A Budget of Paradoxes Volume II Part 22

I append to the foregoing a letter from Dr. Whewell[395] to Mr. James Smith. The Master of Trinity was conspicuous as a rough customer, an intellectual bully, an overbearing disputant: the character was as well established as that of Sam Johnson. But there was a marked difference. It was said of Johnson that if his pistol missed fire, he would knock you down with the b.u.t.t end of it: but Whewell, in like case, always acknowledged the miss, and loaded again or not, as the case might be. He reminded me of Dennis Brulgruddery, who says to Dan, Pacify me with a good reason, and you'll find me a dutiful master. I knew him from the time when he was my teacher at Cambridge, more than forty years. As a teacher, he was anything but dictatorial, and he was perfectly accessible to proposal of objections.

He came in contact with me in his slas.h.i.+ng way twice in our after joint lives, and on both occasions he acknowledged himself overcome, by that change of manner, and apologetic mode of continuance, which I had seen him employ towards others under like circ.u.mstances.

I had expressed my wish to have a _thermometer of probability_, with impossibility at one end, as 2 and 2 make 5, and necessity at the other, as 2 and 2 make 4, and a graduated rise of examples between them. Down came a blow: "What! put necessary and contingent propositions together! It's absurd!" I pointed out that the two kinds of necessity are but such extremes of probability as 0 and [infinity] are of number, and ill.u.s.trated by an urn with 1 white and _n_ black {247} b.a.l.l.s, _n_ increasing without limit. It was frankly seen, and the point yielded; a large company was present.

Again, in a large party, after dinner, and politics being the subject, I was proceeding, in discussion with Mr. Whewell, with "I think"...--"Ugh!

_you_ think!" was the answer. I repeated my phrase, and gave as a reason the words which Lord Grey[396] had used in the House of Lords the night before (the celebrated advice to the Bishops to set their houses in order).

He had not heard of this, and his manner changed in an instant: he was the rational discutient all the rest of the evening, having previously been nothing but a disputant with all the distinctions strongly marked.

I have said that Whewell was gentle with his pupils; it was the same with all who wanted teaching: it was only on an armed enemy that he drew his weapon. The letter which he wrote to Mr. J. Smith is an instance: and as it applies with perfect fidelity to the efforts of unreasoning above described, I give it here. Mr. James Smith is skilfully exposed, and felt it; as is proved by "putting the writer in the stocks."

"The Lodge, Cambridge, September 14th, 1862.

"Sir,--I have received your explanation of your proposition that the circ.u.mference of the circle is to its diameter as 25 to 8. I am afraid I shall disappoint you by saying that I see no force in your proof: and I should hope that you will see that there is no force in it if you consider this: In the whole course of the proof, though the word cycle occurs, there is no property of the circle employed. You may do this: you may put the word _hexagon_ or _dodecagon_, or any other word describing a polygon in the place of _Circle_ in your proof, and the proof would be just as good as before. Does not this satisfy you that you cannot have proved a property of that special figure--a circle? {248}

"Or you may do this: calculate the side of a polygon of 24 sides inscribed in a circle. I think you are a Mathematician enough to do this. You will find that if the radius of the circle be one, the side of this polygon is .264 etc. Now, the arc which this side subtends is according to your proposition 3.125/12 = .2604, and therefore the chord is greater than its arc, which you will allow is impossible.

"I shall be glad if these arguments satisfy you, and

"I am, Sir, your obedient Servant,

"W. WHEWELL."

AN M.P.'S ARITHMETIC.

In the debate of May, 1866, on Electoral Qualifications, a question arose about arithmetical capability. Mr. Gladstone asked how many members of the House could divide 1330l. 7s. 6d. by 2l. 13s. 8d. Six hundred and fifty-eight, answered one member; the thing cannot be done, answered another. There is an old paradox to which this relates: it arises out of the ignorance of the distinction between abstract and concrete arithmetic.

_Magnitude_ may be divided by _magnitude_; and the answer is number: how often does 12d. contain 4d.; answer three times. _Magnitude_ may be divided by _number_, and the answer is _magnitude_: 12d. is divided in four equal parts, what is each part? Answer three _pence_. The honorable objector, whose name I suppress, trusting that he has mended his ways, gave the following utterance:

"With regard to the division sum, it was quite possible to divide by a sum, but not by money. How could any one divide money by 2l. 16s. 8d.?

(Laughter.) The question might be asked, 'How many times 2s. will go into 1l.?' but that was not dividing by money; it was simply dividing 20 by 2.

He might be asked, 'How many times will 6s. 8d. go into a pound?' but it was only required to divide 240 by 80. If the right hon. gentleman were to ask the hon. {249} member for Brighton (Professor Fawcett),[397] or any other authority, he would receive the same answer--viz., that it was possible to divide by a sum, but not by money. (Hear.)"

I shall leave all comment for the second edition, if I publish one.[398] I shall be sure to have something to laugh at. Anything said from a respectable quarter, or supposed to be said, is sure to find defenders. Sam Johnson, a sound arithmetician, comparing himself, and what he alone had done in three years, with forty French Academicians and their forty years, said it proved that an Englishman is to a Frenchman as 40 40 to 3, or as 1600 to 3. Boswell, who was no great hand at arithmetic, made him say that an Englishman is to a Frenchman as 3 to 1600. When I pointed this out, the supposed Johnson was defended through thick and thin in _Notes and Queries_.

I am now curious to see whether the following will find a palliator. It is from "Tristram Shandy," book V. chapter 3. There are two curious idioms, "for for" and "half in half"; but these have nothing to do with my point:

"A blessing which tied up my father's tongue, and a misfortune which set it loose with a good grace, were pretty equal: sometimes, indeed, the misfortune was the better of the two; for, for instance, where the pleasure of harangue was as _ten_, and the pain of the misfortune but as _five_, my father gained half in half; and consequently was as well again off as if it had never befallen him."

This is a jolly confusion of ideas; and wants nothing but a defender to make it perfect. A person who invests five {250} with a return of ten, and one who loses five with one hand and gains ten with the other, both leave off five richer than they began, no doubt. The first gains "half in half,"

more properly "half _on_ half," that is, of the return, 10, the second 5 is gain upon the first 5 invested. "Half _in_ half" is a queer way of saying cent. per cent. If the 5l. invested be all the man had in the world, he comes out, after the gain, twice as well off as he began, with reference to his whole fortune. But it is very odd to say that balance of 5l. gain is _twice_ as good as if nothing had befallen, either loss or gain. A mathematician thinks 5 an infinite number of times as great as 0. The whole confusion is not so apparent when money is in question: for money is money whether gained or lost. But though pleasure and pain stand to one another in the same algebraical relation as money gained and lost, yet there is more than algebra can take account of in the difference.

Next, Ri. Milward[399] (Richard, no doubt, but it cannot be proved) who published Selden's[400] Table Talk, which he had collected while serving as amanuensis, makes Selden say, "A subsidy was counted the fifth part of a man's estate; and so fifty subsidies is five and forty times more than a man is worth." For _times_ read _subsidies_, which seems part of the confusion, and there remains the making all the subsidies equal to the first, though the whole of which they are to be the fifths is perpetually diminished.

Thirdly, there is the confusion of the great misomath {251} of our own day, who discovered two quant.i.ties which he avers to be identically the same, but the greater the one the less the other. He had a truth in his mind, which his notions of quant.i.ty were inadequate to clothe in language. This erroneous phraseology has not found a defender; and I am almost inclined to say, with Falstaff, The poor abuses of the time want countenance.

ERRONEOUS ARITHMETICAL NOTIONS.

"Shallow numerists," as c.o.c.ker[401] is made to call them, have long been at work upon the question how to _multiply_ money by money. It is, I have observed, a very common way of amusing the tedium of a sea voyage: I have had more than one bet referred to me. Because an oblong of five inches by four inches contains 5 4 or 20 _square_ inches, people say that five inches multiplied by four inches _is_ twenty _square_ inches: and, thinking that they have multiplied length by length, they stare when they are told that money cannot be multiplied by money. One of my betters made it an argument for the thing being impossible, that there is no _square money_: what could I do but suggest that postage-stamps should be made legal tender. Multiplication must be _repet.i.tion_: the repeating process must be indicated by _number_ of times. I once had difficulty in persuading another of my betters that if you repeat five s.h.i.+llings as often as there are hairs in a horse's tail, you do not _multiply five s.h.i.+llings by a horsetail_.[402]

I am very sorry to say that these wrong notions have found support--I think they do so no longer--in the University of Cambridge. In 1856 or 1857, an examiner was displaced by a vote of the Senate. The pretext was that he was too severe an examiner: but it was well known that {252} great dissatisfaction had been expressed, far and wide through the Colleges, at an absurd question which he had given. He actually proposed such a fraction as

6s. 3d.

17s. 4d.

As common sense gained a hearing very soon, there is no occasion to say more. In 1858, it was proposed at a college examination, to divide 22557 days, 20 hours, 20 minutes, 48 seconds, by 57 minutes, 12 seconds, and also to explain the fraction

32l. 18s. 8d.

62l. 12s. 9d.

All paradoxy, in matters of demonstration, arises out of muddle about first principles. Who can say how much of it is to be laid at the door of the University of Cambridge, for not taking care of the elements of arithmetical thought?

ON LITERARY BARGAINS.

The phenomena of the two ends of society, when brought together, give interesting comparisons: I mean the early beginnings of thought and literature, and our own high and finished state, as we think it. There is one very remarkable point. In the early day, the letter was matter of the closest adherence, and implied meanings were not admitted.

The blessing of Isaac meant for Esau, went to false Jacob, in spite of the imposition; and the writer of Genesis seems to intend to give the notion that Isaac had no power to p.r.o.nounce it null and void. And "Jacob's policy, whereby he became rich"--as the chapter-heading puts it--in speckled and spotted stock, is not considered as a violation of the agreement, which contemplated natural proportions. In {253} the story of Lycurgus the lawgiver is held to have behaved fairly when he bound the Spartans to obey his laws until he returned--intimating a short absence--he intending never to return. And Vishnoo, when he asked the usurper for three steps of territory as a dwarf, and then enlarged himself until he could bring heaven and earth under the bargain, was thought clever, certainly, but quite fair.

There is nothing of this kind recognized in our day: so far good. But there is a bad contrary: the age is apt, in interpretation, to upset the letter in favor of the view--very often the after thought--of one side only. The case of John Palmer,[403] the improver of the mail coach system, is smothered. He was to have an office and a salary, and 2 per cent for life on the increased _revenue_ of the Post-Office. His rights turned out so large, that Government would not pay them. For misconduct, real or pretended, they turned him out of his _office_: but his bargain as to the percentage had nothing to do with his future conduct; it was payment for his _plan_. I know nothing, except from the debates of 1808 in the two Houses: if any one can redeem the credit of the nation, the field is open.

When I was young, the old stagers spoke of this transaction sparingly, and dismissed it speedily.

The government did not choose to remember what private persons must remember, and are made to remember, if needful. When Dr. Lardner[404] made his bargain with the {254} publishers for the _Cabinet Cyclopaedia_ he proposed that he, as editor, should have a certain sum for every hundred sold above a certain number: the publishers, who did not think there was any chance of reaching the turning sale of this stipulation, readily consented. But it turned out that Dr. Lardner saw further than they: the returns under this stipulation gave him a very handsome addition to his other receipts. The publishers stared; but they paid. They had no idea of standing out that the amount was too much for an editor; they knew that, though the editor had a percentage, they had all the rest; and they would not have felt aggrieved if he had received ten times as much. But governments, which cannot be brought to book before a sworn jury, are ruled only by public opinion. John Palmer's day was also the day of Thomas Fyshe Palmer,[405] and the governments, in their prosecutions for sedition, knew that these would have a reflex action upon the minds of all who wrote about public affairs.

DECLARATION OF BELIEF

1864-65.--It often happens that persons combine to maintain and enforce an opinion; but it is, in our state of society, a paradox to unite for the sole purpose of blaming the opposite side. To invite educated men to do this, and above all, men of learning or science, is the next paradoxical thing of all. But this was done by a small combination in 1864. They got together and drew up a _declaration_, to be signed by "students of the natural sciences," who were to express their "sincere regret that researches into {255} scientific truth are perverted by some in our own times into occasion for casting doubt upon the truth and authenticity of the Holy Scriptures." In words of ambiguous sophistry, they proceeded to request, in effect, that people would be pleased to adopt the views of churches as to the _complete_ inspiration of all the canonical books. The great question whether the Word of G.o.d is _in_ the Bible, or whether the Word of G.o.d is _all_ the Bible, was quietly taken for granted in favor of the second view; to the end that men of science might be induced to blame those who took the first view. The first public attention was drawn to the subject by Sir John Herschel,[406] who in refusing to sign the writ sent to him, administered a rebuke in the _Athenaeum_, which would have opened most eyes to see that the case was hopeless. The words of a man whose _suaviter in modo_ makes his _fort.i.ter in re_[407] cut blocks with a razor are worth preserving:

"I consider the act of calling upon me publicly to avow or disavow, to approve or disapprove, in writing, any religious doctrine or statement, however carefully or cautiously drawn up (in other words, to append my name to a religious manifesto) to be an infringement of that social forbearance which guards the freedom of religious opinion in this country with especial sanct.i.ty.... I consider this movement simply mischievous, having a direct tendency (by putting forward a new s.h.i.+bboleth, a new verbal test of religious partisans.h.i.+p) to add a fresh element of discord to the already too discordant relations of the Christian world.... But no nicety of wording, no artifice of human language, will suffice to discriminate the hundredth part of the shades of meaning in which the most world-wide differences of thought on such subjects may be involved; or prevent the most gentle worded and apparently justifiable expression of regret, so embodied, from grating on the {256} feelings of thousands of estimable and well-intentioned men with all the harshness of controversial hostility."

Other doses were administered by Sir J. Bowring,[408] Sir W. Rowan Hamilton,[409] and myself. The signed declaration was promised for Christmas, 1864: but nothing presentable was then ready; and it was near Midsummer, 1865, before it was published. Persons often incautiously put their names without seeing the _character_ of a doc.u.ment, because they coincide in its _opinions_. In this way, probably, fifteen respectable names were procured before printing; and these, when committed, were hawked as part of an application to "solicit the favor" of other signatures. It is likely enough no one of the fifteen saw that the declaration was, not _maintenance_ of their own opinion, but _regret_ (a civil word for _blame_) that others should _think differently_.

When the list appeared, there were no fewer than 716 names! But a.n.a.lysis showed that this roll was not a specimen of the mature science of the country. The collection was very miscellaneous: 38 were designated as "students of the College of Chemistry," meaning young men who attended lectures in that college. But as all the Royal Society had been applied to, a test results as follows. Of Fellows of the Royal Society, 600 in number, 62 gave their signatures; of writers in the _Philosophical Transactions_, 166 in number, 19 gave their signatures. Roughly speaking, then, only one out of ten could be got to express disapprobation of the free comparison of the results of science with the statements of the canonical books. And I am satisfied that many of these thought they were signing only a declaration of difference of opinion, not of blame for that difference. The number of persons is not small who, when it comes to signing printed doc.u.ments, would put their names to a declaration that the coffee-pot ought to be taken down-stairs, meaning that the teapot ought to be brought {257} up-stairs.

And many of them would defend it. Some would say that the two things are not contradictory; which, with a snort or two of contempt, would be very effective. Others would, in the candid and quiet tone, point out that it is all one, because coffee is usually taken before tea, and it keeps the table clear to send away the coffee-pot before the teapot is brought up.

The original signatures were decently interred in the Bodleian Library: and the advocates of scattering indefinite blame for indefinite sins of opinion among indefinite persons are, I understand, divided in opinion about the time at which the next attempt shall be made upon men of scientific studies: some are for the Greek Calends, and others for the Roman Olympiads. But, with their usual love of indefiniteness, they have determined that the choice shall be argued upon the basis that which comes first cannot be settled, and is of no consequence.

I give the declaration entire, as a curiosity: and parallel with it I give a subst.i.tute which was proposed in the _Athenaeum_, as worthy to be signed both by students of theology, and by students of science, especially in past time. When a new attempt is made, it will be worth while to look at both:

_Declaration._ _Proposed Subst.i.tute._

We, the undersigned Students We, the undersigned Students of the Natural Sciences, of Theology and of Nature, desire to express our sincere desire to express our sincere regret, that researches into regret, that common notions of scientific truth are perverted religious truth are perverted by some in our own times into by some in our own times into occasion for casting doubt occasion for casting reproach upon the Truth and upon the advocates of Authenticity of the Holy demonstrated or highly Scriptures. probable scientific theories.