royal Society upon his system: the {293} answer was very brief. The King was quite right; so was the Society: the fault lay with those who advised His Majesty on a matter they knew nothing about. The writings of M. Demonville in my possession are as follows.[634] The dates--which were only on covers torn off in binding--were about 1831-34:

_Pet.i.t cours d'astronomie_[635] followed by _Sur l'unite mathematique._--_Principes de la physique de la creation implicitement admis dans la notice sur le tonnerre par M. Arago._--_Question de longitude sur mer._[636]--_Vrai systeme du monde_[637] (pp. 92). Same t.i.tle, four pages, small type. Same t.i.tle, four pages, addressed to the British a.s.sociation. Same t.i.tle, four pages, addressed to M. Mathieu. Same t.i.tle, four pages, on M. Bouvard's report.--_Resume de la physique de la creation; troisieme partie du vrai systeme du monde._[638]

Pa.r.s.eY'S PARADOX.

The quadrature of the circle discovered, by Arthur Pa.r.s.ey,[639] author of the 'art of miniature painting.' Submitted to the consideration of the Royal Society, on whose protection the author humbly throws himself. London, 1832, 8vo.

Mr. Pa.r.s.ey was an artist, who also made himself conspicuous by a new view of perspective. Seeing that the sides of a tower, for instance, would appear to meet in a point if the tower were high enough, he thought that these sides ought to slope to one another in the picture. On this {294} theory he published a small work, of which I have not the t.i.tle, with a Grecian temple in the frontispiece, stated, if I remember rightly, to be the first picture which had ever been drawn in true perspective. Of course the building looked very Egyptian, with its sloping sides. The answer to his notion is easy enough. What is called the picture is not the picture from which the mind takes its perception; that picture is on the retina.

The _intermediate_ picture, as it may be called--the human artist's work--is itself seen perspectively. If the tower were so high that the sides, though parallel, appeared to meet in a point, the picture must also be so high that the _picture-sides_, though parallel, would appear to meet in a point. I never saw this answer given, though I have seen and heard the remarks of artists on Mr. Pa.r.s.ey's work. I am inclined to think it is commonly supposed that the artist's picture is the representation which comes before the mind: this is not true; we might as well say the same of the object itself. In July 1831, reading an article on squaring the circle, and finding that there was a difficulty, he set to work, got a light denied to all mathematicians in--some would say through--a crack, and advertised in the _Times_ that he had done the trick. He then prepared this work, in which, those who read it will see how, he showed that 3.14159... should be 3.0625. He might have found out his error by _stepping_ a draughtsman's circle with the compa.s.ses.

Perspective has not had many paradoxes. The only other one I remember is that of a writer on perspective, whose name I forget, and whose four pages I do not possess. He circulated remarks on my notes on the subject, published in the _Athenaeum_, in which he denies that the stereographic projection is a case of perspective, the reason being that the whole hemisphere makes too large a picture for the eye conveniently to grasp at once. That is to say, it is no perspective because there is too much perspective. {295}

ON A COUPLE OF GEOMETRIES.

Principles of Geometry familiarly ill.u.s.trated. By the Rev. W.

Ritchie,[640] LL.D. London, 1833, 12mo.

A new Exposition of the system of Euclid's Elements, being an attempt to establish his work on a different basis. By Alfred Day,[641] LL.D.

London, 1839, 12mo.

These works belong to a small cla.s.s which have the peculiarity of insisting that in the general propositions of geometry a proposition gives its converse: that "Every B is A" follows from "Every A is B." Dr. Ritchie says, "If it be proved that the equality of two of the angles of a triangle depends _essentially_ upon the equality of the opposite sides, it follows that the equality of opposite sides depends _essentially_ on the equality of the angles." Dr. Day puts it as follows:

"That the converses of Euclid, so called, where no particular limitation is specified or implied in the leading proposition, more than in the converse, must be necessarily true; for as by the nature of the reasoning the leading proposition must be universally true, should the converse be not so, it cannot be so universally, but has at least all the exceptions conveyed in the leading proposition, and the case is therefore unadapted to geometric reasoning; or, what is the same thing, by the very nature of geometric reasoning, the particular exceptions to the extended converse must be identical with some one or other of the cases under the universal affirmative proposition with which we set forth, which is absurd."

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On this I cannot help transferring to my reader the words of the Pacha when he orders the bastinado,--May it do you good! A rational study of logic is much wanted to show many mathematicians, of all degrees of proficiency, that there is nothing in the _reasoning_ of mathematics which differs from other reasoning. Dr. Day repeated his argument in _A Treatise on Proportion_, London, 1840, 8vo. Dr. Ritchie was a very clear-headed man. He published, in 1818, a work on arithmetic, with rational explanations. This was too early for such an improvement, and nearly the whole of his excellent work was sold as waste paper. His elementary introduction to the Differential Calculus was drawn up while he was learning the subject late in life. Books of this sort are often very effective on points of difficulty.

NEWTON AGAIN OBLITERATED.

Letter to the Royal Astronomical Society in refutation of Mistaken Notions held in common, by the Society, and by all the Newtonian philosophers. By Capt. Forman,[642] R.N. Shepton-Mallet, 1833, 8vo.

Capt. Forman wrote against the whole system of gravitation, and got no notice. He then wrote to Lord Brougham, Sir J. Herschel, and others I suppose, desiring them to procure notice of his books in the reviews: this not being acceded to, he wrote (in print) to Lord John Russell[643] to complain of their "dishonest" conduct. He then sent a ma.n.u.script letter to the Astronomical Society, inviting controversy: he was answered by a recommendation to study {297} dynamics. The above pamphlet was the consequence, in which, calling the Council of the Society "craven dunghill c.o.c.ks," he set them right about their doctrines. From all I can learn, the life of a worthy man and a creditable officer was completely embittered by his want of power to see that no person is bound in reason to enter into controversy with every one who chooses to invite him to the field. This mistake is not peculiar to philosophers, whether of orthodoxy or paradoxy; a majority of educated persons imply, by their modes of proceeding, that no one has a right to any opinion which he is not prepared to defend against all comers.

David and Goliath, or an attempt to prove that the Newtonian system of Astronomy is directly opposed to the Scriptures. By Wm. Lauder,[644]

Sen., Mere, Wilts. Mere, 1833, 12mo.

Newton is Goliath; Mr. Lauder is David. David took five pebbles; Mr. Lauder takes five arguments. He expects opposition; for Paul and Jesus both met with it.