A game of "Authors," used to exist, in which one player had to guess the authorship of the quotation shot by another at him. Years afterwards, as young players abruptly learned that their sport was simply a veiled literature cram, the game went out of fashion.

One of my mathematical buddies recently happened to resurrect the old game with his own wonderfully devilish enhancement. Suppose that one player's native language is English. Then, no other player needs to have English as his mother tongue in the enhanced game, but everybody has to know English reasonably well. The player whose native language is English asks each of the others to state if there is a real distinction between some slowly read aloud passages of English prose or poetry. In fact, do all the passages make sense, or is there at least one absurdity?

Not so interesting, others are going to say. Try it for a cultivated Chinese community. You'll be stunned by the destruction. If you cherish any masterpiece in particular, do not apply it to the neutral judgement of unbiased orientals. Healthy topics to skip are politics, theology, and the principles of mathematics. It was even safer to leave ethics alone if the game was not to end in an open fight for everyone.

PLAYING THE GAME

All except one of the above three quotes on virtually any group of players can be easily attempted. The dubious specimen gives away the game to those who have grown up in English. Is that it? Place your own bets; there could be a catch somewhere in it. My friend included the quickly guessed one because one of the others appeared to inspire him specifically, and he could find no fitting example to round out the mystic three. The quotes are here.

Author A. "For that which, though created, is divine, a recurring period exists, which is embraced by a perfect number. For that which is human, however, by that one for which it first occurs that the increasings of the dominant and the dominated, when they take three spaces and four boundaries making similar and dissimilar and increasing and decreasing, cause all to appear familiar and expressible.

"Whose base, modified, as four to three, and married to five, three times increased, yields two harmonies: one equal multiplied by equal, a hundred times the same: the other equal in length to the former, but oblong, a hundred of the numbers upon expressible diameters of five, each diminished by one, or by two if inexpressible, and a hundred cubes of three. This sum now, a geometrical number, is lord over all these affairs, over better and worse births; and when in ignorance of them, the guardians unite the brides and bridegrooms wrongly, the children will not be well-endowed, either in their constitutions or in their fates."

I wish that there was space to replicate the whole of the next one. Fair play, however, requires that one rival is not able to overwhelm another by the sheer excess of his bid.

Author B. ". . . . it may be perceived how little dependence may be placed upon algebraic symbolism in ascertaining essential concrete factors such as the foci of the impacting radii of Solar forces, which absolutely demonstrate the super-physical infallibility of Cosmic Energy, as shall be mathematically disclosed herein by spheroidal measurement

"Thus the present tentative spheroidal measuremnt of [the?] physical continuum of but approximately 45 degrees, or of about one-eighth only of the demonstrable universal scope of futurity, is not alone the lax procreating cause of mundane profanation, or transitional criminal imperfections, but it constitutes the provisional introstatic, or perturbed, growing pains of the evoluting Cosmic Paradise, that is now in course of superphysical growth .... in accordance with precise mathematical conjecture."

Author C.

". . . . One, two! One, two! And through and through

The vorpal blade went snicker-snack!

He left it dead, and with its head

He went galumping back . . . ."

I was stumped as these three were bowling at me. Writers A and B, on numerological terms, appeared to be similar. "A thus has undeniable affinities with B and the "lax procreating source of mundane profanation" with his subtle allusion to brides and their usual prospects. But clearly, it seems to me, C could not have written what B did. And yet the numerological rhythm of the contribution of C's first line is almost identical with that of A's, while one is verse and the other prose. Balancing both the pros and cons, I assumed that C was the author of A's initiative, and that in a subsequent attempt, reflected by B's contribution, he had attempted to make it more intelligible. As he wrote the two specimens, the disparity in rhythm or vibration in the prose of A and B may be due to the different states of C's digestion.

How wrong I was is exposed by the right answer. Author A is Plato (429-348 B.C.); for reasons mentioned in the first chapter, author B need not be named; Author C is, of course, Lewis Carroll.

The quotation from B is taken from an open letter to physicists, dated March 4, 1932. It was received by my friend, who had played it off on me quite scurvily. Every year, virtually every scientist gets hundreds of such letters. However, according to my friend, no modern-day science worker has ever received a letter like Plato's from any human who does not appreciate an imposed retreat from the world's treatment.

THE NUPTIAL NUMBER

Numbers also played an immense role in marriage, experiencing monogamy, polyandry, polygyny, adultery, and concubinage, listing only five significant numerological forms in human breeding. But the most full description of numbers' nuptial consequences is arguably Plato's.

The "lord over all these affairs, over better and worse births" of the quotation is not unlikely the number 60X60X60X60, or 12960000. This is staggering. Even King Solomon had all he could manage with a trivial 300 wives of the first kind and 700 of the second. Notice in passing that 300 + 700 = 1000 = 10 X 10 X 10. This numerology is probably at the root of Solomon's harem.

Where did Plato get 12960000, his famous "nuptial number," from? It would fill an incredible book to go through even a fraction of a percent of the guesses. To all numerologists who want to map the roots of essential things back to the wise and mysterious East, one conjecture is of special interest. Plato received his second or third hand numbers from the priests of Babylon. What they did with it, only God knows. The number definitely existed at a time when Greece was scarcely civilized in Babylon.

There is no point in getting through this in depth here, but it can be mentioned that the hint of a Babylonian root shouts equally in the number's arithmetical existence. It's the fourth power of 60, and the Babylonians, who gave us our minute-and-second sexagesimal system, counted 60 as the basis instead of our 10. He should think of Babylon with reverential gratitude and lift 60 to nuptial strength any time a good numerologist looks at his watch.

Under this, there is more than speculation. The number was found on the Babylonian mathematicians' clay tablets. A modern numerologist should focus his hopes on the decimal system, and as a more manageable number for our effete race, take the fourth power of 10, or 10000. Even Bluebeard may have deterred 12960000.

In order to return to Plato. No scope in all of his works has given more difficulty to his commentators than this passage on the nuptial number in Book VIII of the Republic. An unnecessarily practical critic may say that all the enormous brain power that tried to describe what Plato meant was a pointless waste.Even Plato's immediate predecessors could make nothing sense what he was talking about, and in the 2400 years that followed, only one person actually knew what Plato was saying about a mathematician's way of thought.

If what the dark passage means was proven by Plato and not some clever mathematician in the dim past, then Plato was indeed a very fine arithmeticist, centuries ahead of his day. He himself would overtop even the mighty Diophantus. Of course, the same happens to the mysterious X, who really presented the proof, if X happened.

If the statement of Plato was nothing but a bold guess, it was a strikingly fascinating and acute one, at least. Let us see what could have been said by him.

ROPE-STRETCHERS

We would have to go back to the men who constructed the pyramids in order to get a point of view. The Egyptians were very precise regarding the alignment of their temples for religious purposes, something we seem to have inherited.

Simple astronomical discoveries, such as those made by many people less sophisticated than the ancient Egyptians, could easily establish a real north and south baseline. They had to put down a line at just the correct angles to the first one to get a real east-west line. In addition, the east-west line had to be drawn at the defined point of the north-south line when defining the corner of a house. The dilemma is less simple than it seems without surveying methods, even with just the crudest beginnings of geometry.

Prior to 2000 B.C. The Egyptians used an exceedingly functional approach to the problem, which they find in Fig. 1 was also known to the Hindus and the ancient Chinese. They labelled a rope XABCDY at four points A, B, C, D, taking every suitable unit of length, not too short, say a yard, so that the lengths of AB, BC, CD were 3, 4, 5 yards, respectively. Suppose they needed the approximate east-west of the North-South line NS at point P. On NS, they drove pegs at P and Q, four yards apart, and extended the rope at P and Q, so that point B was marked at P and C at Q. Then, still holding B, C on P, Q, and holding the rope close, one man holding the end of X, and one man holding the end of Y, walked toward each other until the marks A and D matched, say at R. A right angled triangle RPQ on the ground was now created by the cord, and RP was the East-West necessary.

Of necessity, this solution was only feasible because all numbers in the 3 to 4 to 5 ratios are the sides of a right-angled triangle. The first to discover the valuable reality has disappeared into oblivion. A deluge of numerology and no end of functional mathematics has begun.

As the first cousins are astrology and numerology, we note in passing that Oenopides was the first Greek to consider the problem of drawing a perpendicular from a given point to a given straight line. Oenopides (500 B.C.-430 B.C.) took an interest in the issue, according to Proclus, since he found it necessary for astrology. This is just as unusual as chemistry's alchemical roots. Although even illegitimate children are not accountable for their parents' lapses.

Another way of stating what made the trick of the rope stretchers possible is to say that the sum of the squares of 3 and 4 is equal to the square of 5, or 3(2) + 4(2) = 5(2), that is 9 + 16 = 25.

After a red herring, this set Plato off like a beagle. What might have been more seductive than this?

We have three consecutive numbers here, 3-, 4-, 5, and the sum of the first two squares is equal to the third square.

In this, the realistic Egyptians saw nothing more than a simple way to build proper angles in their architecture. Far more was seen by the impractical Greeks, including some of the strangest flights that numerology has ever taken, and even the germ of much that is of no value whatsoever in temple orientation, but is invaluable in a period of peace negotiations, research, mass production, overpopulation, and aircraft bombing. So the ranking, whichever way we look at the board, is even.

I have not forgotten the enigmatic nuptial numbers of Plato. But before the promised revelation can be given a truly remarkable property of the numbers 3, 4, 5, 6 must be noticed.

A great deal of circumlocution will be avoided hereafter if we use "powers" as we did at school. Take any number, say 10, multiply it by itself, 10 X 10. Multiply the result again by 10, thus 10 X 10 X 10. Repeat the process, 10 X 10 X 10 X 10. Instead of writing out these clumsy strings of numbers we condense them as follows, 10, 102, 103, 104, and call them the first, second, third, and fourth powers of 10. Similarly for any number and its successive powers. For example, the successive powers of 6 are 6, 62, 63, 64, . . . or 6, 36, 216,

1296, . . . ; the seventh power of 3 is written 37 and is 2187. If n is any number its powers are written n, n2, n3, n*, . . . ; thus »s means nXnXn Xn X n. Here I have to record two exasperating exceptions to the above simple system of naming powers as the first, second, third, fourth, and so on. The second power of a number is called its square, the third power its cube. For instance, the square of 10 is 102, or 100; the cube of 10 is 10s, or 1000.

I would point out in passing that these annoying exceptions are, as we shall see considerably later, one of the direct contributions of Pythagorean numerology to mathematics. They are on a par with the barbarous way of writing ratios and proportions that persist in our school books and that make mud of what it has learned since it started sucking its thumb to the average intelligent boy. Why should the fact that £ = f be disguised as 3:4::6:8? Let some numerologist explain. But we must get back to Plato.

Who can blame him for being fascinated and mystified by the following facts?

3(2) + 4(2) = 52,

3(3) + 4(3) + 5(3) = 6(3) = 2(3) X 3(3).

It would be a rather sluggish numerologist and a sluggish arithmetician whose imagination would not be ignited by such an arithmetical miracle. If someone believes it's commonplace, he is in this universe above arithmetic. The best that can be wished for is that in the next one he would presumably misprove Fermat's "Last Theorem" for this unanswered mystery, as we shall see, is of the same magic as Plato's fuddled miracle.

It is singularly and numerologically fitting that a woman should have captured the enigmatic nuptial number of Plato. Grace Chisholm Young, a name well known to all mathematicians, explained the mystery in a fascinating paper in the 1923 Proceedings of the London Mathematical Society.

To do justice to Mrs. Young's description of what Plato intended would require her whole essay to be reprinted. This being out of the question here, I will only offer what seems to me to be the core gem uncovered by the probing examination of Mrs. Young. She concludes that Plato guessed and probably showed that the only complete numbers x, y, z, w were free of a common element, for which it is possible thatx2 + y2 = z2 and

x3 + y3 + z3 = w3,

are x = 3, y = 4, z = 5, w = 6.

In the preceding section we saw that 3, 4, 5, 6 actually do satisfy the two equations. The clause about "free of a common factor" is inserted to exclude the trivially obvious solutions got by multiplying 3, 4, 5, 6 by the same whole number. For example 6, 8, 10, 12, or 9, 12, 15, 18, or 12, 16, 20, 24, and so on, also are solutions.

The interesting thing about Plato's guess is found only in that word. Ingenuity of a very rare nature is needed to show that 3, 4, 5, 6, apart from those mentioned above, is the only solution. The topic is not one that is acceptable for a university review paper. None, even with all the strong machines of modern mathematics, could solve it within a reasonable time.

If Plato proved the "only" part, he merits a high place among the great pioneers of number theory. To help him, he had no arithmetic, and the Greek way of writing numbers was merely an incredibly primitive form of shorthand. In comparison, being devoid of algebra, in these uncomfortable terms, Plato was compelled to learn about cubes and squares and actually to worry about numbers.

The success of Mrs. Young in making strict mathematical sense of everything in the mysterious passage, except, of course, the strictly numerological sections of which she may not pretend to be an authority, illustrates that even a philosopher may have to take off a woman's hat.

Plato spoke not only of one number, but of many, and all of these were discovered by Mrs. Young and seen by her to dovetail precisely as required by the mathematical importance of the words of Plato. Mrs. Young gave Plato's words the mathematical values to draw her conclusions, which researchers in the history of Greek mathematics have ferreted out in several ways. All makes sense.

One of these dovetailing numbers is the 12960000, or 604, already cited; another is 36, which is the square root of 1296. Another is the square of 4500, or 20250000, and this, according to Mrs. Young, is the nuptial number. It fits all of the second portion of the passage's specifications. Anyhow, it doesn't matter much what one of them is to be blessed with the prestigious title. In the ties of numerological wedlock, the entire tribe is so inextricably interlocked that not even the Grand Lama himself could break the union.

Why?

It is not a dumb question to wonder why Plato, or his pundits, or his final dis-entangler, might have dedicated years of their lives and undoubtedly a lot of hard labor to the sort of thing mentioned above, but simply a lack of tact. What do we think about a man or woman who could walk into the middle of a perfectly innocent bridge game and yell, "What are all you idiots playing cards for?" If the hostess were up to her task, she would rout the attacker with the perfectly courteous unanswerable arm.

The "why" of it all will be plain enough as we proceed. For the moment we may let the hostess speak for herself.

I have been informed through long experiences with mathematicians," writes Mrs. Young, "that their secret allusions to mathematics, such as throwing light on metaphysical or other subjects, are generally as established as their own mathematical knowledge: and some years ago, when I first found myself face to face with the issue of what mathematical truths Plato alluded to in these oracular utterances, I felt that.

I have only one slight dispute with this, and I assume it is justified by the example of Plato and his most baffling excursion into numerology. It is my opinion that not only some prominent Greek mathematicians, but also some prominent English, French, German, Italian, Dutch, and American mathematicians and mathematical physicists, when they attempted to combine numbers with marriage, or with something else besides more numbers, have proven to be very queer fish. The strongest policy is monogamy.

Keep Out!

As Plato's name will recur frequently in our story, I shall dismiss this matter and introduce the next by citing a famous warning of his own.

"Let no one who is ignorant of geometry enter here."

It is reputed that this notice was written above the entrance to Plato's Academy. No wonder, in the general sense of mathematics, he used geometry, much as the French often do nowadays. Now, if an apologetic mathematician with an irrepressible inferiority complex finds it appropriate to justify himself or his trade for being, he sooner or later drags on this Platonic praise until there is a stronger mathematician present.

I assume that few trained mathematicians will ever again cite the popular warning and will take the pains to analyze for themselves what Plato really meant about numbers and geometry. Without the sort of recommendation that Plato was competent to give, mathematics is better off. He was a numerologist at heart, as far as his mathematical convictions go, and this is so, I suppose, considering the great services he provided to mathematics by boasting about other men's work.

Not all of the numerology on the nuptial number was idly used. It is Exhibit A to lend some color to the claims of those modern theorists who argue that Plato did much more harm than good to mathematics.

Is it credible that a mind capable of what Plato said about "better and worse births" could predict the technically meaningful or intellectually useful flowering of mathematics that started only when, after nearly 2000 years of suffocation, irreverent innovators dared to shake off the Platonic incubus? Uniquely enough, in spite of the critics, it is very credible.

As we move on, we will find more than one intellectual giant, both contemporary and ancient, whose numerology is much stranger than that of Plato. Whoever discusses these puzzles will discover that philosophy and numerology are more receptive to our bedeviled race than knowledge and arithmetic.