Notes For CMTHU201History of Mathematics

In this course we will study five early centers of civilization and track thedevelopment of mathematics in those five areas. There are other ancient centers, but therecords for these is not as complete. Our five centers are the Tigris-Euphrates region,the Nile River Valley, the Aegean Sea area, the Yellow River Valley in China and theIndus River in what is now Pakistan and the culture of India that grew from this.

We will then go on to study the first civilization that benefited from themathematical and scientific discoveries of all five of these centers, the combined Persianand Islamic societies.

I Prehistory

The origins of counting in a broad sense are earlier in the evolutionary processthan the emergence of man. Many animal species can tell the difference between thepresence of a predator and the absence of the same. This carries forward to the differencebetween 1 and 2 objects. Wolves and lions know well enough not to attack 2 or 3 of theirkind when they are alone. Experiments have shown that crows and jackdaws can keeptrack of up to six items. (In the Company of Crows and Ravens, Marzluff and Angell)“Pigeons can, under some circumstances, estimate the number of times they have peckedat a target and can discriminate, for instance, between 45 and 50 pecks.” (StanislasDehaene, The Number Sense)

Human infants can discriminate between 2 and 3 objects at the age of 2 or 3 daysfrom birth. This hard-wired and primitive sort of counting is known as digitizing. Itextends to 3 and perhaps 4 objects, but not beyond that. Babies at the age of 6 months cancorrectly associate a number of sounds to a number of objects near them, a sort of graspof abstract integer counting. Five month old babies also have been shown to grasp that1 + 1 is not 1, nor 3, but exactly 2. (Cf. Dehaene, ibid.)

With regard to the earliest records of human representation of numbers, one of theoldest is a piece of baboon fibula found in a cave in the Lebombo Mountains inSwaziland, the Lebombo bone. It has 29 tallying notches on it and has been dated toabout 35,000 B.C.E. Another is a piece of shinbone from a young wolf and was found in1937 in Pekarna, Moravia, Czechoslovakia. This bone has 57 notches, arranged in groupsof 5 and is dated from about 30,000 B.C.E. A third example is known as the Ishangobone. This tiny 10 centimeter curved bone was found by a Professor Heinzelin on theshore of the Semliki River in Zaire. It was on the Ugandan side of the river, but in Zaire.This is just north of Lake Edwards. It has been dated to about 20,000 years of age. Thisbone has notched columns engraved on it, yielding patterned numbers. One interpretationis of certain periods of the moon. Certainly the association of early mathematics and earlyastronomy is a fact. We see that early humans had the ability to count, and that it hadimportance to them, enough to warrant the keeping of records.

II Examples of Prehistoric Civilizations

One very old example of social organization is found in south-central Turkey at asite called

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Chatal −H˙ ̇ o y˙ ̇ u k . The town here had up to 10,000 inhabitants at some times inits 1,300 years of use, from about 7500 B.C.E. to 6200 B.C.E. It had no streets, as thepeople had no wheels yet. The mud houses were clustered together and entrances werethrough the roof. There is no sign of king or ruling classes, as the dwellings are prettyeven in their appearance of owners’ status. There also seems to have been equality ofsexes, without more elaborate burials of one or the other. There are no fortifications orwalls around this site. Why the site was deserted is not known. (The Goddess and TheBull, Michael Balter)

Jericho contrasts with this site in being one of the first walled cities. The ruins atJericho include some dating back to 9000 B.C.E. The city was deserted about 700 yearsafter the building of a tower and a wall around the city, about 7300 B.C.E. The earliestconstruction here is believed to belong to the ancient Natufian culture. We have norecords of any mathematics from these old cultures and sites, but they are large enough torequire organization and planning for harvesting crops and schedules for planting andhunting.

We turn to Mesopotamia, the land between the rivers Tigris and Euphrates, for theemergence of a culture which left us evidence we can decipher. The people known to usas Ubaid people settled along these rivers, and the people in cities in the southern areadid not leave, but began to create even more complex cities and societies. Why we do notknow. The Ubaid period is dated from 5600 to 3900 B.C.E. The early Ubaid peoplebegan making pottery on a slow wheel. The town of Eridu is the largest of their sites,having a population of perhaps 5,000 people by 4500 B.C.E. A large mud-brick templehere was used at least from 4500 to 2000 B.C.E. There is evidence that the temple wasfor Enki, the god of water and city god of Eridu. We thus see a shared religious practice,as opposed to the household ritual in

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Chatal −H˙ ̇ o y˙ ̇ u k . Larger, fancier houses areassociated with the temple. The agriculture included growing of wheat, barley and lentilstogether with the husbandry of sheep, goats and cattle. Towards the end of the Ubaidperiod, many towns associated with the Ubaid culture began building walls around thetowns.

At the same time the pre-Elamite city of Susa, in the watershed of the KarunRiver, east of Mesopotamia, was growing. The city was founded about 4,000 B.C.E. andthe earliest pottery is not of Mesopotamian style. There is a period of Sumerian influence,followed by the proto-Elamite culture from 3200 B.C.E. to 2700 B.C.E. The Elamitelanguage has not been translated and is an isolated language, like Sumerian, not related toany of our remaining language groups. The control of Susa and Mesopotamia shiftedback and forth between Elamite and Sumerian control in the years 2500-2400 B.C.E. TheElamite culture is poorly understood, but remained strong enough to become part of thebasis for the later Achaemenid Persian Empire. It thus forms part of the cultural roots

of such mathematicians as Omar Khayyam and Nasir al-Din al-Tusi, both of whom wewill study later. The Elamites originally came from the Iranian plateau.

The Uruk period extended from 3900 to 3100 B.C.E. and included theinnovations of the fast pottery wheel and of the wheeled cart. The largest early Sumeriansite is at Uruk, which had a population of 10,000 to 20,000 by 3100 B.C.E. Early writingoccurred in a pre-cuneiform style by 3400 B.C.E., before the Sumerian ascendancy. Itwas in the form of pictographs, simplified pictures of a fish, a bird or a man, pressed intosoft clay and baked. It was principally used to account for goods, and evolved to includeletters between the elite and also the recording of myth. We find larger population centersin this era, with the formation of city-states, and conflict between these city states. TheSumerians made real advances in writing up until 2300 B.C.E. They ended up using lotsof sound-based syllables in symbolic form. By 2500 B.C.E. they had scribal schools forthe elite, who also had to practice solving mathematical problems in addition to learningto write.

In order to keep such things as farming organized they had a purely lunar calendarat first. Ultimately they had to combine the sun and the moon to keep better track of theseasons. Since 29.5 days is the approximate period of the moon, they used 12 times thisto create a 354 day year. They were then short by about 11 days and had to add a 13th

month every 3 years. Here is the symbiosis of astronomy and early mathematics in afairly structured social setting. One notices the organized, long-term astronomicalobservations in conjunction with the arithmetic of the calendar. In particular the number12 has acquired importance. The Sumerian hours varied with the seasons. There were sixdaytime hours and six nighttime hours. Thus there were 12 hours in each day. Each hourwas divided into 60 minutes of varying length, and each minute into 60 seconds.

III Mesopotamian Mathematics

There are three main achievements of Mesopotamian mathematics which we willfocus on, and which we will experience first-hand in worksheets #1 and #2 andhomework I. These are:

the sexagesimal, or base 60, number system,

the use of abstract symbols for numbers and

a positional system for writing numbers, including fractions.

The first of these accomplishments was in place in Sumer by 2350 B.C.E. It mayhave been a compromise between various different ways of counting in societies withwhom the Sumerians traded. We shall come to know one of these, the ancient Harappansociety of the Indus valley, in short order. This society was probably as old and civilizedas the Mesopotamian societies, Some ancient cultures counted by 2’s at first. Ten is apopular base, perhaps deriving from our hands. We will see the evidence for the influence

of ten as a base in the mixture of base ten symbols and base 60 calculations inMesopotamian arithmetic.

But, where we in the west count on our fingers using each finger as one, getting ten, someIndians count using their thumb as a counter, and their twelve facing finger joints as thenumbers.

60 is perhaps a compromise between the bases of two, ten and twelve, and alsoserves one well when using fractions. It is better than base ten for this purpose. Sincereligion, astronomy and mathematics were intertwined in the Mesopotamian societies, thefact that 360 days is a good approximation to the number of days in a year might havealso played a role in the choice of 60 as the basis for the number system. It seems naturalto suppose that the 360-degree circle measure comes from the rotation of any oneconstellation about the north star in one year, about 360 days. In our age of lightpollution, we are very unaware of the powerful effect of the heavens on peoples with lessluminescence in their lives. We do, however, still use sexagesimal arithmetic for circlemeasurement, and time. Our electronic marvels, such as microwave ovens, allow us touse sexagesimal arithmetic to time our nuking. We have the same sort of mixed base tenand base 60 in our microwave displays that the Mesopotamians had on their clay tablets.Another possible cause for this adoption of a base 60 number system was the early use of

12 different systems of metrology within Sumerian society itself, all of which the scribeshad to use and relate to each other. See the handout, An Abbreviated SumerianMetrology, gotten out of the book, The Mathematics of Egypt, Mesopotamia, China,India and Islam, edited by Victor J. Katz, Princeton University press, 2007.

We engage with the online Worksheet #1, The Mesopotamian Number System,here.

The use of abstract symbols in arithmetic, (ideograms) instead of pictographs oftwo or three bushels of wheat, is due to the Akkadians, who conquered the Sumerians,but who kept Sumerian intellectual traditions alive. The Akkadians used Sumerianlanguage in much the same way that medieval Europe held on to Latin as a universallanguage. Just as the use of base 60 may have resulted from compromising betweendifferent ways of counting in different cultures, so the rise of abstract symbols, orideograms, may have resulted from the need to unite the Sumerian and Akkadianexpressions for numbers, a language problem. There was still a drawback because theideograms were abstract uses of the many Sumerian pictographs and were so numerousand this made calculating hard to do. In addition to creating ideograms for numbers, theAkkadians invented an abacus by 2100 B.C.E. and were developing systems for additionand subtraction. The Akkadians used a base 10 number system.

By 2000 B.C.E. Amoritic Semites conquered the Akkadians, again keeping theintellectual heritage of Sumer. These peoples were at their highest power underHammurabi (1823-1763 B.C.E.) In this period they developed the positional notationfor writing numbers. The importance of this is hard for us to grasp as we are so used to it.Calculations are much more difficult without it. We have never known anything else. Weowe it to the Amoritic Babylonians. They had a unified positional system for positiveintegers and fractions. This was never achieved again until the Persian mathematicianJamshid Al-Kashi (1406-1437) and Simon Stevin in Europe in 1585 who both used suchsystems. The Babylonians cut down on the number of ideograms, but went over the edgethe other way, using just two symbols repetitively,

one = and ten = , gotten by pressing a cut reed-stem in clay

straight down, or at an angle. Here is the legacy of a base ten number system. Thispresents other difficulties in calculating. They had no sexagesimal point, so the samesymbol would be used for 1, 60 and 3600 for instance, as well as many otherambiguities. One had to know the context of the number being used in order to know thecorrect interpretation of the symbol. They had no zero, as a number, or as a place-holderin positional notation.

The Babylonians had extensive tables of multiplication by various numbers, butinstead of divisions, they calculated 1/n and multiplied by that number to divide by n.Their scribes had to learn the use of such tables, as well as tables of square and cuberoots, tables of comparisons of market rates for different commodities, tables of“Pythagorean” triples of numbers and how to calculate areas as well as how to solvethree dimensional geometry problems

A brief timeline of the history which accompanies these mathematicaldevelopments follows.

• The Sumerian era extended from approximately 3500 B.C.E. to about 2000B.C.E. The first Sumerian king for whom we have evidence is Etana, the king ofthe Sumerian city Kish. Ur, Nippur and Uruk are other important Sumerian cities.

• A century-long period of Akkadian domination occurred from 2335 to 2218.The Akkadian Sargon (true king) conquered the Sumerians in 2335 and absorbed

Sumerian culture into his land of Akkad-Sumer. His capital was Agade, north ofSumer. Its location was where the city of Babylon came to be.

• There were other short conquests and upheavals until the Amoritic Babylonians,came to power around 2000 B.C.E. with Hammurabi rulng from 1823 to 1763B.C.E. The Babylonians kept power for about 1,000 years until the more despoticand harsh Assyrians conquered Mesopotamia. The Babylonians adopted theprevious Sumerian-Akkadian culture.

• The Assyrians ruled from farther north than the Babylonians, and Assurbanipal, the last Assyrian king had his capital at Nineveh from 668 to 626 B.C.E.

Homework I

In base 10, explicit form, the number 385 is written

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3 ⋅102 + 8 ⋅101 + 5 ⋅100

or

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3 ⋅102 + 8 ⋅10 + 5 ⋅1.In base 60, explicit form, 385 is written

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6 ⋅ 601 + 25 ⋅ 600 or

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6 ⋅ 60 + 25 ⋅1

1. Write the following numbers in base 10, explicit form:a. 62 b. 485 c. 8,126 d. 35,423

2. Write the following numbers in base 5, explicit form:a. 62 b. 485 c. 8,126 d. 35,423

We will use a semicolon “ ;” to indicate the sexagesimal point (the “decimal”point in base 60.) We will use commas to separate sexagesimal digits. Thus thedecimal number 623.6 would be written in base 60 as 10,23;36, or

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10 ⋅ 60 + 23 ⋅1 + 3660

3. The following numbers are decimal numbers. Change them m to base 60, explicit form.

a. 372 b. 81.4 c. 3,795.7

4. The following numbers are in sexagesimal form. Change them to base 10, explicit form.

a. 4,3,25 b. 2,0,35;15 c. 4,10,41;3,36

5. Express the fractions in sexagesimal form:

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115

, 140

, 19

The Babylonians knew how to approximate square roots. We do not know howthey did this, but one ancient method is the following.

To find the square root of a number, say 23 you guess an integer, say 5you average the sum of your guess and 23 divided by your guess and take this as

your next guess. Keep on doing the same thing over and over. This is an iterativealgorithm.

In this instance you take (5 + 23/5)/2 = 4.8 as your second guess.Then you repeat, taking (4.8 + 23/4.8)/2 = 4.7958333333 as your third guess.And so on....

6. One can set this up on a calculator very easily. Use this method, writing down yourguess and the next three numbers which this algorithm gives you, to approximate thesquare root of

a. 17 b. 457 c. 89 d. 3,547

7. (Math Credit) The fractions

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13

and 19

have infinite decimal expansions. Find them

and prove that they equal the fractions.

8. (Math Credit) Find the base 5 expansion for

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13

and 16

and prove that they equal the

fractions.

Later Babylonian mathematics includes tablets which list the lengths of the sidesof right triangles, basically compiling Pythagorean triples of numbers. They certainlyhad an empirical knowledge of the Pythagorean Theorem, but perhaps saw no need toprove such things.

A Babylonian tablet of about 1900 B.C.E. records a value of

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Π = 318

= 3.125. Mesopotamian mathematics was more arithmetic and pre-algebraic

than geometric. But they certainly knew elementary geometric formulas.They also had a verbal description of the quadratic formula. They describe how

to solve the equation

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x 2 + 23x = 35

60 using the formula

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x = a2

2

+ b − a2

when

we are solving

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x 2 + ax = b. You should work out why this is the same formula that youlearned in high school. They also solved quadratic equations using the method which nowgoes by the name of completing the square. They did not allow negative numbers.

The Babylonians had knowledge of the five planets (Mercury, Venus, Mars,Jupiter and Saturn) which are visible to the unaided eye as distinguished wanderersamong the stars. They had a set of 12 constellations along what we now call the zodiac,for keeping track of the calendar.

The later Assyrians did not support research into science and mathematics, andastronomy changed into astrology and became more aligned with religion. It is interestingto reflect on the cultures of early Mesopotamia, and notice that larger and much morematerially well-off cultures, such as the Romans, accomplished much less in theadvancement of science and mathematics.

We engage with Worksheet #2, Mesopotamian Calculations, here.