APPENDIX.

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

Forty stripes save one for the smaller Paradoxers.

Hark to the wisdom the sages preachWho never have learnt what they try to teach.We are the lights of the age, they say!We are the men, and the thinkers we!So we build up guess-work the livelong day,In a topsy-turvy sort of way,Some with and some wantingaplusb.Let the British Association fuss;What are theirs to the feats to be wrought by us?Shall the earth stand still? Will the round come square?Must Isaac's book be the nest of a mare?Ought the moon to be taught by the laws of spaceTo turn half round without right-about-face?Our whimsey crotchets will manage it all;Deep! Deep! posterity will them call!Though the world, for the present, lets them fallDown! Down! to the twopenny box of the stall!

Hark to the wisdom the sages preach

Who never have learnt what they try to teach.

We are the lights of the age, they say!

We are the men, and the thinkers we!

So we build up guess-work the livelong day,

In a topsy-turvy sort of way,

Some with and some wantingaplusb.

Let the British Association fuss;

What are theirs to the feats to be wrought by us?

Shall the earth stand still? Will the round come square?

Must Isaac's book be the nest of a mare?

Ought the moon to be taught by the laws of space

To turn half round without right-about-face?

Our whimsey crotchets will manage it all;

Deep! Deep! posterity will them call!

Though the world, for the present, lets them fall

Down! Down! to the twopenny box of the stall!

Thus they—But the marplot Time stands by,With a knowing wink in his funny old eye.He grasps by the top an immense fool's cap,Which he calls a philosophaster-trap:And rightly enough, for while these little menCroak loud as a concert of frogs in a fen,He first singles out one, and then another,Down goes the cap—lo! a moment's pother,A spirit like that which a rushlight uttersAs just at the last it kicks and gutters:When the cruel smotherer is raised againOnly snuff, and but little of that, will remain.

Thus they—But the marplot Time stands by,

With a knowing wink in his funny old eye.

He grasps by the top an immense fool's cap,

Which he calls a philosophaster-trap:

And rightly enough, for while these little men

Croak loud as a concert of frogs in a fen,

He first singles out one, and then another,

Down goes the cap—lo! a moment's pother,

A spirit like that which a rushlight utters

As just at the last it kicks and gutters:

When the cruel smotherer is raised again

Only snuff, and but little of that, will remain.

But thoughuno avulsothus comes every dayNon deficit alteris also in play:For the vacant parts are, one and all,Soon taken by puppets just as small;Who chirp, chirp, chirp, with a grasshopper's glee,We're the lamps of the Universe, We! We! We!But Time, whose speech is never long,—He hasn't time for it—stops the songAnd says—Lilliput lamps! leave the twopenny boxes,And shine in the Budget of Paradoxes!

But thoughuno avulsothus comes every day

Non deficit alteris also in play:

For the vacant parts are, one and all,

Soon taken by puppets just as small;

Who chirp, chirp, chirp, with a grasshopper's glee,

We're the lamps of the Universe, We! We! We!

But Time, whose speech is never long,—

He hasn't time for it—stops the song

And says—Lilliput lamps! leave the twopenny boxes,

And shine in the Budget of Paradoxes!

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

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

"Our brethren of the literary class have not the least idea of the small amount of appearance of knowledgewhich sets up the scientific charlatan. Their world is large, and there are many who have that moderate knowledge, and perception of what is knowledge, before which extreme ignorance is detected in its first prank. There is a public of moderate cultivation, for the most part sound in its judgment, always ready in its decisions. Accordingly, all their successful pretenders havesome pretension. It is not so in science. Those who have a right to judge are fewer and farther between. The consequence is, that many scientific pretenders havenothing but pretension."

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

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

"Sines, tangents, secants, radius, cosines

Subtangents, segments and all those signs;

Enough to prove that he who read 'em

Was just as mad as he who made 'em."

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

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

"Horse-pleas, traverses, demurrers,

Jeofails, imparlances, and errors,

Averments, bars, and protestandos,

And puis d'arreign continuandos."

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

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

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

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

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

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

"Si doceas stultum, lætum non dat tibi vultum;

Odit te multum; vellet te scire sepultum.[658]"

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

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

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

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

THE DOUBLE VAHU PROCESS.

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

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

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

The three paradoxers last named and myself have a pentasyllable convention, under which, though we go far beyond civility, we keep within civilization. Though Mr. James Smith pronounced that I must be dishonest if I did not see his argument, which he knew I should not do [to say nothing of recent accusation]; though Dr. Thorn declared me a competitor for fire and brimstone—and my wife, too, which doubles the joke: though Mr. Reddiewas 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. Buttheyare 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 thehonestumand theutile, to say nothing of thedecorumor thedulce.[662]He is the fourth who has taken elaborate notice of me; and my advice to him would be,Nec quarta loqui persona laboret.[663]According to him, I scorn humanity, scandalize learning, and disgrace the press; it admits of no manner of doubt that my object is to mislead the public and silence truth, at the expense of the interests of science, the wealth of the nation, and the lives of my fellow men. The only thing left to be settled is, whether this is due to ignorance, natural distaste for truth, personal malice, a wish to curry favor with the Astronomer Royal, or mere toadyism. The only accusation which has truth in it is, that I have made myself a "public scavenger of science": the assertion, which is themost false of all is, that the results of my broom and spade are "shot right in between the columns of" theAthenæum. I declare I never in my life inserted a word between the columns of theAthenæum: I feel huffed and miffed at the very supposition. Ihavemade myself a public scavenger; and why not? Is the mud never to be collected into a heap? I look down upon the other scavengers, of whom there have been a few—mere historical drudges; Montucla, Hutton, etc.—as not fit to compete with me. I say of them what one crossing-sweeper said of the rest: "They are well enough for the common thing; but put them to a bit of fancy-work, such as sweeping round a post, and see what a mess they make of it!" Who can touch me at sweeping round a paradoxer? If I complete my design of publishing a separate work, an old copy will be fished up from a stall two hundred years hence by the coming man, and will be described in an article which will end by his comparing our century with his own, and sighing out in the best New Zealand pronunciation—

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

"Dans ces tems-là

C'était déjà comme ça!"[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 theglorious dream, we learn that there is a way to Hell from the gates of Heaven, as well as from the City of Destruction: and that this is true of other things besides Christian pilgrimage is affirmed at the end of the Budget of Paradoxes. If D'Alembert[665]had producedenoughof 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 producedenoughto match his reception of the story that Nausicaa, Homer's Phæacian princess, invented the celestial sphere, followed by his serious surmise that she got it from the Argonauts,—then Newton himself would have had an appearance entered for him, in spite of thePrincipia. In illustration, I may cite a few words fromTristram Shandy:

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

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

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

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

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

"Flen, flyys, and freris, populum domini male cædunt;

Thystlis and breris crescentia gramina lædunt.

Christe nolens guerras qui cuncta pace tueris,

Destrue per terras breris, flen, flyys, and freris.

Flen, flyys, and freris, foul falle hem thys fyften yeris,

For non that her is lovit flen, flyys ne freris."[667]

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

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

"Meum est propositum genti imperitæ

Artes frugi reddere melioris vitæ."[668]

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

For the present I cut and run: a Catiline, pursued by a chorus of Ciceros, withQuousque 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.

I think it right to give the proof that the ratio of the circumference to the diameter is incommensurable. This method of proof was given by Lambert,[672]in theBerlin Memoirsfor 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.,

——

b+c

——

d+e

f+ etc.,

be abbreviated intoa/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 supposeb,d,f, etc.positive;a,c,e, &c. being as required: and all are supposed integers. If this succession be continuedad infinitum, and ifa/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 fractionp/(q+ω), whereωis between ±1: for,p/q, being < 1, andpandqinteger, cannot be brought up to1, by the value ofω. Hence, if we take any of the fractions

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

saya/b+c/d+e/f+g/hwe have,g/hbeing between ±1, so ise/f+g/h, so therefore isc/d+e/f+g/h; and so therefore isa/b+c/d+e/f+g/h.

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

P = Ac/d+e/f+ etc., Q = Pe/f+g/h+ etc., R = Qg/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

A/B =a/b+ P/A, or, P =aB -bAP/A =c/d+ Q/P, or, Q =cA -dPQ/P =e/f+ R/Q, or, R =eP -fQ

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, sincea, 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 suchunlimitedsuccession ofdescendingintegers: consequently, it is impossible thata/b+c/d+, etc. can have a commensurable limit.

It easily follows that the continued fraction is incommensurable ifa/b,c/d, etc., being at first greater than unity, become and continue less than unity after some one point. Say thati/k,l/m,... are all less than unity. Then the fractioni/k+l/m+ ... is incommensurable, as proved: let it beκ. Theng/(h+κ) is incommensurable, sayλ;e/(f+λ) is the same, sayμ; alsoc/(d+μ), sayν, anda/(b+ν), sayρ. Butρis the fractiona/b+c/d+ ... itself; which is therefore incommensurable.

Letφzrepresent

Letzbe positive: this series is convergent for all values ofa, and approaches without limit to unity aszincreases without limit. Changezintoz+ 1, and formφz-φ(z+1): the following equation will result—

ψzbeing (a/z)(φ(z+1)/φz); of which observe that it diminishes without limit aszincreases without limit. Accordingly, we have

And,ψ(z+n) diminishing without limit, we have

Letz= ½; and let 4a= -x2. Then

Again

and the continued fraction is

whence

Or, as written in the usual way,

tanx=x——1 -x2——3 -x2——5 -x2——7 - ...

tanx=x

——

1 -x2

——

3 -x2

——

5 -x2

——

7 - ...

This result may be proved in various ways: it may also be verified by calculation. To do this, remember that if

in the case before us we have

We can use this algebraically, or arithmetically. If we divide Pnby Qn, we shall find a series agreeing with the known series for tanx,as far asnterms. That series is

Take P5, and divide it by Q5in the common way, and the first five terms will be as here written. Now takex= .1, which means that the angle is to be one tenth of the actual unit, or, in degrees 5°.729578. We find that whenx= .1, P6= 1038.24021, Q6= 10347.770999; whence P6divided by Q6gives .1003346711. Now 5°.729578 is 5°43′46½″; and from the old tables of Rheticus[675]—no modern tables carry the tangents so far—the tangent of this angle is .1003347670.

Now letx= ¼π; in which case tanx= 1. If ¼πbe commensurable with the unit, let it be (m/n),mandnbeing integers: we know that ¼π< 1. We have then

Now it is clear thatm2/3n,m2/5n,m2/7n, etc. must at last become and continue severally less than unity. The continued fraction is therefore incommensurable, and cannot be unity. Consequentlyπ2cannot be commensurable: that is,πis an incommensurable quantity, and so also isπ2.

I thought I should end with a grave bit of appendix, deeply mathematical: but paradox follows me wherever I go. The foregoing is—in my own language—from Dr. (now Sir David) Brewster's[676]English edition of Legendre's Geometry, (Edinburgh, 1824, 8vo.) translated by some one who is not named. I picked up a notion, which others had at Cambridge in 1825, that the translator was the late Mr. Galbraith,[677]then known at Edinburgh as a writer and teacher.

But it turns out that it was by a very different person, and one destined to shine in quite another walk; it was a young man named Thomas Carlyle.[678]He prefixed, from his own pen, a thoughtful and ingenious essay on Proportion, as good a substitute for the fifth Book of Euclid as could have been given in the space; and quite enough to show that he would have been a distinguished teacher and thinker on first principles. But he left the field immediately.

(The following is the passage referred to at Vol. II, page54.)

Michael Stifelius[679]edited, in 1554, a second edition of the Algebra (Die Coss.), of Christopher Rudolff.[680]This is one of the earliest works in which + and - are used.

Stifelius was a queer man. He has introduced into this very work of Rudolff his own interpretation of the number of the Beast. He determined to fix the character of Pope Leo: so he picked the numeral letters from LEODECIMVS, and by taking in X from LEO X. and striking out M as standing formysterium, he hit the number exactly. This discovery completed his conversion to Luther, and his determination to throw off his monastic vows. Luther dealt with him as straight-forwardly as with Melanchthon about his astrology: he accepted the conclusions, but told him to clear his mind of all the premises about the Beast. Stifeliusdid not take the advice, and proceeded to settle the end of the world out of the prophet Daniel: he fixed on October, 1533. The parishioners of some cure which he held, having full faith, began to spend their savings in all kinds of good eating and drinking; we may charitably hope this was not the way of preparing for the event which their pastor pointed out. They succeeded in making themselves as fit for Heaven as Lazarus, so far as beggary went: but when the time came, and the world lasted on, they wanted to kill their deceiver, and would have done so but for the interference of Luther.

Pages denoted by numerals of this kind (287) refer to biographical notes, chiefly by the editor. Numerals like426refer to books discussed by De Morgan, or to leading topics in the text. Numerals like 126 indicate minor references.

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