Chapter 7

Ἐννεακαιδέκα κύκλα φαεινοῦ ἠελίοιο.[295]

Some man of letters must have turned Apollo into Phaeinus of Elis; and there he is in the histories of astronomy tothis day. Salmasius[296]will have Aratus to have meant him, and proposes to readἠλείοιο: 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 easierthan those in the French books." He was not aware that theuniversally admittedsuperiority of decimalcalculationmade 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 association 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 Association. 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,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 introducednow, there must be a column for the odd shilling.

"He was glad that some hon. gentleman had derived benefit from the issue of florins. His only experience of their convenience was, that when he ought to have received half-a-crown, he had generally received a florin, and when he ought to have paid a florin, he had generally paid half-a-crown." (Hear, hear, and laughter.)

If the hon. gentleman make this assertion of himself, it is not for us to gainsay it. It only proves that he is one of that class ofmen who are described in the old song, of which one couplet runs thus:

I sold my cow to buy me a calf;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 2-5ths of a penny, but who ever bought anything, who ever reckoned or wished to reckon in such a coin as that?" (Hear, hear.)

Note the sophism of expressing our coin in terms of the penny, which we abandon, instead of the florin, which we retain. Remember that this 2 2-5ths is the hundredth part of the pound, which is called, as yet, acent. Nobody buys anything at a cent, because the cent is not yet introduced. Nobody 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 asthe 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 into generalcirculationbecause it represents nothing which corresponds with any of the wants of the people."

Does 2½d. never pass from hand to hand? And is 2½d. so precisely the modulus of popular wants, that an alteration of 4 per cent. would make it useless? Of all the values which 2½d. 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 adopted, they would all be compelled to live in decimals for ever; if a man dined at a public house he would have to pay for his dinner in decimal fractions. (Hear, hear.) He objected to that, for he thought that a man ought to be able to pay for his dinner in integers." (Hear, hear, and a laugh.)

The confusion of ideas here exhibited is most instructive. The speaker is under the impression thatweare introducing fractions: the truth is, that we only want to abandon themore difficultfractions whichwehave got, and to introduceeasier fractions. Does he deny this? Let us trace his denial to its legitimate consequences. A man ought to pay for his dinner in integers.

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 looksupthe present system, with Cocker,[299]and Walkingame,[300]and always looksdownthe proposed system. The worddecimalis obstinately associated withfractions, 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."

"Dinner he finds too painful an endeavor,

Condemned to pay in decimals for ever."

"The present system, however, had not yet been changed into decimal system. That change might appear very easy to accomplished mathematicians and men of science, but it was one which it would be very difficult to carry out. (Hear, hear). What would have to be done? Every sum would have to be reduced into a vulgar fraction of a pound, and then divided by the decimal of a pound—a pleasant sum for an old applewoman to work out!" (Hear, hear, and laughter.)

A pleasant sum even for an accomplished mathematician. What does divided by the decimal of a pound mean? Perhaps it meansreducedto the decimal of a pound! Mr. Lowe supposes, as many others do, that, after the change, all calculations will beproposed in old money, and thenconverted into new. He cannot hit theidea that the new coins will take the place of the old. This lack of apprehension will presently appear further.

"It would not be an agreeable task, even for some members of that House, to reduce 4½d., or nine half-pence, to mils." (Hear, hear.)

Let the members be assured that nine half-pence will be, for every practical purpose, 18 mils. But now to the fact asserted. 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 eighteenmils 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 constituencies. We do not hold it impossible to give the Members of the House in general a sufficient knowledge of the meaning and consequences of thedecimalsuccession 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—theChiltern 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 4½d. 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 asknownothings, those others cryHearto 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 they went below 6d., that the new scale of coins would not be commensurate in any finite ratio with anything in this new currency of mils."

Fine words, wrongly used. The new coins are commensurable with, and in a finite ratio to, the old ones. The farthing is to the mil as 25 to 24. The 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.

"No less than five of our present coins must be called in, or else—which would be worse—new values must be given to them."

This dreadful change of value consists in sixpence farthing going to the half-shilling instead of sixpence. Whether the new farthings be called mils or not is of no consequence.

"If a poor man put a penny in his pocket, it would come out a coin of different value, which he would not understand. Suppose he owed another man a penny, how was he to pay him ? Was he to pay him in mils? Four mils would be too little, and five mils would be too much. The hon. gentlemen said there would be only a mil between them. That was exactly it. He believed there would be a 'mill' between them." (Much laughter.)

Mr. Lowe, who cannot pass a half-crown for more than a florin, or get in a florin at less than half-a-crown, has such a high faith in the sterner stuff of his fellow countrymen, that he believes any two of them would go to fisty cuffs for the 25th part of a farthing. He reasons thus: He has often heard in the streets, "I'd fight you for the fiftieth part of a farden:" and having (that is, 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, apro formareport of progress. Do the Members of the House think they have all the forms to themselves?

"What would be the present expression for four-pence? Why, 0.166 (a laugh); for threepence? .0125; for a penny? .004166, and so onad infinitum(a laugh); for a half-penny? .002083ad infinitum. (A laugh). What would be the present expression for a farthing? Why, .0010416ad infinitum. (A laugh). And this was the system which was to cause such a saving in figures, and these were the quantities into which the poor would have to reduce the current coin of the realm. (Cheers). With every respect for decimal fractions, of which he boasted no profound knowledge, he doubted whether the poor were equal to mental arithmetic of this kind, (hear, hear) and he hoped the adoption of the system would be deferred until there were some proof that they would be able to understand it; for, after all, this was the question of the poor, and the whole weight of the change would fall upon them. Let the rich by all means have permission to perplex themselves by any division of a pound they pleased; but do not let them, by any experiment like this, impose difficulties upon the poor and compel men to carry ready-reckoners in their pocket to give them all these fractional quantities." (Hear, hear.)

We should hardly believe all this to be uttered in earnest, if we had not knownthat several persons who have not Mr. Lowe's humor, nevertheless have his impressions on this point. It must therefore be answered; but how is this to be done seriously?

Dialogue between a member of Parliament and an orange-boy, three days after the introduction of the complete decimal system. The member, going down to the House, wants oranges to sustain his voice in a two hours' speech on moving that 100000l. be placed at the disposal of Her Majesty, to supply the poor with ready-reckoners.

Boy.Fine oranges! two a penny! two a penny!

Member.Here boy, two! Now, how am I to pay you?

Boy.Give you change, your honor.

Member.Ah! but how? Where's your ready-reckoner?

Boy.I sells a better sort nor them. Mine's real Cheyny.

Member.But you see a farthing is now .0014166666ad infinitum, and if we multiply this by 4——

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.

Member.You don't mean that you do without a book!

Boy.Book be blowed. Come now, old un, here's summut for both on us. I got a florin, you gives me a half-a-crown for it, and I larns you the new money, gives you your oranges, and calls you a brick into the bargain.

Member(to himself). Never had such a chance of getting off half-a-crown for value since that —— fellow Bowring carried his crochet. (Aloud.) Well, boy, it's a bargain. Now!

Boy.Why, look 'e here, my trump, its a farden more to the tizzy—that's what it is.

Member. What's that?

Boy.Why, you knows a sixpence when you sees it. (Aside). Blest if I think he does! Well, its six browns and a farden now. A lady buys two oranges, and forksout a sixpence; well in coorse, I hands over fippence farden astead of fippence. I always gives a farden more change, and takes according.

Member(in utter surprise, lets his oranges tumble into the gutter). Never mind! They won't be wanted now. (Walks off one way. Boy makes a pass of naso-digital mesmerism, and walks off the other way).

To the poor, who keep no books, the whole secret is "Sixpence farthing to the half shilling, twelve pence halfpenny to the shilling." Thenew twopence halfpenny, or cent, will be at once five to the shilling.

In conclusion, we remark that three very common misconceptions run through the hon. Member's argument; and, combined in different proportions, give variety to his patterns.

First, he will have it that we design to bring the uneducated into contact withdecimal fractions. If it be so, it will only be as M. Jourdain was brought into contact with prose. In fact,Quoi! quand je dis, Nicole, apportez-moi mes pantoufles, c'est de la prose?[305]may be rendered: "What! do you mean thatten to the florin is a cent a piecemust be called decimal reckoning?" If we had to comfort a poor man, horror-struck by the threat ofdecimals, we should tell him what manner of fractions had been inflicted upon him hitherto; nothing less awful thanquarto-duodecimo-vicesimals, we should assure him.

Secondly, he assumes that the penny, such as it now is, will remain, as a coin of estimation, after it has ceased to be a coin of exchange; and that the mass of the people will continue to think of prices in old pence, and to calculate them in new ones, or else in new mils. No answer is required to this, beyond the mere statement of the nature of the assumption and denial.

Thirdly, he attributes to the uneducated community a want of perception and of operative power which really does not belong to them. The evidence offered to the Committee of the House shows that no fear is entertained on this point by those who come most in contact with farthing purchasers. And this would seem to be a rule,—that is, fear of the intelligence of the lower orders in the minds of those who are not in daily communication with them, no fear at all in the minds of those who are.

A remarkable instance of this distinction happened five-and-twenty years ago. The Admiralty requested the Astronomical Society to report on the alterations which should be made in theNautical Almanac, the seaman's guide-book over the ocean. The greatest alteration proposed was the description of celestial phenomena inmean(or clock time), instead ofapparent(or sundial) time, till then always employed. This change would require that in a great many operations the seaman should let alone what he formerly altered by addition or subtraction, and alter by addition or subtraction what he formerly let alone; provided always that what he formerly altered by addition he should, when he altered at all, alter by subtraction, andvice versa. This was a tolerably difficult change for uneducated skippers, working by rules they had only learned by rote. The Astronomical Society appointed a Committee of forty, of whom nine were naval officers or merchant seamen [I was on this Committee]. Some men of science were much afraid of the change. They could not trust an ignorant skipper or mate to make those alterations in their routine, on the correctness of which the ship might depend. Had the Committee consisted of men of science only, the change might never have been ventured on. But the naval men laughed, and said there was nothing to fear; and on their authority the alteration was made. The upshot was, that, after the new almanacs appeared, not a word of complaint was ever heard on the matter. Had the House of Commons had todecide this question, with Mr. Lowe to quote the description given by Basil Hall[306](who, by the way, was one of the Committee) of an observation on which the safety of the ship depended, worked out by the light of a lantern in a gale of wind off a lee shore, this simple and useful change might at this moment have been in the hands of its tenth Government Commission.

[Aug. 14, 1866.The Committee was appointed in the spring of 1830: it consisted of forty members. Death, of course, has been busy; there are now left Lord Shaftesbury,[307]Mr. Babbage,[308]Sir John Herschel,[309]Sir Thomas Maclear[310](Astronomer Royal at the Cape of Good Hope), Dr. Robinson[311](of Armagh), Sir James South,[312]Lord Wrottesley,[313]and myself].

THE TONAL SYSTEM.

Project of a new system of arithmetic, weight, measure, and coins, proposed to be called the tonal system, with sixteen to the base. By J. W. Mystrom.[314]Philadelphia, 1862, 8vo.

That is to say, sixteen is to take the place of ten, and to be written 10. The whole language is to be changed; every man of us is to be sixteen-stringed Jack and every woman sixteen-stringed Jill. Our oldone,two,three, up to sixteen, are to be (Nollgoing for nothing, which will please those who dislike the memory ofOld Noll) replaced by An, De, Ti, Go, Su, By, Ra, Me, Ni, Ko, Hu, Vy, La, Po, Fy, Ton; and then Ton-an, Ton-de, etc. for 17, 18, etc. The number which in the system has the symbol

28(13)5(11)7(14)0(15)

(using our present compounds instead of new types) is to be pronounced

Detam-memill-lasan-suton-hubong-ramill-posanfy.

The year is to have sixteen months, and here they are:

Anuary, Debrian, Timander, Gostus,Suvenary, Bylian, Ratamber, Mesudius,Nictoary, Kolumbian, Husamber, Vyctorius,Lamboary, Polian, Fylander, Tonborius.

Anuary, Debrian, Timander, Gostus,

Suvenary, Bylian, Ratamber, Mesudius,

Nictoary, Kolumbian, Husamber, Vyctorius,

Lamboary, Polian, Fylander, Tonborius.

Surely An-month, De-month, etc. would do as well. Probably the wants of poetry were considered. But what are we to do with our old poets? For example—

"It was a night of lovely June,High rose in cloudless blue the moon."

"It was a night of lovely June,

High rose in cloudless blue the moon."

Let us translate—

"It was a night of lovely Nictoary,High rose in cloudless blue the (what, in the name of all that is absurd?)."

"It was a night of lovely Nictoary,

High rose in cloudless blue the (what, in the name of all that is absurd?)."

And again,Fylanderthrown into our December! What isto become of those lines of Praed, which I remember coming out when I was at Cambridge,—

"Oh! now's the time of all the year for flowers and fun, the Maydays;To trim your whiskers, curl your hair, and sinivate the ladies."

"Oh! now's the time of all the year for flowers and fun, the Maydays;

To trim your whiskers, curl your hair, and sinivate the ladies."

If I were asked which I preferred, this system or that of Baron Ferrari[315]already mentioned, proceeding bytwelves, I should reply, with Candide, when he had the option given of running the gauntlet or being shot: Les volontés sont libres, et je ne veux ni l'un ni l'autre.[316]We can imagine a speculator providing such a system for Utopia as it would be in the mind of a Laputan: but to explain how an engineer who has surveyed mankind from Philadelphia to Rostof on the Don should for a moment entertain the idea of such a system being actually adopted, would beat a jury of solar-system-makers, though they were shut up from the beginning of Anuary to the end of Tonborius. When I see such a scheme as this imagined to be practicable, I admire the wisdom of Providence in providing the quadrature of the circle, etc., to open a harmless sphere of action to the possessors of the kind of ingenuity which it displays. Those who cultivate mathematics have a right to speak strongly on such efforts of arithmetic as this: for, to my knowledge, persons who have no knowledge are frequently disposed to imagine that their makers are true brothers of the craft, a little more intelligible than the rest.

SOME SMALL PARADOXERS.

Vis inertiae victa,[317]or Fallacies affecting science. By James Reddie.[318]London, 1862, 8vo.

An attack on the Newtonian mechanics; revolution by gravitation demonstrably impossible; much to be said for the earth being the immovable center. A good analysis of contents at the beginning, a thing seldom found. The author has followed up his attack in a paper submitted to the British Association, but which it appears the Association declined to consider. It is entitled—

Victoria Toto Cœlo; or, Modern Astronomy recast. London, 1863, 8vo.

At the end is a criticism of Sir G. Lewis'sHistory of Ancient Astronomy.

On the definition and nature of the Science of Political Economy. By H. Dunning Macleod,[319]Esq. Cambridge, 1862, 8vo.

A paper read—but, according to the report, not understood—at the British Association. There is a notion that political economy is entirely mathematical; and its negative quantity is strongly recommended for study: it contains "the whole of the Funds, Credit, 32 parts out of 33 of the value of Land...." The mathematics are described as consisting of—first, number, or Arithmetic; secondly, the theory of dependent quantities, subdivided into dependence by cause and effect, and dependence by simultaneous variations; thirdly, "independent quantities or unconnected events, which is the theory of probabilities." I am not ashamed, having the British Association as a co-non-intelligent, to say I do not understand this: there is a paradox in it, and the author should give further explanation, especially of his negative quantity. Mr. Macleod has gainedpraise from great names for his political economy; but this, I suspect, must have been for other parts of his system.

On the principles and practice of just Intonation, with a view to the abolition of temperament.... By General Perronet Thompson.[320]Sixth Edition. London, 1862, 8vo.

Here is General Thompson again, with another paradox: but always master of the subject, always well up in what his predecessors have done, and always aiming at a useful end. He desires to abolish temperament by additional keys, and has constructed an enharmonic organ with forty sounds in the octave. If this can be introduced, I, for one, shall delight to hear it: but there are very great difficulties in the way, greater than stood even in the way of the repeal of the bread-tax.

In a paper on the beats of organ-pipes and on temperament published some years ago, I said that equal temperament appeared to me insipid, and not so agreeable as the effect of the instrument when in progress towards being what is called out of tune, before it becomes offensively wrong. There is throughout that period unequal temperament, determined by accident. General Thompson, taking me one way, says I have launched a declaration which is likely to make an epoch in musical practice; a public musical critic, taking me another way, quizzes me for preferring musicout of tune. I do not think I deserve either one remark or the other. My opponent critic, I suspect, takesequally temperedandin tuneto be phrases of one meaning. But by equal temperament is meant equal distribution among all the keys of the error which an instrumentmusthave, which, with twelve sounds only in the octave, professes to be fit for all the keys. I am reminded of the equal temperament which was once applied to the postmen's jackets. The coats were all made for the average man: theconsequence was that all the tall men had their tails too short; all the short men had them too long. Some one innocently asked why the tall men did not change coats with the short ones.

A diagram illustrating a discovery in the relation of circles to right-lined geometrical figures. London, 1863, 12mo.

The circle is divided into equal sectors, which are joined head and tail: but a property is supposed which is not true.

An attempt to assign the square roots of negative powers; or what is √ -1? By F.H. Laing.[321]London, 1863, 8vo.

If I understand the author, -aand +aare the square roots of -a2, as proved by multiplying them together. The author seems quite unaware of what has been done in the last fifty years.

BYRNE'S DUAL ARITHMETIC.

Dual Arithmetic. A new art. By Oliver Byrne.[322]London, 1863, 8vo.

The plan is to throw numbers into the forma(1.1)b(1.01)c(1.001)d... and to operate with this form. This is an ingenious and elaborate speculation; and I have no doubt the author has practised his method until he could surprise any one else by his use of it. But I doubt if he will persuade others to use it. As asked of Wilkins's universal language, Where is the second man to come from?

An effective predecessor in the same line of inventionwas the late Mr. Thomas Weddle,[323]in his "New, simple, and general method of solving numeric equations of all orders," 4to, 1842. The Royal Society, to which this paper was offered, declined to print it: they ought to have printed an organized method, which, without subsidiary tables, showed them, in six quarto pages, the solution (x=8.367975431) of the equation

1379.664x622+ 2686034 × 10432x152- 17290224 × 10518x60+ 2524156 × 10574= 0.

The method proceeds by successive factors of the form,abeing the first approximation,a× 1.b× 1.0c× 1.00d.... In my copy I find a few corrections made by me at the time in Mr. Weddle's announcement. "It was read before that learned body [the R. S.] and they were pleased [but] to transmit their thanks to the author. The en[dis]couragement which he received induces [obliges] him to lay the result of his enquiries in this important branch of mathematics before the public [, at his own expense; he being an usher in a school at Newcastle]." Which is most satirical, Mr. Weddle or myself? The Society, in the account which it gave of this paper, described it as a "new and remarkably simple method" possessing "several important advantages." Mr. Rutherford's[324]extended value ofπwas read at the very next meeting, and was printed in theTransactions; and very properly: Mr. Weddle's paper was excluded, and very very improperly.

HORNER'S METHOD.

I think it may be admited that the indisposition to look at and encourage improvements of calculation which oncemarked the Royal Society is no longer in existence. But not without severe lessons. They had the luck to accept Horner's[325]now celebrated paper, containing the method which is far on the way to become universal: but they refused the paper in which Horner developed his views of this and other subjects: it was printed by T. S. Davies[326]after Horner's death. I make myself responsible for the statement that the Society could not reject this paper, yet felt unwilling to print it, and suggested that it should be withdrawn; which was done.

But the severest lesson was the loss ofBarrett's Method,[327]now the universal instrument of the actuary in his highest calculations. It was presented to the Royal Society, and refused admission into theTransactions: Francis Baily[328]printed it. The Society is now better informed: "live and learn," meaning "must live, so better learn," ought to be the especial motto of a corporation, and is generally acted on, more or less.

Horner's method begins to be introduced at Cambridge: it was published in 1820. I remember that when I first went to Cambridge (in 1823) I heard my tutor say, in conversation, there is no doubt that the true method of solving equations is the one which was published a few years ago in thePhilosophical Transactions. I wondered it was not taught, but presumed that it belonged to the higher mathematics. This Horner himself had in his head: and in a sense it is true; for all lower branches belong to the higher: but he would have stared to have been told that he, Horner,was without a European predecessor, and in the distinctive part of his discovery was heir-at-law to the nameless Brahmin—Tartar—Antenoachian—what you please—who concocted the extraction of the square root.

It was somewhat more than twenty years after I had thus heard a Cambridge tutor show sense of the true place of Horner's method, that a pupil of mine who had passed on to Cambridge was desired by his college tutor to solve a certain cubic equation—one of an integer root of two figures. In a minute the work and answer were presented, by Horner's method. "How!" said the tutor, "this can't be, you know." "There is the answer, Sir!" said my pupil, greatly amused, for my pupils learnt, not only Horner's method, but the estimation it held at Cambridge. "Yes!" said the tutor, "there is the answer certainly; but itstands to reasonthat a cubic equation cannot be solved in this space." He then sat down, went through a process about ten times as long, and then said with triumph: "There! that is the way to solve a cubic equation!"

I think the tutor in this case was never matched, except by the country organist. A master of the instrument went into the organ-loft during service, and asked the organist to let himplay the congregation out; consent was given. The stranger, when the time came, began a voluntary which made the people open their ears, and wonder who had got into the loft: they kept their places to enjoy the treat. When the organist saw this, he pushed the interloper off the stool, with "You'll never play 'em out this side Christmas." He then began his own drone, and the congregation began to move quietly away. "There," said he, "that's the way to play 'em out!"

I have not scrupled to bear hard on my own university, on the Royal Society, and on other respectable existences: being very much the friend of all. I will now clear the Royal Society from a very small and obscure slander, simply because I know how. This dissertation began withthe work of Mr. Oliver Byrne, the dual arithmetician, etc. This writer published, in 1849, a method of calculating logarithms.[329]First, a long list of instances in which, as he alleges, foreign discoverers have been pillaged by Englishmen, or turned into Englishmen: for example, O'Neill,[330]so called by Mr. Byrne, the rectifier of the semi-cubical parabola claimed by the Saxons under the name ofNeal: the grandfather of this mathematician was conspicuous enough asNeal; he was archbishop of York. This list, says the writer, might be continued without end; but he has mercy, and finishes with his own case, as follows:—"About twenty years ago, I discovered this method of directly calculating logarithms. I could generally find the logarithm of any number in a minute or two without the use of books or tables. The importance of the discovery subjected me to all sorts of prying. Some asserted that I committed a table of logarithms to memory; others attributed it to a peculiar mental property; and when Societies and individuals failed to extract my secret, they never failed to traduce the inventor and the invention. Among the learned Societies, the Royal Society of London played a very base part. When I have more space and time at my disposal, I will revert to this subject again."

Such a trumpery story as this remains unnoticed at the time; but when all are gone, a stray copy from a stall falls into hands which, not knowing what to make of it, make history of it. It is a very curious distortion. The reader may take it on my authority, that the Royal Society played no part, good or bad, nor had the option of playing a part.But I myselfpars magna fui:[331]and when the author has "space and time" at his disposal, he must not take all of them; I shall want a little of both.

ARE ATOMS WORLDS?

The mystery of being; or are ultimate atoms inhabited worlds? By Nicholas Odgers.[332]Redruth and London, 1863, 8vo.

This book, as a paradox, beats quadrature, duplication, trisection, philosopher's stone, perpetual motion, magic, astrology, mesmerism, clairvoyance, spiritualism, homœopathy, hydropathy, kinesipathy, Essays and Reviews, and Bishop Colenso,[333]all put together. Of all the suppositions I have given as actually argued, this is the one which is hardest to deny, and hardest to admit. Reserving the question—as beyond human discussion—whether our particles of carbon, etc. areclustersof worlds, the author produces his reasons for thinking that they are at least single worlds. Of course—though not mentioned—the possibility is to be added of the same thing being true of the particles which make up our particles, and so down, for ever: and, on the other hand, of our planets and stars as being particles in some larger universe, and so up, for ever.

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