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Introduction

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

For many centuries, ancient Egypt was seen as the source of wisdom and knowledge, about mathematics as well as other things. There was a long classical Greek tradition to this effect, and in later centuries the indecipherability of the hieroglyphs did nothing to dispel this belief. But since the early nineteenth century, when the deciphering of the Rosetta Stone by Young and Champollion enabled rapid progress to be made in translating extant Egyptian texts, the picture has changed to reveal a civilisation more pragmatic and down-to-earth. In this unit, we shall investigate what we now know of Egyptian mathematics, and how we know it.

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Learning outcomes

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

After studying this unit you should be able to:

know something about how hieroglyphs were used to represent numbers and the nature of the problems that have survived;
understand that Egyptian calculation was fundamentally additive. Operations such as doubling and halving being used for multiplication and division;
appreciate the advanced understanding of mathematics in Ancient Egypt in relation to the manipulation of fractions;
consider some views of the mathematics of Ancient Egypt in relation to that of the Babylonians.

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1.1 Mathematics in Egyptian history

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

Only a small number of the surviving Egyptian papyri are concerned with mathematical calculations – perhaps a dozen or so in all, of which the earliest dates from about 1850 BC and the most recent from AD 750. The two major ones are the Rhind Papyrus (named after the man who bought it on his holidays in Luxor in 1858), which you can see in the British Museum, and the Golenischev (or Moscow) Papyrus, which is in Moscow. They are dated at around 1650 BC and 1850 BC respectively. So here are authentic primary sources – that is, examples of the foundational artefacts upon which our knowledge of the history of mathematics is constructed.

Figure 1 Rhind Papyrus, Problems 39 and 40

If you look at Figure 1, you will see at once that there is a problem; the text is not meaningful until it has been translated into something comprehensible to us. It will be instructive to spend a few minutes discovering what to do about this. It is not generally practicable for any of us to learn afresh each new language or script of cultures whose mathematics we might be interested in. We are reliant, in this case, on the knowledge of Egyptologists for the material on which we can start to build our own understanding. Even once a translation is provided there is still a process of interpretation to be gone through.

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1.2 The Rhind papyrus

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

For a literate civilisation extending over some 4000 years, that of the ancient Egyptians has left disappointingly little evidence of its mathematical attainments. Even though the classical Greeks believed mathematics to have been invented in Egypt – though their accounts are far from unanimous on how this happened – there are now but a handful of papyri and other objects to convey a sense of Egyptian mathematical activity. The largest and best preserved of these is the Rhind papyrus (Extract 1), now in the British Museum, a copy made in about 1650 BC of a text from two centuries earlier. In Box 3 the Egyptologist Sir Alan Gardiner explains an initially puzzling feature of Egyptian arithmetic, the Egyptian concept of fraction or part. The commentaries given in Box 4 are contrasting perceptions of Egyptian mathematics, from the translator of the Rhind papyrus and from a historian of mathematics.

Extract 1 Two problems from the Rhind papyrus

(a) Problem 24

(b) Problem 40

Question 1

Examine the extract from Chace's edition of the Rhind Papyrus, Extract 1(b), comparing it with the photograph shown in Figure 1. Try to work out what things the Egyptologist has done. Do you yet feel in a position to begin to understand what the Egyptian scribe was doing? (Do not worry if your answer to the last question is ‘no’!)

Discussion

You should have been able to notice, just from the look of the page, that it has three sections. The top one is a hand-written copy of the papyrus text itself, while the bottom section contains a rather literal English translation into our language and numerals. The middle section is a transcription of the text itself (which is written in a script called hieratic) into hieroglyphic, the standard script in which Egyptologists work, with an indication of the sounds and numeral values. (You may have noticed other features too, such as that hieroglyphic and the original hieratic-reads from right to left, so the ordering has been reversed in going from the middle to the bottom section.)

As to understanding the text, I would say that not only is the calculation of an unfamiliar kind — just what is going on there is far from clear – but also, even what the problem is has yet to be put in a form accessible to most of us.

If we were to work through the calculation, it would be possible to infer what the question was. ‘Loaves 100 for man 5, ¹/₇ of the 3 above to man 2 those below. What is the difference of share?’ seems to be saying that a hundred loaves are to be divided between five men in a particular way. This is that each is to receive a different number of loaves so that the two men who receive the least end up with a seventh of what the other three men get between them. Further, it requires that the difference between what they receive is constant: each man receives a fixed number of loaves more than the next man. (So the number of loaves each receives is in what we would call an arithmetical progression.)

You may observe that the scribe's way of putting the problem had at least the virtue of succinctness! We shall leave the actual calculation for now, and return to it once we have looked at the more general principles governing the ways Egyptians handled numbers, as inferred from the evidence of the Rhind and other papyri. As already mentioned, evidence from papyri extends over a long period, although it is rather scanty and is reinforced by evidence from only a few other artefacts, such as bits of pot, tiles and stone inscriptions. From the limited evidence we have, however, the fundamental spirit and mathematical approach of Egyptian mathematics seem to have changed very little over three millennia.

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2 Egyptian calculation

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

Box 1 A note on Egyptian scripts and numerals

The earliest Egyptian script was hieroglyphic, used from before 3000 BC until the early centuries AD. Initially an all-purpose script, it was eventually used only for monumental stone-carving and formal inscriptions. It had been superseded (by about 2000 BC) by a more fluid script called hieratic, used for more rapid writing on papyri. Later still, there developed a cursive, everyday script called demotic, that looks like a mad doctor's handwriting at the end of a bad day.

Most of the handful of extant mathematical papyri are written in hieratic. Because of the great variability of hieratic and demotic in the hands of different scribes, Egyptologists habitually transcribe all texts into the more legible and standardised hieroglyphic.

The representation of numerals in the different scripts shows an interesting development. Hieroglyphic numerals are formed on a straightforward decimal repetitive principle, each symbol for a power of ten being repeated as often as necessary. In hieratic, and even more so in demotic, new symbols were devised for each of two to nine, ten, twenty to ninety, and so on, so there was a great increase in the degree of cipherisation (having separate, concise, independent symbols for different numbers). Some historians have seen this as a significant step in the development of numeration.

Egyptian calculation was fundamentally additive. The most frequent operations were doubling (that is, adding a number to itself) and halving (that is, finding what number can be added to itself to make the number you started with). Here is a sample calculation for you to try (from the Rhind Papyrus problem 69); we start from the transcription into hieroglyphs so you can see the principles more readily.

Question 2

Given that in hieroglyphs | is 1, is 10, is 100 and is 1000, transcribe this calculation into our numerals and try to work out what is being done, and how.

Discussion

The sum looks like a multiplication of eighty by fourteen; the scribe has considered the latter as ten and four, multiplied eighty by each (to give the lines marked by /) and added the results, to obtain 1120. Notice that to multiply 80 by 4, he first doubled SO and then doubled it again. Notice too, that multiplying by ten is very easy in hieroglyphs, simply by changing each symbol into the next one in the ordered sequence of symbols for powers of ten.

You can see from having tried this how it is possible to begin to make sense of Egyptian methods. Division turns out to be rather similar. The problem is seen in terms of discovering by what one number must be multiplied to make another. So, for instance, the calculation above could equally have stood for the division of 1120 by 80! One would scan down selected multiples of 80 to sec which would add up to 1120. This yields the answer 10 and 4 – that is, 14.

Of course, the answer to such an attempt at division may not always be exact if the calculation is restricted to whole numbers. Exploring what happens when a division does not yield a whole-number answer leads us to one of the most striking and influential features of Egyptian mathematics — their fractions.

Question 3

What has the scribe done to compute 19 divided by 8? (This is taken from Problem 24 of the Rhind Papyrus, transcribed into our numerals.)

Discussion

He is trying to find by what 8 must be multiplied to give 19. So he doubles 8, to give 16 which is 3 short of 19; so he must find what multiplies 8 to produce this remaining 3. He halves the 8 (obtaining 4, which is still too much), then halves that and halves it again. Notice that the quarter of 8 and the eighth of S together will make up the 3 he wants. So his answer, as indicated by the / marks, is the sum of 2 and ¹/₄ and ¹/₈.

This, for the Egyptian scribe, is the answer: 2 ¹/₄ ¹/₈. (Or, rather, the equivalent of that in hieratic.) One important thing to notice is that he did not, as we might do, add the fractions together to produce 2³/₈. In Egyptian mathematics, only what we would call unit fractions, that is, ¹/₂, ¹/₃, ¹/₄, and so on, are used, together with the fraction we would write as ²/₃. Why there should have been what seems to us this strange restriction of Egyptian mathematics is an interesting question that we return to shortly.

In view of the importance of the principle of doubling, as well as of the use of unit fractions, it was necessary to know what was the result of doubling fractions. Nearly a third of one side of the Rhind Papyrus (which is some 18 feet long by about 1 foot wide) is taken up with calculating the doubles of the odd fractions ¹/₃, ¹/₅, ¹/₇, …, ¹/₉₉, ¹/₁₀₁, expressed in the equivalent form of dividing 2 by the odd numbers from 3 to 101. We shall write these as 2 : 3, 2 : 5, and so on. For instance:

In modern notation, the first two results read:

Figure 2 The beginning of the Rhind Papyrus Shown here are the divisions by 2 of the odd numbers from 3 to 27. preceded by the scribe's introduction.

We shall not pursue the details here of how these were calculated, but there is an example in Box 2 to illustrate the skill and ingenuity of Egyptian fraction-handling techniques.

Box 2 A note on how 2 : 17 was calculated

The instruction for this problem in the ancient Egyptian reads something like, ‘Get 2 by operating on 17’, which provides a clue to the guiding principle: do things to 17 until you arrive at 2.

The calculation is in two stages:

carry on halving 17 (starting from its two-thirds part) until some number under 2 is reached;
see what this leaves you with, to get back to 2 precisely.

Step 1 of the calculation goes

Thus ¹/₁₂ will be part of the answer; what is the rest?

For step 2 the scribe wrote down the difference between the number reached (1 ¹/₄ ¹/₆) and 2.

Remainder ¹/₃ ¹/₄

Question: How did the scribe know this?

Then, as a subsidiary computation, the scribe worked out what needed to be done to 17 to get this remainder.

There seems to be a tight argument compressed into these figures; he has calculated that 3 times 17 is 51; and thus ¹/₅₁ times 17 is ¹/₃. Similarly 4 times 17 is 68; thus ¹/₆₈ of 17 is ¹/₄.

Since ¹/₃ ¹/₄ is the remainder we needed, as parts of 17 these are ¹/₅₁ ¹/₆₈.

The overall result, then is 2 : 17 = ¹/₁₂ ¹/₅₁ ¹/₆₈.

The one part of this calculation that is not explicitly written down is how the scribe arrived at the remainder, the difference between 2 and the number obtained at the end of the step (i) calculation, A method for doing this appears in other problems on the Rhind Papyrus, so we shall go through how it could have been done in this case.

We had reached 1 ¹/₄¹/₆ and wanted to know how far short this was of 2. The method involves an auxiliary calculation, set up as follows. Express the fractions you have as multiples of some other convenient fraction, say ¹/₁₂. Then ¹/₄ is 3 times ¹/₁₂, and ¹/₆ is 2 times ¹/₁₂. The numbers 3 and 2 are written down (in red ink, on the papyrus, the rest being in black) and added: 5. Now, how far short of 12 are we? 7. So, the remainder we want is however many times 12 goes into 7:

So ¹/₃ ¹/₄ is the remainder, as indeed the scribe wrote down.

Notice one thing, however, about the above results: what might seem to us the ‘obvious’ answer, for instance that 2 : 17 is ¹/₁₇ ¹/₁₇, was never the one reached. The possibility of repeating the fraction in order to double it just did not count as an answer, it seems, for the Egyptian scribe. This observation gives us a clue concerning the question raised earlier above, of why in Egyptian mathematics we find only unit fractions (and ²/₃).

The answer to this question is important and revealing, because thinking it through will inform us not only about Egyptian mathematics, but also about what studying history can involve. The question arose because it seemed to suggest itself from the source material, or from our efforts to understand it. We use fractions, the Egyptians also seemed to do so, yet only in a seemingly highly restricted way which drove them into subtle and complicated contrivances. Why did they not just use fractions as we do ? It would have saved them a lot of trouble! Now notice that the more the problem is spelled out, the more peculiar it begins to look. Imagine yourself asking the scribe, ‘Why do you only use unit fractions?’. His look of bewilderment will be explained if you go on to reflect that that question presupposes him to be making the deliberate choice of not using fractions in the way we do.

Expressed like that, it is surely an unreasonable presupposition. So how have we got into this problem, and how do we get out? The problem seems to have arisen partly conceptually and partly notationally. Conceptually, because the very use of the word fraction carries our set of expectations with it, whereas if we had used another word, such as reciprocal, say (for ¹/₇ can be thought of as ‘the reciprocal of 7’ as well as ‘one-seventh’), then no-one would have expected compound reciprocals. Notationally, the translation of the Egyptian symbol into ‘our’ fractions set up possibly unhelpful expectations. By looking again at the statement of Rhind Papyrus Problem 40 on the previous page, you will see that our ‘¹/₇’ was translating the hieroglyphic which in turn is equivalent to a hieratic squiggle with a dot on top. Translating these marks as , or by some people as , helps preserve better the awareness that Egyptian fractions were conceptually different from ours.

So the upshot of this argument is that our initial question, of why Egyptian fractional usage was ‘restricted’ to unit fractions, was badly posed. It need not arise if we reconsider the question as: what was the Egyptian concept for which we have hitherto used the word fraction? The Egyptian symbol for what we have been thinking of as ¹/₃, ¹/₄,… is better translated by the word part, as in the third part, the fourth part, and so on. The reason this is better is that to say one seventh or use the symbol ¹/₇ (say) invites the idea familiar to us of there being ‘two sevenths’ (²/₇), ‘three sevenths’ (³/₇) and so on, whereas the seventh part is unique. So this would explain why the result of doubling the eleventh part is not the eleventh and the eleventh; for there can be only one eleventh part. Or, to put it another way, to write 2 : 17 as ¹/₁₇ ¹/₁₇ would simply be a restatement of the problem, not an answer to it.

The Egyptian concept of part is explained more fully in the box below.

Box 3 Sir Alan Gardiner on the Egyptian concept of part

The commonest method of expressing fractions in Egyptian was by the use of the word r ‘part’, below which (or partly below it in the case of the higher numbers) was written the number described in English as the denominator. Thus , r-5 ‘part 5’ is equivalent to our ¹/₅, r-276 ‘part 276’ to our ¹/₂₇₆.

For the Egyptian the number following the word r had ordinal meaning; r-5 means ‘part 5’, i.e. ‘the fifth part’ which concludes a row of equal parts together constituting a single set of five. As being the part which completed the row into one series of the number indicated, the Egyptian r-fraction was necessarily a fraction with, as we should say, unity as the numerator. To the Egyptian mind it would have seemed nonsense and self-contradictory to write r-7 4 or the like for ⁴/₇; in any series of seven, only one part could be the seventh, namely that which occupied the seventh place in the row of seven equal parts laid out for inspection. Nor would it have helped matters from the Egyptian point of view to have written r-7( + )r-7( + )r-7( + )r-7, a writing which would likewise have assumed that there could be more than one actual ‘seventh’. Consequently, the Egyptian was reduced to expressing (e.g.) ⁴/₇ by ¹/₂( + )¹/₁₄ For more complex fractions even as many as five terms, all representing fractions with 1 as the numerator and with increasing denominators, might be needed; thus the Rhind mathematical papyrus, dating from the Hyksos period, gives as equivalent of our ²/₆₁ the following complex writing: r-40 r-244 r-488 r-610 ‘¹/₄₀ + ¹/₂₄₄ + ¹/₄₈₈ + ¹/₆₁₀’. It is generally known that the same cumbrous methods of expression were in common use with the Greeks and Romans. It would seem also that a relic of them survives in the use of English ordinals in the names of our fractions, though we speak of ‘one-third’ and ‘three-fifths’ without any qualms. […]

Though the Egyptians were unable to say ‘three-sevenths’ or ‘nine-sixteenths’, yet they made a restricted use of certain fractions which appear, at first sight, to stand on the same footing: a great rôle is played in Egyptian arithmetic by the fraction rwy 'the two parts’ (out of three), i.e. ²/₃, and a very rare sign r-3 (perhaps to be read mt rw) can be quoted for ‘the three parts’ (out of four), i.e. ³/₄. These ‘complementary fractions’ represent the parts remaining over when ‘the third’ or ‘the fourth’ is taken away from a set of three or four, and indeed their existence is practically postulated by the terms r-3, r-4. But we must be careful to note that in r-3=³/₄ the numeral is a cardinal, not an ordinal, and that the expression means ‘the three parts’ and was not construed, as with ourselves, as meaning ‘three fourths’. In ordinary arithmetic the only complementary fraction used was ²/₃. Compare in English ‘two parts full’, i.e. two-thirds full, doubtless a survival of the old Egyptian way of regarding the same fraction.

Thus equipped, we can more easily see how a calculational practice involving parts could form a coherent framework in itself, not needing the overtones of our concept of fractions. That it was a coherent framework is attested by its extraordinary longevity. Two millennia after the Rhind Papyrus, for instance, we find many of the results in Ptolemy's Almagest (c. AD 150), the greatest of Greek astronomical texts, presented in Egyptian fashion, as in this passage.

One [sighting] we made in the year 18 of Hadrian, Egyptianwise Pharmouthi 2–3, according to which the morning Venus was at its greatest elongation from the sun; and, sighted with the star called Antares, it was 11 + ¹/₂ + ¹/₃ + ¹/₁₂° within the Goat, while the mean sun was then 25¹/₂° within the Water Bearer. And so the greatest morning elongation was 43 +¹/₂ + ¹/₁₂°.
Ptolemy, Almagest Book X, ch.3, translated by R. Catesby Taliaferro. The ‘plus’ signs have been inserted by the translator.

Indeed the use of parts can still be found in European writers of the thirteenth and fourteenth centuries AD. Henceforth, we shall use for ¹/₇, and similarly for other parts.

Having absorbed, then, the idea that past mathematical procedures and concepts may actually be more difficult to understand unless we do our best to meet them on their own terms, let us turn back to Egyptian calculation methods. Here is an example of a procedure that was followed to solve a problem. We have chosen Rhind Papyrus Problem 24, as you have already studied one calculation from it (Question 3).

The problem is in Extract 1, so please study the extract briefly now before coming back to the offered explanation which follows.

The numbering of the problems, by the way, is that of a nineteenth-century German editor of the Rhind Papyrus — in the original, the problems are unnumbered.

The problem, ‘A quantity, ¹/₇ of it added to it, becomes it: 19’, can be seen to mean something like, ‘A quantity and the seventh part of it is nineteen. What is the quantity?’. The way this is solved is that the scribe produces the number 7, as a kind of trial guess perhaps, but one having the useful property that its seventh part is easy to determine:

So we can see that had the quantity been seven, it and its seventh part would make 8; but what we want is for it and its seventh part to make 19; so if we knew how many times 8 goes into 19, we would know how many times 7 goes into the quantity we want. This part of the calculation, dividing 19 by 8, is what you did in Question 3, and the answer was 2 . So all that remains is to multiply that by 7 to get the answer, which is done by doubling.

The sum of these is 16 , which is therefore the desired quantity. But just to make sure, the scribe checks the result — ‘The doing as it occurs’. He adds to the quantity its seventh part (2 ), to show that their sum is indeed 19 as required.

The strategy followed in this problem, namely choosing any convenient number and working out the solution in respect to it, then multiplying by the ratio of your answer to the answer you want, is quite a common one in Egyptian mathematics. Indeed, if we generalise this description of the strategy, and call it a systematic method of'trial and error’, then this seems characteristic of the Egyptian style of arithmetic. For example, if you look again at the way division was described (Question 3, Comment) you will notice that it can be seen in terms of getting close to the desired result and then seeing what needs to be done to make up the remainder. The strategy used in Problem 24 is a solution method in common use up to quite recent times, appearing under various names such as the method of false position or the rule of false.

We now return to Problem 40, with which we started. It provides another example of this strategy.

Question 4

Re-read the suggested explanation of what Problem 40 is about, which was given in the paragraph after Question 1, and see if you can work out how the scribe has solved it, in the way that we approached Problem 24. Use the English translation in Extract 1 to work from (unless your hieratic or hieroglyphic skills are especially well developed). It will save you time if you have two further pieces of information: is the symbol we use to denote ²/₃, and the scribe makes a guess or assumption about how many loaves the man who gets least receives, as well as about what the constant difference is.

(Do not spend any longer on this than you feel you want to — go on to read the Comment when you are ready.)

Discussion

The scribe produces for the initial try a share difference of 5¹/₂, and the assumption that the last man receives 1 loaf. So the previous man got 6¹/₂, the one before him 12, then 17¹/₂, then 23. (This comes from adding 5¹/₂ each time.) Adding all of these together gives 60, so since it was actually 100 loaves to start with, another 40 (that is, the two-thirds part of 60) is needed. He says ‘Make thou the multiplication: 1²/₃’ and multiplies up each of his initial shares 23, 17¹/₂, 12, 6¹/₂ and 1 by 1²/₃, and finally checks that they do indeed total 100 by adding the results.

That is what appears on the papyrus. But in checking over what the scribe has done, and comparing it with the statement of the problem, it becomes clear that a substantive part of the solution has taken place off-stage, as it were. The shares do have the property they ought to, namely that the sum of the least two is one seventh of the sum of the largest three. But this is not something he shows, nor does he indicate how the initial ‘guesses’ were arrived at to ensure this. (Nor, indeed, does he explicitly give the answer to the question, ‘What is the difference of share?’.) We might conjecture that either something was left out during his copying the solution (at the start of the Rhind Papyrus the scribe says he is copying an earlier one), or else perhaps that it was only this application of the false position strategy he wanted lo show.

Figure 3 An Egyptian scribe, Ka-Irw-Khufu, who lived c. 2500 BC

This discussion has raised the question of who or what the Rhind Papyrus was for, that is, what the scribe's intention was. His opening words were, in Chace's free translation, ‘Accurate reckoning. The entrance into the knowledge of all existing things and all obscure secrets’, which is interesting, but does not really clarify for us whether this kind of thing was taught to schoolchildren, or whether the Rhind Papyrus represented quite an advanced theoretical work by Egyptian standards. In trying to understand this ancient mathematical activity, it is natural for us to ask how it compares with later mathematics. It would be interesting to know whether it were simply a collection of empirical computational techniques, brought together for applying to everyday practical problems, or whether there are visible traces of, say, the more abstract theoretical tone that you will be seeing in Greek mathematics. Historians have differed in the judgements they have reached on this question. For instance:

The Rhind and Moscow papyri are handbooks for the scribe, giving model examples of how to do things which were a part of his everyday tasks.… The sheer difficulties of calculation with such a crude numeral system and primitive methods effectively prevented any advance or interest in developing the science for its own sake. It served the needs of everyday life …., and that was enough.
G. J. Toomer.

A careful study of the Rhind Papyrus convinced me several years ago that this work is not a mere selection of practical problems especially useful to determine land values, and that the Egyptians were not a nation of shopkeepers, interested only in that which they could use. Rather I believe that they studied mathematics and other subjects for their own sakes.
A. B. Chace.

Question 5

Which of these views do you feel more applicable to what you have seen of the Rhind Papyrus (namely, Problems 24 and 40)?

Discussion

There is something to be said for Toomer's view: the problem of dividing loaves among a group of men could certainly have been an everyday task for the overseers of large building works, for instance; and your experience of Egyptian calculation may well have inclined you to the view that ‘crude’ and ‘primitive’ are understandable epithets.

But there are features of the problems which are harder to fit with everyday life in quite so straightforward a way. For instance, the circumstances under which real loaves are to be divided among a group of people in arithmetical progression are hard to imagine. Again, the fact that Problem 24 is posed in terms of'a quantity’ (the Egyptian word can also be translated as 'heap' – something rather unspecific) does imply a certain level of abstraction, a recognition that the same techniques and rules apply to any of a range of real-world objects. So there is some recognisable mathematical activity ‘for its own sake’, which was Chace's claim.

However, this does not quite resolve the matter. If the Rhind Papyrus were primarily a teaching document, and if it were the calculation methods that one wanted to teach, then the fact that Problem 40 is quite unrealistic may be a tribute to the scribe's imaginative teaching style as much as to an exploration of mathematics for its own sake. Quite what significance should be attached to the formulation of Problem 24 is problematical. Some historians have seen in this an early technique foreshadowing the later development of algebra, insofar as that is concerned with operating on unknown quantities. On the other hand, the level of abstraction in this problem could be seen as no greater than that implicit in the very use of numbers. That one can have things-numbers-that apply indifferently to a range of real-world objects is quite a sophisticated concept, but one reached long before the Egyptian records to which we have access.

Let us reflect on the situation where historians reach apparently contradictory views. How is this possible, and how do we decide which, if any, we agree with? Question 5 implied a response to the latter question, that it is immediately helpful to check out the judgements against whatever relevant source material you are aware of. We did not reach any very firm conclusion, though, either because we have not yet looked at enough evidence, or perhaps because moving from the evidence to a judgement upon that evidence is more difficult than appears at first sight. So we should go on to investigate more fully what evidence, and what arguments from that evidence, these historians used to reach their conclusions. (Note that what we are doing at present has more general import. Our interest is not only in the pros and cons of this particular question, bul also in trying to work towards a method which can apply in other contexts in order to evaluate historical judgement.) The fuller passages from which the above views of Chace and Toomer are taken are dealt with on the next page. You will also learn more about the contents of the Rhind Papyrus from these extracts.

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3 More information about the Rhind papyrus

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

Box 4

Arnold Buffum Chace on Egyptian mathematics as pure science

A careful study of the Rhind papyrus convinced me several years ago that this work is not a mere selection of practical problems especially useful to determine land values, and that the Egyptians were not a nation of shopkeepers, interested only in that which they could use. Rather I believe that they studied mathematics and other subjects for their own sakes. In the Rhind papyrus there are problems of area and problems of volume that might be of use to the farmer who owns land and raises grain. There are pyramid problems that might furnish specifications to the builders, or enable an interested observer to determine the dimensions of a pyramid before him. Many of the arithmetical problems concern a division of loaves or of a quantity of grain among a certain number of men, or the relative values of different amounts of food or drink. But when we come to examine the conditions laid down and the numbers involved in these various problems as well as the purely numerical ones, we see that they are more like theoretical problems put in concrete form. In one (Problem 63) 700 loaves are divided among four men in shares that are proportional to the four fractions ²/₃, ¹/₂, ¹/₃ and ¹/₄, the first four terms of their fraction-series. In two (Problems 40 and 64) there is a dividing into shares that form an arithmetical progression, in Problem 67 the tribute for cattle is determined as ¹/₆ ¹/₁₈ of the herd and the problem asks for the number of the herd when the number of tribute cattle is given, and Problem 31 is a problem whose answer is

14 ¹/₄ ¹/₅₆ ¹/₉₇ ¹/₁₉₄ ¹/₃₈₈ ¹/₁₆₉ ¹/₇₇₆.

Such problems and such quantities were not likely to occur in the daily life of the Egyptians. Thus we can say that the Rhind papyrus, while very useful to the Egyptian, was also ‘an example of the cultivation of mathematics as a pure science, even in its first beginnings’ [H. Wieleitner, ‘Zur ägyptischen Mathematik’, Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, 56 (1925) pp. 129–137].

G. J. Toomer on Egyptian mathematics as strictly practical

The Rhind and Moscow papyri are handbooks for the scribe, giving model examples of how to do things which were a part of his everyday tasks. This is confirmed, if confirmation were needed, by a papyrus in the form of a satirical letter in which a scribe ridicules a colleague for his inability to do his job, and cites among other examples of his failures calculations of the rations of soldiers and of the number of bricks required for building a ramp of given dimensions. A further indication of the origin of these texts is the kind of expression used to introduce problems, for instance: ‘If a scribe says to you …, let him hear …’. The texts are in one respect similar to the Babylonian mathematical texts, in that these too are in the form not of treatises but of specific problems with solutions. But there the similarity ends: the cuneiform texts have a claim to be called mathematical in a fully scientific sense. The problems are only formally about the measurement of areas, determination of lengths, etc. Many of them are not of a kind which could conceivably ever occur in actual mensuration, and the whole point of them is the algebraic procedure involved. They are really ‘pure’ mathematics. However, this difference from the Egyptian texts is not the important one; mathematics can be applied to practical ends without losing any of its scientific quality. What really distinguishes Babylonian mathematics is the systematic development of intricate algebraic techniques which we can deduce from the working of the problems. These techniques could never have been created by mere empiricism, and we must posit an order of mathematical reasoning of which there is no trace in the Egyptian sources.

To illustrate the elementary and practical nature of Egyptian mathematics, we set out Problem 42 of the Rhind papyrus in full below.

Problem 42

Find the volume of a cylindrical granary of diameter 10 and height 10.

Take away ¹/₉ of 10, namely 1 ¹/₉ the remainder is 8 ²/₃ ¹/₆ ¹/₁₈ . Multiply 8 ²/₃ ¹/₆ ¹/₁₈ times 8 ²/₃ ¹/₆ ¹/₁₈; it makes 79 ¹/₁₀₈ ¹/₃₂₄. Multiply 79 ¹/₁₀₈ ¹/₃₂₄ times 10; it makes 790 ¹/₁₈ ¹/₂₇ ¹/₅₄ ¹/₈₁ cubed cubits. Add ¹/₂ of it to it; it makes 1185 ¹/₆ ¹/₅₄, its contents in khar. 1/20 of this is 59 ¹/₄ ¹/₁₀₈. 59 ¹/₄ ¹/₁₀₈ times 100 hekat of grain will go into it.

Method of working out:

The problem is to determine the cubic content of a cylinder of diameter (D) 10 cubits and height (h) 10 cubits. This is complicated by the fact that for the Egyptian cubic content means how much it will hold of some specific thing, so an answer in cubic cubits is not satisfactory. It is therefore necessary to convert to hundreds of quadruple-hekat of corn by way of the equivalences:

1 cubic cubit = 1¹/₂ khar.

1 khar = 20 hundreds of quadruple-hekat.

In the working, some of the steps, which would require the use of auxiliary fractions, have certainly been omitted. But what is set down is enough to show that the real difficulty for the Egyptian scribe was the mastering of elementary arithmetical calculations; we can see how hemmed in he was by his numerical system, his crude methods, and his concrete mode of thought.

The truth is that Egyptian mathematics remained at much too low a level to be able to contribute anything of value. The sheer difficulties of calculation with such a crude numeral system and primitive methods effectively prevented any advance or interest in developing the science for its own sake. It served the needs of everyday life (it is only a relatively advanced technology, such as was never achieved in the ancient world, which demands more than the most elementary mathematics), and that was enough. Its interest for us lies in its primitive character, and in what it reveals about the minds of its creators and users, rather than in its historical influence.

Question 6

Do you now feel able lo form a view on which, if either, of these judgements on Egyptian mathematics is the better justified?

Discussion

My answer is ‘yes’ — I can now see better on what grounds the historians were putting forward their views, which helps me to decide which, if either, is preferable. In fact, I think neither can be endorsed wholeheartedly.

Chace's argument seems flawed by the point made in the previous Comment (Question 5), that unrealistic problems and answers do not in themselves imply a study of mathematics ‘for its own sake’,

Toomer's argument is difficult because he is making his judgement of Egyptian mathematics partly by means of a comparison with the Babylonians. Indeed, he seems engaged in a broader question than Chace does, one more to do with assessing how much the Egyptians contributed to the development or advance of mathematics, (This difference in the questions addressed by the two historians may go some way towards accounting for the difference in their conclusions.) It is also hard to separate out the validity of Toomer's judgement from the vigour of his language: ‘mere empiricism’, ‘hemmed in’, ‘crude methods’, ‘crude numeral system’, ‘primitive methods’, ‘primitive character’, leave us in no doubt about his feelings, certainly. In fact, Toomer's rhetoric is curiously stronger than that needed to make his point that the Egyptian influence on other cultures was not great. That might well be agreed independently of whether or not the Egyptians studied mathematics ‘for its own sake’.

What this exercise has also shown is that the endeavour to understand past mathematics naturally leads to forming judgements about it, in much the same way as art history and criticism are cognate activities. There is a disappointment in the reaction of some historians that a civilisation favoured by the gods in so many ways did not contribute more advanced mathematics. There was perhaps no need for more sophisticated mathematical investigation, nor any perception of such a possibility. As the historian Otto Neugebaucr has agreeably remarked:

Of all the civilisations of antiquity, the Egyptian seems to me to have been the most pleasant. The excellent protection which desert and sea provide for the Nile valley prevented the excessive development of the spirit of heroism which must often have made life in Greece hell on earth.
The Exact Sciences in Antiquity (Dover, 1969) p. 71.

Figure 4 Set square with plumb-line, c.1300 BC

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References

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

A.B. Chace (tr. and ed.), The Rhind Mathematical Papyrus, Mathematical Association of America, 1927.

A. Erman, The Literature of the Ancient Egyptians, Mathematical Association of America, 1927.

A. Gardiner, Egyptian Grammar, Oxford, 1957 (third edition), pp. 196-197.

Aristotle, Metaphysics 981^b 20-25; as 1.A1, p.195.

G.J Toomer, 'Mathematics and Astronomy', in J.R. Harris (ed.), The Legacy of Egypt, Oxford, 1971, pp. 37-40, 45.

Herodotus, History, II, 109; tr. A. D. Godley, Heinemann, 1920.

Plato, Phaedrus, 274 cd; tr. H. Cary, Bell & Daldy, 1872.

Proclus, On Euclid, I; tr. I. Thomas, Greek Mathematical Works I, Heinemann, 1939, pp. 145-147.

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Introduction

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This unit is from our archive and it is an adapted extract from Topics in the history of mathematics (MA290) which is no longer in presentation. If you wish to study formally at The Open University, you may wish to explore the courses we offer in this curriculum area.

This unit looks at Babylonian mathematics. You will learn how a series of discoveries have enabled historians to decipher stone tablets and study the various techniques the Babylonians used for problem-solving and teaching. The Babylonian problem-solving skills have been described as remarkable and scribes of the time received a training far in advance of anything available in medieval Christian Europe 3000 years later.

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Learning outcomes

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

After studying this unit you should be able to:

know something about cuneiform how it was used to represent numbers for mathmatical problem solving and computation;
understand the relationship between a decimal place-value system and a sexagesimal one;
appreciate the advanced understanding of mathematics in Acient Mesopotamia in relation to anyone in medieval Christian Europe 3000 years later.

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1 Babylonian mathematics

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

In Mesopotamia, the scribes of Babylon and the other big cities were impressing on clay tablets economic and administrative records, literary, religious and scientific works, word-lists, and mathematical problems and tables. Nearly all of the texts that give us our fullest understanding of Babylonian mathematics—indeed, of any mathematics before the Greeks—date from about 1800—1600 BC. During this period, King Hammurabi unified Mesopotamia out of a rabble of small city-states into an empire whose capital was Babylon, which was on the river Euphrates sixty miles south of present-day Baghdad. It is in this area, the valley of the two rivers Euphrates and Tigris which lead into the Persian Gulf, that there is the most ancient evidence for writing, some 1500–2000 years earlier still (see Figure 1).

Figure 1

Long description

By the middle of the third millennium BC, the writing style had evolved into the highly abstract and unrepresentational cuneiform (‘wedge-shaped’) script. This script, which was used initially for writing down words in the Sumerian language, was later also adopted by neighbouring peoples. All of the Babylonian tablets are written in Akkadian, a Semitic language quite different from Sumerian, although some mathematical tablets do use a few Sumerian words.

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1.2 A Babylonian mathematical problem

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

Before seeing how our knowledge has been acquired, let us get into the spirit of things by ascertaining what a problem looks like once the modern cuneiform scholar has translated a tablet. The following example is taken from a tablet (see Figure 2), now at Yale University, translated by Otto Neugebauer and Abraham Sachs. Words in square brackets are their suggested reconstructions of what the tablet presumably says (where it is damaged), and words in parentheses are the translator's additions so that the English is (relatively!) more understandable.

I found a stone, (but) did not weigh it; (after) I subtracted one-seventh, added one-eleventh, (and) subtracted one-thir[teenth], I weighed (it): 1 ma-na. What was the origin(al weight) of the stone? [The origin(al weight)] of the stone was 1 ma-na, 9½ gin, (and) 2½ se.

This tablet contained 22 such problems-and-answers, none indicating how the answer was reached, and all involving a stone of 1 ma-na when weighed.

We can make little progress without knowing how the units of weight are related (there are in fact 60 gin to 1 ma-na, and 180 se to 1 gin), but it is possible to reach some conclusions from your experience.

Figure 2 Three views of a tablet from the Yale Babylonian Collection

Long description

Question 1

Do you think this is an actual practical problem? Have you seen anything like it before? Can you suggest what the tablet might have been for?

Discussion

It is clear this is not a practical problem—he would have done better to have weighed the stone when he found it, if he were really interested directly in its weight!

You probably noticed that it was formulated in terms of unit fractions. So, on the evidence of this tablet at least, similar things seem to have been taking place mathematically in Egypt and Babylon, at much the same time.

The facts that there are so many similar problems on the tablet and that no working is shown, both suggest that it may have been for teaching purposes in an oral teaching situation where the method was explained verbally. Just what was being taught is unclear, however, as several possibilities spring to mind: it could have been the method of solving problems like this; it could have been the learning of units of weight (for the solution comes out only if these are understood correctly). It could also have been a question of how to handle these unit fractions—notice that they are all awkward ones in that they do not divide into the weights (one-seventh of a ma-na does not come out as a whole number of gin or of se).

There is one further point that we should mention, in case you tried to work out the problem but could not obtain his answer. The fractions in the question are not all parts of the original stone, but are parts of whatever the previous step has been. So it is the stone less its seventh, plus the eleventh of that, and so on. This makes for a slightly more complicated calculation than most of the otherwise similar Rhind Papyrus problems that exist from this period in Egypt.

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1.3 The historical study of cuneiform

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Now, how did historical study reach the stage where Neugebauer and Sachs could pick up a tablet in a library and translate it so as to provide a fair degree of understanding? As with Egyptian hieroglyphs, cuneiform studies date from the last century. Their equivalent of the Rosetta Stone—a trilingual inscription for which one of the languages could be partially understood—was a sheer rock-face at Behistun in south-western Iran into which a text was carved in three languages, Old Persian, Elamite and Babylonian, proclaiming the victories of Darius the Great (520 BC). It was the British Consul in Baghdad, Henry Rawlinson, who rediscovered this inscription and between 1835 and 1851 copied it (at the risk to his life that any amateur mountaineer faces 300 feet up a precipice) and began to decipher both the script and the languages. Shortly thereafter, the burgeoning science of archaeology resulted in excavations of cuneiform tablets from ancient sites in Mesopotamia. These have sometimes been unearthed in vast quantities, with the result that there are now many more tablets available, in museums and universities throughout the world, than have been translated or even catalogued. It is only a small proportion of these that have been shown to have mathematical content, perhaps five hundred or so, compared with the several hundred thousand extant tablets. The results of studying these emerged in the 1920s and 1930s, and led to a considerable re-evaluation of the Babylonians, who within a decade changed from being a bare footnote to biblical studies (as in the Tower of Babel), to being a culture whose mathematical attainments put those of the Greeks of 1200 years later into a fresh perspective.

The earliest understanding to emerge was that of the Babylonians' remarkable numeration system. This discovery was due, once again, to Henry Rawlinson, who in 1855 was studying a tablet from the ancient city of Larsa. Look at the illustration and see if you can identify some of its main features, then come back to the description here.

Figure 3 Tablet from Larsa

Long description

It seems to consist of four columns, of which the second and fourth do not change, but the first and third do. The third, especially, changes in so regular a way that it is fair to infer that this is a column of successive numbers, constructed on a principle like that of the Egyptian hieroglyphic numbers. If represents 1, and is 10, then the third column would be the numbers 49, 50,51,…, 58, 59, and then 1, for a reason that is not yet clear. The first column, though, is not so regular and has the curious feature that like symbols (if that is what they are) are not all collected together. The third line, for instance, has four 0s, then three 1s, then two 10s, then one 1. Rawlinson realised that all could be consistently explained if the assumption were dropped that a number sign could represent only one number value. So he suggested that the 1-symbol at the foot of the third column was to be understood as 60, and that the third line's first-column number was forty-three 60s and twenty-one 1s, This is, then, a place-value system (see Box 1), in which the value of each component number symbol depends on its place in the numeral as a whole.

Box 1: A note on numeration systems

We write numerals in what is called the decimal place-value system: in ‘88’, for example, the first 8 has a value which is ten times that of the 8 in the units place.
We also have distinct symbols, 1,2,…, 9, 0, to put in each place without involving repetition; we have enough distinct symbols to avoid having to repeat ‘1’ eight times to signify 8, for instance.

The Babylonians had a numeration system as in A, except that it was sexagesimal—each place has value sixty times the next, compared with our ten times.

For constructing numbers within each place, the Babylonians used a repetitive system as with the Egyptian hieroglyphs. If there were no value in some place (which is what our zero symbol signifies) a space was sometimes left, but otherwise meant 1 or 60 or 3600 (or, indeed, or etc.) according to context. In much later sources, mainly astronomical texts dating from c. 300 BC onwards, a zero symbol is found to mark empty places within numerals; but not at the end of a numeral, so the absolute value of the whole is still left ‘floating’.

Question 2

Try to transcribe the Larsa tablet. Can you suggest what the cuneiform words (columns two and four) might mean? (Hint: You may find that it helps to form some initial hypothesis about relationships among the numbers, and see if this is borne out elsewhere. So try first to work out the relationship of numbers in the second line (what relation does ‘forty-one sixties and forty’ have to fifty?), then see if the third line confirms this, and so to the whole tablet.)

Discussion

If you followed the hint, you should have found that

(forty-one sixties and forty) comes to 2500 in our numerals, which is the square of

(fifty). So we should try to see if this also works for other numbers. The next line we might hope would be the square of fifty-one, (that is, 2601), which indeed is forty-three sixties and twenty-one. It looks as though our hypothesis is on the right lines. Now we can go back to the first line, (which the hint steered you away from as it is slightly trickier), and note that if the first number is to be 49 squared (2401), then the cuneiform symbols cannot be forty-one anything—even though that is what they look like—but must be forty sixties and one unit.

It seems safe, then, to infer that column two carries the meaning ‘is’ or ‘equals’, and column four means ‘squared’, so the first line would read

2401 equals 49 squared

and so on, down to

3600 equals 60 squared.

There is one further thing to notice about our interpretation of the tablet. Suppose that all the numbers but one, say, had fitted our conjectured pattern: how should we respond to the inconsistent entry? It is just possible, number patterns being indefinitely many, that some other much more complicated interpretation could be found to cover every number without exception. Historians generally adopt the simpler view that the scribe must have made a mistake. Primary sources are not necessarily ‘correct’ merely by virtue of being old! Note that our confidence in sometimes changing the content of a tablet is only possible because of the mathematical structuring we presume it to have. Indeed, if a tablet is quite badly damaged it may be only that presumption that enables it to be reconstructed at all. (Perhaps this is a distinction between the history of mathematics and that of more empirical subjects.)

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1.4 A remarkable numeration system

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The Babylonian numeral system was described in Section 3 as ‘remarkable’. It is worth spelling out the reasons for this judgement. Although what we notice first is that it was a place-value system (see Box 1), what is perhaps more striking is the coupling of this feature with a ‘floating sexagesimal point’; that is, the lack of any indication about the absolute value of the number. This makes life hard for us in reading the tablets initially, but seems to have given the Babylonians unprecedented flexibility in calculations, because, among other things, there was no symbolic distinction between ‘whole numbers' and ‘fractions’. could be 30, or 1800 (=30×60), or ½ (=30×), or (=30×), and so on. This approach completely sidesteps the relatively cumbersome Egyptian technique of handling fractional parts, and, together with the use of multiplication tables, leads arguably to computations even smoother than our own (at least before pocket calculators). We presume that, in any case where the absolute value of the number was significant, this would be clear to the scribe from the context. Also he would have needed to have kept his wits about him in doing addition or subtraction, where the places need to be lined up correctly. This system was used consistently only within mathematics, as far as we know. In dating, weights and measures, economic records and the like, there seems to have been a wide mixture of units with many local variations. (You saw an example earlier, in the ma-na to gin to se ratios of our first problem.)

It follows that there is a translation problem even in the task of finding equivalents in our system for what the Babylonian scribe wrote down. A helpful notation has been devised by Otto Neugebauer. He represents the value within each sexagesimal place in our numerals, separating the places by commas, (So the entries in the first column of the Larsa tablet would be transcribed as 40,1 41,40 43,21 and so on.) This leaves unspecified, just as the scribe did, the absolute value of a number. However, if we have reason to believe that we know where the ‘integer part’ of the number ends and the ‘fractional part’ begins, then a semi-colon is used to separate them. So, for instance, 1,10;30 would represent 60+10+30/60, which is 70½, or 70.5 in decimal fractions.

Now let us try this notation, and see more of the flexibility of Babylonian calculations. Below is an array of numbers from a different tablet, transcribed in Neugebauer's notation.

2	30	16	3,45	45	1,20
3	20	18	3,20	4X	1,15
4	15	20	3	50	1,12
5	12	24	2,30	54	1,6,40
6	10	25	2,24		1
8	7,30	27	2,13,20	1.4	56,15
9	6,40	30	2	1,12	50
10	6	32	1,52,30	1,15	48
12	5	36	1,40	1,20	45
5	4	40	1,30	1,21	44,26,40

Question 3

What do you think the table is about? (Hint: Try multiplying the paired numbers together and see if a pattern emerges.)
Observe that in the left-hand columns, certain numbers such as 7, 11, 13, etc. do not appear. Which ones are missing? Can you suggest a reason for this?

Discussion

Let us take line 6, and multiply 7,30 by 8. Now, depending on whether we take 7,30 to be 7;30 (=7½), or 0;7,30 (i.e. 7/60 + 30/3600, =450/3600, =0.125 in decimal fractions), or 7,30;0 (= 450), when we multiply by 8 we get 60, or 1, or 3600 respectively. But these are all represented by the same cuneiform symbol, , and so it does not matter which we choose. You can reach an identical conclusion by looking at any other pair. In short, the table shows numbers which multiply together to give 1 (say), so we could call it a reciprocal table. The numbers in the right-hand columns are, in effect, 1 divided by the left-hand numbers (and vice versa—notice, for instance ‘45 1,20’ at the top right, and then ‘1,20 45’ lower down the same column).
If you try to divide 7, 11 or 13 into 60, and to express the result in sexagesimal fractions, you will find that it does not work exactly; the process is never-ending. More generally, notice that the only numbers appearing in the table are ones which are the result of multiplying together 2s, 3s and 5s in various combinations, and that 2, 3 and 5 are the only prime factors of 60 (=2×2×3×5). Thus 50 (=2×5×5) is in the table, but 21 (=3 × 7) is not. It is an arithmetical fact that only numbers whose prime factors are 2, 3 and 5 have reciprocals which can be expressed as finite (i.e. terminating) sexagesimal fractions. Such numbers we will call, following Neugebauer, regular numbers.

There are a couple of further points to make at this stage. There is some debate among historians about whether this is to be interpreted as a table of reciprocals, as suggested above, or whether its function was as a conversion table of fractional parts into their sexagesimal equivalent. (So 2 would stand for the second part, 3 the third part, and so on down to 1,21 as the eighty-first part, in something like the Egyptian mode.) In this interpretation, the columns would be not of numbers related reciprocally, but of the same number expressed in two different ways, as a unit fraction and as a sexagesimal fraction. This is an attractive idea, illustrating once again that it is harder than we should like to identify definitively even simple-looking tables. Fortunately, for our purposes, we do not need to resolve this point, so we shall continue to refer to the table as a reciprocal table.

Secondly, the Babylonians were quite able to divide by numbers other than regular ones, and approximated the results to three or four sexagesimal places. There are reciprocal tables for complete sequences of numbers (i.e. containing both regular and non-regular ones) from the period. Any division problem, therefore, could be converted into an equivalent multiplication one by using such tables. In order to divide by a number, you multiply by its reciprocal. If the number you are trying to divide by is regular, then the answer will be exact; otherwise, it will be approximate, (Note that the Babylonians had more regular numbers in the above sense than we have—any number with prime factors other than 2 and 5 lands us with an unending decimal fraction: e.g. ⅓=0.333… .)

We have now seen how the Babylonian numerals work, and also two examples of tables, one of squares and one of reciprocals, which give an idea of the level, spirit and flexibility of Babylonian computations. We did not try to ascertain how the results on the tables were arrived at, as that seemed either obvious or not very interesting. But thus equipped, we are ready to tackle a table whose method of construction does turn out to be rather interesting.

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1.5.1 Uncertain origins

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

The tablet is called Plimpton 322, and is described by Neugebauer (The Exact Sciences in Antiquity (Dover, 1969) p. 40) as ‘one of the most remarkable documents of Old-Babylonian mathematics’. The name arises simply from the fact that the tablet has catalogue number 322 in the George A. Plimpton collection at Columbia University, New York. Plimpton bought it in about 1923 from a Mr Banks who lived in Florida; it is not certain where he obtained it, but it may have been dug up at Larsa in Mesopotamia. The left hand side of the original tablet appears to have been broken off, and traces of modern glue suggest that this has happened since its excavation. (All of this is a fairly typical example of the random, not to say slapdash, way in which things have emerged from under the sand into the eventual light of public knowledge.)

Figure 4 Plimpton 322

Long description

Look at the photograph and notice the main features of what remains: four columns of numbers, with words at the head of each column. Now look at the transcription (we have labelled the columns A, B, C for ease of reference), and see if any pattern is evident to you

A	B	C
[1;59,0,]15	1,59	2,49	1
[1;56,56,]58,14,5O,6,15	56,7	3,12,1	2
[1;55,7,]41,15,33,45	1,16,41	1,50,49
[1;]5[3,1]0,29,32,52,16	3,31,49	5.9,1	4
[1;]48,54,1,40	1,5	1,37	5
[1;]47,6,41,40	5,19	8,1	6
[1 ;]43,11,56,28,26.4O	38,11	59,1	7
[1;]41,33,45,14,3,45	13,19	20,49	8
[1;]38,33,36,36	9,1	12,49	9
1;35,10,2,28,27,24,26,40	1,22,41	2,16,1	10
1 ;33,45	45	1,15	11
1;29,21,54,2,15	27,59	48,49	12
[1;]27,0,3,45	7,12,1	4,49	13
1;25,48,51,35,6,40	29,31	53,49	14
[1;]23.13,46,40	56	53	15

At first sight, this is not very promising! Something seems to be being listed, as the lines are numbered (final column); and the numbers in column A do diminish fairly regularly, from just under 2, down to just over 1⅓. (But if you did manage to notice that it was partly the effect of the editorially informed semi-coIons, and Neugebauer's reconstructions in square brackets, of course.) Otherwise, the numbers look fairly random, and there is little to tell that this is not a Babylonian supermarket till receipt. (Until Neugebauer studied the tablet, it was, in fact, catalogued as a ‘commercial account’.) But Neugebauer discovered—presumably after a considerable amount of conjecture and refutation—that the numbers in each line can be related as:

and this was the basis for his reconstruction of the illegible entries. Let us check this out on the simplest looking complete case, line 11.

So it is possible to find a relation, albeit a somewhat devious one, between the columns of the tablet. (In fact, for this to hold consistently throughout, the underlined numbers have to be considered as mistakes by the scribe, a point to which we shall return.) Before trying to decide what this is all about, let us investigate the numbers a little more. If we calculate C²−B² for each entry, and then take its square root, something rather surprising emerges. Look at the next table, in which we have calculated the values (and the scribe's ‘errors’ have been corrected’). Ignore for the time being the final two columns which we have labeled p and q.

B	C			p	q
		(decimal)	(sexagesimal)
119	169	120	2,0	12	5
3367	4825	3456	57,36	64	27
4601	6649	4800	1,20,0	75	32
12709	18541	13500	3,45,0	125	54
65	97	72	1,12	9	4
319	481	360	6,0	20	9
2291	3541	2700	45,0	54	25
799	1249	960	16,0	32	15
481	769	600	10,0	25	12
4961	8161	6480	1,48,0	81	40
45	75	60	1,0
1679	2929	2400	40,0	48	25
161	289	240	4,0	15	8
1771	3229	2700	45,0	50	27
56	106	90	1,30	9	5

Notice that, compared wilh columns B and C, the values computed in D are remarkably simple looking numbers (this is particularly noticeable in their sexagesimal representation). You may indeed have recognised some as old friends, from the reciprocal table. In fact, all of them are regular numbers (whereas B and C are all, except for line 1L, non-regular), which explains why their squares could divide into C² and yield an exact, finite sexagesimal expression.

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1.5.2 What is the significance of the numbers?

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

In seeking the significance of these numbers, there is more information on the tablet that we have not yet taken into account, namely the text of the column headings themselves. The heading of column A is partly destroyed, but the text headings for B and C are clearer. B says something like ‘ib-sa of the front’, and C ‘ib-sa of the diagonal’, where ib-sa is a Sumerian word whose significance here is not precisely known. The geometrical cue provided by the headings suggests that we try viewing the numbers on the tablet as relating to geometrical figures. Indeed, as the numbers in the table above have the property B²+D²=C² (which follows immediately from the way D was defined), it appears that the middle two columns of Plimpton 322 contain the shortest side and diagonal respectively of right-angled triangles, whose third side is a simple regular number. Successive lines in the table can then be seen to be ordered so that the triangles gradually change shape from B being almost equal to C, to it being rather shorter. (This is equivalent to what is shown in column A, where the ratio C²/D² diminishes as one goes down the column—note that the actual numerical size of the triangles changes randomly, it is just the shape that varies in a smooth ordered way.)

We should think carefully about what these geometrical connotations of the numbers on the tablet might mean. We know from several other tablets that the Babylonians were well aware that adding the squares of the lengths of the two shorter sides of a right-angled triangle gives the same number as squaring the length of the diagonal (one version of the result that has come to be called, for no very sound historical reason, ‘Pythagoras' Theorem’). But why were whole number examples of such triangles listed on this tablet? (If that is indeed what the numbers signify.) Why ordered in this curious way? Why these particular triangles? What is the purpose of column A? The successful decipherment of Plimpton 322 has managed to raise rather more questions than when it was safely catalogued as a ‘commercial account’!

Historians have reached various conclusions about what this tablet signifies. Neugebauer and Sachs, the original decipherers, regarded it as ‘the oldest preserved document in ancient number theory’ and that it concerned ‘the fundamental laws of the numbers themselves’. (O. Neugebauer and A. Sachs, Mathematical Cuneiform Texts (American Oriental Society, 1945) pp. 37, 41.) So, on this interpretation, it has little to do with geometry as such, except incidentally, but more to do with properties of numbers. For what struck them especially, when trying to work out how the table had been constructed, was that each line can arise from a pair of small regular numbers (this is the significance of the columns labelled p and q in the table above). In fact,

B = p²; C = p² + q²; D = 2pq,

as you can see by trying this out for yourself on any of the lines of the above table. So it is possible (no less, no more) that the Babylonian scribe was aware of these formulae though he would not, of course, have written them in anything like the modern algebraic form just given. These formulae generate Pythagorean triples, i.e. sets of whole numbers B, D, C, such that B² + D²=C². Was Plimpton 322 part of an investigation of number relationships?

It is evident, certainly, that the numbers on Plimpton 322 were the result of some mathematical procedure carried out off-stage, on another tablet perhaps. Apart from common-sense grounds, this can be inferred from the nature and occurrence of the errors on the tablet.

Figure 5

Long description

Question 4

Look again at the original translation of Plimpton 322, noting where the errors by the scribe were made. Are these compatible with column A having been calculated directly from columns B and C, as our ‘formula’ A=C²/(C² – B² ) might have tended to suggest?

Discussion

If column A had been calculated directly from columns B and C, then any mistake in B or C wouid carry over to A. But this has not happened, so A must have been calculated independently of B and C.

It is also worth noting that the scribe made four errors in computing the relatively simple B and C numbers, but none of consequence in the more complicated A column (there were in fact two minor confusions of place value, here ‘corrected’ in the transcription). As this is contrary to what one would expect, it seems a significant observation about the computations that went in to the tablet, even if it is not clear just what the significance is.

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1.5.3 Errors in Plimpton 322

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

The presence of errors on the tablet is of further benefit to the historian, in that trying to discover how they could have arisen provides strong clues about how the computations were done. So, for instance, the last entry in column C is half what it ‘should’ be. The scribe wrote 53, where 1,46 is what is needed to preserve the pattern of the rest. This tells us that some stage in the computation must have involved a doubling or halving, which on this occasion the scribe overlooked.

Finally on this issue, it should be remarked that the very detection of ‘errors’ is bound up with the historian's analysis and understanding of the tablet. Although there is clearly an error somewhere along the bottom line, our pattern is equally preserved by taking it to be the column B entry that is in error, as double what it should be. This, however, disrupts the analysis in terms of generation by means of p,q pairs. The choice, then, between these two alternative errors depends on one's confidence that the Babylonian scribe was indeed generating the numbers on the tablet from something analogous to the formulae in terms of p and q given above.

Because of the tablet's evident importance, many historians have considered this question. Not all historians have followed Neugebauer and Sachs in their interpretation of Plimpton 322 as an investigation of number theory. An interesting suggestion has been put forward by Jöran Friberg, who argues that the tablet was a ‘teacher's aid’ for setting and solving problems involving right-angled triangles. So, for example, if the teacher wanted to set a problem like, ‘a ladder of length C leans against a wall, its base is distance B from the wall, how high up the wall does it reach?’. Then choosing the numbers B and C from the Plimpton tablet would ensure that the answer (D) would come out nicely, and not leave the poor student with an unending sexagesimal fraction to cope with. We have no space here to investigate Friberg's justification for this view, but it does sound plausible, especially because it is attentive to the situation in which we believe the tablets to have been produced and used. Please read the short extract from Friberg attached as a pdf. We shall then move on to consider this wider aspect of the social context of Babylonian mathematical activity.

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The purpose of Plimpton 322 by Joran Frïberg

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1.6 The social context of Babylonian mathematical activity

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The extant mathematical tablets from the Old Babylonian period fall broadly into two categories, table texts and problem texts. You have seen examples of both of these. The weighing-the-stone problem with which we started is from a problem text, while all the others—the table of squares, the reciprocal table and Plimpton 322—are table texts, tablets consisting solely of tables of numbers. Several hundred table texts have been found, and many types of calculations appear to have been carried out by means of them. As well as squares and reciprocals, there are multiplication tables, tables of cubes and cube roots, tables of the sums of squares and cubes, combined tables where several of these are present, tables for working out compound interest, tables of weights and measures, and others. Numerical tables seem to have been a staple constituent of Babylonian life, as ubiquitous for them as is the pocket calculator for us today. Problem texts, by contrast, are rarer—only a hundred or so tablets have been found—and they seem to relate to an educational context of advanced scribal training. Early Mesopotamian culture had seen the development of specialised occupations, as a part of the newly-developing and highly-complex urban structuring of the community, and the profession of scribe was central to the running of economic, bureaucratic and other aspects of the state. There were special institutions, schools, for training future scribes in the arts of writing, counting and accounting, and other necessary skills. (The attached pdf affords revealing glimpses into the scribal art and its training.)

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Scribal art and its training

It is in this educational context that the problem texts seem to have been written and used. Some merely give the problem and the answer (the one you saw in Question 1 was of this sort); others are fortunately more forthcoming on what to do to reach the answer. Let us look at one of these now.

Please read through the extract below to gain an impression of its style.

I have subtracted the side of my square for the area: 14, 30. You write down 1, the coefficient. You break off half of 1. 0; 30 and 0; 30 you multiply. You add 0; 15 to 14, 30. Result 14, 30; 15. This is the square of 29; 30. You add 0; 30, which you multiplied, to 29; 30. Result 30, the side of the square.

The problem is given in the first sentence, the rest is its solution. It turns out that after doing various things to the numbers initially given, the result 30 is reached. This indeed solves the original problem, for a square of side 30 has area 30² = 900 (= 15,0 in sexagesimal), and subtracting 30 from 900 gives 870 (= 14,30). This is a problem of the kind we would call quadratic, that is, involving the square of some unknown number which is to be found. As a way in to understanding the Babylonian computational procedure for this problem, let us follow the instructions through in a modern algebraic format (and take into account later how far this may have distorted the Babylonian scribe's activity).

Let us call the unknown, the side, ‘x’; the coefficient, (which here is 1), call ‘b’; and the number in the statement of the problem (here, 14,30) call ‘c’. Then the problem is; to find x, where x² − bx = c. The solution consists of taking the coefficient b, halving it, ½b squaring that, (½b)²; adding this to the result, c+(½b)²; taking the square root of that, √(c + (½b)²); and finally adding the result to the halved coefficient ½b + √(c + (½b)²); which is the answer. Indeed, this algebraic formula is just the same as we would reach (see Box 2 below), which is most satisfactory—or perhaps, on another consideration, somewhat alarming. If our method of understanding what the Babylonian scribe might have been doing is to turn him into a twentieth-century algebraist, it is possible that there has been some misunderstanding.

Figure 6 A Babylonian tablet now in the British Museum (BM 13901)

Long description

Box 2: A note on solving quadratic equations

If we have a quadratic equation (a problem with one unknown, involving the square of that unknown but no higher power) in its standard form

then we find a solution by putting the values given for a, b and c inlo the formula

which can be obtained from the standard form of the original equation through a succession of algebraic operations called completing the square.

In the particular case of a quadratic equation of the form

into which the problem specified on the tablet fits, the procedure of completing the square goes as follows.

Take the coefficient b; halve it (½b); square that and then add it to both sides:

Then the left hand side is the square of (x − ¹/₂b that is,

so take the square root of both sides, giving

Finally, add ½b to both sides to give

Question 5

How does the symbolic description given above compare with what the Babylonian scribe did? Approach this by considering separately: (i) what similarities there are, and (ii) what differences there are.

Discussion

The sequence of instructions given by the scribe seems to follow quite closely the procedure called (in Box 2) completing the square, in terms of actions on theparticular numbers specified at the outset.

You may have noticed another similarity if you pondered the way in which the Babylonian problem was formulated. As with the stone of Question 1, the scribe seems to be labelling the unknown in an abstract, symbolic way. To see this, consider the alternative possibility that this is a realistically geometrical problem, as the words side and area seem, on the face of it, to imply. Surely the formulation of taking a side away from an area does not really make sense 10 our way of thinking about geometry (apart from conjuring up imagery of farmers ploughing up hedgerows). They are different kinds of things–taking away a hedge does not alter the numerical measure of the area that was previously enclosed. So either Babylonian geometry was quite different from ours, or they were using the terms side and area not as meaning geometrical entities particularly, but more as unknown and square of the unknown. In other words, side referred to some number to be discovered, with no connotations of where it came from or what, if anything, it measured. This is similar, perhaps, to the way we speak of x squared without necessarily imagining a square.
I expect you observed that there are major notational differences. It is worth noting that these are of three sorts: our ‘x’ for unknown, our ‘b’ and ‘c’ for fixed numbers, where we do not want to be specific about what they are, and all the other symbols (+, √, =, and so on). The latter seem fairly harmless translations of Babylonian words and operations in this case.

It has been argued above that our x and their side are conceptually more similar than they may first sight. But in changing their numbers into our ‘b’ and ‘c’ we effected a rather dramatic conceptual change. There is nothing in the Babylonian text paralleling our formula, a structure in which all the contributions of the original coefficients are still evident.

But bear in mind that our ‘formula’ only makes sense through our understanding the conventions about the order in which the operations it embodies are to be performed. Given a formula like √(c + (½b)²), we are taught to interpret it as, ‘square ½b, then add c, before taking the square root…’, which is beginning to sound like Babylonian instructions again. Indeed, the parallel is all the more marked in the computational techniques developed over the last couple of decades Solving a quadratic equation on a pocket calculator or computer involves carrying out a sequenced program of operations closely mirroring the instructions on the Babylonian tablet, even down to pushing the ‘square root’ button at just the stage the scribe would have leant over to consult his square root tables.

All the problem texts that have solutions are of this sort, apparently instructing about a general approach through particular instances. In some cases, all the answers on a particular tablet turn out to be the same, which seems a clear indication that it was the journey rather than the destination that mattered. This is confirmed by details within the calculations. For instance, sometimes a number is explicitly multiplied by 1, which seems pointless until one realises that, in general, it might be some number other than 1 at that stage. This serves as a reminder that some multiplication is to be done there.

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1.7 Babylonian mathematical style

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

Not only should you have learnt through this exercise more about the Babylonian mathematical style, but also, on another level, you should have gained more experience in the endeavour of trying to understand past mathematics. The model that we have been trying out can be characterised thus: use any means, any symbolism or notation that occurs to you, to find your way into the problem, then check rigorously to see how much of your new understanding is more a projection backwards from your own time and techniques. First, try to understand what they might have been doing. Then address the harder questions of how and why. As this process becomes more familiar, you will find it increasingly easy both to respond to past mathematics on its own terms, and to understand and evaluate historical questions and concerns. (Doing this may also produce the added bonus of a better understanding of your own mathematics.)

Figure 7 Babylon

Long description

The Babylonian ‘quadratic’ problem above is fairly characteristic of problem texts; some have two unknowns (length and width) and two conditions connecting them, and a few even have three unknowns and three conditions. Some involve cubes or higher powers of unknown numbers which are to be found. The problems are expressed sometimes fairly abstractly, as we have seen, sometimes in terms of more concrete imagery that seems direct training for practical problems which a scribe might be called upon to solve professionally. (See, for example, the collection of problems in the attached pdf concerning a number of workers digging a volume of earth for so many days at a certain expected productivity.) Yet even apparently practical problems can have flamboyantly unrealistic solutions—on one tablet (see the attched pdf), the problem is to discover what area of field can be irrigated by a particular volume of water, and the answer turns out to be a field some 3½ kilometres square, covered with water to a uniform depth of one finger's breadth, a procedure that would tax even the most diligent of Babylonian farmers!

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Problem (e)

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Problem (b)

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1.8 Conclusion

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In conclusion, what is Babylonian mathematics about? Although it is not easy to answer this question precisely, because of the difficulties of interpretation such as you saw with Plimpton 322, the overwhelming impression is of the study and use of numbers, and various techniques for solving problems involving numbers. Where the numbers arise from—whether land measurement, economic questions, idealised geometrical objects (cubes, triangles and so on), or just fairly abstractly—seems a relatively secondary matter. As Otto Neugebauer has put it:

The central problem in the early development of mathematics lies in the numerical determination of the solution which satisfies certain conditions. At this level there is no essential difference between the division of a sum of money according to certain rules and the division of a field of given size into, say, parts of equal area.
O. Neugebauer, The Exact Sciences in Antiquity pp. 44–5.

The Babylonian problem-solving skills, as we infer them from the problem texts, were remarkable. The Babylonian scribe, for example, received a training in these matters far in advance of anyone in medieval Christian Europe 3000 years later. The Old Babylonian knowledge was preserved somehow through all the alarums and excursions of subsequent Mesopotamian history, and filtered through in some measure to the later Alexandrian Greeks. Just what knowledge, however, and when, is not precisely known at present.

Babylonian techniques are sometimes described as algebraic, although other historians would reject the applicability of the term. You can begin to see that the style of mathematics used in some culture or period can be a better way of trying to understand and empathise with past mathematical activity than is the desire to fit topics of concern into pre-ordained slots. For, as we have already seen, the ostensible subject matter, the language used, and the techniques applied do not always fit together in past cultures in the way they do in ours. It is some combination of all these things that is characterised by the word style.

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1.9 Further reading

The Open University — Tue, 05 Jul 2011 09:40:41 GMT

Doblhofer, Ernst, Voices in Stone (Paladin, 1973; orig. edn. 1957). Not especially mathematical, but a good account of the decipherment of hieroglyphs and cuneiform texts if you want to follow that up.

Flegg, Graham, Numbers (Penguin, 1984; orig. edn. 1983). A book packed with much more information about numbers and their history than we have time for during the course.

Friberg, Jöran, ‘Methods and Traditions of Babylonian Mathematics’, Historia Mathematics 8 (1981) pp. 277–318. The most recent, full discussion of Plimpton 322, a little more advanced than ours, and what we can learn about how and why Babylonians used mathematics.

MacKie, Euan, Science and Society in Prehistoric Britain (Elek, 1977). A useful and scholarly attempt to see what picture emerges from the results of archaeological and other recent research.

Menninger, Karl, Number Words and Number Symbols (MIT Press, 1969; orig. edn. 1958). More full detail of the history of numbers, particularly attentive to linguistic evidence.

Neugebauer, Otto, The Exact Sciences in Antiquity (Dover, 1969; orig. edn. 1949).A highly readable account by one of the most eminent of modern historians of the period, van der Waerden, B. L., Science Awakening I (Oxford University Press, 1961). A full and carefully considered account of what is known of ancient mathematics, with much more detail about Egyptian and Babylonian problems than we have been able to cover.

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