The Golden Ratio: The Story of Phi, the World's Most Astonishing Number - Mario Livio (2003)

2. THE PITCH AND THE PENTAGRAM

As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.

—ALBERT EINSTEIN (1879–1955)

I see a certain order in the universe and math is one way of making it visible.

—MAY SARTON (1912–1995)

No one knows for sure when humans started to count, that is, to measure multitude in a quantitative way. In fact, we do not even know with certainty whether numbers like “one,” “two,” “three” (the cardinal numbers) preceded numbers like “first,” “second,” “third” (the ordinal numbers), or vice versa. Cardinal numbers simply determine the plurality of a collection of items, such as the number of children in a group. Ordinal numbers, on the other hand, specify the order and succession of specific elements in a group, such as a given date in a month or a seat number in a concert hall. Originally it was assumed that counting developed specifically to address simple day-to-day needs, which clearly argued for cardinal numbers appearing first. However, some anthropologists have suggested that numbers may have first appeared on the historical scene in relation to some rituals that required the successive appearance (in a specified order) of individuals during ceremonies. If true, this idea suggests that the ordinal number concept may have preceded the cardinal one.

Clearly, an even bigger mental leap was required to move from the simple counting of objects to an actual understanding of numbers as abstract quantities. Thus, while the first notions of numbers might have been related primarily to contrasts, associated perhaps with survival—Is it one wolf or a pack of wolves?—the actual understanding that two hands and two nights are both manifestations of the number 2 probably took centuries to grasp. The process had to go through the recognition of similarities (as opposed to contrasts) and correspondences. Many languages contain traces of the original divorce between the simple act of counting and the abstract concept of numbers. In the Fiji Islands, for example, the term for ten coconuts is “koro,” while for ten boats it is “bolo.” Similarly, among the Tauade in New Guinea, different words are used for talking about pairs of males, pairs of females, and mixed pairs. Even in English, different names often are associated with the same numbers of different aggregations. We say “a yoke of oxen” but never “a yoke of dogs.”

Surely the fact that humans have as many hands as they have feet, eyes, or breasts helped in the development of the abstract understanding of the number 2. Even there, however, it must have taken longer to associate this number with things that are not identical, such as the fact that there are two major lights in the heavens, the Sun and the Moon. There is little doubt that the first distinctions were made between one and two and then between two and “many.” This conclusion is based on the results of studies conducted in the nineteenth century among populations that were relatively unexposed to mainstream civilization and on linguistic differences in the terms used for different numbers in both ancient and modern times.

THREE IS A CROWD

The first indication of the fact that numbers larger than two were once treated as “many” comes from some five millennia ago. In the language of Sumer in Mesopotamia, the name for the number 3, “es,” served also as the mark of plurality (like the suffix s in English). Similarly, ethnographic studies in 1890 of the natives of the islands in the Torres Strait, between Australia and Papua New Guinea, showed that they used a system known as two-counting. They used the words “urapun” for “one,” “okosa” for “two,” and then combinations such as “okosa-urapun” for “three” and “okosa-okosa” for “four.” For numbers larger than four, the islanders used the word “ras” (many). Almost identical forms of nomenclatures were found in other indigenous populations from Brazil (the Botocudos) to South Africa (Zulus). The Aranda of Australia, for example, had “ninta” for “one,” “tara” for “two,” and then “tara mi ninta” for “three” and “tara ma tara” for “four,” with all other numbers expressed as “many.” Many of these populations were also found to have the tendency to group things in pairs, as opposed to counting them individually.

An interesting question is: Why did the languages used in these and other counting systems evolve to “four” and then stop (even though three and four were already expressed in terms of one and two)? One explanation suggests that this may simply reflect the fact that we happen to have four fingers in a similar position on our hands. Another, more subtle idea proposes that the answer lies in a physiological limit on human visual perception. Many studies show that the largest number we are able to capture at a glance, without counting, is about four or five. You may remember that in the movie Rain Man, Dustin Hoffman plays an autistic person with an unusual (in fact, highly exaggerated) perception of and memory for numbers. In one scene, all the toothpicks but four from a toothpick box scatter all over the floor, and he is able to tell at a glance that there are 246 toothpicks on the floor. Well, most people are unable to perform such feats. Anyone who ever tried to tally votes of any kind is familiar with this fact. We normally record the first four votes as straight lines, and then we cross those with a fifth line once a fifth vote is cast, simply because of the difficulty to perceive at a glance any number of lines that is larger than four. This system has been known in English pubs (where the barman counts the beers ordered) as the five barred gate. Curiously, an experiment described by the historian of mathematics Tobias Dantzig (1884–1956) in 1930 (in his wonderful book Number, the Language of Science) suggests that some birds also can recognize and discriminate among up to four objects. Dantzig's story goes as follows:

A squire was determined to shoot a crow which made its nest in the watch-tower of his estate. Repeatedly he had tried to surprise the bird, but in vain: at the approach of man the crow would leave its nest. From a distant tree it would watchfully wait until the man had left the tower and then return to its nest. One day the squire hit upon a ruse: two men entered the tower, one remained within, the other came out and went on. But the bird was not deceived: it kept away until the man within came out. The experiment was repeated on the succeeding days with two, three, then four men, yet without success. Finally, five men were sent: as before, all entered the tower, and one remained while the other four came out and went away Here the crow lost count. Unable to distinguish between four and five it promptly returned to its nest.

More pieces of evidence suggest that the initial counting systems followed the “one, two,… many” philosophy. These come from linguistic differences in the treatments of plurals and of fractions. In Hebrew, for example, there is a special form of plural for some pairs of identical items (e.g., hands, feet) or for words representing objects that contain two identical parts (e.g., pants, eyeglasses, scissors) that is different from the normal plural. Thus, while normal plurals end in “im” (for items considered masculine) or “ot” (for feminine items), the plural form for eyes, breasts, and so on, or the words for objects with two identical parts, end in “ayim.” Similar forms exist in Finnish and used to exist (until medieval times) in Czech. Even more important, the transition to fractions, which surely required a higher degree of familiarity with numbers, is characterized by a marked linguistic difference in the names of fractions other than a half. In Indo-European languages, and even in some that are not (e.g., Hungarian and Hebrew), the names for the fractions “one-third” (), “one-fifth” (), and so on generally derive from the names of the numbers of which these fractions are reciprocals (three, five, etc.). In Hebrew, for example, the number “three” is “shalosh” and “one-third” is “shlish.” In Hungarian “three” is “Hàrom” and “one-third” is “Harmad.” This is not true, however, for the number “half,” which is not related to “two.” In Romanian, for example, “two” is “doi” and “half is “jumate;” in Hebrew “two” is “shtayim” and “half is “hetsi;” in Hungarian “two” is “kettö” and “half is “fel.” The implication may be that while the number ½ was understood relatively early, the notion and comprehension of other fractions as reciprocals (namely, “one over”) of integer numbers probably developed only after counting passed the “three is a crowd” barrier.

COUNTING MY NUMBERLESS FINGERS

Even before the counting systems truly developed, humans had to be able to record some quantities. The oldest archaeological records that are believed to be associated with counting of some sort are in the form of bones on which regularly spaced incisions have been made. The earliest, dating to about 35,000 B.C., is a part of a baboon's thigh bone found in a cave in the Lembedo Mountains in Africa. That bone was engraved with twenty-nine incisions. Another such “bookkeeping” record, a bone of a wolf with fifty-five incisions (twenty-five in one series and thirty in another, the first series grouped in fives), was found by archaeologist Karel Absolon in 1937 at Dolné Vestonice, Czechoslovakia, and has been dated to the Aurignacian era (about 30,000 years ago). The grouping into 5, in particular, suggests the concept of a base, which I will discuss shortly. While we do not know the exact purpose of these incisions, they may have served as a record of a hunter's kills. The grouping would have helped the hunter to keep tally without having to recount every notch. Similarly marked bones, from the Magdalenian era (about 15,000 years ago), were also found in France and in the Pekarna cave in the Czech Republic.

A bone that has been subjected to much speculation is the Ishango bone found by archaeologist Jean de Heinzelin at Ishango near the border between Uganda and Zaire (Figure 6). That bone handle of a tool, dating to about 9000 B.C., displays three rows of notches arranged, respectively, in the following groups: (i) 9, 19, 21, 11; (ii) 19, 17, 13, 11; (iii) 7, 5, 5, 10, 8, 4, 6, 3. The sum of the numbers in the first two rows is 60 in each, which led some to speculate that they may represent a record of the phases of the Moon in two lunar months (with the possibility that some incisions may have been erased from the third row, which adds up only to 48). More intricate (and far more speculative) interpretations also have been proposed. For example, on the basis of the fact that the second row (19, 17, 13, 11) contains sequential primes (numbers that have no divisors except for 1 and the number itself), and the first row (9, 19, 21, 11) contains numbers that are different by 1 from either 10 or 20, de Heinzelin concluded that the Ishango people had some rudimentary knowledge of arithmetic and even of prime numbers. Needless to say, many researchers find this interpretation somewhat far-fetched.

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Figure 6

The Middle East has produced another interesting recording system, dating to the period between the ninth and second millennia B.C. In places ranging from Anatolia in the north to Sudan in the south, archaeologists have discovered hoards of little toylike objects of different shapes made of clay. They are in the form of disks, cones, cylinders, pyramids, animal shapes, and others. University of Texas at Austin archaeologist Denise Schmandt-Besserat, who studied these objects in the late 1970s, developed a fascinating theory. She believes that these clay objects served as pictogram tokens in the market, symbolizing the types of objects being counted. Thus, a small clay sphere might have stood for some quantity of grain, a cylinder for a head of cattle, and so on. The mideastern prehistoric merchants could therefore, according to Schmandt-Besserat's hypothesis, conduct the accounting of their business by simply lining up the tokens according to the types of goods being transacted.

Whatever type of symbols was used for different numbers—incisions on bones, clay tokens, knots on strings (devices called quipu, used by the Inca), or simply the fingers—at some point in history humans faced the challenge of being able to represent and manipulate large numbers. For practical reasons, no symbolic system that has a uniquely different name or different representing object for every number can survive for long. In the same way that the letters in the alphabet represent in some sense the minimal number of characters with which we can express our entire vocabulary and all written knowledge, a minimal set of symbols with which all the numbers can be characterized had to be adopted. This necessity led to the concept of a “base” set—the notion that numbers can be arranged hierarchically, according to certain units. We are so familiar in everyday life with base 10 that it is almost difficult to imagine that other bases could have been chosen.

The idea behind base 10 is really quite simple, which does not mean it did not take a long time to develop. We group our numbers in such a way that ten units at a given level correspond to one unit at a higher level in the hierarchy. Thus 10 “ones” correspond to 1 “ten,” 10 “tens” correspond to 1 “hundred,” 10 “hundreds” correspond to 1 “thousand,” and so on. The names for the numbers and the positioning of the digits also reflect this hierarchical grouping. When we write the number 555, for example, although we repeat the same cipher three times, it means something different each time. The first digit from the right represents 5 units, the second represents 5 tens, or 5 times ten, and the third 5 hundreds, or 5 times ten squared. This important rule of position, the place-value system, was first invented by the Babylonians (who used 60 as their base, as described below) around the second millennium B.C., and then, over a period of some 2,500 years, was reinvented, in succession, in China, by the Maya in Central America, and in India.

Of all Indo-European languages, Sanskrit, originating in northern India, provides some of the earliest written texts. In particular, four of the ancient scriptures of Hinduism, all having the Sanskrit word “veda” (knowledge) in their title, date to the fifth century B.C. The numbers 1 to 10 in Sanskrit all have different names: eka, dvau, trayas, catvaras, pañca, sat, sapta, astau, náva, dasa. The numbers 11 to 19 are all simply a combination of the number of units and 10. Thus, 15 is “pañca-dasa,” 19 is “náva-dasa,” and so on. English, for example, has the equivalent “teen” numbers. In case you wonder, by the way, where “eleven” and “twelve” in English came from, “eleven” derives from “an” (one) and “lif” (left, or remainder) and “twelve” from “two” and “lif” (two left). Namely, these numbers represent “one left” and “two left” after ten. Again as in English, the Sanskrit names for the tens (“twenty,” “thirty,” etc.) express the unit and plural tens (e.g., 60 is sasti), and all Indo-European languages have a very similar structure in their vocabulary for numbers. So the users of these languages quite clearly adopted the base 10 system.

There is very little doubt that the almost universal popularity of base 10 stems simply from the fact that we happen to have ten fingers. This possibility was already raised by the Greek philosopher Aristotle (384–322 B.C.) when he wondered (in Problemata): “Why do all men, barbarians and Greek alike, count up to ten and not up to any other number?” Base 10 really offers no other superiority over, say, base 13. We could even argue theoretically that the fact that 13 is a prime number, divisible only by 1 and itself, gives it an advantage over 10, because most fractions would be irreducible in such a system. While, for example, under base 10 the number 36/100 also can be expressed as 18/50 or 9/25, such multiple representations would not exist under a prime base like 13. Nevertheless, base 10 won, because ten fingers stood out in front of every human's eyes, and they were easy to use. In some Malay-Polynesian languages, the word for “hand,” “lima,” is actually the same as the word for “five.” Does this mean that all the known civilizations chose 10 as their base? Actually, no.

Of the other bases that have been used by some populations around the world, the most common was base 20, known as the vigesimal base. In this counting system, which was once popular in large portions of Western Europe, the grouping is based on 20 rather than 10. The choice of this system almost certainly comes from combining the fingers with the toes to form a larger base. For the Inuit (Eskimo) people, for example, the number “twenty” is expressed by a phrase with the meaning “a man is complete.” A number of modern languages still have traces of a vigesimal base. In French, for example, the number 80 is “quatre-vingts” (meaning “four twenties”), and an archaic form of “six-vingts” (“six twenties”) existed as well. An even more extreme example is provided by a thirteenth-century hospital in Paris, which is still called L'Ôpital de Quinze-Vingts (The Hospital of Fifteen Twenties), because it was originally designed to contain 300 beds for blind veterans. Similarly, in Irish, 40 is called “daichead,” which is derived from “da fiche” (meaning “two times twenty”); in Danish, the numbers 60 and 80 (“tresindstyve” and “firsindstyve” respectively, shortened to “tres” and “firs”) are literally “three twenties” and “four twenties.”

Probably the most perplexing base found in antiquity, or at any other time for that matter, is base 60—the sexagesimal system. This was the system used by the Sumerians in Mesopotamia, and even though its origins date back to the fourth millennium B.C., this division survived to the present day in the way we represent time in hours, minutes, and seconds as well as in the ° of the circle (and the subdivision of ° into minutes and seconds). Sixty as a base for a number system taxes the memory considerably, since such a system requires, in principle, a unique name or symbol for all the numbers from 1 to 60. Aware of this difficulty, the ancient Sumerians used a certain trick to make the numbers easier to remember—they inserted 10 as an intermediate step. The introduction of 10 allowed them to have unique names for the numbers 1 to 10; the numbers 10 to 60 (in units of 10) represented combinations of names. For example, the Sumerian word for 40, “nišmin,” is a combination of the word for 20, “niš,” and the word for 2, “min.” If we write the number 555 in a purely sexagesimal system, what we mean is 5 × (60)2+5 × (60)+5, or 18,305 in our base 10 notation.

Many speculations have been advanced as to the logic or circumstances that led the Sumerians to choose the unusual base of 60. Some are based on the special mathematical properties of the number 60: It is the first number that is divisible by 1, 2, 3, 4, 5, and 6. Other hypotheses attempt to relate 60 to concepts such as the number of months in a year or days in a year (rounded to 360), combined somehow with the numbers 5 or 6. Most recently, French math teacher and author Georges Ifrah argued in his superb 2000 book, The Universal History of Numbers, that the number 60 may have been the consequence of the mingling of two immigrant populations, one of which used base 5 and the other base 12. Base 5 clearly originated from the number of fingers on one hand, and traces for such a system can still be found in a few languages, such as in the Khmer in Cambodia and more prominently in the Saraveca in South America. Base 12, for which we find many vestiges even today—for example, in the British system of weights and measures—may have had its origins in the number of joints in the four fingers (excluding the thumb; the latter being used for the counting).

Incidentally, strange bases pop up in the most curious places. In Lewis Carroll's Alice's Adventures in Wonderland, to assure herself that she understands the strange occurrences around her, Alice says: “I'll try if I know all the things I used to know. Let me see: four times five is twelve, and four times six is thirteen, and four times seven is—oh dear! I shall never get to twenty at that rate!” In his notes to Carroll's book, The Annotated Alice, the famous mathematical recreation writer Martin Gardner provides a nice explanation for Alice's bizarre multiplication table. He proposes that Alice is simply using bases other than 10. For example, if we use base 18, then 4×5 = 20 will indeed be written as 12, because 20 is 1 unit of 18 and 2 units of 1. What lends plausibility to this explanation is of course the fact that Charles Dodgson (“Lewis Carroll” was his pen name) lectured on mathematics at Oxford.

OUR NUMBERS, OUR GODS

Irrespective of the base that any of the ancient civilizations chose, the first group of numbers to be appreciated and understood at some level was the group of whole numbers (or natural numbers). These are the familiar 1, 2, 3, 4,… Once humans absorbed the comprehension of these numbers as abstract quantities into their consciousness, it did not take them long to start to attribute special properties to numbers. From Greece to India, numbers were accredited with secret qualities and powers. Some ancient Indian texts claim that numbers are almost divine, or “Brahma-natured.” These manuscripts contain phrases that are nothing short of worship to numbers (like “hail to one”). Similarly, a famous dictum of the Greek mathematician Pythagoras (whose life and work will be described later in this chapter) suggests that “everything is arranged according to number.” These sentiments led on one hand to important developments in number theory but, on the other, to the development of numerology—the set of doctrines according to which all aspects of the universe are associated with numbers and their idiosyncrasies. To the numerologist, numbers were fundamental realities, drawing symbolic meanings from the relation between the heavens and human activities. Furthermore, essentially no number that was mentioned in the holy writings was ever treated as irrelevant. Some forms of numerology affected entire nations. For example, in the year 1240 Christians and Jews in Western Europe expected the arrival of some messianic king from the East, because it so happened that the year 1240 in the Christian calendar corresponded to the year 5000 in the Jewish calendar. Before we dismiss these sentiments as romantic naïveté that could have happened only many centuries ago, we should recall the extravagant hoopla surrounding the ending of the last millennium.

One special version of numerology is the Jewish Gematria (possibly based on “geometrical number” in Greek), or its Muslim and Greek analogues, known as Khisab al Jumal (“calculating the total”), and Isopsephy (from the Greek “isos,” equal, and “psephizéin,” to count), respectively. In these systems, numbers are assigned to each letter of the alphabet of a language (usually Hebrew, Greek, Arabic, or Latin). By adding together the values of the constituent letters, numbers are then associated with words or even entire phrases. Gematria was especially popular in the system of Jewish mysticism practiced mainly from the thirteenth to the eighteenth century known as cabala. Hebrew scholars sometimes used to amaze listeners by calling out a series of apparently random numbers for some ten minutes and then repeating the series without an error. This feat was accomplished simply by translating some passage of the Hebrew scriptures into the language of Gematria.

One of the most famous examples of numerology is associated with 666, the “number of the Beast.” The “Beast” has been identified as the Antichrist. The text in the Book of Revelations (13:18) reads: “This calls for wisdom: let anyone with understanding calculate the number of the beast, for it is the number of a man. Its number is six hundred and sixty-six.” The phrase “it is the number of a man” prompted many of the Christian mystics to attempt to identify historical figures whose names in Gematria or Isopsephy give the value 666. These searches led to, among others, names like those of Nero Caesar and the emperor Diocletian, both of whom persecuted Christians. In Hebrew, Nero Caesar was written as (from right to left): xqw oexp, and the numerical values assigned in Gematria to the Hebrew letters (from right to left)—50, 200, 6, 50; 100, 60, 200—add up to 666. Similarly, when only the letters that are also Roman numerals (D, I, C, L, V) are counted in the Latin name of Emperor Diocletian, DIOCLES AVGVSTVS, they also add up to 666 (500+ 1+ 100+ 50+ 5+ 5+ 5). Clearly, all of these associations are not only fanciful but also rather contrived (e.g., the spelling in Hebrew of the word Caesar actually omits a letter, of value 10, from the more common spelling).

Amusingly, in 1994, a relation was “discovered” (and appeared in the Journal of Recreational Mathematics) even between the “number of the Beast” and the Golden Ratio. With a scientific pocket calculator, you can use the trigonometric functions sine and cosine to calculate the value of the expression [sin 666°+ cos (6 × 6 × 6)°]. Simply enter 666 and hit the [sin] button and save that number, then enter 216 (= 6 × 6 × 6) and hit the [cos] button, and add the number you get to the number you saved. The number you will obtain is a good approximation of the negative of phi. Incidentally, President Ronald Reagan and Nancy Reagan changed their address in California from 666St. Cloud Road to 668 to avoid the number 666, and 666 was also the combination to the mysterious briefcase in Quentin Tarantino's movie Pulp Fiction.

One clear source of the mystical attitude toward whole numbers was the manifestation of such numbers in human and animal bodies and in the cosmos, as perceived by the early cultures. Not only do humans have the number 2 exhibited all over their bodies (eyes, hands, nostrils, feet, ears, etc.), but there are also two genders, there is the Sun-Moon system, and so on. Furthermore, our subjective time is divided into three tenses (past, present, future), and, due to the fact that Earth's rotation axis remains more or less pointed in the same direction (roughly toward the North Star, Polaris, although small variations do exist, as described in Chapter 3), the year is divided into four seasons. The seasons simply reflect the fact that the orientation of Earth's axis relative to the Sun changes over the course of the year. The more directly a part of the Earth is exposed to sunlight, the longer the daylight hours and the warmer the temperature. In general, numbers acted in many circumstances as the mediators between cosmic phenomena and human everyday life. For example, the names of the seven days of the week were based on the names of the celestial objects originally considered to be planets: the Sun, the Moon, Mars, Mercury, Jupiter, Venus, and Saturn.

The whole numbers themselves are divided into odd and even, and nobody did more to emphasize the differences between the odd and even numbers, and to ascribe a whole menagerie of properties to these differences, than the Pythagoreans. In particular, we shall see that we can identify the Pythagorean fascination with the number 5 and their admiration for the five-pointed star as providing the initial impetus for the interest in the Golden Ratio.

PYTHAGORAS AND THE PYTHAGOREANS

Pythagoras was born around 570 B.C. in the island of Samos in the Aegean Sea (off Asia Minor), and he emigrated sometime between 530 and 510 to Croton in the Dorian colony in southern Italy (then known as Magna Graecia). Pythagoras apparently left Samos to escape the stifling tyranny of Polycrates (died ca. 522 B.C.), who established Samian naval supremacy in the Aegean Sea. Perhaps following the advice of his presumed teacher, the mathematician Thales of Miletus, Pythagoras probably lived for some time (as long as twenty-two years, according to some accounts) in Egypt, where he would have learned mathematics, philosophy, and religious themes from the Egyptian priests. After Egypt was overwhelmed by Persian armies, Pythagoras may have been taken to Babylon, together with members of the Egyptian priesthood. There he would have encountered the Mesopotamian mathematical lore. Nevertheless, the Egyptian and Babylonian mathematics would prove insufficient for Pythagoras' inquisitive mind. To both of these peoples, mathematics provided practical tools in the form of “recipes” designed for specific calculations. Pythagoras, on the other hand, was one of the first to grasp numbers as abstract entities that exist in their own right.

In Italy, Pythagoras began to lecture on philosophy and mathematics, quickly establishing an enthusiastic crowd of followers, which may have included the young and beautiful Theano (daughter of his host Milo), whom he later married. The atmosphere in Croton proved extremely fertile for Pythagoras' teachings, since the community there was composed of a plethora of semimystic cults. Pythagoras established a strict routine for his students, paying particular attention to the hour of waking and the hour of falling asleep. Students were advised upon rising to repeat the verses:

As soon as you awake, in order lay
the actions to be done the coming day.

Similarly, at nightfall, they were to recite:

Allow not sleep to close your eyes
Before three times reflecting on
Your actions of the day. What deeds
Done well, what not, what left undone?

Most of the details of Pythagoras' life and the reality of his mathematical contributions remain veiled in uncertainty. One legend has it that he had a golden birthmark on his thigh, which was taken by his followers to indicate that he was a son of the god Apollo. None of the biographies of Pythagoras written in antiquity have survived, and biographies written later, such as the Lives of the Eminent Philosophers, written by Diogenes Läertius in the third century, often rely on many sources of varying reliability. Pythagoras apparently wrote nothing, and yet his influence was so great that the more attentive of his followers formed a secretive society, or brotherhood, and were known as the Pythagoreans. Aristippus of Cyrene tells us in his Account of Natural Philosophers that Pythagoras derived his name from the fact that he was speaking (agoreuein) truth like the God at Delphi (tou Pythiou).

The events surrounding Pythagoras' death are as uncertain as the facts about his life. According to one story, the house in which he was staying at Croton was set on fire by a mob, envious of the Pythagorean elite, and Pythagoras himself was murdered during his escape, upon reaching a place full of beans on which he wouldn't trample. A different version is provided by the Greek scientist and philosopher Dicaearchus of Messana (ca. 355–280 B.C.), who states that Pythagoras managed to escape as far as the Temple of the Muses at Metapontum, where he died following forty days of self-imposed starvation. A completely different story is told by Hermippus, according to which Pythagoras was slain by the Syracusans in their war against the Agrigentine army, which Pythagoras joined.

Even though it is almost impossible to attribute with certainty any specific mathematical achievements either to Pythagoras himself or to his followers, there is no question that they have been responsible for a mingling of mathematics, philosophy of life, and religion unparalleled in history. In this respect it is perhaps interesting to note the historical coincidence that Pythagoras was a contemporary of Buddha and Confucius.

Pythagoras is in fact credited with having coined the words “philosophy” (“love of wisdom”) and “mathematics” (“that which is learned”). To him, a “philosopher” was someone who “gives himself up to discovering the meaning and purpose of life itself… to uncover the secrets of nature.” Pythagoras emphasized the importance of learning above all other activities, because, in his words, “most men and women, by birth or nature, lack the means to advance in wealth and power, but all have the ability to advance in knowledge.” He was also famous for introducing the doctrine of metempsychosis—that the soul is immortal and is reborn or transmigrated in human and animal incarnations. This doctrine resulted in a strong advocacy of vegetarianism, since animals to be slaughtered could represent reincarnated friends. To purify the soul, the Pythagoreans established strict rules, which included, for example, a prohibition on eating beans and an extreme emphasis on the training of the memory. In his treatise On the Pythagoreans, the famous Greek philosopher Aristotle gives several possible reasons for the abstention from beans: They resemble genitals; being plants without parts they are like the gates of hell; beans were supposed to arise simultaneously with humans in the act of universal creation; or beans were used in elections in oligarchical governments.

Pythagoras and the Pythagoreans are best known for their presumed role in the development of mathematics and for the application of mathematics to the concept of order, whether it is musical order, the order of the cosmos, or even ethical order. Every child in school learns the Pythagorean theorem of a triangle that has a right (90-degree) angle (a right triangle). According to this theorem (Figure 7, on the right), the area of the square constructed on the longest side (the hypotenuse) equals the sum of the areas of the squares constructed on the two shorter sides. In other words, if the length of the hypotenuse is c,then the area of the square constructed on it is c;2 the areas of the squares constructed on the other two sides (of lengths a and b) are a2 and b2 respectively. The Pythagorean theorem can therefore be stated as: c2=a2+b2 in every right triangle. In 1971, when the republic of Nicaragua selected the ten mathematical equations that changed the face of the Earth as the theme for a series of stamps, the Pythagorean theorem appeared on the second stamp. The numbers 3, 4, 5, or 7, 24, 25, for example, form Pythagorean triples, because 32 +42 = 52 (9+16 = 25); 72+242 = 252 (49+576 = 625), and they can be used as the lengths of the sides of a right triangle.

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Figure 7

Figure 7 also suggests what is perhaps the easiest proof of the Pythagorean theorem: On one hand, when one subtracts from the square whose side equals a + b the area of four identical triangles, one gets the square built on the hypotenuse (middle figure). On the other, when one subtracts from the same square the same four triangles in a different arrangement (left figure), one gets the two squares built on the shorter sides. Thus, the square on the hypotenuse is clearly equal in area to the sum of the two smaller squares. In his 1940 book The Pythagorean Proposition, mathematician Elisha Scott Loomis presented 367 proofs of the Pythagorean theorem, including proofs by Leonardo da Vinci and by the twentieth president of the United States, James Garfield.

Even though the Pythagorean theorem was not yet known as a “truth” characterizing all right-angle triangles, Pythagorean triples actually had been recognized long before Pythagoras. A Babylonian clay tablet from the Old Babylonian period (ca. 1600 B.C.) contains fifteen such triples. The Babylonians discovered that Pythagorean triples can be constructed using the following simple procedure, or “algorithm.” Choose any two whole numbers p and qsuch that p is larger than q. You can now form the Pythagorean triple of numbers p2 – q2; 2pq; p2 – q2. For example, suppose q is 1 and p is 4. Then p2 – q2 = 42 − 12 = 16 − 1 = 15; 2pq = 2 × 4 × 1 = 8; p2 + q2 = 42 + 12 = 16 + 1 = 17. The set of numbers 15, 8, 17 is a Pythagorean triple because 152 + 82 = 172 (225 + 64 = 289). You can easily show that this will work for any whole numbers p and q. (For the interested reader, a brief proof is presented in Appendix 1.) Therefore, there exists an infinite number of Pythagorean triples (a fact proven by Euclid of Alexandria).

However, in the Pythagorean world, orderly patterns were far from being restricted to triangles and geometry. Pythagoras is traditionally said to have discovered the harmonic progressions in the notes of the musical scale, by finding that the musical intervals and the pitch of the notes correspond to the relative lengths of the vibrating strings. He observed that dividing a string by consecutive integers yields (up to a point) harmonious and pleasing (consonant) intervals. When two arbitrary musical notes are made to sound together, the resulting sound isusually harsh (dissonant) to our ear. Only a few combinations produce pleasant sounds. Pythagoras discovered that these rare consonances are obtained when the notes are produced by similar strings whose lengths are in ratios given by the first few whole numbers. Unison is obtained when the strings are of equal length (a 1:1 ratio); the octave is obtained by a 1:2 ratio of string lengths; the fifth by 2:3; and the fourth by 3:4. In other words, you can pluck a string and sound a note. If you pluck an equally taut string that is one-half the length, you will hear a note that is precisely one harmonic octave above the first. Similarly, image of a C-string gives the note A, image of it gives G, image of it gives F, and so on. These remarkable early findings formed the basis for the more advanced understanding of musical intervals that developed in the sixteenth century (in which, incidentally, Vincenzo Galilei, Galileo's father, was involved). A wonderful illustration by Franchinus Gafurius, which appeared as a frontispiece in Theorica Musice in 1492, shows Pythagoras experimenting with the sounds of various devices, including hammers, strings, bells, and flutes (Figure 8; the upper left depicts the biblical figure of Jubal or Tubal, “the father of all such as handle the harp and organ”). But, wondered the Pythagoreans, if musical harmony can be expressed by numbers, why not the entire cosmos? They therefore concluded that all objects in the universe owed their characteristics to the nature of number. Astronomical observations suggested, for example, that the motions in the heavens also were extremely regular and subject to a specific order. This led to the concept of a beautiful “harmony of the spheres”—the notion that in their regular motions, heavenly bodies also create harmonious music. The philosopher Porphyry (ca. A.D. 232–304), who wrote more than seventy works dealing with history, metaphysics, and literature, also wrote (as a part of his four-volume work History of Philosophy) a brief biography of Pythagoras entitled Life of Pythagoras. In it, Porphyry says about Pythagoras: “He himself could hear the harmony of the Universe, and understood the music of the spheres, and the stars which move in concert with them, and which we cannot hear because of the limitations of our weak nature.” After enumerating more of Pythagoras' exquisite qualities, Porphyry continues: “Pythagoras affirmed that the Nine Muses were constituted by the sounds made by the seven planets, the sphere of the fixed stars, and that which is opposed to our earth, called the ‘counter-earth’ (the latter, according to the Pythagorean theory of the universe, revolved in opposition to Earth, around a central fire). The concept of the “harmony of the spheres” was elaborated upon again, more than twenty centuries later, by the famous astronomer Johannes Kepler (1571–1630). Having witnessed in his own life much agony and the horrors of war, Kepler concluded that Earth really created two notes, mi for misery (“miseria” in Latin) and fa for famine (lames in Latin). In Kepler s words: “the Earth sings MI FA MI, so that even from the syllable you may guess that in this home of ours Misery and Famine hold sway.”

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Figure 8

The Pythagorean obsession with mathematics was mildly ridiculed by the great Greek philosopher Aristotle. He writes in Metaphysics (in the fourth century B.C.): “The so-called Pythagoreans applied themselves to mathematics, and were the first to develop this science; and through penetrating it, they came to fancy that its principles are the principles of all things.” Today, while we may be amused by some of the Pythagorean fanciful ideas, we have to recognize that the fundamental thought behind them is really not very different from that expressed by Albert Einstein (in Letters to Solovine): “Mathematics is only a means for expressing the laws that govern phenomena.” Indeed, the laws of physics, sometimes referred to as the “laws of nature,” simply represent mathematical formulations of the behavior that we observe all natural phenomena to obey. For example, the central idea in Einstein's theory of general relativity is that gravity is not some mysterious, attractive force that acts across space but rather a manifestation of the geometry of the inextricably linked space and time. Let me explain, using a simple example, how a geometrical property of space could be perceived as an attractive force, such as gravity. Imagine two people who start to travel precisely northward from two different points on Earth's equator. This means that at their starting points, these people travel along parallel lines (two longitudes), which, according to the plane geometry we learn in school, should never meet. Clearly, however, these two people will meet at the North Pole. If these people did not know that they were really traveling on the curved surface of a sphere, they would conclude that they must have experienced some attractive force, since they arrived at the same point in spite of starting their motions along parallel lines. Therefore, the geometrical curvature of space can manifest itself as an attractive force. The Pythagoreans were probably the first to recognize the abstract concept that the basic forces in the universe may be expressed through the language of mathematics.

Due perhaps to the simple harmonic ratios found in music, 1:2, 2:3, 3:4, the Pythagoreans became particularly intrigued by the differences between the odd and even numbers. They associated with the odd numbers male attributes and, rather prejudiciously, also light and goodness, while they gave the even numbers female attributes and associated with them darkness and evil. Some of these prejudices toward even and odd numbers persisted for centuries. For example, the Roman scholar Pliny the Elder, who lived A.D. 23 to 79, wrote in his Historia Naturalis (a thirty-seven-volume encyclopedia on natural history): “Why is it that we entertain the belief that for every purpose odd numbers are the most effectual?” Similarly, in Shakespeare's Merry Wives of Windsor (Act V, Scene I), Sir John Falstaff says: “They say there is divinity in odd numbers, either in nativity, chance or death.” Mideastern religions produced a similar attitude. According to the Muslim tradition, the prophet Muhammad ate an odd number of dates to break his fast, and Jewish prayers often have an odd number (3, 7) of repetitions associated with them.

Besides the roles that the Pythagoreans assigned to the odd and even numbers in general, they also attributed special properties to some individual numbers. The number 1, for example, was considered the generator of all other numbers and thus not regarded as a number itself. It was also assumed to characterize reason. Geometrically, the number 1 was represented by the point, which in itself was assumed to be the generator of all dimensions. The number 2 was the first female number and also the number of opinion and of division. Somewhat similar sentiments are expressed in the yin and yang of the Chinese religious cosmology, with the yin representing the feminine and negative principle, like passivity and darkness, and the yang the bright and masculine principle. The number 2 is associated to this very day in many languages with hypocrisy and unreliability, as manifested by such expressions as “two-faced” (in Iranian) or “double-tongued” (in German and Arabic). The original identification of the number 2 with feminine and of 3 with masculine may have been inspired by the configurations of female breasts and male genitalia. This tentative conclusion is supported by the fact that the Konso of East Africa make the same identification. In everyday life, division into two categories is the most common: good and bad, up and down, right and left. Geometrically, 2 was expressed by the line (which is determined by two points), which has one dimension. Three was supposed to be the first real male number and also the number of harmony, since it combines unity (the number 1) and division (the number 2). To the Pythagoreans, 3 was in some sense the first real number because it has a “beginning,” a “middle,” and an “end” (unlike 2, which does not have a middle). The geometrical expression of 3 was the triangle, since three points not on the same line determine a triangle, and the area of the triangle has two dimensions.

Interestingly, 3 was also the basis for the construction of military units in the Bible. For example, in 2 Samuel 23, there is a story on the very basic unit, the “three warriors” that King David had. In the same chapter, there is a detailed count of the “thirty chiefs” who “went down to join David at the cave of Adulam,” but at the end of the count the biblical editor concludes that they were “thirty-seven in all.” Clearly, “thirty” was the definition of the unit, even if the actual number of members was somewhat different. In Judges 7, when Gideon needs to fight the Midianites, he chooses three hundred men, “all those who lap the water with their tongues.” Moving to yet larger units, in 1 Samuel 13, “Saul chose three thousand out of Israel” to fight the Philistines, who at the same time “mustered to fight with Israel, thirty thousand chariots.” Finally, in 2 Samuel 6, “David again gathered all the chosen men of Israel, thirty thousand” to fight the Philistines.

The number 4, for the Pythagoreans, was the number of justice and order. On the surface of Earth, the four winds or directions provided the necessary orientation for humans to identify their coordinates in space. Geometrically, four points that are not in the same plane can form a tetrahedron (a pyramid with four triangular faces), which has a volume in three dimensions. Another consideration that gave the number 4 a somewhat special status for the Pythagoreans was their attitude toward the number 10, or the holy tetractys. Ten was the most revered number, because it represented the cosmos as a whole. The fact that 1+2+3+4 = 10 generated a close association between 10 and 4. At the same time, this relation meant that 10 not only united the numbers representing all dimensions but also combined all the properties of uniqueness (as expressed by 1), polarity (expressed by 2), harmony (expressed by 3), and space and matter (expressed by 4). Ten was therefore the number of everything, with properties best expressed by the Pythagorean Philolaus around 400 B.C.: “sublime, potent and all-creating, the beginning and the guide of the divine concerning life on Earth.”

The number 6 was the first perfect number, and the number of creation. The adjective “perfect” was attached to numbers that are precisely equal to the sum of all the smaller numbers that divide into them, as 6 = 1+2+3. The next such number, incidentally, is 28 = 1+2+4+7+14, followed by 496 = 1+2+4+8+16+31+62+124+248; by the time we reach the ninth perfect number, it contains thirty-seven digits. Six is also the product of the first female number, 2, and the first masculine number, 3. The Hellenistic Jewish philosopher Philo Judaeus of Alexandria (ca. 20 B.C.–ca. A.D. 40), whose work brought together Greek philosophy and Hebrew scriptures, suggested that God created the world in six days because six was a perfect number. The same idea was elaborated upon by St. Augustine (354 − 430) in The City of God: “Six is a number perfect in itself, and not because God created the world in six days; rather the contrary is true: God created the world in six days because this number is perfect, and it would remain perfect, even if the work of the six days did not exist.” Some commentators of the Bible regarded 28 also as a basic number of the Supreme Architect, pointing to the 28 days of the lunar cycle. The fascination with perfect numbers penetrated even into Judaism, and their study was advocated in the twelfth century by Rabbi Yosef ben Yehudah Aknin in his book, Healing of the Souls.

I have deliberately left the number 5 for last in giving these examples of the Pythagoreans' attitude to numbers, because this number also leads us to the origins of the Golden Ratio. Five represented the union of the first female number, 2, with the first male number, 3, and as such it was the number of love and marriage. The Pythagoreans apparently used the pentagram—the five-pointed star (Figure 3)—as the symbol of their brotherhood, and they called it “Health.” The second-century Greek writer and rhetorician Lucian writes (in In Defense of a Slip of the Tongue in Greeting):

At any rate all his [Pythagoras'] school in serious letters to each other began straightway with “Health to you,” as a greeting most suitable for both body and soul, encompassing all human goods. Indeed the Pentagram, the triple intersecting triangle which they used as a symbol of their sect, they called Health.”

An imaginative (though perhaps not altogether sound) explanation for the association of the pentagram with health was suggested by A. de la Fuÿe in his 1934 book, Le Pentagramme Pythagoricien, Sa Diffusion, Son Emploi dans le Syllabaire Cuneiform (The Pythagorean pentagram, its distribution, its usage in the cuneiform spelling book). De la Fuÿe proposed that the pentagram symbolized the Greek goddess of health, Hygeia, through a correspondence of the five points of the star to a cartoon-like representation of the goddess (Figure 9).

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Figure 9

image

Figure 10

The pentagram is also closely related to the regular pentagon—the plane figure having five equal sides and equal angles (Figure 10). If you connect all the vertices of the pentagon by diagonals, you obtain a pentagram. The diagonals also form a smaller pentagon at the center, and the diagonals of this pentagon form a pentagram and a yet smaller pentagon (Figure 10). This progression can be continued ad infinitum, creating smaller and smaller pentagons and pentagrams. The striking property of all of these figures is that if you look at line segments in order of decreasing lengths (the ones marked a, b, c, d, e, fin the figure), you can easily prove using elementary geometry that every segment is smaller than its predecessor by a factor that is precisely equal to the Golden Ratioö. That is, the ratio of the lengths of a to b is phi; the ratio of b to c is phi; and so on. Most important, you can use the fact that the process of creating a series of nested pentagons and pentagrams can be continued indefinitely to smaller and smaller sizes to prove rigorously that the diagonal and the side of the pentagon are incommensurable, that is, the ratio of their lengths (which is equal to phi) cannot be expressed as a ratio of two whole numbers. What this means is that the diagonal and the side of the pentagon cannot have any common measure, such that the diagonal is some integer multiple of that measure and the side is also an integer multiple of the same measure. (For the more mathematically inclined reader, the proof is presented in Appendix 2.) Recall that numbers that cannot be expressed as ratios of two whole numbers (namely as fractions, or rational numbers) are known as irrational numbers. This proof therefore establishes the fact that phi is an irrational number.

Several researchers (including Kurt von Fritz in his article entitled “The Discovery of Incommensurability by Hippasus of Metapontum” published in 1945) suggested that the Pythagoreans are the ones who first discovered the Golden Ratio and incommensurability. These historians of mathematics argued that the Pythagorean preoccupation with the pentagram and the pentagon, coupled with the actual geometrical knowledge in the middle of the fifth century B.C., make it very plausible that the Pythagoreans, and in particular perhaps Hippasus of Metapontum, discovered the Golden Ratio and, through it, incommensurability. The arguments appear to be at least partially supported by the writings of the founder of the Syrian school of Neoplatonism, Iamblichus (ca. A.D. 245–325). According to one of Iamblichus' accounts, the Pythagoreans erected a tombstone to Hippasus, as if he were dead, because of the devastating discovery of incommensurability. In another place, however, Iamblichus reports that:

It is related of Hippasus that he was a Pythagorean, and that, owing to his being the first to publish and describe the sphere from the twelve pentagons, he perished at sea for his impiety, but he received credit for the discovery, though really it all belonged to HIM (for in this way they refer to Pythagoras, and they do not call him by his name).

In the phrase “describe the sphere from the twelve pentagons,” Iamblichus refers (somewhat vaguely, since the figure is not really a sphere) to the construction of the dodecahedron, a solid with twelve faces, each of which is a pentagon, which is one of the five solids known as Platonic solids. The Platonic solids are intimately related to the Golden Ratio, and we shall return to them in Chapter 4. In spite of the somewhat mythical flavor of these accounts, the mathematical historian Walter Burkert concludes in his 1972 book Lore and Science in Ancient Pythagoreanism that “the tradition about Hippasus, though surrounded by legend, makes sense.” The main reason for this statement is provided by Figure 10 (and Appendix 2). The conclusion that the diagonal and the side in a regular pentagon are incommensurable is based on the very simple observation that the construction of smaller and smaller pentagons can be continued indefinitely. The proof therefore would have definitely been accessible to the fifth-century B.C. mathematicians.

TO THE RATIONAL BEING ONLY THE IRRATIONAL IS UNENDURABLE

While it is certainly possible (and perhaps even likely) that incommensurability and irrational numbers were first discovered via the Golden Ratio, the more traditional view is that these concepts were discovered through the ratio of the diagonal and the side of the square. Aristotle writes in Prior Analytics: “the diagonal [of a square] is incommensurable [with the side] because odd numbers are equal even if it is assumed to be commensurate.” Aristotle alludes here to a proof of incommensurability, which I now present in full detail, because it is a beautiful example of a proof by the logical method known as reductio ad absurdum (reduction to absurdity). In fact, when in 1988 the journal The Mathematical Intelligencer invited its readers to rank a selection of twenty-four theorems according to their “beauty,” the proof I am about to present was ranked seventh.

The idea behind the ingenious method of reductio ad absurdum is that you prove a proposition simply by proving the falsity of its contradictory. The most influential Jewish scholar of the Middle Ages, Mai-monides (Moses Ben Maimon; 1135–1204), even attempted to use this logical device to prove the existence of a creator. In his monumental work, Mishne Torah (The Torah reviewed), which attempts to encompass all religious subject matter, Maimonides writes: “The basic principle is that there is a First Being who brought every existing thing into being, for if it be supposed that he did not exist, then nothing else could possibly exist.” In mathematics, reductio ad absurdum is used as follows. You start by assuming that the theorem you seek to prove true is in fact false. From that, by a series of logical steps you derive something that represents a clear logical contradiction, such as 1 = 0. You thus conclude that the original theorem could not have been false; therefore, it must be true. Note that for this method to work, you have to assume that a theorem or statement has to be either true or false—you are either reading this page right now or you are not.

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Figure 11

Examine first the square in Figure 11, in which the length of the side is one unit. If we want to find the length of the diagonal, we can use the Pythagorean theorem in any of the two right triangles into which the square is divided. Recall that the theorem states that the square of the hypotenuse (the diagonal) is equal to the sum of the squares of the two shorter sides of the triangle. If we call the length of the hypotenuse d, we have that: d2 = 12 +12, or d2 = 1+1 = 2. If we know the square of a number, we find the number itself by taking the square root. For example, if we know that the square of x is equal to 25, then x = 5 = image. From d2 = 2 we therefore find that d = image. The ratio of the diagonal to the side of a square is therefore the square root of 2. (A pocket calculator will show you that the value of the latter is equal to 1.41421356….) What we want to show now is that image cannot be expressed as a ratio of anytwo whole numbers (and therefore that it is an irrational number). Think about this for a moment: What we are about to prove is that even though we have an infinite collection of whole numbers at our disposal, no matter for how long we will search, we will never find two of them that have a ratio that is precisely equal to image. Isn't this mind-boggling?

The proof (by reductio ad absurdum) goes as follows: We start by assuming the opposite of what we want to prove, namely, we assume that image is actually equal to some ratio of two whole numbers a and bimage = a/b. If a and bhappen to have some common factors (as the numbers 9 and 6 both have a common factor of 3), then we would simplify the fraction by dividing both numbers by those factors until we get two numbers, p and q, which have no common factors. (In the above example, this will turn image into image.) Clearly, p and q cannot both be even. (If they were, they would contain a common factor 2.) Our assumption is therefore that p/q = image, where p and q are whole numbers that have no common factors. We can take the square of both sides to obtain: p2/q2 = 2. We will now multiply both sides by q2 to obtain: p2 = 2q2. Notice that the right-hand side of this equation is clearly an even number, since it is some number (q2) multiplied by 2, which always gives an even number. Since p2 is equal to this even number, p2 is an even number. But if the square of a number is even, then the number itself also must be even. (The square is simply the number multiplied by itself, and if the number were odd, its multiplication by itself would also be odd.) We therefore find that p itself must be an even number. Recall that this means that q must be odd, because p and q had no common factors. If, however, p is even, then we can write p in the form p = 2r (since any even number has 2 as a factor). The previous equation p2= 2q2can therefore be written as (simply substituting 2 r for p): (2 r)2= 2 q2, which is [since (2 r)2= (2 r) × (2 r)] 4 r2= 2 q2. Dividing both sides by 2 gives: 2 r2q2. By the same arguments we used before, this says that q2is even (since it is equal to 2 times another number) and therefore that q must be even. Note, however, that above we showed that q must be odd! Thus we have reached something that is clearly a logical contradiction, since we showed that q must be odd and even at the same time. This fact demonstrates that our initial assumption, that there exist two whole numbers, p and q, the ratio of which is equal to image, is false, thus completing the proof. Numbers like image represent a new kind of number —irrational numbers.

We can prove in a very similar way that the square root of any number that is not a perfect square (such as 9 or 16) is an irrational number. Numbers like image are irrational numbers.

The magnitude of the discovery of incommensurability and irrational numbers cannot be overemphasized. Before this discovery, mathematicians had assumed that if you have any two line segments, one of which is longer than the other, then you can always find some smaller unit of measure so that the lengths of both segments will be exact whole-number multiples of this smaller unit. For example, if one segment is precisely 21.37 inches long and the other is 11.475 inches long, then we can measure both of them in units of one thousandth of an inch, and the first one will be 21,370 such units and the second 11,475 units. Early scholars therefore believed that finding such a common smaller measure was merely a matter of patient search. The discovery of incommensurability means that for the two segments of a line cut in a Golden Ratio (such as AC and CB in Figure 2), or for the diagonal and the side of a square, or for the diagonal and side of the pentagon, a common measure is never to be found. Steven Cushing published in 1988 a short poem (in the Mathematics Magazine) that describes our natural reaction to irrationals:

Pythagoras
Did stagger us
And our reason encumber
With irrational number

We can appreciate better the intellectual leap that was required for the discovery of irrational numbers by realizing that even fractions, or rational numbers such as image represent by themselves an extremely important human discovery (or invention). The nineteenth-century mathematician Leopold Kronecker (1823–1891) expressed his opinion on this matter by saying: “God created the natural numbers, all else is the work of man.”

Much of our knowledge about the familiarity of the ancient Egyptians with fractions, for example, comes from the Rhind (or Ahmes) Papyrus. This is a huge (about 18 feet long and 12 inches high) papyrus that was copied around 1650 B.C. from earlier documents by a scribe named Ahmes. The papyrus was found at Thebes and bought in 1858 by the Scottish antiquary Henry Rhind, and it is currently in the British Museum (except for a few fragments, which turned up unexpectedly in a collection of medical papers, and which are currently in the Brooklyn Museum). The Rhind Papyrus, which is in effect a calculator's handbook, has simple names only for unit fractions, such as ½, ⅓, ¼ etc., and for ⅔ A few other papyri have a name also for ¾The ancient Egyptians generated other fractions simply by adding a few unit fractions. For example, they had image to represent ⅘ and image+image to represent imageTo measure fractions of a capacity of grain called hekat, the ancient Egyptians used what were known as “Horus-eye” fractions. According to legend, in a fight between the god Horus, the son of Osiris and Isis, and the killer of his father, Horus' eye got torn away and broke into pieces. The god of writing and of calculations, Thoth, later found the pieces and wanted to restore the eye. However, he found only pieces that corresponded to the fractions imageimage and image. Realizing that these fractions only add up to image, Thoth produced the missing fraction of image by magic, which allowed him to complete the eye.

Strangely enough, the Egyptian system of unit fractions continued to be used in Europe for many centuries. For those during the Renaissance who had trouble memorizing how to add or subtract fractions, some writers of mathematical textbooks provided rules written in verse. An amusing example is provided by Thomas Hylles's The Art of Vulgar Arithmetic, both in Integers and Fractions (published in 1600):

Addition of fractions and likewise subtraction
Requireth that first they all have like bases
Which by reduction is brought to perfection
And being once done as ought in like cases,
Then add or subtract their tops and no more
Subscribing the base made common before.

In spite of, and perhaps (to some extent) because of, the secrecy surrounding Pythagoras and the Pythagorean Brotherhood, they are tentatively credited with some remarkable mathematical discoveries that may include the Golden Ratio and incommensurability. Given, however, the enormous prestige and successes of ancient Babylonian and Egyptian mathematics, and the fact that Pythagoras himself probably learned some of his mathematics in Egypt and Babylon, we may ask: Is it possible that these civilizations or others discovered the Golden Ratio even before the Pythagoreans? This question becomes particularly intriguing when we realize that the literature is bursting with claims that the Golden Ratio can be found in the dimensions of the Great Pyramid of Khufu at Giza. To answer this question, we will have to mount an exploratory expedition in archaeological mathematics.