A Budget of Paradoxes Volume II Part 30

(n + 1) (4n^2 + 2n + 1)

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

The three paradoxers last named and myself have a pentasyllable convention, under which, though we go far beyond civility, we keep within civilization.

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

I will not give the name of the poor fellow who has fallen so far below both the _honestum_ and the _utile_, to say nothing of the _decorum_ or the _dulce_.[662] He is the fourth who has taken elaborate notice of me; and my advice to him would be, _Nec quarta loqui persona laboret_.[663] According to him, I scorn humanity, scandalize learning, and disgrace the press; it admits of no manner of doubt that my object is to mislead the public and silence truth, at the expense of the interests of science, the wealth of the nation, and the lives of my fellow men. The only thing left to be settled is, whether this is due to ignorance, natural distaste for truth, personal malice, a wish to curry favor with the Astronomer Royal, or mere toadyism. The only accusation which has truth in it is, that I have made myself a "public scavenger of science": the a.s.sertion, which is the {363} most false of all is, that the results of my broom and spade are "shot right in between the columns of" the _Athenaeum_. I declare I never in my life inserted a word between the columns of the _Athenaeum_: I feel huffed and miffed at the very supposition. I _have_ made myself a public scavenger; and why not? Is the mud never to be collected into a heap? I look down upon the other scavengers, of whom there have been a few--mere historical drudges; Montucla, Hutton, etc.--as not fit to compete with me.

I say of them what one crossing-sweeper said of the rest: "They are well enough for the common thing; but put them to a bit of fancy-work, such as sweeping round a post, and see what a mess they make of it!" Who can touch me at sweeping round a paradoxer? If I complete my design of publis.h.i.+ng a separate work, an old copy will be fished up from a stall two hundred years hence by the coming man, and will be described in an article which will end by his comparing our century with his own, and sighing out in the best New Zealand p.r.o.nunciation--

"Dans ces tems-la C'etait deja comme ca!"[664]

ORTHODOX PARADOXERS.

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

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

I now proceed to die out. Some prefatory remarks will follow in time.[666]

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

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

I am strongly reminded of the monks in the darker part of the Middle Ages.

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

"Flen, flyys, and freris, populum domini male caedunt; Thystlis and breris crescentia gramina laedunt.

Christe nolens guerras qui cuncta pace tueris, Destrue per terras breris, flen, flyys, and freris.

Flen, flyys, and freris, foul falle hem thys fyften yeris, For non that her is lovit flen, flyys ne freris."[667]

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

"Meum est propositum genti imperitae Artes frugi reddere melioris vitae."[668]

To this end I have spoken with freedom of books as books, of opinions as opinions, of ignorance as ignorance, of {366} presumption as presumption; and of writers as I judge may be fairly inferred from what they have written. Some--to whom I am therefore under great obligation--have permitted me to enlarge my plan by a.s.saults to which I have alluded; a.s.saults which allow a privilege of retort, of which I have often availed myself; a.s.saults which give my readers a right of partners.h.i.+p in the amus.e.m.e.nt which I myself have received.

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

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APPENDIX.

I think it right to give the proof that the ratio of the circ.u.mference to the diameter is incommensurable. This method of proof was given by Lambert,[672] in the _Berlin Memoirs_ for 1761, and has been also given in the notes to Legendre's[673] Geometry, and to the English translation of the same. Though not elementary algebra, it is within the reach of a student of ordinary books.[674]

Let a continued fraction, such as

a ----- b + c ----- d + e - f + etc.,

be abbreviated into a/b+ c/d+ e/f+ etc.: each fraction being understood as falling down to the side of the preceding sign +. In every such fraction we may suppose b, d, f, etc. {368} positive; a, c, e, &c. being as required: and all are supposed integers. If this succession be continued ad infinitum, and if a/b, c/d, e/f, etc. all lie between -1 and +1, exclusive, the limit of the fraction must be incommensurable with unity; that is, cannot be A/B, where A and B are integers.

First, whatever this limit may be, it lies between -1 and +1. This is obviously the case with any fraction p/(q + [omega]), where [omega] is between 1: for, p/q, being < 1,="" and="" p="" and="" q="" integer,="" cannot="" be="" brought="" up="" to="" 1,="" by="" the="" value="" of="" [omega].="" hence,="" if="" we="" take="" any="" of="" the="" fractions="">

a/b, a/b+ c/d, a/b+ c/d+ e/f, etc.

say a/b+ c/d+ e/f+ g/h we have, g/h being between 1, so is e/f+ g/h, so therefore is c/d+ e/f+ g/h; and so therefore is a/b+ c/d+ e/f+ g/h.

Now, if possible, let a/b+ c/d+ etc. be A/B at the limit; A and B being integers. Let

P = A c/d+ e/f+ etc., Q = P e/f+ g/h+ etc., R = Q g/h + i/k + etc.

P, Q, R, etc. being integer or fractional, as may be. It is easily shown that all must be integer: for

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A/B = a/b+ P/A, or, P = aB - bA

P/A = c/d+ Q/P, or, Q = cA - dP

Q/P = e/f+ R/Q, or, R = eP - fQ

etc., etc. Now, since a, B, b, A, are integers, so also is P; and thence Q; and thence R, etc. But since A/B, P/A, Q/P, R/Q, etc. are all between -1 and +1, it follows that the unlimited succession of integers P, Q, R, are each less in numerical value than the preceding. Now there can be no such _unlimited_ succession of _descending_ integers: consequently, it is impossible that a/b+ c/d+, etc. can have a commensurable limit.

It easily follows that the continued fraction is incommensurable if a/b, c/d, etc., being at first greater than unity, become and continue less than unity after some one point. Say that i/k, l/m,... are all less than unity.

Then the fraction i/k+ l/m+ ... is incommensurable, as proved: let it be [kappa]. Then g/(h + [kappa]) is incommensurable, say [lambda]; e/(f + [lambda]) is the same, say [mu]; also c/(d + [mu]), say [nu], and a/(b + [nu]), say [rho]. But [rho] is the fraction a/b+ c/d+ ... itself; which is therefore incommensurable.

Let [phi]z represent

a a^2 a^3 1 + - + ------- + -------------- + ....

z 2z(z+1) 23z(z+1)(z+2)

{370} Let z be positive: this series is convergent for all values of a, and approaches without limit to unity as z increases without limit. Change z into z + 1, and form [phi]z - [phi](z+1): the following equation will result--

a [phi]z-[phi](z+1) = ------([phi](z+2)) z(z+1)

a [phi](z+1) a [phi](z+1) a [phi](z+2) or a = - ---------- z + - ---------- --- ---------- z [phi]z z [phi]z z+1 [phi](z+1)