Ancient Advances in Mathematics
Ancient knowledge of the sciences was often wrong and wholly
unsatisfactory by modern standards. However not all of the knowledge of the
more learned peoples of the past was false. In fact without people like Euclid
or Plato we may not have been as advanced in this age as we are. Mathematics is
an adventure in ideas. Within the history of mathematics, one finds the ideas
and lives of some of the most brilliant people in the history of mankind’s’
populace upon Earth.
First man created a number system of base 10. Certainly, it is not just
coincidence that man just so happens to have ten fingers or ten toes, for when
our primitive ancestors first discovered the need to count they definitely would
have used their fingers to help them along just like a child today. When
primitive man learned to count up to ten he somehow differentiated himself from
other animals. As an object of a higher thinking, man invented ten number-
sounds. The needs and possessions of primitive man were not many. When the
need to count over ten aroused, he simply combined the number-sounds related
with his fingers. So, if he wished to define one more than ten, he simply said
one-ten. Thus our word eleven is simply a modern form of the Teutonic ein-lifon.
Since those first sounds were created, man has only added five new basic
number-sounds to the ten primary ones. They are “hundred,” “thousand,” “
million,” “billion” (a thousand millions in America, a million millions in
England), “trillion” (a million millions in America, a million-million millions
in England). Because primitive man invented the same number of number-sounds as
he had fingers, our number system is a decimal one, or a scale based on ten,
consisting of limitless repetitions of the first ten number sounds.
Undoubtedly, if nature had given man thirteen fingers instead of ten,
our number system would be much changed. For instance, with a base thirteen
number system we would call fifteen, two-thirteen’s. While some intelligent and
well-schooled scholars might argue whether or not base ten is the most adequate
number system, base ten is the irreversible favorite among all the nations.
Of course, primitive man most certainly did not realize the concept of
the number system he had just created. Man simply used the number-sounds
loosely as adjectives. So an amount of ten fish was ten fish, whereas ten is an
adjective describing the noun fish.
Soon the need to keep tally on one’s counting raised. The simple
solution was to make a vertical mark. Thus, on many caves we see a number of
marks that the resident used to keep track of his possessions such a fish or
knives. This way of record keeping is still taught today in our schools under
the name of tally marks.
The earliest continuous record of mathematical activity is from the
second millennium BC When one of the few wonders of the world were created
mathematics was necessary. Even the earliest Egyptian pyramid proved that the
makers had a fundamental knowledge of geometry and surveying skills. The
approximate time period was 2900 BC
The first proof of mathematical activity in written form came about one
thousand years later. The best known sources of ancient Egyptian mathematics in
written format are the Rhind Papyrus and the Moscow Papyrus. The sources
provide undeniable proof that the later Egyptians had intermediate knowledge of
the following mathematical problems: applications to surveying, salary
distribution, calculation of area of simple geometric figures’ surfaces and
volumes, simple solutions for first and second degree equations.
Egyptians used a base ten number system most likely because of biologic
reasons (ten fingers as explained above). They used the Natural Numbers
(1,2,3,4,5,6, etc.) also known as the counting numbers. The word digit, which
is Latin for finger, is also another name for numbers which explains the
influence of fingers upon numbers once again.
The Egyptians produced a more complex system then the tally system for
recording amounts. Hieroglyphs stood for groups of tens, hundreds, and
thousands. The higher powers of ten made it much easier for the Egyptians to
calculate into numbers as large as one million. Our number system which is both
decimal and positional (52 is not the same value as 25) differed from the
Egyptian which was additive, but not positional.
The Egyptians also knew more of pi then its mere existence. They found
pi to equal C/D or 4(8/9) whereas a equals 2. The method for ancient peoples
arriving at this numerical equation was fairly easy. They simply counted how
many times a string that fit the circumference of the circle fitted into the
diameter, thus the rough approximation of 3.
The biblical value of pi can be found in the Old Testament (I Kings
vii.23 and 2 Chronicles iv.2)in the following verse
“Also, he made a molten sea of ten cubits from
brim to brim, round in compass, and five cubits
the height thereof; and a line of thirty cubits did
compass it round about.”
The molten sea, as we are told is round, and measures thirty cubits
round about (in circumference) and ten cubits from brim to brim (in diameter).
Thus the biblical value for pi is 30/10 = 3.
Now we travel to ancient Mesopotamia, home of the early Babylonians.
Unlike the Egyptians, the Babylonians developed a flexible technique for dealing
with fractions. The Babylonians also succeeded in developing more
sophisticated base ten arithmetic that were positional and they also stored
mathematical records on clay tablets.
Despite all this, the greatest and most remarkable feature of Babylonian
Mathematics was their complex usage of a sexagesimal place-valued system in
addition a decimal system much like our own modern one. The Babylonians counted
in both groups of ten and sixty. Because of the flexibility of a sexagismal
system with fractions, the Babylonians were strong in both algebra and number
theory. Remaining clay tablets from the Babylonian records show solutions to
first, second, and third degree equations. Also the calculations of compound
interest, squares and square roots were apparent in the tablets.
The sexagismal system of the Babylonians is still commonly in usage
today. Our system for telling time revolves around a sexagesimal system. The
same system for telling time that is used today was also used by the Babylonians.
Also, we use base sixty with circles (360 degrees to a circle).
Usage of the sexagesimal system was principally for economic reasons.
Being, the main units of weight and money were mina,(60 shekels) and talent (60
mina). This sexagesimal arithmetic was used in commerce and in astronomy.
The Babylonians used many of the more common cases of the Pythagorean
Theorem for right triangles. They also used accurate formulas for solving the
areas, volumes and other measurements of the easier geometric shapes as well as
trapezoids. The Babylonian value for pi was a very rounded off three. Because
of this crude approximation of pi, the Babylonians achieved only rough estimates
of the areas of circles and other spherical, geometric objects.
The real birth of modern math was in the era of Greece and Rome. Not
only did the philosophers ask the question “how” of previous cultures, but they
also asked the modern question of “why.” The goal of this new thinking was to
discover and understand the reason for mans’ existence in the universe and also
to find his place. The philosophers of Greece used mathematical formulas to
prove propositions of mathematical properties. Some of who, like Aristotle,
engaged in the theoretical study of logic and the analysis of correct reasoning.
Up until this point in time, no previous culture had dealt with the negated
abstract side of mathematics, of with the concept of the mathematical proof.
The Greeks were interested not only in the application of mathematics
but also in its philosophical significance, which was especially appreciated by
Plato (429-348 BC). Plato was of the richer class of gentlemen of leisure. He,
like others of his class, looked down upon the work of slaves and craftsworker.
He sought relief, for the tiresome worries of life, in the study of philosophy
and personal ethics. Within the walls of Plato’s academy at least three great
mathematicians were taught, Theaetetus, known for the theory of irrational,
Eodoxus, the theory of proportions, and also Archytas (I couldn’t find what made
him great, but three books mentioned him so I will too). Indeed the motto of
Plato’s academy “Let no one ignorant of geometry enter within these walls” was
fitting for the scene of the great minds who gathered here.
Another great mathematician of the Greeks was Pythagoras who provided
one of the first mathematical proofs and discovered incommensurable magnitudes,
or irrational numbers. The Pythagorean theorem relates the sides of a right
triangle with their corresponding squares. The discovery of irrational
magnitudes had another consequence for the Greeks: since the length of
diagonals of squares could not be expressed by rational numbers in the form of
A over B, the Greek number system was inadequate for describing them.
As, you might have realized, without the great minds of the past our
mathematical experiences would be quite different from the way they are today.
Yet as some famous (or maybe infamous) person must of once said “From down here
the only way is up,” so you might say that from now, 1996, the future of
mathematics can only improve for the better.
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Hooper, Alfred. Makers of Mathematics. Random House. New York, N.Y. 1948
Morley, S.G. The Ancient Maya. Stanford University Press. 1947.
Newman, J.R. The World of Mathematics. Simon and Schuster. New York, N.Y. 1969.
Smith, David E. History of Mathematics. Dover Publications Inc. Mineola, N.Y.
Struik, Dirk J. A Concise History of Mathematics. Dover Publications Inc.
Mineola, N.Y. 1987