Notes For CMTHU201 History of Mathematics In this course we will

Notes For CMTHU201
History of Mathematics
In this course we will study five early centers of civilization and track the
development of mathematics in those five areas. There are other ancient centers, but the
records 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 the
Indus 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 the
mathematical and scientific discoveries of all five of these centers, the combined Persian
and Islamic societies.
I Prehistory
The origins of counting in a broad sense are earlier in the evolutionary process
than the emergence of man. Many animal species can tell the difference between the
presence of a predator and the absence of the same. This carries forward to the difference
between 1 and 2 objects. Wolves and lions know well enough not to attack 2 or 3 of their
kind when they are alone. Experiments have shown that crows and jackdaws can keep
track 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 pecked
at a target and can discriminate, for instance, between 45 and 50 pecks.” (Stanislas
Dehaene, The Number Sense)
Human infants can discriminate between 2 and 3 objects at the age of 2 or 3 days
from birth. This hard-wired and primitive sort of counting is known as digitizing. It
extends to 3 and perhaps 4 objects, but not beyond that. Babies at the age of 6 months can
correctly associate a number of sounds to a number of objects near them, a sort of grasp
of abstract integer counting. Five month old babies also have been shown to grasp that
1 + 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 the
oldest is a piece of baboon fibula found in a cave in the Lebombo Mountains in
Swaziland, the Lebombo bone. It has 29 tallying notches on it and has been dated to
about 35,000 B.C.E. Another is a piece of shinbone from a young wolf and was found in
1937 in Pekarna, Moravia, Czechoslovakia. This bone has 57 notches, arranged in groups
of 5 and is dated from about 30,000 B.C.E. A third example is known as the Ishango
bone. This tiny 10 centimeter curved bone was found by a Professor Heinzelin on the
shore 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. This
bone has notched columns engraved on it, yielding patterned numbers. One interpretation
is of certain periods of the moon. Certainly the association of early mathematics and early
astronomy is a fact. We see that early humans had the ability to count, and that it had
importance 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 a
site called Chatal − H˙o˙y˙u˙k . The town here had up to 10,000 inhabitants at some times in
its 1,300 years of use, from about 7500 B.C.E. to 6200 B.C.E. It had no streets, as the
people had no wheels yet. The mud houses were clustered together and entrances were
through the roof. There is no sign of king or ruling classes, as the dwellings are pretty
€ in their appearance of owners’ status. There also seems to have been equality of
even
sexes, without more elaborate burials of one or the other. There are no fortifications or
walls around this site. Why the site was deserted is not known. (The Goddess and The
Bull, Michael Balter)
Jericho contrasts with this site in being one of the first walled cities. The ruins at
Jericho include some dating back to 9000 B.C.E. The city was deserted about 700 years
after the building of a tower and a wall around the city, about 7300 B.C.E. The earliest
construction here is believed to belong to the ancient Natufian culture. We have no
records of any mathematics from these old cultures and sites, but they are large enough to
require organization and planning for harvesting crops and schedules for planting and
hunting.
We turn to Mesopotamia, the land between the rivers Tigris and Euphrates, for the
emergence of a culture which left us evidence we can decipher. The people known to us
as Ubaid people settled along these rivers, and the people in cities in the southern area
did not leave, but began to create even more complex cities and societies. Why we do not
know. The Ubaid period is dated from 5600 to 3900 B.C.E. The early Ubaid people
began 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 temple
here was used at least from 4500 to 2000 B.C.E. There is evidence that the temple was
for Enki, the god of water and city god of Eridu. We thus see a shared religious practice,
as opposed to the household ritual in Chatal − H˙o˙y˙u˙k . Larger, fancier houses are
associated with the temple. The agriculture included growing of wheat, barley and lentils
together with the husbandry of sheep, goats and cattle. Towards the end of the Ubaid
period, many towns associated with the Ubaid culture began building walls around the
€
towns.
At the same time the pre-Elamite city of Susa, in the watershed of the Karun
River, east of Mesopotamia, was growing. The city was founded about 4,000 B.C.E. and
the 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 Elamite
language has not been translated and is an isolated language, like Sumerian, not related to
any of our remaining language groups. The control of Susa and Mesopotamia shifted
back and forth between Elamite and Sumerian control in the years 2500-2400 B.C.E. The
Elamite culture is poorly understood, but remained strong enough to become part of the
basis 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 we
will study later. The Elamites originally came from the Iranian plateau.
The Uruk period extended from 3900 to 3100 B.C.E. and included the
innovations of the fast pottery wheel and of the wheeled cart. The largest early Sumerian
site is at Uruk, which had a population of 10,000 to 20,000 by 3100 B.C.E. Early writing
occurred in a pre-cuneiform style by 3400 B.C.E., before the Sumerian ascendancy. It
was in the form of pictographs, simplified pictures of a fish, a bird or a man, pressed into
soft clay and baked. It was principally used to account for goods, and evolved to include
letters between the elite and also the recording of myth. We find larger population centers
in this era, with the formation of city-states, and conflict between these city states. The
Sumerians made real advances in writing up until 2300 B.C.E. They ended up using lots
of sound-based syllables in symbolic form. By 2500 B.C.E. they had scribal schools for
the elite, who also had to practice solving mathematical problems in addition to learning
to write.
In order to keep such things as farming organized they had a purely lunar calendar
at first. Ultimately they had to combine the sun and the moon to keep better track of the
seasons. Since 29.5 days is the approximate period of the moon, they used 12 times this
to 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 a
fairly structured social setting. One notices the organized, long-term astronomical
observations in conjunction with the arithmetic of the calendar. In particular the number
12 has acquired importance. The Sumerian hours varied with the seasons. There were six
daytime hours and six nighttime hours. Thus there were 12 hours in each day. Each hour
was 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 will
focus on, and which we will experience first-hand in worksheets #1 and #2 and
homework 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 may
have been a compromise between various different ways of counting in societies with
whom the Sumerians traded. We shall come to know one of these, the ancient Harappan
society of the Indus valley, in short order. This society was probably as old and civilized
as the Mesopotamian societies, Some ancient cultures counted by 2’s at first. Ten is a
popular 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 in
Mesopotamian arithmetic.
But, where we in the west count on our fingers using each finger as one, getting ten, some
Indians count using their thumb as a counter, and their twelve facing finger joints as the
numbers.
60 is perhaps a compromise between the bases of two, ten and twelve, and also
serves one well when using fractions. It is better than base ten for this purpose. Since
religion, astronomy and mathematics were intertwined in the Mesopotamian societies, the
fact that 360 days is a good approximation to the number of days in a year might have
also played a role in the choice of 60 as the basis for the number system. It seems natural
to suppose that the 360-degree circle measure comes from the rotation of any one
constellation about the north star in one year, about 360 days. In our age of light
pollution, we are very unaware of the powerful effect of the heavens on peoples with less
luminescence in their lives. We do, however, still use sexagesimal arithmetic for circle
measurement, and time. Our electronic marvels, such as microwave ovens, allow us to
use sexagesimal arithmetic to time our nuking. We have the same sort of mixed base ten
and 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 scribes
had to use and relate to each other. See the handout, An Abbreviated Sumerian
Metrology, 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 of
two or three bushels of wheat, is due to the Akkadians, who conquered the Sumerians,
but who kept Sumerian intellectual traditions alive. The Akkadians used Sumerian
language in much the same way that medieval Europe held on to Latin as a universal
language. Just as the use of base 60 may have resulted from compromising between
different ways of counting in different cultures, so the rise of abstract symbols, or
ideograms, may have resulted from the need to unite the Sumerian and Akkadian
expressions for numbers, a language problem. There was still a drawback because the
ideograms were abstract uses of the many Sumerian pictographs and were so numerous
and this made calculating hard to do. In addition to creating ideograms for numbers, the
Akkadians invented an abacus by 2100 B.C.E. and were developing systems for addition
and subtraction. The Akkadians used a base 10 number system.
By 2000 B.C.E. Amoritic Semites conquered the Akkadians, again keeping the
intellectual heritage of Sumer. These peoples were at their highest power under
Hammurabi (1823-1763 B.C.E.) In this period they developed the positional notation
for 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. We
owe it to the Amoritic Babylonians. They had a unified positional system for positive
integers and fractions. This was never achieved again until the Persian mathematician
Jamshid Al-Kashi (1406-1437) and Simon Stevin in Europe in 1585 who both used such
systems. The Babylonians cut down on the number of ideograms, but went over the edge
the 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. This
presents other difficulties in calculating. They had no sexagesimal point, so the same
symbol would be used for 1, 60 and 3600 for instance, as well as many other
ambiguities. One had to know the context of the number being used in order to know the
correct interpretation of the symbol. They had no zero, as a number, or as a place-holder
in positional notation.
The Babylonians had extensive tables of multiplication by various numbers, but
instead 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 cube
roots, 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 solve
three dimensional geometry problems
A brief timeline of the history which accompanies these mathematical
developments follows.
•
The Sumerian era extended from approximately 3500 B.C.E. to about 2000
B.C.E. The first Sumerian king for whom we have evidence is Etana, the king of
the 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 of
Sumer. 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 1763
B.C.E. The Babylonians kept power for about 1,000 years until the more despotic
and harsh Assyrians conquered Mesopotamia. The Babylonians adopted the
previous 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 3⋅10 2 + 8 ⋅101 + 5 ⋅10 0
or 3⋅10 2 + 8 ⋅10 + 5 ⋅1.
In base 60, explicit form, 385 is written 6 ⋅ 601 + 25 ⋅ 60 0 or 6 ⋅ 60 + 25 ⋅1
€
1. Write the following numbers in base 10, explicit form:
€
a. 62
b. 485
c. 8,126
€
€
2. Write the following numbers in base 5, explicit form:
a. 62
b. 485
c. 8,126
d. 35,423
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 the
decimal number 623.6 would be written in base 60 as 10,23;36, or
36
10 ⋅ 60 + 23⋅1 +
60
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:
1
1 1
,
,
15 40 9
The Babylonians knew how to approximate square roots. We do not know how
they did this, but one ancient method is the following.
€
To find the square root of a number, say 23 you guess an integer, say 5
you 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 iterative
algorithm.
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 your
guess and the next three numbers which this algorithm gives you, to approximate the
square root of
a. 17
b. 457
c. 89
d. 3,547
1
1
have infinite decimal expansions. Find them
and
3
9
and prove that they equal the fractions.
7. (Math Credit) The fractions
8. (Math Credit) Find€the base 5 expansion for
1
1
and prove that they equal the
and
3
6
fractions.
€
Later Babylonian mathematics includes tablets which list the lengths of the sides
of right triangles, basically compiling Pythagorean triples of numbers. They certainly
had an empirical knowledge of the Pythagorean Theorem, but perhaps saw no need to
prove such things.
A Babylonian tablet of about 1900 B.C.E. records a value of
1
Π = 3
= 3.125 . Mesopotamian mathematics was more arithmetic and pre-algebraic
8
than geometric. But they certainly knew elementary geometric formulas.
They also had a verbal description of the quadratic formula. They describe how
2
€
 a
a
2
35
to solve the equation x + x =
using the formula x =   + b −
when
 2
2
3
60
we are solving x 2 + ax = b . You should work out why this is the same formula that you
learned in high school. They also solved quadratic equations using the method which now
goes 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 wanderers
among the stars. They had a set of 12 constellations along what we now call the zodiac,
for keeping track of the calendar.
2
The later Assyrians did not support research into science and mathematics, and
astronomy changed into astrology and became more aligned with religion. It is interesting
to reflect on the cultures of early Mesopotamia, and notice that larger and much more
materially well-off cultures, such as the Romans, accomplished much less in the
advancement of science and mathematics.
We engage with Worksheet #2, Mesopotamian Calculations, here.