Changing attitudes towards ancient arithmetic: reconciling mathematics with Egyptology

Oxford Mathematician Christopher Hollings and Oxford Egyptologist Richard Bruce Parkinson explain how our interpretation of Egyptian Mathematics has changed over the past two centuries and what that says about how historians of mathematics approach their subject.

"When Egyptian hieroglyphs were deciphered in the nineteenth century, scholars were finally able to read the millennia-old texts that provided a key to understanding the ancient Egyptian civilisation.  The documents that could now be read dealt with a range of subjects, from administrative, legal, and religious matters, to medicine – and mathematics.  Examples of numerals may be found in many surviving accounts papyri, which record numbers of goods and workers, etc., but only a very small collection of sources provide an insight into the mathematics that ancient Egyptian scribes actually knew.

One of the most complete surviving sources is the Rhind Mathematical Papyrus, dating from c. 1537 BCE, and now held in the British Museum (P. BM EA 10057 and P. BM EA 10058 - see image above © the Trustees of the British Museum).  The papyrus consists of over 80 arithmetical and geometrical problems and solutions, ranging from the distribution of rations among workers, to the calculation of areas (such as of triangles) and volumes (of cylinders, for example).  The papyrus was probably placed in a tomb, as part of a display of the tomb-owner’s social and cultural status.

Within a couple of decades of the discovery of the Rhind Papyrus in Luxor in 1858, it had provided a much more complete picture of the nature of Egyptian mathematics than had previously been available.  Several translations of parts of the Rhind Papyrus appeared in print, but perhaps the most celebrated was that published in 1923 by the Egyptologist Thomas Eric Peet (1882–1934), then Professor of Egyptology at the University of Liverpool.  Peet was ideally placed to produce this new edition, having studied both mathematics and classics in Oxford.  His edition was much praised by both mathematicians and Egyptologists, and reignited an academic interest in Egyptian mathematics, which had stagnated slightly in the decades prior to Peet’s publication.  The subject was now to be studied on its own terms, rather than standing in the shadow of ancient Greek mathematics, which had traditionally been taken as the start of ‘true’ mathematics.

One scholar who was particularly inspired by Peet’s work was Otto Neugebauer (1899–1990), a young student in Göttingen.  Neugebauer had already studied mathematics and physics, but was now cultivating an interest in ancient science.  By the end of 1926, he had completed a doctoral dissertation on the principles of Egyptian fraction reckoning (as reflected in the Rhind Papyrus).  Whilst completing the dissertation, he was in correspondence with Peet, and two of his letters to Peet have recently come to light in Oxford in the Peet Memorial Library of The Queen’s College – Peet ended his career in Oxford (as a Fellow of Queen’s), with the result that some of his personal papers and books survive there in archives and libraries.

The letters shed light on the way in which ancient Egyptian mathematics was being re-evaluated in the 1920s.  In particular, they show up a contrast between the attitudes of two scholars who approached the subject from different directions.  Both were competent mathematicians and Egyptologists, and yet one (Neugebauer) put the mathematics first, and made general assertions about the nature of ancient Egyptian mathematics that arguably owed more to modern ideas about how mathematics ‘should’ be than to the direct evidence of papyri; the other (Peet) brought Egyptological considerations to the fore, drawing conclusions that were more firmly embedded in a knowledge of the cultural context of surviving sources – where Neugebauer saw lacunae in the historical record as gaps to be filled with educated speculation, Peet tip-toed cautiously around them, confining his commentary largely to what was clearly and unequivocally present in the original texts.

Perhaps partly because of his early death, Peet’s influence on the later study of ancient Egyptian mathematics was minimal.  Neugebauer, on the other hand, went on to become one of the most prominent historians of mathematics of the twentieth century; his work helped to turn the history of mathematics into an academic discipline.  It was not until the 1970s that Peet’s culturally-sensitive approach to the history of mathematics began to gain ground once again.  And yet the essential tension between the attitudes of Peet and Neugebauer can still be found in the study of ancient mathematics today."

Christopher Hollings is Departmental Lecturer in Mathematics and its History, and Clifford Norton Senior Research Fellow in the History of Mathematics at The Queen’s College.  Richard Bruce Parkinson is Professor of Egyptology and a Fellow of The Queen’s College.  Their paper on the correspondence between Peet and Neugebauer, ‘Two letters from Otto Neugebauer to Thomas Eric Peet on ancient Egyptian mathematics’ (Historia Mathematica, 2020) can be found here.