The Mathematics of Ancient Mesopotamia Background

• Mesopotamia: Greek , “between the rivers,” specifically the Tigris and Euphrates. This area occupies most of what is present-day Iraq, and parts of Syria, Turkey, Lebanon, and Iran.

Background

• Thought to be the (or at least a) “cradle of civilization.” • Delta region extremely fertile – The “Fertile Crescent” • Semi-arid climate required extensive irrigation projects Four Empires • Four civilizations flourished here, from 3100 BCE to 539 BCE. These included the early Sumerian (3100 – 2400 BCE) and Akkadian (2400-2100 BCE) empires, and the later Old Babylonian (1800-1200 BCE) and Assyrian (1200 -612 BCE; Ashurbanipal) empires. There followed a brief Neo-Babylonian period from 612 – 539 BCE. Then Persia. Then Alexander the Great. Then…. Timeline

Archaic Old Kingdom Int Middle Kingdom Int New Kingdom EGYPT

3000 BCE 2500 BCE 2000 BCE 1500 BCE 1000 BCE

Sumaria Akkadia Int Old Babylon Assyria MESOPOTAMIA Some names you might recognize • Hammurabi, founder of the Old Babylonian Empire • Code of Hammurabi - 232 laws, lex talionus, an eye for an eye – If anyone strikes the body of a man higher in rank than he, he shall receive sixty blows with an ox-whip in public. Some names you might recognize • The Epic of Gilgamesh – Poem relates story of Gilgamesh, ruler of Uruk, who seeks out survivor of great flood in quest of immortality. • Ur of the Chaldees, Birthplace of Abraham. • King Nebuchadnezzar (Neo- Babylonian Empire) Sources

• Most of what we know about Mesopotamian mathematics comes from several hundred clay tablets belonging to the Old Babylonian kingdom, around roughly 1800-1600 BCE. • Tablets are of two kinds: – Table texts – Problem texts But Before We Go There…

• We need to understand a little about the number system used in that Old Babylonian era. The theories about how it evolved the way it did are interesting in themselves. Babylonian Number System

• A base-60 positional system with individual numbers formed by two different wedge-shaped marks: a horizontal wedge  worth 10 and a vertical wedge  worth 1. • Numbers less than 60 were written using these two symbols in a purely additive fashion. Babylonian Number System Notational Aside:

• Notice that the marks from the previous table don’t look exactly like the  and the  that I used a while ago. They look even less like the marks I’ll end up using from here on because they are easier: ‹ and ˅. There is considerable variation in both the original texts and the modern interpretations. Babylonian Number System

• These 59 symbols would be written in a place value system based on powers of 60. Powers of 60 increased from right to left, just as powers of 10 increase from right to left in our system. Babylonian Number System

• Thus, writing ‹˅ ‹‹‹˅˅˅˅ ‹‹˅˅˅ would most likely represent ଶ or 41,663. The First of Two Problems

• There was no “0” or placeholder so we really can’t be sure which power of 60 is being used. Thus, ‹˅˅ ˅˅‹ could represent either: , or ଶ , or ଶ , or many other possibilities. The Second of Two Problems • Even though the Babylonians used this system to write fractions as sexagecimals, there was no “sexagecimal point” or other way of marking where the fractional part began. So, again, ‹˅˅ ˅˅‹ could mean any of: , or ିଵ , or ିଵ ିଶ Resolution of Problems

• These two problems were usually quite easily resolved by the context of the arithmetic being done, so it bothers us much more than it did the Babylonians. Also, there were very frequently units attached. For example, any ambiguity in writing 1 1 is resolved if we say $1 1₵. Resolution of Problems

• In about 300 BC there was a placeholder symbol invented and used, but only between symbols, never at the end. • In our notation, it used for 604 but never 640 or 6400. Our Babylonian Notation

• We will use a comma to separate place values, use a 0 when we need it, and use a semicolon as a “sexagecimal point.” Thus,

ଵ But Why 60? Why? Why?

Some suggested reasons: • Lots of non-repeating sexagecimals, since 60 as more divisors than 10 (btw, how do you tell if one of our fractions will terminate or repeat when converted to a decimal?). • Sacred or Mystical numbers • Combination of two number cultures. Why 60?

• Well, actually, we aren’t sure. • But we’ll talk about one suggested solution. • According to Peter Rudman in his book How Mathematics Happened: The First 50,000 Years, it’s probably more like 6’s and 10’s than 60. Example:

60x60 60 1 1/60

› ٧ › ٧ (carrying row)

٧٧٧ ›››› ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧٧٧ › (1st number)

٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧ ›››› ٧٧٧٧ › + (2nd number)

٧ ›› ٧٧٧٧ ›› ٧٧ ›› (Sum)

60x60 60 1 1/60

› ٧ › ٧ (carrying row)

٧٧٧ ›››› ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧٧٧ › (1st number)

٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧ ›››› ٧٧٧٧ › + (2nd number)

٧ (Sum)

60x60 60 1 1/60

› ٧ › ٧ (carrying row)

٧٧٧ ›››› ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧٧٧ › (1st number)

٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧ ›››› ٧٧٧٧ › + (2nd number)

٧ ›› (Sum)

60x60 60 1 1/60

› ٧ › ٧ (carrying row)

٧٧٧ ›››› ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧٧٧ › (1st number)

٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧ ›››› ٧٧٧٧ › + (2nd number)

٧ ›› ٧٧٧٧ (Sum)

60x60 60 1 1/60

› ٧ › ٧ (carrying row)

٧٧٧ ›››› ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧٧٧ › (1st number)

٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧ ›››› ٧٧٧٧ › + (2nd number)

٧ ›› ٧٧٧٧ ›› (Sum)

60x60 60 1 1/60

› ٧ › ٧ › (carrying row)

٧٧٧ ›››› ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧٧٧ › (1st number)

٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧ ›››› ٧٧٧٧ › + (2nd number)

٧ ›› ٧٧٧٧ ›› ٧٧ (Sum)

60x60 60 1 1/60

› ٧ › ٧ › (carrying row)

٧٧٧ ›››› ٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧٧٧ › (1st number)

٧٧٧٧٧٧٧٧ ››› ٧٧٧٧٧ ›››› ٧٧٧٧ › + (2nd number)

٧ ›› ٧٧٧٧ ›› ٧٧ ››› (Sum)

Alternating 10-for-1 and 6-for -1 Exchanges Ok, so…..

• We can understand using groups of 10. But we have to ask: “Why the freak are there groups of 6?” • Well, let’s look at Ancient Sumer: • First, realize that these folks used different measures for different things, and that these measures had different “exchanges” from larger to smaller units. • We did this too: Weight

16 ounces = 1 pound 14 pounds = 1 stone 8 stone = 1 hundredweight 20 hundredweight = 1 ton

(except for us 100 pounds = 1 hundredweight, and 20 hundredweight = 1 ton = 2000 pounds) Capacity

8 (fluid) ounces = 1 cup 2 cups = 1 pint 2 pints = 1 quart 4 quarts = 1 gallon Length:

12 inches = 1 foot 3 feet = 1 yard 22 yards = 1 chain 10 chains = 1 furlong 8 furlongs = 1 mile 3 miles = 1 league And then you have rods and links and thous. . . . Land Measures

• Originally, communal plots of land were laid out in rectangular plots of 1 furlong by 1 chain (660 by 66 feet), or 10 chains by 1 chain (= 1 acre). The furrows ran in the long direction, so the plots were a “furrow long.” So actually furlongs were an agricultural measure that were independent of feet, which was a body-part measure. • By the way, a cricket pitch is still 66 feet long, or 1 chain, or a tenth of a furlong. Moving on….

• Eventually small measures based on body parts had to be reconciled with large agricultural measures like furlongs, so things were shifted and fudged in the measures so that everything was an integral multiple of everything else. The Same Thing Happened in Babylon: • A body-part measure called a kush, about 1 2/3 feet, was the basis for a nindan, which needed to be reconciled with two agricultural measures, the eshe and the USH. The eshe and the USH came pre- loaded with a 6-to-1 exchange, and the nindan and the eshe became an easy 10-to-1 exchange. 10-for-1 and 6-for-1

• Because units of both length and area were exchanged for larger units in both groups of 10 and groups of 6, using counters that reflected those exchanges greatly facilitated calculations with lengths and areas. And the number system went along for the ride. 10-for-1 and 6-for-1

• So Rudman claims that units of length and area that came pre-loaded with exchanges gave rise to a system of arithmetic with alternating 10-to-1 and 6-to-1 exchanges, and then to a base 60 system. In any event, there was real genius in moving to a place-value system. • Now, back to Babylon: Babylonian Tablet Texts:

• Table Texts • Problem Texts Table Texts

• Multiplication tables, of which about 160 are known. • Single tables have the form: p a-rá 1 p a-rá 2 2p a-rá 3 3p And so on, up to 20p, then: a-rá 30 30p a-rá 40 40p a-rá 50 50p Table Texts

• Combined tables (of which there are about 80) have several single tables included on one tablet. • One of them (A 7897) is a large cylinder containing an almost complete set of tables written in 13 columns. There is a hole through the center of the cylinder so that it could be turned on some kind of peg. Table Texts

• Reciprocal Tables had reciprocals of numbers from 2 to 81 (provided their sexagecimal representations did not repeat. • These were used to divide, which they did by multiplying by reciprocals. Table Texts

• There are a few tables of squares, square roots, cube roots, powers, sums of squares and cubes, … .

• Also some conversions and a few special tables used for particular business transactions (finding market rates). Table Texts

• It is likely that many of the table texts we have are “exercises” from students learning to be scribes, or perhaps tables copied and made by students for use in computations. Problem Texts • Also probably intended for educational purposes. • Story problems aimed at computing a number. • Often contrived or “tricky” problems: – If camel A leaves Phoenicia travelling at  nindan per day. . . . • Kinda like our modern story problems or recreational math problems. Problem Texts

• Largely algebraic problems, focusing on what we would call on linear and quadratic equations (though that’s not necessarily how the Babylonians thought about this). • Mainly focused on algorithms, but not on general procedures. Instead, they gave several worked examples. A Little Arithmetic

• The book mentions that we don’t really know how the Babylonians did arithmetic like adding and subtracting and we don’t know their algorithms for multiplication and division, except that they divided by multiplying by reciprocals. Multiplication: 43 x 1,15 Babylonian Style • This would be broken into 43 x 1 and 43 x 15. The 43 x 1 would be easy; it would just be 43 but moved one place value over, in the 60’s place. The 43 x 15 would become 40 x 15 + 3 x 15, both of which would be available from tables; 40 x 15 = 10, 0; 3 x 15 = 45; so 43 x 1,15 = 43, 0 +10, 0 + 0, 45 = 53, 45 • This is why the Babylonian multiplication tables had, for the principle number p, multiples of p up to 20p, then 30p, 40p, and 50p. The distributive property was used to split up larger multiples so these were all that was necessary. (Why 20p instead of 10p? Who knows?) Division

• We’ll do 1029 divided by 64: • In our language, we multiply 1029 by 1/64, or

• 17, 9 by 0; 0, 56,15 0; 0, 56, 15 x 17, 9 2, 15 9 x 15 from table 7,30 9 x 50 from table; shift 54 9 x 6 from table; shift 4,15, 17 x 15 from table; shift 14, 10, 17 x 50 from table; shift2 1, 42, 17 x 6 from table; shift2 16; 4, 41, 15

Or in our system, 16.078125 What About Division by 7? ଵ ଻ • As a Babylonian scribe would say, “7 does not divide.” • So, they used an approximation: ଵ ଻ ଵ ଻ ଵ ଵ ଵଷ ଽଵ ଽଵ ଽ଴

ଵ ( ) ଵଷ Story Problems, 1

• I have added the area and two-thirds of [ the side of ] my square and it is 0;35. What is the side of my square? ଶ

• The solution follows the standard procedure for completing the square: Babylon, 2000 BC Provo, 2011 AD • You take 1. Two-thirds of 1 is 0;40. • Half of this, 0; 20, you multiply by 0;20 and it is 0;6,40, • you add to 0;35

• and the result 0;41;40 has 0;50 as its square root. Babylon, 2000 BC Provo, 2011 AD • The 0;20 which you have multiplied by itself, you subtract from 0;50, and 0;30 is the side of the square. Is This Algebra?

• Another textbook author, Victor Katz, suggests the method was mainly geometric. • I believe most scholars assume the methods were geometric, since algebraic symbolism was not common to any ancient culture – with the exception of Diophantus, perhaps. Is This Algebra?

• You take 1. Two- thirds of 1 is 0;40.

x ;40 • Thus, the area on the right represents x2 ;40x the situation; 0;40 x is two-thirds. And the area of the figure is 0;35. Is This Algebra?

x ;40 • Half of this, 0; 20,

2 ;40x • The idea is to take x x half of the rectangle and rearrange it to form a gnomon – a x square with a smaller

2 ;20x square missing from x x the corner. It still has area 0;35. ;20x Is This Algebra?

• you multiply by 0;20 and it is 0;6,40, • Here, you find the x small square that is

2 ;20x missing, and… x x • you add to 0;35 • Because the gnomon ;20x is still 0.35, you now missing 0;6,40 have the area of the big square. Is This Algebra?

• and the result 0;41;40 has 0;50 as

its square root. x • The areas of the big

square is 0;41,50, 2 ;20x x x and its side is 0;50. • We are now finding 0;20 ;20x the side of the big square, which is (x + 0;20). Is This Algebra?

• The 0;20 which you have multiplied by

itself, you subtract x from 0;50, and 0;30 is the side of the 2 ;20x square. x x • Since we know that , 0;20 ;20x we now subtract the 0;20 to find x. Story Problems, 2 (2/3)(2/3)x+100=x

• First multiply two-thirds by two thirds: result 0;26,40 • Subtract 0;26,40 from 1: result 0;33,20 • Take the reciprocal of 0;33,20: result 1;48 • Multiply 1;48 by 1,40: result 3,00. • 3,00 (qa) is the original quantity. Story Problems, 3

• I found a stone but did not weigh it. After I weighed out 8 times its weight, added 3 gin. [Then] one-third of one- thirteenth I multiplied by 21, added it and then I weighed it. Result 1 mana. What was the original weight of the stone? • The weight was 4;30 gin. (1 mana = 60 gin).

Babylonian “Algebra”

• Whether or not their “algebra” was geometric or not, they were skilled in solving quadratic equations. We should note, however, that – All quantities were positive – Problems were often given in terms of areas and perimeters of rectangles Some Geometry-YBC 7289 Error: 0.000000423847 ish

30

1; 24, 51, 10 = 1.414212963...

42; 25, 35 = 42.42638889... Babylonian Astronomy

• Ancient peoples of Mesopotamia could easily track the movement of the celestial sphere as it revolved around the earth every year. They could also track the movement of the sun in a wiggly path (the ecliptic) against the celestial sphere. Babylonian Astronomy Babylonian Astronomy Babylonian Astronomy, 600 BC • Early version of the Zodiac, 12 areas of 30 ush each; the sun travels 1 ush per day. • So, there were 360 ush in a full circuit of the sun. • The beginning of there being 360 degrees in a circle. Babylonian Astronomy, 600 BC • Two different descriptions of how the sun (and moon) changed speeds along their path. One was a step function (two speeds); the other had a linear change over time and was quite accurate. Babylonian Astronomy, 600 BC • The Babylonians divided the day into twelve intervals called "kaspu". The solar kaspu was the span of thirty degrees which the Sun travels in two hours of daily motion across Earth's sky. Babylonian Astronomy, 600 BC • The Babylonians also predicted certain celestial phenomena, such as eclipses and lunar periods. They began their studies with the eclipse of March 19, 721 BC. Calculations were difficult because the astronomers had no instruments of high accuracy. Babylonian Astronomy, 600 BC • Both the Chaldeans and Babylonian eclipse records are used in studying long-term variations in the lunar orbit in modern theories. • Records of new moons, eclipses, and the rising of Venus were kept from very early times. Some Astrology

• 2. If in Nisannu the sunrise (looks) sprinkled with blood and the light is cool: rebellion will not stop in the country, there will be devouring by Adad. • 3. If in Nisannu the normal sunrise (looks) sprinkled with blood: battles Some Astrology

• 4. If in Nisannu the normal sunrise (looks) sprinkled with blood: there will be battles in the country. • 5. If on the first day of Nisannu the sunrise (looks) sprinkled with blood: grain will vanish in the country, there will be hardship and human flesh will be eaten. Some Astrology

• 6. If on the first day of Nisannu the sunrise (looks) sprinkled with blood and the light is cool: the king will die and there will be mourning in the country. • 7. If it becomes visible on the second day and the light is cool: the king's ... high official will die and mourning will not stop in the country. Oh Happiness!

• 8. If a normal disk is present and one disk stands to the right (and) one to the left: if the king treats the city and his people kindly for reconciliation and they become reconciled, Oh Happiness!

• 8. If a normal disk is present and one disk stands to the right (and) one to the left: if the king treats the city and his people kindly for reconciliation and they become reconciled, the cities will start vying with each other, city walls will be destroyed, the people will be dispersed. Constellations

• On the 1st of Nisannu the Hired Man becomes visible. On the 20th of Nisannu the Crook becomes visible. • On the 20th of Ayyaru the Jaw of the Bull becomes visible. • On the 10th of Simanu the True Shepherd of Anu and the Great Twins become visible. • On the 5th of Du'uzu the Little Twins and the Crab become visible. Secret Knowledge

• "Secret tablet of Heaven, exclusive knowledge of the great gods, not for distribution! He may teach it to the son he loves. To teach it to a scribe from Babylon or a scribe from Borsippa or any other scholar is an abomination to Nabu and Nisaba. • ...a Babylonian or a Borsippan or any other scholar...... whoever speaks... • [Nabu and] Nisaba will not confirm him as a teacher. In poverty and deficiency may they put an end to his ...... ; may they kill [him] with dropsy." Babylonian Calendar

• The problem with calendars is coordinating the different cycles: days, months, years, and seasons. They don’t come in nice integral multiples. • By the way, there were two seasons in Babylon, Summer (barley harvest) and Winter (roughly our fall/winter). Babylonian Calendar

• Months in Babylon started when a new moon (actually, a visible crescent) first appeared. So the priest-astronomers would watch and announce the beginning of the month. • This was common in other cultures, too. “Calendar”

• In Rome, a Pontifex (priest) observed the sky and announced a new moon and therefore the new month to the king. For centuries afterward Romans referred to the first day of each new month as Kalends from their word calare (to announce solemnly, to call out). The word calendar derived from this custom. Babylonian Calendar

• Calendar based on cycles of the moon, and needed to be reconciled with the solar year. • Alternated 29- and 30-day months, and added an extra month three times in every 8 years. • This still necessitated the King adding an extra month every now and then when the seasons shifted too far. Babylonian Calendar

• In the reign of king Nabû-Nasir, the astronomers of Babylon recognized that 235 lunar months are almost identical to 19 solar years. (The difference is only two hours.) They concluded that seven out of nineteen years ought to be leap years with an extra month. Babylonian Calendar

• In the reign of king Nabû-Nasir, the astronomers of Babylon recognized that 235 lunar months are almost identical to 19 solar years. (The difference is only two hours.) They concluded that seven out of nineteen years ought to be leap years with an extra month. Babylonian Calendar

• By about 500 BC, there were six years when a second month Addaru is added, and one year with an extra Ululu. The result is that the first day of the month Nisanu (New year's day) was never far (< 27 days) from the vernal equinox, so that the civil calendar and the seasons were never far out of step. Wrapping Up

• Babylonian mathematics was often practically-oriented, aimed at solving problems of commerce, calendaring, and so forth. However, there is also evidence that Scribes developed a culture of doing difficult problems to show off their skill, or just to have “good clean fun.” (Think about the “I found a stone” problems.) Wrapping Up

• We don’t know exactly how the Scribes came up with their solutions, since they only wrote down the numerical steps of a solution. • They could solve linear and quadratic equations, some cubic equations, understood right triangle relations, had some efficient and accurate arithmetic capabilities. They were also pretty good astronomers. Vestiges of Babylon in Our Culture • Zodiac • 360 degrees, 60 minutes, 60 seconds • 12 hour clocks • Decimal numbers • Others?