Predynastic Egypt (c. 5000—4000 BC)
Predynastic Egypt of the 5th millennium BC pictorially represented geometric spatial designs.^{[1]}
Ancient Egypt (c. 3000—1000 BC)
Overview
Early Dynastic Period (c. 3000—2700 BC)
Written evidence of the use of mathematics dates back to at least 3000 BC with the ivory labels found a Tomb U-j at Abydos. These labels appear to have been used as tags for grave goods and some are inscribed with numbers.^{[2]} Further evidence of the use of the base 10 number system can be found on for instance the Narmer Macehead which depicts offerings of 400,000 oxen, 1,422,000 goats and 120,000 prisoners.^{[3]}
Circa 2700 BC Egyptians introduced the earliest fully developed base 10 numeration system. Though it was not a positional system, it allowed the use of large numbers and also fractions in the form of unit fractions and Eye of Horus fractions, or binary fractions. ^{[4]}
Old Kingdom (c. 2700—2100 BC)
The evidence of the use of mathematics in the Old Kingdom (ca 2690–2180 BC) is scarce, but can be deduced from for instance inscriptions on a wall near a mastaba in Meidum which gives guidelines for the slope of the mastaba.^{[5]} The lines in the diagram are spaced at a distance of one cubit and show the use of that unit of measurement.^{[2]}
By 2700 BC, Egyptian construction techniques included precision surveying, marking north by the sun's location at noon. Clear records began to appear by 2000 BC citing approximations for π and square roots. Exact statements of number, written arithmetic tables, algebra problems, and practical applications with weights and measures also appeared after 2050 BC in hieratic script, with several problems solved by abstract arithmetic methods.
Middle Kingdom (c. 2000—1600 BC)
The earliest true mathematical documents date to the Middle Kingdom period, specifically the 12th dynasty (c. 1990–1800 BC). The Moscow Mathematical Papyrus, the Egyptian Mathematical Leather Roll, the Lahun Mathematical Papyri which are a part of the much larger collection of Kahun Papyri and the Berlin Papyrus 6619 all date to this period. The Rhind Mathematical Papyrus which dates to the Second Intermediate Period (ca 1650 BC) is said to be based on an older mathematical text from the 12th dynasty.^{[6]}
The Moscow Mathematical Papyrus and Rhind Mathematical Papyrus are mathematical problem texts. They consist of a collection of problems with solutions. These texts may have been written by a teacher or a student engaged in solving typical mathematics problems.^{[2]}
An interesting feature of Ancient Egyptian mathematics is the use of unit fractions. The Egyptians used some special notation for fractions such as and and in some texts for , but other fractions were all written as unit fractions of the form or sums of such unit fractions. Scribes used tables to help them work with these fractions. The Egyptian Mathematical Leather Roll for instance is a table of unit fractions which are expressed as sums of other unit fractions. The Rhind Mathematical Papyrus and some of the other texts contain tables. These tables allowed the scribes to rewrite any fraction of the form as a sum of unit fractions.^{[2]}
The Akhmim Wooden Tablet (AWT) listed five divisions of a unit of volume called a hekat beginning with a hekat unity valued at (64/64). The hekat unity was divided by 3, 7, 10, 11 and 13, and recorded by exact unit fraction answers. The first half of the answers cited binary quotients, (64/64/n). For example (64/64) was divided by 3, with a missing intermediate steps reporting a quotient 21/64 and a remainder 1/192. The scribe wrote the quotient 21/64 as (16 + 4 + 1)/64 obtaining (16 + 4 + 1)/64, or (1/4 + 1/16 + 1/64) hekat. The remainder scaled 1/192 to ro units, 1/320 of a hekat, scaling (1/192)*(5/5) by writing (5/3)*(1/320) and finally (1 + 2/3)ro.
The scribe combined quotients and remainders into one statement. The 1/3rd of a hekat was recorded as: (1/4 1/16 1/64)hekat (1 2/3)ro). Scribal addition and multiplication signs were not recorded. The 1/3 of a hekat series was written from right to left. The scribe proved answers by multiplying by initial divisors, finding the initial hekat unity value of (64/64) five times. The AWT scribe used an exact partitioning method, a method that was shortened by Ahmes and other Middle Kingdom scribes by omitting the proof step. However, Ahmes' partitioning steps did follow the AWT's two-part structure, using it 29 times in Rhind Mathematical Papyrus #81, and additional times in other problems.
Hana Vymazalova published in 2002 a fresh copy of the AWT that showed that all five AWT divisions had been exact, by parsing the proof steps, and returning the five division answers to (64/64). Vymazalova thereby updated Daressy's 1906 incomplete discussion of the subject that had only found 1/3, 1/7 and 1/10 answers as exact.
Beyond the (64/64)/n = (Q/64) hekat + (5R/n)ro (with Q a quotient, and R a remainder) formula two additional formula reveal early scribal thinking. The first formula allowed divisors to exceed n the limit 1/64 < n < 64 = 64. Sub-units of the hekat were developed,i.e. hin, dja and ro. In RMP 81 data 29 examples contrast two-part statements to equivalent (10/n)hin and (320/n) statements. The medical texts and 2,000 prescriptions extended the one-part method to (64/n)dja ingredients.
Ahmes went beyond the divisor 1/64 < n < 64 limit by writing quotient and remainder arithmetic in one other way. A second method increased the size of the numerator. A two-part hekat method used in RMP 35 divided 100 hekat by 70. Ahmes wrote (6400/64)/70 as (91/64 + 30/4480)hekat. The quotient was written (64 + 16 + 8 + 2 + 1)/64 or (1 1/4 1/8 1/32 1/64)hekat. Ahmes scaled the remainder (30/4480) by (5/5) obtaining (150/70)*(1/320) by writing (2 1/7)ro. The combined (1 1/4 1/8 1/32 1/64)hekat (2 1/7) ro answer was written down from right to left and used no arithmetic addition or multiplication signs, thereby following the Akhmim Wooden Tablet methodology.
After 2050 BC, a form of Egyptian multiplication doubled answers to problems. The doubled answers reported the arithmetic correctness of answers. Scholars noted that many initial and intermediate calculations were missing, and searched for the ab initio information. Finding few ciphered hieratic letters were transliterated into modern numbers leaving many hieratic arithmetic statements untranslated into modern arithmetic. Ahmes, for example, doubled the RMP 38 answer, substituting 22/7 for pi, to correct inventory losses of grain. The traditional pi value of 256/81 over-estimated grain in hekats, a context that scholars reported little. Scribes showed off skills with RMP 2/n table unit fraction series within finite arithmetic statements, another topic that scholars reported little before 2000 AD. Ahmes solved 87 problems, several complex ones, in the Rhind Mathematical Papyrus (RMP) by using an Egyptian fraction notation that early scholars understood on the additive level, but not on the deeper scribal level until the 21st century. The RMP included selections of least common multiples that scaled 2/n table answers, algebra answers, geometry answers, arithmetic progression answers, hekat, hin, dja, ro, and pesu inventory controls of grain, bread, and beer calculated in the notation, topics that drew many scholarly discussions.
False position was suggested by 20th century scholars to decode the division aspect of the mathematical texts. Scribes commonly used Old Kingdom doubling as proofs during the Middle Kingdom. In the 21st century scholars have decoded several ab initio fragments, i.e. newly parsing RMP 38, that pointed out hieratic multiplication and division operations were inverse operations in a manner that followed modern-like conventions, thereby refuting the false position suggestion.
Traditional 20th century scholars reported that Egyptian Middle Kingdom scribes were confined to applications of practical arithmetic with problems additively addressing how a number of loaves were equally shared, and little more. One exception was Richard Gillings. He reported rhetorical algebra's alternando [(y/x) =(q/p)], and dividendo [(y -x)/x = (q -p)/p] as proportions that mixed beer raw materials in RMP 73 and 75, and a harmonic mean that mixed ingredients in sacrificial bread in RMP 76 pesu formulas. Problems in the Moscow Mathematical Papyrus and Rhind Mathematical Papyrus (RMP) also expressed instructional views of bread and beer production beyond scribal practical limitations. Four ancient texts cover abstract definitions of number, and higher forms of arithmetic. Abstract definitions appear in the Akhmim Wooden Tablet, the Egyptian Mathematical Leather Roll, the Kahun Papyrus, and the Rhind Mathematical Papyrus. The abstract arithmetic scaled hekat and other weights and measures units. The hekat used Eye of Horus quotients and Egyptian fraction remainders scaled to ro, 1/320 of a hekat, and other sub-units. Five hekat two-part statements were used in the Akhmim Wooden Tablet and applied 30 times in the Rhind Mathematical Papyrus, and additional times in the Ebers Papyrus clarifying the arithmetic operations of Egyptian fraction arithmetic.
New Kingdom (c. 1600—1000 BC)
During the New Kingdom (ca 1550–1070 BC) mathematical problems are mentioned in the literary Papyrus Anastasi I, and the Papyrus Wilbour from the time of Ramesses III records land measurements. In the worker's village of Deir el-Medina several ostraca have been found that record volumes of dirt removed while quarrying the tombs.^{[2]}^{[6]}
Sources
Our understanding of ancient Egyptian mathematics has been impeded by the reported paucity of available sources. The most famous such source is the Rhind Mathematical Papyrus, a text that can be read by comparing many of its elements against other texts, i.e., the Egyptian Mathematical Leather Roll and the Akhmim Wooden Tablet. The Rhind papyrus dates from the Second Intermediate Period (circa 1650 BC), but its author, Ahmes, identifies it as a copy of a now lost Middle Kingdom papyrus. The Rhind papyrus contains a table of 101 Egyptian fraction expansions for numbers of the form 2/n, and 84 word problems, the answers to which were expressed in Egyptian fraction notation.
The RMP also includes formulas and methods for addition, subtraction, multiplication and division of sums of unit fractions. The RMP contains evidence of other mathematical knowledge, ^{[7]} including composite and prime numbers; arithmetic, geometric and harmonic means; and understanding of both the Sieve of Eratosthenes and perfect number theory^{[7]}. It also shows how to solve first order linear equations ^{[8]} as well as summing arithmetic and geometric series. ^{[9]}
Henry Rhind's estate donated the Rhind papyrus to the British Museum in 1863. Also included in the donation was the Egyptian Mathematical Leather Roll, dating from the Middle Kingdom era. Like the Rhind papyrus, the Egyptian Mathematical Leather Roll contains a table of Egyptian fraction expansions.
The Berlin papyrus, written around 1300 BC, shows that ancient Egyptians had solved two second-order, one unknown, equations that some have called Diophantine equations. The Berlin method for solving has not been confirmed in a second hieratic text, though it has been confirmed by a second Berlin Papyrus problem. ^{[10]}
List
Sources other than the ones mentioned above include the Moscow Mathematical Papyrus, the Reisner Papyrus, and several other texts including medical prescriptions found in the Ebers Papyrus.
Our understanding of ancient Egyptian mathematics has been impeded by the reported paucity of available sources. The sources we do have include the following texts generally dated to the Middle Kingdom and Second Intermediate Period:
- The Moscow Mathematical Papyrus ^{[11]}
- The Egyptian Mathematical Leather Roll ^{[11]}
- The Lahun Mathematical Papyri ^{[11]}
- The Berlin Papyrus 6619 was written around 1800 BC
- The Akhmim Wooden Tablet.^{[6]}
- The Reisner Papyrus dates to the early Twelfth dynasty of Egypt and was found in Nag el-Deir, the ancient town of This.^{[11]}
- The Rhind Mathematical Papyrus (RMP) dates from the Second Intermediate Period (circa 1650 BC), but its author, Ahmes, identifies it as a copy of a now lost Middle Kingdom papyrus. The RMP is the largest mathematical text.^{[11]}
From the New Kingdom we have a handful of mathematical texts and inscription related to computations:
- The Papyrus Anastasi I is a literary text from the New Kingdom. It is written as a (fictional) letter written by a scribe named Hori and addressed to a scribe named Amenemope. A segment of the letter describes several mathematical problems.^{[6]}
- Ostracon Senmut 153 is a text written in hieratic.^{[6]}
- Ostracon Turin 57170 is a text written in hieratic.^{[6]}
- Ostraca from Deir el-Medina contain computations. Ostracon IFAO 1206 for instance shows the calculations of volumes, presumably related to the quarrying of a tomb.^{[6]}
Numerals
Two number systems were used in ancient Egypt. One, written in hieroglyphs, was a decimal based tally system with separate symbols for 10, 100, 1000, etc, as Roman numerals were later written, and hieratic unit fractions. The second, written in a new ciphered one-number-to-one-symbol system was a digital system that was not similar to hieroglyphic system. The hieroglyphic number system existed from at least the Early Dynastic Period. The hieratic system differed from the hieroglyphic system beyond a use of simplifying ligatures for rapid writing and began around 2150 BC. Hieratic numerals used one symbol for each number replacing the tallies that had been used to denote multiples of a unit. For example, two symbols had been used to write three, thirty, three hundred, and so on, in a system that was superseded by the hieratic method. Later hieroglyphic numeration was modified and adopted by the Romans for official uses, and Egyptian fractions in everyday situations.
The Rhind Mathematical Papyrus was written in hieratic. It contains examples of how the Egyptians did their mathematical calculations. Fractions were denoted by placing a line over the letter n associated with the number being written, as 1/n. This method of writing numbers came to dominate the Ancient Near East, with Greeks 1,500 years later using two of their alphabets, Ionian and Doric, to cipher all of their numerals, alpha = 1, beta = 2 and so forth. Concerning fractions, Greeks wrote 1/n as n', so Greek numeration and problem-solving adopted or modified Egyptian numeration, arithmetic and other aspects of Egyptian math.
Example from the Rhind Papyrus^{[12]} <hiero>Z2:Z1*Z1 Aa16 r:Z1*Z1*Z1*Z1:Z2 r:10 Z1-Z1-Z1-Z1</hiero> </br> 5 + ^{1}⁄_{2} + ^{1}⁄_{7} + ^{1}⁄_{14} (= 5 ^{5}⁄_{7})
Ancient Egyptian texts could be written in either hieroglyphs or in Hieratic. In either representation the number system was always given in base 10. The number 1 was depicted by a simple stroke, the number 2 was represented by two strokes, etc. The numbers 10, 100, 1000, 10,000 and 1,000,000 had their own hieroglyphs. Number 10 is a hobble for cattle, number 100 is represented by a coiled rope, the number 1000 is represented by a lotus flower, the number 10,000 is represented by a finger, the number 100,000 is represented by a frog and a million was represented by a god with his hands raised in adoration.^{[11]}
1 | 10 | 100 | 1000 | 10,000 | 100,000 | 1,000,000 |
---|---|---|---|---|---|---|
<hiero>Z1</hiero> | <hiero>V20\</hiero> | <hiero>V1</hiero> | <hiero>M12</hiero> | <hiero>D50</hiero> | <hiero>I8</hiero> | <hiero>C11</hiero> |
Egyptian numerals date back to the Predynastic period. Ivory labels from Abydos record the use of this number system. It is also common to see the numerals in offering scenes to indicate the number of items offered. The King's Daughter Neferetiabet is shown with an offering of 1000 oxen, bread, beer, etc.
The Egyptian number system was additive. Large numbers were represented by collections of the glyphs and the value was obtained by simply adding the individual numbers together.
Fractions
Rational numbers could also be expressed, but only as sums of unit fractions, i.e. sums of reciprocals of positive integers, 2/3, and 3/4. The hieroglyph indicating a fraction looked like a mouth, which meant "part", and fractions were written with this fractional solidus, i.e. the numerator 1, and the positive denominator below. Special symbols were used for 1/2 and for two non-unit fractions, 2/3 (used often) and 3/4 (used less often).
Problem 25 on the Rhind Papyrus may have used the method of false position to solve the problem "a quantity and its half added together become 16; what is the quantity?" (i.e., in modern algebraic notation, what is x if x+½x=16).
Assume 2
1 2 / ½ 1 / Total 1½ 3
As many times as 3 must be multiplied to give 16, so many times must 2 be multiplied to give the answer.
1 3 / 2 6 4 12 / 2/3 2 1/3 1 /
Total 5 1/3 16
So:
1 5 1/3 (1 + 4 + 1/3) 2 10 2/3
The answer is 10 2/3.
Check -
1 10 2/3 ½ 5 1/3
Total 1½ 16
A more likely and direct approach to solve this class of problem is given by: x + (1/2)x = 16, using these steps
1. (3/2)x = 16, 2. x = 32/3, 3. x = 10 2/3.
Problem 31 sets the problem "q quantity, its 2/3, its 1/2 and its 1/7, added together, become 33; what is the quantity?" In modern algebraic notation, "what is x if x + 2/3 x + 1/2 x + 1/7 x =33?" The answer is 14 1/4 1/56 1/97 1/194 1/388 1/679 1/776, or 14 and 28/97. To solve the problem as Ahmes wrote his answer 28/97 had to be broken up into 2/97 and 26/97, and solved the two separate vulgar fraction conversion problems using Hultsch-Bruins (without using false position, as other algebra problem may have been solved).
The remainder arithmetic solution, the historical method that is most likely, for x + (2/3)x + (1/2)x + (1/7)x = 33 looks like this:
1. 97/42 x = 33, 2. x = 1386/97, and 3. x = 14 + 28/97.
with, 2/97 - 1/56 = (112 - 97)/(56*97) = (8 + 7)/(56*97) = 1/679 1/776,
and 26/97 - 1/4 = (104-97/(4*97) = (4 + 2 + 1)/(4*97)= 1/97 1/194 1/388,
or,
2/97 = 1/56 1/670 1/776,
26/97 = 1/4 1/97 1/194 1/388
such that, writing out x = 14 + 28/97 in an ordered unit fraction series
4. x = 14 1/4 1/56 1/97 1/194 1/388 1/679 1/776, as written by Ahmes.
The Egyptians almost exclusively used fractions of the form 1/n. One notable exception is the fraction 2/3 which is frequently found in the mathematical texts. Very rarely a special glyph was used to denote 3/4. The fraction 1/2 was represented by a glyph that may have depicted a piece of linen folded in two. The fraction 2/3 was represented by the glyph for a mouth with 2 (different sized) strokes. The rest of the fractions were always represented by a mouth super-imposed over a number.^{[11]}
<hiero>Aa13</hiero> | <hiero>r:Z2</hiero> | <hiero>D22</hiero> | <hiero>r:Z1*Z1*Z1*Z1</hiero> | <hiero>r:Z1*Z1*Z1*Z1*Z1</hiero> |
Multiplication & Division
Egyptian multiplication was done by repeated doubling of the number to be multiplied (the multiplicand), and choosing which of the doublings to add together (essentially a form of binary arithmetic), a method that links to the Old Kingdom. The multiplicand was written next to the figure 1; the multiplicand was then added to itself, and the result written next to the number 2. The process was continued until the doublings gave a number greater than half of the multiplier. Then the doubled numbers (1, 2, etc.) would be repeatedly subtracted from the multiplier to select which of the results of the existing calculations should be added together to create the answer.
As a short cut for larger numbers, the multiplicand can also be immediately multiplied by 10, 100, etc.
For example, Problem 69 on the Rhind Papyrus (RMP) provides the following illustration, as if Hieroglyphic symbols were used (rather than the RMP's actual hieratic script).
To multiply 80 × 14 | ||||
Egyptian calculation | Modern calculation | |||
Result | Multiplier | Result | Multiplier | |
<hiero>V20*V20*V20*V20:V20*V20*V20*V20</hiero> | <hiero>Z1</hiero> | 80 | 1 | |
<hiero>V1*V1*V1*V1:V1*V1*V1*V1</hiero> | <hiero>V20</hiero> | / | 800 | 10 |
<hiero>V20*V20*V20:V20*V20*V20-V1</hiero> | <hiero>Z1*Z1</hiero> | 160 | 2 | |
<hiero>V20:V20-V1*V1:V1</hiero> | <hiero>Z1*Z1*Z1*Z1</hiero> | / | 320 | 4 |
<hiero>V20:V20-V1-M12</hiero> | <hiero>V20*Z1*Z1*Z1*Z1</hiero> | 1120 | 14 |
The / denotes the intermediate results that are added together to produce the final answer.
The table above can also be used to divide 1120 by 80. We would solve this problem by finding the quotient (80) as the sum of those multipliers of 80 that add up to 1120. In this example that would yield a quotient of 10+4=14.^{[11]} A more complicated example of the division algorithm is provided by Problem 66. A total of 3200 ro of fat are to be distributed evenly over 365 days.
1 | 365 | |
2 | 730 | |
4 | 1460 | |
8 | 2920 | |
2/3 | 243Template:Fraction | |
1/10 | 36Template:Fraction | |
1/2190 | 1/6 |
First the scribe would double 365 repeatedly until the largest possible multiple of 365 is reached, which is smaller than 3200. In this case 8 times 365 is 2920 and further addition of multiples of 365 would clearly give a value greater than 3200. Next it is noted that times 365 gives us the value of 280 we need. Hence we find that 3200 divided by 365 must equal .^{[11]}
Hieratic and Middle Kingdom math followed this form of hieroglyphic multiplication.
Subtraction defined in the Egyptian Mathematical Leather Roll (EMLR), an 1800 BC document, included four additive or identity methods, followed by one non-additive, abstract, method that was used five to fifteen times for the 26 EMLR series listed, that looked like this:
1/pq = (1/A)* (A/pq)
with A = 3, 4, 5, 7, 25, citing A = (p + 1) 10 times.
1/8 was written using A = (2 + 1)= 3, the A = (p + 1) case, as used in the RMP 24 times, seeing p = 2, q = 4 and A = 25, following
A = 3: 1/8 = (1/3)*(3/8) = 1/3*(1/4 + 1/8) = 1/12 + 1/24
A = 25: 1/8 = 1/25*(25/8) = 1/5*(25/40)= 1/5 *(24/40 + 1/40)
= 1/5*(3/5 + 1/40) = 1/5*(1/5 + 2/5 + 1/40) = 1/5 *(1/5 + 1/3 + 1/15 + 1/40) = 1/25 + 1/15 + 1/75 + 1/200
with the out-of-order 1/25 + 1/15 sequence marking the scribal method of partition.
Confirmation of the EMLR (1/A)* (A/pq), with A = (p + 1) rule is found 24 times in the RMP 2/nth table, using the form
2/pq = (2/A)* (A/pq), with A = (p + 1)
example, 2/27, a = 3, q = 9
2/27 = 2/(3 + 1)*(3 + 1)/9 = 1/4*(1/3 + 1/9) = 1/12 + 1/36
Another subtraction method is seen in the RMP 2/nth table as first suggested by F. Hultsch in 1895, and confirmed by E.M. Bruins in 1944, or
2/p - 1/A = (2A - p)/Ap
or,
2/p = 1/A + (2A -p)/Ap
where the divisors of A, from the first partition, were used to additively find (2A - p), thereby exactly solving (2A -p)/Ap.
For example,
2/19 - 1/12 = (24 - 19)/(12*19)
with the divisors of 12 = 6, 4, 3, 2, 1 being inspected to find (24 - 19) = 5 taken only from the divisors of 12. Optimally (3 + 2) was selected, by Ahmes and other scribes, over (4 + 1) such that,
2/19 = 1/12 + (3 + 2)/(12*19) = 1/12 + 1/76 + 1/114
Geometry
There are a number of problems from ancient Egypt that concern geometry. Geometric problems appear in both the Moscow Mathematical Papyrus (MMP) and in the Rhind Mathematical Papyrus (RMP). The examples demonstrate that the Ancient Egyptians knew how to compute areas of several geometric shapes and the volumes of cylinders and pyramids.
Problem 50 of the Ahmes papyrus uses these methods to calculate the area of a circle, according to a rule that the area is equal to the square of 8/9 of the circle's diameter. This assumes that π is 4×(8/9)² (or 3.160493...), with an error of slightly over 0.63 percent. This value was slightly less accurate than the calculations of the Babylonians (25/8 = 3.125, within 0.53 percent), but was not otherwise surpassed until Archimedes' approximation of 211875/67441 = 3.14163, which had an error of just over 1 in 10,000. Interestingly, Ahmes knew of the modern 22/7 as an approximation for pi, and used it to split a hekat, hekat x 22/x x 7/22 = hekat; however, Ahmes continued to use the traditional 256/81 value for pi for computing his hekat volume found in a cylinder.
Problem 48 involved using a square with side 9 units. This square was cut into a 3x3 grid. The diagonal of the corner squares were used to make an irregular octagon with an area of 63 units. This gave a second value for π of 3.111...
The two problems together indicate a range of values for Pi between 3.11 and 3.16.
RMP 53 calculated the areas of two triangles by the formula: 1/2 base times the altitude, and the area of a third shape by another method. Cubit, khet lengths were used to find areas in setat and mh units. The setat was 100 by 100 cubits and a mh was 1/100 of a setat, one cubit by 100 cubits. RMP 54 partitioned a setat into 7/10, 14/10, 28/10 and 56/10 into setat and mh segments. RMP 55 divided 5 setat by 3/5 to obtain 3 setat, taking three multiples of 1/8 setat and remainder mh units and summing 1/2 setat and 3 1/4 1/8 setat plus 1/8 setat written in mh units.
Problem 14 in the Moscow Mathematical Papyrus gives the only ancient example finding the volume of a frustum of a pyramid, describing the correct formula:
- Area:
- Triangles: The scribes record problems computing the area of a triangle (RMP and MMP).^{[11]}
- Rectangles: Problems regarding the area of a rectangular plot of land appear in the RMP and the MMP.^{[11]} A similar problem appears in the Lahun Mathematical Papyri in London.^{[13]}^{[14]}
- Circles: Problem 48 of the RMP compares the area of a circle (approximated by an octagon) and its circumscribing square. This problem's result is used in problem 50, where the scribe finds the area of a round field of diameter 9 khet.^{[11]}
- Hemisphere: Problem 10 in the MMP finds the area of a hemisphere.^{[11]}
- Volumes:
- Cylindrical granaries: Several problems compute the volume of cylindrical granaries (RMP 41–43), while problem 60 RMP seems to concern a pillar or a cone instead of a pyramid. It Is rather small and steep, with a seked (slope) of four palms (per cubit).^{[11]} In section IV.3 of the Lahun Mathematical Papyri the volume of a granary with a circular base is found is using the same procedure as RMP 43.
- Rectangular granaries: Several problems in the Moscow Mathematical Papyrus (problem 14) and in the Rhind Mathematical Papyrus (numbers 44, 45, 46) compute the volume of a rectangular granary.^{[11]}^{[13]}
- Truncated pyramid (frustum): The volume of a truncated pyramid is computed in MMP 14.^{[11]}
The Seqed
Problem 56 of the RMP indicates an understanding of the idea of geometric similarity. This problem discusses the ratio run/rise, also known as the seqed. Such a formula would be needed for building pyramids. In the next problem (Problem 57), the height of a pyramid is calculated from the base length and the seked (Egyptian for slope), while problem 58 gives the length of the base and the height and uses these measurements to compute the seqed. In Problem 59 part 1 computes the seqed, while the second part may be a computation to check the answer: If you construct a pyramid with base side 12 [cubits] and with a seqed of 5 palms 1 finger; what is its altitude?^{[11]}
The ancient Egyptians knew that they could approximate the area of a circle as follows:^{[15]}
- Area of Circle ≈ [ (Diameter) x 8/9 ]^{2}. ^{[15]}
Algebra
Egyptian algebra problems appear in both the Rhind mathematical papyrus and the Moscow mathematical papyrus as well as several other sources.^{[11]}
Aha problems involve finding unknown quantities (referred to as Aha) if the sum of the quantity and part(s) of it are given. The Rhind Mathematical Papyrus also contains four of these type of problems. Problems 1, 19, and 25 of the Moscow Papyrus are Aha problems. For instance problem 19 asks one to calculate a quantity taken 1 and ½ times and added to 4 to make 10.^{[11]} In other words, in modern mathematical notation we are asked to solve the linear equation:
Solving these Aha problems involves a technique called Method of false position. The technique is also called the method of false assumption. The scribe would substitute an initial guess of the answer into the problem. The solution using the false assumption would be proportional to the actual answer, and the scribe would find the answer by using this ratio.^{[11]}
The mathematical writings show that the scribes used (least) common multiples to turn problems with fractions into problems using integers. The multiplicative factors were often recorded in red ink and are referred to as Red auxiliary numbers.^{[11]}
The use of the Horus eye fractions shows some (rudimentary) knowledge of geometrical progression. Knowledge of arithmetic progressions is also evident from the mathematical sources.^{[11]}
Quadratic equations
The ancient Egyptians were the second civilization (after the Babylonians) to develop and solve second-degree (quadratic) equations. This information is found in the Berlin Papyrus fragment. Additionally, the Egyptians solve first-degree algebraic equations found in the Rhind Mathematical Papyrus.^{[16]}
Hellenistic Egypt (c. 300 BC—AD 400)
- Further information: Egyptian mathematicians
The most important centre of Hellenistic learning during this period was Alexandria in Egypt, which attracted scholars from across the Hellenistic world, including Egyptian, Greek, Jewish, Persian, Phoenician and even Indian scholars.^{[17]}
Most of the mathematical texts written in Greek have been found in Egypt, as well as Greece, Asia Minor, Mesopotamia, and Sicily.
Demographics
Under Greco-Roman rule, Egypt hosted several Greek settlements, mostly concentrated in Alexandria, but also in a few other cities, where Greek settlers lived alongside some seven to ten million native Egyptians.^{[18]} Faiyum's earliest Greek inhabitants were soldier-veterans and cleruchs (elite military officials) who were settled by the Ptolemaic kings on reclaimed lands.^{[19]}^{[20]} Native Egyptians also came to settle in Faiyum from all over the country, notably the Nile Delta, Upper Egypt, Oxyrhynchus and Memphis, to undertake the labor involved in the land reclamation process, as attested by personal names, local cults and recovered papyri.^{[21]} It is estimated that as much as 30 percent of the population of Faiyum was Greek during the Ptolemaic period, with the rest being native Egyptians.^{[22]} By the Roman period, much of the "Greek" population of Faiyum was made-up of either Hellenized Egyptians or people of mixed Egyptian-Greek origins.^{[23]} By the time of Roman emperor Caracalla in the 2nd century CE, the only way to differentiate Alexandria's "Greeks" from "genuine" ethnic Egyptians was "by their speech."^{[24]}
While commonly believed to represent Greek settlers in Egypt,^{[25]}^{[26]} the Faiyum mummy portraits instead reflect the complex synthesis of the predominant Egyptian culture and that of the elite Greek minority in the city.^{[27]} According to Walker, the early Ptolemaic Greek colonists married local women and adopted Egyptian religious beliefs, and by Roman times, their descendants were viewed as Egyptians by the Roman rulers, despite their own self-perception of being Greek.^{[28]} The dental morphology^{[29]} of the Roman-period Faiyum mummies was also compared with that of earlier Egyptian populations, and was found to be "much more closely akin" to that of ancient Egyptians than to Greeks or other European populations.^{[30]}
The mathematics historian Victor J. Katz has criticized the modern misconception portraying the ancient Hellenistic scholars of Egypt as ethnically Greek or European, writing:^{[31]}
But what we really want to know is to what extent the Alexandrian mathematicians of the period from the first to the fifth centuries C.E. were Greek. Certainly, all of them wrote in Greek and were part of the Greek intellectual community of Alexandria. And most modern studies conclude that the Greek community coexisted [...] So should we assume that Ptolemy and Diophantus, Pappus and Hypatia were ethnically Greek, that their ancestors had come from Greece at some point in the past but had remained effectively isolated from the Egyptians? It is, of course, impossible to answer this question definitively. But research in papyri dating from the early centuries of the common era demonstrates that a significant amount of intermarriage took place between the Greek and Egyptian communities [...] And it is known that Greek marriage contracts increasingly came to resemble Egyptian ones. In addition, even from the founding of Alexandria, small numbers of Egyptians were admitted to the privaleged classes in the city to fulfill numerous civic roles. Of course, it was essential in such cases for the Egyptians to become "Hellenized," to adopt Greek habits and the Greek language. Given that the Alexandrian mathematicians mentioned here were active several hundred years after the founding of the city, it would seem at least equally possible that they were ethnically Egyptian as that they remained ethnically Greek. In any case, it is unreasonable to portray them with purely European features when no physical descriptions exist.
Ptolemaic Egypt (c. 300—30 BC)
In Ptolemaic Egypt, the Hellenistic Egyptian mathematician Euclid (c. 300 BC) is an early example of the format still used in mathematics today, definition, axiom, theorem, proof. He also studied conics. His book, Elements, was known to all educated people in the West until the middle of the 20th century.^{[33]} In addition to the familiar theorems of geometry, such as the Pythagorean theorem, Elements includes a proof that the square root of two is irrational and that there are infinitely many prime numbers.
The Sieve of Eratosthenes (ca. 230 BC), by Eratosthenes (Libyan mathematician active in Egypt), was used to discover prime numbers.
Roman Egypt (c. 30 BC—AD 400)
In Roman Egypt, the Hellenized Egyptian/Phoenician geometer Heron described a method of iteratively computing the square root.^{[34]} It is called the Babylonian method, because the Babylonians knew of it before Heron wrote it down.
The Hellenized Egyptian mathematician Menelaus of Alexandria (c. 100 AD) wrote in three books his Sphaerica. In Book I, he established a basis for spherical triangles analogous to the Euclidean basis for plane triangles.^{[35]} He establishes a theorem that is without Euclidean analogue, that two spherical triangles are congruent if corresponding angles are equal, but he did not distinguish between congruent and symmetric spherical triangles.^{[35]} Another theorem that he establishes is that the sum of the angles of a spherical triangle is greater than 180°.^{[35]} Book II of Sphaerica applies spherical geometry to astronomy. And Book III contains the "theorem of Menelaus".^{[35]} He further gave his famous "rule of six quantities".^{[36]}
The Hellenized Egyptian astronomer Claudius Ptolemy (ca. 90 - ca. 168 A.D.) expanded upon Hipparchus' Chords in a Circle in his Almagest, or the Mathematical Syntaxis. The thirteen books of the Almagest are the most influential and significant trigonometric work of all antiquity.^{[37]} A theorem that was central to Ptolemy's calculation of chords was what is still known today as Ptolemy's theorem, that the sum of the products of the opposite sides of a cyclic quadrilateral is equal to the product of the diagonals. A special case of Ptolemy's theorem appeared as proposition 93 in Euclid's Data. Ptolemy's theorem leads to the equivalent of the four sum-and-difference formulas for sine and cosine that are today known as Ptolemy's formulas, although Ptolemy himself used chords instead of sine and cosine.^{[37]} Ptolemy further derived the equivalent of the half-angle formula .^{[37]} Ptolemy used these results to create trigonometric tables.^{[37]}
Diophantus was a Hellenized Babylonian/Egyptian mathematician who lived in Alexandria, Egypt, circa 250 AD, but the uncertainty of this date is so great that it may be off by more than a century. He is known for having written Arithmetica, a treatise that was originally thirteen books but of which only the first six have survived.^{[38]} Arithmetica has very little in common with traditional Greek mathematics since it is divorced from geometric methods, but resembles Babylonian mathematics to a much greater extent.^{[39]} In Arithmetica, Diophantus used symbols for unknown numbers as well as abbreviations for powers of numbers, relationships, and operations;^{[39]} thus he used what is now known as syncopated algebra. The main difference between Diophantine syncopated algebra and modern algebraic notation is that the former lacked special symbols for operations, relations, and exponentials.^{[40]}
Islamic Egypt (c. 639—1600 AD)
- Further information: Egyptian mathematicians
Arabic Egypt (639—1517)
Arabic mathematicians were the first to treat irrational numbers as algebraic objects.^{[41]} The Egyptian mathematician Abū Kāmil Shujā ibn Aslam (c. 850-930) was the first to accept irrational numbers (often in the form of a square root, cube root or fourth root) as solutions to quadratic equations or as coefficients in an equation.^{[42]} He was also the first to solve three non-linear simultaneous equations with three unknown variables.^{[43]}
In the late 10th to early 11th centuries, the Egyptian astronomer Ibn Yunus performed many careful trigonometric calculations and demonstrated the following trigonometric identity:
In integral calculus, Ibn al-Haytham (965-1039) was the first mathematician to derive the formula for the sum of the fourth powers, using a method that is readily generalizable for determining the general formula for the sum of any integral powers. He performed an integration in order to find the volume of a paraboloid, and was able to generalize his result for the integrals of polynomials up to the fourth degree. He thus came close to finding a general formula for the integrals of polynomials, but he was not concerned with any polynomials higher than the fourth degree.^{[6]}
Ottoman Egypt (c. 1517—1600)
In the 16th century, Taqi al-Din contributed to trigonometry in his Sidrat al-Muntaha, in which he was the first mathematician to extract the precise value of Sin 1°. He discusses the values given by his predecessors, explaining how Ptolemy used an approximate method to obtain his value of Sin 1° and how Abū al-Wafā, Ibn Yunus, Jamshīd al-Kāshī, Qāḍī Zāda al-Rūmī, Ulugh Beg and Mirim Chelebi improved on the value. Taqi al-Din then solves the problem to obtain the precise value of Sin 1°:^{[44]}
See also
Notes
- ↑ Thom, Alexander, and Archie Thom, 1988, "The metrology and geometry of Megalithic Man", pp 132-151 in C.L.N. Ruggles, ed., Records in Stone: Papers in memory of Alexander Thom. Cambridge Univ. Press. ISBN 0-521-33381-4.
- ↑ ^{2.0} ^{2.1} ^{2.2} ^{2.3} ^{2.4} Imhausen, Annette, Ancient Egyptian Mathematics: New Perspectives on Old Sources, The Mathematical Intelligencer, Vol 28, Nr 1, 2006, pp 19–27
- ↑ ^{3.0} ^{3.1} Burton, David, The History of Mathematics: An Introduction , McGraw–Hill, 2005, ISBN 978-0-07-305189-5
- ↑ Egyptian Mathematical Papyri - Mathematicians of the African Diaspora
- ↑ Rossi, Corinna Architecture and Mathematics in Ancient Egypt Cambridge University Press. 2007 ISBN 978-0-521-69053-9
- ↑ ^{6.0} ^{6.1} ^{6.2} ^{6.3} ^{6.4} ^{6.5} ^{6.6} ^{6.7} Katz, Victor J. (editor), Imhausen, Annette et al. The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook, Princeton University Press. 2007 ISBN 978-0-691-11485-9
- ↑ ^{7.0} ^{7.1} MathPages - Egyptian Unit Fractions
- ↑ Egyptian Papyri
- ↑ Egyptian Algebra - Mathematicians of the African Diaspora
- ↑ Egyptian Mathematical Papyri - Mathematicians of the African Diaspora
- ↑ ^{11.00} ^{11.01} ^{11.02} ^{11.03} ^{11.04} ^{11.05} ^{11.06} ^{11.07} ^{11.08} ^{11.09} ^{11.10} ^{11.11} ^{11.12} ^{11.13} ^{11.14} ^{11.15} ^{11.16} ^{11.17} ^{11.18} ^{11.19} ^{11.20} ^{11.21} ^{11.22} Clagett, Marshall Ancient Egyptian Science, A Source Book. Volume Three: Ancient Egyptian Mathematics (Memoirs of the American Philosophical Society) American Philosophical Society. 1999 ISBN 978-0-87169-232-0
- ↑ Gardiner (1957), p. 197
- ↑ ^{13.0} ^{13.1} R.C. Archibald Mathematics before the Greeks Science, New Series, Vol.73, No. 1831, (Jan. 31, 1930), pp. 109–121
- ↑ Anette Imhausen Digitalegypt website: Lahun Papyrus IV.3
- ↑ ^{15.0} ^{15.1} Ray C. Jurgensen, Alfred J. Donnelly, and Mary P. Dolciani. Editorial Advisors Andrew M. Gleason, Albert E. Meder, Jr. Modern School Mathematics: Geometry (Student's Edition). Houghlin Mifflin Company, Boston, 1972, p. 52. ISBN 0-395-13102-2. Teachers Edition ISBN 0-395-13103-0.
- ↑ Moore, Deborah Lela (1994). The African roots of mathematics (2nd ed. ed.). Detroit, Mich.: Professional Educational Services. ISBN 1884123007.
- ↑ George G. Joseph (2000). The Crest of the Peacock, p. 7-8. Princeton University Press. ISBN 0-691-00659-8.
- ↑ Adams, Winthrope L in Bugh, Glenn Richard. ed. "The Hellenistic Kingdoms". The Cambridge Companion to the Hellenistic World. Cambridge: Cambridge University Press. 2006, p. 39
- ↑ Stanwick, Paul Edmund. Portraits of the Ptolemies: Greek Kings as Egyptian Pharaohs. Austin: University of Texas Press. 2003, p. 23
- ↑ Adams, op cit.
- ↑ Bagnall, R.S. in Susan Walker, ed. Ancient Faces : Mummy Portraits in Roman Egypt (Metropolitan Museum of Art Publications). New York: Routledge, 2000, p. 27
- ↑ Bagnall, op cit.
- ↑ Bagnall, pp. 28-29
- ↑ qtd. in Alan K. Bowman, Egypt after the Pharaohs, 332 BC − AD 642, Berkeley: University of California Press, 1996, p. 126: "genuine Egyptians can easily be recognized among the linen-weavers by their speech."
- ↑ Egyptology Online: Fayoum mummy portraits accessed on January 16, 2007
- ↑ Encyclopædia Britannica Online - Egyptian art and architecture - Greco-Roman Egypt accessed on January 16, 2007
- ↑ Bagnall, op cit.
- ↑ Walker, Susan, op cit., p. 24
- ↑ Dentition helps archaeologists to assess biological and ethnic population traits and relationships
- ↑ Irish JD (2006). "Who were the ancient Egyptians? Dental affinities among Neolithic through postdynastic peoples.". Am J Phys Anthropol 129 (4): 529-43
- ↑ Victor J. Katz (1998). A History of Mathematics: An Introduction, p. 184. Addison Wesley, ISBN 0-321-01618-1
- ↑ Bill Casselman. "One of the Oldest Extant Diagrams from Euclid". University of British Columbia. http://www.math.ubc.ca/~cass/Euclid/papyrus/papyrus.html. Retrieved 2008-09-26.
- ↑ Howard Eves, An Introduction to the History of Mathematics, Saunders, 1990, ISBN 0030295580 p. 141 "No work, except The Bible, has been more widely used... ."
- ↑ Heath, Thomas (1921). A History of Greek Mathematics, Vol. 2. Oxford: Clarendon Press. pp. p.323-324. http://books.google.com/books?id=LOA5AAAAMAAJ&pg=PR323.
- ↑ ^{35.0} ^{35.1} ^{35.2} ^{35.3} Boyer (1991). "Greek Trigonometry and Mensuration". p. 163. "In Book I of this treatise Menelaus establishes a basis for spherical triangles analogous to that of Euclid I for plane triangles. Included is a theorem without Euclidean analogue - that two spherical triangles are congruent if corresponding angles are equal (Menelaus did not distinguish between congruent and symmetric spherical triangles); and the theorem A + B + C > 180° is established. The second book of the Sphaerica describes the application of spherical geometry to astronomical phenomena and is of little mathematical interest."
- ↑ Needham, Volume 3, 108.
- ↑ ^{37.0} ^{37.1} ^{37.2} ^{37.3} Boyer (1991). "Greek Trigonometry and Mensuration". pp. 164–166. "The theorem of Menelaus played a fundamental role in spherical trigonometry and astronomy, but by far the most influential and significant trigonometric work of all antiquity was composed by Ptolemy of Alexandria about half a century after Menelaus. [...] Of the life of the author we are as little informed as we are of that of the author of the Elements. We do not know when or where Euclid and Ptolemy were born. We know that Ptolemy made observations at Alexandria from A.D. 127 to 151 and, therefore, assume that he was born at the end of the first century. Suidas, a writer who lived in the tenth century, reported that Ptolemy was alive under Marcus Aurelius (emperor from A.D. 161 to 180)."
- ↑ (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 178) Uncertainty about the life of Diophantus is so great that we do not know definitely in which century he lived. Generally he is assumed to have flourished about A.D. 250, but dates a century or more earlier or later are sometimes suggested[...] If this conundrum is historically accurate, Diophantus lived to be eighty-four-years old. [...] The chief Diophantine work known to us is the Arithmetica, a treatise originally in thirteen books, only the first six of which have survived.}"
- ↑ ^{39.0} ^{39.1} (Boyer 1991, "Revival and Decline of Greek Mathematics" pp. 180-182) "In this respect it can be compared with the great classics of the earlier Alexandrian Age; yet it has practically nothing in common with these or, in fact, with any traditional Greek mathematics. It represents essentially a new branch and makes use of a different approach. Being divorced from geometric methods, it resembles Babylonian algebra to a large extent."
- ↑ (Boyer 1991, "Revival and Decline of Greek Mathematics" p. 178) "The chief difference between Diophantine syncopation and the modern algebraic notation is the lack of special symbols for operations and relations, as well as of the exponential notation."
- ↑ O'Connor, John J.; Robertson, Edmund F., "Arabic mathematics: forgotten brilliance?", MacTutor History of Mathematics archive, University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/HistTopics/Arabic_mathematics.html.
- ↑ Jacques Sesiano, "Islamic mathematics", p. 148, in Selin, Helaine; D'Ambrosio, Ubiratan (2000), Mathematics Across Cultures: The History of Non-western Mathematics, Springer, ISBN 1402002602
- ↑ Berggren, J. Lennart (2007). "Mathematics in Medieval Islam". The Mathematics of Egypt, Mesopotamia, China, India, and Islam: A Sourcebook. Princeton University Press. p. 518. ISBN 9780691114859.
- ↑ "Taqi al Din Ibn Ma’ruf's Work on Extracting the Cord 2° and Sin 1°". FSTC Limited. 30 May 2008. http://muslimheritage.com/topics/default.cfm?ArticleID=941. Retrieved 2008-07-04.
External links
- [1] 2005 "ab inito" recovery of modern-looking ancient Egyptian division operation
- Akhmim Wooden Tablet Akhmim Wooden Tablet
- [2] Egyptian geometry: RMP 53, 54, 55
- [3] Egyptian Mathematical Leather Roll
- [4] Rhind Mathematical Papyrus
- [5] Updated Egyptian mathematics references available for download on Open University
- History Topics: Ancient Egyptian mathematics
- Egyptian Numerals
- Egyptian Arithmetic
- Egyptian Mathematics - Aldokkan
Further reading
- Boyer, Carl B. 1968. History of Mathematics. John Wiley. Reprint Princeton U. Press (1985).
- Chace, Arnold Buffum. 1927–1929. The Rhind Mathematical Papyrus: Free Translation and Commentary with Selected Photographs, Translations, Transliterations and Literal Translations. 2 vols. Classics in Mathematics Education 8. Oberlin: Mathematical Association of America. (Reprinted Reston: National Council of Teachers of Mathematics, 1979). ISBN 0-87353-133-7
- Clagett, Marshall. 1999. Ancient Egyptian Science: A Source Book. Volume 3: Ancient Egyptian Mathematics. Memoirs of the American Philosophical Society 232. Philadelphia: American Philosophical Society. ISBN 0-87169-232-5
- Couchoud, Sylvia. 1993. Mathématiques égyptiennes: Recherches sur les connaissances mathématiques de l'Égypte pharaonique. Paris: Éditions Le Léopard d'Or
- Daressy, G. "Ostraca," Cairo Museo des Antiquities Egyptiennes Catalogue General Ostraca hieraques, vol 1901, number 25001-25385.
- Gillings, Richard J. 1972. Mathematics in the Time of the Pharaohs. MIT Press. (Dover reprints available).
- Neugebauer, Otto (1969) [1957]. The Exact Sciences in Antiquity (2 ed.). Dover Publications. ISBN 978-048622332-2. http://books.google.com/books?id=JVhTtVA2zr8C.
- Peet, Thomas Eric. 1923. The Rhind Mathematical Papyrus, British Museum 10057 and 10058. London: The University Press of Liverpool limited and Hodder & Stoughton limited
- Robins, R. Gay. 1995. "Mathematics, Astronomy, and Calendars in Pharaonic Egypt". In Civilizations of the Ancient Near East, edited by Jack M. Sasson, John R. Baines, Gary Beckman, and Karen S. Rubinson. Vol. 3 of 4 vols. New York: Charles Schribner's Sons. (Reprinted Peabody: Hendrickson Publishers, 2000). 1799–1813
- Robins, R. Gay, and Charles C. D. Shute. 1987. The Rhind Mathematical Papyrus: An Ancient Egyptian Text. London: British Museum Publications Limited. ISBN 0-7141-0944-4
- Sarton, George. 1927. Introduction to the History of Science, Vol 1. Willians & Williams.
- Strudwick, Nigel G., and Ronald J. Leprohon. 2005. Texts from the Pyramid Age. Brill Academic Publishers. ISBN 90-04-13048-9.
- Struve, Vasilij Vasil'evič, and Boris Aleksandrovič Turaev. 1930. Mathematischer Papyrus des Staatlichen Museums der Schönen Künste in Moskau. Quellen und Studien zur Geschichte der Mathematik; Abteilung A: Quellen 1. Berlin: J. Springer
- Van der Waerden, B.L. 1961. Science Awakening". Oxford University Press.
- Vymazalova, Hana. 2002. Wooden Tablets from Cairo...., Archiv Orientalni, Vol 1, pages 27-42.
- Wirsching, Armin. 2006. Die Pyramiden von Giza - Mathematik in Stein gebaut. Books on Demand. ISBN 978-3-8370-2355-8.