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“If a man's wit be wandering, let him study the mathematics.” Francis Bacon

Mathematicians

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PYTHAGORAS

(c. 570 – c. 495 BC)

Pythagoras of Samos was a famous Greek mathematician and philosopher. He is known

best for the proof of the important Pythagorean theorem, which is about right triangles. He

started a group of mathematicians, called the Pythagoreans, who worshiped numbers and

lived like monks. He was an influence for Plato.

He had a great impact on mathematics, theory of music and astronomy. His theories are still

used in mathematics today. He was one of the greatest thinkers of his time.

Pythagoras was born in Samos, a little island off the western coast of Asia Minor. There is

not much information about his life. He was said to have had a good childhood. Growing up

with two or three brothers, he was well educated. He did not agree with the government

and their schooling, so he moved to Croton and set up his own cult (little society) of

followers under his rule. His followers did not have any personal possessions, and they were

all vegetarians. Pythagoras taught them all, and they had to obey strict rules.

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Graphical demonstration of the Pythagorean theorem

Some say he was the first person to use the term philosophy. Since he worked very closely

with his group, the Pythagoreans, it is sometimes hard to tell his works from those of his

followers.

Religion was important to the Pythagoreans. They swore their oaths by "1+2+3+4" (which

equals 10). They also believed the soul is immortal and goes through a cycle of rebirths until

it can become pure. They believed that these souls were in both animal and plant life.

His beliefs

Pythagoras' most important belief was that the physical world was mathematical and that

numbers were the real reality.

1. that at its deepest level, reality is mathematical in nature,

2. that philosophy can be used for spiritual purification,

3. that the soul can rise to union with the divine,

4. that certain symbols have a mystical significance, and

5. that all brothers of the order should observe strict loyalty and secrecy.

Pythagorean theorem

Pythagoras is most famous for his theorem to do with right triangles. He said that the length

of the longest side of the right-angled triangle called the hypotenuse (C) squared would

equal the area of the other sides squared. And so (a x a) + (b x b) = (c x c) was born. There

are many different proofs for this Pythagorean theorem.

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EUCLID

(about 325 BC–265 BC)

Euclid of Alexandria was a Greek mathematician who lived in Alexandria, Egypt and worked

at the Library of Alexandria. Little is known about this person, but people think he lived

there when Ptolemy I was Pharaoh. It is not known where or when he was born.

The Elements

Euclid collected together all that was known of geometry, which is part of mathematics.

His Elements is the main source of ancient geometry. Textbooks based on Euclid have been

used up to the present day. In the book, he starts out from a small set of axioms (that is, a

group of things that everyone thinks are true). Euclid then shows the properties of

geometric objects and of whole numbers, based on those axioms.

The Elements also includes works on perspective, conic sections, spherical geometry, and

possibly quadric surfaces. Apart from geometry, the work also includes number theory.

Euclid came up with the idea of greatest common divisors. They were in his Elements. The

greatest common divisor of two numbers is the greatest number that can fit evenly in both

two numbers.

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The geometrical system described in the Elements was long known simply as geometry and

was considered to be the only geometry possible. Today that system is referred to

as Euclidean geometry to distinguish it from other so-called non-Euclidean

geometries which mathematicians developed in the 19th century.

In addition to the Elements, at least five works of Euclid have survived to the present day.

They follow the same logical structure as Elements, with definitions and proved

propositions.

Data deals with the nature and implications of "given" information in geometrical problems;

the subject matter is closely related to the first four books of the Elements.

On Divisions of Figures, which survives only partially in Arabic translation, concerns the

division of geometrical figures into two or more equal parts or into parts in given ratios. It is

similar to a third century AD work by Heron of Alexandria.

Catoptrics, which concerns the mathematical theory of mirrors, particularly the images

formed in plane and spherical concave mirrors. The attribution to Euclid is doubtful. Its

author may have been Theon of Alexandria.

Phaenomena, a treatise on spherical astronomy, survives in Greek; it is quite similar to On

the Moving Sphere by Autolycus of Pitane, who flourished around 310 BC.

Tributes to Euclid

An asteroid, is named after Euclid.

A crater on the moon is named after Euclid.

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ERATOSTHENES(276 BC–194 BC)

Eratosthenes of Cyrene Was a Greek mathematician, geographer and astronomer. He was

head of the Library of Alexandria from 240 BC until his death: this was the most important

library of the ancient world.

According to the Suda (an ancient Greek encyclopaedia), his contemporaries nicknamed

him Beta, (the second letter of the Greek alphabet), because he was the second best in the

world in almost any field. Eratosthenes was a friend of Archimedes, who also lived and

worked in Alexandria. Archimedes was the greatest mathematician and inventor of the age,

so perhaps the Beta nickname was not unjust.

The works Eratosthenes wrote are known to us only indirectly: the great Library was

destroyed, and no copies survived. Strabo (~63BC–24AD) wrote about geography in

antiquity. He tells us that the works of Eratosthenes were On the measurement of the

Earth and Geographica.

Eratosthenes made several remarkable discoveries and inventions. He was the first person

to calculate the circumference of the Earth (with remarkable accuracy), and he invented a

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system of latitude and longitude. He calculated the tilt of the earth's axis (again with

remarkable accuracy); he may also have accurately calculated the distance from the earth to

the sun and invented the leap day. He created a map of the world based on the available

geographical knowledge of the era. Eratosthenes was also the founder of scientific

chronology; he wanted to fix the dates of the chief literary and political events from the

conquest of Troy.

MEASURING THE EARTH’S CIRCUMFERENCE

Measurements taken at Alexandria (A) and Syene (S)

Eratosthenes measured the circumference of the Earth without leaving Egypt. He knew that

on the summer solstice at local noon at the city now called Aswan on the Tropic of Cancer,

the sun would appear at the zenith, directly overhead. He also knew, from measurement,

that in Alexandria, the angle of elevation of the sun would be 1/50 of a full circle (7.2°) south

of the zenith at the same time.

Assuming that Alexandria was due north of Syene he concluded that the distance from

Alexandria to Syene must be 1/50 of the total circumference of the Earth (360 degrees). His

estimated distance between the cities was 5000 stadia (about 500 geographical miles or

800 km) by estimating the time that he had taken to travel from Syene to Alexandria

by camel. He rounded the result to a final value of 700 stadia per degree, which implies a

circumference of 252,000 stadia. The exact size of the stadion he used is frequently argued.

The common Attic stadium was about 185 m, which would imply a circumference of

46,620 km, which is 16.3% too large. However, if we assume that Eratosthenes used the

"Egyptian stadium" of about 157.5 m, his measurement turns out to be 39,375 km, an error

of less than 1%. The method is an early application of trigonometry (relationships of sides

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and angles of triangles) in the measurement science of geodesy (the science of accurately

measuring and understanding three fundamental properties of the Earth).

Eratosthenes went further and computed the tilt of the Earth's axis to within a degree. This

is the tilt which is the main cause of the annual climate cycle of spring, summer, autumn,

winter. He also deduced the length of the year as 365¼ days. He suggested that calendars

should have a leap day every fourth year, an idea taken up two centuries later by Julius

Caesar.

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GEROLAMO CARDANO

(English Jerome Cardan)

(September 24, 1501 - September 21 1576)

He was a celebrated Italian Renaissance mathematician, physician, astrologer and gambler.

As he was an illegitimate child, he could not enter the college of physicians. Despite this he

was a relatively successful physician. The first description of Typhoid fever is attributed to

him.

Today he is mainly known as a mathematician. He made some discoveries in algebra.

Cardano also speaks about imaginary numbers in the same book, Ars Magna. He did not

completely understand what they were like though.

Cardano was often short of money. He gambled. A book of his, called Liber de ludo

aleae (aproximately translates to as Book on the Game of Chances) contains the first

systematic treatment of probability, as well as a section on how to cheat. He wrote it in the

1560s, but it was only published after his death, in 1663.

LEONHARD EULERKiddle.coEnrichment activity

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Swiss mathematician(April 15, 1707 – September 7, 1783)

Euler made important discoveries in fields like calculus (study of continuous change)

and topology (the study of geometrical properties and spatial relations unaffected by the

continuous change of shape or size of figures). He also made many of the words used in

maths today. He came up with the idea of a mathematical function. He is also known for his

works in mechanics, optics, and astronomy.

Euler is considered to be the most important mathematician of the 18th century, one of the

greatest mathematicians of all time, and also one of the mathematicians who wrote the

most. His collected works fill 60–80 volumes. Another mathematician, Pierre-Simon

Laplace said, "Read Euler, read Euler, he is a master for us all".

Euler was featured on the sixth series of the Swiss 10-franc bill and on numerous Swiss,

German, and Russian postage stamps. The asteroid 2002 Euler was named in his honour.

CHARLES BABBAGEKiddle.coEnrichment activity

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(26 December 1791 – 18 October 1871)

He was the first person to come up with the idea of a computer that could be programmed.

Unfinished parts of his mechanisms are on display in the London Science Museum.

His Brain is on display in the Science Museum in London.

Knowing that there were lots of errors in the calculation of mathematical tables, Babbage

wanted to find a method by which they could be calculated mechanically, removing errors

made by humans. Three different factors seem to have influenced him: a dislike of

untidiness; his experience working on logarithmic tables; and existing work on calculating

machines carried out by Wilhelm Schickard, Blaise Pascal, and Gottfried Leibniz. He first

talked about the principles of a calculating engine in a letter to Sir Humphrey Davy in 1822.

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Babbage's engines were among the first mechanical computers. His engines were not

actually completed because he did not have enough money. Babbage realised that a

machine could do the work better and more reliably than a human being. Babbage

controlled building some steam-powered machines that more or less did their job;

calculations could be mechanized to an extent. Although Babbage's machines were large

machines they were organized in a way similar to modern computer architecture. The data

and program memory were separated, operation was instruction based, control unit could

make conditional jumps and the machine had a separate I/O unit. Ada Lovelace studied how

to program them.

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The Science Museum's Difference Engine No. 2, built from Babbage's design

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ALAN TURING

(London, 23 June 1912 – Wilmslow, Cheshire, 7 June 1954)

Turing was interested in artificial intelligence. He proposed the Turing test, to say when a

machine could be called "intelligent". A computer could be said to "think" if a human talking

with it could not tell it was a machine.

During World War II, Turing worked to break German ciphers (secret messages).

Using cryptanalysis, he helped to break the codes of the Enigma machine. After that, he

solved other German codes.

From 1945 to 1947, Turing worked on the design of the ACE (Automatic Computing Engine)

at the National Physical Laboratory. He presented a paper on 19 February 1946. That paper

was "the first detailed design of a stored-program computer". Although it was possible to

build ACE, there were delays in starting the project. In late 1947 he returned to Cambridge

for a sabbatical year. While he was at Cambridge, the Pilot ACE was built without him. It ran

its first program on 10 May 1950.

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1. What is Pythagorean theorem about?

2. Where was Pythagoras born?

3. What was Pythagoras’ most important belief?

4. Where did Euclid of Alexandria work?

5. What branch of mathematics is Euclid best known for?

6. What tributes have been paid to Euclid?

7. Why was Eratosthenes nicknamed Beta?

8. What was Eratosthenes the first person to calculate (without leaving

Egypt)?

9. What did Cardano’s book entitled Game of Chances contain?

10.What was Euler’s image famously used for?

11.What was Charles Babbage famous for?

12.What part of his body is on display in the Science Museum in London?

13.What was Alan Turing interested in?

14.During WW11 what was Alan Turing’s work?

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ANSWERS

1. Right-angled triangles2. Samos (Greece)3. Pythagoras' most important belief was that the physical world was mathematical and that

numbers were the real reality.4. Euclid of Alexandria worked at the Library of Alexandria.5. Geometry6. An asteroid and a crater on the moon have been named after Euclid. 7. He was the second best in the world in almost any field.8. He was the first person to calculate the circumference of the Earth.9. It contains the first systematic treatment of probability, as well as a section on how to cheat.10. Euler was featured on the sixth series of the Swiss 10-franc bill and on numerous Swiss,

German, and Russian postage stamps.11. He was the first person to come up with the idea of a computer that could be programmed.12. Brain13. Turing was interested in artificial intelligence.14. During World War II, Turing worked to break German ciphers

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