Pickett Barret Professor Becker Mathematical Methods 1 November 2007 Mathematics in Classical Civilizations Mathematics, it is the fundamental knowledge that binds the people of the world into a common system centered on concepts such as quantity, structure, space, and charge. This system in involves number, an abstract idea, used for counting and measuring. Today, the field of mathematics is extremely complex including multidimensional equations, integral calculus, complex and imaginary numbers . Mathematics has evolved quite extraordinarily throughout history. Ancient civilizations like the Egyptians and Babylonians built some of the world's greatest structures but only because of the advances in math. Later Greek and Roman added their own contributions, with their Pythagorean triples, Euclid geometry, and Eudoxus' method of exhaustion, an early form of integration. Ancient civilizations relied on mathematics for their superiority and used it to ensure they prospered. Probably the earliest major civilization to implement some higher level of math is the Babylonians. These were the people of Mesopotamia from the days of the early Sumerians until the beginning of the Hellenistic period . Our knowledge of Babylonian mathematics is derived from more than 400 clay tablets discovered since the 1850s. These tablets inscribed while moist and then left out in the sun or baked in an over to harden. All of these tablets were written in Cuneiform script, one of the earliest known forms or written expression. The majority of these tablets these tablets discovered date
back to around 1800 to 1600 BC, and cover various subjects such as, fractions, algebra, quadratic and cubic equations, and the Pythagorean theorem (SOME #) . The Babylonian number system was a sexagesimal (base-60) numeral system. They used this system because 60 is a highly composite number, having divisors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. From this we derive the modern day usage of 60 minutes to an hour, 60 seconds to a minute, and 360 degrees in a circle. This system makes calculations with fractions much easier, allowing for solutions to complex equations. The number system also used the place-value system, a system where digits written in the left column represent larger values, common to our base ten. The Babylonians used their sexagesimal system to give lists of square numbers up to 59 and the cubes up to 32 (SOME #). One key fact is that they did not have long division but instead used method of multiplication of the reciprocal. From this knowledge they could estimate fractions with great accuracy. Many tablets have been found with tables showing numbers and their reciprocals . The Babylonians combined their knowledge of multiplication and reciprocals to write thousands of tablets. As well as arithmetical calculations, Babylonian mathematicians also developed algebraic methods of solving equations. They used the standard quadratic formula to solve a quadratic equation. If they ever needed help finding a square root, they would just use their tables in reverse. On the topic of geometry, the Babylonian mathematicians were somewhat knowledgeable. They took the volume of a cylinder as the product of the base and height, however miscalculated the volume of a cone or pyramid as half the sums of the bases times the height. They calculated the circumference of a circle as three times the diameter and the area as one-twelfth the square of the
circumference. Some tablets also show that they were aware of the Pythagorean theorem and the use that is related to triangles. After the city of Babylon and the Mesopotamia area became a center for intellectual development, cultural diffusion helped spread new ideas and concepts to other parts of the world. The Egyptians became renowned for their proficiency in math. They were the first to develop a base 10 number system. It allowed the use of large numbers and also fractions in the form of unit fractions and Eye of Horus fractions, or binary fractions By . 2700 BC, Egyptian construction techniques included precision surveying, marking north by the sun's location at noon. By 2000 BC, records began to show approximations for pi and square roots. Abstract statements of numbers, algebra problems, written arithmetic tables, and functional applications with real weights and measure also began to appear around 2000 BC. Most of the information we know about Egyptian math comes from several major sources. The biggest and most famous source is the Rhind Mathematical Papyrus, a text containing many different elements . The Rhind includes hundreds of tables with methods and formulas for addition, subtraction, multiplication, and division of sums of unit fractions. It also contains evidence of other mathematical data, including arithmetic, geometric and harmonic means; prime and composite numbers; and how to solve first order linear equations as well as summing arithmetic and geometric series . Other sources with similar information include the Moscow Mathematical Papyrus, the Reisner Papyrus, and the Berlin papyrus. Egyptian arithmetic was achieved by employing the method of doubling and halving a known number to approach a solution. One of their units was a called a hekat, valued as 64/64. It was a volume unit used to measure grain,
bread, and beer. They also had a unit called ro used to describe remainders of quotients. The used their proficient knowledge in both mathematics and astronomy to build some of the worlds greatest pieces of architecture. The pyramids of Giza, temples at Karnak and Luxor, and the hall of Medinet Habu demonstrate the full use of the Egyptian mathematics. The Egyptians set the standard for successful civilizations and helped spread new mathematical concepts and discoveries to the rest of the world . Another ancient civilization that developed higher mathematical skills was the Greeks. Greek mathematics contributed significantly to ideas on number theory, mathematical analysis, applied mathematics, and concepts similar to integral calculus . The origins go back to the 7th century BC. It is believed that computational methods were built from earlier Babylonian and Egyptian mathematics. Famous Greek mathematicians include Pythagoras, a vague figure from the isle of Samos known for his work with number mysticism and numerology, but more commonly with this theorem, and Euclid, who is known for his Elements, a canon of geometry of many centuries . One of the most characteristic products of Greek mathematics is the theory of conic sections, largely developed in the Hellenistic period. The methods used made no explicit use of algebra or trigonometry. Developments in math took place in Greek-speaking centers as far away as Egypt and Sicily. Greek mathematics and astronomy reached a rather advanced stage during Hellenism, with scholars such as Hipparchus, Posidonius and Ptolemy, capable of the construction of simple analogue computers such as the Antikythera mechanism. Greeks also had their own famous architecture including the Parthenon, the Hephaesteum at Athens, the group at Paestum, the temple complex at Slinunte, and the sanctuaries at
Agrigentum. The Greeks calculated angles and measurements vital to the structure's sturdiness. Without math, many of the Greek's accomplishments would be quite impossible. These three ancient civilizations demonstrated the power of mathematics and what it can do to help society. Today our understanding of mathematics is way beyond what's its ever been and it is still continuing to evolve. With the explosion of technology in the 21st century, I'm certain math will continue to be a vital part of our daily life and help our world achieve goals that know ones even thought of.
Works Cited Boyer, C. B.. A History of Mathematics. 2nd Ed. New York: Wiley, 1989. Clagett, Marshall. Ancient Egyptian Science: A Source Book. Philadelphia: American Philosophical Society, 1999. Neugebaur, O.. Exact Sciences of Antiquity. 2nd Ed. Dover: Courier Dover, 1969.