Mathematics

Sources

Mathematical Cuneiform Tablets. The history of mathematics begins with ancient Babylonian mathematics recovered by modern scholars from cuneiform texts. These texts are plentiful and can be found in museums, collections, and libraries all over the world. They are the product of the Mesopotamian scribal schools, where mathematics was an intrinsic part of the curriculum. Whereas surviving astronomical tablets come almost exclusively from just three sites—Nineveh, Babylon, and Uruk—mathematical cuneiform texts come from many different locations across Mesopotamia. With the exception of a small number of school exercise texts from the late third millennium B.C.E., most mathematical tablets fall into two periods of Mesopotamian history. The earlier group comes from the Old Babylonian scribal schools of the first half of the second millennium B.C.E., and the later texts were composed by the Late Babylonian scholars working in the astronomical archive in Babylon and the Resh temple at Uruk in the last centuries of the first millennium B.C.E. Nothing can be said about mathematics during the large gap in time between these two distinct groups, other than that no mathematical texts have been found for this time span—almost certainly the result of the vagaries of preservation and recovery. From the texts that have survived, modern scholars can see how the Mesopotamian number system worked; how they computed numerical quantities as diverse as interest on loans and lengths, widths, and areas; and how they solved mathematical problems. Babylonian mathematics was made up of arithmetic, elementary algebra, number theory, and geometry. As in astronomical texts, theories were never stated, and there are no texts setting forth mathematical rules. For example, they knew and used the concept of the “Pythagorean Theorem” a millennium before Pythagoras’s birth in the sixth century B.C.E., but they never wrote it down as a principle or offered proof. The underlying procedure for solving problems, however, can be found by modern scholars in the details of the solution.

The Number System. The Mesopotamian system of numbers has two distinguishing characteristics. It is, first of all, a sexagesimal system; that is, it uses a base of 60; second, it uses place-value notation so that the place position of a digit in the representation of a number is crucial. In sexagesimal numbering, a digit can be any number from 1 to 59, so even signs made up of two characters, such as 25, represent one sexagesimal digit. Moving a digit one place to the left or right signifies a change in its value by a factor of sixty. When one moves a digit one place to the left, its value becomes sixty times as large, so it must be multiplied by sixty. If a digit is moved two places, it must be multiplied by sixty squared. Similarly, if a digit is moved one place to the right, it must be divided by sixty. The principle is the same as in the modern decimal system, but in that system a digit is multiplied or divided by ten instead of sixty. The idea that the value of a digit varies according to the position it occupies in the entire representation of a number allowed for an easy and economical way of writing large numbers. Using multiplication, division (performed by multiplying by the reciprocal 1/x), addition, and subtraction, this system was a vast improvement over the earlier purely additive system, in which all the individual values contained in a representation of a number were added together to determine the aggregate value.

Sexagesimal Fractions. Like the modern decimal system, the sexagesimal number system was used to write both whole numbers and fractions. Modern people are used to thinking of the decimal fraction 1/8 as 0.125 where the 1 is 1/10; the 2 is 2/100 or 2/10², the 5 is 5/1000 or 5/10³. In the Babylonian system, the units place, when zero (the 0 in 0.125), was not indicated as such in the cuneiform, and there was no decimal point, or anything like one. Sexagesimal fractions were determined by their position, that is, their place value. When one reads numbers in cuneiform, one determines their place value by context. Then modern scholars write them with the units place followed by a semicolon and the remaining digits separated by commas to show that they are fractions. For example, the number of days of a synodic (lunar) month in the nineteen-year cycle is 29;6,22,58 ... in sexagesimal numbers. This number is equal to 29 days plus 6/60, 22/60², and 58/60³. Sexagesimal fractions were a most useful invention, because, with multiplication and reciprocal tables at hand, the work of arithmetical computations was greatly reduced. The Babylonian number system, with its place-value notation and sexagesimal system, was the linchpin of Babylonian mathematics and mathematical astronomy. It was also an important tool in later Greek and Roman astronomy. The Babylonian system of expressing fractions was later used by astronomers, such as Hipparchus (second century B.C.E.) and Ptolemy (second century C.E.), when dealing with complicated calculations with fractions. Even today, hours and time and degrees and angle are subdivided in sexagesimal fractions.

Place-Value Notation

The sexagesimal number system is based on sixty and uses place-value notation. Thus any numeral x can represent itself, or itself multiplied by sixty (x · 60), or by sixty squared (x · 60²), and so on, depending on the place position of the digits. Similarly, the same numeral can represent itself divided by sixty (x/60), or divided by sixty squared (x/60²), and so on. Context alone determines the correct reading. For example,

Source: Asger Aaboe, Episodes From The Early History of Mathematics (Washington, D.C.: Mathematical Association of America, 1964), pp. 8–9.

Babylonian Numerals. Two basic signs were used to write all Babylonian numerals. One was a vertical wedge, which could represent 1 or 60; multiples of the vertical wedge were used to produce the digits 2 through 9. The other fundamental sign was a corner wedge (“Winkelhaken”), which was equal to 10; multiples of this wedge were used to produce the numerals for 20, 30, 40, and 50. These signs could be combined to produce all of the numerals from 1 to 59; then the vertical wedge, which had been used for 1, became equal to 60. The reading of 1 or 60, or for that matter, 1/60, can be determined only by context. There were special signs for 100

Cuneiform Numerals

The numerals employed by cuneiform scribes to express numbers may be traced directly back to the sexagesimal numerals of the earliest proto-cuneiform tablets at the end of the fourth millennium B.C.E. As exemplified by the typical Old Babylonian period (circa 1894 - circa 1595 B.C.E.) numerals below, in the sexagesimal cuneiform system every number from 1 to 599 can be written with the correct relative placement of combinations of just two types of wedges, the vertical wedge for digits and the corner wedge (“Winkelhaken”) for tens. Additional signs were used for 600 and 3600.

In nonmathematical texts special numerals represent 100 and 1000:

In both mathematical and nonmathematical texts from the Seleucid and Parthian periods during the last three centuries b.c.e., a sign formerly used as a word separator occasionally came to represent “zero”:

Source: John Huehnergard, A Grammar of Akkadian (Atlanta: Scholars Press, 1997), pp. 235-237.

A Table of Multiplication

The following table text for the principle number 9 shows how multiplication was done in the sexagesimal, or base 60, number system.

On the front (obverse) and the back (reverse) of this Old Babylonian school tablet, cuneiform signs are written in two columns. In the left-hand column of the obverse are the signs for 1 through 14, and on the reverse, 15 through 19 (here written “20 minus 1” instead of the more usual “10 plus 9”), then 20, 30, 40, 50, and 500. In the right-hand column (beginning on the obverse and continuing on the reverse) are the numbers 9, 18, 27, 36, 45, 54, 63, and so on, each 9 times the number opposite it in the left-hand column. In order to get 63, for example, the vertical wedge and the three that follow must be read 1,3 or 1 · 60 + 3 = 63. The following lines would be 1,12 = 72; 1,21 = 81; and so on.

In the fourteenth line, the two vertical wedges and a 6 must be read 2,6 or 2 · 60 + 6; then 2,15 = 135 and 2,51 = 171. The next line must read 180, so the three vertical wedges must be read 3 · 60 = 180. This numeral might have been written 3,0 = 3 · 60 + 0 = 180 had the Babylonians in this period used a sign for zero at the end of a line, but they did not, so it must simply be assumed. The last line reads “8,20 times 1 (=) 8,20” or 500 · 1 = 500; this line is the “catch line,” the first line of the next multiplication tablet in the series.

Source: Asger Aaboe, Episodes From The Early History of Mathematics (Washington, D.C.: Mathematical Association of America, 1964), p. 7.

(a vertical wedge followed by an horizontal wedge) and for 1000 (a combination of the corner wedge representing 10 and the sign for 100); so 10 did play a role, perhaps as an auxiliary base. Although 100 might be expressed as 1,40 (60 + 40) instead of by the sign for 100, and 19 might be written (20 - 1) rather than with one corner wedge for 10 and nine vertical wedges for 9, mathematical and astronomical texts used the sexagesimal system in a consistent manner, but economic and metrical texts show variations in the writing of numbers in different times and places.

The Concept of Zero. Once a system of place-value notation is used, it becomes desirable to indicate the absence of a value in a particular position so that units of 60, or 60², or 60³, and so on (in a sexagesimal number system) are not missing. A symbol standing for “nothing” was a necessity to show the absence of anything in a particular position in a number. In the Old Babylonian texts, zero was assumed but not written; that is, it was represented by an open space in the middle of a number. At the end of a number, it was omitted, leaving it to the reader to assume an empty place. For example, 180 was written with three vertical wedges each equal to 60. (In another context those same vertical wedges could be read as 3.) In this way, 3 · 60 = 180, and they chose this practice instead of alternatively writing (3 · 60) + 0 = 180. In texts from the fourth century b.c.e. on, a cuneiform sign previously used to separate words in a line of text began to be used to signify nothing in place of the empty space. The Babylonian zero arrived late in the history of Babylonian mathematics; nevertheless it was the first in world history. At a later date, a zero for the base-ten system was developed in India; similarly, the Maya in Central America developed a zero for their base-twenty system.

Table of Reciprocals

On tablets with tables of reciprocals the signs are arranged in two columns. At first glance, it appears that the numbers in column i on the left, all less than sixty, all factors of some power of 60, when multiplied by values in the right column, yield a product of 60 or a power of 60 (for example, 2 · 30 = 60; 3 · 20 = 60). However, there is another way to explain the use of this table. The reciprocal of column i (1/x where x’s are the values in column 1) is equal to the value in column ii on the right (y) divided by 60, or an appropriate power of 60, or y/60ⁿ, where n = 1,2,3,.... Thus, 1/x =y/60ⁿ or ½ · 60 = 30.

Column ii therefore gives the reciprocals of column i. If one thinks of time and angle; that is, hours/minutes, degrees/seconds, the meaning is clear: 1/2 hour = 1/2 · 60 minutes = 30 minutes. This division descends from the Babylonian system. Larger, more extensive, reciprocal tables include reciprocals of regular and irregular numbers. Some tablets have a section of reciprocals followed by a collection of consecutive mathematical tables. A reciprocal table followed by a series of related multiplication tables, which in turn are followed by tables for squares or square roots, suggests that the table texts were intended to perform the operations of multiplication, division, and fractions.

Source: Asger Aaboe, Episodes From The Early History of Mathematics (Washington, D.C.: Mathematical Association of America, 1964), p. 10.

Table Texts. The Mesopotamian mathematical tablets are of two kinds: table texts and problem texts. They were arranged in fixed sequences of mathematical operations, and some were designated as tablets within a series. The table texts are characterized by columns of numbers that are related to each other in some way: products, reciprocals, squares, square roots, cubes, and cube roots. Table texts were the means by which all computations were calculated. There are no tables of division. Instead, reciprocal tables were used for this function. The reciprocal of a number x is 1/x. Instead of dividing by x, the Mesopotamians multiplied by 1/x, using the reciprocal tables of 1/x. A large number of such tables has survived from both the Old Babylonian period and the Seleucid period (311-129 B.C.E.). Other table texts include “Pythagorean” Numbers, co-efficient lists, compound interest, and price equivalencies.

Problem Texts. Problem texts contain one or more problems. The problems on a given tablet may have the same theme or yield the same answer and typically become progressively more difficult. There are many different kinds of problems in these texts. There are solutions to first and second degree equations, cubic and quadratic equations, and arithmetic progressions. The scribes and students correctly computed areas and volumes of simple polygons and solids. They calculated interest rates and worked out construction problems. Some of the problems obviously had no practical value, such as those adding areas to lengths. They were included for intellectual exercise. Some problem texts demonstrate that the Mesopotamians used the so-called Pythagorean Theorem a thousand years before Pythagorus’s birth. These texts showed that they knew that the length of the diagonal of a square is equal to the length of its side multiplied by the square root of two (?2), using an excellent approximation in sexagesimal numbers to ?2. A text now known as Plimpton 322 suggests fifteen solutions, in whole numbers, some quite large, to the “Pythagorean” equation: (a² + b² = c²; that is, the square of the hypotenuse (longest side) of a right triangle (c) is equal to the sums of the squares of the two remaining sides (a and b). The use of place-value notation was crucial. In fact, both the table texts and problem texts make sense today only if the number signs (that is, the digits) change their value with their place notation. The goal of the problems on the problem tablets was always a numerical answer. Procedures and proofs were never stated, instead examples were given.

A Mathematical Problem

This Old Babylonian school tablet shows a problem dealing with the ratio of the length of the diagonal of a square to the length of its side. It demonstrates that the Babylonians knew that the length of the diagonal of a square is equal to the length of its side multiplied by the square root of two (?2), and it shows that they had an excellent approximation of the value of ?2.

Three numbers are inscribed:

a = 30, the length of the side of the square
b = 1,24,51,10
c = 42,25,35, the length of diagonal

It can be shown that c = a · b if b and c are written with semicolons in their correct places in order to show place-value notation and thereby differentiate whole numbers from fractions:

b = 1;24,51,10 (= 1 + 24/60 + 51/60² + 10/60³)
c = 42;25,35 (= 42 + 25/60 + 35/60²)
That is, 30 · 1;24,51,10 = 42;25,35

Using the “Pythagorean Theorem,” c² = 2 a², and c = a ?2, the value of b can be shown to be close to ?2: (1;24,51,10)² = 1;59,59,38,1,40

Therefore b is a very good approximation of ?2.

Source: Asger Aaboe, Episodes From The Early History of Mathematics (Washington, D.C.: Mathematical Association of America, 1964), pp. 26–27.

Cryptography. Numerals could also stand for words and phrases or divine and geographical names in omen, astronomical, land, and medical texts. This use of cryptography reached its peak in the first millennium B.C.E., especially during the Seleucid period. Numerals were used as shorthand for divine names, in particular, from the late third millennium B.C.E. through the end of cuneiform writing in the early Common Era (C.E.). Some gods were represented by more than one numeral, especially in different periods of time. The numeral representing the god’s name was preceded by the Sumerian sign dingir, “god,” for example, dingir 30 (conventionally written^d30), for the moon god Sin.

CRYPTOGRAPHY

Cuneiform scribes could use numerals to stand for syllables, words, and phrases. Such numerals most commonly served as a form of shorthand, but they might also be deliberately employed to conceal the true meaning of the text. As a shorthand for divine names, the practice was in use from the late third millennium B.C.E. through the end of cuneiform writing in the early Common Era (C.E.), with some gods represented by more than one numeral, particularly in different periods of time. Typically the name of a god was written with a numeral preceded by the cuneiform sign read dingir, Sumerian for “god,” and rendered in modern transliterations as a superscripted letter d. The following is a list of numerals for some of the gods:

^d1 = Ea, god of subterranean fresh waters.
^d10 = Adad, storm god.
^d15 = Ishtar, goddess of love and war; planet Venus.
^d20 = Shamash, god of justice, sun god, the sun.
^d21 = Anu, sky god, sky, the heavens.
^d30 = Sin, moon god, the moon.
^d40 = Ea (see^d1)
^d50 = Enlil, god of the wind, the wind.
^d60 = Anu (see ^d21)

The cardinal directions could be rendered numerically, the cuneiform sign IM is Sumerian for “wind.”

IM.1 = south
IM.2 = north
IM.3 = east
IM.4 = west

Similarly, the left and right sides of a person or an object could be rendered:

2,30 (= 150) = left
15 = right

Sources: Jeremy A. Black and Anthony Green, Gods, Demons and Symbols of Ancient Mesopotamia: An Illustrated Dictionary (Austin: University of Texas Press, 1992).

Laurie E. Pearce, “Cuneiform Cryptography: Numerical Substitutions for Syllabic and Logographic Signs”, dissertation, Yale University, 1982.

Sources

Asger Aaboe, Episodes From The Early History of Mathematics (Washington, D.C.: Mathematical Association of America, 1964).

Jens Høyrup, “Babylonian Mathematics,” in Companion Encyclopedia of the History and Philosophy of the Mathematical Sciences, 2 volumes, edited by I. Grattan-Guiness (London: Routledge, 1994), I: 21-29.

Høyrup, Lengths, Widths, Surfaces: A Portrait of Old Babylonian Algebra and Its Kin (New York & Heidelberg: Springer, 2002).

Karen Rhea Nemet-Nejat, Cuneiform Mathematical Texts as a Reflection of Everyday Life in Mesopotamia, American Oriental Series, volume 75 (New Haven: American Oriental Society, 1993).

Otto Neugebauer and Abraham Joseph Sachs, eds., Mathematical Cuneiform Texts, American Oriental Series, volume 29 (New Haven: American Oriental Society/American Schools of Oriental Research, 1945).

Laurie E. Pearce, “Cuneiform Cryptography: Numerical Substitutions for Syllabic and Logographic Signs,” dissertation, Yale University, 1982.

Eleanor Robson, Mesopotamian Mathematics, 2100-1600 B.C.: Technical Constants in Bureaucracy and Education (Oxford: Clarendon Press, 1982.