Today a child learns to distinguish between 25, 205, and 2005 through the base ten position of the zeros. When performing the operations of addition, subtraction, multiplication, and division without a calculator, the vertical alignment of the digits into neat columns of units’ digits, tens’ digits, hundreds’ digits, etc., makes the general procedure for these basic operations seem exceptionally simple. In today's society we take this simplicity for granted. But archaeological remains of calculations by different ancient civilizations reveals that very few ancient cultures had a concept of a base value (such as 10) in which the same symbol (such as 2) in a different position would have a different value (such as 2, 20, 200, et cetera). The written biblical examples of numbers in the Hebrew language show no knowledge of a base ten positional number system with a symbol for zero to define the position and hence the value. Without this positional base concept using a zero, general long division becomes very cumbersome and time consuming.

For example, if the reader attempts to use the symbolism of the Roman number system (with “L” for 50, “XL” for 40, “C” for 100, “M” for 500, etc.), and then attempts to do general long division in this system, it will be a significant chore. Although ancient societies had a concept of a fraction and they knew how to divide by 10 (obtaining a tithe) because the language used words that were multiples of 10, this certainly does not imply that they had a simple general method for long division that could be done quickly. Dividing by 5 was twice a tithe, so that was easy. Dividing by 20 was half a tithe, so that was easy. But these are special examples rather than a general method for long division that would work for all numbers.

Try dividing the Roman equivalent of 237892.21 by the Roman equivalent of 542.37 using only the Roman number system and see how far you get without our modern symbolism for numbers with a zero. Without a positional base number system using a zero, the method for general long division that elementary school children are taught today would not even exist because that very method depends on position. The reference RMP (= Rhind mathematical papyrus) is an explanatory book concerning ancient Egyptian mathematics published by the British Museum. It provides a detailed analysis of a papyrus from ancient Egypt that gives examples of how to solve a wide variety of mathematical problems.

Page 16 of van der Waerden 1961 dates this papyrus after 1800 BCE, which is after the time of the building of the great pyramids at Giza. Page 12 of RMP states, “The hieroglyphic script had distinct signs for units, tens, hundreds, etc., the numbers of each being indicated by repetition of the sign. There was no sign for zero and no positional notation, so that the representation of large numbers became extremely cumbersome.” Page 5 of Gillings states that the ancient Egyptian method for writing the number 1967 required 23 characters while the method for writing 20,000 required only two characters. This ancient Egyptian method for the representation of numbers does not enable the simple methods of general long division used by modern elementary school children or the equivalent simple methods used by the ancient Babylonians. Pages 16-18 of RMP give examples of how long division was performed by the Egyptians, and page 19 of van der Waerden explains the Egyptian methods for long division in a slightly different way. The methods are laborious and cumbersome by today's standards, and if there were a need for many general long division computations, it would be discouraging to have to use the methods of the ancient Egyptians. Mathematical astronomy would require extensive use of general methods of long division where the divisor may be a whole number plus a fraction.

Page 36 of van der Waerden raises the question of whether the ancient Egyptians had more advanced mathematical methods than those that have survived until today. By the word “ancient”, he means before the time of Alexander the Great, after which the city of Alexandria was founded and the Greek astronomers emigrated to Alexandria where they used the mathematical methods of the Babylonians, but dressed in the Greek language rather than the Akkadian language of the Babylonian pagan priests.

He gives two reasons against this. One reason is that there are both elementary mathematical Egyptian texts and advanced texts, and the general character of the mathematics remains the same in both kinds of texts. The second reason is that the Greeks had access to ancient Egyptian mathematical and geometrical methods. The Egyptians successfully used the geometrical methods in a practical way for building purposes, and the Greeks did use selected geometrical methods of the ancient Egyptians. If the Egyptians had developed good methods for doing arithmetic, we would also find some trace of this among the many Greek writings in mathematics. But the Greeks only show use of the Babylonian methods in arithmetic. The ancient Egyptians did not use the positional base 60 number system of the Babylonians or the Babylonian multiplication tables up to 60 times 60.

Pages 353-356 of Ruggles discusses the pyramids of Giza, which are the most impressive pyramids of Egypt. Ruggles makes it clear that we do not know the methods by which the Egyptians constructed these massive monuments. In modern times several writers have made guesses concerning how this may have been done. The largest pyramid required over two million blocks, each weighing about 15 tons, and it is not known how the blocks were transported to such a height. They must have had an excellent knowledge of applied levers and pulleys, but even this supposition does not explain how they could have done it. Our lack of knowing how this marvelous feat of construction occurred is not evidence that it required advanced methods of mathematics that differs significantly from the examples we already possess. The mathematics needed for building construction is different from the mathematics that is needed for mathematical astronomy.

On pages 128-129 of Clagett, he wrote the following:

“It should be clear from my summary account that the ancient Egyptian documents do not employ any kinematic models, whether treated geometrically or arithmetically. However they did use tabulated lists of star risings and transits (as is revealed clearly in Documents III.11, III.12, and III.14), all tied to their efforts to measure time by means of the apparent motions of celestial bodies.”

“On more than one occasion in this chapter, I have remarked on the absence in early Egyptian astronomy of the use of degrees, minutes, and seconds to quantify angles or arcs, though slopes were copiously used in the construction of buildings, water clocks and shadow clocks, such slopes were measured by linear ratios.”

Otto Neugebauer (1899-1990) is unquestionably considered to be the greatest historian of ancient mathematical astronomy in the 20th century. He studied the ancient Egyptian language as well as the ancient Assyrian language known as Akkadian (see pp. 289-290 of Swerdlow 1993), and his pioneering studies were based on his own readings of the original texts. Neugebauer first studied how to read Egyptian hieroglyphics so that he could study ancient Egyptian mathematics from the original documents. Before he began his studies on ancient Egyptian and Babylonian astronomy, he made a detailed study of their mathematics. His doctoral dissertation was on ancient Egyptian mathematics, primarily based on the Rhind Papyrus from ancient Egypt.

After repeated efforts Neugebauer convinced Richard Anthony Parker, the most acclaimed expert on ancient Egyptian science and calendation, to leave the University of Chicago and join him as a professor at Brown University in 1949. Neugebauer and Parker published three volumes of ancient Egyptian astronomical texts from before the time of Alexander the Great (see Neugebauer and Parker). These many texts from ancient Egypt show that we have an understanding of their ancient knowledge of astronomy. These texts show no indication of the abilities later achieved by the Babylonians and Greeks in predictive astronomy, as Clagett pointed out.

On page 559 of HAMA, Neugebauer wrote, “Egypt has no place in a work on the history of mathematical astronomy. Nevertheless I devote a separate ‘Book’ on this subject [10 pages] in order to draw the reader's attention to its insignificance which cannot be too strongly emphasized in comparison with the Babylonian and the Greek contribution to the development of scientific astronomy.”

Concerning the extremely high accuracy of aligning the largest ancient Egyptian pyramids with the east-west direction, and hence a precise knowledge of the time of the equinoxes by the ancient Egyptians, Neugebauer 1980 wrote on pages 1-2, “It is therefore perhaps permissible to suggest as a possible method a procedure which combines greatest simplicity with high accuracy, without astronomical theory whatsoever beyond the primitive experience of symmetry of shadows in the course of one day.” A diagram and further discussion by Neugebauer explain how the Egyptians could have achieved the accurate alignments without any mathematically sophisticated theory. The reason he sought and proposed this method is simply that his studies into ancient Egyptian mathematics and astronomy did not hint at any Egyptian ability to accurately predict the time of the equinoxes.

Ronald Wells wrote a chapter titled “Astronomy in Egypt”, which concerns the time before Alexander the Great and his command to build the most modern city of ancient civilization, Alexandria. On page 40 of this chapter, Wells provides the following summary: “Historians of science concede only two items of [astronomical] scientific significance bequeathed to us by the ancient Egyptians: the civil calendar of 365 days used by astronomers even as late as Copernicus in the Middle Ages, and the division of the day and night into 12 hours each. These fundamental contributions may seem meager to many; engineering of the pyramids and surviving temples notwithstanding.” Page 7 of this book edited by Walker states, “Ronald A. Wells was a Fulbright scholar in Egypt at the University of Cairo and at Helwan Observatory in 1983-4, and again at the Institute of Archaeology, Egyptology Division, University of Hamburg, in 1987-8."

Otto Neugebauer wrote (1945) on page 11, “It will be clear from this discussion that the level reached by Babylonian mathematics was decisive for the development of such methods [for the numerical study of astronomy]. The determination of characteristic constants (e.g., period, amplitude, and phase in periodic motions) not only requires highly developed methods of computation but inevitably leads to the problem of solving systems of equations corresponding to the outside conditions imposed upon the problem by the observational data. In other words, without a good stock of mathematical tools, devices of the type which we find everywhere in the Babylonian lunar and planetary theory could not be designed. Egyptian mathematics would have rendered hopeless any attempt to solve problems of the type needed constantly in Babylonian astronomy.”

On page 8 he wrote, “It is a serious mistake to try to invest Egyptian mathematical or astronomical documents with the false glory of scientific achievements or to assume a still unknown science, secret or lost, not found in the extant texts.”

Neugebauer wrote (1969) on page 78, “The handling of fractions always remained a special art in Egyptian arithmetic. Though experience teaches one very soon to operate quite rapidly within this framework, one will readily agree that the methods exclude any extensive astronomical computations comparable to the enormous numerical work which one finds incorporated in Greek and late Babylonian astronomy. No wonder that Egyptian astronomy played no role whatsoever in the development of this field.”

From the many ancient texts of the Egyptians we conclude that they did not apply mathematics to astronomy before the time of Alexander the Great. After that time, the city of Alexandria was founded and the leading Greek mathematicians and astronomers settled in that city of Egypt, so that it became the world's leading center of Greek astronomy. But this was not part of ancient Egyptian culture; instead, it was the transplanting of Greek science into Egypt by foreigners due to the newly constructed city of Alexandria with its modern marble streets and its grand marble museum and library. This combination museum and library with its many lecture halls became the best ancient equivalent to a modern university, and its library became the greatest one in ancient times. The attention devoted to ancient Egypt serves the purpose of showing that ancient Israel could not have obtained knowledge of mathematical astronomy from Egypt because Egypt did not possess knowledge of mathematical astronomy.