The Saga of Mathematics A Brief History Greece � In 700 BC, Greece consisted of a collection of Those Incredible independent city-states covering a large area including modern day Greece, Turkey, and a Greeks! multitude of Mediterranean islands. � The Greeks were great travelers. Chapter 3 � Greek merchant ships sailed the seas, bringing them into contact with the civilizations of Egypt, Phoenicia, and Babylon. Lewinter & Widulski The Saga of Mathematics 1 The Saga of Mathematics 2 Lewinter & Widulski Greece Greece � Also brought cultural influences like Egyptian � This prosperous Greek society accumulated geometry and Babylonian algebra and enough wealth to support a leisure class. commercial arithmetic. � Intellectuals and artists with enough time on their � Coinage in precious metals was invented around hands to study mathematics for its own sake, 700 BC and gave rise to a money economy and generally, seeking knowledge for its own based not only on agriculture but also on sake. movable goods. � They realized that non-practical activity is � This brought Magna Greece (“greater Greece”) important in the advancement of knowledge. prosperity. 3 4 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Greece The Greeks � As noted by David M. Burton in his book � Made mathematics into one discipline. The History of Mathematics, � More profound, more rational, and more � “ The miracle of Greece was not single but abstract (more remote from the uses of twofold—first the unrivaled rapidity and variety everyday life). and quality of its achievement; then its � In Egypt and Babylon, mathematics was a success in permeating and imposing its tool for practical applications or as special values on alien civilizations .” knowledge of a privileged class of scribes. 5 6 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski 1
The Saga of Mathematics A Brief History The Greeks The Greeks � Best seen in the attitude toward the square root � Made mathematics a detached intellectual of 2. subject for the connoisseur instead of being � The Babylonians computed it with high accuracy monopolized by the powerful priesthood. � The Greeks proved it was irrational � They weren’t concerned with triangular fields, � Changed the nature of the subject of but with “triangles” and the characteristics of mathematics by applying reasoning to it ⇒ “triangularity.” Proofs! � The Greeks had a preference for the abstract. � Mathematical ‘truths’ must be proven! � Mathematics builds on itself. The Saga of Mathematics 7 The Saga of Mathematics 8 Lewinter & Widulski Lewinter & Widulski The Greeks Thales of Miletus � Plato’s inscription over the door of his academy, � Born in Miletus, and lived “ Let no man ignorant of geometry enter here. ” from about 624 BC to about 547 BC. � The Greeks believed that through inquiry and logic one could understand their place in the � Thales was a merchant in universe. his younger days, a statesman in his middle � The rise of Greek mathematics begins in the life, and a mathematician, sixth century BC with Thales and Pythagoras. astronomer, and � Later reaching its zenith with Euclid, philosopher in his later Archimedes, and Apollonius. years. � Followed by Ptolemy, Pappus, and Diophantus. � Extremely successful in his business ventures. 9 10 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Thales of Miletus Thales � Thales used his skills to � Traveled to Egypt, and probably Babylon, on deduce that the next commercial ventures, studying in those places season’s olive crop would and then bringing back the knowledge he be a very large one. learned about astronomy and geometry to � He secured control of all the oil presses in Miletus Greece. and Chios in a year when � He is hailed as the first to introduce using logical olives promised to be proof based on deductive reasoning rather than plentiful, subletting them at his own rental when experiment and intuition to support an argument. the season came. 11 12 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski 2
The Saga of Mathematics A Brief History Thales Thales � Proclus states, � Founded the Ionian (Milesian) school of Greek astronomy. � “ Thales was the first to go into Egypt and bring back this learning [geometry] into � Considered the father of Greek astronomy, Greece. He discovered many propositions geometry, and arithmetic. himself and he disclosed to his successors � Thales is designated as the first the underlying principles of many others, in mathematician. some cases his methods being more general, in others more empirical. ” � The first of the Seven Sages of Greece. The Saga of Mathematics 13 The Saga of Mathematics 14 Lewinter & Widulski Lewinter & Widulski Thales Thales � His philosophy was that “Water is the � “Know thyself” and “nothing overmuch” principle, or the element, of things. All were some of Thales philosophical ideas. things are water.” � Asked what was most difficult, he said, “To � He believed that the Earth floats on water know thyself.” and all things come to be from water. � Asked what was easiest, he answered, � For him the Earth was a flat disc floating “To give advice.” on an infinite ocean. 15 16 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Thales Thales Thales is credited with proving six propositions � Thales measured the height of pyramids. � of elementary geometry: � Thales discovered how to obtain the height of A circle is bisected by its diameter. 1. pyramids and all other similar objects, namely, by The base angles of an isosceles triangle are equal. 2. measuring the shadow of the object at the time when If two straight lines intersect, the opposite angles 3. a body and its shadow are equal in length. are equal. � Thales showed how to find the distances of Two triangles are congruent if they have one side 4. and two adjacent angles equal. ships from the shore necessarily involves the The sides of similar triangles are proportional. 5. use of this theorem (iv). An angle inscribed in a semicircle is a right angle. (*) 6. 17 18 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski 3
The Saga of Mathematics A Brief History A circle is bisected by its Vertical Angles Are Equal diameter � Thales supposedly The angles between two intersecting � demonstrated that a straight lines are equal. circle is bisected by its diameter. a + b = 180º ⇒ a = 180º – b � � But Euclid did not even b + c = 180º ⇒ c = 180º – b prove this, rather he only � stated it. ∴ a = c. � b � It seems likely that Thales also only stated it rather c a than proving it. The Saga of Mathematics 19 The Saga of Mathematics 20 Lewinter & Widulski Lewinter & Widulski Alternate Interior Angles Interior Angles in a Triangle � Thales immediately drew forth new truths The sum of the angles of any triangle is � from these six principles. He observed that 180º. a line crossing two given parallel lines Draw a line through the upper vertex � makes equal angles with them. parallel to the base obtaining two pairs of alternate interior angles. b c ∴ a + b + c = 180º . c a � b a a c 21 22 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Thales Thales � An angle in a semicircle is a right angle. � One night, Thales was gazing at the sky as he walked and fell into a ditch. � α + β = 180° γ δ � A pretty servant girl lifted him out and said � 2γ + α = 180° to him “ How do you expect to understand � 2δ + β = 180° γ α β δ what is going on up in the sky if you do not � 2(δ + γ) + (α + β) = 360° even see what is at your feet .” � ∴ δ + γ = 90°. 23 24 Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski The Saga of Mathematics Lewinter & Widulski 4
Recommend
More recommend
