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.
Bibliography
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Mineloa, N.Y. 1985
Beckmann, Petr. A History of Pi. St. Martin's Press. New York, N.Y. 1971
De Camp, L.S. The Ancient Engineers. Double Day. Garden City, N.J. 1963
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. 1991.
Struik, Dirk J. A Concise History of Mathematics. Dover Publications Inc. Mineola, N.Y.
1987
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