A History of Pi Page 3

  Add 4 to 100, multiply by 8, and add 62,000. The result is approximately the circumference of a circle of which the diameter is 20,000.

  This makes

  as in the Siddhanta. The same value is also given by Bashkara (born 1114 A.D.), who calls the above value “exact,” in contrast to the “inexact” value 3 1/7.

  It is highly likely9 that the Hindus arrived at the value above by the Archimedean method of polygons, to which we shall come in Chapter 6. If the length of the side of a regular polygon with n sides inscribed in a circle is s(n), then the corresponding length for 2n sides is

  Starting (naturally) with a hexagon, progressive doubling leads to polygons of 12, 24, 48, 96, 192 and 384 sides. On setting the diameter of the circle equal to 100, the perimeter of the polygon with 384 sides is found to be the square root of 98694, whence

  which is the value given by Arayabatha.

  The Hindu mathematician Brahmagupta (born 598 A.D.) uses the value

  which is probably also based on Archimedean polygons. It has been suggested10 that since the perimeters of polygons with 12, 24, 48 and 96 sides, inscribed in a circle with diameter 10, are given by the sequence

  the Hindus may have (incorrectly) assumed that on increasing the number of sides, the perimeter would ever more closely approach the value √1000, so that

  * * *

  WHAT has been said about the ancient history of mathematics in India, and our ignorance of it, applies equally well to the Pacific end of the Belt, China. However, study of this subject has recently been facilitated by the impressive work of Needham.11

  As in the West (see here), the value π ≈ 3 was used for several centuries; in 130 A.D., Hou Han Shu used π = 3.1622, which is close to π = √10. (The Chinese were singular among the ancient peoples in that they used the decimal system from the very beginning.) A document in 718 A.D. takes π = 92/29 = 3.1724 … Liu Hui (see figure here), in 264 A.D., used a variation of the Archimedean inscribed polygon; using a polygon of 192 sides, he found

  and with a polygon of 3,072 sides, he found π = 3.14159.

  18th century explanation of Liu Hui’s method (264 A.D.) of calculating the approximate value of π.12

  Determination of the diameter and circumference of a walled city from a distant observation point (1247 A.D.).13

  In the 5th century, Tsu Chung-Chih and his son Tsu Keng-Chih found

  an accuracy that was not attained in Europe until the 16th century.

  Not too much should be made of this, however. The number of decimal places to which π could be calculated was, from Archimedes onward, purely a matter of computational ability and perseverance. Some years ago, it was only a matter of computer programming know-how; and today it is, in principle, no more than a matter of dollars that one is willing to spend for computer time. The important point is that the Chinese, like Archimedes, had found a method that, in principle, enabled them to calculate π to any desired degree of accuracy.

  Yet the high degree of accuracy that the Chinese attained is significant in demonstrating that they were far better equipped for numerical calculations than their western contemporaries. The reason was not that they used the decimal system; the decimal system by itself proves nothing except that nature was a poor mathematician in giving us ten fingers (instead of a number that has more integral factors, like twelve). As human language shows, everybody used base ten for numeration (or its multiple, 20, as in French and Danish).

  But the Chinese discovered the equivalent of the digit zero. Like the Babylonians, they wrote numbers by digits multiplying powers of the base (10 in China, 60 in Mesopotamia), just like we do. But where the corresponding power of 10 was missing (102 has no tens), they left a space. The Hindus later used a circle for the digit zero (0), and this reached Europe via the Arabs and Moors only in the late Middle Ages (Italy) and the early Renaissance (Britain). An edict of 1259 A.D. forbade the bankers of Florence to use the infidel symbols, and the University of Padua in 1348 ordered that price lists of books should not be prepared in “ciphers,” but in “plain” letters (i.e., Roman numerals).14 But until the infidel digit 0 was imported, few men in Europe had mastered the art of multiplication and division, let alone the extraction of square roots which was needed to calculate π in the Archimedean way.

  * * *

  THE Belt, as we have argued, was the region conducive to the Great Agricultural Revolution that turned human communities from packs of hunters to societies with surplus productivity, freeing some of its members for activities other than provision of food. If this Belt stretched from the valley of the Nile to the Pacific, why should it not also cross America?

  Indeed, it did. The agricultural revolution had taken place, at times comparable to the origins of the Afro-Asian Belt, in parts of Central and South America, where the impressive civilizations of the Aztecs, the Maya, the Chibcha, the Inca and others had grown up. A Maya ceremonial center in Guatemala, by carbon 14 tests, dates back to 1182 ( ± 240) B.C,15 and agriculture had of course begun much earlier; in fact, it was the American Indian who first domesticated two of our most important crops — corn (maize) and the potato.16

  The most advanced of these cultures was that of the Maya, who established themselves in the Yucatan peninsula of Mexico, parts of Guatemala, and western Honduras. Judging by the Maya calendar, Maya astronomy must have been as good as that of early Egypt, for by the first century A.D. (from which time we have some dated documents), they had developed a remarkably accurate calendar, based on an ingenious intermeshing of the periods of the Sun, the Moon and the Great Star noh ek (Venus). The relationship between the lunar calendar and the day count was highly accurate: The error amounted to less than 5 minutes per year. The Julian calendar, which had been introduced in Rome in the preceding century, and which some countries of the Old World retained up to the 20th century, was in error by more than 11 minutes per year. From this alone it does not follow that the Maya calendar was more accurate than the Julian calendar (as some historians have concluded), but the point has some bearing, albeit very circumstantial, on the value that the Maya might have used for π, and we shall digress to examine the history of our calendar.

  A calendar is a time keeping device. If it is to be of practical use beyond a prediction of when to observe a religious holiday, it must be matched to the seasons of the year (i.e., to the earth’s orbit round the sun) very accurately: If we lose an hour or two per calendar year with respect to the time it takes the earth to return to the same point on its orbit, this does not seem a significant error; yet the difference will accumulate as the years go by, and eventually one will find that at 12 noon on a summer day (by this calendar) it is not only dark, but it is snowing as well.

  Astronomy for calendar making was therefore one of the earliest activities involving mathematics in all ancient societies. In Egypt and Babylon, as well as in Maya society, the priests had a monopoly of learning. In all three societies the priests were the astronomers, calendar makers and time keepers.

  The Babylonians first took a year equal to 360 days (presumably because of their sexagesimal system and their 360° circle), but later they corrected this by 5 additional days. This value was also adopted by the Egyptians, and it was to be the value used by the Maya. To correct this value to a fraction of a day, it was necessary to observe the movement of the other stars on the celestial sphere. The Maya watched the planet Venus, the Egyptians the fixed star Sirius. Owing to the precession and nutation of the earth’s axis, Sirius is not really fixed on the celestial sphere (tied to the coordinates of the terrestrial observer), but has, as viewed from the earth, a motion of its own. The Egyptians found that Sirius moved exactly one day ahead every 4 years, and this enabled them to determine the length of one year as 365¼ days. One of the Greco-Egyptian rulers of Egypt in the Hellenic age, Ptolemy III Euergetes, a mathematician whom we shall meet again, issued the following decree in 238 B.C.:

  Since the Star [Sirius] advances one day every four years, and in order that the hol
idays celebrated in the summer shall not fall into winter, as has been and will be the case if the year continues to have 360 and 5 additional days, it is hereby decreed that henceforth every four years there shall be celebrated the holidays of the Gods of Euergetes after the 5 additional days and before the new year, so that everyone might know that the former shortcomings in reconing the seasons of the year have henceforth been truly corrected by King Euergetes.17

  But by this time Egyptian priesthood had become more interested in religious mumbo jumbo than in science, and they sabotaged the enforcement of Ptolemy’s decree. Ironically, it fell to the Roman adventurer, warlord and vandal Gaius Julius Caesar to bring about the adoption of the leap year. Alexandria, in the 3rd to 1st centuries B.C., was the intellectual center of the ancient world, the like of which had not been seen before and was not to be seen again until the rise of Cambridge and the Sorbonne. In 47 B.C., Caesar’s hordes ransacked the city and burned its libraries, and Caesar, during whatever time he had left between his romance with Cleopatra and contemplating further conquests, managed to get acquainted with the astounding achievements of Alexandrian astronomers. He took one of them, Sosigenes, back to Rome and inaugurated the Julian calendar as of January 1, 45 B.C. The new calendar introduced Ptolemy’s leap year, giving a year an average duration of 365.25 days. But the true duration of the earth’s orbit about the sun is about 365.2422 days. (Actually, the earth is subject to all kinds of perturbations, and today we do not use its orbit as a standard any more. “Atomic clocks” run much more accurately than the earth. When such an atomic clock appears to be x seconds late at the end of a tropical year, we say that the earth has completed its orbit x seconds early.)

  The difference between the two values, a little over 11 minutes per year, accumulated over the centuries and once more threatened to put the date out of date. In 1582, Pope Gregory XIII decreed that the extra day of a leap year was to be omitted in years that are divisible by 100, unless also divisible by 400 (i.e., omitted in 1800, 1900, but not in 2000). This is the calendar we are using now. It was soon adopted by the Catholic countries, but the others thought it “better to disagree with the Sun than to agree with the Pope,” and Britain, for example, did not adopt it until 1752. By that time the British had slipped behind 11 days, and when they were simply omitted to catch up with the rest of Europe, many Britons were outraged, accusing the government of conspiring to shorten their lives by 11 days and to rob them of the interest on their bank accounts. Russia held on to the Julian calendar until the October Revolution, which took place in November (1917).

  Getting further and further away from the Maya, we may note that our present calendar is far from satisfactory. Not because it is still off by 2 seconds per year, for these accumulate to a full day only once in 3,300 years; but because our present calendar is highly irregular: A date falls on a different day of the week every year, and the months (even the quarters) have different lengths, which complicates, among other things, accounting and other business administration. In 1923 the League of Nations established a Committee for Calendar Reform, which, predictably, achieved nothing, and the same result has hitherto been achieved by the United Nations. Of the hundreds of submitted proposals, UNESCO in 1954 recommended the so-called “World Calendar” for consideration by the UN General Assembly. The proposal does away with both of the mentioned disadvantages of our present calendar, yet does not change it so radically as to cause widespread confusion. Most governments agreed to the proposal in principle, but some (including the US) considered it “premature” and the matter is still being “considered.” The UN, a grotesque assembly of propaganda-bent hacks, has found itself unable to condemn international terrorism by criminals, much less to reform the calendar.

  It is against this background of difficulties with calendar making that the achievements of the Maya must be viewed. True, their calendar year, like that of the Babylonians and Egyptians, amounted to 365 days, so that their religious festivals drifted with respect to the natural seasons. Like the Babylonians and Egyptians, they must have known that they were gaining one day every four years, but like the Egyptians, they evidently preferred keeping their religious holidays intact to using the calendar as a time keeper for sowing, harvesting and other activities geared to the periodicity of nature.

  Maya glyph denoting position of month in half-year period.

  If we do not require a calendar to be geared to a tropical year (earth’s orbit), but only that it be geared to some part of the celestial clock, then the Maya calendar was more accurate than the Julian calendar, more accurate than the Egyptian (solar) calendar, and more accurate than the Babylonian (solar-lunar) calendar; it intermeshed the “gear wheels” of Sun, Moon and Venus, and was based on a more accurate “gear ratio” than the other calendars, repeating itself only once in 52 years.

  It is unthinkable that a people as advanced in astronomy should not have come across the problem of calculating the circle ratio. If their astronomy was as advanced as that of the Egyptians, is it not reasonable to assume that the Maya value of π was as good as that of the Egyptians?

  It is not. It is reasonable to assume that it was far better. For the Maya were incomparably better equipped for numerical calculations than the Egyptians were; they had discovered the zero digit and the positional notation that had escaped the genius of Archimedes, and that held up European arithmetic for a thousand years after the Maya were familiar with it. They used the vigesimal system (base 20), the digits from 1 to 19 being formed by combinations of ones (dots) and fives (bars), as shown in the figure below.

  Maya digits

  Maya numerical notation (read from top to bottom)

  Any positionally expressed number abcd.ef using base x denotes the number

  The Maya notation fits this pattern with x = 20 and the digits shown here, with an exception for the second-order digit. In a pure vigesimal system, this would multiply 202 = 400, but for reasons connected with their calendar, the Maya used this position to multiply the number 18 × 20 or 202 – 40. Also, it is not known whether they progressed beyond a “vigesimal point” to negative powers of 20, i.e. to vigesimal fractions. Examples of the Maya notation are given in the figure above. However this may be, it is clear that with a positional notation closely resembling our own of today, the Maya could out-calculate the Egyptians, the Babylonians, the Greeks, and all Europeans up to the Renaissance.

  The Chinese, who had also discovered the digit zero and the positional notation utilizing it, had found the value of π to 8 significant figures a thousand years before any European. The Maya value might have been close to that order.

  But we can only guess. The Maya civilization went the way of other civilizations; after it peaked, it decayed. About the 7th century A.D., the Maya started to desert their temple cities, and within a century or two these splendid cities were abandoned, no one knows why. Civil strife set in, and later they were conquered by the Aztecs. By the time the Spaniards arrived, they were far down from their classical age.

  There were some records, of course. The Maya wrote books on long strips of bark or parchment, folded like a screen. How many of these dealt with their mathematics, geometry and the circle ratio, we shall never know. In the 1560’s, Diego de Landa, Bishop of Yucatan, burned the literature of the Maya on the grounds that “they contained nothing in which there were not to be seen superstition and lies of the devil.”18 What remained was burned by the natives who had been converted to the Bishop’s religion of love and tolerance.

  Today the American Indians, in their quest for identity and self-respect, complain that even the name of their people was given them “by some honkey who landed here by mistake.”

  They might add that the Red Man had made the great discovery of positional notation employing the digit zero a thousand years before the Palefaces; at a time when Spain was a colony of a benighted empire, and when the ancestors of the Anglo-Saxons were illiterate hunters in the virgin forests of continental Europe, no one knows exactly
where.

  3

  THE EARLY GREEKS

  O King, for traveling over the country, there are royal roads and roads for common citizens; but in geometry there is one road for all.

  MENAECHMUS

  (4th century B.C.), when his pupil Alexander the Great asked for a shortcut to geometry.

  IN following the Belt along its length, we have lost the thread of time; and we now return to the Eastern Mediterranean, where the history of mankind went through a few inspiring centuries associated with the ancient Greeks. The Golden Age of this time was the age of the University of Alexandria; but just as the Alexandrian sun continued to glimmer some centuries after the Romans had sacked its places of learning and burned its libraries, so there was already some light before the city and its University were founded.

  This was a time when the Greeks first held their own against the then all-mighty Persian Empire at Marathon (490 B.C.) and then defeated the Persians at Plataea (479 B.C.). This was also the time when democratic government developed; a slaveowners’ democracy, yes, but a democracy. The next 150 years saw the confrontation of Athens and Sparta, the thinkers against the thugs. The thugs always win, but the thinkers always outlast them.

  In mathematics, which is but a mirror of the society in which it thrives or suffers, the pre-Athenian period is one of colorful men and important discoveries. Sparta, like most militaristic states before and after it, produced nothing. Athens, and the allied Ionians, produced a number of works by philosophers and mathematicians; some good, some controversial, some grossly overrated.

  As far as the history of π is concerned, there were four men of this period who had some bearing on the problem: Anaxagoras, Antiphon, Hippocrates and Hippias.