A Budget of Paradoxes Volume II Part 15

(P. 154). The periodic times of the five planets were stated by Eudoxus,[286] as we learn from Simplicius;[287] the following is his statement, to which the true times are subjoined, for the sake of comparison:

STATEMENT OF EUDOXUS TRUE TIME Mercury 1 year -- 87d. 23h.

Venus 1 " -- 224d. 16h.

Mars 2 " 1y. 321d. 23h.

Jupiter 12 " 11y. 315d. 14h.

Saturn 30 " 29y. 174d. 1h.

Upon this determination two remarks may be made. First, the error with respect to Mercury and Venus is considerable; with respect to Mercury, it is, in round numbers, 365 instead of 88 days, more than four times too much. Aristotle remarks that Eudoxus distinguishes Mercury and Venus from the other three planets by giving them one sphere each, with the poles in common. The proximity of Mercury to the sun would render its course difficult to observe and to measure, but the cause of the large error with respect to Venus (130 days) is not apparent.

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Sir G. Lewis takes Eudoxus as making the planets move round the sun; he has accordingly compared the _geocentric_ periods of Eudoxus with our _heliocentric_ periods. What greater blunder can be made by a writer on ancient astronomy than giving Eudoxus the Copernican system? If Mercury were a black spot in the middle of the sun it would of course move round the earth in a year, or appear to do so: let it swing a little on one side and the other of the sun, and the average period is still a year, with slight departures both ways. The same for Venus, with larger departures.

Say that a person not much accustomed to the distinction might for once write down the mistake; how are we to explain its remaining in the mind in a permanent form, and being made a ground for such speculation as that of the difficulty of observing Mercury leading to a period four times what it ought to be, corrected in proof and published by an industrious and thoughtful person? Only in one way: the writer was quite out of his depth.

This one case is conclusive; be it said with all respect for the real staple of the work and of the author. He knew well the difference of the systems, but not the effect of the difference: he is another instance of what I have had to ill.u.s.trate by help of a very different person, that it is difficult to reason well upon matter which is not familiar.

(P. 254). Copernicus, in fact, supposed the axis of the earth to be always turned towards the Sun.^{(169)} [(169). See Delambre, _Hist. Astr. Mod._, Vol. I, p. 96]. It was reserved to Kepler to propound the hypothesis of the constant parallelism of the earth's axis to itself.

If there be one thing more prominent than another in the work of Copernicus himself, in the popular explanations of it, and in the page of Delambre[288] cited, it is that the _parallelism of the earth's axis_ is a glaring part of the {166} theory of Copernicus. What Kepler[289] did was to throw away, as unnecessary, the method by which Copernicus, _per fas et nefas_,[290] secured it. Copernicus, thinking of the earth's...o...b..tal revolution as those would think who were accustomed to the _solid orbs_--and much as the stoppers of the moon's rotation do now: why do they not strengthen themselves with Copernicus?--thought that the earth's axis would always incline the same end towards the sun, unless measures were taken to prevent it. He _did_ take measures: he invented a _compensating_ conical motion of the axis to preserve the parallelism; and, which is one of the most remarkable points of his system, he obtained the precession of the equinoxes by giving the necessary trifle more than compensation. What stares us in the face at the beginning of the paragraph to which the author refers?

"C'est donc pour arriver a ce parallelisme, ou pour le conserver, que Copernic a cru devoir recourir a ce mouvement egal et oppose qui detruit l'effet qu'il attribue si gratuitement au premier, de deranger le parallelisme."[291]

Parallelism at any price, is the motto of Copernicus: you need not pay so dear, is the remark of Kepler.

The opinions given by Sir G. Lewis about the effects of modern astronomy, which he does not understand and singularly undervalues, will now be seen to be of no authority. He fancies that--to give an instance--for the determination of a ship's place, the invention of chronometers has been far more important than any improvement in astronomical theory (p. 254). Not to speak of lat.i.tude,--though the omission is not without importance,--he ought to have known that longitude is found by the difference between what o'clock it is at Greenwich and at the ship's place, at {167} one absolute moment of time. Now if a chronometer were quite perfect--which no chronometer is, be it said--and would truly tell Greenwich mean time all over the world, it ought to have been clear that just as good a watch is wanted for the time at _the place of observation_, before the longitude of that place with respect to Greenwich can be found. There is no such watch, except the starry heaven itself: and that watch can only be read by astronomical observation, aided by the best knowledge of the heavenly motions.

I think I have done Sir G. Lewis's very excellent book more good than all the reviewers put together.

I will give an old instance in which literature got into confusion about astronomy. Theophrastus,[292] who is either the culprit or his historian, attributes to Meton,[293] the contriver of the lunar calendar of nineteen years, which lasts to this day, that his solstices were determined for him by a certain Phaeinus of Elis on Mount Lycabettus. n.o.body else mentions this astronomer: though it is pretty certain that Meton himself made more than one appointment with him for the purpose of observing solstices; and we may be sure that if either were behind his time, it was Meton. For _Phaeinus Helius_ is the shining sun himself; and in the astronomical poet Aratus[294] we read about the nineteen years of the shining sun:

[Greek: Enneakaideka kukla phaeinou eelioio].[295]

Some man of letters must have turned Apollo into Phaeinus of Elis; and there he is in the histories of astronomy to {168} this day. Salmasius[296]

will have Aratus to have meant him, and proposes to read [Greek: eleioio]: he did not observe that Phaeinus is a very common adjective of Aratus, and that, if his conjecture were right, this Phaeinus would be the only non-mythical man in the poems of Aratus.

[When I read Sir George Lewis's book, the points which I have criticized struck me as not to be wondered at, but I did not remember why at the time.

A Chancellor of the Exchequer and a writer on ancient astronomy are birds of such different trees that the second did not recall the first. In 1855 I was one of a deputation of about twenty persons who waited on Sir G. Lewis, as Chancellor of the Exchequer, on the subject of a decimal coinage. The deputation was one of much force: Mr. Airy, with myself and others, represented mathematics; William Brown,[297] whose dealings with the United States were reckoned by yearly millions, counted duodecimally in England and decimally in America, was the best, but not the only, representative of commerce. There were bullionists, accountants, retailers, etc. Sir G. L.

walked into the room, took his seat, and without waiting one moment, began to read the deputation a smart lecture on the evils of a decimal coinage; it would require alteration of all the tables, it would impede calculation, etc. etc. Of those arguments against it which weighed with many of better knowledge than his, he obviously knew nothing. The members of the deputation began to make their statements, and met with curious denials. He interrupted me with "Surely there is no doubt that the calculations of our books of arithmetic are easier {169} than those in the French books." He was not aware that the _universally admitted_ superiority of decimal _calculation_ made many of those who prefer our system for the market and the counter cast a longing and lingering look towards decimals. My answer and the smiles which he saw around, made him give a queer puzzled look, which seemed to say, "I may be out of my depth here!" His manner changed, and he listened. I saw both the slap-dash mode in which he dealt with subjects on which he had not thought, and the temperament which admitted suspicion when the means of knowledge came in his way. Having seen his two phases, I wonder neither at his more than usual exhibition of shallowness when shallow, nor at the intensity of the contrast when he had greater depth.]

DECIMAL COINAGE.

Among the paradoxers are the political paradoxers who care not how far they go in debate, their only object being to carry the House with them for the current evening. What I have said of editors I repeat of them. The preservation of a very marked instance, the a.s.sociation of political recklessness with cyclometrical and Apocalyptic absurdity, may have a tendency to warn, not indeed any hardened public-man and sinner, but some young minds which have yearnings towards politics, and are in formation of habits.

In the debate on decimal coinage of July 12, 1855, Mr. Lowe,[298] then member for Kidderminster, an effective speaker and a smart man, exhibited himself in a speech on which I wrote a comment for the Decimal a.s.sociation.

I have seldom seen a more wretched attempt to distort the points of a public question than the whole of this speech. Looking at the intelligence shown by the speaker on other occasions, {170} it is clear that if charity, instead of believing all things, believed only all things but one, he might tremble for his political character; for the honesty of his intention on this occasion might be the incredible exception. I give a few paragraphs with comments:

"In commenting on the humorous, but still argumentative speech of Mr. Lowe, the member for Kidderminster, we may observe, in general, that it consists of points which have been several times set forth, and several times answered. Mr. Lowe has seen these answers, but does not allude to them, far less attempt to meet them. There are, no doubt, individuals, who show in their public speaking the outward and visible signs of a greater degree of acuteness than they can summon to guide their private thinking. If Mr. Lowe be not one of these, if the power of his mind in the closet be at all comparable to the power of his tongue in the House, it may be suspected that his reserve with respect to what has been put forward by the very parties against whom he was contending, arises from one or both of two things--a high opinion of the arguments which he ignored--a low opinion of the generality of the persons whom he addressed. [Both, I doubt not].

"Did they calculate in florins In the name of common sense, ?" how can it be objected to a system that people do not use it before it is introduced?

Let the decimal system be completed, and calculation shall be made in florins; that is, florins shall take their proper place. If florins were introduced _now_, there must be a column for the odd shilling.

"He was glad that some hon. If the hon. gentleman make gentleman had derived benefit this a.s.sertion of himself, it from the issue of florins. His is not for us to gainsay it.

only experience of their It only proves that he is one convenience was, that when he of that cla.s.s of {171} men who ought to have received are described in the old song, half-a-crown, he had generally of which one couplet runs received a florin, and when he thus: ought to have paid a florin, he had generally paid I sold my cow to buy me a half-a-crown." (Hear, hear, calf; and laughter.) I never make a bargain but I lose half, With a etc. etc. etc.

But he cannot mean that Englishmen in general are so easily managed. And as to Jonathan, who is but John lengthened out a little, he would see creation whittled into chips before he would even split what may henceforth be called the Kidderminster difference. The House, not unmoved--for it laughed--with sly humor decided that the introduction of the florin had been "eminently successful and satisfactory."

The truth is that Mr. Lowe here attacks nothing except the coexistence of the florin and half-crown. We are endeavoring to abolish the half-crown.

Let Mr. Lowe join us; and he will, if we succeed, be relieved from the pressure on his pocket which must arise from having the turn of the market always against him.

"From a florin they get to 2 Note the sophism of expressing 2-5ths of a penny, but who our coin in terms of the ever bought anything, who ever penny, which we abandon, reckoned or wished to reckon instead of the florin, which in such a coin as that?" we retain. Remember that this (Hear, hear.) 2 2-5ths is the hundredth part of the pound, which is called, as yet, a _cent_. n.o.body buys anything at a cent, because the cent is not yet introduced. n.o.body reckons in cents for the same reason.

Everybody wishes to reckon in cents, who wishes to combine the advantage of decimal reckoning with the preservation of the pound as {172} the highest unit of account; amongst others, a majority of the House of Commons, the Bank of England, the majority of London bankers, the Chambers of Commerce in various places, etc. etc. etc.

"Such a coin could never come Does 2d. never pa.s.s from hand into general circulation to hand? And is 2d. so because it represents nothing precisely the modulus of which corresponds with any of popular wants, that an the wants of the people." alteration of 4 per cent.

would make it useless? Of all the values which 2d.

measures, from three pounds of potatoes down to certain arguments used in the House of Commons, there is not one for which a cent would not do just as well. Mr. Lowe has fallen into the misconception of the person who admired the dispensation of Providence by which large rivers are made to run through cities so great and towns so many. If the cent were to be introduced to-morrow, straightway the buns and cakes, the soda-water bottles, the short omnibus fares, the bunches of radishes, etc. etc. etc., would adapt themselves to the coin.

"If the proposed system were The confusion of ideas here adopted, they would all be exhibited is most instructive.

compelled to live in decimals The speaker is under the for ever; if a man dined at a impression that _we_ are public house he would have to introducing fractions: the pay for his dinner in decimal truth is, that we only want to fractions. (Hear, hear.) He abandon the _more difficult_ objected to that, for he fractions which we _have got_, thought that a man ought to be and to introduce _easier able to pay for his dinner in fractions_. Does he deny this?

integers." (Hear, hear, and a Let us trace his denial to its laugh.) legitimate consequences. A man ought to pay for his dinner in integers.

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Now, if Mr. Lowe insists on it that our integer is the pound, he is bound to admit that the present integer is the pound, of which a shilling, etc., are fractions. The next time he has a chop and a pint of stout in the city, the waiter should say--"A pound, sir, to you," and should add, "Please to remember the waiter in integers." Mr. Lowe fancies that when he pays one and sixpence, he pays in integers, and so he does, if his integer be a penny or a sixpence. Let him bring his mind to contemplate a mil as the integer, the lowest integer, and the seven cents five mils which he would pay under the new system would be payment in integers also. But, as it happens with some others, he looks _up_ the present system, with c.o.c.ker,[299] and Walkingame,[300] and always looks _down_ the proposed system. The word _decimal_ is obstinately a.s.sociated with _fractions_, for which there is no need. Hence it becomes so much of a bugbear, that, to parody the lines of Pope, which probably suggested one of Mr. Lowe's phrases--

"Dinner he finds too painful an endeavor, Condemned to pay in decimals for ever."

"The present system, however, A pleasant sum even for an had not yet been changed into accomplished mathematician.

decimal system. That change What does divided by the might appear very easy to decimal of a pound mean?

accomplished mathematicians Perhaps it means _reduced_ to and men of science, but it was the decimal of a pound! Mr.

one which it would be very Lowe supposes, as many others difficult to carry out. (Hear, do, that, after the change, hear). What would have to be all calculations will be done? Every sum would have to _proposed in old money_, and be reduced into a vulgar then _converted into new_. He fraction of a pound, and then cannot hit the {174} idea that divided by the decimal of a the new coins will take the pound--a pleasant sum for an place of the old. This lack of old applewoman to work out!" apprehension will presently (Hear, hear, and laughter.) appear further.

"It would not be an agreeable Let the members be a.s.sured task, even for some members of that nine half-pence will be, that House, to reduce 4d., or for every practical purpose, nine half-pence, to mils." 18 mils. But now to the fact (Hear, hear.) a.s.serted. Davies Gilbert[301]

used to maintain that during the long period he sat in the House, he never knew more than three men in it, at one time, who had a tolerable notion of fractions. [I heard him give the names of three at the time when he spoke: they were Warburton,[302] Pollock,[303]

and Hume.[304] He himself was then out of Parliament.]

Joseph Hume affirmed that he had never met with more than ten members who were arithmeticians. But both these gentlemen had a high standard.

Mr. Lowe has given a much more damaging opinion. He evidently means that the general run of members could not do his question. It is done as follows: Since farthings gain on mils, at the rate of a whole mil in 24 farthings (24 farthings being 25 mils), it is clear that 18 farthings being three-quarters of 24 farthings, will gain three-quarters of a mil; that is, 18 farthings are eighteen {175} mils and three-quarters of a mil. Any number of farthings is as many mils and as many twenty-fourths of a mil. To a certain extent, we feel able to protest against the manner in which Kidderminster has treated the other const.i.tuencies. We do not hold it impossible to give the Members of the House in general a sufficient knowledge of the meaning and consequences of the _decimal_ succession of units, tens, hundreds, thousands, etc.; and we believe that there are in the House itself competent men, in number enough to teach all the rest. All that is wanted is the power of starting from the known to arrive at the unknown. Now there is one kind of decimals with which every member is acquainted--the _Chiltern Hundreds_. If public opinion would enable the competent minority to start from this in their teaching, not as a basis, but as an alternative, in three weeks the fundamentals would be acquired, and members in general would be as fit to turn 4d. into mils, as any boys on the lower forms of a commercial school.

For a long period of years, allusion to the general ignorance of arithmetic, has been a standing mode of argument, and has always been well received: whenever one member describes others as _knownothings_, those others cry _Hear_ to the country in a transport of delight. In the meanwhile the country is gradually arriving at the conclusion that a true joke is no joke.

"The main objection was, if Fine words, wrongly used. The they went below 6d., that the new coins are commensurable new scale of coins would not with, and in a finite ratio be commensurate in any finite to, the old ones. The farthing ratio with anything in this is to the mil as 25 to 24. The new currency of mils." speaker has something here in the bud, which we shall presently meet with in the flower; and fallacies are more easily nipped in flower than in bud. {176} "No less than five of our This dreadful change of value present coins must be called consists in sixpence farthing in, or else--which would be going to the half-shilling worse--new values must be instead of sixpence. Whether given to them." the new farthings be called mils or not is of no consequence.

"If a poor man put a penny in Mr. Lowe, who cannot pa.s.s a his pocket, it would come out half-crown for more than a a coin of different value, florin, or get in a florin at which he would not understand. less than half-a-crown, has Suppose he owed another man a such a high faith in the penny, how was he to pay him ? sterner stuff of his fellow Was he to pay him in mils? countrymen, that he believes Four mils would be too little, any two of them would go to and five mils would be too fisty cuffs for the 25th part much. The hon. gentlemen said of a farthing. He reasons there would be only a mil thus: He has often heard in between them. That was exactly the streets, "I'd fight you it. He believed there would be for the fiftieth part of a a 'mill' between them." (Much farden:" and having (that is, laughter.) for a Member) a notion both of fractions and logic, he infers that those who would fight for the 50th of a farthing would, _a fortiori_, fight for a 25th. His mistake arises from his not knowing that when a person offers to fight another for 1/200d., he really means to fight for love; and that the stake is merely a matter of form, a feigned issue, a _pro forma_ report of progress. Do the Members of the House think they have all the forms to themselves?

"What would be the present We should hardly believe all expression for four-pence? this to be uttered in earnest, Why, 0.166 (a laugh); for if we had not known {177} that threepence? .0125; for a several persons who have not penny? .004166, and so on _ad Mr. Lowe's humor, nevertheless infinitum_ (a laugh); for a have his impressions on this half-penny? .002083 _ad point. It must therefore be infinitum_. (A laugh). What answered; but how is this to would be the present be done seriously?

expression for a farthing?

Why, .0010416 _ad infinitum_. _Dialogue between a member of (A laugh). And this was the Parliament and an orange-boy, system which was to cause such three days after the a saving in figures, and these introduction of the complete were the quant.i.ties into which decimal system. The member, the poor would have to reduce going down to the House, wants the current coin of the realm. oranges to sustain his voice (Cheers). With every respect in a two hours' speech on for decimal fractions, of moving that 100000l. be placed which he boasted no profound at the disposal of Her knowledge, he doubted whether Majesty, to supply the poor the poor were equal to mental with ready-reckoners._ arithmetic of this kind, (hear, hear) and he hoped the _Boy._ Fine oranges! two a adoption of the system would penny! two a penny! {178} be deferred until there were some proof that they would be _Member._ Here boy, two! Now, able to understand it; for, how am I to pay you?

after all, this was the question of the poor, and the _Boy._ Give you change, your whole weight of the change honor.

would fall upon them. Let the rich by all means have _Member._ Ah! but how? Where's permission to perplex your ready-reckoner?

themselves by any division of a pound they pleased; but do _Boy._ I sells a better sort not let them, by any nor them. Mine's real Cheyny.

experiment like this, impose difficulties upon the poor and _Member._ But you see a compel men to carry farthing is now .0014166666 ready-reckoners in their _ad infinitum_, and if we pocket to give them all these multiply this by 4---- fractional quant.i.ties." (Hear, hear.) _Boy._ Hold hard, Guv'ner; I sees what you're arter. Now what'll you stand if I puts you up to it? which Bill Smith he put me up in two minutes, cause he goes to the Ragged School.