. Underwood Dudley : “Numerology or, what Pythagoras wrought”, Mathematical Association of America, 1997, pages 215 bottom and 216 top.
 Otto Neugebauer: “The Exact Sciences in Antiquity”, 1957, edition consulted Dover, New York, 1969, see page 91.
 André Pichot “La naissance de la science”, Gallimard, Paris, 1991, edition consulted “Die Geburt der Wissenschaft: Von den Babyloniern zu den frühen Griechen”,
(= “The Birth of Science, from the Babylonians to the early Greeks”), Wissenschaftliche Buchgesellschaft, Darmstadt, 1995, page 173.
 André Pichot: “The Birth of Science...”, cited above, page 193.
 Marshall Clagett: “Ancient Egyptian Science: A Source Book; Volume Three: Ancient Egyptian Mathematics”, American Philosophical Society, Philadelphia, 1999, see pages 205 and 206 for Mr. Golenischeff’s account about his acquisition of this payrus.
 Richard J. Gillings: “Mathematics in the Time of the Pharaohs”, Massachusetts Institute of Technology, 1972, edition consulted Dover, New York, 1982, pages 246 and 247 for description of contents and 196 top for poor writing, 187 to 193 for a discussion of the truncated pyramid formula, and 193 to 201 of the hemisphere area.
 Gay Robins and Charles Shute: “The Rhind Mathematical Papyrus: an ancient Egyptian Text”, British Museum Publications, London, 1987, see page 58.
 Gay Robins & Charles Chute: “The Rhind Mathematical Papyrus ...”, cited above, pages 10 and 11.
 André Pichot: “The Birth of Science...”, cited above, page 180.
 S. R. K. Glanville: “The Mathematical Leather Roll in the British Museum”, Journal of Egyptian Archaeology, 1927, pp. 237-9, as quoted in Richard J. Gillings: “Mathematics in the Time of the Pharaohs”, cited above, page 89.
 Richard J. Gillings: “Mathematics in the Time of the Pharaohs”, cited above, pages 218 to 231, see page 218 for description, and page 229 for errors.
 Richard J. Gillings: “Mathematics in the Time of the Pharaohs”, cited above, pages 91 and 176 to 180.
 Marshall Clagett: “Ancient Egyptian Science: A Source Book; Volume Three: Ancient Egyptian Mathematics”, American Philosophical Society, Philadelphia, 1999, see pages 239 to 241.
 André Pichot: “The Birth of Science...”, cited above, page 197 bottom.
 Richard J. Gillings: “Mathematics in the Time of the Pharaohs”, cited above, pages 124 to 127.
 André Pichot: “The Birth of Science...”, cited above, pages 181 and 183.
 Richard J. Gillings: “Mathematics in the Time of the Pharaohs”, cited above, page 161.
Numerals and constants
Our meager sources about ancient Egyptian mathematics
Some modern scholars believe that much, if not all, of the pre-Hellenic Egyptian mathematical knowledge was already firmly established in or before the Old Kingdom .
This is hardly surprising because by then, the agricultural economy of the Nile valley had been based for several thousand years on its surveyors’ ability to properly compute the areas of the annually inundated and re-measured fields. The farmers could only begin to plant once the mud-obliterated boundaries of their properties were securely and verifiably re-determined -- for who would want to waste his carefully saved seed grain on what might turn out to be a neighbor’s land?
Yet, despite this reliance of the ancient Egyptians on geometry for their basic livelihood, despite their well attested interest in matters of the mind, and despite their equally well attested contacts with Mesopotamia, one of the cradles and nurseries for number mysticism, their math skills are belittled or denied by many mainstream scholars today. Some current authors try to tell us that the Egyptian scribes had no interest in non-practical mathematics. In the words of one modern mathematician, they were
“ ... clever but very, very primitive. With no algebra, no trigonometry, hardly any geometry, and laborious arithmetic, the Egyptians were not going to make any deep mathematical discoveries and in fact they did not.”
This reflects the views of an earlier leading scholar in the field who concluded his discussion of Egyptian mathematics and astronomy with this assertion:
“Ancient science was the product of a very few men; and these few happened not to be Egyptians.”
Most of these detractors typically omit to tell us how little we know about Egyptian mathematics, and that the minuscule sampling of surviving papyri with mathematical content does not enable anyone to judge what the ancient Egyptians did or did not know.
Whereas the inhabitants of Mesopotamia, the land between the Two Rivers, left us several thousand clay tablets with mathematical contents, only eight clearly mathematics-related documents survive from the Two Lands along the Nile. These are :
, and these are now at the British Museum in London. Additional fragments from the same papyrus came up for sale a decade later and wound up in the Brooklyn Museum, New York.
· An inscription in the tomb of Methen, a Third Dynasty noble who died around 2,600 BCE, which gives a calculation for the area of a rectangle.
· The Moscow Mathematical Papyrus with 25 problems from late Middle Kingdom times, bought in the 1890s from a notorious tomb robber and now at the Museum of Fine Arts in Moscow. Seven of these problems are not clear because, according to Richard Gillings, the author of “Mathematics in the Time of the Pharaohs”, its scribe was “a very bad writer” whose hieratic sign forms were “criminally inconsequent” and who had copied from “a faulty original, or an original which he did not understand”.
The comprehensible parts of that papyrus deal with routine divisions of bread and beer, the areas of triangles and rectangles, plus two formulae that have no counterpart in the other surviving documents: the correct volume of a truncated pyramid which Gillings says “has not been improved upon in 4,000 years”, and a controversial calculation which some scholars believe to give the surface area of a hemisphere. Regarding the latter, Gillings adds that “if this is so, it becomes the outstanding Egyptian achievement in the field of mathematics”.
· The famous Rhind Mathematical Papyrus, a roll of 14 papyrus sheets, out of the 20 sheets that usually made a complete roll at its time. Mr. Rhind bought most the surviving parts in the 1850s on the Luxor antiquities market
That papyrus dates from around 1550 BCE and is about two centuries younger than the original from which a scribe named Ahmose said he had copied it. The content itself seems to be still older. For instance, Gay Robins and Charles Shute point out that the pyramid slopes in its problems 57 to 59 were used in the Fourth Dynasty and became universal during the Sixth but had been abandoned by the Twelfth when the original for the Rhind’s surviving copy was purportedly written.
The Rhind parts we have contain 84 problems in basic arithmetic and elementary geometry with their solutions, plus three more that remain enigmatic. Although this papyrus is our most extensive single source of information about ancient Egyptian numeracy, it was only a study aid for apprentice scribes; moreover, its copyist did not always fully grasp what he wrote, and he left out some important elements.
· The Egyptian Mathematical Leather Roll, a now extremely brittle piece of leather 10 inches by 17 inches, now in the British Museum. This roll was also bought on the antiquities market. It is said to have been found near the Rhind Papyrus in the ruins of the Ramessseum at Thebes and to date back to the Middle Kingdom. It contains a duplicate collection of 26 sums done in unit fractions, as in the Rhind Papyrus, and was apparently a “handy table for popular use, probably the work of a junior official, not of a schoolboy because the writing is far too good”.
· The Reisner Papyri: four badly worm- eaten rolls found in Upper Egypt by the archaeologist George A. Reisner that are now in the Museum of Fine Arts in Boston. As Gillings describes them, these Papyri were for the most part the official registers of a Middle Kingdom dockyard workshop. They contain long lists of workers, cargo, and foodstuffs, as well as calculations for the building of a rectangular temple in terms of foundation volume to be excavated, labor required, and stone supplies. Some other sections give various totals and many large numbers without any context. Its writer made many, although slight, errors in the more complicated operations.
· The Kahun Papyrus was found by the archaeologist Flinders Petrie at the workers’ town Kahun near the pyramid of Sesostris II and dates also from around 1800 BCE. It “contains six mathematical fragments, not all of which have been penetrated”. The three of these fragments that have been explained repeat a portion of a list from the Rhind Papyrus, and one of the unclear fragments seems to deal with a progression that resembles Problem 40 from the Rhind Papyrus.
Another one of the fragments lists pairs of squares that produce squares as their sum, all of them variations of the well- known 32 + 42 = 52 equation we learned in school as an example of the so- called Pythagorean theorem about the squares over the sides of a right- angled triangle. This list has been interpreted by some as possibly an early expression of that theorem although it does not mention the sides of a triangle.
· The salary list for personnel of the Illahun temple, also found near Kahun and assigned to the Middle Kingdom. This relatively short list contains several errors and omissions. It is concerned with distributing loaves of bread and jugs of beer in which some of the parts expressed as fractions amount to no more than crumbs and drops.
· The Berlin Papyrus, a group of unprovenanced fragments now in Berlin that are believed to date from about that same time. The largest of these scraps is about the size of your palm; it deals with square roots and the decomposition of a given square into sums of two squares. Gillings describes the condition of these fragments as “mutilated, so that the restorations, although quite reasonable and plausible, perhaps still remain open to some slight reinterpretation”.
This is it.
These few bits and pieces are all the known written traces that remain from the daily mathematical activities of millions of people in a region a thousand kilometers long, during more than three thousand years of pharaonic civilization.
Add to this time span the about equally long period of Nile valley farming during which field geometry developed, long before the invention of writing, and you can appreciate how little we really know about ancient Egyptian mathematics.