No one knows when our ability to vocalise numbers began, and at what stage and how much later we learned to make representational marks to symbolise these numbers. Many, including the economist Peter Jay in the BBC programme `The Road to Riches', believe that the need to formalise numbers was responsible for the development of writing as we know it. The programme even going so far as to state that accountants were responsible for the discovery of writing.

What academics are agreed upon, is that both of these skills arose to meet practical needs over thousands of years. It is hypothesised that the earliest expression of numbers was by reference to parts of the body, which led over a period of time and over a number of stages of development to the adoption of specific names for certain numbers. The earliest Hunter Gatherers generally had, and in certain remote areas of the world today, still have, specific names for only the numbers one and two. Every number then became an expression of these two numbers. One, two, two-one for three, two-two for four, two-two-one, for five, etc. Where this practice was used these peoples generally referred to any number over six as `many'.¹

Mathematics distinguishes three different methods of forming or referring to numbers; these are Additive, Multiplicative, and Positional notation. All modern number systems use positional notation, which is itself a variant of multiplicative notation, which in turn arose to make easier the complexities of number notation. Archaeologists have discovered evidence for various counting systems, and although it is impossible to say which additive method was first, finger counting, and representational marking must be two of the earliest. Tying knots, or marking cuts on pieces of wood or bone, all clearly imply a need to keep an accurate record. These crude tally systems at some stage in the distant past, evolved into the marking of numerals, on bone, stone, clay, and papyrus, and although the earliest known numerals were written on clay by the Sumerians some 5000 years ago, we cannot assume they were the first to do so.

There is archaeological evidence for the use of Tally sticks (carving grooves, or notches) going back over 30,000 years.² The oldest example unearthed so far dates back to Palaeolithic times, and is a wolf bone found in 1937 in Moravia, Czechoslovakia, that had fifty-five deep notches carved into it. Twenty-five notches of similar length, arranged in-groups of five, followed by a single notch twice as long which appears to terminate the series. Then starting from the next notch, also twice as long, a new set of notches runs up to thirty. This wolf radius bone clearly denotes the concept of five, and corresponds to the digits on each hand, which doubtless led Diluvial man to think in terms of numbers. Dr Karl Absolon, who discovered the bone, comments that "We have here direct proofs of reckoning by fossil man."³

We tend to associate mathematics with learning and there is a tendency to falsely assume that the use of numbers was the occupation of scribes and learned scholars. If nothing else the example of the wolf bone indicates that the use of counting for a purpose was widespread. In Palaeolithic times man was a hunter-gatherer, not a scholar. If he found it necessary to indent onto a bone to form a tally stick, it was because he had an express need to count. What he chose to count we can never know, but was likely to be flint, or food, or animals. Later when we learned to domesticate animals, early man would use a tally stick to keep stock of his crops, livestock and harvests.⁴ At some distant point in time, someone in some tribe or culture, for which there is no archaeological evidence, decided that a tally stick was inadequate. This scientific breakthrough resulted in the emergence of written numbers.

The earliest Egyptian system, which belongs to the additive category, can be found on inscriptions dating back to 3000BC⁵ and was a number system based on hieroglyphic symbols. (See figure 1).

Hieroglyphic numbers were generally, but not exclusively, written from right to left. However in the Egyptian system there was no need for a zero symbol, which resulted almost uniquely in that numbers would mean the same whichever way round they were written. For example the number for 357 appears as:

They are both the same number. Seven made up of seven vertical lines, fifty being made up of five heel bone signs and three hundred represented by three coiled ropes, or spirals.

Both versions of the same number are equally unambiguous in contrast to our number system where 357 cannot be written as 753.⁶

However only a limited amount of arithmetical information can be obtained from the hieroglyphic symbols found on Egyptian monuments, these in the main dealing with tally's of bounty captured by Pharaoh's soldiers in their many campaigns. Captives, animals, weapons and the like. For our present knowledge of Egyptian mathematics we are indebted to an early antiquarian called Henry Rhind.

Henry Rhind was a Scottish lawyer with an interest in antiquity. In 1858 at the age of twenty-five he purchased a document that had been found some years earlier in a small building at Thebes. ⁷ The manuscript known thereafter as the Rhind Papyrus currently resides in the British Museum, and is written from left to right in heiratic script. Heiratic is a younger version of the more formal hieroglyphic script and in essence is a form of shorthand, where hieroglyphic symbols are written more simply and are therefore much quicker to scribe.⁸ The heiroglyphic forms were reserved largely for formal inscriptions, leaving the Heiratic and Demotic forms to be used in everyday life. A large quantity of Papyrus scrolls have been identified as being principally concerned with numeration, two others of note being the Moscow Papyrus,⁹ and the Kahun papyrus, but our present knowledge of how the ancient Egyptians carried out mathematics, is principally derived from the Rhind Papyrus.¹⁰

The Rhind Papyrus when found, was a scroll 18 feet long and 13 inches wide, but by the time it reached the British Museum, some parts were missing. Much later these parts were found in the possession of the New York Historical society and thereafter scholars had the whole roll to work with. ¹¹ The Rhind Papyrus, sometimes called the Ahmes Papyrus, for reasons that will become self evident shortly, is a collection of 84 mathematical problems, dealing with arithmetic, primitive algebra, and geometry, and three text entries unrelated to mathematics, sometimes termed problems 85 to 87.¹² The Heiratic script is in two colours, red and black and has drawings of geometric shapes interspersed in the script, much in the same way that a modern publisher would illustrate a manuscript, for ease of understanding. A scribe called A'h-mose, more commonly called Ahmes, copied it from an earlier work, as he himself states in the Papyrus introduction.

This book was copied in the year 33, in the fourth month of the inundation season, under the majesty of the king of Upper and Lower Egypt, `A-user-Re', endowed with life, in likeness to writings of old made in the time of the king of Upper and Lower Egypt, `Ne-ma' et-Re'. It is the scribe A'h-mose who copies this writing. ¹³

The first king mentioned has been identified as a member of the Hyksos dynasty who lived around 1650 BC, whilst the second king has been identified as Amenem-het III, who reigned from 1849 to 1801 BC, during the Middle Kingdom.¹⁴Written nearly four thousand years ago, it is the most extensive ancient mathematical text known, but certainly not the only one. Another important source is the Mathematical Leather roll, similarly held in the British Museum, and reported to have been found with the Rhind Papyrus near the Ramesseum at Thebes.¹⁵ Where the Rhind Papyrus has the look and feel of a maths textbook, the leather roll appears to be a pupil's notebook. Between the two of them, scholars have unravelled many of the mysteries of ancient Egyptian mathematics.

The 84 problems deal with straight arithmetic, problems of finding an unknown quantity (algebra), area and volume calculations, and arithmetic and geometric progressions. Anyone familiar with the formal structure of the later and more advanced Greek mathematics will find the Papyrus disappointing. There are no general rules, and the results do not derive from previously established facts. The problems are a series of examples dealing with matters such as finding the volume of a granary, or how to distribute a number of loaves amongst so many men.¹⁶  This collection of exercises has the feel of a child's textbook, but whether this papyrus denotes the `state of the art' of Ancient Egyptian mathematics, it is impossible to say. It appears that it was a textbook intended for use in a school for scribes. The question we may never know is whether in our terminology, it was aimed at `graduate' level students, or at adolescents. In either case we must assume that Egyptian mathematics followed the same basic structure and premises.

The papyrus itself begins with two tables, a division table of 2 by all odd integers from 3 to 101, and a division table of the integers 1 through 9 divided by 10.

The results are given in unit fractions with the denominator always 1.¹⁷

In today's speech certain simple fractions such as ¥, ½, ¼, have special names. A third, a half, a quarter; we do not say `one-fourth'. Fractions of this nature are termed natural fractions, and serve the needs of everyday life. In ancient Egypt the situation was the same. ¹⁸ The Egyptians had special words and symbols for ¼, ½, ¾, ¥, and ¥.

The fraction ¥ and the whole, which was sometimes referred to as `the third part', was similarly

In Greek it is interesting to note that the terms `the two parts', `the third part', `the three parts' and `the fourth part' are also used to denote ¥ , ¥ , ¾, and ¼. In addition to their notation for natural fractions, the Egyptians had a standard symbol for unit fractions of the type 1/n

They wrote the denominator n below the hieroglyph r as follows:

In the Rhind Papyrus a division of a divided by b is introduced with the words `reckon with b so as to obtain a'. For example, in Problem 24, students are asked to `reckon with 8 so as to obtain 19' (Divide 19 by 8).

 Problem 30 amounts to solving the equation

The problem reads, `What is the quantity of which two thirds plus one tenth will make ten.'

The answer is given as 13 + 1/23.

 Equations of this type were solved by the `rule of false position', whereby a convenient value for x is assumed and substituted into the equation. If x = 30 for example, the left side of the equation becomes 23 instead of the required 10. Since 23 has to be multiplied by 10/23 to obtain 10, the correct value will be 10/23 times the assumed value of 30, or x = 300/23 = 13 + 1/23. Therefore, as Eli Maor in Trigonometric Delights, comments, some 3,500 years before the creation of symbolic algebra, the Egyptians were in possession of a method that allowed them to solve linear equations.²⁰

All textbooks are concerned with imparting knowledge and technique. The Rhind Papyrus is no exception, being a textbook of applied mathematics, as all of its examples deal with the day to day management of the Egyptian culture. Problems 41 through to 60 are generic in nature. For example, problem 41 requires the student to find the volume of a cylindrical granary of diameter 9 and height 10. The solution reads `Take away 1/9 of 9, namely 1: the remainder is 8. Multiply 8 times 8; it makes 64. Multiply 64 times 10; it makes 640 cubed cubits.' Ahmes then multiplied the result by 15/2 to convert to the hekat, which was the standard unit of measure for a volume of grain. ²¹ To find the area of a circular base, the scribe replaced it with a square of side 8/9 of the diameter. Denote the diameter by d and you get the formula

                                                                      Area A = [(8/9) d]²

                                 If you compare to the formula A = p d² / 4

It becomes a simple calculation to find that in this example the Egyptians were using p = 256/81, or 3 + 13/81. In decimal this equals 3.160494 as opposed to the correct value of 3.141593. An error of 0.6 percent of the true value.²² However all Egyptian fractions had to be written as a sum of Unit Fractions, and 256/81 written as a sum of unit fractions, is 3 + 1/9 + 1/27 +1/81. You might equally say that if the Egyptians were aware of p , then why didn't they use the fraction 22/7, which in their notation would be written much simpler and easier as 3 plus the unit fraction 1/7. This would have given them a result of 3.142857143, which would have been 1.0004025 of the true value of p . I will return to this issue in another essay, but for now it is sufficient to say that the Egyptians had a method of using a constant we call Pi . Whether they knew they were using it is a hotly disputed debate.

From the previous examples you will have determined that the Egyptian method of calculation appears very laborious and alien to modern man. That they made such a system work appears to be quite an achievement, but in order to have a better understanding of their computation technique, it is worth considering problem 56.

Problems 56 to 60 deal with pyramids, or in other words 5 out of 84 problems, or 6% of the total. As it goes without saying that the pyramid held a special religious and cultural significance for the ancient Egyptians, it is a further indication that the Rhind Papyrus was intended to deal with day to day problems of a mathematical nature. Indeed it is possible to infer that the paucity of problems dealing with pyramids is an indication that pyramid geometry was not uppermost in the Egyptian mind.

Problem 56 reads: `If a pyramid is 250 cubits high and the side of its base 360 cubits long, what is its seked?'

Ahmes's solution is listed as:

Take ½ of 360; it makes 180. Multiply 250 so as to get 180; it makes ½, 1/5, 1/50 of a cubit. A cubit is 7 palms. Multiply 7 by ½, 1/5, 1/50:

1                          7

                          ½                         3                     ½

                         1/5                        1                     1/3                1/15

                         1/50                                             1/10              1/25

The seked is 5+1/25 palms [that is (3+1/2) +(1+1/3+1/15) +(1/10+1/25) = 5+1/25] ²³

As this method of computation is totally alien to modern man, some explanation is necessary.

The problem is essentially asking the pupil to compute the cotangent for angle a. as in Fig 2.

Clearly ½ of the base side of 360 (length l in Fig 3) is 180. `Multiply 250 so as to get 180' you will recall means find a number x where 250 times x equals 180. This gives x = 180/250 or 18/25. Converting this answer into the customary form of unit fractions, and you will find that the sum of the fractions ½, 1/5, and 1/50, is indeed 18/25. (The pupil was aided in his calculation by the presence of the division tables at the start of the Papyrus, enabling a quick and ready reckoner for many of the numbers he would need to formulate his answer.)

This number is therefore the ratio of half the side of the base of the pyramid to its height. This is the Run to Rise ratio, or what we would term the Cotangent of angle a.

Having found his Run to Rise ratio, Ahmes goes on to multiply his answer by 7. Why?

Horizontal distances were apparently measured in `palms'. It is reported that there were four digits to a palm, and seven palms to a cubit. More about this in a further essay, but for now, it is sufficient to note that 28 digits make one cubit.

If you consider the table below, which is given as Ahmes's answer, there are two columns. One headed by the number 1 and one headed by the number 7.

                                                    1                           7

                          ½                         3                     ½

                         1/5                        1                     1/3                1/15

                         1/50                                             1/10              1/25

The cotangent of angle a was found to be ½, + 1/5, + 1/50. Each of these in turn is multiplied by 7 to give the horizontal result. ½ times 7 gives 3 + ½. The next line indicates 1/5 times 7 which gives 1 + 1/3 + 1/15, and so on. To arrive at the seked you add up all of the derivatives. I.e.: - (3, + ½, + 1, + 1/3, +1/15, + 1/10, +1/25), which gives you the answer 5 +1/25.

It is clear by the fact that the Egyptians gave this run to rise ratio its own name, seked, that it held particular significance. The answer suggested is straightforward. To build a pyramid you need to maintain the angle of the pyramid. To do this you would use a run to rise ratio. In the above example you would move in 5 and 1/25 palms and then go up one cubit (or seven palms), then in a further 5 and 1/25 palms, then up one cubit, etc. This would maintain the slope of the pyramid.

This is all very well in principle, but in reality has one glaringly obvious difficulty. To build a pyramid as in the previous example, it would certainly be convenient to measure 5 and 1/25 palms in, then up a cubit, etc, to maintain the slope of the pyramid, but exactly how do you measure 5 and 1/25 palms? The Egyptians used cubits as their main measure, which were divided up into 7 palms of four digits, or 28 digits to a cubit. 5 and 1/25 palms equals 20.16 digits. To measure 0.16 of a digit requires either that the artisans guess the length, or presupposes that the digit was subdivided into units of even smaller measure.

A number of wooden cubit sticks have been found, with the earliest dating to around the time of Ahmes in the Middle Kingdom. Some, but by no means all, have sub divisions into unit fractions. One that does is a measuring stick held in the Turin museum. Along one edge the stick is divided up into 28 digits, with each division being represented by a deity. For instance the first digit has the symbol for Ra, followed by, and in sequential order, the symbols for Shou, Tefnout, Geb, Nout, Osiris, etc, with Osiris denoting the number 6, and Isis and Set denoting the numbers 7 and 8 respectively. The first fifteen of the digits have additional symbols referring to various fractions of a digit, up to and including the fractional number 16. Using one of these sticks it would therefore have been possible to measure fractions of a digit up to sixteenths. The earliest known measuring sticks however, do not show such fractional notations, leading to the possibility that the use of fractions of digits was a Middle Kingdom innovation.

As no evidence for such a sub division of the digit before the Middle Kingdom exists, this leaves the digit as the smallest unit, known to have been used at around the time the Pyramids were built. One cubit is approximately equal to 20.6 inches, making the digit equivalent to 0.736 inches, or 18.68 millimetres. Roughly three-quarters of an inch. If the artisans rounded down to 20 digits, the slope of the pyramid would be wrong, and would give the wrong build height. It is possible to adopt the view that the example is merely a student's exercise, but one need only look at existing pyramids to see that the majority deliver a seked ratio that does not divide down into a whole number of digits. One that does give you a whole number is the Great Pyramid, or Khufu's pyramid. This pyramid closely approximates to 280 cubits in height, and has base sides of 440 cubits. If you apply the same computation as in the Rhind Papyrus, the following applies: -

Find ½ of 440, which is 220 and divide 220 by 280 giving 22/28. This breaks down to ½, + ¼, + 1/28. Multiply each of these numbers by 7 and you obtain, [(3 + ½), +(1, +1/2, +1/4), + (1/4)], or 5 + ½ palms. This equals 22 digits exactly. If the `building' hypothesis were correct, it would certainly be possible to construct the Great Pyramid by having a run of 22 digits and a rise of 28 digits. Hey Presto, you would end up with the right height and the right outside slope.

However this in itself leads to an obvious problem, that anyone who has ever stood on, or by, the granite blocks that make up the Khufu pyramid will instantly recognise. Although by definition the pyramid has a seked ratio of 5 + ½, the individual blocks are much bigger than 22 digits in and 28 digits up. Therefore it is clear that if a seked ratio was used to build the pyramids, there would need to be a smaller unit of measure than the digit. Otherwise no granite blocks bigger than 1 cubit in height could have been used without loss of accuracy, and if nothing else is certain; nobody disputes that the Great Pyramid was built to incredible tolerances, making it one of the most accurately executed designs ever built.

So we appear to be back where we started. A mathematical method for computing the slope of a pyramid that appears to be limited in practicality, without the presumption of a standardised division of the digit.

So are we missing something? Based on the evidence of the mathematical papyrus available for study, it would be impossible to postulate that the Egyptians understood trigonometry. There are no records that indicate they understood the concept of an angle, but it is clear that they had some method of comparing a right-angled triangle to a similar one. To quote Eli Maor, "We may therefore be justified in crediting the Egyptians with a crude knowledge of practical trigonometry – perhaps `proto-trigonometry' would be a better word – some two thousand years before the Greeks…."²⁴

We have seen then, that the ancient Egyptians had a method for computing the height to base ratio for a pyramid of given seked. The seked found in problem 56, of 18/25, corresponds to a pyramid angle, or slope, of 54° 15'. Compare this slope to the main pyramids of the Giza plateau. That of Khufu, 51° 52', that of Khafre, 52°20', and that of Menkaure, 50° 47'. Although closely related it is clear that no uniform angle was chosen when building a pyramid. Far less obvious, with the exception of the Great Pyramid of Khufu, is that none of these angles gives a seked that corresponds to an exact number of digits. It is therefore unlikely that the pyramid slope was the governing factor when deciding on the design dimensions of a pyramid. The consensus of opinion is that pyramid angles were chosen as a compromise between aesthetic principles and the practicalities of construction. A smaller slope produces a less appealing pyramid, whilst one with a greater slope is more difficult to construct, becoming increasingly prone to subsidence. The Bent pyramid of Sneferu is a good example. Evidence within the core of the pyramid infers that it began as a smaller pyramid with a slope of around 60°. Structural problems with subsidence resulted in the addition of extra stone around the base giving a slope of around 55°. This was only partially successful, as mid way up the pyramid, the builders had to make the angle even shallower to avoid further problems. This new angle of about 43° gives the pyramid its unique bend.²⁵

If not the angle, could it be the height? Certainly not in isolation, as a base to height ratio is clearly required to compute the seked slope. The base dimension then? Certainly the most logical, and therefore the most likely. Proponents of the run to rise theory would conclude that the final build height of a pyramid would differ from its design height, by the cumulative accuracy of each masonry levels seked. Hence the variation of pyramid slope in the examples given, of between 50° and 54°. It is equally possible that some combination of height and base dimension was required, raising the possibility that either cross sectional area, or face area was a governing factor. The only other possibility is volume. Choosing the external characteristics of a pyramid by its volume appears unlikely, but not outside the bounds of possibility.

The Ancient Egyptians were well aware that the volume of a square based pyramid is one third the area of its base multiplied by its height.²⁶ This could have been easily demonstrated by comparing the amount of water or sand needed to fill pyramid and prism shaped boxes, and was well within the capabilities of ancient Egypt. More difficult would be the application of this principle to determine the volume of an incomplete pyramid. Thomas F Banchoff sites this example in his book `Beyond the Third Dimension' aimed at showing that algebraic, and geometric patterns work in higher dimensions.²⁷ Consider an incomplete pyramid, (fig 3.) of side a with height h, and the incomplete part sitting on top, of side b, and assumed height x. Algebra reveals the volume of the incomplete pyramid of height h, as: -

Clever indeed, but the Rhind Papyrus has more clues to reveal about the condition of mathematics in the time of Ahmes, the most interesting being problem 79. This problem is one of geometric progression and states `seven houses, 49 cats, 343 mice, 2,301 ears of spelt, 16807 hekats.'

The answer is the sum of 7 + 7² + 7³ + 7⁴ + 7⁵ , or 19,607.

This problem has been used by mathematicians to demonstrate that Egyptians could undertake any multiplication by repeated doubling and adding, and was as fundamental to the Egyptian Scribe as the multiplication table to a pupil today. For a fuller explanation, I refer you to the excellent `Trigonometric Delights, by Eli Maor,²⁹ but for our purposes the problem itself is of sufficient interest. One part of the problem translates as 2,301 ears of spelt. This figure is wrong, 7 x 7 x 7 x 7 equals 2,401. Clearly there is a mistake in the Rhind Papryus, that raises a few interesting questions. You will recall that the Scribe Ahmes appeared to copy the work from an earlier document. What he wrote at the beginning of the Papyrus was: -

This book was copied in the year 33, in the fourth month of the inundation season, under the majesty of the king of Upper and Lower Egypt, `A-user-Re', endowed with life, in likeness to writings of old made in the time of the king of Upper and Lower Egypt, `Ne-ma' et-Re'. It is the scribe A'h-mose who copies this writing.

Now whilst I accept that deciphering hieroglyphics is a mixture of art with science; if we accept this translation at face value, there is a curious phrase within this introduction, `in likeness to writings of old', which could have a number of meanings. In his opening words he states that `this book was copied…' This is taken literally to mean that it is the work of an earlier Scribe or Scribes of unknown origin. Delving into history is the realm of the historian or archaeologist, and therefore this work introduces Ahmes as either the first documented archaeologist, or historian. The introduction could be taken to mean that it was an amalgamation of more than one Papyrus, just as easily as it has been assumed to be a copy of just one. Indeed the error in problem 79 strongly suggests that Ahmes was merely a copyist and not a mathematician. If Ahmes copied 2,401 wrongly, then why didn't he correct the error? If Ahmes was a mathematician, or intended the Papyrus to be used as a working textbook, surely the error would have been discovered. If he copied the work exactly, then this lack of knowledge is perfectly understandable, for he would not have needed to know that the figure was wrong. Equally why does the introduction not say `this book was copied from writings of old'? By using the words `in likeness', Ahmes could just as well mean that he was copying an uncommon form of writing exactly as written, or that he was editing a number of works into one. In either case it would not have been necessary for him to understand what was written, and in no way qualifies him as a mathematician. To use the plural ` writings of old', is a further indication that the Rhind Papyrus is an edited version of a number of earlier works.

To speculate on what was included, or excluded, and why, is an interesting if fruitless occupation. For now let us merely state that the available historical evidence does not support the notion that the Ancient Egyptians had any understanding of Pi; that their quite considerable arithmetical expertise was limited in exactitude to triangles, and that their knowledge of circular constructs was at best inexact.

1       Mathematics from the Birth of Numbers, Jan Gullberg, W W Norton & Company, 1997.

2       Numbers through the ages, edited by G Flegg, MacMillan education 1989

3       Science history journal Isis, xxvii, 462-3, as cited by G Flegg, Numbers through the ages.p37.

4       Number words and Number symbols , K Menninger, MIT press 1969, as cited by G Flegg, Numbers through the ages

5       Numbers through the ages, edited by G Flegg, MacMillan education 1989

6        I.b.i.d.

7       Trigonometric Delights , Eli Maor, Princeton University Press 1998.

8      Fundamental Steps in the Development of Numeration, C.B.Boyer, Isis 35, as cited by G Flegg, Numbers through the ages

9       Author unknown, but containing 25, mainly practical examples. Its origin is thought to be around 1700BC. It resides in the Moscow Museum of Fine Art. It is 15 feet long and about 3 inches wide.

10     Numbers through the ages, edited by G Flegg, MacMillan education 1989

11     Trigonometric Delights , Eli Maor, Princeton University Press 1998

12      The Rhind Mathematical Papyrus: Free translation and Commentary with selected Photographs, Transcriptions, Transliterations and Literal Translations, Arnold Chase, Reston VA National Council of Teachers of Mathematics 1979, as cited by Eli Maor, Trigonometric Delights.

13     Rhind Mathematical Papyrus, A B Chase, published by the Mathematical Association of America in two volumes in 1927 and 1929.

14     Trigonometric Delights , Eli Maor, Princeton University Press 1998.

15     Numbers through the ages, edited by G Flegg, MacMillan education 1989

16     Trigonometric Delights, Eli Maor, Princeton University Press 1998.

17      I.b.i.d.

18     Numbers through the ages, edited by G Flegg, MacMillan education 1989

19      I.b.i.d

20     Trigonometric Delights , Eli Maor, Princeton University Press 1998.

21      One hekat has been determined to equal 292.24 cubit inches or 4.789 lites. See Mathematics , Gillings, as cited by Eli Maor, Trigonometric Delights.

22     Trigonometric Delights , Eli Maor, Princeton University Press 1998.

23     Rhind Mathematical Papyrus, A B Chase, published by the Mathematical Association of America in two volumes in 1927 and 1929, as cited by Eli Maor, Trigonometric Delights.

24     Trigonometric Delights , Eli Maor, Princeton University Press 1998.

25      The Complete Pyramids, Mark Lehner, Thames & Hudson, 1997.

26     Beyond the third Dimension, Thomas F Banchoff, Scientific American library, 1990.

27      I.b.i.d

28      I.b.i.d

29      Trigonometric Delights, Eli Maor, Princeton University Press 1998.