*Die Grundlagen der Ägyptischen* *Bruchrechnung (The Principles of Egyptian Calculation by Fractions*), Berlin, 1926

### by Otto Neugebauer

The Austrian historian of mathematics and astronomy, Otto Neugebauer (1899-1990), is one of the commanding scholarly figures in the modern history of ancient science and mathematics. In the field of Egyptology, he is best known for his monumental work co-authored with the Egyptologist Richard Anthony Parker (1905-1993), *Egyptian Astronomical Texts*, in four volumes (1960). To historians of science in general, his best known work is possibly his single volume, *The Exact Sciences in Antiquity* (1952). And to Assyriologists, he was the master scholar who published the definitive *Cuneiform Astronomical Texts* in three volumes (1956). His book *Die Grundlagen der Ägyptischen Bruchrechnung* (The Principles of Egyptian Calculation by Fractions), Berlin, 1926, was written during Neugebauer’s years as Assistant at the Mathematical Institute of the University of Göttingen in Germany. The book was actually his thesis, and contains an analysis of the arithmetic of fractions found in the Rhind Papyrus. His studies of ancient Egyptian mathematics thus preceded his more widely known studies of ancient Babylonian mathematics, concerning which he commenced publishing only in the following year, 1927. Neugebauer’s later work, especially his spectacular work on ancient astronomy, became so famous that it tended to eclipse his earlier work. Many people are unaware that at the beginning of his career, he made a special and intensive study of the curious arithmetic of the ancient Egyptians, which as everyone familiar with the subject knows very well, was based entirely upon the ingenious manipulation of fractions, mostly with a numerator of one, in the absence of a system of decimal fractions. Anyone who has dabbled in the ancient Egyptian world of fractions soon comes to realize that it is a kind of parallel mental universe to that with which we are familiar today in our modern mathematics. At first, the fractions seem cumbersome, but as one ‘plays’ with them, they become strangely attractive, and can even begin to become rather addictive. It is difficult to describe the special charm they can exert as one enters their world and leaves behind our familiar mathematical universe of decimal fractions. Even though there is some slight evidence suggesting that the Egyptians may have thought of or been somehow passingly familiar with decimal fractions, fractions conceived in that manner clearly held no attraction for them. It is possible that the decimal system was not really invented for the very first time by the Chinese in the fourteenth century BC (the period of the Shang Dynasty), with decimal fractions being invented by the Chinese in the first century BC, as widely accepted today (since the researches of the late Joseph Needham have established those upper limit dates in China with certainty). It may be that in fact decimals had been used before that for many centuries by the people of the Harāppan, or Indus Valley, Civilisation of North-West India (today’s Pakistan), who traded regularly with the peoples residing in today’s Yemen. From there, knowledge of them could easily have spread across the Red Sea to Egypt. But assuming that anything such as this could have happened, decimal fractions certainly did not find favour there and would probably have been regarded as a strange and exotic mathematical curiosity, much as we regard the endless series of Egyptian fractions with numerators of unity today. So much concerning the earliest history of mathematics is inevitably lost in the mists of time, for want of sufficient surviving texts, that our ignorance of origins is fated always to haunt us. It is because any serious study of the subject, especially one by such a great scholar as Otto Neugebauer, holds such value and importance for us as we try to piece together from the little evidence we have the story of the most ancient mathematics, that we can never know too much about the Egyptian fractions. This largely forgotten work by Neugebauer on this subject is thus placed online for the better dissemination of his ideas and conclusions about that subject. The Foundation’s own copy of this extremely rare publication has therefore been scanned and made available here as a pdf download, so that it may become known to all those who have not had the good fortune ever to have access to a copy, or may not have known of its existence at all.