Civilisation reached a high level in Egypt at an early period. The country was well suited for the people, with a fertile land thanks to the river Nile yet with a pleasing climate. It was also a country which was easily defended having few natural neighbours to attack it for the surrounding deserts provided a natural barrier to invading forces. As a consequence Egypt enjoyed long periods of peace when society advanced rapidly.

By 3000 BC the Egyptians had already developed their hieroglyphic writing (see our article Egyptian numerals for some more details). This marks the beginning of the Old Kingdom period during which the pyramids were built. For example the Great Pyramid at Giza was built around 2650 BC and it is a remarkable feat of engineering. This provides the clearest of indications that the society of that period had reached a high level of achievement.

Hieroglyphs for writing and counting gave way to a hieratic script for both writing and numerals. Details of the numerals themselves are given in our article Egyptian numerals. Here we are concerned with the arithmetical methods which they devised to work with these numerals

The Egyptian number systems were not well suited for arithmetical calculations. We are still today familiar with Roman numerals and so it is easy to understand that although addition of Roman numerals is quite satisfactory, multiplication and division are essentially impossible. The Egyptian system had similar drawbacks to that of Roman numerals. However, the Egyptians were very practical in their approach to mathematics and their trade required that they could deal in fractions. Trade also required multiplication and division to be possible so they devised remarkable methods to overcome the deficiencies in the number systems with which they had to work. Basically they had to devise methods of multiplication and division which only involved addition.

Early hieroglyphic numerals can be found on temples, stone monuments and vases. They give little knowledge about any mathematical calculations which might have been done with the number systems. While these hieroglyphs were being carved in stone there was no need to develop symbols which could be written more quickly. However, once the Egyptians began to use flattened sheets of the dried papyrus reed as "paper" and the tip of a reed as a "pen" there was reason to develop more rapid means of writing. This prompted the development of hieratic writing and numerals.

There must have been a large number of papyri, many dealing with mathematics in one form or another, but sadly since the material is rather fragile almost all have perished. It is remarkable that any have survived at all, and that they have is a consequence of the dry climatic conditions in Egypt. Two major mathematical documents survive.

You can see an example of Egyptian mathematics written on the Rhind papyrus and another papyrus, the Moscow papyrus, with a translation into hieratic script. It is from these two documents that most of our knowledge of Egyptian mathematics comes and most of the mathematical information in this article is taken from these two ancient documents.

Here is the Rhind papyrus

The Rhind papyrus is named after the Scottish Egyptologist A Henry Rhind, who purchased it in Luxor in 1858. The papyrus, a scroll about 6 metres long and ¹/₃ of a metre wide, was written around 1650 BC by the scribe Ahmes who states that he is copying a document which is 200 years older. The original papyrus on which the Rhind papyrus is based therefore dates from about 1850 BC.

 

Here is the Moscow papyrus

The Moscow papyrus also dates from this time. It is now becoming more common to call the Rhind papyrus after Ahmes rather than Rhind since it seems much fairer to name it after the scribe than after the man who purchased it comparatively recently. The same is not possible for the Moscow papyrus however, since sadly the scribe who wrote this document has not recorded his name. It is often called the Golenischev papyrus after the man who purchased it. The Moscow papyrus is now in the Museum of Fine Arts in Moscow, while the Rhind papyrus is in the British Museum in London.

The Rhind papyrus contains eighty-seven problems while the Moscow papyrus contains twenty-five. The problems are mostly practical but a few are posed to teach manipulation of the number system itself without a practical application in mind. For example the first six problems of the Rhind papyrus ask how to divide n loaves between 10 men where n =1 for Problem 1, n = 2 for Problem 2, n = 6 for Problem 3, n = 7 for Problem 4, n = 8 for Problem 5, and n = 9 for Problem 6. Clearly fractions are involved here and, in fact, 81 of the 87 problems given involve operating with fractions. Rising, in [37], discusses these problems of fair division of loaves which were particularly important in the development of Egyptian mathematics.

Some problems ask for the solution of an equation. For example Problem 26: a quantity added to a quarter of that quantity become 15. What is the quantity? Other problems involve geometric series such as Problem 64: divide 10 hekats of barley among 10 men so that each gets ¹/₈ of a hekat more than the one before. Some problems involve geometry. For example Problem 50: a round field has diameter 9 khet. What is its area? The Moscow papyrus also contains geometrical problems.

Unlike the Greeks who thought abstractly about mathematical ideas, the Egyptians were only concerned with practical arithmetic. Most historians believe that the Egyptians did not think of numbers as abstract quantities but always thought of a specific collection of 8 objects when 8 was mentioned. To overcome the deficiencies of their system of numerals the Egyptians devised cunning ways round the fact that their numbers were poorly suited for multiplication as is shown in the Rhind papyrus.

We examine in detail the mathematics contained in the Egyptian papyri in a separate article Mathematics in Egyptian Papyri. In this article we next examine some claims regarding mathematical constants used in the construction of the pyramids, in particular the Great Pyramid at Giza which, as we noted above, was built around 2650 BC.

Joseph [8] and many other authors gives some of the measurements of the Great Pyramid which make some people believe that it was built with certain mathematical constants in mind. The angle between the base and one of the faces is 51° 50' 35". The secant of this angle is 1.61806 which is remarkably close to the golden ratio 1.618034. Not that anyone believes that the Egyptians knew of the secant function, but it is of course just the ratio of the height of the sloping face to half the length of the side of the square base. On the other hand the cotangent of the slope angle of 51° 50' 35" is very close to π/4. Again of course nobody believes that the Egyptians had invented the cotangent, but again it is the ratio of the sides which it is believed was made to fit this number. Now the observant reader will have realised that there must be some sort of relationship between the golden ratio and π for these two claims to both be at least numerically accurate. In fact there is a numerical coincidence: the square root of the golden ratio times π is close to 4, in fact this product is 3.996168.

In [38] Robins argues against both the golden ratio or π being deliberately involved in the construction of the pyramid. He claims that the ratio of the vertical rise to the horizontal distance was chosen to be 5 ¹/₂ to 7 and the fact that (¹¹/₁₄) × 4 = 3.1428 and is close to π is nothing more than a coincidence. Similarly Robins claims the way that the golden ratio comes in is also simply a coincidence. Robins claims that certain constructions were made so that the triangle which was formed by the base, height and slope height of the pyramid was a 3, 4, 5 triangle. Certainly it would seem more likely that the engineers would use mathematical knowledge to construct right angles than that they would build in ratios connected with the golden ratio and π.

Books

1. A Aaboe, Episodes from the Early History of Mathematics (1964).
2. R Calinger, A conceptual history of mathematics (Upper Straddle River, N. J., 1999).
3. A B Chace, L S Bull, H P Manning and R C Archibald, The Rhind Mathematical Papyrus (Oberlin, Ohio, 1927-29).
4. M Clagett, Ancient Egyptian science : a source book. Vol. 3. Ancient Egyptian mathematics (Philadelphia, PA, 1999).
5. S Couchoud, Mathématiques égyptiennes: Recherches sur les connaissances mathématiques de l'égypte pharaonique (Paris, 1993).
6. R J Gillings, Mathematics in the Time of the Pharaohs (Cambridge, MA., 1982).
7. G Ifrah, A universal history of numbers : From prehistory to the invention of the computer (London, 1998).
8. G G Joseph, The crest of the peacock (London, 1991).
9. E Lundsgaard, Egyptisk Matematik (Copenhagen, 1945).
10. B L van der Waerden, Science Awakening (Groningen, 1954).
11. B L van der Waerden, Geometry and Algebra in Ancient Civilizations (New York, 1983).
12. B L van der Waerden, Ontwakende Wetenschap. Egyptische, Babylonische en Griekse Wiskunde (Groningen, 1950).

Articles:

1. M H Ahmadi, On Egyptian fractions, in Proceedings of the 21st. Annual Iranian Mathematics Conference, Isfahan, 1990 (Isfahan, 1992), 1-20.
2. E M Bruins, Ancient Egyptian arithmetic: 2/N, Nederl. Akad. Wetensch. Proc. Ser. A. Indagationes Math. 14 (1952), 81-91.
3. E M Bruins, On the approximation to π/4 in Egyptian geometry (Dutch), Nederl. Akad. Wetensch. Indagationes Math. 7 (1945), 11-15.
4. E M Bruins, The Egyptian shadow clock, Janus 52 (1965), 127-137.
5. E M Bruins, Egyptian arithmetic, Janus 68 (1-3) (1981), 33-52.
6. E M Bruins, Reducible and trivial decompositions concerning Egyptian arithmetics, Janus 68 (4) (1981), 281-297.
7. E M Bruins, The part in ancient Egyptian mathematics, Centaurus 19 (4) (1975), 241-251.
8. M Caveing, Le statut arithmétique du quantième égyptien, in Histoire de fractions, fractions d'histoire (Basel, 1992), 39-52; 403; 409.
9. L de Freitas, Notes on some pentagonal "mysteries" in Egyptian and Christian iconography, in Fivefold symmetry (River Edge, NJ, 1992), 307-332.
10. M Dunton and R E Grimm, Fibonacci on Egyptian fractions, Fibonacci Quart. 4 (1966), 339-354.
11. P Ernest, On the adequacy of the Egyptian representation of fractions, Bull. Inst. Math. Appl. 16 (10) (1980), 219-221.
12. S Gandz, A few notes on Egyptian and Babylonian mathematics, in Studies and Essays in the History of Science and Learning Offered in Homage to George Sarton on the Occasion of his Sixtieth Birthday, 31 August 1944 (New York, 1947), 449-462.
13. P Gerdes, Three alternate methods of obtaining the ancient Egyptian formula for the area of a circle, Historia Math. 12 (3) (1985), 261-268.
14. R J Gillings, The Egyptian 2/3 table for fractions, The Rhind mathematical papyrus (B.M. 10057-8), Austral. J. Sci. 22 (1959), 247-250.
15. R J Gillings, The Recto of the Rhind Mathematical Papyrus and the Egyptian mathematical leather roll, Historia Math. 6 (4) (1979), 442-447.
16. R J Gillings, The recto of the Rhind Mathematical Papyrus : How did the ancient Egyptian scribe prepare it?, Arch. History Exact Sci. 12 (1974), 291-298.
17. R J Gillings, The Egyptian Mathematical Leather Role - line 8 : How did the scribe do it?, Historia Math. 8 (4) (1981), 456-457.
18. M Guillemot, De l'arithmetique égyptienne à l'arithmetique arabo-islamique, in Deuxième Colloque Maghrebin sur l'Histoire des Mathématiques Arabes, Tunis, 1988 (Tunis, 1990), 95-105.
19. M Guillemot, Les méthodes de simple fausse position dans les mathématiques égyptiennes et les mathématiques arabes, in Histoire des mathématiques arabes, Vol. 2 Tipaza, 1990 (Algiers, 1998), 125-145.
20. M Guillemot, Les notations et les pratiques opératoires permettent-elles de parler de "fractions égyptiennes"?, in Histoire de fractions, fractions d'histoire (Basel, 1992), 53-69; 403; 409.
21. R Lehti, Geometry of the Egyptians (Finnish), Arkhimedes (2) (1971), 15-28.
22. B Lumpkin, Note: the Egyptians and Pythagorean triples, Historia Math. 7 (2) (1980), 186-187.
23. A E Raik, On the theory of Egyptian fractions (Russian), Istor.-Mat. Issled. No. 23 (1978), 181-191; 358.
24. C S Rees, Egyptian fractions, Math. Chronicle 10 (1-2) (1981/82), 13-30.
25. G R Rising, The Egyptian use of unit fractions for equitable distribution, Historia Math. 1 (1) (1974), 93-94.
26. G Robins and C C D Shute, Mathematical bases of ancient Egyptian architecture and graphic art, Historia Math. 12 (2) (1985), 107-122.
27. C S Roero, Egyptian mathematics, in I Grattan-Guinness (ed.), Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences (London, 1994), 30-45.
28. G J Toomer, Mathematics and Astronomy, in J R Harris (ed.), The Legacy of Egypt (Oxford, 1971), 27-54.
29. I N Veselovskii, Egyptian science and Greece, Akad. Nauk SSSR. Trudy Inst. Istorii Estestvoznaniya 2 (1948), 426-498.
30. W Wendrich, Ancient Egyptian rope and knots, in History and science of knots (River Edge, NJ, 1996) 43-68.
31. S A Yanovskaya, On the theory of Egyptian fractions (Russian), Akad. Nauk SSSR. Trudy Inst. Istorii Estestvoznaniya 1 (1947), 269-282.