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MATH 520 Fall 2007 Historical Background MATH 520 – Foundations of Geometry Historical Background The word geometry is derived from the Greek words meaning “earth measure” implying that geometry involves measuring earthly things. Ancient geometry in part has its beginnings in the practical need necessary for the agriculture of the Babylonians (modern region is southern Iraq from around Baghdad to the Persian Gulf) and Egyptians, civilizations “known for their engineering prowess in marsh drainage, irrigation, flood control, and the erection of great edifices and structures” (Eves, 1969). There is evidence from clay tablets that the Babylonians (~ 1900 BC) were familiar with Pythagorean Triples long before Pythagoras lived. 1 The Egyptians used the formula ++= dbcaA ))(( to calculate the area of an arbitrary quadrilateral with 4 successive sides a, b, c, and d. This formula proves correct for rectangles but not for quadrilaterals in general. Not until the 6th century B.C. did mathematicians begin to question whether such empirical results were always true. The contemporary study of geometry bears little resemblance to its historical beginnings in that it does not necessarily require that we measure anything or even restrict ourselves to the earth. People/Characters Thales of Miletus (ca. 640 – 546 B.C.) Greek Most often credited with initiating the formal study of demonstrative geometry as a discipline – The Father of Demonstrative Geometry; A Wise Man of Antiquity Well traveled throughout Babylonia, Egypt, and the Middle East. Thales knew the work and culture of the regions. He began to generalize and demonstrate geometric propositions by using logical reasoning to argue in favor of propositions rather than simply going with the “it works so use it” method. Proclus (411-485 A.D.) stated that Thales was the first to demonstrate that a circle is bisected by the diameter. The indirect proof of this assertion by Thales is not acceptable by modern standards – Euclid avoided it – but is significant as a first attempt to justify geometric statements using reason, instead of intuition and experimentation. The work of Thales began to take hold and provided the stimulation for those who followed. “Without Thales there would not have been a Pythagoras – or such a Pythagoras; and without Pythagoras there would not have been a Plato – or such a Plato” (Smith, 1958). Pythagoras (ca. 560 – 480 B.C.) Greek Genius or Villain?? Born on the Greek Island of Samos and was probably a student of Thales. Again, very well traveled throughout the Mediterranean region and possibly even India since his philosophical orientation more aligned with Indian civilization and with the Greek. Established the quasi-religious brother hood called the Pythagoreans which existed for about 200 years. Much of the mathematics attributed to Pythagoras may in fact have been developed by the brotherhood. It is likely that while Pythagoras may have stated the theorem bearing his name, he could not prove it. The Pythagorean philosophy was to develop mathematical results exclusively as the result of deduction. Upon his death, the brotherhood split into two groups one of which helped transform mathematics into a deductive science (Calinger, 1995) and “chains of propositions were developed in which each successive proposition was derived from earlier ones” (Eves, 1969). “All is Number” - The Pythagoreans believed that Natural numbers were at the center of the universe – Male #s and Female #s, 10 planets, do not pick up what has fallen, do not to stir the fire with iron, do not look in a mirror beside a light, avoided beans, all kinds of odd stuff. Carl Sagan blames Pythagoras for setting science back hundreds of years because often results contradictory to their philosophy were stifled i.e. the square root of 2. This led to incommensurable quantities (Eudoxus dealt with this). Had Pythagoras acknowledged / dealt with the existence of irrational numbers he most likely would be viewed a great today. Hippocrates of Chios (ca. 460 – 380 B.C.) Greek The island of Chios is located near Samos so Hippocrates was most likely influenced by the Pythagorean brotherhood. Spent most of his adult life in Athens studying / teaching geometry and is credited with preparing the textbook Elements of Geometry in which he arranged theorems in a logical sequence so that later ones could be proved on the basis of earlier ones. This text, lost to us now, foreshadowed Euclid’s Elements. Plato (ca. 427 – 348 B.C.) Born into an aristocratic family in Athens. He served in the Athenian cavalry during the Peloponnesian Wars fought by Athens and its empire against the Peloponnesian League, led by Sparta. Best known as a philosopher. Established the Academy sometime around 388 B.C. “Let no man ignorant of geometry enter” Plato was an idealist; mathematics seen as a pure abstract subject which was perfect. At the Academy, mathematicians were not concerned with applications of their work only the development of mathematical thought. Thus by 350 B.C., mathematics had taken on the nature of a pure science. Mathematics is today often referred to as the Queen of the Sciences. The study of mathematics at the Academy was confined to pure mathematics with the emphasis placed on soundness of reason. Contributions to theoretical mathematics included the necessity of framing sound definitions, developing the theory of irrationals, the study of regular polyhedra, and the derivation of a formula for Pythagorean triples. Plato believed “Mathematics purifies and elevates the soul.” In his most famous treatise, the Republic, he stated “geometry will draw the soul towards the truth and create the spirit of philosophy … nothing else will be more likely to have such an effect.” 2 Eudoxus (ca. 400 – 347 B.C.) Born on the island of Cnidus in the Black Sea. Attended Plato’s Academy. None of his writings survive today though it is believed his mathematical works became the basis for Books V, VI, and XII of Euclid’s Elements. Eudoxus’ most notable achievement was his resolution of the Pythagoreans’ difficulty with incommensurable (irrational) quantities. Eudoxus was the first to formally systematize Aristotle’s theory of statements with its axioms, postulates, and definitions into what has come to be known as the Axiomatic Method. Euclid (ca. 330 – 270 B.C.) Greek May have been Greek, may have been Egyptian – not much is known of his background. Likely to have received his mathematical education at Plato’s Academy. No evidence to suggest Euclid and Plato ever met. Believed to have been the first mathematics professor at the great Museum of Alexandria in Egypt. By the time of Euclid, the development of rational thought had progressed sufficiently to allow for and demand a systematic study of geometry. Best known for Elements a collection of 13 books with 465 propositions (theorems) organized in succession from simple to complex. Elements, based on the use of a minimal set of assumptions, was the prototype for organizing mathematics into a deductive system. Elements was not perfect but it was powerful. Elements lacked undefined terms, contained proofs which used unstated postulates, and had some diagrammatical issues. When King Ptolemy I of Egypt asked Euclid if there was a shorter way to study geometry than through the Elements, Euclid replied “There is no royal road to geometry.” 3.
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