Hellenistic Mathematics

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HELLENISTIC MATHEMATICS By the 3rd Century BC, in the wake of the conquests of Alexander the Great, mathematical breakthroughs were also beginning to be made on the edges of the Greek Hellenistic empire. In particular, Alexandria in Egypt became a great centre of learning under the beneficent rule of the Ptolemies, and its famous Library soon gained a reputation to rival that of the Athenian Academy. The patrons of the Library were arguably the first professional scientists, paid for their devotion to research. Among the best known and most influential mathematicians who studied and taught at Alexandria were Euclid, Archimedes, Eratosthenes, Heron, Menelaus and Diophantus. During the late 4th and early 3rd Century BC, Euclid was the great chronicler of the The Sieve of Eratosthenes mathematics of the time, and one of the most influential teachers in history. He virtually invented classical (Euclidean) geometry as we know it. Archimedes spent most of his life in Syracuse, Sicily, but also studied for a while in Alexandria. He is perhaps best known as an engineer and inventor but, in the light of recent discoveries, he is now considered of one of the greatest pure mathematicians of all time. Eratosthenes of Alexandria was a near contemporary of Archimedes in the 3rd Century BC. A mathematician, astronomer and geographer, he devised the first system of latitude and longitude, and calculated the circumference of the earth to a remarkable degree of accuracy. As a mathematician, his greatest legacy is the Sieve of Eratosthenes algorithm for identifying prime numbers.

It is not known exactly when the great Library of Alexandria burned down, but Alexandria remained an important intellectual centre for some centuries. In the 1st century BC, Heron (or Hero) was another great Alexandrian inventor, best known in mathematical circles for Heronian triangles (triangles with integer sides and integer area), Herons Formula for finding the area of a triangle from its side lengths, and Herons Method for iteratively computing a square root. He was also the first mathematician to confront at least the idea of -1 (although he had no idea how to treat it, something which had to wait for Tartaglia and Cardano in the 16th Century). Menelaus of Alexandria, who Menelaus of Alexandria introduced the concept of spherical triangle lived in the 1st - 2nd Century AD, was the first to recognize geodesics on a curved surface as the natural analogues of straight lines on a flat plane. His book Sphaerica dealt with the geometry of the sphere and its application in astronomical measurements and calculations, and introduced the concept of spherical triangle (a figure formed of three great circle arcs, which he named "trilaterals"). In the 3rd Century AD, Diophantus of Alexandria was the first to recognize fractions as numbers, and is considered an early innovator in the field of what would later become known as algebra. He applied himself to some quite complex algebraic problems, including what is now known as Diophantine Analysis, which deals with finding integer solutions to kinds of problems that lead to equations in several unknowns (Diophantine equations). Diophantus Arithmetica, a collection of problems giving numerical solutions of both determinate and indeterminate equations, was the most prominent work on algebra in all Greek mathematics, and his problems exercised the minds of many of the world's best mathematicians for much of the next two millennia.

But Alexandria was not the only centre of learning in the Hellenistic Greek empire. Mention should also be made of Apollonius of Perga (a city in modern-day southern Turkey) whose late 3rd Century BC work on geometry (and, in particular, on conics and conic sections) was very influential on later European mathematicians. It was Apollonius who gave the ellipse, the parabola, and the hyperbola the names by which we know them, and showed how they could be derived from different sections through a Conic cone.

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Hipparchus, who was also from Hellenistic Anatolia and who live in the 2nd Century BC, was perhaps the greatest of all ancient astronomers. He revived the use of arithmetic techniques first developed by the Chaldeans and Babylonians, and is usually credited with the beginnings of trigonometry. He calculated (with remarkable accuracy for the time) the distance of the moon from the earth by measuring the different parts of the moon visible at different locations and calculating the distance using the properties of triangles. He went on to create the first table of chords (side lengths corresponding to different angles of a triangle). By the time of the great Alexandrian astronomer Ptolemy in the 2nd Century AD, however, Greek mastery of numerical procedures had progressed to the point where Ptolemy was able to include in his Almagest a table of trigonometric chords in a circle for steps of which (although expressed sexagesimally in the Babylonian style) is accurate to about five decimal places. By the middle of the 1st Century BC and thereafter, however, the Romans had tightened their grip on the old Greek empire. The Romans had no use for pure mathematics, only for its practical applications, and the Christian regime that followed it even less so. The final blow to the Hellenistic mathematical heritage at Alexandria might be seen in the figure of Hypatia, the first recorded female mathematician, and a renowned teacher who had written some respected commentaries on Diophantus and Apollonius. She was dragged to her death by a Christian mob in 415 AD.

HELLENISTIC MATHEMATICS - EUCLID The Greek mathematician Euclid lived and flourished in Alexandria in Egypt around 300 BC, during the reign of Ptolemy I. Almost nothing is known of his life, and no likeness or firsthand description of his physical appearance has survived antiquity, and so depictions of him (with a long flowing beard and cloth cap) in works of art are necessarily the products of the artist's imagination. He probably studied for a time at Plato's Academy in Athens but, by Euclid's time, Alexandria, under the patronage of the Ptolemies and with its prestigious and comprehensive Library, had already become a worthy rival to the great Academy. Euclid is often referred to as the Father of Geometry, and he wrote perhaps the most important and successful mathematical textbook of all time, the Stoicheion or Elements, which Euclid (c.330-275 BC, fl. c.300 represents the culmination of the mathematical revolution which BC) had taken place in Greece up to that time. He also wrote works on the division of geometrical figures into into parts in given ratios, on catoptrics (the mathematical theory of mirrors and reflection), and on spherical astronomy (the determination of the location of objects on the "celestial sphere"), as well as important texts on optics and music. The "Elements was a lucid and comprehensive compilation and explanation of all the known mathematics of his time, including the work of Pythagoras, Hippocrates, Theudius, Theaetetus and Eudoxus. In all, it contains 465 theorems and proofs, described in a clear, logical and elegant style, and using only a compass and a straight edge. Euclid reworked the mathematical concepts of his predecessors into a consistent whole, later to become known as Euclidean geometry, which is still as valid Euclids method for constructing of an equilateral triangle from a today as it was 2,300 years ago, given straight line segment AB using only a compass and straight even in higher mathematics edge was Proposition 1 in Book 1 of the "Elements" dealing with higher dimensional spaces. It was only with the work of Bolyai, Lobachevski and Riemann in the first half of the 19th Century that any kind of non-Euclidean geometry was even considered.

The "Elements remained the definitive textbook on geometry and mathematics for well over two millennia, surviving the eclipse in classical learning in Europe during the Dark Ages through Arabic translations. It set, for all time, the model for mathematical argument, following logical deductions from inital assumptions (which Euclid called axioms and "postulates") in order to establish proven theorems. Euclids five general axioms were: 1. 2. 3. 4. 5. Things which are equal to the same thing are equal to each other. If equals are added to equals, the wholes (sums) are equal. If equals are subtracted from equals, the remainders (differences) are equal. Things that coincide with one another are equal to one another. The whole is greater than the part.

His five geometrical postulates were: 1. It is possible to draw a straight line from any point to any point. 2. It is possible to extend a finite straight line continuously in a straight line (i.e. a line segment can be extended past either of its endpoints to form an arbitrarily large line segment). 3. It is possible to create a circle with any center and distance (radius). 4. All right angles are equal Euclids Postulates (1 5) to one another (i.e. "half" of a straight angle). 5. If a straight line crossing two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which the angles are less than the two right angles.

Among many other mathematical gems, the thirteen volumes of the Elements contain formulas for calculating the volumes of solids such as cones, pyramids and cylinders; proofs about geometric series, perfect numbers and primes; algorithms for finding the greatest common divisor and least common multiple of two numbers; a proof and generalization of Pythagoras Theorem, and proof that there are an infinite number of Pythagorean Triples; and a final definitive proof that there can be only five possible regular Platonic Solids. However, the Elements also includes a series of theorems on the properties of n