ANCIENT GREEK MATHEMATICS

BY:FARHANIS SYAMIMI ZOOLKAFLILAILI AZLIN ISMAILNOR AZURA MOHD HASHIMNOR IZZAH ISANORMALA AHMAD

Map of Greece

The Greeks had only very clumsy ways of writing down numbers, they didn't like algebra. Instead, Greek mathematicians were more focused on geometry, and used geometric methods to solve problems.

Greek mathematicians were also very interested in proving that certain mathematical ideas were true. So they spent a lot of time using geometry to prove that things were always true.

The Greeks in general were very interested in rationality, in things making sense and hanging together. They wanted to tie up the loose ends. They liked music, because music followed strict rules to produce beauty. So did architecture, and so did mathematics.

Some famous Greek mathematicians were Pythagoras, Aristotle, Anaxagoras, Euclid, Archimedes, Thales, and Aristarchus.

Though the Greeks certainly borrowed from other civilizations, they built a culture and civilization on their own which is The most impressive of all civilizations, The most influential in Western culture, The most decisive in founding mathematics as we

know it.

By far, the most significant development in mathematics was giving it firm logical foundations. This took place in ancient Greece in the centuries preceding Euclid. See Euclid's Elements. Logical foundations give mathematics more than just certainty-they are a tool to investigate the unknown.

CURRENCY

Before 600 B.C. there was no monetary system in Greece, so they utilized the barter system. This was a system of trading goods or services.

By 500 B.C., each city-state began minting their own coin. A merchant usually only took coins from their own city. Visitors had to find a moneychanger to exchange their coins.

Typically a 5 or 6 percent fee was charged to exchange foreign currency to the local currency.

Athens used a currency known as the drachma (from the Greek word for "a handful").

Their currency was widely used because of the large trade network that they developed.

Athenian coin could be used in other Greek cities and not have to be exchanged for the local currency.

The Athenian monetary system was set up in the following way: 6 obols = 1 drachma 100 drachma = 1 mina 600 minae = 1 talent

(or the equivalent of 57 pounds of silver) A worker in Athens could earn about two drachmas

a day. Sculptors and doctors were able to make up to six drachmas daily. An unskilled worker would make around half of a drachma a day.

The typical costs of goods in ancient Greek: loaf of bread 1 obol lamb 8 drachmas gallon of olive oil 5 drachmas shoes 8 to 12 drachmas slaves 200 to 300 drachmas houses 400 to 1000 drachmas

Development of the Athenian Silver Tetradrachma and the Greek 1 Euro version today 

NUMERAL SYSTEM

The ancient Greeks had different systems for cardinal numbers and ordinal numbers, but none of them was very efficient.

The first Greek number system is acrophonic system which was use in the first millennium BC.

'Acrophonic' means that the symbols for the numerals come from the first letter of the number name, so the symbol has come from an abbreviation of the word which is used for the number.

The system was based on the additive principle in a similar way to Roman numerals. This means that 8 is simply V|||, the symbol for five followed by three symbols for one.

This system was used for more than money. A very similar system was also used in dealing with weights and measures which is not surprising since the value of money would certainly have evolved from a system of weights.

This is confirmed by the fact that the drachma was also the name of the unit of weight.

Acrophonic System

The second ancient Greek number system, the alphabetical numerals or alphanumeral is sometimes called, the 'learned' system. As the name 'alphabetical' suggests the numerals are based on giving values to the letters of the alphabet.

The Greeks were one of the first to adopt a system of writing based on an alphabet.

There are 24 letters in the classical Greek alphabet and these were used together with 3 older letters which have fallen out of use.

Alphanumeric System

The Greeks did not have a symbol for zero. They could string these symbols together to represent any number up to 1000. By putting a comma in front of any symbol in the first row, they could now write any number up to 10,000.

This works great for smaller numbers, but what about larger numbers? Here the Greeks used the symbol M for 10,000. And used multiples of 10,000 by putting symbols above M. 

The number 20000.

The number 1230000.

Aristarchus wrote the number 71755875 as:

MEASUREMENT

Jakob Köbel, Geometry, Frankfurt, Germany in1616, Definition of

the rute as16 feet used as a length standard.

Greeks used different systems of measuring distances and weights, partly taken by Egyptian such as in the Hellenistic age.

The same word could mean different lengths such as the stadion length depending when and where it is used.

This produced confusion of what the earth radius was estimated by Eratosthenes.

It is difficult to give real accurate distances and lengths and the following gives only a rough idea of what units were used.

LENGTH

Greek measures of length were based on the relative lengths of body parts, such as the foot and finger segment.

The specific values assigned to these units varied according to location and epoch

(e.g., in Aegina a foot or pous was approximately 13 inches or 333 mm, whereas in Athens it was about 11.6 inches or 296 mm).

The relative proportions, however, were generally the same throughout the Greek world.

The cubit was a commonly used unit of length in many kingdoms.

It was the distance from the peak of the elbow to the tip of the middle finger, and was the basis of other units based on the human body.

Wood and stone cubit standards dating from 2400 BC to the first century AD and the variation over a long period of these standards for example for the cubit varied less than 5 %.

Units derived from the dactylos (plural: dactyloi):

Larger units derived from the pous (plural: podes):

AREA

One plethron was traditionally the amount of land a yoke of oxen could plough in one day (approximately 4 acres); more specifically, it was any area equal to the area of a square each of whose sides is 100 podes or 1 plethron in length

TRIANGULATION

Triangles are the basis of many measurement techniques, from basic geodesic measurements performed in ancient Greece to more modern laser-based 3D cameras.

Dioptra is a surveying device using triangulation which was used in navigation, surveying and civil engineering to determine the direction of roads, tunnels or other structures.

Odometer, a device used in the late Hellenistic time for indicating distance traveled by a vehicle.

The number of miles traveled is given simply by counting the number of pebbles.

LEVELING

Chorobates an instrument that in Greek means “land racer” or “land ranger”.

WEIGHT

The Greeks used a wheat seed as the smallest unit of weight, a standard that was very uniform and accurate for the times.

Weights are often associated with currency since units of currency involve prescribed amounts of a given metal.

There were two dominant standards of weight in the eastern Mediterranean - a standard that originated in Euboea and that was subsequently introduced to Attica by Solon, and also a standard that originated in Aegina.

The Attic/Euboean standard was supposedly based on the barley corn, of which there were supposedly twelve to one obol.

VOLUME

Greeks measured dry capacity by the medimnos (25 kilogram) and liquid capacity by the metretes (34 liters).

Each measure was based on the common unit of the kotyle (plural kotylai) so that the liquid metretes was 4.5 times greater in volume than the dry medimnos.

Greeks measured volume according to either dry or liquid capacity, suited respectively to measuring grain and wine.

DRY MEASURES

LIQUID MEASURES

TIME

Ancient Greeks also used the principle of liquid measures for time measurements since time “flows”.

Athenians measured the day by sundials. Periods during night or day could be measured by a

water clock (clepsydra) that dripped at a steady rate. The Greek day began just after sunset.

Athenians named each year after the Archon Eponymos for that year, and in Hellenistic times years were reckoned in quadrennial epochs according to the Olympiad.

The Athenian year was divided into 12 months, with one additional month (poseideon deuteros, 30 days) being inserted between the sixth and seventh months every second year.

The start of the year was at the summer solstice and months were named after Athenian religious festivals

Clepsydra One could define the time

by the amount of water that is required to empty a filled container with a hole in the bottom that allowed the water to escape which is approximately 4 minutes for 3 liters.

Sun Dials

Greek scaphe dial The days are divided

into temporal hours. Their length is not fixed but instead the time between sunrise and sunset is divided into 12 intervals of equal length.

Universal Ring dial A portable clock that

was also an astronomical instrument. It was used to measure approximate latitude, azimuth and zenith in the distances of the stars.

Intercalary Month

Athens Month

CONTRIBUTIONS

Year Name Contributions

582-507 BC Pythagoras Discovered irrational numbers like √2, √5, Л,…Discovered Pythagoras Theorem

440 BC Hippocrates Produced the earliest geometrical

textbook.

427-347 BC Plato Used mathematics to explain the system of logic.Turned geometry into a science of abstract reasoning.

300 BC Euclid Wrote 13 book of The Elements containing theorems, geometrical problems to form principles of deduction.

.

287-212 BC Archimedes Discovered formulae of area and volume of pyramid, cone, cylinder and sphere.Calculated the value of Л up to 3 significant figures.Discovered the Principle of Floatation

275-195 BC Eratosthenes Discovered Sieve of Eratosthenes to find prime numbers.

250 BC Appollonius Wrote Arithmetica for solving problems in algebra.

220 BC Diophantus Constructed a number of geometrical theorems related to cross-section of cone, circle and loci.