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Mathematics

 Introduction Between 800 BCE and 800 CE, the world witnessed a shift in the intellectual centres of the time from Egypt and Babylon to the cities of the Mediterranean - Greece and Rome. The first portion of this time period, now known as the Thalassic Age, belonged to the Greeks. Initially, Greece kept itself isolated from its neighbours. Ships were used only for war, not for trade. Only when they opened their ports - and their minds - to other civilisations like the Egyptians and Babylonians did they begin to develop mathematics as a science. Unfortunately, their literary skills showed much less development and accounts of the advancements in science and technology were passed down orally for generations before being recorded. The two pioneers of Greek mathematics, Thales and Pythagoras, therefore have no surviving texts to their credit. Much of what we know from them is obtained from a summary by Proclus (fifth century CE) of the lost History of Geometry by Eudemus (fourth century BCE), who is regarded as the first mathematical historian. The reader must be warned that there is little physical evidence of these men and their deeds; in fact, they could be regarded as folklore. Early Greek Mathematics (650-400 BCE) Thales Of Miletus (ca. 624-548 BCE) Thales was, by legend, a clever man, who was said to have learned much from the Egyptians and Babylonians. He is reputed to have demonstrated that the angle inscribed in a semicircle is a right angle (Theorem of Thales) and put down a series of rules regarding the angles of triangles. He was reported to have measured the height of the Pyramids by comparing the length of their shadows to that of a vertical stick. At the moment the length of the stick’s shadow was equal to its height, the length of the Pyramids’ shadow would indicate their height. Though much of the knowledge attributed to Thales originated from Egypt and Babylon, he is credited with organising them in a rational manner. Thales also broke away from the rigidity of using geometry solely for measurement and tried to apply it in practical methods. This logical structure he provided to geometry set forth a great idea followed by later mathematicians such as Pythagoras and Plato. Pythagoras of Samos (ca. 580-500 BCE) Pythagoras was one of the most influential persons of his period, having a say not only in mathematics but also in astronomy and religion. The school he founded in 530 BCE in Croton (in present-day Italy) gave rise to a whole new school of thought, the members of which, the Pythagoreans, would later be responsible for a barrage of new discoveries and ideas in mathematics, astronomy and philosophy. Pythagoras probably proved that the plane space around a point could be divided into six equilateral triangles, four squares or three regular hexagons; and that the sum of angles of a triangle was half the central angle of a circle. However, the theorem that is most often credited to Pythagoras (the sum of the squares of the shorter sides of a right triangle is equal to the square of the third side) is in fact not his own. The Babylonians had knowledge of this fact as far back as 1900 BCE. Perhaps Pythagoras was able to provide a proof for it. Though few mathematical discoveries can be directly attributed to Pythagoras, it was the Pythagoreans who linked mathematics with everything else in the universe. They saw numbers in life, nature and religion and help make mathematics a liberal science. Early Greek Mathematics (After 400 BCE) This period saw a group of Greek mathematicians, many of the Pythagoreans, who laid the foundation for modern geometry. Hippocrates of Chios (ca. 530 BCE) described a method for constructing a quadrilateral equal in area to a lune - a figure bound by two circular arcs of unequal radii. This was the first ever representation of a curvilinear area as a quadrilateral of equal area. Around 430 BCE, Hippias of Elis discovered the quadratrix, the first curve that could be defined but could not be constructed with a straightedge and compass. Hippasus, a Pythagorean, defined the dodecahedron, a regular polyhedron with twelve faces. Some also attribute to him the proof that certain line segments (the diagonal of a unit square, for example) are incommensurable. Democritus (born ca. 460 BCE) was the first to show the relation between the volume of a cone and that of a cylinder of equal base and height. He also computed the volume of a pyramid for the Egyptians. During this time also appeared a crude method of integral calculus called the method of reduction, which involved increasing the number of sides of a rectilinear figure (such as a polygon) to approximate it to the area of a curvilinear one (such as a circle). Plato and Aristotle Plato (born ca. 420 BCE), Socrates’ famed pupil, was himself not involved in technical mathematics, but his interest in the subject as a military and philosophical tool led him to establish his Academy at Athens, which was a training ground for many pioneering mathematicians. His powerful influence helped Greece survive its defeat at the hands of the Spartans (404 BCE) and revive itself to its Old Glory. Plato’s teacher, Theodorus of Cyrene, proved around 400 BCE that the square roots of non-square integers from 3 to 17 are irrational. Eudoxus of Cnidus wrote a work on ratio and proportion. Menaechmus, a pupil of Eudoxus, was the first to discover and state the properties of conics - curves formed by the intersection of planes with cones. Dinostratus, his brother, succeeded in using Hippias’ quadratrix to “square” the circle; that is, to construct a square equal in area to a circle. Aristotle, Plato’s brightest student, was a prolific writer. Two of his books are mathematical in nature, dealing with indivisible lines and mechanics. Aristotle’s discussions on what was truly infinite in magnitude and on the importance of hypotheses in mathematics influenced many later writers. The University of Alexandria After the death of Alexander the Great (323 BCE), Egypt passed under the rule of Ptolemy I who made Alexandria a commercial and a scientific centre. Here were established the greatest library of the ancient world and its first international university. To this university is linked the names of the mathematicians who would make the few centuries from 300 BCE to the beginning of the common era the Golden Age of Greek mathematics. Undoubtedly the most prominent of these mathematicians is Euclid (born ca.300 BCE) who succeeded in encompassing the essential parts of all known mathematical knowledge of his time in one book, The Elements. The book also contained many original propositions, but it became famous for its simple and logical explanation. The book had the effect of implying to mathematicians that the field of basic geometry was all but exhausted; it was time to move on to higher geometry and mensuration. Archimedes Archimedes (born 287 BCE) is famous for having set besieging ships on fire with the aid of mirrors and the sun’s rays, and for discovering the principle of buoyancy in his bathtub. Heralded as a “genius more divine than human” by an Italian historian, his contributions to mathematics are as far-reaching as those in the field of mechanics. Among his mathematical achievements are the summation of squares of consecutive numbers, the solution of cubic equations and the computation of the area of a parabolic segment. He put down twenty-eight propositions about spiral curves in his work On Spirals. Archimedes also proved that the volume of a sphere is one-third the volume of a cylinder with the same radius and height, and that the surface area of a sphere is four times the area of its greatest circle. As the Roman invasion of Greece began during the first century of the Common Era, Greek mathematics suffered, but it did not perish. Apollonius of Pegra studies conics in detail and wrote on plane loci. Heron (ca. 50 CE) developed the famous Heron’ fomula for the area of a triangle (sq.root.(s(s-a)(s-b)(s-c))). During this time, the Greeks mastered the solution of quadratic equations. Diophantus (ca. 250 CE) was one of the last great Greek mathematicians. His principal work, the Arithmetica, covers much that is known in modern algebra. He described various methods of solving determinate and indeterminate equations. When the whole of Western Europe came under the Roman Empire, the centres of mathematical intellect shifted towards the east - China, India and the Arabs. But the contribution of the Greeks to mathematics cannot be overlooked. They set down the basics of algebra, geometry and mensuration which form a large part of mathmatics even today.

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