*Mathematics in Ancient Egypt*, Imhausen looks in some detail at the mathematical texts and the methods that they reveal to us, but also at the broader context of mathematical activity.

"Despite only twenty-five mathematical texts being extant from all of Egyptian history, it is still possible to get an idea of Egyptian mathematics and its role within Egyptian culture."

Her approach is chronological, surveying the evidence roughly in order. She looks at details of individual manuscripts and other evidence, but intersperses that with more general mathematical and cultural background.

There is little evidence from the Predynastic period or even the Old Kingdom, but we can discern "a set of seven distinct hieroglyphic signs to represent powers of 10 in a decimal, nonpositional number system", units of measurement, and the distinctive Egyptian treatment of fractions as sums of "unit fractions" with numerator 1.

The Middle Kingdom brings the first major mathematical texts that survive, such as the Rhind Papyrus and the Moscow Papyrus.

"The Moscow mathematical papyrus is the second-largest extant source text. While its total length is approximately 5.44 m, its height is only 8 cm. It consists of one big piece (containing thirty-eight columns of text) and nine little fragments. The papyrus was bought by W. Golenischeff between 1892 and 1894. In 1912, it was acquired by the Museum of Fine Arts in Moscow, where it is now held under the inventory number E4676."

Imhausen looks at the formal structure of problems and procedure texts, and their role in the training of scribes, before giving a brief introduction to the foundations of arithmetic: multiplication and division by addition of repeated doublings and halvings, manipulation of fractions, and so forth.

"for an operation to be performed as often as doubling, it is too cumbersome to find a solution again each time the problem is encountered. Thus a table is needed from which the result can readily be taken and used. The2 ÷ ntable is probably the most important table in Egyptian mathematics. Two copies of this table are extant. One of them can be found at the beginning of the Rhind papyrus, and the other is found on one of the Lahun fragments. The table of the Rhind papyrus includes representations of the divisions2 ÷ nfor oddnfrom 3 to 101. The Lahun fragment holds the same for oddnfrom 3 to 21."

"Despite the scarcity of mathematical texts, the New Kingdom offers
a variety of sources that allow us insights into the uses and roles
of mathematics." Papyrus Anastasi I is a satirical letter set in
the context of a competition between two scribes, one challenging
another with mathematical problems involving digging lakes,
constructing brick ramps, provisioning an expedition, and so forth.
And aspects of mathematics or metrology appear in documents such
as *Teaching of Amenemope* and *Duties of the Vizier*.

We have a significant corpus of demotic mathematical papyri from the Greco-Roman period. The focus in those is still on problems, but while some are realistic others lack direct practical application. And the passage of 1500 years from the main hieratic sources shows some changes, but also clear continuation of the existing tradition. (Included here are some contrasting modern evaluations of the unit fraction system.)

Imhausen includes enough background history for her account to be followed without prior knowledge of ancient Egypt, but doesn't get stuck doing too much of that. She also emphasizes that ancient Egyptian culture was not static and unchanging, and that our sources are hardly representative.

"Due to natural conditions, the majority of sources for our study of ancient Egypt is found in tombs and temples — cities were located then as today along the Nile and hence have often been superseded by modern cities (and are, therefore, unavailable for excavations). Moreover, the humidity found near the river is likely to have destroyed ancient evidence, especially in the form of papyri. ... This imbalance has led to the presumably wrong impression that the Egyptians were constantly focused on death and afterlife, and, therefore, even mathematical practices can be found in religious texts, such as the weighing of the heart described in a section of theBook of the Dead.

... I would like to propose a different scenario: mathematics not only played a huge role within the education and work life of a scribe, but it also held an important role within ancient Egyptian culture in general."

Anyone interested in Egyptian mathematics only as mathematics
might be better off reading something like Reimer's *Count Like an
Egyptian*, but for those who want the broader context, Imhausen
also offers a feel for the detail of the mathematical texts, the
mathematical elements of literary and practical texts, and the
cultural background of scribal numeracy.

March 2021

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*Count Like An Egyptian: A Hands-On Introduction to Ancient Mathematics*

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