Primitive Mathematics

As we have seen, the civilization of the valley of the Nile was extremely complex, based on a formidable social, economic, and religious structure.

All this would suggest that science was quite rich in content, and, in particular, the construction of so many monumental buildings suggests a deep knowledge of geometry, mathematics, and engineering. Strangely, however, according to the classic works on the history of science, this seems not to be the case. For example, one of the most renowned experts on the science of the ancient times, Otto Neugebauer, has often called the mathematics of ancient Egypt "primitive,'' and did so especially in his famous book The Exact Sciences in Antiquity, considered by many a fundamental reference.

How the primitive Egyptian mathematics worked is something that we will discuss shortly. But first, it would be interesting to know on what documents and evidence Neugebauer—and the science historian Boyer (1991) and many others—based his lapidary assessment of the Egyptian mathematics. He based it on only two pieces of evidence: one papyrus, the Rhind Papyrus, and one piece of a papyrus, the Moscow Papyrus. The available written documentation ends here. The reason, I believe, is simple, although I cannot prove it. The papyri containing scientific and technical information, such as astronomical observations, mathematics and geometric notions, and building plans, were not usually placed in tombs as an "endowment" for the afterworld, and yet the great majority of the papyri that we have today do come from the tombs. As a result, we have many copies of the Book of the Dead, but only two little pieces of papyrus addressing mathematics. To base our conclusions on just these two fragments, as Neugebauer and other science historians have done, is ludicrous; in this case, we cannot rely on just these brief written sources for our knowledge (I will address this problem in depth in later chapters).

Further, even the interpretation of these sources is quite a delicate issue. The Rhind papyrus, preserved today in the British Museum in London, gets its name from the Egyptologist Henry Rhind, who bought it in Luxor in 1858. It is about 6 meters long and 30 centimeters wide, and was written at the beginning of the New Kingdom or shortly before that by a man named Ahmes, who acknowledges that he is copying from a document that is 200 years older (the Moscow Papyrus belongs more or less to the same period). The Rhind Papyrus addresses 87 problems, the Moscow Papyrus, 25. Some of those problems are of a theoretical nature and deal with the manipulation of fractions or with the solution of simple equations similar to the famous brick riddle ("A brick weighs 1 kilo plus a half of a brick; how much does a brick weigh?") for instance: "A quantity added to a quarter of that quantity gives 15. What is the quantity?'' Other problems are of a geometric nature, such as the calculation of the volume of a pyramidal frustum (truncated pyramid). The elementary operations between integer numbers involved are

Figure 4.2: Examples of exercises of the Rhind papyrus

performed following a method often vilified and considered primitive in the mathematics literature. It is, however, basically equivalent to the binary system, the one used by computers today. Rational fractions are used as well, but always changing them to the form of unit fractions (for example, 2/5 = 1/ 15 + 1/3).

All in all, the choice of problems looks quite didactic. It is, therefore, difficult to determine the level of knowledge required to solve the problems contained in the Rhind papyrus.

Thus, the science historians' claim about the poor knowledge of the ancient Egyptians is quite suspicious. Further, we have no document regarding mathematical knowledge in the Old Kingdom, and no one can affirm that such knowledge was inferior to that in later times; it is very hard to believe that somebody like Ahmes would have been able to orient within 3 arc-minutes to true north an object 150 meters tall and weighing some seven million tons. However, this has really been done, and it has been done in the Old Kingdom, as we shall see. Perhaps, Neugebauer should have given it a little bit of extra thought.

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