The publication of Chace's edition of the Rhind papyrus is an event of importance to all mathematicians interested in the history of their science.

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The Rhind Mathematical Papyrus, British Museum 10057 and 10058. Vol. I.
Free Translation and Commentary, by A. B. Chace, Chancellor, Brown
University, with the assistance of H. P. Manning, Associate Professor of
Mathematics, Brown University, Retired. Bibliography of Egyptian
Mathematics, by R. C. Archibald, Professor of Mathematics, Brown University.
Mathematical Association of America, Oberlin, Ohio, U. S. A.,
1927. Pages 1-210. Volume 11, Photographs, Transcription, Transliteration,
Literal Translation, by A. B. Chace . . . L. Bull, Associate Curator in the
Egyptian Department, Metropolitan Museum, New York, H. P. Manning,
. . Bibliography of Egyptian and Babylonian Mathematics (Supplement),
by R. C. Archibald . . . . The Mathematical Leather Roll in the British
Museum, by S. R. K. Glanville, Department of Egyptian and Assyrian
Antiquities, British Museum, . 1929.
The publication of Chace's edition of the Rhind papyrus is an event of importance
to all mathematicians interested in the history of their science. It
marks the culmination of the efforts of many students of Egyptian mathematics,
endeavoring to present to the modern reader the complete and exact form of the
ancient papyrus, to supply an exact translation of it and to give a philosophic
interpretation of Egyptian processes in mathematics. The Chace edition represents
all that painstaking study and liberality in financial expenditure could
accomplish at the present time.
Before the discovery of the Rhind papyrus, little was known of Egyptian
mathematics, except the fragmentary statements of Greek writers. Thales and
Pythagoras had travelled in Egypt and acquired Egyptian lore. But the
Egyptian mathematics which percolated into Greek books which have come
down to the present time seem unworthy of a people which had erected the
Egyptian pyramids. The Rhind papyrus was found about 1858, in a building
near the great monument Ramesseum, at Thebes. It came into the possession
of A. Henry Rhind from whom its name is derived. In 1864 it was purchased
by the trustees of the British Museum. The papyrus roll was broken in some
places and parts were missing. The trustees of the British Museum, in 1869,
authorized the preparation of a facsimile edition and a descriptive text of the
papyrus. Samuel Birch, Keeper of the Egyptian antiquities of this Museum,
prepared plates, but the progress of the work was delayed. In the Spring of
1872, the archeologist August Eisenlohr of Heidelberg visited England and
secured from Birch proofs taken from these plates. After five years of study,
in which he was assisted by Moritz Cantor, the historian of mathematics,
Eisenlohr brought out his edition of the Rhind papyrus in 1877. He did this,
however, without securing the consent of the trustees of the British Museum to
use Birch's plates. Besides the reproduction of the papyrus itself, Eisenlohr gave
a commentary, as well as a translation of the papyrus into German and a
transcription of the hieratic script in which the papyrus is written into the more
picturesque hieroglyphic script. During the half century following, Eisenlohr's
very creditable publication was the only source from which mathematicians
could obtain a knowledge of the contents of the Rhind papyrus. Meanwhile, in
1898, the British Museum issued a beautiful lithographic "facsimile" of the
papyrus. By "facsimile" we are not to understand an exact reproduction by
the aid of photography; it was simply an imperfect hand copy, containing many
little deviations from the original. As Eisenlohr used the British Museum plates,
his representation of the papyrus was no better; besides he made some slight
alterations of his own.
During the fifty years following 1877, great strides were made in the mastery
of the languages of ancient Egypt. It became evident that a new translation
and commentary of the Rhind papyrus was desirable. In 1923 this was accomplished
in a masterly way by T. Eric Peet, professor of Egyptology in the University
of Liverpool. By a strange good fortune some of the missing parts of the
papyrus were found in 1922, in the possession of the New York Historical
Society. Copies of these fragments were prepared, and Peet succeeded in finding
the proper places in the papyrus for 25 of the 40 fragments. Using the papyrus
itself, not the "facsimile" of it, Peet gave a hieroglyphic transcription and
a translation into English. Critics found this a very able piece of work, notwithstanding
occasional slips of perhaps minor importance.
For some years preceding the appearance of Peet's translation and commentary,
Arnold B. Chace, Chancellor of Brown University, had made an intensive
study of the Rhind papyrus. In some details his conclusions were not in entire
agreement with those of Peet. Chace and his co-workers felt that there was
still room for a new edition of the papyrus, especially as no real facsimile of the
papyrus, and of the fragments possessed by the New York Historical Society,
were available to the general reader. As the outcome of Chace's enthusiasm and
liberality, Lye have the present magnificent edition, of which the second volume
contains the photographic reproduction of the papyrus as it appears at the
present: time in the British Museum, and of the New York fragments. There
is a satisfaction in examining an exact copy. When told that a pair of legs
walking forward represents "addition," and when walking backward represents
"subtraction," we like to see how much of this statement is due to the imagination
of the modern investigator, and how much to the graphic technique of the
ancient scribe. Our inspection leads to the conclusion that the modern imaginative
faculty operated in full force.
The different parts of the papyrus are shown by 31 photographs. In making
the hieroglyphic transcription, the transliteration, and literal translation,
109 plates are used. This is not the place to enumerate the many minute new
interpretations. Suffice it to say that a dozen small fragments not located by
Peet, are in the Chace edition assigned to what seem to be their proper places
in the papyrus.
As indicated on the title page Chancellor Chace had as collaborators, H. P.
Manning and R. C. Archibald, both of Brown University, and Ludlow Bull of
the Egyptian department of the Metropolitan Museum in New York. Mr. S.
R. K. Glanville of the British Museum supplied a brief account of a mathematical
leather roll in that Museum. I t is an Egyptian document which appears
to be of greater significance than at first thought.
A valuable feature of the Chace edition is the bibliography of Egyptian
mathematics prepared by Archibald. The many informative and critical remarks
are particularly valuable to readers who have not the time or opportunity
to consult the original sources. The very last entry is a pre-publication
notice of V. IT. Struve's account of the Golenishchev mathematical papyrus of
the Moscow Museum of Fine Arts, which will appear in Quellen und Studien
zur Geschichte der il.i'atherilatik, Abteilztng A : Quellen, Berlin, 1930. Parts of this
Egyptian papyrus have been deciphered and published before. It contains
startling revelations on early Egyptian mathematics, and may rival in importance
the Rhind papyrus.
FLORIAN CA JORI

Mathematics in the Times of the Pharaohs Mathematics in Egyptian Papyri from :Mathematical Atlas: A Gateway to Mathematics In the article An overview of Egyptian mathematics we looked at some details of the major Egyptian papyri which have survived. In this article we take a detailed look at the mathematics contained in them. The Rhind papyrus Ahmes, in the Rhind papyrus, illustrates the Egyptian method of multiplication in the following way. Assume that we want to multiply 41 by 59. Take 59 and add it to itself, then add the answer to itself and continue:- 41 59 _________________ 1 59 2 118 4 236 8 472 16 944 32 1888 _________________ Since 64 > 41, there is no need to go beyond the 32 entry. Now go through a number of subtractions 41 - 32 = 9, 9 - 8 = 1, 1 - 1 = 0 to see that 41 = 32 + 8 + 1. Next check the numbers in the right hand column corresponding to 32, 8, 1 and add them. 41 59 _________________ 1 59 2 118 4 236 8 472 16 944 32 1888 _________________ 2419 Notice that the multiplication is achieved with only additions, notice also that this is a very early use of binary arithmetic (see below). Reversing the factors we have: 59 41 _________________ 1 41 2 82 4 164 8 328 16 656 32 1312 _________________ 2419 Notice that for this method to work we need to know that ever number is the sum of powers of 2. The ancient Egyptians would not have had a proof of this, nor would have appreciated that a proof was necessary. They would just know from practical experience that it could always be done. Basically we can think of the method as writing one of the numbers to base 2. In the examples above we have written 41 = 1.20 + 0.21 + 0.22 + 1.23 + 0.24 + 1.25 and 59 = 1.20 + 1.21 + 0.22 + 1.23 + 1.24 + 1.25. Division works also using doubling. For example to divide 1495 by 65 we proceed as follows: 1 65 2 130 4 260 8 520 16 1040 We stop at this point because the next doubling will take us beyond 1495. Now we look for numbers in the right hand column which add up to 1495. We see that 1040 + 260 + 130 + 65 = 1495 and we tick the rows in which these numbers occur: 1 65 2 130 4 260 8 520 16 1040 Now add the numbers in the left hand column which are in ticked rows: 16 + 4 + 2 + 1 = 23, so 1495 divided by 65 is 23. What happens if the numbers do not divide exactly? Then the Egyptian method will yield fractions as the following example shows. To divide 1500 by 65 proceed as before: 1 65 2 130 4 260 8 520 16 1040 Again we stop since the next doubling takes us beyond 1500. Now look for the numbers in the right hand column which add to a number n with 1500-65 < n ≤ 1500. [The Egyptians knew that this was always possible: can you prove that this is so?] In this case we have 1040 + 260 + 130 + 65 = 1495 and we are 5 short of our sum. Again tick the rows with these entries: 1 65 2 130 4 260 8 520 16 1040 Now add the numbers in the left hand column which are in ticked rows: 16 + 4 + 2 + 1 = 23, so 1500 divided by 65 is 23 and 5/65 = 1/13 remaining. Hence the answer is 23 1/13. We have cheated a little here for the fraction obtained is a unit fraction, that is a number of the form 1/n for n an integer. In fact the Egyptians only had fractions of this type and if the answer had not involved a unit fraction then the Egyptians would have written the fractional part as the sum of unit fractions. We see below how this was done but we examine a more general case. The next problem is how to multiply and divide numbers involving fractions. The first important point is that the Egyptians only used unit fractions, and to be able to calculate a table was needed to convert twice a unit fraction into a sum of unit fractions. Now it might be supposed that doubling the unit fraction 1/5 would be easy and yield the sum of the unit fractions 1/5 + 1/5. However, for reasons which we do fully understand, this was not their approach. They wrote twice a unit fraction as the sum of distinct unit fractions. For example twice 1/5 would be written as 1/3 + 1/15. The Rhind papyrus gives a table for doubling unit fractions 1/n for n odd, n between 5 and 101. Note that Ahmes did not need to give the double of 1/n for n even since it is just 1/m where m = 2n. The doubling table for unit fractions begins Unit fraction Double unit fraction ____________________________________ 1/5 1/3 + 1/15 1/7 1/4 + 1/28 1/9 1/6 + 1/18 1/11 1/13 1/15 1/10 + 1/30 1/17 1/12 + 1/51 + 1/68 .... .................. It is remarkable that there are no errors in the table. Certainly Ahmes would have been expert at calculating and this would not have been simply a copying exercise for him. There are few errors in the Rhind papyrus but those which there are appear to be errors of calculation, not of copying, since the incorrect result is carried forward rather than a return to the correct path which would happen from an error in copying. There is the fascinating question of how these decompositions were found, and why some decompositions were chosen in preference to others. This is discussed in and further ideas, adding and correcting information from , is given in and . The favourite rules which many historians such as Gillings believe guided the scribes in their choice of decomposition of 2/n into unit fractions are (1) prefer small numbers (2) the fewer terms the better, and never more than four (3) prefer even to odd numbers. However other historians such as Bruins argues against such rules. His argument is essentially that before applying these rules one would need to work out all unit decompositions of 2/n and there is no evidence that the Egyptians had any methods to do this. As an example of how to use the table, let us examine Problem 21 of the Rhind papyrus. Note that 2/3 was an allowable Egyptian fraction despite not being a unit fraction. Problem 21: Complete 2/3 and 1/15 to 1. In modern terms, this asks for a fraction x such that 2/3 + 1/15 + x = 1. The method of solution was to "get rid of" the fractions by multiplying through. In this case multiply each fraction by 15 to obtain 10 + 1 + y = 15. This is called the "red auxiliary" equation since the scribe wrote this equation in red ink. [Of course it would not appear in this form but rather "complete 10 and 1 to 15".] Now the answer to the red auxiliary equation is 4 so the original equation had solution twice (twice 1/15). From the doubling table we see that double 1/15 is 1/10 + 1/30. Doubling this gives 1/5 + 1/15 which is the required solution to Problem 21. Another example of solving an equation is Problem 24 which asks: Problem 24: A quantity added to a quarter of that quantity become 15. What is the quantity? Ahmes uses the "method of false position" which was still a standard method three thousand years later. In modern notation the problem is to solve x + x/4 = 15. Ahmes guesses the answer x = 4. This is to remove the fraction in the x/4 term. Now with x = 4 the expression x + x/4 becomes 5. This is not the correct answer, for the expression is required to equal 15. However, 15 is 3 times 5 so taking 3 times his guess of x = 4, namely x = 12, gives Ahmes the correct result. Another interpretation, favoured by some historians, is that Ahmes thought of the method as dividing x into 4 equal pieces of a size to be determined. Now Ahmes computes x + x/4 getting 5 of these equal pieces. Each piece must now be three so that 5 pieces equals 15. Not very different to our previous way of thinking, but one which is likely to come closer to Ahmes' way of thinking than our former description. Finally Ahmes checks his solution, or proves his answer is correct. He takes x = 4 3 = 12. Then x/4 = 3, so x + x/4 = 15 as required. The methods of false position is used in Problems 24 to 29 of the Rhind Papyrus. However, in Problem 31 of the Papyrus Ahmes uses the simpler method of pure division. This is discussed in detail in Let us now see how to multiply, using Egyptian methods, 1 + 1/3 + 1/5 by 30 + 1/3. 1 1 + 1/3 + 1/5 2 2 + 2/3 + 1/3 + 1/15 = 3 + 1/15 4 6 + 1/10 + 1/30 8 12 + 1/5 + 1/15 16 24 + 1/3 + 1/15 + 1/10 + 1/30 2/3 2/3 + 1/6 + 1/18 + 1/10 + 1/30 1/3 1/3 + 1/12 + 1/36 + 1/20 + 1/60 Now here the row beginning 2/3 has been computed from 2/3 of 1 is 2/3, 2/3 of 1/3 is double 1/9 which is 1/6+1/18, 2/3 of 1/5 is double 1/15 which is 1/10 + 1/30. Next find the numbers in the left hand column which add to 30+1/3. These are the rows marked with a tick: 1 1 + 1/3 + 1/5 2 3 + 1/15 4 6 + 1/10 + 1/30 8 12 + 1/5 + 1/15 16 24 + 1/3 + 1/15 + 1/10 + 1/30 2/3 2/3 + 1/6 + 1/18 + 1/10 + 1/30 1/3 1/3 + 1/12 + 1/36 + 1/20 + 1/60 Add the entries in the right hand column of the rows which are ticked to get the result of the multiplication 46 + 1/5 + 1/10 + 1/12 + 1/15 + 1/30 + 1/36. As a final look at the Rhind papyrus let us give the solution to Problem 50. A round field has diameter 9 khet. What is its area? Here is the solution as given by Ahmes. Take away 1/9 of the diameter, namely 1; the remainder is 8. Multiply 8 times 8; it makes 64. Therefore it contains 64 setat of land. Do it thus: 1 9 1/9 1 this taken away leaves 8 1 8 2 16 4 32 8 64 Its area is 64 setat. Notice that the solution is equivalent to taking π = 4(8/9)2 = 3.1605. This is a remarkable result if one considers the date at which this approximation must have been discovered. The intriguing question is raised as to how such a discovery might have been made. Although we have no way of ever knowing this with certainty, several interesting conjectures have been suggested. In Gerdes gives three ideas which might have led the Egyptians to this result. Two such conjectures suggested in concern African crafts where a snake curve and a set of equidistant concentric rings are often seen. These two geometric designs are widespread in Africa and Gerdes shows how these could have led to a formula for the area of a circle. The third conjecture in relates to a board game "mancala" which was popular throughout Africa and ancient Egypt. The game involves comparing small circles with larger circles and may have provided the motivation for the area formula. The Moscow papyrus Although the mathematical methods we have described are found in various Egyptian documents, all the actual examples we have given so far have come from Rhind papyrus. Let us finish this article by looking at an example from the Moscow papyrus which many historians argue is the most impressive achievement of Egyptian mathematics. The problem is number 14 from the papyrus and it concerns the geometrical figure visible in the portion of Moscow papyrus seen in this image. Example 14. Example of calculating a truncated pyramid. The base is a square of side 4 cubits, the top is a square of side 2 cubits and the height of the truncated pyramid is 6 cubits. First we remark that by "calculate a pyramid" the author means "calculate the volume of the pyramid". Also not how appropriate this calculation is for the civilisation which today is universally known for the remarkable construction of pyramids. The calculation begins by working out the area of the base: 4 4 = 16. Then the area of the top is worked out: 2 2 = 4. Next the product of the side of the base with the side of the top is computed: 4 2 = 8. These three are then added: 16 + 4 + 8 = 28. Now 1/3 of the height is taken, namely 2. Finally the product of 1/3 of the height with the previous sum of 28 is taken and the scribe writes:- Behold it is 56. This example means that the Egyptian knew the formula for the volume (although of course not in the algebraic sense which we now think of formulas). If the base square has side a, the top square has side b, and the height is h then V = h (a2 + ab + b2)/3. References (43 books/articles) An overview of Egyptian mathematics Civilisation reached a high level in Egypt at an early period. The country was well suited for the people, with a fertile land thanks to the river Nile yet with a pleasing climate. It was also a country which was easily defended having few natural neighbours to attack it for the surrounding deserts provided a natural barrier to invading forces. As a consequence Egypt enjoyed long periods of peace when society advanced rapidly. By 3000 BC two earlier nations had joined to form a single Egyptian nation under a single ruler. Agriculture had been developed making heavy use of the regular wet and dry periods of the year. The Nile flooded during the rainy season providing fertile land which complex irrigation systems made fertile for growing crops. Knowing when the rainy season was about to arrive was vital and the study of astronomy developed to provide calendar information. The large area covered by the Egyptian nation required complex administration, a system of taxes, and armies had to be supported. As the society became more complex, records required to be kept, and computations done as the people bartered their goods. A need for counting arose, then writing and numerals were needed to record transactions. By 3000 BC the Egyptians had already developed their hieroglyphic writing (see our article Egyptian numerals for some more details). This marks the beginning of the Old Kingdom period during which the pyramids were built. For example the Great Pyramid at Giza was built around 2650 BC and it is a remarkable feat of engineering. This provides the clearest of indications that the society of that period had reached a high level of achievement. Hieroglyphs for writing and counting gave way to a hieratic script for both writing and numerals. Details of the numerals themselves are given in our article Egyptian numerals. Here we are concerned with the arithmetical methods which they devised to work with these numerals The Egyptian number systems were not well suited for arithmetical calculations. We are still today familiar with Roman numerals and so it is easy to understand that although addition of Roman numerals is quite satisfactory, multiplication and division are essentially impossible. The Egyptian system had similar drawbacks to that of Roman numerals. However, the Egyptians were very practical in their approach to mathematics and their trade required that they could deal in fractions. Trade also required multiplication and division to be possible so they devised remarkable methods to overcome the deficiencies in the number systems with which they had to work. Basically they had to devise methods of multiplication and division which only involved addition. Early hieroglyphic numerals can be found on temples, stone monuments and vases. They give little knowledge about any mathematical calculations which might have been done with the number systems. While these hieroglyphs were being carved in stone there was no need to develop symbols which could be written more quickly. However, once the Egyptians began to use flattened sheets of the dried papyrus reed as "paper" and the tip of a reed as a "pen" there was reason to develop more rapid means of writing. This prompted the development of hieratic writing and numerals. There must have been a large number of papyri, many dealing with mathematics in one form or another, but sadly since the material is rather fragile almost all have perished. It is remarkable that any have survived at all, and that they have is a consequence of the dry climatic conditions in Egypt. Two major mathematical documents survive. You can see an example of Egyptian mathematics written on the Rhind papyrus and another papyrus, the Moscow papyrus, with a translation into hieratic script. It is from these two documents that most of our knowledge of Egyptian mathematics comes and most of the mathematical information in this article is taken from these two ancient documents. Here is the Rhind papyrus The Rhind papyrus is named after the Scottish Egyptologist A Henry Rhind, who purchased it in Luxor in 1858. The papyrus, a scroll about 6 metres long and 1/3 of a metre wide, was written around 1650 BC by the scribe Ahmes who states that he is copying a document which is 200 years older. The original papyrus on which the Rhind papyrus is based therefore dates from about 1850 BC. Here is the Moscow papyrus The Moscow papyrus also dates from this time. It is now becoming more common to call the Rhind papyrus after Ahmes rather than Rhind since it seems much fairer to name it after the scribe than after the man who purchased it comparatively recently. The same is not possible for the Moscow papyrus however, since sadly the scribe who wrote this document has not recorded his name. It is often called the Golenischev papyrus after the man who purchased it. The Moscow papyrus is now in the Museum of Fine Arts in Moscow, while the Rhind papyrus is in the British Museum in London. The Rhind papyrus contains eighty-seven problems while the Moscow papyrus contains twenty-five. The problems are mostly practical but a few are posed to teach manipulation of the number system itself without a practical application in mind. For example the first six problems of the Rhind papyrus ask how to divide n loaves between 10 men where n =1 for Problem 1, n = 2 for Problem 2, n = 6 for Problem 3, n = 7 for Problem 4, n = 8 for Problem 5, and n = 9 for Problem 6. Clearly fractions are involved here and, in fact, 81 of the 87 problems given involve operating with fractions. Rising, in discusses these problems of fair division of loaves which were particularly important in the development of Egyptian mathematics. Some problems ask for the solution of an equation. For example Problem 26: a quantity added to a quarter of that quantity become 15. What is the quantity? Other problems involve geometric series such as Problem 64: divide 10 hekats of barley among 10 men so that each gets 1/8 of a hekat more than the one before. Some problems involve geometry. For example Problem 50: a round field has diameter 9 khet. What is its area? The Moscow papyrus also contains geometrical problems. Unlike the Greeks who thought abstractly about mathematical ideas, the Egyptians were only concerned with practical arithmetic. Most historians believe that the Egyptians did not think of numbers as abstract quantities but always thought of a specific collection of 8 objects when 8 was mentioned. To overcome the deficiencies of their system of numerals the Egyptians devised cunning ways round the fact that their numbers were poorly suited for multiplication as is shown in the Rhind papyrus. We examine in detail the mathematics contained in the Egyptian papyri in a separate article Mathematics in Egyptian Papyri. In this article we next examine some claims regarding mathematical constants used in the construction of the pyramids, in particular the Great Pyramid at Giza which, as we noted above, was built around 2650 BC. Joseph and many other authors gives some of the measurements of the Great Pyramid which make some people believe that it was built with certain mathematical constants in mind. The angle between the base and one of the faces is 51° 50' 35". The secant of this angle is 1.61806 which is remarkably close to the golden ratio 1.618034. Not that anyone believes that the Egyptians knew of the secant function, but it is of course just the ratio of the height of the sloping face to half the length of the side of the square base. On the other hand the cotangent of the slope angle of 51° 50' 35" is very close to π/4. Again of course nobody believes that the Egyptians had invented the cotangent, but again it is the ratio of the sides which it is believed was made to fit this number. Now the observant reader will have realised that there must be some sort of relationship between the golden ratio and π for these two claims to both be at least numerically accurate. In fact there is a numerical coincidence: the square root of the golden ratio times π is close to 4, in fact this product is 3.996168. In Robins argues against both the golden ratio or π being deliberately involved in the construction of the pyramid. He claims that the ratio of the vertical rise to the horizontal distance was chosen to be 5 1/2 to 7 and the fact that (11/14) 4 = 3.1428 and is close to π is nothing more than a coincidence. Similarly Robins claims the way that the golden ratio comes in is also simply a coincidence. Robins claims that certain constructions were made so that the triangle which was formed by the base, height and slope height of the pyramid was a 3, 4, 5 triangle. Certainly it would seem more likely that the engineers would use mathematical knowledge to construct right angles than that they would build in ratios connected with the golden ratio and π. Finally we examine some details of the ancient Egyptian calendar. As we mentioned above, it was important for the Egyptians to know when the Nile would flood and so this required calendar calculations. The beginning of the year was chosen as the heliacal rising of Sirius, the brightest star in the sky. The heliacal rising is the first appearance of the star after the period when it is too close to the sun to be seen. For Sirius this occurs in July and this was taken to be the start of the year. The Nile flooded shortly after this so it was a natural beginning for the year. The heliacal rising of Sirius would tell people to prepare for the floods. The year was computed to be 365 days long and this was certainly known by 2776 BC and this value was used for a civil calendar for recording dates. Later a more accurate value of 365 1/4 days was worked out for the length of the year but the civil calendar was never changed to take this into account. In fact two calendars ran in parallel, the one which was used for practical purposes of sowing of crops, harvesting crops etc. being based on the lunar month. Eventually the civil year was divided into 12 months, with a 5 day extra period at the end of the year. The Egyptian calendar, although changed much over time, was the basis for the Julian and Gregorian calendars.