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'Counting is a symbolic process employed only by man, the sole symbol-creating animal.' R. L. Wilder
Palaeolithic man undoubtedly possessed a rudimentary sense of number, as indeed do some mammals, birds and insects. He could recognize that something had changed when a pebble was added to or removed from a small heap of pebbles. His next step, a major intellectual achievement, was to create the abstract concept of number: to realize that three goats, three fingers and three days shared a common property of 'threeness'. This gave rise to the idea of matching the members of one collection against those of another. If there is perfect matching, the two collections are the same size: they have the same number of members. There is a one-to-one correspondence between the members of one collection and those of the other. True counting entails a further step: that of setting up a correspondence between the objects to be counted and an ordered sequence of symbols. Such symbols may take a variety of forms – spoken number-words, notches on a tally-stick, knots on a cord, the position of the fingers (as in finger-counting) or written characters known as numerals. To count efficiently we need a system of numeration to specify rules of succession so that large numbers can be expressed using only a small number (ten in our system) of different numerals. We see here the genesis of our dual concept of number. We make a distinction between cardinal numbers, based on the one-to-one correspondence principle, and ordinal numbers, based on the principle of ordered succession. Although both principles are at the heart of mathematical thinking, it is the creation of the ordinal number concept that may be taken to mark the beginning of mathematics as we understand it today.
It seems that what we loosely call 'civilization' first came into being along the river valleys of Egypt, Mesopotamia, India and China. The high fertility of the soil enabled the communities to produce more than was required for bare subsistence, so allowing some people to be released from primary production to perform a variety of specialized functions: as craftsmen, administrators, priests, scribes, surveyors – and, eventually, mathematicians. There are no reliable records of the early oriental civilizations, so we shall start our story on the banks of the Nile, and then move to the land of the two great rivers – the Tigris and the Euphrates.
Before doing so, however, we should point out that some authorities would date the beginnings of mathematics far back in prehistoric times. In the 1950s in the village of Isango in Zaïre a bone was dug up which has a large number of notches carved on it. To some scholars – but by no means all – the patterning of the notches suggests the use of a decimal counting system, and even a knowledge of a few prime numbers. On the geometrical side, the situation is broadly similar. Statistical analysis of recent surveys of megalithic sites in Western Europe has led some enthusiasts to claim a very high level of geometrical sophistication for the Stone Age builders of the fourth and third millennia BC. Once again, there is much scholarly disagreement over the interpretation of the evidence. An excellent survey of the main facts and contending views may be found in Reference 2.
A distinctive Egyptian civilization began to emerge in the fifth millennium BC, and lasted until the Roman conquests of the first century BC. During most of that time Egyptian society was largely self-contained, resistant to change, practical rather than speculative, and intensely religious. Why, we may ask, did the ancient Egyptians concern themselves with mathematical matters at all?
Herodotus, 'the Father of History', believed that Egyptian mathematics arose from the practical need to resurvey the land every year after the annual flooding of the Nile Valley. Aristotle, however, put the emphasis on the existence of a priestly leisured class with intellectual interests. The question remains open.
Most of our knowledge of Egyptian mathematics comes from two primary sources: the Rhind Papyrus and the Moscow Papyrus. The first of these is a practical handbook written in about 1700 BC by a scribe called Ahmes. It contains 84 worked problems which include a variety of arithmetical calculations with whole numbers and fractions, the solution of linear equations, and the mensuration of simple areas and volumes.
The Egyptians' system of numeration was based on the scale of ten, but they had no concept of place value. There was simple repetition within each decade, with separate symbols for 10, 100, etc., up to a million. Thus, if we denote the symbols for 1, 10 and 100 by I, T and H, then 234 might be written as HHTTTIIII. No symbol for zero was needed, the numerals could be written in any order, and special arrangements had to be made to represent very large numbers.
Egyptian arithmetic was based on the operations of adding, doubling and halving; multiplication was achieved by successive doubling. So, to multiply 29 by 13, Ahmes would proceed thus:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].
This method, known as duplation, has a long history: a hundred years ago it was known as the Russian peasant's method of multiplication. More recently it has been given a new lease of life with the arrival of the electronic computer, in which internal operations are carried out in the binary system of numeration.
Division entails operating with fractions, and here we meet a distinctive feature of Egyptian arithmetic. All fractions were reduced to sums of Unit fractions, i.e. fractions of the form 1/n. The only exception was 2/3, which was accorded a special symbol. In Problem 24 Ahmes wishes to divide 19 by 8; in Problem 25 to divide 16 by 3. The calculations are carried out as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
The objective is, starting with 8 (or 3), to produce a set of numbers that add to 19 (or 16). In Problem 24 Ahmes uses the sequence 1/2, 1/4, 1/8, ...; in Problem 25 the sequence 2/3, 1/3, 1/6,.... These are, indeed, the two basic 'halving sequences' of Egyptian arithmetic.
When carrying out such calculations it is often necessary to double a unit fraction (i.e. to compute 2/n as a sum of unit fractions). If n is even, the operation is trivial; if n is odd, it may be quite difficult. (Egyptian scribes were not allowed, for some reason, simply to write 2/7, for example, as 1/7 + 1/7.) No less than a third of the complete Rhind Papyrus consists of a table for expressing fractions of the form 2/(2n + 1), for values of n from 2 to 50, as sums of unit fractions, each with a different denominator. The method of calculation, as well as the final result, is given for each value of n.
Thus to evaluate 2/7, Ahmes proceeds as follows:
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
With the aid of the table, any division by 7 becomes a fairly simple matter. Thus, for example, if the problem is to divide 16 loaves of bread equally among 7 men, we have
[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
as the share of each man. Here are three more examples:
2/19 = 1/12 + 1/76 + 1/114
2/59 = 1/36 + 1/236 + 1/531
2/97 = 1/56 + 1/679 + 1/776
Multiplication of sums of unit fractions was also covered. Problem 13, for example, asks for the product of 1/16 + 1/112 and 1 + 1/2 + 1/4 and obtains the correct result of 1/8. Some of the Rhind problems require considerable skill and ingenuity for their solution: for example, the division of 7 by 29 to yield
7/29 = 1/6 + 1/24 + 1/58 + 1/87 + 1/232
It is worth noting that 7/29 may be expressed more simply as 1/5 + 1/29 + 1/145, but the use of the Ahmes 2/(2n + 1) table leads to the former result.
The algebraic problems in the papyrus are mainly concerned with the solution of linear equations. Problem 23 calls, in present-day notation, for the solution of the equation x + x /7 = 19. Ahmes uses the method of false position: he guesses a value of x which he knows to be incorrect, and then scales the result appropriately. Here he assumes that x = 7, giving a value of 8 for the left side of the equation. He needs, therefore, to multiply his guessed value by 19/8, i.e. by 2 + 1/4 + 1/8. The answer is found to be 16 + 1/2 + 1/8, which Ahmes checks by augmenting his answer by 1/7 to obtain 19. The Papyrus also contains a few geometrical problems, but there is no mention of any numerical instance of Pythagoras' theorem. However, the 'rope stretchers', as the Egyptian surveyors were called, almost certainly used knotted ropes with lengths in the ratio of 3:4:5 to lay out a right angle. In Problem 50, Ahmes equates the area of a square field of side 8 units with that of a circle of diameter 9 units. This leads to a value of ? (to use the modern symbol) of 4 × (8/9)2, or very nearly 3.16. More information on the Rhind Papyrus is given in References 2, 4 and 5.
The Moscow Papyrus, which was written about 1900 BC, contains 25 problems. One of these is of exceptional interest in that it contains – albeit in somewhat disguised form – the correct formula for the volume of a truncated pyramid, namely V = 1/3h (a2 + ab + b2), where a and b are the sides of the two parallel square faces, and h is the distance between them. We would, of course, expect the Egyptians to know a lot about pyramids, and we shall return to the subject in Chapter 3.
In marked contrast to Ancient Egypt, Mesopotamia was the scene of incessant conflict: invasions, pillage, forced migrations and dynastic upheavals. Even so, the cultural pre-eminence of the region lasted for some 4000 years, during which time the indigenous Sumerians were succeeded by Akkadians, Hittites, Medes, Persians and Chaldeans, among others. The convenient term 'Babylonian' is commonly applied to both the region and its culture during the period of greatest achievement – roughly from. 2200 to 500 BC. The Babylonian Empire actually came to an end in 538 BC when Babylon fell to Cyrus of Persia, but a distinctive Babylonian culture survived for another five centuries.
In our study of Babylonian mathematics we are fortunate in having an abundance of primary source material in the form of inscribed clay tablets. The writing is known as cuneiform ('wedge shaped') because it was produced by pressing a stylus into a block of soft clay which was then baked or allowed to harden in the sun. Most of the surviving tablets containing mathematical material come from one of two widely separated periods: either the first half of the second millennium BC (the 'Old Babylonian' age) or the last third of the first millennium BC (the 'Seleucid' period). However, the oldest tablets go back to about 2100 BC, when the region was ruled by a Sumerian dynasty centred on Ur, the city of Abraham. Babylonian mathematics seems to have reached its zenith very early, around 1800 BC, about the time of the reign of Hammurabi, the great lawgiver.
The Babylonian number system is entirely different from the Egyptian. It is sexagesimal and positional, and uses only two numerals, made by pressing the triangular end of a stylus into the soft clay in two different ways. Let us denote these numerals by V and X; they basically stand for 1 and 10. The positional (or place value) feature means that V can also represent 1 × 60n, and × can represent 10 × 60n, where n can take any integral value – either positive or zero for whole numbers, or negative for fractions. The correct value of n in a particular case must be deduced (or guessed) from the context. Thus, for example, the ordered sequence of numerals VXXVVV could represent 60 + 2(10) + 3 = 83, or 3600 + 2(10 × 60) + 3/60 = 4800 1/20, or 1/60 + 20/602 + 3/603. It will be convenient to transcribe Babylonian numbers into a more readable form, so let us write the numbers quoted above as 1,23 and 1,20,0;3 and 0;1,20,3. The cause of the ambiguity is the lack of a symbol for zero, a symbol which in our number language enables us to distinguish between, say, 32, 302 and 320. The omission was eventually rectified, but only in part. During the Seleucid period a separator symbol was introduced to indicate an empty space inside a number, but it was not used in a terminal position. In spite of its deficiencies, the Babylonian system offered so many advantages, especially for dealing with large numbers and fractions, that it was used by Greek and Arab astronomers for many centuries. This is why we still retain a sexagesimal system to measure time and angles.
Many of the mathematical tablets contain tables for multiplication, and tables of reciprocals, squares, square roots, etc. Here are the first few entries in a table of reciprocals:
2 30 8 7,30 The reciprocals of the 'irregular'
3 20 9 6,40 numbers 7 and 11 are omitted
4 15 10 6 because they are non-terminating
5 12 12 5 in a sexagesimal system – as is 1/3
6 10 in our decimal system.
By 1900 BC the Babylonians had a well-established algebra. They could solve quadratic equations (positive coefficients and positive solutions only) and some types of higher-degree equation as well. Thus one problem asks for the side of a square whose area exceeds the side by 14,30. This leads to the equation x2 — x = 870. The solution is set out thus:
Take half of 1 and multiply it by itself to give 0;15. Add the result to 14,30 to give 14,30; 15. This is the square of 29;30. To this add 0;30 to give 30, which is the desired result.
The procedure is exactly equivalent to our method of solving quadratic equations by 'completing the square'.
Some Babylonian mathematics reached a high level of sophistication. For example, the tablet known as Plimpton 322 contains a table of 15 rows of which the first 5 and the last 2 are (with a couple of obvious clerical errors corrected):
1,59,0,15 1,59 2,49 1
1,56,56,58,14,50,6,15 56,7 1,20,25 2
1,55,7,41,15,33,45 1,16,41 1,50,49 3
1,53,10,29,32,52,16 3,31,49 5,9,1 4
1,48,54,1,40 1,5 1,37 5
. . . .
. . . .
. . . .
1,25,48,51,35,6,40 29,31 53,49 14
What, we may ask, does this table mean? Let us consider a right-angled triangle ABC with sides x, y and z (Figure 1.1). If the numbers in the second and third columns are taken to be x and z, then the first column turns out to be z2/y2 – or sec2A, in modern notation. (A more logical notation would be (sec A)2, but we shall adhere to the established convention for denoting powers of the trigonometric functions. We define sec A as the reciprocal of cos A.) Clearly, not only were the Babylonians familiar with Pythagoras' theorem, but they also knew how to construct what are known as Pythagorean number triples. These are solutions (x,y,z) in positive integers of the equation x2 + y2 = z2. To form such a triple we take any two positive integers p and q, where p >q, such that p and q have no common factor and are of opposite parity, i.e. one is odd and the other even. (The symbols > and < denote 'greater than' and 'less than' respectively.) Then the expressions
x = p2 - q2, y = 2pq and z = p2 + q2
will produce all primitive number triples with no repetitions. By 'primitive' we mean that the numbers comprising the triple have no common factor. Thus p = 2, q = 1 gives the well-known (3, 4, 5), while p = 3, q = 2 gives (5, 12, 13). The triple corresponding to the first row of our table is (119, 120, 169), derived from p = 12, q = 5; that corresponding to the fifth row is (65, 72, 970), from p = 9, q = 4. We don't know how or for what purpose the table was constructed, but the numbers are so large that they must have been determined by some rule. The numbers in the first column decrease steadily, with the first nearly equal to sec2 45° (=2) and the last to sec231°. We have a table of sec2A where the angle A in Figure 1.1 decreases from 45° to 31° in 1° steps: an impressive example of what could be achieved nearly 4000 years ago. However, not all authorities accept this interpretation of the significance of the Plimpton tablet. One rival theory is that it is a 'teacher's aid' for setting and solving problems involving right-angled triangles which would 'come out' nicely in integers. Another suggestion is that the tablet was computed, not from pairs of numbers (p,q), but from a single parameter s = p/q. Writing sR for the reciprocal 1/s, we see that
x/y = 1/2(s - sR) and z/y = 1/2(s + sR)
Excerpted from Makers of Mathematics by Stuart Hollingdale. Copyright © 1994 Stuart Hollingdale. Excerpted by permission of Dover Publications, Inc..
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