THE MATHEMATICS OF ANCIENT EGYPT

THE FRACTION TWO-THIRDS

THE EMLR

TABLES FOR THE ADDITION OF UNIT FRACTIONS, AND THE G RULE

AN INTERESTING OSTRACON

THE SCRIBAL RULE FOR RECIPROCALS

DEMOTIC MATHEMATICAL PAPYRI

AN INTERPRETATION OF RMP 79

PESU PROBLEMS

EGYPTIAN MEASURES

EQUATIONS

ADDENDA

NOTES

BIBILOGRAPHY

ADDITIONAL BIBLIOGRAPHICAL NOTES

R. J. Gillings

When one refers to mathematics in the time of the pharaohs, the significance of the word “mathematics” inevitably comes into consideration. In the long period of ancient Egypt, some three millennia, mathematics meant first arithmetic, then some elementary geometry, then varied problems that by modern standards had an algebraic flavor. The historian would be wise not to judge too hastily the very few genuinely mathematical papyri and ostraca that have come down to us, by making critical comparisons with the more numerous and detailed works of the Greeks, which have been known and studied for over two thousand years. Pharaonic mathematics has been available to the student and the historian for barely a century, although it originated nearly four thousand years ago. One had to wait for a Champollion and a Rosetta Stone before interpretation of hieroglyphs, and the cursive hieratic writing and numbers, became possible. When it appeared that Egyptian arithmetic was based solely upon a complete knowledge of the “two-times” table and an ability to find two-thirds of any number, integral or fractional; that their geometry dealt almost wholly with areas and volumes; and that problems were solved by a kind of literal algebraic reasoning, the question arose whether this can be called mathematics.

It is a matter of semantics; and to help resolve it, one should note what Ernst Mach wrote in 1898: “There is no problem in all mathematics that cannot be solved by direct counting, but with present implements, many operations can be performed in a few minutes, which without mathematical methods, would take a lifetime.” And Comte wrote: “There is no inquiry which is not finally reducible to a question of numbers.” Acceptance of these statements would perhaps justify saying that Egyptians at the time of the pharaohs did have mathematics, in their own particular way.

Of all the ancient papyri and ostraca that have been recovered from ancient Egypt and are preserved in universities, museums, and other institutions, relatively few are of a mathematical nature. About a dozen of these, often referred to in a study of the history of mathematics, are briefly listed below, not in order of their importance but alphabetically, with their abbreviations that will be used throughout the essay. More detail regarding these papyri will be found following the bibliographical notes.

THE FOUR OPERATIONS

Addition . Modern arithmetic books give special techniques for addition, subtraction, multiplication, and division. Combinations like 3 + 5 = 8 and 2 + 7 = 9 are taught first by simple counting, and then they are memorized; at the same time the

subtractive operations of 8 – 5 = 3 and 9 – 7 = 2, being related, are also learned. Subtraction of large numbers is taught in several different ways, however, such as the “borrow and pay back” method or the “equal additions” method. For multiplication and division, the present technique is that all multiplication tables must be learned by heart up to that for twelve —a time-consuming operation for students, many of whom never really learn them.

We are to examine the probable techniques of the Egyptian scribes for the four operations with their hieratic notation, which, although it was based on a decimal system, is still unlike our modern Hindu-Arabic form. Many historians have glossed over considerations of how the ancient scribes performed any of the four operations, with statements like the following: “Addition is simple counting. Subtraction is merely counting backwards. Multiplication is a special form of counting, and division is the reverse of multiplication.”¹ How easy it all sounds! “People who could count beyond a million had no difficulty about the addition and subtraction of whole numbers.”² But they did, and we still do, which explains the invention of the abacus and desk computers. “For ordinary additions and subtractions, nothing needs be said.”³ Oh, but it does!

Keeping in mind that hieratic numbers are written from right to left, note that in the first addition, A, the third addend 5 is written in the tens column, and the total is not placed underneath but is to the left of the 5. There is nothing to be “carried.” In the second addition, B, the addend 192 is slightly out of alignment and the “carrying” is easily enough done, with the total at the bottom. In both C and D the arrangements of the addends and the totals are closer to what we would expect to see. Note, however, that 320 is not to be included in the addition, because it does not have a check mark (or tick) alongside it. In E, the 6 is in the tens’ column instead of the units’ column, and again the total is not underneath. Finally, in F, note that the sign for five thousands is a dual one and, like the third addend, does not have a check mark and thus is not to be included in the total. There are three “carryings” in this sum.

The proper alignment of digits in their correct columns was not so necessary to the Egyptian scribes as it is to us, because the hieratic sign for (say) three units was , for three tens it was , and for three hundreds it was –symbols that are quite different. The scribe would therefore know at once not to add units to hundreds merely because they appeared to be in the same column. Nevertheless, all these additions need careful attention, especially the last; and to say that “there are no difficulties about addition and subtraction,”and leave it at that, is clearly evading the issue.

Many tables for the addition of fractions have come down to us, notably the EMLR. These will be considered in a separate section.

Subtraction . Tables for the addition of integers at least up to 10, and possibly further, prepared by simple counting, were most surely available, although none has come down to us in the papyri. Such tables would be equally useful for subtraction, for the scribe did not say “Subtract 5 from 8”or “Subtract 7 from 9,”but “5, how many to make

8 “and” 7, how many to make 9 “Thus we find in MMP9,” Complete thou the excess of these 10 over these 4. Result 6.”And in KP LV, 4. “Make thou the excess of 100 over 45. The result thereof is 55.”

Subtraction of unit fractions was done in the same way; and in this case tables of the addition of fractions have certainly come down to us, the best example of which is the EMLR. Examples of such subtractions, taken from the Recto of the RMP, are given below.

Consider the subtrahend of the last subtraction. To it add

In the RMP Recto there are more than twenty such subtractions, some much more complicated than those shown here; but in no case is the working shown. Only the answer is written, and there are no errors.

Multiplication . This process was done by repeated multiplications of 2, and a thorough knowledge of the “two-times”table was all that the scribal arithmetician required. Preferred or favored multipliers were 3, 7, 15, 31, 63,· · ·. Thus, to multiply 29 by 7, the scribe proceeded as shown below.

The sums of terms of the series 1, 2, 4, 8, 16, · · · produce, in order, the totals 3, 7, 15, 31, 63, · · ·, which, if they are used as multipliers, means that every number in the second column is included in the final addition. The scribes early found, however, that any multiplier at all could be expressed uniquely as the sum of certain terms of this geometrical progression, a minor difficulty being to locate which terms these were. Thus, suppose that the multiplier of 29 was 21 instead of 7.

The scribe located the numbers in the first column that totaled 21 in one way only – 1 + 4 + 16 = 21–and placed a check mark alongside them, indicating that the product was the sum of the corresponding numbers in the second column: 29 + 116 + 464 = 609. This particular property of the progression 1, 2, 4, 8, 16, … is nowhere referred to in any Egyptian papyri known to me; but there are various other properties of both arithmetical and geometrical progressions that arise in certain problems and will be considered later.

Sometimes other multipliers were used, such as 10. which meant that the hieratic digits of the multiplicand were merely rewritten, the units as tens. the tens as hundreds, and so on, since the notation is of course a decimal one. The multiplier 1/10 was used in a similar manner. The fraction two-thirds (written γ, but here ) was frequently used, and tables giving of numbers were available. The scribal rule for expressing of any fraction, in unit fractions, is given in RMP 61 B. Two-thirds of quite large numbers occurs often. For example, in RMP 33 we find of 5,432 written as 3,621 . The rule given for of the odd fraction was to add the double of the number to six times the number, giving (. and similarly for all odd fractions. The rule holds equally for even fractions, so that of would be : it was seldom used, however, because the value was simpler, from the addition to 6 of its half 3. That the reciprocal of 1 was was well-known to the scribes; indeed, the hieroglyph for was originally , one vertical line being half the length of the other. Or, again, would be written as , as indicated by line 3 of the EM LR, , which becomes on multiplication of the numbers by 3.

For the ordinary multiplication of integers the work was simple enough; but with fractions in the multiplicand, it sometimes became difficult just to multiply by 2. For example, the double of was , and the double of was ; but the double of or any other odd unit fraction was quite another matter. The doubling of the fifty “odd”unit fractions , is performed by the scribe in the first portion of the RMP, called the Recto; and it takes up almost one-third of the eighteen feet of the papyrus.

Division . Little further needs to be said about division, because the method of dividing was the same as that of multiplying. To calculate 297 divided by 11, the scribe thought of it as “By what must I multiply 11 to give me 297?”Thus 11 became the multiplicand, and he kept on doubling it until 297 was reached, just like a multiplication sum.

A further doubling would give 352, which is beyond 297. The problem is now to determine which numbers of the second column will add to 297, or come close to it –a task not always easy. In this case the numbers are 11 + 22 + 88 + 176 = 297, indicated by check marks, so that the required quotient is the sum of the corresponding numbers in the first column: 1 + 2 + 8 + 16 = 27. When the divisor contained one or more odd unit fractions, then trouble brewed, just as it did when the division of two integers did not produce an integral quotient, as 297÷11 does. To illustrate this, divide 297 by 12.

A study of the right-hand column shows that 192 + 96 = 288 is the closest sum to 297; and these, and/ or 16 and 8, are ticked. Also, 288 + 9 = 297. Then of 12 = 6, and of 12 = 3, so that we now have

The Rmp Recto

The Recto of the RMP gives the ancient Egyptian values chosen and accepted by the scribes for the expression of 2÷(2n – 1) as the sum of not more than four unit fractions, where n had the fifty values, 2, 3, 4, 5, …, 51. Since the latter part of the nineteenth century, historians of mathematics have discussed and debated how and why the scribes determined the values given in this table, and still no general agreement has been reached. In 1967, with the purpose of examining more closely the various equalities possible, I was able to enlist the services of Prof. C. L. Hamblin of the University of New South Wales to program the computer KDF-9 at Sydney University to produce all the possible answers for these equivalents. The restrictions were such that not more than four unit fractions should be included in any one equality, and that none of the denominators should exceed 1,000.

On this basis KDF–9 produced the grand total of 22,295 equivalents. But for the purpose of proper comparison of these values with the specific value given in each case by A’hmosé, the scribe of the RMP , 22,295 must be reduced by the number of redundancies properly included by KDF – 9, according to its instructions, such as

and

because ancient Egyptian scribes would never accept two equal fractions in any stated number. Therefore the total of 22,295 must be reduced by 2,024. Further, the program for KDF – 9 stated. “No fraction less than 1/1,000 should be included,”when it should have been instructed to “include only fractions greater than 1/1,000.”This is a fine distinction, but it means that the nine values in which the unit fraction 1/1,000 occurs should be subtracted, so that the total for comparison purposes is reduced to 20,262.

The purpose of this table, giving the numbers of the possible decompositions of the RMP Recto fractions, is to make clear the nature of the problems historians have puzzled over for some years. Some general agreement appears to have been reached that the scribal values for 2 divided by the odd numbers are the very best available, although in some cases there are differences of opinion. On what are these differences based? Some critics appear not to be clear on the purpose of the table, which is primarily to simplify ordinary multiplication and division, and not just an interesting operation in the theory of numbers. Attempts to explain how the scribe arrived at his values have varied considerably; indeed, there is little agreement over what precepts guided the scribe in choosing his specific

values, although some are of course obvious.

The scribe first states what solution he has selected and then, by ordinary multiplication, proves that the equality is correct. In these proofs there are no scribal errors. Nowhere in any papyrus known to me is there any indication of the scribal technique used by the scribe.⁴ Some theories regarding this technique seem to be based on modern mathematics; and attempts to reproduce the scribal values, following these theories but using only the techniques of which the scribe was capable, often are not possible. J. J. Sylvester’s treatment is a classic example.⁵ I suggest the following as the precepts that guided the scribe in his choice of equalities; and to assist in understanding their significance. I precede them with some of the Recto equalities for reference (bars over the numbers of the unit fractions have been omitted).

Canon for the RMP Recto .

Precept No. 1: Of the possible equalities, those with the smaller numbers are preferred, but none as large as 1,000.

Precept No. 2: An equality of two terms is preferred to one of three terms, and one of three terms to one of four terms; but an equality of more than four terms is never to be used.

Precept No. 3: The unit fractions are set down in descending order of magnitude –that is, smaller numbers come first –but never the same number twice.

Precept No. 4: The smallness of the first number is the main consideration, but the scribe will accept a slightly larger first number if it will greatly reduce the last number.

Precept No. 5: Even numbers are preferred to odd numbers, even though they might be larger, and even though the number of terms might thereby be increased. (There are more than one hundred even numbers used in the table, but only twenty-four odd numbers.)

There can be no doubt about the general acceptance among authorities of the first three precepts, and I am sure that competent critics will agree with the last two. It is of course possible that they might wish to add further precepts.

Historians’ Views on the Recto Equalities .

The following statements represent a cross section of opinions:

O. Becker and Hofmann – “The principle of calculation does not seem to be uniform.”

L. Hogben —“They went to extraordinary pains to split up fractions like 2/43 into a sum of unit fractions, a procedure that was as useless as it was ambiguous. The Greeks and Alexandrians continued this extraordinary performance.”

A. B. Chace –“Of the discussions which I have seen, the clearest is that by Loria, but no formula or rule has been discovered that will give all the results of the table.”

F. O. Hultsch – “Attempts to explain it have hitherto not succeeded.”

P. Mansion –“The decompositions are always, from one point of view or another, simpler than any other decompositions.”

J. J. Sylvester (1882) –“The very beautiful ancient Egyptian method of expressing all fractions under the form of a sum of the reciprocals of continually increasing integers.”

THE FRACTION TWO-THIRDS

In hieratic papyri all fractions had only unity as numerator, although the unit 1 was not written. A number became its own reciprocal by putting the sign (r, an open mouth), above it in the hieroglyphs, or a large dot in the cursive hieratic. The solitary exception to this notation was their largest fraction, 2/3, written as a hieroglyph but γ in hieratic; and it was used whenever possible, (There is evidence that a hieroglyph for 3/4 was used.) Tables for finding two-thirds (here written ) of both integers and fractions were available to the scribes, and in RMP 61 there is such a table giving of seventeen different fractions. As late as the seventh century of the Christian era, the Greek AMP had an extensive table giving of integers up to ten thousand, shortened and condensed in some obvious ways. There are similar tables in Coptic of an even later date (see Crum’s Catalogue). In the RMP alone there are more than seventy occasions on which the scribe writes of both integers and fractions, including seventeen where is written with the intermediary value omitted for brevity.

In RMP 61 B , the scribe A’mosé states the rule for odd unit fractions. Chace translates it as follows: “The making of two-thirds of a fraction uneven. If it is said to thee. What is two-thirds of , make thou times of it 2, and times of it 6, two-thirds of it this is. Behold does one according to the like, for fraction every uneven, which may occur.”Although the rule applies equally well to even unit fractions, it was seldom used for them, because the scribe knew that was the reciprocal of 1 (see the hieroglyph for ). and it was easier and simpler merely to increase the even number by its half, like of is .

Examples of the two-thirds rule are shown below.

RMP 32;

RMP 33;

RMP42;

THE EMLR

Because of its brittle condition, the leather roll remained unopened for sixty years. When A. Scott and H. R. Hall finally succeeded, it was found to contain a table of twenty-six unit-fraction equalities, in duplicate. The disappointment of archaeologists over the contents was not shared by historians of mathematics, who have found it of great interest. In common with other scribal tables, no methods of calculation are shown, nor is there even a heading or title that might indicate the use to which the entries could be put. The following selection of entries from the EMLR will interest historians.

A close study of these equalities shows that the scribe clearly knew that any equality–forexample, 9 18 = 6 – can be multiplied by 2 or 3 or 4, and so on, to produce other equalities, such as 18 36 = 12; therefore division would do the same, but only if the denominators of the fractions had a common factor, so that division by 3 would produce 3 6 = 2. Other possibilities at once suggest themselves, such as combining 4 12 = 3 and 5 20 = 4, to produce, by simple substitution, 5 12 20 = 3 —and so on indefinitely, as Sylvester’s treatment suggests. This short table shows how the scribe of the EMLR utilized the two most elementary of his unit fraction equalities, 3 6 = 2 and 2 3 6 = 1, to produce seven of the thirteen equalities shown.

A study of the papyri, particularly the RMP and the MMP , will show how frequently the use of these equalities was necessary. To illustrate this, I choose a rather extreme example from RMP 70, which, on the evidence, the scribe did in his head.

Summation of Sixteen Unir Fractions From KMP 70

There are seven groups of unit fractions within parentheses in this summation. Their respective sums, shown above, are derived from scribal tables like the EMLR and are listed below.

In the above expressions the occurrence of the same fraction twice is an accident of the detail of the calculations involved. Duplication would never occur in a scribal answer to the problem.

An ingenious and competent scribe might conceivably devise a more expeditious method of deriving the answer, unity.⁶ Or he could use the “red auxiliaries,”When using the red auxiliaries, the scribes invariably chose the largest number of the set of fractions for use as their “common denominator.”Each number was then divided into this largest number, and the sum of all the quotients was found as shown below.

Quite by chance, 1,008 is here what we call the least common multiple; but usually in such summations this does not happen, and there may be several unit fractions occurring in the quotients. The division of the quotients’ total by the “common denominator”chosen is seldom as simple as 1,008 divided by 1,008. Furthermore, the scribes seldom show the actual divisions (sixteen of them in RMP 70), which probably were performed on an odd piece of scribbling papyrus or ostracon; this work might well have been quite voluminous, compared with the apparent brevity and conciseness of the table above.

The detail shown above has been included because this summation is a very small portion of RMP 70, so small that the scribe merely writes the total as I, with no further comment. But he could never have done this mentally. He is concerned to find the amount of meal in each of 100 loaves made from 7 hekats of meal, as well as their pesu – and this is quite a calculation. Just how the scribe performed the summation of these sixteen fractions, the historian can never truly know.

TABLES FOR THE ADDITION OF UNIT FRACTIONS, AND THE G RULE

Of the twenty-six equalities in the EMLR table, ten are examples of, and are derivable from, the simplest of all the additions of unit fractions known to the scribes — —which fundamental dual sum I have designated as having the generator (1,2) — that is, the second term is double the first. A scribe preparing a table of such equalities would naturally begin with it and, by simple multiplication or simple addition, produce the following, which could be extended indefinitely.

The equality of EMLR line 3 is , which I refer to as of generator (1, 3); and it produces the table below, again by either multiplication or addition, which also could be extended indefinitely.

The third table would of course be that resulting from the generator (1, 4), as on lines 2 and 1 of the EMLR.

The G Rule . An observant scribe looking at the equalities of generator (1, 2) could well have summed up the situation briefly in some such manner as “For adding two fractions, if one is double the other, divide it by three.”A little further observation would enable him to change the word “double”to “three times,”“four times,”and so on, and to divide by four, five, and so on for the generators (1, 3), (1, 4), and so on. This is in essence what I have called the G rule: but using modern terms and introducing some further detail, I express it as follows:” If of two unit fractions, one is K times the other, then their sum is found by dividing the larger number by (K + 1) if, and only if, the quotient is an integer. If it is not an integer, then a unit-fraction sum is not possible.”So far as the scribe was concerned, his rule, if he used it as suggested, did not apply to equalities from generators such as (2, 3), (2, 5),· · ·, (3, 4), (3, 5), · · ·, (4, 5), (4, 7), · · ·. Mathematically the G rule still held, although with his notation he could not apply it. It would have been clear to him, however, that pairs of unit fractions like , which evolve from the generator (1, 2), cannot add up to another unit fraction,

Let us look at line 1 of the EMLR, which is . It is related to line 2, which is . while line 4 is .

Now the only other dual equalities with as the first term are and . The first of these clearly follows the G rule but is not among the twenty-six equalities of the EMLR, although it might well have been. The student of ancient Egyptian mathematics may now ask, “If the scribe knew that , how did he find it?”

Three-Term Equalities . The simplest three-term equalities of unit fractions have the generators (1, 2, 4), (1, 3, 6), (2, 3, 6), which produce the following equalities:

Eight of these are included in the EMLR table, five of them from the generator (2, 3, 6) forming lines 14– 18;

Line 12 of the EMLR is , one of the more interesting of all the equalities, which could have been derived in many ways. We look here at one method. The Egyptian table of length was simple enough:

4 digits = 1 palm
7 palms = 1 cubit.

Therefore,

And by addition,

Thus

Four-Term Equalities . Only two four-term equalities occur in the EMLR:

One finds it hard to understand why these two particular examples should have been chosen, when so many simpler and more useful ones might have been included:

which have generators (1, 2, 3, 6), (1, 3, 4, 6), (2, 3, 4, 6), and (2, 3, 5, 6), compared with generator (5, 3, 15, 40). An explanation why the first two terms are out of normal order would make very interesting reading for the reflective scholar.

An Interesting Ostracon

Ostracon 153, dating from the early New Kingdom and thus somewhat later than the RMP, contains abbreviated divisions of 2 by 7 and 4 by 7, as in the RMP Recto, but uses the red auxiliaries and a reference number.⁷ What the scribe in fact does is to consider the divisions of 6 by 21 and 12 by 42; 6 is partitioned as , written in red. When divided by 21, these auxiliaries produce the answer . which is not as simple or convenient as the Recto value of . The scribe proceeded by partitioning 12 as . again in red; when divided by 42, these auxiliaries produce the answer .

Of the fifty divisions of 2 by the odd numbers in the Recto, only one, that of 2 ÷ 35, discloses anything of the scribe’s methods–and it happens to be the same as that of the scribe of Ostracon 153. Thus A’hmosé considers 2 ÷ 35 as 12 ÷ 210, and then partitions 12 as (7 5), which auxiliaries, on division by 210, produce the answer . The only other pair of red auxiliaries he could have chosen is (10 2), which would have given –not so acceptable as the first answer.

THE SCRIBAL RULE FOR RECIPROCALS

The following examples show that the scribes were familiar with the operation that in modern notation may be written as “If a × K = b, then 1/b × K = 1/a.”In RMP 33 it is clearly shown that the reciprocal of is 1 .

RMP 34. “A quantity, its half, and its quarter, added together, become 10. What is this quantity?”Multiply 1 so as to obtain 10.

Line 3 states that , and therefore line 4 drives that of . Line 5 is line 4 multiplied through by 2, so that from the Recto the scribe finds that 2×7, which is 2 / 7, is . Again, multiplying by 2, line 6 shows that of 1 = 1. The answer is 5 .

RMP 70.

The scribe finds that

He then writes that of

RMP 32.

The scribe finds that

He then writes that .

RMP 38.

The scribe finds that .

He then writes that .

RMP 33.

The scribe finds that of 42 = 28.

He then writes that of .

DEMOTIC MATHEMATICAL PAPYRI

For the translations of the DMP that follow, I am indebted to R. A. Parker of Brown University. (See the bibliography and “Mathematical Papyri and Ostraca.”)

Cairo Papyrus (JE 89127) , Discovered at Tuna el-Gebel in 1938 and first examined by Parker in 1962, this papyrus contains forty problems of a mathematical nature. Of these, eight are obscure or fragmentary, and fourteen have counterparts in the much earlier hieratic papyri. The Cairo papyrus dates from 300 B.C. or possibly earlier, and it is interesting to examine what advances or developments are to be found in its mathematical techniques.

First we note that there are nine problems dealing solely with Pythagoras’ theorem –for instance, “A ladder of ten cubits has its foot six cubits from a wall; to what height will it reach?”With one exception, the numbers used are those of the triads (3, 4, 5), (5, 12, 13), and (20, 21, 29); and no such problems have been noted in the earlier papyri. The one exception requires the calculation of an approximate square root.

Two problems deal with rectangles having areas of sixty square cubits and diagonals of thirteen and fifteen cubits, respectively. It is required to find the sides. The directions for their solution are identical with the steps of the following algebra, but of course they are entirely descriptive.

The second problem is the same except for the numbers, and so the square roots of 345 and 105 are to be found. The formula used is the approximation usually attributed to Archimedes or Hero, . Thus we find the square roots to be and from

correct to .05 percent correct to .03 percent

There are two problems on the areas of circles in which the equivalent of ダ is 3, the Old Testament value, as compared with the RMP value of more than a millennium earlier, 256/81, or 3.16. The problems are to find the diameters of circles with areas of 100 square cubits and 10 square cubits. The answers given are 11 and , as further approximations from the formula of Hero given above.

Finally, there are three problems concerning the areas of circles that circumscribe two equilateral triangles and a square. Rather surprisingly, the scribe finds the areas of the triangles and the square separately, and then the areas of the segments of the circles. The arithmetic for the areas of the segments is equivalent to using the formula A = ½ s(s+ c), where s, where s is the height (sagitta) and c is the chord length of the segment. This is unexpected, and the earliest reference I can find to it is the Chui-chang suan-shu from China (ca. 300 B.C.).⁸

In the Cairo papyri the old notation for unit fractions is clearly retained, but there are signs of slight changes developing. In the forty problems there are demotic signs for 5/6 and 2/3. In addition, the fractions 6/47, 39/47, 17/53, 35/53, and at least a dozen others are used, written with both numerator and denominator on the same line, but the numerator underlined to distinguish it. These are the first nonunit fractions.

P. Dem. Heidelberg 663 . Parker’s translation of the fragmentary demotic papyrus P. Dem. Heidelberg 663 (Journal of Egyptian Archaeology, 61 [1975], 189–196) deals with four problems concerning the dimensions of isosceles trapezoidal fields. Of great interest is the division of one such field into two equal areas, by a straight line parallel to the top and bottom sides, and the determination of its length. Parker remarks that the details of the papyrus are “so scanty that it is rarely possible to do more than suggest a connected translation, but with the help of the four figures, it is possible to reconstruct the aim of each problem.”He then offers a careful translation of the papyrus, which dates from the Ptolemaic period of the first or second century B.C.

The second problem deals with the length of the parallel line x that divides the field into two equal areas.

The detail of this calculation is not clear in the papyrus, but Parker suggests that the formula or its equivalent in demotic terms could have been used and that the square root, if not integral, could be found from Hero’s approximation,

For problem 2 the formula gives

Because 164 is not a perfect square, its square root from Heron’s formula would be approx. Because of the demotic units of measure, the value of x had to be expressed 12 ½ ¼ 1/6, a reasonably close approximation. One naturally wonders whether the dimensions of the trapezoid could have been chosen so that the bisector x would turn out to be a whole number, and it so happens that this can be done.

Yule Babylonian Collection clay tablet 4675, dating some centuries earlier than Heidelberg 663, contains a similar problem in which the parallel sides of the trapezoid are 7 and 17. From Mathematical Cuneiform Texts, American Oriental Society, 1975, by Neugebauer and Sachs, The translation of this tablet indicates in the cuneiform sexagesimal number system that the method for calculating x using only

the values of a and b would have followed in modern terms the steps indicated by the formula

in which there is no reference to perpendicular heights or to sides or areas. With the sides a and b given as 17 and 7, we have

We ask ourselves if the Babylonian scribes had some method of their own in this type of problem for avoiding awkward square roots, even though they may have had some sort of a square root rule like Hero’s many centuries before he had made it known. Indeed, we find another related problem in the clay tablet VAT 7535³ at the Berlin Muséum, dating from ancient Babylonia.

Using the same formula we have

so that

There are many ways of finding the length of the area-bisecting parallel of a trapezoid, isosceles or otherwise; and methods or formulas may be devised to include sides, perpendicular heights, and areas. The simplest formula in modern terms is

A further problem that presented itself to the Babylonian and Egyptian scribes was, what values should be given to a and b to ensure that the value of x should be an integer or an exact number?

The apparently simple problem suggested by the ancients is thus conveniently shown in Figure 5, in which it is required to devise integral values of a, b, and x, so that A₁ = A₂

A solution of the problem treated purely as theory of numbers was given by A. Guibert in 1862; reference to it is made in L. E. Dickson’s History of the Theory of Numbers. There the general solution in positive relatively prime integers of a² + c² = 2b² is staled to be

a = ±(p²—q²—2pq),
b = (p² + q²),
c = (p²—q² + 2pq),

where p and q are relatively prime and one even.

Thus solutions for values of a, b, and x all less than 100 can be found for the trapezoid.

We note that the scribe of VAT 7535 utilized the first trio (1, 5, 7) multiplied through by 15, while the scribe of YBC 4675 used the second trio (7, 13, 17). It would be of great interest to the historian of mathematics if it were ever determined exactly how these values were determined.

British Muséum Papyrus (10399, 10520) . These papyri date from the period of the Ptolemies and the early Romans, say 300 B.C. They contain twenty-seven problems, of which about half are similar to those found in the hieratic papyri. Thus, six deal with the four operations, mostly operations with unit fractions: two with the areas of rectangles; three are multiplication tables —64 times the numbers 1 to 16, times the numbers 1 to 10, and times the numbers 1 to 10; and two are not sufficiently preserved to be translated. But the other half are of interest to the historian.

There are four similar problems on frustums of cones and the calculations of their volumes. This is done by averaging the top and bottom diameters; finding the area of the circle with this diameter, using the formula A = 3/4 d²; and then multiplying by the height. This is a better approximation than that of averaging the areas of the top and bottom circles. Then there are six consecutive very closely allied problems, the first of which is worded “The fraction is added to 1. Determine what fraction of 1 must be subtracted from it to give 1 again.”The succeeding questions are similarly worded, with the fractions to be added to unity changed to , and 5/6. The answers given are , and 5/11. There are two questions requiring the square roots of 10 and . The controversial (only regarding its origin) formula used is and is found as but requires preliminary treatment. The scribe replaces by 18/36 and thus finds . The division of 2 by 35. almost exactly as A’hmosé indicated it should be done in RMP Recto, is calculated to be , as in the Recto. Parker, like Peet, expected to find ⁹

The last remaining problem is of particular interest. It deals with the sum of a number of terms of certain progressions, the first of which is the arithmetical progression formed by the natural numbers. In order to understand clearly what series the scribe is summing to ten terms, I write them first, because the scribe did not so write them but merely described them in his own way. He plans to find the sum of the first ten terms of the following series, but actually does so for only the first two, which are 1 “filled once up”to 10, and 1 “filled twice up”to 10. The third (and fourth, and so on), which would have been 1 “filled thrice (and so on) up”to 10, are not included,

The scribe writes:

The scribe’s directions hold, of course, for any number in the natural number series, merely by replacing the 10 that he chose for his example. Let us first rewrite the above as follows.

First,

10² = 100

10²+ 10 = 110

1 is “filled once up”to 10 1/2; (10² + 10)= 55

Then,

10 + 2 = 12

⅓(10+2) = 4

1 is “filled twice up”to 10 4×55 = 220

Now we replace the 10 by n, representing any term of the natural number series.

1 is “filled once up”to n S_n = ½ (n² + n)

1 is “filled twice up”to n, S_n = 1/3 (n+2) ½ (n²+n)

And if he proceeded to the next series, he would have written

I is “filled thrice up”to, n, S_n =

¼(n+3)1/3 (n+2) ½ (n²+n).

We may now say that in our modern notation, we have the formulas for the sum to n terms of the series, as follows.

Series (O) S_n = ½ n(n+1)

for instance, S₅ = ½ 5×6 = 15

Series(A)

for instance,

Series (B)

for instance,

AN INTERESTING OSTRACON

RMP 79 consists of a house inventory.

In the second column the scribe starts with 7, multiplies it by 7, then multiplies this product by 7, repeats these multiplications by 7 a certain number of times, then adds all the products for a grand total. This operation is self-explanatory. In order to explain the multiplication of 2,801 by 7 (1 use the index notation for brevity), we take the foregoing step by step.

1 term No 7=7 =7 =7
additions

2 terms 7+7² = 7[1+ (7)] =7(1 + 7) =56
One addition

3 terms 7 + 7² + 7³ = 7[1+ (7 + 7²)] =7(1 + 56) =399
Two additions

4 terms 7+ 7² + 7³ + 7 = 7[1 + (7 + 7² + 7³)] =7(1+399) =2,800
Three additions

5 terms 7 + 7² + 7³+ 7⁴ + 7⁵ = 7[ 1 + (7 + 7² + 7³+ 7⁴)) = 7(1 + 2,800) = 19,607
Four additions

If the scribe wished to go one step further, he would need only to add unity to the total 19,607, and then multiply by 7. In the scribe’s setting for RMP 79, the shorter method was shown first —that of multiplying 2.801 by 7 — and then its correctness proved by the longer, detailed calculation. It would appear that in this problem, the scribe is concerned only with a property of certain geometrical progressions, an exercise in the field of numbers. The title “A House Inventory”(or “The Contents of a House”; there is some doubt in the translation) and the names given to the successive totals —Houses, Cats (in the houses). Mice (that the cats would destroy), Spelt (ears of grain the mice would eat). and Hekats (the number of particles of grain that would be saved) — are all put in to give the problem an air of verisimilitude. But if this was the scribe’s plan, it failed, because the grand total he is seeking cannot have any sensible meaning in regard to the contents of a house.

If, as is elsewhere claimed, this problem is the origin of the Mother Goose nursery rhyme “As I was going to St. Ives, I met a man with seven wives … how many were coming from St. Ives?”(not “going to St. Ives.”which is the storyteller’s trick), then the change to seven wives, seven sacks, seven cats, seven kits removes this disability and makes the problem sensible. Not only that, the multiplications are reduced and the problem slightly simplified to the multiplication of 400 by 7.

PESU PROBLEMS

Pesu is a number indicating the strength of bread or of beer made from grain. If one hekat (about 1/8 bushel) of grain were used to produce one loaf of bread or one des jug of beer, then its pesu was one. If one hekat of grain produced two loaves or two jugs of beer, their pesu was two, and so on, so that the higher the pesu number, the weaker the bread or beer. The relation was expressed as

There are twenty problems concerning pesus in the RMP and MMP. Generally speaking, their arithmetic is simple, as the examples below indicate.

RMP 69. If 3 hekats of meal are made into 80 loaves, find the amount of meal in each loaf, and the pesu.

RMP 73. 100 loaves of pesu 10 are exchanged for loaves of pesu 15. How many of the latter will there be?

RMP 78. 100 loaves of pesu 10 are exchanged for des jugs of beer of pesu 2. How many des jugs of beer will there be? (A des jug of beer is roughly 7/8; pint.)

Two problems. RMP 74 and 76, merit closer attention than the others, even though at first glance they appear simple enough and, indeed, very similar.

RMP 74. 1,000 loaves of pesu 5 are exchanged, half of them for loaves of pesu 20 and half for loaves of pesu 10. How many of each will there be?

RMP 76. 1,000 loaves of pesu 10 are exchanged for a number of loaves of pesu 20 and the same number of loaves of pesu 30. How many of each will there be?

The scribe’s solutions are shown below.

RMP 74. 1,000 loaves of pesu 5 required 200 hekats, and half of 200 is 100 hekats. Multiply 100 by 10; it makes 1,000, the number of loaves of pesu 10. Multiply 100 by 20; it makes 2,000, the number of loaves of pesu 20. The answer is 1,000 and 2,000 loaves.

RMP 76. For loaves of pesu 20, hekat produces one loaf. For loaves of pesu 30, hekat produces one loaf. Then, hekat produces two loaves, one of each kind. The number of hekats in 1,000 loaves of pesu 10 is 100. Multiply 100 by 12; the result is 1,200, the number of loaves of each kind for the exchange.

We now restate the scribe’s processes for these two problems in modern terms, for our own information.

RMP 74. 1,000 loaves of pesu 5 required 200 hekats of grain. The arithmetic mean of”the two new pesus is ½ (20 + 10) = 15. Then the number of loaves received in exchange is greater, in the ratio of 15 to 5 —that is, 3 to 1. Therefore the number of loaves received is 1,000 × 3, or 3,000 total. The new pesus are in the ratio of 20 to 10, or 2 to 1. Hence there must be 2,000 loaves of pesu 20 and 1,000 loaves of pesu 10.

RMP 76. 1,000 leaves of pesu 10 required 100 hekats of grain. The harmonic mean of the two new pesus is 2(20 × 30) / (20 + 30) = 24. Then the number of loaves received in exchange is greater in the ratio of 24 to 10, that is, 12 to 5. Therefore the number of loaves received is 1,000 × 12 / 5 = 2,400 total. Hence there must be 1,200 loaves of pesu 20 and 1,200 loaves of pesu 30.

Now consider the problems RMP 73 and 72, in that order.

RMP 73. 100 loaves of pesu 10 are exchanged for loaves of pesu 15. How many of the latter are there?

RMP 72. 100 loaves of pesu 10 are exchanged for loaves of pesu 45. How many of the latter are there?

RMP 73 Solution. The number of hekats in 100 loaves of pesu 10 is 100÷10= 10. Multiply 10 by 15; it makes 150, the number of loaves exchanged.

RMP 72 Solution. Instead of a simple solution like the above, of 10 × 45 = 450 loaves, the scribe has the following:

The excess of 45 over 10 is 35

45– 10=35 Q-P

Divide this 35 by 10; it makes 3

45 - 10 = 35 Q - P

Divide this 35 by 10; it makes 3

35 ÷ 10 = 3½

Multiply 3 by 100; it makes 350

3½ × 100 = 350

Add 100 to 350; it makes 450

350+100 = 450

The historian desiring to examine the reasoning in this “round-about way,”as A. B. Chace put it, of solving RMP 72 would write L for the original number of loaves, P for the original pesu, N for the number of new loaves, and Q for the new pesu. The only data available to the scribe is the relation

and since the number of hekats is constant, he would have

Therefore,

or (alternando),

and (dividendo),

so that

Hence,

and, finally, adding L to both sides, which is the formula of the method the scribe adopted for RMP 73 and all the other pesu problems of a like nature. The incidence of the step that we today term “dividendo”is unavoidable, because of the instruction to find the excess of the final pesu of 45 over the original pesu of 10.

Although the pesu problems of the RMP and the MMP are in general similar, there are one or two (as we have seen) that call for special mention. The scribe of the MMP was not as careful, nor as good a writer of hieratic as was A’hmosé; and MMP 21 is one such problem, which may resemble RMP 76. It is included here for interest and inquiry.

MMP 21. This problem involves the method of calculating the mixing of sacrificial bread.

If one names 20 measured as of a hekat.
And 40 measured as of a hekat.
Compute of 20. Result 2 .
Compute of 40. Result 2 .
The total of both these halves is 5.
Compute the sum of both halves. Result 60.
Divide thou 5 by 60.
Result . Lo! The mixture is .
You have correctly found it.

EGYPTIAN MEASURES

Listed below are some of the Egyptian measures.

Areas and Volumes . The areas of triangles and rectangles were found in the usual way, half the base times the height, and length times breadth: but the areas of circles were something quite different. RMP 50 states the rule as “Take away one-ninth of the diameter, and square the remainder,”In modern terms we would express this as

We may therefore say that the ancient Egyptian equivalent of π was 256/81 or 3.1605, an unexpectedly close value, in excess of the true value by 0.6 percent.

Those problems concerned with finding the contents of cylindrical granaries, expressed in khars, where 1.5 khars equals a cubic cubit, were solved by multiplying the area of the circular base by the height in cubits, and then multiplying by 1.5. The area of the circular base was found by the formula above.

There are two problems, KP IV, 3 and RMP 43, where this calculation was done differently. The directions read: “Add to the diameter its one-third part, square the total, and then multiply by two-thirds of the height. This is the contents in khar.”How the scribes thought out this variation is not explained in the papyri; but it worked correctly, as we can check by setting down both methods in modern form.

METHOD I

METHOD 2

A challenge remains for the historian of mathematics: to explain the scribal thought process of transforming the first method into the second, without using any symbolic algebra.

Pyramids . The problems that concern pyramids show that the scribes were able to calculate the slope or inclination of the sides of a pyramid, the volume of a pyramid, and, perhaps surprisingly, the volume of a truncated pyramid or the frustum of a pyramid. The slope or seked of the side of a pyramid was stated as the relation between the vertical height and half the base, and was given as so many palms horizontally for each cubit measured vertically–usually 5 palms, or 5 palms, 1 finger, or 5 palms, possibly because of the seked of Cheops was 5 palms.

Struve’s translation of MMP 14 is as follows:

Method of calculating a truncated pyramid.

If it is said to thee, a truncated pyramid of 6 cubits in height,

Of 4 cubits of the base by 2 cubits of the top,

Reckon thou with this 4, its square is 16.

Multiply this 4 by 2, Result 8.

Reckon thou with this 2, its square is 4.

Add together this 16, with this 8, and with this 4, Result 28.

Calculate thou 3̄ of 6. Result 2.

Calculate thou with 28 twice. Result 56.

Lo! It is 56! You have correctly found it.

If we replace the base of 4 cubits by a, the top of 2 cubits by b, and the height of 6 cubits by h, we find that the scribe has calculated the volume of the frustum, according to the modern formula V = 1/3h (a² + ab + b²). It is generally accepted that the Egyptians knew that the volume of a right square pyramid was one-third that of a right square prism of equal base and height, although this is nowhere specifically attested (to my knowledge) and although there are several simple ways in which the scribes may have found the equivalent of the modern formula V = 1/3a²h.¹⁰ Even with the powerful help of such a formula or its equivalent, however, it is still not easy to establish just how the scribes arrived at this relatively erudite method for finding the frustum of a pyramid, an achievement that, in the words of B. G. Gunn and T. E. Peet, “has not been improved upon in 4,000 years.”

Area of the Surface of a Hemisphere . Struve’s translation of MMP 10 is as follows.

Method of calculating a basket.

If it is said to thee, a basket with an opening,

Of 4 in its containing, Oh!

Let me know its surface.

Calculate thou of 9, because the basket

Is the half of an egg. There results 1.

Calculate thou the remainder as 8.

Calculate thou of 8.

There results .

Calculate thou the remainder of these 8 left

After taking away these . There results 7 .

Reckon thou with 7 , 4 times.

There results 32. Lo! This is its area.

You have correctly found it.

In modern terms, we would express the operations above in the form , which is the equivalent of A = 2π r².

W. W. Struve (d. 1965), the original translator of the MMP, was thoroughly convinced that the above is the correct interpretation of MMP 10, T. E. Peet (d, 1934) thought that a semicylinder may have been concerned, but had to introduce two new terms, “diameter and heights,”into the translation. B. L. van der Waerden and R. J. Gillings agree with Struve: and T. G. H. James, while acknowledging the translation difficulties raised by Peet. inclines to concur with Struve’s conclusion.¹¹ O. Neugebauer still has some doubts that the paleography is definite enough to convince him beyond question.¹²”Thus far, no authorities have expressed any views or opinions, in scientific or historical journals, contrary to those expressed here; and MMP 10 and MMP 14 remain the outstanding mathematical achievements of the ancient Egyptians, A detailed discussion of the views of the above-mentioned writers is given by Gillings.¹³

EQUATIONS

In this section we consider problems requiring for their solution what we might call algebraic reasoning, but which are treated by the scribes quite literally and perhaps termed “rhetorical algebra.”For clarity and brevity, I will not give complete translations from the hieratic writing of the papyri: and so that we may examine the thought processes involved, I use the standard x, y, z for unknowns, and a, b, c for knowns. For obvious reasons, however, I retain the standard notation for the ancient Egyptian unit fractions. Thus, in a limited space we may cover a considerable field of study of the mathematical techniques of the scribes.

First-Degree Equations, One Unknown . Below are several first-degree equations, with one unknown.

These are straightforward enough, largely arithmetical operations with unit fractions. In each case the scribe gives the proof that his answer is correct.

This second group of equations represents a slight advance on the first four, each of which was solved by a method that today is called “false position”or “false assumption.”Methods for the second four vary.

Answers

If the scribe A’hmosé faltered, he did so here. His method of solution was that of division: and if he was teaching a technique, his choice of fractions and of integers, or both, was unfortunate, for the answers, all of which are correct, show that the problems were not of a practical nature. So here he was, lost in a maze of four-digit unit fractions, in one instance (RMP 33) finding it necessary to add sixteen unit fractions, the last twelve of which have the denominators

which he found by using red auxiliaries. This could all have been neatly avoided had he chosen 97 instead of 37, and his answer would have been 42. And in RMP 31, had he written instead of . he would have simplified it so greatly that the answer would have been . Finally, the answer to RMP 32 could have been simplified from to in the scribal working, as the G rule shows, since

Second-Degree Equations, Two Unknowns . In the discussion of the demotic papyri, the solution of simultaneous second-degree equations is shown, so they are not repeated here. DMP 34 and 35 were

x²+ y² = 169

xy = 60,

x²+ y² = 225

xy = 60.

DMP 34, in exactly the form shown above and with the same numbers, occurs in W. E. Paterson’s School Algebra ([London, 1916], 250), where the textual work on solutions is the same as that of the scribe. Two problems from BP 6619, restored and translated by H. Schack-Schackenburg, are

x²+ y² = 100

4x-3y = 0,

x²+ y² = 400

4x-3y = 0.

F. L. Griffith, in his translation of a problem in KP IV, 4 (1897), was notable to understand certain incomplete lines, which have since been restored with reasonable surety and produce another problem of simultaneous equations, details of which may be studied in Mathematics in the Time of the Pharaohs (162 – 165):

xy = 12

Progressions. RMP 40 speaks of an arithmetic progression of five terms, in which the sum of the three largest terms is seven times the sum of the two smallest terms. Chace’s translation reads:“100 loaves for 5 men, one-seventh of the total of the largest 3 shares shall equal the total of the smallest 2 shares. What is the common difference of the shares?”¹⁴

KP IV, 3, is solely a column of eleven decreasing numbers, and next to it is a short multiplication. There are no words! I have judged the calculations shown to be an answer to the problem “The sum of 12 terms of an arithmetic progression is 110, and the common difference is 5/6. What is this series?”¹⁵

RMP 64 gives the sum of ten terms of an arithmetic progression as 10, and the common difference as one-eighth. What are the terms of this series? Since the scribal statement of the problem speaks of dividing ten hekats of barley among ten men, the fractions used are all Horus-eye fractions, which were used for grain: , of a heket, written in a special way. The scribe’s method of solving RMP 64 is quite unexpected; and if one follows each step of the solution, and replaces each word and number with the modern notation used in algebra texts–such as S for the sum, d for the difference, a and l for the first and last terms, n for the number of terms —one has, at the conclusion,

The problem stated in RMP 79 has been treated elsewhere in some detail. Here we can state it as “Find the sum of five terms of a geometrical progression of which the first term is 7, and of which the common ratio is also 7.”

Think of a Number . RMP 28 reads in Chace’s translation:

Two-thirds is to be added, then one-third is to be subtracted.

[The foregoing is in red, all else is in black.]

There remains 10.

Make one-tenth of this, there becomes 1. The remainder is 9.

Two-thirds of it, namely 6, is to be added. The total is 15.

One-third of this is 5. Lo! 5 is that which goes out, and the remainder is 10.

The doing as it occurs.¹⁷

Chace concludes, “The solution does not seem to be complete. The words ’ The doing as it occurs,’ are usually put at the beginning, and in no other problem are they at the end. Peet has suggested that in copying, the scribe let his eye pass to the same words in the next question.”There are no such words in the next question!

I draw attention to the title page of the RMP, where the first sentence (again in red) is “Accurate reckoning of entering into the knowledge of all existing things, and all mysteries and secrets.”RMP 28 is the first problem in which the scribe discloses one of his secrets and shows how his magic works. Having done this, he writes, “The doing this it occurs.”it is in its proper place. I restate the problem in modern terms. “Think of a number. Add to it its two-thirds part. From this number take away its third part. What is your answer? Suppose you are told the answer is 10. Then you (mentally) subtract its tenth part, and say that the number first thought of was nine! That is how you do it. That is your magic!”

RMP 29 is a second example of “think of a number”problems given by A’hmosé, and they are probably the earliest in recorded history: “Think of a number. Add to it its two-thirds part. To this number add its third part. Find one-third of this number. What is your answer? Suppose you are told the answer is 10. Then you (mentally) add a quarter and a tenth part to this 10, and say that the number you first thought of was 13½ That is how you do it. That is your magic!”

In the section on DMP, the summation of the terms of the natural number series is treated, as are certain other series derived therefrom.

ADDENDA

J. J. Sylvester’s Formula . This formula is quoted in the additional bibliographical notes. From his formula the mathematician can derive unit fraction equalities very similar to those found in the EMLR, when it was successfully unrolled. I have found the following:

Other sets are possible. A careful scrutiny will show that they are interrelated.

Two-Term, Three-Term, and Four-Term Equalities for Unit Fractions . Note that the overbars have been omitted.

Hekat Measures for Grain . One hekat of grain was approximately one-eighth of a bushel. For stating the contents of larger containers, the unit used was sometimes a double hekat, or even a quadruple

hekat. For granaries, an even larger unit was used, “100 quadruple hekats.”A cubic cubit of grain was equal to thirty hekats. For smaller quantities, fractions of a hekat were used–but only 1/2, 1/4, 1/8, 1/16, 1/32, 1/64, which were called Horus-eye fractions. They were not written like ordinary unit fractions, but as portions of the eye of the god Horus, which he lost in battle. The very smallest measure used was the ro, which was 1/320 of a hekat–about a tablespoonful. Some other non-Horus-eye units were also sometimes used: the hinu. which was one-tenth of a hekat, and the khar. which was twenty hekats.

The RMP Recto . In the section dealing with the RMP Recto, the views of some historians are given on how they thought the scribes calculated the fifty values of 2 ÷ (2n— 1), expressed in unit fractions. My own theory has been published in Archive for the History of Exact Sciences.

I do not consider it proper, however, to discuss that article in detail until it has been read and studied or reviewed by competent judges, and its merits evaluated. It is therefore sufficient to state that I considered the one equality of the Recto that was entirely unique in its context (as verified by the computer KDF–9): the very last, 2 ÷ 101 = , which is the only possible four-term unit fraction value for this division. From this equality, expressed generally . I derive every other equality of the Recto, using those methods and techniques accepted as being attributable to the scribes. I direct the attention of those interested in the problem to my article.

NOTES

1. R. W. Sloley, “Science,”166.

2. E. T. Peel, “Mathematics in Ancient Kgypt,”412.

3. O. Neugcbauer, The Exact Sciences in Antiquity, 73.

4. In the papyrus KP IV, 2, a portion of the Recto is included: 2 divided by 3, 5, 7, · · ·, 21. Proofs are identical bill briefer.

5. Following Archibald: If U_x+1 – U_x² + U_x – 1 = 0, then Thus, for example, 1/5 + 1/21 = 1/4 - 1/420.

6. See, for example, R.J. trillings, “The Addition of Egyptian Unit Fractions.”in Journal of Egyptian Archaeology, 51 1956), 95.

7. See W. C. Hayes, Ostraca From the Tomb of Sen-Mut.

8. Further references are in Heronis Alexundrini Opera quae supersunt omnia, 5 vols. (Leipzig, 1899-1914), III, 73; IV, 357; V, 187.

9. Compare the Canon for the RMP Recto. See R. J. G tilings, Mathematics in the Time of the Pharaohs, 99.

10.Ibid., ch. 17, “Pyramids and Truncated Pyramids,”185–193.

11. B. L. van der Waerden, letter (1967); R. J. Gillings, “The Area of the Curved Surface of a Hemisphere in Ancient Egypt,”in Australian Journal of Science, 30, no, 4 (1967), 113–116; T. G. H. James, letter (1970).

12. O, Neugebauer, letter (1967).

13. Gillings, Mathematics in the Time of the Pharaohs, ch, 18.

14. A. B. Chace, L. Bull, H. P. Manning, and R. C. Archibald, The Rhine Mathematical Papyrus. II (Oberlin, Ohio, 1929), prob. 40, pl, 62.

15. Gillings, Mathematics in the Time of the Pharaohs, 176–180.

16. Ibid., 173–175.

17. A. B. Chace el al., The Rhind Mathematical Papyrus. II, pl. 51.

BIBILOGRAPHY

This bibliography, which should not be regarded as complete, may suggest further sources of information to students of the history of ancient Egyptian mathematics.

R. C. Archibald, Bibliography of Egyptian Mathematics, I and supp. (Oberlin, Ohio. 1927– 1929).

O. Becker and J. E. Hofmann, Geschichte der Mathematik (Bonn, 1951).

V. V. Bobynin. Bibliotheca mathematics, 2nd ser., 3 (1889). 104– 108; 4 (1890), 109– 112; 8 (1894), 55–60; 10(1896), 97–101.

L. Borchardt, Gegen die Zahienmystik an der grossen Pyramide bei Gise (Berlin. 1922).

C. B. Boyer, A History of Mathematics (New York, 1968).

J. H. Breasted, A History of Egypt (London, 1946).

E. M, Bruins, “Ancient Egyptian Arithmetic: 2/N,”in Indagationes mathematicae, 14 (1952), 81–91.

E. A. W. Budge, The Rosetta Stone (London, 1922; rev. eds., 1950, 1957).

E. A. W. Budge, “The Rosetta Stone,”ch. 3, pt. 3,of M. Wheeler, A Second Book of Archaeology (London, 1959).

A. B. Chace, L. S. Bull, H. P. Manning, and R. C. Archibald. The Rhind Mathematical Papyrus, 2 vols. (Oberlin, Ohio, 1927–1929).

J. F, Champollion, Précis du Système hiéroglyphique (Paris, 1824).

L. Cottrell, Life Under the Pharaohs (London, 1957).

W. E. Crum. Coptic Ostraca From the Collations of the Egypt Exploration Fund, Cario Muséum (London, 1905).

W. E. Crum, Catalogue of Coptic Manuscripts in the British Muséum (London, 1905).

G. Daressy, “Calculs égyptiens du moyen-empire.”in Recueil de travaux relatifs è l’archéologie (Paris, 1906).

D. Davidson and H. Aldersmith, The Great Pyramid, Its Divine Message (London. 1924).

L. E. Dickson, History of the Theory of Numbers, 3 vols. (Washington, D.C., 1919–1923; repr. New York, 1934), 437–438.

J. Edgar and M. Edgar, The Great Pyramid Passages and Chambers (London, 1923).

I. E. S. Edwards. The Pyramids of Egypt (London, 1952; rev. ed., 1961).

A. Eisenlohr, Ein mathematisches Handbuch der alten Ägypter. RMP (Leipzig. 1877).

R. Engelbach. “The Volume of a Truncated Pyramid,”in Journal of Egyptian Archaeology, 14 (1927).

H. Eves, An Introduction to the History of Mathematics (New York, 1964).

A. H. Gardiner, Egyptian Grammar, 3rd ed., rev. (London, 1969).

O. Gillain. La science égyptienne, L’arithmethpie an moyen empire (Paris, 1927).

R. J. Gillings, Mathematics in the Time of the Pharaohs (Cambridge. Mass., 1972).

R.J. Gillings, “The Recto of the Rhind Mathematical Papyrus. How Did the Ancient Egyptian Scribe Prepare It?”in Archive for the History of Exact Sciences, 12, no. 4(1974), 291–298.

R. J. Gillings, “What Is the Relation Between the EMLR and the RMP Recto?” ibid., 14, no. 3 (1975), 159–167.

S. R. K. Glanville, “The Mathematical Leather Roll in the British Muséum,”in Journal of Egyptian Archaeology. 13(1927), 232–239.

S. R. K. Glanville. ed., The Legacy of Egypt (Oxford, 1942; repr., 1963; 2nd ed., 1971).

F. L. Griffith, Hieratic Papyri From Kahun und Gurob (London, 1898).

W. C. Hayes, Ostracon 153. Osttaca and Name Stones From the Tomb of Sen-Mat at Thebes, Metropolitan Muséum of Fine Art Publication 15 (New York, 1942).

L. Hogben, Mathematics for the Million (London, 1945), 261, 297, 555.

F, Hultsch, Die Elemente der ägyptischen Theilungs-rechuung (Leipzig, 1895).

P. E. B. Jourdain. “The Nature of Mathematics,”1 ch. 1, of J. R. Newman, ed., The World of Mathematics (New York, 1956).

L. C. Karpinski, “Algebraical Developments Among the Egyptians and the Babylonians,”in American Mathematical Monthly, 24 (1917), 257–265.

M. Kline, Mathematics, a Cultural Approach (Reading, Mass., 1962).

G. Loria, “Studi intorno alla logística grecoegiziana.”in Giornale di matematiche di Battaglini, 32 (1894), 28–57, decompositions of 2/(2n-1).

P. Luckey, “Anschauliche Summierung der Quadrat-zahlen und Berechnung des Pyramideninhalts.”in Zeitschrift für mathematischen und naturwissenschaftlichen Unterricht, 61 (1930), 145–158.

P. Mansion. “Sur une table du papyrus Rhind,”in Annales de ia Société scientifique de Bruxelles, 12 (1888), 44–46.

E. K. Milliken, Cradles of Western Civilisation (London, 1955), 86–98.

O. NeÜbert, The Valley of the Kings (London, 1957).

O. Neugebauer, The Exact Sciences in Antiquity (Copenhagen, 1951; New York. 1962), 91–96.

J. R. Newman. “The Rhind Papyrus,”1, ch. 2, of J. R. Newman, ed., The World of Mathematics (New York, 1956).

C. F. Nims. “The Bread and Beer Problems of the Moscow Mathematical Papyrus,”in Journal of Egyptian Archaeology, 44 (1958), 56–65.

R. A. Parker, “A Demotic Mathematical Papyrus Fragment,”in Journal of Near Eastern Studies, 18, no. 4 (1959), 275–279.

R. A, Parker, Demotic Mathematical Papyri (Providence, R.I.-London. 1972), prob. 56 (pl, 20; C8–16), pp. 65–66.

T. E. Peet, The Rhind Mathematical Papyrus (London, 1923).

T. E. Peet, “Mathematics in Ancient Egypt,” in Bulletin of the John Rylands Library, 15, no. 2 (1931), 409–441, esp. 412.

W. M. F. Petrie, The Pyramids and Temples of Gizeh (London, 1883).

G. Posener, Dictionary of Egyptian Civilization (New York, 1959).

V. Sanford, A Short History of Mathematics (London, 1930), 225.

G. Sarton, The Study of the History of Mathematics (Cambridge, Mass., 1936; New York. 1957).

H. Schack-Schackenhurg. “Der Berliner Papyrus 6619,”in Zeitschrift für ägyptische Sprache, 38 (1900), 135–140.

A. Scott and H. R. Hall, “Egyptian Leather Roll of the 17th Century B.C.,”in British Muséum Quarterly, 2 (1927), 56–57.

K. H. Sethe. “Von Zahlen und Zahlworten bei den alten Ägyptern,”in Schriften der Wissenschaftlichen Gesellschaft in Strassburg, no. 25 (1916), 85– 119.

J. W. S. Sewell, “The Calendars and Chronology.”ch. I of S. R. K. Glanville, ed., The Legacy of Egypt (London, 1942; repr. 1963; 2nd ed., 1971).

W, K. Simpson, The Papyrus Reisner, 3 vols. (Boston, 1963–1969).

R. W. Sloley, “Science.”ch. 6 of S. R. K. Glanville, ed., The Legacy of Egypt (London, 1942; repr, 1963; 2nd ed., 1971).

Piazzi Smyth, Our inheritance in the Great Pyramid (London, 1864 and repr. eds.)

D. J. Struik, A Concise History of Mathematics (New York, 1948), 19–23.

W. W. Struve, Mathematischer Papyrus des Staatlichen Muséums der schönen Künste in Moskau, which is Quellen und Studien zur Geschichte der Mathematik, Abt. A, I (Berlin, 1930), esp. 98 ff.

D. P. Tsinzerling, “Geometria v drevnikh egiptyan”(“Geometry in Ancient Egypt”), in Izvestiya Rossyskoy Akademii nauk SSSR, 6th ser., 19 (1925), 541–568.

B. A. Turaev, “The Volume of the Truncated Pyramid in Egyptian Mathematics,”in Ancient Egypt (1917), 100–102.

B. L. van der Waerden, Science A wakening, translated by Arnold Dresden (Groningen, 1954), 15–36.

K. Vogel, “Erweitert die Lederolle unsere Kenntniss ägyptischer Mathematik?”in Archiv für Geschichte der Mathematik (1929), 386 ff.

K. Vogel, “The Truncated Pyramid in Egyptian Mathematics,”in Journal of Egyptian Archaeology. 16 (1930), 242–249.

K. Vogel, Vorgriechische Mathematik, I, Vorgeschichte und Ägypten (Hanover, 1958).

N. F. Wheeler, “Pyramids and Their Purpose,”in Antiquity. 9 (1935), 5–21.

J. A. Wilson, Signs and Wonders Upon Pharaoh. A History of American Egyptology (Chicago, 1964).

ADDITIONAL BIBLIOGRAPHICAL NOTES

Names included here may be referred to in general reading on the history of ancient Egypt. Such references may concern articles on specific topics, notes on archaeological work or Egyptology generally, or a published text from which lines may have been quoted. However brief the notes, the names included are of interest to the historian.

Abd el-Rasoul. Dealer in Egyptian antiquities who sold a papyrus to Golenischev in 1893, “pour une somme assez modique.”that was early referred to as the Golenischev Papyrus.

J. Baitlet, The Akhmim Papyrus (Paris, 1892). A Greek mathematical papyrus of a.d. 750, that still used Egyptian unit fractions, as in RMP Recto.

G. B. Belzoni. A six-foot, six-inch strong man who discovered the tomb of Seti I at Thebes and the entrance to the Cephren pyramid at Giza in 1818.

L. Borchardt. Archaeologist who excavated in Egypt. Found the now famous head of Queen Nefertiti at Tellel-Amarna in 1907.

J. H. Breasted. First professor of Egyptology in the United States, 1895. Author of many books. Original director of the Oriental Institute at the University of Chicago, 1919. Died 1935.

H. K. Brugsch. Wrote a dictionary of demotic, 7 vols., and Aegyptische Studien (Leipzig, 1855). Articles on RMP. Professor of Egyptology, Göttingen, 1868. Died 1894.

L. S. Bull. One of the authors of vol. II of RMP: photographs, transcriptions, transliterations, and literal translations of the papyrus.

H. Carter. Associated with Lord George Carnarvon from 1907 in Egyptian excavations. Responsible for the discovery and clearance of the tomb of Tutankhamen, Valley of the Kings, in 1922. Died 1939.

J. Cerny. Wrote on the religion of the ancient Egyptians in Annales du Service des antiquités de l’Égypte, 43 (1943), 179 ff.

E. Collignon. Discussed the decomposition of fractions, of the form 2 /(2n - 1), as in the RMP. Recto (Paris, 1881).

W. R. Dawson. Who Was Who in Egyptology (London. 1951).

G. M, Ebers. Professor of Egyptology, Leipzig. Obtained a medical papyrus at Luxor (1872) that bears his name. Died 1898.

J. P. A. Erman. Professor of Egyptology at Berlin. Produced Egyptian dictionary (1926). A major figure. Died 1937.

M. Eyth. A pyramid mystic. Wrote Der Kampf um die Cheops Pyramide, 2 vols. (Heidelberg, 1902). Quotes value of ダ to forty decimal places.

A. Fakhry. Details the history of Cheops in The Pyramids (Chicago. 1961).

A. Favaro. An early disputant on Eisenlohr’s translation of the RMP Recto and unit fraction values of 2/(2n -1). See “Sulla interpretazione matemática del papiro Rhind pubblicato ed illustrato dal Prof. Augusto Eisenlohr,”in Memorie della R. Accademia di scienze, lettere ed arti in Modena, 2nd ser., 19 (1879), 89–143.

V. S. Golenischev, Member of the tsarist nobility who became professor of Egyptology at Cairo University. His collection of antiquities, loaned to the Moscow Museum in 1912 “contre une rente viagère,”included the famous Golenischev Papyrus. After the 1917 Revolution, payments ceased; the collection became government property; and the roll became known as the Moscow Mathematical Papyrus. Died 1947.

H. Grapow. Edited an Egyptian dictionary in association with Erman.

B. G. Gunn. Worked for the Service of Antiquities and for the Cairo Muséum. Professor of Egyptology at Oxford in 1934. Wrote article on finger numbering for Zeitschrift far ägyptische Spruche. 57 (1922). 71–72, dealing with Horus-eye fractions. Died 1950.

H. R. Hall. Keeper of Egyptian antiquities in the British Muséum. 1924. Hin Ancient History of the Near East (London, 1913) went through many editions. With Dr. A, Scott he unrolled the Egyptian Mathematical Leather Roll, which had remained rolled in the museum for sixty years, owing to its brittle condition. The EMLR came to the trustees, with the RMP, from A, H. Rhind in 1864.

Herodotus. Greek historian (ca. 500— ca. 424 B.C.) who visited Egypt and recorded stories of the pyramids 2.000 years after they were built, H. W. Turnbull refers to one of his “obscure”passages, “which implies that the area of the triangular face of Cheops equals the square of the vertical height.”This led to the Sectio aurea, or the golden section, in geometry.

E. Iversen. In his book The Myth of Egypt and Its Hieroglyphs in European Tradition (Copenhagen, 1961), Iversen discusses some of the misconceptions regarding the deciphering of the hieroglyphic writings.

T. G. H. James. Editor of Journal of Egyptian Archaeology, published annually by the Egypt Exploration Society, Manchester Square, London.

A. Jarolimek, Die Rätsel der Cheops Pyramide (Berlin, 1910). supports the pyramid mysticism of Taylor and Smyth, particularly with regard to π and the golden section, and surveys the works of other mystics: Neikes, and Eyth.

K. R. Lepsius. Curator of Egyptian antiquities at Berlin, 1865. Produced Denkmäler aus Aegypten und Aethiopien. 12 vols, (plus) (Berlin. 1859). An authority on the Egyptian language and monuments. Died 1884.

A. F. F. Mariette. A famous Egyptologist. Wrote Les papyrus egyptiens du Musée de Boalag (Paris, 1872), Discoverer of the Scrapeum at Sakkarah. Became curator of Egyptian monuments at Cairo. Died 1881.

G. C. C. Maspero. Professor of Egyptology at Paris, 1869. Publications include Dawn of Civilization (London. 1894): Manual of Egyptian Archaeology (London. 1895): Guide to Cairo Muséum (Cairo. 1895); Struggle of the Nations (London., 1896); fussing of the Empires (London. 1900); Art in Egypt (New York, 1912). Ranks with Erman and Petrie. Received a British knighthood in 1909. Died 1916.

G. Möller. In “Die Zeichen für die Bruchteile des Hohlmasses und das Uzatauge,”in Zeitschrift für ägyptische Sprache, 48(1911). 99– 106, Möller discusses the unit fractions used as portions of the hekat, the measure of capacity, referred to as Horus-eye fractions.

G. A. Reisner. Professor of Egyptology at Harvard, 1914. Excavator of renown who died at Giza in 1942. Many publications, including History of the Giza Necropolis, 2vols. (Cambridge, Mass., 1942).

A. H. Rhind. A Scotsman who went to Thebes for health reasons and became interested in excavating. In 1858 he purchased the RMP and the EMLR, which after his death (1863) came to the British Muséum. He wrote Thebes, its Tombs and Their Tenants (London, 1862).

M. Simon. Wrote at least four articles on the mathematics of the RMP-1904, 1905, 1907, 1909–dealing with π and with the unit fractions of the Recto.

E. Smith. A dealer in Egyptian antiquities and hieratic papyri, including the Ebers Papyrus and the Edwin Smith Papyrus. Died 1906.

J.J. Sylvester, Considered the 2 ÷(2n - 1) of the RMP Recto as a problem in number theory, to which he gave the name “fractional sorites.”In “On the Theory of Vulgar Fractions,”in American Journal of Mathematics, 3(1880), 332–385. 388–389; and Educational Times, 37(1882), 42–43, 80; he wrote: “It was their [the ancient Egyptians] curious custom to resolve every fraction into a sum of unit fractions according to a certain traditional method.”

J. Taylor. One of the earliest pyramid mystics, perhaps the first. His théories were expressed in The Great Pyramid, Why It Was Built, and Who Built It? (London, 1859; 2nd ed., 1864). They were supported and fürther developed by the astronomer royal for Scotland, Piazzi Smyth (see Bibliography).

H. W. Turnbull. In The Great Mathematicians (London, 1951), 2–5, Turnbull refers to the geometry of the Great Pyramid, to the golden section, and to some of the théories of the pyramid mystics.

Q. Vetter. Discussed the methods of Egyptian division, especially with regard to 2–(2n -1) of the RMP Recto, in Egypliské deleni (Prague, 1921).

H. E. Winlock. One of the more successful excavators in Egypt. Discovered statues and relics of Queen Hat-shepsut at Deir-el-Bahri in 1927. Became director of the New York Metropolitan Muséum of Art in 1932. Highly regarded in the field of Oriental scholarship. Died 1950.

T. Young. British physicist, originator of the wave theory of light. An early decipherer of certain hieroglyphs. in 1814 he recognized that the cartouches contained the names of pharaohs and queens, but did not pursue these studies. Died 1829.