are the values most often met in antiquity.
For example, in the Old Testament (I Kings vii.23, and 2 Chronicles iv.2), we find the following verse:
“Also, he made a molten sea of ten cubits from brim to brim, round in compass, and five cubits the height thereof; and a line of thirty cubits did compass it round about.”
The molten sea, we are told, is round; it measures 30 cubits round about (in circumference) and 10 cubits from brim to brim (in diameter); thus the biblical value of π is 30/10 = 3.
The Book of Kings was edited by the ancient Jews as a religious work about 550 B.C., but its sources date back several centuries. At that time, π was already known to a considerably better accuracy, but evidently not to the editors of the Bible. The Jewish Talmud, which is essentially a commentary on the Old Testament, was published about 500 A.D. Even at this late date it also states “that which in circumference is three hands broad is one hand broad.”
The molten sea as reconstructed by Gressman from the description in 2 Kings vii.2
In early antiquity, in Egypt and other places, the priests were often closely connected with mathematics (as custodians of the calendar, and for other reasons to be discussed later). But as the process of specialization in society continued, science and religion drifted apart. By the time the Old Testament was edited, the two were already separated. The inaccuracy of the biblical value of π is, of course, no more than an amusing curiosity. Nevertheless, with the hindsight of what happened afterwards, it is interesting to note this little pebble on the road to confrontation between science and religion, which on several occasions broke out into open conflict, and about which we shall have more to say later.
Returning to the determination of π by direct measurement using primitive equipment, it can probably safely be said that it led to values no better than (4).
From now on, man had to rely on his wits rather than on ropes and stakes in the sand. And it was by his wits, rather than by experimental measurement, that he found the circle’s area.
* * *
THE ancient peoples had rules for calculating the area of a circle. Again, we do not know how they derived them (except for one method used in Egypt, to be described in the next chapter), and once more we have to play the game “How do you do it with their knowledge” to make a guess. The area of a circle, we know, is
where r is the radius of the circle. Most of us first learned this formula in school with the justification that teacher said so, take it or leave it, but you better take it and learn it by heart; the formula is, in fact, an example of the brutality with which mathematics is often taught to the innocent. Those who later take a course in the integral calculus learn that the derivation of (5) is quite easy (see figure below). But how did people calculate the area of a circle almost five millenia before the integral calculus was invented?
Calculation of the area of a circle by integral calculus. The area of an elementary ring is dA =2πρdρ; hence the area of the circle is
They probably did it by a method of rearrangement. They calculated the area of a rectangle as length times width. To calculate the area of a parallelogram, they could construct a rectangle of equal area by rearrangement as in the figure below, and thus they found that the area of a parallelogram is given by base times height. The age of rigor that came with the later Greeks was still far away; they did not have to know about congruent triangles to be convinced by the “obvious” validity of the rearrangement.
The parallelogram and the rectangle have equal areas, as seen by cutting off the shaded triangle and reinserting it as indicated.
Determination of the area of a circle by rearrangement. The areas of the figures (b), (c), (d) equal exactly double the area of circle (a).
So now let us try to use the general idea of rearrangement as in the figure above to convert a circle to a parallelogram of equal area. We are still using sticks to draw pictures in the sand, but this time we do this only to help our imagination, not to perform an actual measurement.
We first cut up a circle into four quadrants as in (a) above, and arrange them as shown in figure (b). Then we fill in the spaces between the segments by four equally large quadrants. The outline of the resulting weird figure is vaguely reminiscent of a parallelogram. The length of the figure, measured along the circular arcs, is equal to the circumference of the original circle, 2πr. What we can say with certainty is that the area of this figure is exactly double the area of the original circle.
If we now divide the circle not into four, but into very many segments, our quasi-parallelogram (c) will resemble a parallelogram much more closely; and the area of the circle is still exactlly one half of the quasi-parallelogram (c).
The rearrangement method used in a 17th century Japanese document.4
On continuing this process by cutting up the original circle into a larger and larger number of segments, the side formed by the little arcs of the segments will become indistinguishable from a straight line, and the quasi-parallelogram will turn into a true parallelogram (a rectangle) with sides 2πr and r. Hence the area of the circle is half of this rectangle, or πr2.
The same construction can be seen in the Japanese document above (1698). Leonardo da Vinci also used this method in the 16th century. He did not have much of a mathematical education, and in any case, he could use little else, for Europe in his day, debilitated by more than a millenium of Roman Empire and Roman Church, was on a mathematical level close to that achieved in ancient Mesopotamia. It seems probable, then, that this was the way in which ancient peoples found the area of the circle.
And that should be our last speculation. From now on, we can rely on recorded history.
2
THE BELT
Accurate reckoning — the entrance into the knowledge of all existing things and all obscure secrets.
AHMES THE SCRIBE
17th century B.C.
MAN is not the only animal that uses tools on his environment; so do chimpanzees and other apes (also, some birds). As long as man was a hunter, the differences between the naked ape and the hairy apes was not very radical. But roughly about 10,000 B.C., the naked ape learned to raise crops and to tame other animals, and thereby he achieved something truly revolutionary: Human communities could, on an average, produce so much more food above the subsistence minimum that they could free a part of their number for activities not directly related to the provision of food and shelter.
This Great Agricultural Revolution first took place where the geographical conditions were favorable: Not in the north, where the winters were long and severe, and the conditions for farming generally adverse; nor in the tropics, where food was plentiful, clothing unnecessary, shelter easily available, and therefore no drastic need for improvement; but in the intermediate belt, where conditions were sufficiently adverse to create pressures for change, yet not so adverse as to foil the attempts of farming and livestock raising.
This intermediate belt stretched from the Mediterranean to the Pacific. The Great Revolution first took place in the big river valleys of Mesopotamia; later the Belt stretched from Egypt through Persia and India to China. States developed. Specialists came into being. Soldiers. Priests. Administrators. Traders. Craftsmen. Educators.
And Mathematicians.
The hunters had neither time nor need for ratios, proportionalities or conic sections. The new society needed surveyors and builders, navigators and timekeepers (astronomers), accountants and stock-keepers, planners and tax collectors, and, yes, mumbo-jumbo men to impress and bamboozle the uneducated and oppressed. This was the fertile ground in which mathematics flourished; and it is therefore not surprising that the cradle of mathematics stood in this Belt.
* * *
SINCE Mesopotamia was the first region of the Belt where the agricultural revolution occurred and a new society took hold, one would expect Babylonian mathematics to be the first and most advanced. This was indeed the case; the older literature on the history of mathematics ofte
n saw the Egyptians as the founders of mathematics, but this was due to the fact that more and earlier Egyptian documents than any others used to be available. The research of the last few decades has changed this, and as a small sidelight, we find a better approximation for π in Mesopotamia than in Egypt.
One of the activities for which the new society freed some of its members was, unfortunately, organized warfare, and the various peoples inhabiting the region at different times, Sumerians, Babylonians, Assyrians, Chaldeans and others, warred against each other as well as against outsiders such as Hittites, Scythians, Medes and Persians. The city of Babylon was not at all times the center of this culture, but the mathematics coming from this region is simply lumped together as “Babylonian.”
In 1936, a tablet was excavated some 200 miles from Babylon. Here one should interject that the Sumerians were first to make one of man’s greatest inventions, namely, writing; through written communication, knowledge could be passed from one person to others, and from one generation to the next and future ones. They impressed their cuneiform (wedge-shaped) script on soft clay tablets with a stylus, and the tablets were then hardened in the sun. The mentioned tablet, whose translation was partially published only in 1950,5 is devoted to various geometrical figures, and states that the ratio of the perimeter of a regular hexagon to the circumference of the circumscribed circle equals a number which in modern notation is given by 57/60 + 36/(60)2 (the Babylonians used the sexagesimal system, i.e., their base was 60 rather than 10).
The Babylonian value of π.
The Babylonians knew, of course, that the perimeter of a hexagon is exactly equal to six times the radius of the circumscribed circle, in fact that was evidently the reason why they chose to divide the circle into 360 degrees (and we are still burdened with that figure to this day). The tablet, therefore, gives the ratio 6r/C, where r is the radius, and C the circumference of the circumscribed circle. Using the definition π = C/2r, we thus have
which yields
i.e., the value the Babylonians must have used for π to arrive at the ratio given in the tablet. This is the lower limit of our little thought experiment here, and a slight underestimation of the true value of π.
* * *
MORE is known about Egyptian mathematics than about the mathematics of other ancient peoples of the pre-Hellenic period. Not because they had more of it, nor because more of their documents have been found, but because the key to the Egyptian hieroglyphs was discovered much earlier than for the other cultures. In 1799, the Napoleonic expedition to Egypt found a trilingual tablet at Rosetta near Alexandria, the so-called Rosetta stone. Its message was recorded in Greek, Demotic and Hieroglyphic. Since Greek was known, the code was cracked, and decipherment of the Egyptian hieroglyphs proceeded rapidly during the last century. The Babylonian tablets are more durable, and tens of thousands have been unearthed; the university libraries of Columbia and Yale, for example, have large collections. However, an analogous trilingual stone in Persian, Medean and Assyrian was not deciphered until about 100 years ago (with Persian known, and its writing system related to Babylonian), and as far as the history of mathematics is concerned, substantial progress in deciphering tablets in cuneiform script was not made until the 1930’s. Even now, a large quantity of the available tablets await investigation.
The oldest Egyptian document relating to mathematics, and for that matter, the oldest mathematical document from anywhere, is a papyrus roll called the Rhind Papyrus or the Ahmes Papyrus. It was found at Thebes in a room of a ruined building and bought by a Scottish antiquary, Henry Rhind, in a Nile resort town in 1858. Four years later, it was purchased from his estate by the British Museum, where it is now, except for a few fragments which unexpectedly turned up in 1922 in a collection of medical papers in New York, and which are now in the Brooklyn Museum.
Histories like these make one wonder how many such priceless documents have been used up by the Arabs as toilet papyri. The sad story of how some of these papyri and tablets have survived millenia only to be rendered worthless by the excavators of our time is told by Neugebauer.6
The Ahmes Papyrus contains 84 problems and their solutions (but often no hint on how the solution was found). The papyrus is a copy of an earlier work and begins thus:7
Accurate reckoning. The entrance into the knowledge of all existing things and all obscure secrets. This book was copied in the year 33, in the 4th month of the inundation season, under the King of Upper and Lower Egypt “A-user-Re,” endowed with life, in likeness to writings made of old in the time of the King of Upper and Lower Egypt Ne-mat’et-Re. It is the scribe Ahmes who copies this writing.
From this, Egyptologists tell us, we know that Ahmes copied the “book” in about 1650 B.C. The reference to Ne-mat’et-Re dates the original to between 2,000 and 1800 B.C., and it is possible that some of this knowledge may have been handed down from Imhotep, the man who supervised the building of the pyramids around 3,000 B.C.
A typical problem (no. 24) runs like this:
A heap and its 1/7 part become 19. What is the heap?
The worked solution accompanying the problem is this:
Assume 7.
(Then) 1 (heap is) 7
(and) 1/7 (of the heap is) 1
(making a) Total 8
(But this is not the right answer, and therefore)
As many times as 8 must be multiplied to give 19, so many times 7 must be multiplied to give the required number.
The text then finds (by rather complicated arithmetic) the solution
Today we would solve the problem by the equation
where x is the “heap.” The Egyptian method, called regula falsi, i.e., assuming a (probably wrong) solution and then correcting it by proportionality, is no longer used in high school algebra. However, one version of regula falsi, called scaling, is still very advantageous for some electrical circuits (see figure below).
But the problem of interest here is no. 50. Here Ahmes assumes that the area of a circular field with a diameter of 9 units is the same as the area of a square with a side of 8 units. Using the formula for the area of a circle A = πr2, this yields
and hence the Egyptian value of π was
a value only very slightly worse than the Mesopotamian value 3 1/8, and in contrast to the latter, an overestimation. The Egyptian value is much closer to 3 1/6 than to 3 1/7, suggesting that it was neither obtained nor checked by experimental measurement (which, as here, would have given a value between 3 1/7 and 3 1/8).
20th century version of Ahmes’ method. The calculation of the currents due to a known voltage V in the network above becomes very cumbersome when the circuit is worked from left to right. So one assumes a (probably wrong) current of 1 ampere in the last branch and works the circuit from right to left (which is much easier); this ends up with the wrong voltage, which is then corrected and all currents are scaled in proportion. Just as was done by the Egyptians in 2,000 B.C.
The circuit above is also a mathematical curiosity for another reason. If all its elements are 1 ohm resistors, and the current in the last branch is 1 ampere, then the voltages across the resistors (from right to left) are Fibonacci numbers (1, 1, 2, 3, 5, 8, 13, 21, 34, 55,…, each new member being the sum of the last two).
Egyptian method of calculating π.
How did the Egyptians arrive at this weird number? Ahmes provides a hint in Problem 48. Here the relation between a circle and the circumscribed square is compared. Ahmes forms an (irregular) octagon by trisecting the sides of a square with length 9 units (see figure above) and cutting off the corner triangles as shown. The area of the octagon ABCDEFGH does not differ much from the area of the circle inscribed in the square, and equals the area of the five shaded squares of 9 square units each, plus the four triangles of 4½ square units each. This is a total of 63 square units, and this is close to 64 or 82. Thus the area of a circle with diameter 9 approximately equals 64 square units, i.e., the area of a square with side 8, which, as before, will lead to th
e value π = 4 × (8/9)2.
Ahmes has, of course, swindled twice: once in setting the area of the octagon equal to the area of the circle, and again in setting 63 ≈ 64. It is, however, noteworthy that these two approximations partially compensate for each other. Indeed, for a square of side a, the area of the octagon is 7(a/3)2, and this is to equal p2, where p is the side of the square with equal area. If we now swindle only once by setting the area of the octagon equal to that of the circle, then
and from the first and last expressions we find π = 28/9 = 3 1/9. This is a much worse approximation that Ahmes’, confirming the time-honored rule that the last error in a calculation should cancel out all the preceding ones.
* * *
THE agricultural revolution in the Indian river valleys probably took place at roughly the same time as along the Nile valley and the valley of the Euphrates and Tigris (Mesopotamia). The apparent delay of Indian mathematics behind that of Babylon, Egypt and Greece may well be simply due to our ignorance of early Indian history. There is much indirect evidence of Hindu mathematics. Like everybody else in the Belt, they knew Pythagoras’ Theorem long before Pythagoras was born, and there is evidence that Hindu astronomy was at a highly advanced level. Direct records, however, have been lost, and the first available documents are the Siddhantas or systems (of astronomy), published about 400 A.D., though the knowledge contained in them is of course much older.
One of the Siddhantas, published in 380 A.D., uses the value
which differs little from the sexagesimal value
used by the Greeks much earlier.
Early Hindu knowledge was summarized by Aryabhata in the Aryabhatiya, written in 499 A.D. This gives the solutions to many problems, but usually without a hint of how they were found. One statement is the following:8
A History of Pi Page 2
