Delphi Complete Works of Stephen Leacock, page 347
In the second place stream-lining is apt to go so far that there’s nothing left. You get the pupil so flattened out that he has become caseless, voiceless, tenseless, and moodless. For him there is no joy in the ethical dative or the genitive of value or the accusative of nearer definition. They’re all one to him. He may be all right in an aeroplane because there’s no friction left in him. But his “Latin” has just turned into a bunch of “roots,” like the language of pigeon Chinese, or Pottawatamie English. When a Chinaman speaking “pigeon” (that is, business language) wishes to indicate the Episcopal Bishop of Hong Kong he calls him the “A.1. top-side, Heaven-pigeon man.” That is exactly how Latin must look to the stream-lined student, just a collection of chunks of language to be sorted out for significance.
Nor does ready-made translation help. If we read Pliny in English and Virgil in prose, that’s English not Latin. But never mind. There must be some way of compromise, to set us free from the dilemma, that steers between the Scylla of too much and the Charybdis of too little. The problem is to preserve enough of the old rigid discipline of moods and tenses, rules and exceptions to strengthen the mind without arresting it.
I revert again in conclusion to the objection that so many of us might make that we learnt Latin for years in school and college and never got anywhere with it, never got to be able to read it straight off. Of course we didn’t. Nobody ever does. Not even the professors; no, nor the Romans themselves, not in the way that we read English. This is of course a sort of official secret handed down for generations, and I am really violating here the obligation of my profession by divulging it. But perhaps the time has come to remove the veil. Hitherto it has been better for the world to pretend that at least somebody could “read Latin straight off.” Now it is better to have the truth. The Romans themselves couldn’t.
I am prepared to support this statement. Written language in Roman days, before printing and newspapers, was on a quite different footing from what it is now. There was the ordinary speech of ordinary people, jabbering away all day, just as we do now, with fragments of sentences, exclamations, phrases, false starts and short circuits. Except on the stage, conversation is never done out in full periods, unless by old maids, professors, and garrulous village philosophers; and done thus it is always either ludicrous or tiresome. But we moderns have a written language also of easy and rapid comprehension because we need it for the daily news and the love romances and the crime stories of which the Romans had no current supply. For them one love story had to last a thousand years, from Dido till the fall of Rome. So when they wrote, it was different. They took up the pen as a man puts on his Sunday clothes; they were not trying to be easily intelligible. They wanted to get the full effect, and expected it to take a few moments’ reflection to grasp it. I am quite sure that if one read out an oration of Cicero to a Roman who had never heard it he would soon get mixed and interrupt to say, “Read that last paragraph again, will you?” Just as you yourself would do, if someone read you a section of Browning without fair warning. The parallel is exact. Browning and Cicero were doing the same thing, proposing to sacrifice immediate comprehension for the sake of deeper comprehension when comprehended.
But, you say, some of Cicero’s writings were speeches made in court? Not at all; in court they didn’t sound like that, and they were retouched afterwards. A glance at the pages of the Congressional Record will show what is meant.
I remember once when I was a master at school giving a prize of a cake, specially made, with all sorts of icing and emblems — a joy to look at. The baker showed it to me and received my congratulations with obvious pleasure. But he was an honest man, and he said, “I’m not saying, sir, that it will be much of a cake for eating.” I assured him that no one would think of doing anything as brutal as that. So with the Roman writings — not much of writings for understanding.
All this I say by way of comforting those who, like myself, studied Latin for years and never were able to read it — unless we had read it already.
CHAPTER IV. MATHEMATICS VERSUS PUZZLES
ARE MATHEMATICAL JUDGMENTS synthetically a priori — The multiplication table a fair hand-to-hand fight — Puzzles a fraud — Mr. Brown and the equation — Mathematics and mystifications; Two gazinta four — Can we improve our mathematical sense?
I remember being taken as a boy of twelve years old to listen to a “paper” at the University of Toronto Literary and Debating Society, on the question, “Are mathematical judgments synthetically a priori?” In those simple days before “pictures” and radio and motor-cars and emancipated girls, to go and listen to a “paper” or to a debate between two black-robed students, sipping water off a table, was presumed to be first-class fun. When they discussed mathematical judgments and whether or not a priori, I felt that I didn’t understand it, but that I would when I grew up. That’s where I was wrong.
I am still very vague as to what mathematical judgments being synthetically a priori means. I imagine it refers to the question how do we know that one and one makes two, and if it does, what do we mean by it? But at any rate it bears witness to the profundity of mathematics — I mean, its reach toward the infinite and the unknowable.
This element of fundamental mystery has been expanded in our own day by the glorious confusion introduced by Professor Einstein into all our notions of distance, time and magnitude. How far is one thing from another? The question becomes unknowable. It may be twice as far away as something else is, or half as far; but, beyond the relative number, there seems no such thing as solid distance. What is a foot? Twelve inches. What is an inch? One twelfth of a foot. Similarly where is here? And when is now?
I only refer to these mysteries in order to explain why I still have to speak of mathematics in a reverential whisper, like a Christian entering a Mohammedan mosque, in wicker slippers. He knows it’s a reverend place though he doesn’t understand it.
My attitude toward mathematics, indeed, is that of nine out of ten of educated people — a sense of awe, something like horror, a gratitude for escape but at times a wistful feeling of regret, a sense that there might have been more made of it. Everything, therefore, that I say about mathematics is tempered by so great a humility as to rob it of all controversial aspect. But I do think that as far as a practical school curriculum goes I could shorten it by at least one half. What I would do, to express it in a single phrase, would be to separate true mathematics from mathematical puzzles.
If mathematics is for many students the dragon in the path, these puzzles are the dragon’s teeth. Take them out and the dragon is as easy to handle as a cow. Children learn to count and add and multiply, and feel that it is all plain and straightforward; the multiplication table may be tricky, but it’s fair. Then presently comes a “puzzle” problem. “What number,” says the teacher to the child, “is made up of two figures, the second meaning twice as many as the first, and the two adding up to nine?”
Now, this is not mathematics in the proper sense; this is a puzzle. The only true mathematical operation here would be to set down all the numbers of two digits, from 10 to 99 in turn, and see which one fitted it. But when it comes to guessing and choosing, to ingenuity, that’s a puzzle. Half our school mathematics in algebra and geometry consist of “puzzles,” freak equations and inventive geometry. Students are not discoverers. Pythagoras solved the problem of the squares on the right-angled triangle. I’m willing to “take it as read” and learn it in ten minutes.
This puzzle “bunker” is built right across the mathematical fairway and down the middle of it. “Scholars” pound the sand in it and wonder why they can’t do mathematics. True mathematics means a process learned and used; hard to learn, but later, second nature. Show me how to extract a square root and I’ll extract it as neatly as a dentist. Tell a ship’s captain how to calculate the angle of the sun’s declination and show a broker’s clerk how to use logarithms for compound interest. But don’t expect a student to be a discoverer, working out “problems” which Isaac Newton or Copernicus might solve or miss.
Now at the present time all school-books on mathematics are mixtures of what may be called “sums,” “problems” and “mathematical puzzles.” A sum is an operation dealing with numbers and following a definite and known routine of calculation. When a waiter adds up a restaurant cheque he performs a sum. A calculating machine can do a sum. But it can’t do a problem. For a problem is an operation involving a selection of methods of calculation, of which only certain ones will fit the case. A school-boy calculating when the hour hand of a clock will overtake the minute hand is working out a problem. There are plenty of wrong ways of working at it, as when Achilles tried to overtake a tortoise, and kept the Greeks guessing for generations. But the school-boy soon finds that there are a whole lot of problems dealing with motion and time which all fall into a definite and known method of solution that becomes itself as familiar as the waiter’s addition table. Now the extension of a problem in difficulty and intricacy, to where only one method of many will bring a solution, turns it, at some point, into a puzzle.
When Archimedes jumped out of his bath and shouted “Eureka,” what he had solved was not a problem but a puzzle. He had been asked by some king or other, had he not? how to tell whether a gold crown was really a gold crown or was made of two metals melted together. A modern chemist would find this out with an acid. But Archimedes found a way without chemistry. Yet a professor of mathematics might take a bath every morning for years and never think of it. Since there was no way of forcing a solution by an inevitable method, the thing was not a problem but a puzzle.
Such a puzzle is legitimate enough, though it is no true test of mathematical knowledge. But further out on the field are puzzles that may be called illegitimate, since they present the added difficulty of misleading or paradoxical language. For the information and perhaps the diversion of the reader, let me illustrate the difference. Here is a legitimate puzzle. A man wishes to buy a piece of linoleum that is to cover a space 12 feet by 12. A dealer offers him a piece that is 9 feet by 16 feet. Obviously each piece contains 144 square feet. The dealer tells the customer that all he needs to do is to cut the piece that is 9 feet by 16 feet into two separate pieces that can then be fitted together to cover 12 feet by 12 feet. This of course — or rather not of course, for few people can do it — is done by drawing lines across the 12x12 piece, 3 feet apart in one direction and 4 feet in the other. Start 9 feet east from the top north-west corner and cut along the lines alternately south and west, and there you are. But such a puzzle does not belong in mathematical education although it corresponds in nature to a lot of the things called “problems” that wreck the lives of students.
Here however is a sample of an illegitimate puzzle. A man has 17 camels. He leaves them in his will to his three sons, 1/2 to the eldest, 1/3 to the next and 1/9 to the youngest. But these fractions won’t divide unless you cut up the camels themselves. When the sons are still in perplexity a Dervish happens to pass by, riding on a camel. Dervishes always ride by on camels at convenient moments in these Arabian problems. The sons tell him of their dilemma. After deep thought — Dervishes always think deeply — he says, “Let me lend you my camel to make eighteen instead of seventeen. Now take one half which is nine, and one third which is six, and one ninth which is two, and you each have your proper share. And as nine and six and two only add up to seventeen, you may kindly return my camel.” With which the Dervish departed, and the sons no doubt told the story all the rest of their lives.
Now this problem is of course as full of fallacies as a sieve is full of holes. In the first place the sons didn’t get one half and one third and one ninth of 17 but of something else: and when the father left them these fractions, a little arithmetic — beyond them, no doubt — would have shown that 1/2 and 1/3 and 1/9 of a thing don’t add up to the whole thing but only to 17/18 of it. There was still 1/18 of each camel coming to somebody.
Here is another type of puzzle problem turning on misleading suggestion. Three men at a summer hotel were going fishing and were told they must pay 10 dollars each for a license. They each put up 10 dollars and sent it by a hotel-boy to the inspector’s office. The boy came back with 5 dollars and said that the inspector had made a rebate of 5 dollars out of 30, because it was understood they were all one party in the same boat. The men, greatly pleased, gave the boy 2 dollars out of the 5 and kept one each. One of them then said: “Look here! This is odd. We expected our fishing to cost ten dollars each (thirty dollars) and it has only cost us nine dollars each, and two to the boy. Three times nine is twenty-seven, and two makes twenty-nine; where has the other dollar of the thirty gone?”
The reader no doubt sees the fallacy instantly; but some people wouldn’t.
Now I admit that text-books on mathematics never push the problems quite as far as this on illegitimate puzzle ground — unless indeed they do it on purpose, as in the book of Mathematical Recreations once compiled by the celebrated Professor Ball. But what I claim is that the element of the problem, and even of the puzzle, looms far too large in mathematics as we have it. Indeed for most people it overshadows the subject and ends their advance.
The ordinary straight “discipline” of school mathematics should consist of plain methods of calculation, like division, square root, highest common factor and so on, or such problems as conform to a recognized method of regular solution. All that goes in arithmetic under the name of the “unitary method” is of this class. If A in one hour can do twice as much work as B does in two hours, then — well, we know all about them. Yet few people realize that this beautiful and logical unitary method is quite new — I mean belongs only in the last two generations. When I first learned arithmetic it was just emerging from the “rule-of-three” in the dim light of which all such calculations appeared something like puzzles.
In algebra also a vast part of the subject can be studied as regular calculation, or at least as a problem of regular order, such as the motion and time illustration mentioned above. I gather, also, that another large section of algebraical calculation, though capable of being effected by short, ingenious, or individual methods, can always, if need be, be submitted to a forced operation, clumsy but inevitable — as if a person wanting to know how many squares there are on a chess-board counted them one by one.
To illustrate what I mean, let me call back, from nearly sixty years ago, the recollection of our Sixth Form class in mathematics at Upper Canada College. Our master, Mr. Brown, was a mathematician, the real thing, with a gold medal in proof of it, and gold spectacles through which he saw little but x and y — gentle, simple and out of the world. The class had early discovered that Mr. Brown, with a long equation on the black-board and his back to the class, would stay there indefinitely, in his academic cap and gown, lost in a reverie in which the bonds of discipline fell apart. So the thing was to supply him with a sufficiently tough equation.
This became the special business of the farceur of the class, a large and cheerful joker called Donald Armour, later on the staff of the Rush Medical College and a distinguished Harley Street surgeon. Armour would approach Mr. Brown in the morning and say: “I was looking over some Woolwich examination papers last night, Mr. Brown, and I found this equation. I can’t make anything of it.” “Oh!” said Mr. Brown with interest. He accepted without question the idea that Armour spent his evenings in mathematics. “Let me look at it, Armour.” Then another spirit in collusion would call out, “Won’t you put it on the board, Mr. Brown?” And in a minute there it was, strung out along the black-board, a tangled mass of x’s and y’s and squares and cubes, with Mr. Brown in front of it, as still as Rodin’s penseur.
Meanwhile the class relaxed into easy conversation, and Armour threw paper darts with pins in the end to try and hit Mr. Brown in the yoke of his gown. Presently, without turning round, Mr. Brown spoke. “Of course, I could force it...”
“Oh, please, Mr. Brown,” pleaded Armour, “don’t force it!” and there came a chorus from the class, “Don’t force it, Mr. Brown,” and subdued laughter, because we didn’t know what forcing it was, anyway. “I assure you, gentlemen, I shall not force it until I have tried every expedient.” A chorus of thanks and a renewed reverie. Then presently Mr. Brown would suddenly turn toward us and say excitedly:
“Did you try a function of m, Armour?”
“I never thought of it.”
“It may resolve it.” And away rattled Mr. Brown’s chalk, line upon line, till there stretched the equation, solved! To us it looked bigger than ever.
I won’t swear that it was a function of m that did the trick. It may have been one of the other mystic agents such as a “coefficient of x,” or perhaps pi, a household word to us, as vague as it was familiar.
But what I mean is that when Mr. Brown said he could “force” an equation he referred to a definite mathematical process, as certain as extracting a square root and needing only time and patience.
What I am saying, then, is that school mathematics, and college mathematics as far as made compulsory, should be made up in great proportion, in overwhelming proportion, of straight calculation. I admit that the element of ingenuity, of individual discovery, must also count for something; but for most people even the plainest of plain calculations contain something of it. For many people the multiplication table is still full of happy surprises: and a person not mathematical but trained to calculate compound interest with a logarithm can get as much fun out of it as Galileo could with the moon.
Now to many people, mathematicians by nature, all that I have said about problems and puzzles is merely a revelation of ignorance. These things, they say, are the essence of mathematics. The rest of it is as wooden as a Chinese abacus. They would tell me that I am substituting a calculating machine for a calculating mind. I admit it, in a degree. But the reason for it is perhaps that that is all most of us are capable of. We have not been made “mathematically minded,” and hence the failure of our mathematics.
