Ancient Advances in MathematicsAncient knowledge of the sciences was often wrong and whollyunsatisfactory by modern standards.

However not all of the knowledge of themore learned peoples of the past was false. In fact without people like Euclidor Plato we may not have been as advanced in this age as we are. Mathematics isan adventure in ideas. Within the history of mathematics, one finds the ideasand 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 justcoincidence that man just so happens to have ten fingers or ten toes, for whenour primitive ancestors first discovered the need to count they definitely wouldhave used their fingers to help them along just like a child today.

Whenprimitive man learned to count up to ten he somehow differentiated himself fromother animals. As an object of a higher thinking, man invented ten number-sounds. The needs and possessions of primitive man were not many. When theneed to count over ten aroused, he simply combined the number-sounds relatedwith his fingers. So, if he wished to define one more than ten, he simply saidone-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 basicnumber-sounds to the ten primary ones. They are “hundred,” “thousand,” “million,” “billion” (a thousand millions in America, a million millions inEngland), “trillion” (a million millions in America, a million-million millionsin England). Because primitive man invented the same number of number-sounds ashe 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 thirteennumber system we would call fifteen, two-thirteen’s. While some intelligent andwell-schooled scholars might argue whether or not base ten is the most adequatenumber system, base ten is the irreversible favorite among all the nations.

Of course, primitive man most certainly did not realize the concept ofthe number system he had just created. Man simply used the number-soundsloosely as adjectives. So an amount of ten fish was ten fish, whereas ten is anadjective describing the noun fish. Soon the need to keep tally on one’s counting raised.

The simplesolution was to make a vertical mark. Thus, on many caves we see a number ofmarks that the resident used to keep track of his possessions such a fish orknives. This way of record keeping is still taught today in our schools underthe name of tally marks. The earliest continuous record of mathematical activity is from thesecond millennium BC When one of the few wonders of the world were createdmathematics was necessary.

Even the earliest Egyptian pyramid proved that themakers had a fundamental knowledge of geometry and surveying skills. Theapproximate time period was 2900 BCThe first proof of mathematical activity in written form came about onethousand years later. The best known sources of ancient Egyptian mathematics inwritten format are the Rhind Papyrus and the Moscow Papyrus. The sourcesprovide undeniable proof that the later Egyptians had intermediate knowledge ofthe following mathematical problems: applications to surveying, salarydistribution, calculation of area of simple geometric figures’ surfaces andvolumes, simple solutions for first and second degree equations. Egyptians used a base ten number system most likely because of biologicreasons (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, whichis Latin for finger, is also another name for numbers which explains theinfluence of fingers upon numbers once again. The Egyptians produced a more complex system then the tally system forrecording amounts. Hieroglyphs stood for groups of tens, hundreds, andthousands. The higher powers of ten made it much easier for the Egyptians tocalculate into numbers as large as one million. Our number system which is bothdecimal and positional (52 is not the same value as 25) differed from theEgyptian which was additive, but not positional.

The Egyptians also knew more of pi then its mere existence. They foundpi to equal C/D or 4(8/9) whereas a equals 2. The method for ancient peoplesarriving at this numerical equation was fairly easy. They simply counted howmany times a string that fit the circumference of the circle fitted into thediameter, thus the rough approximation of 3. The biblical value of pi can be found in the Old Testament (I Kingsvii.

23 and 2 Chronicles iv. 2)in the following verse”Also, he made a molten sea of ten cubits frombrim to brim, round in compass, and five cubitsthe height thereof; and a line of thirty cubits didcompass it round about. “The molten sea, as we are told is round, and measures thirty cubitsround 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 dealingwith fractions. The Babylonians also succeeded in developing moresophisticated base ten arithmetic that were positional and they also storedmathematical records on clay tablets. Despite all this, the greatest and most remarkable feature of BabylonianMathematics was their complex usage of a sexagesimal place-valued system inaddition a decimal system much like our own modern one. The Babylonians countedin both groups of ten and sixty. Because of the flexibility of a sexagismalsystem with fractions, the Babylonians were strong in both algebra and numbertheory.

Remaining clay tablets from the Babylonian records show solutions tofirst, second, and third degree equations. Also the calculations of compoundinterest, squares and square roots were apparent in the tablets. The sexagismal system of the Babylonians is still commonly in usagetoday. Our system for telling time revolves around a sexagesimal system. Thesame 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 (60mina). This sexagesimal arithmetic was used in commerce and in astronomy. The Babylonians used many of the more common cases of the PythagoreanTheorem for right triangles.

They also used accurate formulas for solving theareas, volumes and other measurements of the easier geometric shapes as well astrapezoids. The Babylonian value for pi was a very rounded off three. Becauseof this crude approximation of pi, the Babylonians achieved only rough estimatesof the areas of circles and other spherical, geometric objects. The real birth of modern math was in the era of Greece and Rome. Notonly did the philosophers ask the question “how” of previous cultures, but theyalso asked the modern question of “why. ” The goal of this new thinking was todiscover and understand the reason for mans’ existence in the universe and alsoto find his place.

The philosophers of Greece used mathematical formulas toprove 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 negatedabstract side of mathematics, of with the concept of the mathematical proof. The Greeks were interested not only in the application of mathematicsbut also in its philosophical significance, which was especially appreciated byPlato (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 philosophyand personal ethics. Within the walls of Plato’s academy at least three greatmathematicians were taught, Theaetetus, known for the theory of irrational,Eodoxus, the theory of proportions, and also Archytas (I couldn’t find what madehim great, but three books mentioned him so I will too). Indeed the motto ofPlato’s academy “Let no one ignorant of geometry enter within these walls” wasfitting for the scene of the great minds who gathered here. Another great mathematician of the Greeks was Pythagoras who providedone of the first mathematical proofs and discovered incommensurable magnitudes,or irrational numbers.

The Pythagorean theorem relates the sides of a righttriangle with their corresponding squares. The discovery of irrationalmagnitudes had another consequence for the Greeks: since the length ofdiagonals of squares could not be expressed by rational numbers in the form ofA over B, the Greek number system was inadequate for describing them. As, you might have realized, without the great minds of the past ourmathematical 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 herethe only way is up,” so you might say that from now, 1996, the future ofmathematics can only improve for the better. BibliographyBall, W. W.

Rouse. A Short Account of The History of Mathematics. DoverPublications Inc. Mineloa, N. Y.

1985Beckmann, Petr. A History of Pi. St. Martin’s Press.

New York, N. Y. 1971De Camp, L. S. The Ancient Engineers. Double Day.

Garden City, N. J. 1963Hooper, Alfred. Makers of Mathematics. Random House.

New York, N. Y. 1948Morley, 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

However not all of the knowledge of themore learned peoples of the past was false. In fact without people like Euclidor Plato we may not have been as advanced in this age as we are. Mathematics isan adventure in ideas. Within the history of mathematics, one finds the ideasand 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 justcoincidence that man just so happens to have ten fingers or ten toes, for whenour primitive ancestors first discovered the need to count they definitely wouldhave used their fingers to help them along just like a child today.

Whenprimitive man learned to count up to ten he somehow differentiated himself fromother animals. As an object of a higher thinking, man invented ten number-sounds. The needs and possessions of primitive man were not many. When theneed to count over ten aroused, he simply combined the number-sounds relatedwith his fingers. So, if he wished to define one more than ten, he simply saidone-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 basicnumber-sounds to the ten primary ones. They are “hundred,” “thousand,” “million,” “billion” (a thousand millions in America, a million millions inEngland), “trillion” (a million millions in America, a million-million millionsin England). Because primitive man invented the same number of number-sounds ashe 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 thirteennumber system we would call fifteen, two-thirteen’s. While some intelligent andwell-schooled scholars might argue whether or not base ten is the most adequatenumber system, base ten is the irreversible favorite among all the nations.

Of course, primitive man most certainly did not realize the concept ofthe number system he had just created. Man simply used the number-soundsloosely as adjectives. So an amount of ten fish was ten fish, whereas ten is anadjective describing the noun fish. Soon the need to keep tally on one’s counting raised.

The simplesolution was to make a vertical mark. Thus, on many caves we see a number ofmarks that the resident used to keep track of his possessions such a fish orknives. This way of record keeping is still taught today in our schools underthe name of tally marks. The earliest continuous record of mathematical activity is from thesecond millennium BC When one of the few wonders of the world were createdmathematics was necessary.

Even the earliest Egyptian pyramid proved that themakers had a fundamental knowledge of geometry and surveying skills. Theapproximate time period was 2900 BCThe first proof of mathematical activity in written form came about onethousand years later. The best known sources of ancient Egyptian mathematics inwritten format are the Rhind Papyrus and the Moscow Papyrus. The sourcesprovide undeniable proof that the later Egyptians had intermediate knowledge ofthe following mathematical problems: applications to surveying, salarydistribution, calculation of area of simple geometric figures’ surfaces andvolumes, simple solutions for first and second degree equations. Egyptians used a base ten number system most likely because of biologicreasons (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, whichis Latin for finger, is also another name for numbers which explains theinfluence of fingers upon numbers once again. The Egyptians produced a more complex system then the tally system forrecording amounts. Hieroglyphs stood for groups of tens, hundreds, andthousands. The higher powers of ten made it much easier for the Egyptians tocalculate into numbers as large as one million. Our number system which is bothdecimal and positional (52 is not the same value as 25) differed from theEgyptian which was additive, but not positional.

The Egyptians also knew more of pi then its mere existence. They foundpi to equal C/D or 4(8/9) whereas a equals 2. The method for ancient peoplesarriving at this numerical equation was fairly easy. They simply counted howmany times a string that fit the circumference of the circle fitted into thediameter, thus the rough approximation of 3. The biblical value of pi can be found in the Old Testament (I Kingsvii.

23 and 2 Chronicles iv. 2)in the following verse”Also, he made a molten sea of ten cubits frombrim to brim, round in compass, and five cubitsthe height thereof; and a line of thirty cubits didcompass it round about. “The molten sea, as we are told is round, and measures thirty cubitsround 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 dealingwith fractions. The Babylonians also succeeded in developing moresophisticated base ten arithmetic that were positional and they also storedmathematical records on clay tablets. Despite all this, the greatest and most remarkable feature of BabylonianMathematics was their complex usage of a sexagesimal place-valued system inaddition a decimal system much like our own modern one. The Babylonians countedin both groups of ten and sixty. Because of the flexibility of a sexagismalsystem with fractions, the Babylonians were strong in both algebra and numbertheory.

Remaining clay tablets from the Babylonian records show solutions tofirst, second, and third degree equations. Also the calculations of compoundinterest, squares and square roots were apparent in the tablets. The sexagismal system of the Babylonians is still commonly in usagetoday. Our system for telling time revolves around a sexagesimal system. Thesame 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 (60mina). This sexagesimal arithmetic was used in commerce and in astronomy. The Babylonians used many of the more common cases of the PythagoreanTheorem for right triangles.

They also used accurate formulas for solving theareas, volumes and other measurements of the easier geometric shapes as well astrapezoids. The Babylonian value for pi was a very rounded off three. Becauseof this crude approximation of pi, the Babylonians achieved only rough estimatesof the areas of circles and other spherical, geometric objects. The real birth of modern math was in the era of Greece and Rome. Notonly did the philosophers ask the question “how” of previous cultures, but theyalso asked the modern question of “why. ” The goal of this new thinking was todiscover and understand the reason for mans’ existence in the universe and alsoto find his place.

The philosophers of Greece used mathematical formulas toprove 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 negatedabstract side of mathematics, of with the concept of the mathematical proof. The Greeks were interested not only in the application of mathematicsbut also in its philosophical significance, which was especially appreciated byPlato (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 philosophyand personal ethics. Within the walls of Plato’s academy at least three greatmathematicians were taught, Theaetetus, known for the theory of irrational,Eodoxus, the theory of proportions, and also Archytas (I couldn’t find what madehim great, but three books mentioned him so I will too). Indeed the motto ofPlato’s academy “Let no one ignorant of geometry enter within these walls” wasfitting for the scene of the great minds who gathered here. Another great mathematician of the Greeks was Pythagoras who providedone of the first mathematical proofs and discovered incommensurable magnitudes,or irrational numbers.

The Pythagorean theorem relates the sides of a righttriangle with their corresponding squares. The discovery of irrationalmagnitudes had another consequence for the Greeks: since the length ofdiagonals of squares could not be expressed by rational numbers in the form ofA over B, the Greek number system was inadequate for describing them. As, you might have realized, without the great minds of the past ourmathematical 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 herethe only way is up,” so you might say that from now, 1996, the future ofmathematics can only improve for the better. BibliographyBall, W. W.

Rouse. A Short Account of The History of Mathematics. DoverPublications Inc. Mineloa, N. Y.

1985Beckmann, Petr. A History of Pi. St. Martin’s Press.

New York, N. Y. 1971De Camp, L. S. The Ancient Engineers. Double Day.

Garden City, N. J. 1963Hooper, Alfred. Makers of Mathematics. Random House.

New York, N. Y. 1948Morley, 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