by Colin Pask
It remains to explain the choice of values for p. Britton, Proust, and Shnider explore various published possibilities and conclude that the table is based on p values given by the regular numbers obtained by dividing the number r by the number s, r/s, with the condition that
Regular numbers are those having an inverse that is a sexagesimal fraction with a finite number of places, something these mathematicians would have readily tabulated (see Neugebauer's text for further details). Working up through these choices for p produces the table set out on Plimpton 322. It also shows how the detected errors would naturally arise.
If the above procedure is continued through all suitable r and s values (those values satisfying the above conditions), it will produce a total of 38 sets of triples. Britton, Proust, and Shnider give a picture of the reverse side of Plimpton 322 showing vertical lines and space (they argue) for the 23 triples which could not be fitted onto the front side of the tablet. They also suggest that the part broken off from the left edge of the tablet might contain the values of x and y.
We will never know for certain whether the above construction was actually the one used for Plimpton 322, but we can be sure that it was a systematic and beautiful piece of mathematics showing an appreciation for numbers and their properties. Mathematical thinking had begun in ancient Mesopotamia and has continued ever since. There is no historical speculation involved; the evidence is clearly written down for all to see.
2.1.4 Comments
One other related calculation is worth mentioning. It is natural to ask about the right-angled triangle with side lengths 1 which leads to the triple (1, 1, √2). Tablets similar to Plimpton 322 show that this problem was considered and that the mathematicians involved knew how to calculate √2. According to Neugebauer, they found for √2 the sexagesimal value 1,24,51,10, which in our decimal system gives 1.414213 (as against 1.414214 correct to 7 figures).
These calculations suggest the use of the “false position” or iterative methods, which allow a guess for the answer to a problem to be refined. Such methods were widespread in early mathematics, and they remain part of numerical analysis to this very day. (See chapter 3 in the book by Chabert.)
It would be wrong to see Plimpton 322 as an instigator of what today we call number theory, but it does suggest questions we might expect to find in later work. For example, we can ask: Does the above construction method generate all possible Pythagorean triples? The answer is no. The complete formalism was probably not known until the time of Euclid. Coming right up to the present, we can now answer another question suggested by the study of triples of numbers in particular relationships: What happens if we use cubes or other powers instead of squares? So we ask about a3 + b3 = c3. In fact, there are no solutions (suitable integer values for a, b, and c) for the case of cubes or any higher powers; that result is embodied in the famous Fermat's last theorem.
Finally, we should note that the ancient Egyptian civilization was developing alongside that of Mesopotamia. There too a start was made on a range of mathematical investigations. Many examples can be seen on the famous Rhind papyrus (see the books by Calinger and Gillings), but in my opinion none of them could replace Plimpton 322 in the list of highly significant calculations. Interestingly, Richard Gillings presents evidence suggesting that the ancient Egyptian mathematicians had no knowledge of Pythagoras's theorem.
2.2 ARCHIMEDES GIVES US CALCULATION 3
The circle is the simplest of curves, and it is also the most symmetric, beautiful, mysterious, and captivating. As an example of the power of these properties, in chapter 5 we shall see how some of them actually set back science for over a thousand years. Evidence of the interest in circles can be found in ancient civilizations, and the references given for the previous section show that to be true for the Mesopotamian and Egyptian mathematicians. Moving on to around 300 BCE, we find Euclid in his Elements specifying only one curve (beyond the straight line) in his basic definitions:
15. A circle is a plane figure contained by one line such that all the straight lines falling upon it from one point among those lying within the figure are equal to one another.
16. And the point is called the center of the circle.
18. A diameter of the circle is any straight line drawn through the center and terminated in both directions by the circumference of the circle, and such a straight line also bisects the circle.1
We say that the distance from the center to any point on the circle is the radius r, and then the diameter has length d = 2r. We also define C as the length of the circumference and A as the area enclosed by the circle. (Incidentally, if we seek the closed curve of a given length enclosing the maximum area, the answer is a circle with C equal to the given length—just one of those wonderful properties of the circle.)
The natural question is now: How are a circle's circumference C and area A related to its basic size parameter r or d? Ancient mathematicians had empirical methods for estimating the area of a circle (see Gillings for Egyptian examples), but there is no evidence of a true mathematical relationship.
In the Elements, Euclid writes:
Book XII. Proposition 2. Circles are to one another as the squares on the diameters.
In the Elements, area is dealt with by comparing figures and showing when their areas are equal; there is no algebraic or numerical approach as we use today when saying, for example, a rectangle with side lengths s and p has area s times p. Euclid obtained his circles result by considering polygons inscribed in the circles and letting the number of sides increase indefinitely in an application of the method of exhaustion.
Today we say Euclid showed that the area of a circle is proportional to the square of its diameter, or A ∝ d2. (Readers will probably want to say that A = πr2 and C = πd = 2πr, but that is jumping ahead in the story!)
2.2.1 Enter Archimedes
Archimedes was an intellectual giant and certainly the greatest mathematician of ancient times. He was born in Syracuse in Sicily around 287 BCE. His extensive writings cover an enormous range of mathematical and scientific topics, but here we are concerned with his “Measurement of the Circle.” The original no longer exists, but rough versions (which were perhaps part of a larger document) do survive. “Measurement of the Circle” contains just three propositions, and they brilliantly answer our question about the relationship between r (or d) and C and A. All three are linked together in
Proposition 1. The area of any circle is equal to a right-angled triangle in which one of the sides about the right-angle is equal to the radius, and the base [the other side about the right-angle] is equal to the circumference.2
Notice that Archimedes works in the style of Euclid and expresses the area of a circle in terms of the area of another figure, this time a right-angled triangle. Today we express Proposition 1 as
If we can relate the circumference to the radius, then the whole problem is solved, since then we can use the above result to give A just in terms of the radius. This is exactly what Archimedes achieves, but for some reason the propositions are given in the reverse order.
2.2.2 The Great Step
Archimedes gives
Proposition 3. The circumference of any circle is three times the diameter and exceeds it by less than one-seventh of the diameter and by more than ten-seventyoneths.3
We write this as
or symbolically,
Many readers will not be able to resist saying that C = πd, so, in effect, Archimedes is telling us that
However you wish to view this result, it is a startling and brilliant one. Archimedes has told us that the circumference of a circle is a certain number times its diameter, and he has given bounds on that number. The upper bound of 22/7 has always been commonly used as a good, simple approximation for π. In modern terms, Archimedes has discovered that 3.1408 < π < 3.1429, so he has fixed the first two decimal places of π.
This defines calculation 3, Archimedes bounds π. Archimedes has essentially introduced what we now call th
e physical constant π (although he did not explicitly say that or use the symbol π), and he has shown how to find bounds on its value. The idea of finding upper and lower bounds for a quantity when an exact calculation is not available is an important one in mathematics and science. Archimedes's method of calculation is also highly innovative. All in all, this is a remarkable result, clearly worthy of inclusion in any list of important calculations.
Archimedes completes the story in “Measurement of the Circle” with
Proposition 2. The area of the circle is to the square on its diameter as 11 to 14.4
This result (A = 11⁄14d2) follows simply by using the 22/7 upper bound in equation (2.4). Thus Archimedes has shown how to relate a circle's circumference C and its area A to its diameter d.
2.2.3 Some Computational Details
Archimedes made his calculation by comparing the circumference of a circle with the perimeters of polygons fitting inside and outside the circle as in the examples shown in figure 2.2. For the four-sided polygon (a square) the reader may easily derive the bounds 2√2 < π < 4 or 2.83 < π < 4. For the six-sided polygon, or hexagon, the bounds are 3 < π < 2√3 or 3 < π < 3.3642. These are not very useful results, and it took the genius of Archimedes to see a clever way to go beyond them.
Figure 2.2. Approximating a circle using polygons with four sides (a square) and six sides (a hexagon). A circle with radius 1 has a circumference equal to 2π. Figure created by Annabelle Boag.
Archimedes began with hexagons, which give the quite-poor bounds mentioned above. Archimedes realized that he needed to use polygons with many more sides to generate accurate results. His idea was to go in steps in which the number of sides would be doubled each time. He increased the number of sides from 6 to 12, then to 24, to 48, and finally to 96, at which point he was satisfied with the results, which are given in equations (2.5) and (2.6).
What makes Archimedes's calculation so brilliant is that he showed how to use the answer for a polygon with n sides to generate the answer for a polygon with 2n sides; it was not necessary to go right back to the beginning each time. The angles in a polygon with 2n sides are half those in a polygon with n sides and this fact, together with Euclid's:
Book VI, Proposition 3. If an angle of a triangle is bisected and the straight line cutting the angle cut the base also, the segments of the base will have the same ratio as the remaining sides of the triangle,5
allowed Archimedes to find the length of a side of a 2n-sided polygon simply from the length of the n-sided one. Along the way he had to calculate square roots, and the manipulations used are equally impressive. (The reader wishing to follow the complete details should see the original working in the books by Heath and Dijksterhuis or the more modern version as set out by Stein.)
Recall that Archimedes started with a hexagon whose side length involves √3. To proceed beyond that he needed a measure of that square root and he came to
These are extremely accurate bounds on √3, and their origin is discussed in the references just given.
The innovative conceptual design and the method of working produced by Archimedes surely tell us that his calculation of circle properties is truly one of the great calculations.
Another of Archimedes's innovative calculations will be referred to in chapter 3.
2.2.4 Archimedes Has Begun the Race
With his calculation, Archimedes started what we can view as a race to determine the ratio of a circle's circumference to its diameter—a number which today we call π, following the suggestion of William Jones in 1706. The following highly selective list gives an idea of the progress over the centuries. (The results have been converted to decimal form for comparison with π = 3.14159265358…)
Date Author Approximation for π
150 Ptolemy 377/120 or 3.14167
263 Liu Hui 3.14159
480 Tsu C’ung Chi 3.1415926
800 Al-Khowarizimi 3.1416
1593 Viete to 9 decimal places
1596 Van Ceulen to 20 decimal places
1615 Van Ceulen to 35 decimal places
1700 Machin to 100 decimal places
1844 Strassnitzky & Dase to 200 decimal places
1949 Smith & Wrench to 1,120 decimal places
1973 Guilloud & Bouyer to 1,001,000 decimal places
1989 Kanada & Tamura to 1,071,741,799 decimal places
(You should consult the bible for this subject—the source book edited by Berggren, Borwein, and Borwein—for details of these and many other calculations, and in fact, for anything you might ever want to know about π!)
The above (highly selective) list tells you that the search for π was an international affair. Archimedes's polygon method was carried to extremes by people like Van Ceulen whose result was reputedly carved onto his tombstone. Calculations by Machin and others represent the start of a new era in which mathematical analysis and calculus were used to present new expressions for calculating π, which has led to a whole research field providing super-convergent series. I just cannot leave out two early beautiful results (even though they are hopeless from a computational point of view):
John Wallis in 1655:
James Gregory in 1670:
You will notice that early results gave π as a fraction like 22/7. This raises the question of whether there is actually an exact expression for π of the form n/m where n and m are both integers; that is, whether π is a rational number. A negative answer was proven by Lambert in 1768, and in 1882 Lindeman proved that π is not just irrational, it is also transcendental. That also solved an old problem, taking us back to Euclid and his method of dealing with areas in terms of equivalent figures. Using the algebraic theory related to geometric constructions, Lindeman's result tells us that Euclid could never give a ruler-and-compass construction of a square with the same area as a given circle. Thus the old squaring-of-the-circle problem was finally put to rest (although cranks and eccentrics have never given up the quest).
Look at almost any area of science, and you will find π in the equations and formulas. Few constants could be more important or better known, and Archimedes told us how to evaluate it. In so doing he introduced the idea of bounding a quantity and demonstrated the power of iterative methods, two innovative and outstanding contributions to mathematics.
2.3 A DIFFICULT DECISION
I suspect that many readers will be asking: Is that it? Nothing more from the ancient world? I confess that it has been difficult to stop at those two calculations, but for me they are supreme examples of what early mathematicians were achieving. I feel particularly bad about leaving out material from the ancient Chinese and Indian mathematicians, but I did set a difficult limit of fifty calculations for the whole book. (I suggest that readers interested in this period consult the books listed in the bibliography; I particularly recommend the book by Joseph, who has done much to restore balance in the history of mathematics.)
in which we look at four steps on the way to our modern mathematical world.
Mathematics from the ancient world was preserved by the Arabs, and after hundreds of years, it became available to mathematicians in what we now call “the West.” The following four examples show how that process began a progression of thinking and ideas that finally led to the mathematics we know today.
3.1 CALCULATION 4: FIBONACCI SHOWS US HOW TO DO IT
The calculations in this book are presented using our familiar representation of numbers, which uses the symbols 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, and arithmetic to base 10. Thus 463 means 4 hundreds, 6 tens, and 3 units. Early mathematicians used other approaches, and in section 2.1 we saw the Mesopotamians using base 60. Archimedes used the very clumsy Greek representation of numbers involving a system of letters to indicate numbers (see Dijksterhuis, chapter 3, section 0.6). Equally awkward to use were the Roman numerals, which today we use in a decorative way as when we write MMXIV for the year 2014. Our modern number system has its origins in India and, like much other ancient knowledge, it was recorded and n
urtured in the Arab world. After a long period of comparative inaction, the growth of learning and science in most of Europe only began again when that knowledge was translated from Arabic into Latin, ready for western scholars to make use of it. For arithmetic and the art of calculation, we can assign two vitally important dates: 1202 and 1228.
The Prologue of the great book Liber Abbaci opens with:
Here begins the Book of Calculation Composed by Leonardo Pisano, Family Bonaci In the Year 1202.
It is this book, Liber Abbaci, The Book of Calculation, that changed the way people did their mathematical work. The second edition, published in 1228, is still in use today.
3.1.1 The Author
The exact details of Leonardo's birth are not known, but as his name indicates, he grew up in Pisa. His father was involved in commerce, and naturally, Leonardo was schooled in basic mathematics and its uses. However, the event that ultimately changed the course of history was his father's posting to Bugia (in modern Algeria) as a diplomatic representative of Pisa and contact for trade with Africa. The young Leonardo went to live with his father in Bugia and also traveled around the region. So it was that he came to learn about and appreciate the accounting and mathematical methods used by the Arabs and trading nations around the Mediterranean and far beyond. Obviously Leonardo was a talented young man because he absorbed all he could and became the foremost mathematician of his time. The publication of his Liber Abbaci marks a turning point in the spread of mathematics and its applications.
Today Leonardo Pisano is better known as Fibonacci, a nickname introduced in 1838 by the historian Guillaume Libri. (I will use Fibonacci here, as most people are more familiar with that name. His story is given in the recent biography by Keith Devlin and in the older, little book by Joseph Gies and Frances Gies.)