Math Problems from Ancient Egypt

Math Problems From Ancient Egypt

Note: This article has been excerpted from a larger work in the public domain and shared here due to its historical value. It may contain outdated ideas and language that do not reflect TOTA’s opinions and beliefs.

From Adolf Erman’s Life in Ancient Egypt, 1894.

Every department of Egyptian intellectual activity that we have as yet examined, we have found to be overgrown with superstition and magic. One branch of science, viz. mathematics, remains however untouched, so far as we know, by these noxious weeds. Thanks to a papyrus in the British Museum, we are now pretty well informed on this subject. This book, which is a copy made under one of the Hyksos kings of an older work, is a collection of specimen examples of all kinds of arithmetical and geometrical problems, and thus it gives us a good idea of the proficiency the Egyptians had then attained.

Their knowledge of this science at that time was not very great, and we doubt whether they carried their studies much further even under the New Empire, for, more than a century and a half later, we find in the agricultural lists of the temple of Edfu the same primitive ideas of geometry which are contained in our old book.

Mathematics as well as medicine seems to have remained stationary at the same stage that it had reached under the Old Empire; progress was made in certain details, but a genius seems never to have arisen to give a fresh impetus to this science. There was indeed no need. The problems presented to the skill of the arithmetician were ever the same, and if the solution, which was often only an approximate one, had contented the government of the Old Empire, it sufficed also for that of the New Empire.

Mathematics served merely a practical purpose for the ancient Egyptians,— they only solved the problems of everyday life, they never formulated and worked out problems for their own sake. How certain eatables were to be divided as payments of wages; how, in the exchange of bread for beer, the respective value was to be determined when converted into a quantity of corn; how to reckon the size of a field; how to determine whether a given quantity of corn would go into a granary of a certain size— these and similar problems were taught in the arithmetic book.

In pure arithmetical examples there are no errors as far as I can see, at most a small fraction is sometimes purposely disregarded. Everything is worked out in the slowest and most cumbrous manner—even the multiplication of the most simple numbers. If, for instance, the schoolboy had to find out in his sum the product of 8 times 8, this difficult problem would be written out thus:

1 8

2 16

4 32

8 64

Evidently in mental arithmetic, he was only equal to multiplication by 2. 1 Strange to say also, he had no proper method for division; he scarcely seems to have had a clear idea of what division meant. He did not ask how many times 7 was contained in 77, but with which number 7 was to be multiplied that the product might be 77. In order to discover the answer he wrote out the multiplication table of 7, in the various small numbers, and then tried which of these products added together would give 77:

1 7

2 14

4 28

8 56

In this instance the multiplicators belonging to 7 and 14 and 56 are marked by the pupil with a stroke and give the numbers wanted. Therefore it is necessary to multiply 7 by 1 + 2 + 8, i.e. by 11, in order to obtain the 77, or 7 goes 11 times in 77. If the question were how often is 8 contained in 19, or in other words what number must 8 be multiplied by to obtain 19, the result of the sum:

1 8

2 16

½ 4

¼ 2

⅛ 1

would be that 2 and ¼ and ⅛ are the desired numbers, for the addition of the numbers belonging to them would exactly make 19. We should say: 8 goes 2 ⅜ times in 19.

In connection with this imperfect understanding of division, it is easy to understand that the Egyptian student had no fractions in our arithmetical sense. He could quite well comprehend that a thing could be divided into a certain number of parts, and he had a special term for such a part, e.g. re-met= mouth often, i.e. a tenth. This part however was always a unit to him, he never thought of it in the plural; they could speak of "one tenth and a tenth and a tenth" or "of a fifth and a tenth," but our familiar idea of 3/10 did not exist in the mind of the Egyptian.

Two thirds was an exception; for ⅔ he possessed a term and a sign,— it was his only fraction which was not of the most simple kind. When he had to divide a smaller number by a greater, for instance 5 by 7, he could not represent the result as we do by the fraction 5/7 but had to do it in the most tiresome roundabout fashion. He analysed the problem either by the division of 1 by 7 five times, so that the result would be 1/7 + 1/7 + 1/7 + 1/7 + 1/7, or more usually he took the division of 2 by 7 twice and that of 1 by 7 once. There were special tables which gave him the practical result of the division of 2 by the odd numbers of the first hundred. He thus obtained ¼, 1/28, ¼, 1/28. 1/7, which he then knew how to reduce to ½ + 1/7 + 1/14.

If by this awkward mechanism they obtained sufficiently exact results, it was owing exclusively to the routine of their work. The range of the examples which occurred was such a narrow one, that for each there was an established formula. Each calculation had its special name and its short conventional form, which when once practised was easily repeated. The following example giving the calculation of a number may illustrate what has been said:

a. | A number together with its fifth part makes 21.

b. | 1 5

| 5 1 together 6

c. | 1 6

| 2 12 | ½ 3 together 21

d. | 1 3 ½

| 2 7

| 4 15 (read 14)

e. | The number 17 ½

⅕ 3 ½ together 21

I scarcely think that the most expert arithmetician will grasp what these figures mean; it is only by the comparison of similar calculations that we can understand all these abbreviations. The proposition formulated by a corresponds to the equation x + ⅕ = 21, the result of which x = 17 ½is given quite correctly in e. As the Egyptian could not very well calculate in fractions, he had next to create this wretched ⅕ x; this he did thus: in b he multiplies the number and the fifth of the number by 5, which together makes 6. In c 21 is divided according to the cumbrous Egyptian method by this 6, the result being 3 ½. This 3 ½ would have been the desired number if we had not beforehand in b changed the fraction 6/5 into the number 6 by multiplication with 5; the result of our division must therefore be made five times greater. This multiplication takes place in d, and gives the final result 17 ½. In e we find the example is proved by adding together this 17 ½ with the ⅕ obtained above, namely 3 ½, which correctly gives the 21 of our problem.

Written after our fashion, the whole would stand thus :

(a) 6/5 x = 21 (b) 6x = 21 x 5 (c) x = 21/6 x 5 (d) x = 3 ½ x 5 (e) x = 17 ½. Proof: 17 ½ + 3 ½ =21.

The Egyptians knew still less of geometry than they did of arithmetic, though surface-measurement was most necessary to them, because of the destruction of so many field boundaries in the inundation every year. All their calculations were founded on the right angle, the content of which they correctly determined as the product of the two sides.

But in the strangest way they quite overlooked the fact that every quadrilateral figure, in which the opposite sides are of the same length, could not be treated in the same manner. Now as they treated each triangle as if it were a quadrangle, in which two sides are identical, and the others are half the size, they carried this error into the calculation of triangles also. To them also an isosceles triangle equalled half the product of its short and its long side, because they would in all cases determine the quadrangle corresponding to it by the multiplication of its two sides, though it were nothing but a right angle. The error which would arise from this kind of misconception might be considerable under some circumstances.

The calculation of the trapezium suffered also from the same error; in order to find its content, they would multiply the oblique side by half the product of the two parallel sides. As we see, the fundamental mistake of these students of surface-measurement was that they never realised the value of the perpendicular; instead of the latter they used one of the oblique sides, and therewith from the outset they excluded themselves from the correct manner of working. It is remarkable that, with such errors, they should have rightly determined approximately the difficult question of the area of a circle; in this case they deducted a ninth of the diameter, and multiplied the remainder of the same by itself. Thus if the diameter of a circle amounted to 9 rods, they would calculate its area to be 8 x 8 = 64 square rods, a result which would deviate from the correct result by but about ⅔ of a square rod. 4

Amongst the volumetric problems which they attempted, they would calculate, for instance, the quantity of corn which would go into a granary of a certain size; the little that we can as yet understand with any certainty of these problems seems to indicate right conceptions, but the conditions are here too complicated for us to give a decided opinion.

Probably however if we understood them, it would not much alter our general impression as to the mathematics of the ancient Egyptians: our conclusion on this subject is that there is little to be said for their theoretic knowledge of this science, but their practical knowledge sufficed very well for the simple requirements of daily life.

Bibliography

1. Adolf Erman, Life in Ancient Egypt, trans. H. M. Tirard (New York, NY: Macmillan and Co., 1894), 364-68.

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