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Home , Pythagoras, Zeno of Citium

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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 FROM Set of 13 books titled “ITS POSSIBLEONE 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”

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“ITS POSSIBLE TO DRAW A 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 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 , Sun, , and stars orbited . Earth, in the centre of the , did not move at all.

GEOCENTRIC THEORYGEOCENTRIC 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 I T T E - M I L T

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02 “EARTH IS CENTRAL TO THE UNIVERSE” 0302

EARTH AS ECCENTRIC EARTH AS ECCENTRIC CENTER CENTER EQUANT EQUANT PYTHAGORIAN THEORUM

01 “In , the , also knownPYTHAGORIAN as Pythagoras' theorem, is a fundamental relationTHEORUM in among the three sides of a right triangle. It states that the square of the 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 . Taught by Thales. L HYPOTENISEb E G Founded Pythagoras 2 EG 2 School, Cambridge L c a Mathematics Theory Astronomy 02 “The (Greek:HYPOTENISE τετρακτύς), or tetrad, or the tetractys of the decad2 is a triangular figure 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 .” 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.”

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S RID Y C A A W S

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10 “10 INTERSECTING POINTS MAKE UP A SMALLER TRIANGLE” “10 INTERSECTING POINTS MAKE UP A “ is sometimes creditedSMALLER with the TRIANGLE” discovery of the existence of irrational . 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 SING A MUSIC IF ONLY WE HAD EARS TO HEAR ” 01

01 DIOGENES 01 01

404 BC - 323 BC

A Greek 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

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He modeled himself on the example of , and believed that 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 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 (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” 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.

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“A horse is a thing of ... None will tire of looking at him as long as he displays himself in his splendour.”

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“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 and harmonics. He was born in Gerasa, in the Roman province of Syria (now Jerash, Jordan), and was strongly influenced by Ar- istotle. He was a Pythag- orean, who wrote about the mystical properties of numbers.

A square whose side length is a triangular number can be partitioned into squares and half-squares whose areas add to cubes.

3 3 3 3 2 1 + 2 + 3 + ...+n = (1+2+3+...+n)

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T1 =1 T2 =3 T3 =6 T4 =10

The first six triangular numbers A triangular number or triangle number counts objects arranged in an equilateral Triangle, as in the diagram above. The nth triangular number is the number of dots in the triangular arrangement with n dots on a side, and is equal to the sum of the n natural numbers from 1 to 1 , the sum of the first n cubes is the square of the nth triangular number. 03 Fundamental 1st Harmonic

1st Overtone 2nd harmonic

2nd Overtone 3rd harmonic

3rd Overtone 4th harmonic

In thought, the musical scale discovered by the philosopher Pythagoras was seen as a utopian model of the harmonic order behind the structure of the and human existence. Through proportion and harmony, the musical scale bridges the gap between two extremes. It encapsulates the most fundamental pattern of harmonic and demonstrates how the phenomena of are inseparably related to one another through the principle of reciprocity. Because of these relationships embodied in its structure, the musical scale was seen as an ideal metaphor of human society by Plato and other Pythagorean thinkers, for it is based on the cosmic principles of harmony, reciprocity, and proportion, whereby each part of the whole receives its just and proper share. STRABO 01

63 BC - 23 AD

Strabo was a Greek Geographer, Philosopher, and Historian who Lived in Asia Minor During the transitional Period of the Roman into the Roman Empire

Statue of Strabo in his hometown (modern-day Amasya, Turkey), beside the River

Strabo who wrote his famous Geographia at The beginning of the Christian era and compiled his map from Travellers’ reports and the “writings” of ancients. The now lost Map by Strabo represented the sum total of cartographic before the Christian era.

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The contribution of Strabo as a scholar of great stature as philosopher, historian, And geographer, epitomizes the continuing importance of the Greek intellectual heritage - And contemporary practice - to the development of cartography in the early Roman World. As the reviser of (#112), he also illustrates the continuous way later Generations had built on the cartographic concepts first clearly set out in the Helle

03 THE WORLD ACCORDING TO

STRABO(25 BC)

EUROPA

ASIA

LIBYA

ERYTHREAN SEA ATLANTIC OCEAN ARCHIMEDES 01

287BC - 212 BC

Archimedes of Syracuse. He was a Greek , physicist, engineer, inventor, and astronomer. Although few details of his life are known, he is regarded as one of the Leading scientists in Antiquity. He proved geometrical Theorems, including the , the surface area and volume Syracusia was designed by Archimedes and built around 240 BC by Archias of Corinth of a sphere, and the area on the orders of Heron II of Syracuse. Syracusia was a 110 m ancient Greek ship some- under a parabola. times claimed to be the largest transport ship of antiquity. She was reportedly too big for any port in , and thus only sailed once from Syracuse in Sicily to in the of Egypt, whereupon she was given as a present to Ptolemy II

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15 5 15 5

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10kg

4 kg displaced

10kg

Buoyant Force Kg

Archimedes’ principle states that the upward buoyant force that is exerted on a body immersed in a fluid, whether fully or partially submerged, is equal to the weight of the fluid that the body displaces and acts in the upward direction at the centre of mass of the displaced fluid. 03

The Archimedes screw consists of a screw (a helical surface surrounding a central cylindrical shaft) inside a hollow pipe. The screw is usually turned by windmill, manual labour, cattle, or by modern means, such as a motor. As the shaft turns the bottom end scoops up a volume of water. This water is then pushed up the tube by the rotating heli- con until it pours out from the top of the tube. HERACLITUS 01

535 BC - 475BC

Heraclitus was a pre-Socratic Greek philosopher, and a native of the city of , then part of the Persian Empire. The nature of his philosophy and his stress upon the needless unconsciousness of humankind affirmed his titles as: “The Obscure” and the “Weeping Philosopher”.

Bust of Heraclitus, “The Weeping Philosopher” by Johann Christoph Ludwig Lücke ca. 1757

Heraclitus was famous for his insistence on ever-present change as being the Fundamental essence of the universe, as stated in the famous saying, “No man ever steps in the same river twice” This is commonly considered to be one of the first digressions into the philosophical concept of becoming, and has been contrasted with statement that “what is-is” as one of the first Digressions into the philosophical concept of being.

The transformation is a replacement of one element by another: “The death of fire is the birth of air, and the death of air is the birth of water.”

Heraclitus considered fire as the most fundamental element. He believed fire gave rise to the other elements and thus to all things. He regarded the as being a mixture of fire and water, with fire being the noble part of the soul, and water the ignoble part. ZENO OF 01 CITIUM

334BC - 262 BC

Zeno was the founder of the Stoic school of philos- ophy, which he taught in Athens from about 300 BC. Based on the moral ideas of the Cynics, Stoi- cism laid great emphasis on goodness and peace of mind gained from living a life of Virtue in accordance with Nature. It proved very popular, and flourished as one of the major schools of philosophy from the

Zeno, portrayed as a medieval scholar in the Nuremberg Chronicle

Following the ideas of the Academics, Zeno divided philosophy into three parts: (a very wide subject including , grammar, and the of and thought); (not just , but the divine nature of the universe as well); and , the end goal of which was to achieve happiness through the right way of living according to Nature. 02

During his lifetime, Zeno received appreciation for his philosophical and pedagogical teachings. Among other things, Zeno was honored with the golden crown and a tomb was built in honor of his moral influence on the youth of his era. 03

A crater Zeno on the Moon is named in his honour.