RAPHAEL’S SCHOOL OF ATHENS 1509 - 1511 05 10 03 09 06 01 EUCLID 02 PTOLEMY 03 PYTHAGORAS 04 DIOGENES 05 XENPHON 06 NICOMACHUS 07 STRABO 08 ARCHIMEDES 09 HERACLITUS 10 ZENO OF CITIUM 07 08 02 04 01 5 AXIOM (FOUNDATION)5 AXIOM EUCLID (FOUNDATION)5 AXIOMS 5(FOUNDATION) 5AXIOM AXIOM - 285 BC 5 AXIOM (FOUNDATION)01(FOUNDATION) “ITS(FOUNDATION) POSSIBLE TO JOIN A STRAIGHT LINE FROM Set of 13 books titled “ITS POSSIBLEONE POINT TO TOJOIN ANY A STRAIGHT OTHER POINT” LINE FROM ‘Elements’. These are a ONE POINT TO ANY OTHER POINT” series of statements & proofs arranged into a “IT IS“ITS POSSIBLE“ITS POSSIBLE POSSIBLE TO JOIN A TO STRAIGHT TO JOIN JOIN LINE A A STRAIGHTFROM STRAIGHT ONE POINT LINE LINE TO ANYFROM FROM OTHER POINT” formal system. “ITS POSSIBLEONEONE POINT POINT TO TO JOINTO ANY ANY A OTHER STRAIGHT OTHER POINT” POINT” LINE FROM 02 “ITS POSSIBLEONE POINT TO EXTEND TO ANY A OTHER LINE IN POINT” ANY Known as plane “ITS POSSIBLE TO EXTEND A LINE IN ANY Geometry these rules DIRECTION INFINATELY” apply to Planar space. DIRECTION INFINATELY” “ITS“ITS POSSIBLE POSSIBLE TO TO EXTEND EXTEND A ALINE LINE IN IN ANY ANY “IT IS POSSIBLE“ITS POSSIBLE TODIRECTION EXTENDDIRECTION A TOSTRAIGHT EXTENDINFINATELY” INFINATELY” LINE IN A ANY LINE DIRECTION IN ANY INFINITELY” DIRECTION INFINATELY” 03 “ITS POSSIBLE TO DRAW A CIRCLE WITH A “ITS POSSIBLECENTRE POINTTO DRAW & ANY A CIRCLE RADIUS” WITH A CENTRE POINT & ANY RADIUS” “IT IS POSSIBLE TO DRAW A CIRCLE WITH A CENTRE POINT & ANY RADIUS” “ITS“ITS POSSIBLE POSSIBLE TO TO DRAW DRAW A ACIRCLE CIRCLE WITH WITH A A 04 “ITS CENTREPOSSIBLECENTRE POINT POINTTO DRAW & &ANY ANY A RADIUS”CIRCLE RADIUS” WITH A CENTRE POINT & ANY RADIUS” “ALL“ALL RIGHT RIGHT AGLESANGLES ARE ARE EQUAL EQUAL OR CONGRUENT” OR “ALL RIGHTCONGRUENT” AGLES ARE EQUAL OR 05 CONGRUENT” “ALL“ALL RIGHT RIGHT AGLES AGLES ARE ARE EQUAL EQUAL OR OR “ALL RIGHTCONGRUENT”CONGRUENT” AGLES ARE EQUAL OR CONGRUENT” SS CRO S ILL OS S W CR INE ILL L ES W LIN SOSSS C RCOR S ILLIL L OS S WS W CR INIENE ILL L L ES W LIN “SUM OF THE INTERIOR ANGLE IS LESS THAN 90 (X)” “SUM OF THE INTERIOR ANGLE IS LESS THAN “SUM OF THE INTERIOR90 (X)” ANGLE IS LESS THAN 90 (X)” “SUM“SUM OF OF THE THE INTERIOR INTERIOR ANGLE ANGLE IS IS LESS LESS THAN THAN “SUM OF THE INTERIOR9090 (X)” (X)” ANGLE IS LESS THAN 90 (X)” PTOLEMY 01 100 AD - 160 AD Earth-centered, or Geocentric. Ptolemy thought that all celestial Objects including the Planets, Sun, Moon, and stars orbited Earth. Earth, in the centre of the Universe, did not move at all. GEOCENTRIC THEORYGEOCENTRIC THEORY 01 “A representation“EARTH IS of CENTRALPtolemy fromthe TO Blaeu THE Atlas UNIVERSE” Maior (1662)” 01 “EARTH IS CENTRAL TO THE UNIVERSE” LE C 02 IR E C L C O IR T C E O M T I T E - M I L T - L 02 “EARTH IS CENTRAL TO THE UNIVERSE” 0302 EARTH AS ECCENTRIC EARTH AS ECCENTRIC CENTER CENTER EQUANT EQUANT PYTHAGORIAN THEORUM 01 “In mathematics, the Pythagorean theorem, also knownPYTHAGORIAN as Pythagoras' theorem, is a fundamental relationTHEORUM in Euclidean geometry among the three sides of a right triangle. It states that the square of the hypotenuse is equal to the sum of the 01 “In mathematics, the Pythagorean theorem, also squares of the other two sides” known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of2 a right triangle.L It states that the square PYTHAGORAS PYTHAGOREAN b E THEOREM of the hypotenuseG is equal to the sum of the EG 2 570BC - 495BC L squares of thea other two sides” 01 Lived on the Greek isle of 2 Samos. Taught by Thales. L HYPOTENISEb E G Founded Pythagoras 2 EG 2 School, Cambridge L c a Mathematics Music Theory Astronomy 02 “The tetractys (Greek:HYPOTENISE τετρακτύς), or tetrad, or the Philosophy tetractys of the decad2 is a triangular figure Religion consisting of ten pointsc arranged in four rows: “In Mathematics,one, the two, Pythagorean three, Theorem,and four also points known inas eachPythagoras row, Theorem is a fundamentalwhich relation is in the Euclidean geometrical Geometry representationamong the three sides of ofthe a triangle . It states that02 the square“The of the tetractys hypotenuse (Greek: is equal to τετρακτύς the sum of), the or squares tetrad, of theor theother fourth triangular number.” tetractys oftwo the side’s.” decad is a triangular figure 02 consisting of ten points arranged in four rows: one, two, three, and four points in each row, which is the geometrical representation of the fourth triangular number.” RID C A S Y R E V S RID Y C A A W S Y 10 R E V S Y A W 10 “10 INTERSECTING POINTS MAKE UP A SMALLER TRIANGLE” “10 INTERSECTING POINTS MAKE UP A “Hippasus is sometimes creditedSMALLER with the TRIANGLE” discovery of the existence of irrational numbers. Pythagorean preached that all numbers could be expressed as the ratio of integers and the discovery of irrational numbers is said to have shocked them” 03 “Irrational numbers. Hippasus is sometimes “The tetracyclic,credited or tetrad, with or the the tetractys discovery of the decad of the is aexistence triangular figure of consisting of ten points arranged in four“10 rows: INTERSECTING one, two, three, andPOINTS four points MAKE in each UP row, A which is theirrational geometrical numbers, representation following of the fourth which triangular he wasnumber.” drowned at sea. PythagoreansSMALLER TRIANGLE” preached that all numbers could be expressed as the ratio of 03 “Irrational numbers. Hippasus is sometimes 03 integers, and the discovery of irrational numbers credited with the discovery of the existence of is said to have shocked them..” irrational numbers, following which he was drowned at sea. Pythagoreans preached that all numbers could be expressed as the ratio of integers, and the discovery of irrational numbers is said to have shocked them..” “THE STARS AND THE HEAVENS SING A MUSIC IF ONLY WE HAD EARS TO HEAR ” 01 01 DIOGENES 01 01 404 BC - 323 BC A Greek Philosopher and one of the founders of Cynic Philosophy. He was born in Si-nope, an Ionian colony on the Black Sea. Diogenes was a controversial figure. His father minted coins for a “FOUND BEGGING IN A CIRAMIC JAR” living, and Diogenes was banished from Sinope when he took to Containers used among the civilizations that bordered the Debasement of Currency. Mediterranean“FOUND Sea inBEGGING the Neolithic, IN A CIRAMIC the Bronze JAR” Age and the After being exiled, he succeeding Iron Age. Pithoi had been used for bulk storage, moved to Athens and primarily for“FOUND fluids andBEGGING grains; IN they A CIRAMIC were comparable JAR” to the criticized many cultural Containers“FOUNDdrums, used BEGGING barrels among andIN the A caskscivilizationsCERAMIC of recent JAR that ” times. bordered the conventions of the city. Mediterranean Sea in the Neolithic, the Bronze Age and the Containers used among the civilizations that bordered the move at all. Containers used succeedingamong the civilisationsIron Age. Pithoi that hadbordered been theused Mediterranean for bulk storage, Sea in the Mediterranean Sea in the Neolithic, the Bronze Age and the Neolithic, the Bronzeprimarily Age and for thefluids succeeding and grains; Iron they Age. were Pithoi comparable had been toused the for bulk succeeding Iron Age. Pithoi had been used for bulk storage, storage, primarily02 for fluids drums,and grains; barrels they and were casks comparable of recent to times. the drums, barrels and primarily for fluidscasks and of recent grains; times” they were comparable to the drums, barrels and casks of recent times. 02 02 02 A B A B A B He modeled himself on the example of Heracles, and believed that virtue was better revealed in action than in theory 03He modeled himself on the example of Heracles, and believed “HE MODELLED HIMSELFthat virtue ON was THE better EXAMPLE revealed OF CORACLES,in action than AND in theoryBELIEVED THAT VIRTUEHe modeledWAScriticized BETTER himself Plato REVEALED (A), on disputedthe example IN ACTION his interpretationof Heracles, THAN IN andTHEORY” of believedSocrates (B), andthat sabotagedvirtue was betterhis lectures,sometimes revealed in action distracting than in theory attenders 03 by bringing food and eating during the discussions. 03 03He criticized Plato (A), disputed his interpretation of Socrates (B), and sabotaged his lectures,sometimes distracting attenders He criticizedby bringing Plato food (A), and disputed eating his during interpretation the discussions. of Socrates (B), and sabotaged his lectures,sometimes distracting attenders by bringing food and eating during the discussions. He criticized Plato (A), disputed his interpretation of Socrates (B), and sabotaged his lectures, sometimes distracting attenders by bringing food and eating during the discussions” XENOPHON 01 430 BC - 354 BC SCYTHIANS BACK SEA Greek Historian, Soldier & Mercenary Student of Socrates. Born in Athens. Sardes Trapezus MED SEA SYRIA LIBYA “In 401 BC he was convinced by his Boeotian friend Pyroxenes to participate in the military expedition led by Cyrus the Younger against his elder brother, King Artaxerxes II of Persia. 01 “A horse is a thing of beauty... None will tire of looking at him as long as he displays himself in his splendour.” 03 “Excess of grief for the dead is madness; for it is an injury to the living, and the dead know it not.’ NICOMACHUS THEOREM 01 60 AD - 100AD Mathematician best known for Arithmetic and harmonics. He was born in Gerasa, in the Roman province of Syria (now Jerash, Jordan), and was strongly influenced by Ar- istotle.