[Math Lair] The Search for Truth: Chapter III: The Great Pyramid

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Chapter III



ANYONE who has never yielded to the impulse which we all have at times of indulging some crazy idea has missed one of the richest experiences in life. Give the thing its head and let it run; there will always be a stout telephone pole half a mile or so up the road to stop it. My own pet idea came to me suddenly in the silence of the night nearly three years ago, after a most enjoyable evening spent in reading about the decipherment of the Moscow papyrus.

To come out with the worst at once, it is simply this. If anyone were to ask me what I considered the most important thing, person, or event in the history of rational or mystic thought, I would say the pyramid. It does not have to be the Great Pyramid of Cheops; that one is selected merely to give, the idea a marker worthy of it; but it does have to be an Egyptian pyramid. That is my idea—or delusion, if you like— and I shall stick to it.

It will be necessary to go a long way back, to yzyi B.C., and a longer way forward before the reasonableness of the pyramid's claim can be established.

Before we start on our way I should like to say that I am fairly well acquainted with Piazzi Smyth's monumental insanity on the mystic significance of the Great Pyramid. His book was given to me as a Christmas present many years ago, so I could not in decency refuse to read it. My own theory has but little in common with Smyth's—I trust. Again, friends who belong to certain lodges and secret societies have told me much about the pyramid, but I have always been content to take their word for what they tell me.

Smyth's book is now somewhat of a rarity, so unless the reader is near a large library or a larger secondhand bookstore he probably will be unable to refresh his fainting spirit by a dip into that deep well of mysticism. I almost drowned in it myself when I was a boy and that, possibly, is what is the matter with me now. Nevertheless I hope before this book is ended to dig a deeper well than Smyth's and to drown in it something that will be deader than a dead cat, for all time. For I shall endeavor to show that the first steps upward toward that mystical universe over and above what is given by reputable science, which is revealed to the credulous by the prophets of scientific mysticism, were laid by the ancient Egyptians in their pyramids.

As it is no exaggeration to say that Smyth's particular brand of reasoning is still being sold, we may as well glance at it in passing. Only last week a bewildered psychologist sent me an enormous chart, beautifully engraved, showing exactly how and when the world is going to perdition as prophesied by the architect of the Great Pyramid.

Smyth started from numbers. So far, fair enough; many professed scientists also start from numbers. Suppose you were

given the fraction 36195682. What would it bring to mind> If you

believe in the creative power of numbers, you will at once fall into step with Smyth and hail this awkward fraction as the soul of the circle and the eternal spirit of the Great Pyramid. Here is the simple "proof":

11, 12, 23, 711, 100157, 207325, 36195682.

The first of this string of numbers is 1, the last is the fraction we are trying to decipher, and the middle one is 711. Now, either take it for granted (it is so), or verify it by turning the fractions into decimals, that the first six numbers in the string are alternately greater than 36195682, and less than this fraction. It is

also easily verified that each of the first six is a closer approximation to 36195682 than is any one of the fractions preceding it. All this dovetails so beautifully that whatever truths we may wish to infer from this remarkable "mathematical harmony" cannot possibly be due to chance, but must be the expression of eternal verities. This is how Smyth reasoned. Let us see what he got out of it.

Concentrate on the first, the middle, and the last:

11, 711, 36195682.

The first is easily recognizable as the "Number of the Universe." For is not 1 unity, and is not the Universe the all-inclusive, the stupendous Unity of all unities? What about the middle? Turn it upside down, and lo and behold! we have 117, This is one-half the 227 which we learned at school as the crude approximation to π (pi, the ratio of the circumference of a circle to its diameter).

Smyth was a deeply mystical man. "The first shall be last and the last shall be first" had a profounder significance for him than it had for some of his critics. He started with the last, and he got it from the Egyptians, whom he considered the first. The ratio of the height of the Great Pyramid to the length of the base is our fraction 36195682. Now for Smyth's conclusions—or a few of them.

First, the Egyptians "squared the circle" long before Greece was civilized, and they immortalized this impossible feat in the proportions of the Great Pyramid. This pyramid was built solely to confound skeptical mathematicians who were to be born 3000 or 4000 years after the builders were mummies or dust. These skeptics of course deny that pi is exactly 227. But the pyramid proves them to be wrong; for is not the Universe of necessity right> Smyth here echoes Pope's hardboiled assertion that "Whatever is, is right." Second—but the list is too long. It includes such things as the British system of weights and measures, the nautical almanac, Antichrist, a pope or two, the precession of the equinoxes, the multiplication table, and a scathing denunciation of all mathematicians who do not believe everything they are asked to believe by circle-squarers and others. All of this, according to Smyth, is plainly set out in the mystic arithmetic of the Great Pyramid. What is more, the ancient Egyptians buried all this wisdom in their massive conundrum in order that we degenerate descendents of the great, vanished race should rediscover the forgotten lore of the ancients and not perish utterly in folly and ignorance.

This is a sufficient sample of Smyth's theory. There are many like it. Once we grant Smyth's assumptions all the rest follows by strict deductive reasoning precisely as in a mathematical theory of the universe. Yet we know that the conclusions are absurd. I state it as a simple assertion of fact, and not in any spirit of perverseness, that Smyth's theory has its equals in some of the speculations on the nature of God and the destiny of the universe that have been put forth within the past five years in the name of sober science. The technicalities, of course, are more recondite and the language is more elevated. But the spirit which animates the speculations and the meaning behind the debased scientific symbols are merely more sophisticated disguises, suited to our skeptical age, of the hoary mysticism which poor old Piazzi Smyth honestly but mistakenly read into the meaningless arithmetic of the pyramids.

What follows is of more importance. It is the steps next described which count, and we must try to keep a simple picture of them in mind. We see them first in Egypt. In early Greece we shall stumble over them again. The Middle Ages tried to scale these steps, and failed. The Seventeenth, Eighteenth, and Nineteenth Centuries struggled to climb where their predecessors had only slipped. They also failed. The Twentieth Century is still climbing.

If you mill look at a good photograph of the Great, Pyramid, you mill see that each side of it is like a tapering flight of stone stairs. Those stairs are the steps by which our race began its ascent to the mystical heaven of Cloudcuckooland. Those stairs, literally, brought the Infinite into human reasoning. Let us try to remember what they look like.

before the pyramids were systematically robbed and ruined by successive hordes of archaeologists, in ancient times, the steps, as everyone knows, were concealed beneath a beautifully smooth surfacing of stone. The surfacing has disappeared, and today the pyramids brood over the desert as the most enduring monuments ever raised by man to the ingenuity and gullibility of our ingenious and gullible race. The steps heavenward are what give these monuments their unique importance. That each pyramid was planted like a mountain on the sands as a superfluous precaution against the possible resurrection from the dead of some unwanted king, may be of historical interest, but it is of no human significance. The crude stairway alone proved to be of human importance. Our main business in this chapter will be to see how those steps suggested strictly practical questions to the Egyptian builders which cannot be answered by finite reasoning. In a later chapter we shall see how the Greeks attempted to answer them, and later still, what we ourselves have done with them.

The king has been unswathed, plundered, and desecrated, and as likely as not his sacred relics are all neatly ticketed in glass cases in some museum that will be bombed to smithereens in the next war; but for all the indignities that have been heaped upon his defenceless head he. still is king, for the superstitions which rule our minds derive from him. His own absurd mythologies have been forgotten for thousands of years; the grosser mysticisms of speculative science were started by his slaves when they piled tier upon stone tier to build his tomb. The king has the last laugh at us morbidly gaping at his eviscerated husk. His builders bequeathed to us the riddles of the infinite which make us doubtful of all our reasoning.

2. B.C. 4241

Those who have thoroughly enjoyed all the disadvantages of an old-fashioned classical miseducation with plenty of Latin and too much Greek, will remember the insistence with which the pedagogues told them that Greece, wonder-child of the ages, sprang full armed from the head of Zeus. Not content with admiring the marvellous things the Greeks did in their brief burst of glory, in the arts, in philosophy, in science, and in mathematics, these unscientific and thoroughly un-Greek gentlemen insisted that we believe in the male equivalent of a virgin birth, and credit the extremely unlikely theory that the Greek brilliance had no mother.

Our enthusiastic teachers even ordered us to disbelieve what some of the more impartial Greeks themselves—not many, it is true—said of the indebtedness of Greek culture to the much older civilizations of Babylon and Egypt. The plain statement of Pythagoras (Sixth Century B.C.), for instance, that a prime object of his great school was the preservation of past knowledge, was pooh-poohed as the blasphemous lie of an unreliable mystic who probably was only half Greek anyway, and who could not have known what he was talking about if he had been two


Historians of mathematics, especially those who have been through the modern mathematical mill themselves, always have looked with a certain suspicion on the spontaneous combustion theory of Greek brilliance. Anyone who has traced the excessively slow emergence of the really great and simple ideas of mathematics in times nearer our own, finds it difficult to believe that the Greek pioneers accomplished all that they did

without a considerable body of fairly close thinking inherited from the past to build upon. Once they were well started the story of course was different and credible. Starting from almost nothing, modern science has done as much, if not more, in the past three centuries. But those first slow steps—Who had taken them? And why?

Until quite recently all the historical evidence which is worth anything supported the fabulous legend. A considerable amount of ancient Egyptian mathematics had been recovered and deciphered, but most of it was grievously disappointing. An untrained but intelligent child of ten or twelve might easily do better than some of the absurd geometry—for example the incorrect rule for finding the area of a certain simple figure. It seemed incredible that a race which had produced magnificent art, which governed itself efficiently and which had built in stone as no other race has, could have been so hopelessly stupid as to miss an obvious fact about a simple figure which stared them in the face in scores of their own designs. Yet they had apparently done the impossible, thereby immortalizing themselves in granite, basalt, copper, linen, terra cotta, papyrus and leather as the greatest blockheads in history.

But again, there was that awkward matter of the first reasonably serviceable calendar in history, with its twelve months of thirty days each and its five feast days at the end, to give a total of 365 days. The Egyptians did that in 4241 B.C.— over 6100 years ago—incidently hanging up the world record for the earliest definitely dated event in history.

Now, no fool ever devised a sensible calendar. Compared to some of the clumsy schemes now current for the reform of our own somewhat crazy calendar, the Egyptian's effort of 4241 B.C. is perfection itself. The obvious implications of close astronomical observation, straight thinking, and correct arithmetic behind that great invention need no elaboration.

And then, after this magnificent start, they seem to have a terrible slump. Historians of Egypt discover many such ups

and downs in its four or five thousand years, in everything from art to war. Their "depressions" lasted centuries: ours has just begun. Petrie's attempted explanation of these successive dark ages is interesting, but is not favored by other Egyptologists, I believe. Nevertheless as it is suggestive, especially for Americans, we may take a look at it in passing.

Egypt had its share of "barbaric" invasions and slave raids. After some of these the invaders stayed on and intermarried with theirs hosts, and we know that some of the Israelites came to serve and stayed to be served. To get a picture of what happened as the years lapsed into centuries, let us imagine a few handfuls of black beans tossed into a small sack of white beans. By vigorous shaking the black beans will finally be interspersed among the white, and there will be a stage at which the mixing of black and white is as thorough as possible. If the shaking is quite random—as it probably is—we can calculate by the mathematics of probabilities how long it will take for the mixing to reach the thorough stage. In Egypt the black beans were the invaders, the white the invaded; the shaking was the process of more or less random mating, and the final mixture (which would be gray or tan instead of black and white) the inter-bred population at the stage when all of it was as thoroughly inter-related as possible by ties of mixed blood and mixed ancestry. Now it is only a hypothesis, but a striking one, that in this final stage the population will be better intellectually and physically than either of the populations whose thorough mixing produced the final one. Up to this stage there will be an accelerating improvement; after this stage has been reached, there will be an accelerated decline. The half period (the time taken to reach the best) comes out at something between 500 and 700 years; 1300 years is about the time required for a complete up and down. Professional historians and biologists will have to say whether the hypothesis can be trimmed to fit the facts. If there is anything at all in it, we may expect the lid of our own colossal Melting Pot to blow o& 5I sometime in the next three centuries with a roar of ebullient genius that will be heard round the universe. Till then we can only sit and simmer.

For the present it is enough to remember that 4241 B.C. is the earliest authentic date which records the ability of human beings to reason and think abstractly. Of course they "must" have reasoned abstractly long before 4241 B.C., but we have no indisputable evidence that they did. That date then, in addition to being the first in history, is also one of the most important. The last assertion can be disputed, but not by anyone who will think of the part played by abstract reasoning in scientific discovery, and of the impact of scientific discoveries on our own attempt at civilization in the past 300 years.


"Our purpose is the preservation of past knowledge," Pythagoras declared in the Sixth Century B.C. Things which were of no importance to posterity, like the records of battles, interminable prayers for the, dead, names and honorific titles of a mob of kings, were incised into the hardest stone the Egyptians could find, or locked up in hermetically sealed chambers under thousands of tons of masonry, safe from corruption (but not from thieves or archaeologists) with gold ornaments, beads, bushels of toy images of the gods, and other imperishable trinkets. The records of their achievements in the art of thinking were confided to perishable papyrus or leather to resist the onslaught of successive waves of barbaric ignorance, and only the arid Egyptian climate preserved what few of them have Survived.

Possibly there was a motive in their madness. The king ruled the people, the gods ruled the king, and the priests ruled the gods. The "dark things"—the crude beginnings of science and mathematics—which the priesthood found so potent for their own purposes, were not the sort of stuff to be put into the hands of Tom, Dick, and Harry, lest those humble citizens discover for themselves how simple, how almost trivial, were the elementary facts behind all the mumbo-jumbo business of incantations and inspired rules of thumb.

The similar fight of our own Reformation is not so far behind us but that we can appreciate the point of view of the Egyptian priesthood. To prevent "the vulgar" from "misunderstanding" the wisdom without which they perish, our own sacred books were sealed up for centuries in a tongue which only the priests

or the learned could read. When the jealously guarded knowledge became the common property of at least a part of humanity at large, much of the pernicious mysticism in which it had been shrouded was dissipated in the ribald light of common sense and sane incredulity.

So with the ancient custodians of natural knowledge. It was to the selfish advantage of them and of no one else to make the simple and natural appear mysterious and supernatural. Being shrewd cynics they knew how to play upon ignorance to their own glory, and they realized that the difFusion of such elementary natural knowledge as they possessed would be their own end. In the light of reason and nature all of their mysteries were moonshine, and they knew it. But they could not afford to let their dupes know it, so they kept them as ignorant as they could with threats of a hellish hereafter.

In our own decade we have heard an echo of this all but forgotten struggle. Dean Inge—or one of his sort—proposed that science decree a moratorium on discovery. It takes no psychologist to see through the thin pretense of humanitarianism which clothed this bizarre proposal in a robe of shining light, and to discern the rationalization of what it really is beneath the specious fraud. And if we keep our eyes open we may see in the next decade the other side of the picture: the self-appointed high-priests of speculative science intruding where they know nothing, and where there are no grounds whatever for believing that the scientific method as they know it will work, all "for the good of humanity." But if we remember the poet, we shall probably see for whose good these noble spirits are sacrificing themselves—perhaps not consciously—and if we have any sense left we shall put them under heavy bond before we let them begin spending our heritage for us.

On the whole we must admit that the wise men of Egypt knew what they were about when they decreed that papyrus and leather were durable enough for the preservation of the only records of any human value which they possessed.


From 4241 B.C. to 1801 B.C. we lose the thread completely—a blank of 2440 years.

To get some idea of the length of that blank let us project it from the birth of Christ onto our own era. It takes us to the year 2440 A.D., ora little more than 500 years ahead of where we now are. In even that fraction 500 of the whole blank 2440, half a dozen revolutions might easily occur, each sufficient to obliterate whatever culture might survive to be destroyed. The prophets, of course, tell us that our present civilization will have gone to the devil long before goo years have passed, and that 2440 A.D. will have forgotten everything about us except that we were a particularly moronic species of homo sap. Our printed books are less likely than the Egyptian papyri to survive, and it may be reasonably doubted whether they will have any influence on the Utopia of 3000 A.D., unless possibly they are used as fuel to warm the bath water of the barbarians who are to overwhelm us. There seems to be nothing very surprising then in the disappearance of the thread in the Egypt of 4241 B.C, to 1801 B.C.

From evidence to be presented before long we infer that somewhere in the blank 2440 years Egyptian thought experienced a golden age. By reasonable inference we guess that somewhere in that period lived men who knew what straight abstract thinking is, and who practised the art of it to advance their civilization. The question naturally arises, why should the Egyptians ever have bothered their heads about abstract thinking? For that matter, why should any human being ever bother his head about abstract thought? A simple hypothesis to account for the Egyptians' peculiar taste in the matter may also dispose of the broader question.

From what we know of them, the Egyptians of 4241 B.C. to 1801 B.C. were an industrious, gifted, on the whole peaceable, many-god-fearing, temperate, artistic, and beer-loving agricultural people. This oldest true civilization lived on the land, and the River Nile was its lifeblood. The rich alluvial silt brought down once a year with the regularity of clockwork and deposited by the swollen Nile on the fields of the farmers, did more than make it possible for them to fill their bellies without fail, and have plenty to spare for the famished Israelite when he came a-borrowing and got his foot caught in the door; it awakened their abstract reasoning. And this is how.

Before the annual flood a particular farmer had, let us say, ten acres which it was his special privilege to cultivate. His ten acres were marked off from the fields of his neighbors by boundary posts or stones. After some unusually rich deposit of silt the markers would be obliterated, and in justice to all concerned the first man must get back ten acres, and no more, in the new parcelling out of land. At first it may have been done by guesswork and an appeal to the gods through the local priest. It takes no very vivid imagination to picture the rows that such a method of land surveying would stir up among the farmers of the Mississippi valley, and we have reason for believing that the ancient Egyptian farmer was as intelligent and as individualistic as his modern American successor. Something better had to be divised. But what? And how?

The answer, I think, indicates the most important steps ever taken by human beings toward civilization, sane reasoning, and science.

First, they had to agree that the problem was worth attacking. Second, they had to seek a solution on which all sane men could agree. Third, the solution had to be such that any other sane men, either then or in the future, proceeding by the rules agreed upon, would reach precisely the same conclusion from the given facts of the problem.

We do not know when human beings first sat down together to make such an agreement, or whether they ever did so consciously. I used the inundations of the Nile merely to suggest that such a solution as that indicated of finding how many ncres there are in a given field was necessary before any civilization as far along as the Egyptian could survive and advance.

Let us suppose that neither they nor any other people ever made the effort consciously, but somehow or another built up by slow trial and error a set of working rules. Then, unless they were inconceivably lucky and far more intelligent than human beings are today, sooner or later they must find themselves in lawsuits and wrangles, and they would be forced to make the effort at common agreement and common solution. Their advancing civilization would soon carry them beyond measuring fields to measuring the bins and granaries in which they stored the produce of their fields, and ultimately to barter and trade.

Now, how would a modern, to say nothing of an ancient Egyptian, proceed to find how much corn a granary shaped like a beehive will hold? If anyone does not know, he would have to use the integral calculus, or take it to the Lord in prayer. But, with this powerful aid a formula can be worked out once and for all, and the use of the formula will ensure that the buyer of the grain pays for not a bushel more than he gets.

The Egyptians, of course, did not get as far as that with their granaries, but they made remarkable progress. Before leaving this particular point, I should like to emphasize again that the two steps of agreeing to agree, and producing a set of rules on which all sane men could agree and get the same results, at all times, in all places, were the most important ever taken by our race in its search for truth—with or without a capital T, as you please, or with or without quotation marks. Those two steps led to mathematics, to logic and to science; they also led to no end of quibbling, metaphysics, mysticism, theology, speculation, philosophy, and superstition. They are at the bottom of all our muddles. If we can understand something about them, we shall be that much nearer an escape from some of the nonsenses which make our lives wretched and our theories largely futile. The whole matter is so simple that it is difficult to put clearly, but we shall try to see into it as we go on.

Why could they not have settled the whole business by passing a code of laws to fix what the area of a triangular piece of ground of given dimensions was, or to fix the cubical contents of a beehive, and so on? Possibly they tried that way first and found it would not work. It has been tried in the United States within the past thirty years, when a certain Representative tried to orate a bill through the House fixing the circumference of a circle as three times the diameter. The bill got to the second reading, but was finally killed in the most shocking manner. It is the second of the steps italicized above which conflicts with the legal way out: not all sane men in all places in all times have ever agreed on any code of laws. Nor is there any reason why they should.

Is there any reason why all sane men in all ages and in all places should get 6 as the result of multiplying 2 by 3? I believe that there is none, but this is perhaps merely an excess of caution on my part. Few mathematicians would agree, so the reader may believe if he wishes that "2 times 3 is 6" is one of the "eternal verities" vouchsafed to mankind by an omniscient providence, and not a mere tautological by-product of his own verbalizing. However, we shall all be in a position to use what brains our parents have given us if we follow the thread to the end of the maze.

Somewhere between 4241 B.C. and 1801 B.C. the Egyptians took the decisive steps on the long, steep paths which lead up from chaos to the twin peaks of science and mysticism. These steps were taken not later than the Twelfth Dynasty, the first of them in the reign of Amenemhet the Third, 1849 B.C. to 1801 B.C. In that reign some forgotten priest committed a part of the Egyptian wisdom concerning arithmetic and geometry to writing. The writing was lost, but a copy of it—or of a part, made about 1650 B.C.— survived, made its way into the British Museum, and was first deciphered in 1877. Since then it has been much studied by scholars in the history of mathematics; the best edition is that of Chace, 1927, The Rhind Papyrus. It was on this uninspired copy of the lost original that most of our knowledge and estimates of Egyptian mathematics were based until about three years ago. The arithmetic is remarkably ingenious in some respects; the geometry seems trivial. The whole effect is disappointing.

If my case for the Great Pyramid rested here, I should have to exchange my theory for Smyth's. The damning efFect of the copied papyrus is that it is little more than a miscellaneous collection of rules of thumb for doing certain simple kinds of mathematical problems, or bald statements of results, without any real hint as to how the rules were obtained..For all that the papyrus tells us, some imaginative priest might have dreamed all these rules in his sleep. No case for the capacity of the ancient Egyptians to reason abstractly could be made out on any such evidence as this. And if they could not reason abstractly they had no hope of ever understanding what truth is—again with or without a capital T, etc. Nor could they ever have taken the first step toward its attainment.

The immeasurable superiority of the Greeks over the Egyptians, were the latter to be left in the rule of thumb stage just described, was their recognition of the fact that deductive reasoning was indicated as a workable way to take the steps toward a common agreement, and that all such reasoning rests on postulates, sometimes called axioms, or in plain English, out-and-out assumptions. The postulates are not questioned. Considerably later we shall see in what sense they are "true"—they aren't. Belief in the "absolute truth" of assumptions which are so simple that it takes genius of the highest order to doubt them has been one of the most fruitful sources of superstition from the ancient Greeks to the present day. We shall see, I think, that much of what the Greeks believed about "truth" is vitiated by their incapacity to doubt.

Now, did the Egyptians ever conceive of the idea of proof, and of the human necessity for it? We do not know definitely, and perhaps we never shall. But if they did not, their intuition was so inhumanly good that they could get on without bothering about anything even faintly resembling proof as we understand the term. And this brings us to the apex of it all. It is so important for everything that is to follow that it deserves a separate section to itself.


We can easily understand why it was of importance to the Egyptians to know how many cubic yards of stone there are in a pyramid of given dimensions. They were always building the monstrous things, and the king, contractor or slave driver would have to know how many slaves to put on the job in order to get it finished before the inevitable state funeral. The courtiers might shout "O King, live forever!" but in his heart the King knew that he would do well to last a hundred years.

Suppose all but the top go feet had been built. The unfinished pyramid is called a truncated pyramid. Most of us have proved the formula for calculating its volume when we were in school, but few of us could do it off-hand now. Yet the impatient King would want to know, to see how fast or how slow the stubborn Israelites were really working. His wise men, at some time prior to 1801 B.C., were able to tell him, and what they told him was right, in the sense that it was, and is, consistently workable.

Now, this was an achievement of the very first magnitude. It started, among other things, some of the hottest purely scientific and philosophical controversies of the present day.

The only known way of getting the correct formula for the truncated pyramid is through another pyramid: it is necessary first to find the volume of a pyramid standing on a triangular base of given dimensions. The answer to the last is the familiar formula taught in grammar school: multiply the area of the triangular base by the height of the pyramid and take one(bird of the result.

To get that last conclusion—to prove it—demands abstract, deductive reasoning of a very advanced type. It demands in fact the ability to reason correctly about the infinite—that mysterious word or concept beloved of the mathematicians, theologians, mystics, and philosophers. It can be proved that there is no other way of arriving at the result. As this is a large subject in itself we shall not go into it here, but defer what is necessary to say about it in our attempt to silence Pilate to succeeding chapters. For the moment a very rough hint as to how this problem introduced the infinite into human thought will suffice. An approximation to the volume of the smooth, surfaced, truncated pyramid is obtained by adding up the volumes of all the tiers in the crude, unfinished step stage of the building. The top and base of each tier are equal squares. To get the volume of a tier, multiply the area of its base by the thickness of the tier. Now, it is easily imaginable that if we knew how to do all this when the tiers are "infinitely thin," we should get the exact volume of the smooth, surfaced, truncated pyramid. But, to do this, we should have to add, in imagination at least, an infinite number of tiers. So long as we do not pass to the limit, and take the tiers "infinitely thin," we shall get only an approximation to the smoothed volume, for there will be the jaggedness of the steps still not smoothed out to make the approximate volume less than the correct result.

Did the Egyptians have any conception of this proof, even the faintest? Or did they get the answer by guessing? If so, they were much better at guessing than some of the moderns. However they got it, the fact that they did has caused us to revise our estimate of their intellectual capacity. This revision is due to the decipherment, about three years ago, of a mathematical papyrus which had been reposing in Moscow for a number of years. Mathematical history can thank its luckiest star that no scholarly Bolshevik devoured the evidence of Egyptian genius in the hungry days of the Russian revolution.

The Greatest Pyramid of Egypt is not that of Cheops, but the insubstantial essence of all pyramids which some forgotten seeker after truth saw and measured only with his mind's eye. Here we leave Egypt, and set out on our hazardous journey across the sands to Greece. Nothing definite is known (except for Babylon, which we reluctantly leave aside) of the influence on the development of deductive reasoning of the 1200 years between 1801 B.C. and 600 B.C.

The first important milestone has been passed. Already, in the problem of the pyramid, the paradoxes, the controversies and the useful contributions which the concept of the infinite brought into rigorous thinking, are in sight. Some of the questions thus raised are unsettled even today. Again and again we shall see these same difficulties coming up to torment successive ages of reason. But although the paradoxes and the rest played a major part in sharpening the processes of deductive reasoning, the main advances (as we shall see toward the end of the story) of the past few years could have been made independently of successive attempts to "settle infinity." Historically, however, it was these early problems which awakened the critical faculties of one age after another, including our own.