The Ishango, living on the shores of what is now known as Lake Edward on the borders of modern-day Uganda and Zaire, live by catching fish and hunting. The Ishango bone is marked with notches. The notches are grouped in 4 rows. In two rows they sum to 60. One row contains the prime numbers up to 20.
Swaziland, southern Africa
-35000 The bone of a baboon found in a cave in the Lebembo mountains on the borders of Swaziland and South Africa has 29 clear notches. Similar to "calendar sticks" found in Namibia and dated much later.
Ishango
Ishango, central Africa
-20000 Czechoslovakia, 30,000 bc… the bone of a wolf found with 57 deeply cut notches, grouped in fives. Probably the record of "kills" by a hunter… who could count in fives.
What was the Ishango bone used for? A game? A calendar recording the phases of the moon? Religious festivals based on the seasons?
Prime numbers as written over 20,000 years ago
Lebembo, Swaziland
©
Ancient Egypt
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BABYLON… between the Euphrates and Tigris. The start of a civilisation which will last 3,000 years and leave its mark on mathematics. This is the land of Noah's Arc and Abraham, the founding father of three world religions.
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Herodotus: "The Egyptian people and culture come from the lands of the 'long-lived Ethiopians'… the vast tract of inner Africa ."
Egypt
© pifactory.net
Egyptian hieroglyphics recorded numbers with pictures. 1 was drawn as a short piece of rope. A 10 was a longer piece of rope. 100 was a coil of rope. 1000 looked like a lotus flower. 10,000 was represented by a crooked finger. 100,000 looked like a tadpole. A man with arms outstretched, as if saying "Wow!", was 1,000,000. The rising sun as 10 million. Ancient Egypt was famous for its "ropestretchers" who surveyed the land.
Pharoah Narmer… captured 120,000 prisoners, 400,000 oxen and 1,422,000 goats. Exactly.
Indus Valley, India
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The Abacus is developed in China and the Middle East
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The Indus Valley, west India, Pakistan
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Babylon, the land between the Euphrates and the Tigris
For the ancient Babylonians the number 60 was magic. Today we count in tens... But the Babylonians counted in 60s. It was great. They never had any headaches with fractions! We still use some Babylonian numbers today. Look at your watch.
60! The early Babylonians wrote their numbers in clay using sticks with sharpened points. Their numbers were made up of just two symbols, T and < shapes , pressed into the clay. We call it cuneiform writing.
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Egyptian calendars
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For 1,500 years a highly civilised people lived in the Indus Valley, now in India and Pakistan. Even to this day the code of their sophisticated writing, the Indus script, has not been cracked. The maths of the Indus Valley is the foundation for the maths of India.
-2600
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Babylon at this time is a number of small city-states. The Babylonians use their clay tablets and numbers to develop multiplication and division tables. Their base number is 60. We call it a sexagesimal system. So, the important times tables were for the numbers 2 to 20 and 30, 40 and 50. With these the Babylonians could carry out all possible multiplications. Today our base number is 10, the decimal system. Our important times tables are 2 to 10.
Fractions… were easy! 60 can be divided by 2, 3, 4, 5, 6, 10, 15, 20 and 30. So, 1/2 is written as 30. Until the arrival of decimal fractions, 1/2 is 0.5, the Babylonian way of working with fractions stayed the best.
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The Indus Valley… developed intricate brickwork patterns, and uniform decimal weights and measures
The Indus Valley civilisation lasts another 500 years… before disappearing
-2100
Babylon… quadratic equations solved
-2000
Hey!... These guys got this Pythagoras stuff before me!
A clay tablet of Babylonian cuneiform writing.
The Babylonians knew about right-angled triangles and similar triangles. This diagram appeared on a clay tablet with a proof that showed the Babylonians knew the square of the hypotenuse is equal to the sum of the squares of the opposite two sides. Other tablets listed what we now call Pythagorean triples, the lengths of the sides of right-angled triangles.
-1900
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Babylon is united and the city on the banks of the Euphrates becomes the prosperous centre of a highly cultured empire. The most famous king of the time was King Hammurabi.
The Babylonians wrote lots of tables in clay - tables of multiplications, reciprocals, squares and cubes as well as square and cube roots. They even had tables for n3 + n2
Egyptian fractions in the Eye of Horus
-1600
-1700
Egyptians are using symbols, hieratic numerals, by now. They do multiplication and division by repeatedly multiplying or dividing by 2. In their calculations for instance, 23 is 2 0 + 21 + 22 + 24. They break fractions into complicated lists of unit fractions, for instance, 2/5 = 1/3 + 1/15 and 2/47 = 1/30 + 1/141 + 1/470. -1800
-1900
pi is 4(8/9)2
-1600
We Babylonians spend the next 600 years gazing at the stars and doing the early stuff on astronomy
The Vedic people
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China… decimal numbers but with no zero
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Vedic people enter India from what is now Iran. Over the next 1000 years they develop Vedic maths based on rules called Sulbasutras written for building sacrificial altars
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Rameses II divided the land into lots and gave a square piece of equal size, from the produce of which he exacted an annual tax. The king… would send inspectors to measure the extent of the loss… Perhaps this was the way in which geometry was invented, and passed afterwards into Greece. Heroditus
Plato, Athens, 427BC founder of The Academy
Egypt, 1650BC - Rhind papyrus contains tables of fractions and about 80 maths problems.
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China… counting boards used for calculation
'The rope which is stretched across the diagonal of a square produces an area double the size of the original square' -1000
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Hi, I'm Sri Bharati Krsna Tirthaji and I rediscovered Vedic maths in ancient Sanskrit texts
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Manava, 750BC… more sulbasutras pi = 25/8
Chapter1, verse 61 states: 2=577/408… or, 1.41422
India, 600BC… Apastamba writes a remarkable sulbasutra
Zero Babyloni use three hooks to show the empty sp in their number system
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India, 800BC… Baudhayana writes one of the first sulbasutras… all the maths is written in words and verses
Hey! That's my theorem!
-800
The School of Athens - by Raphael
Pythagoras,
427BC e Academy
Euclid, Alexandria, 325BC - author of The Elements
Elements
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China… counting rods introduced… zero was a space. Adding, subtraction, multiplication and division as done now
China… decimal notation with symbols for 1 to 9, then 10, 100, 1000 10000...
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China
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Greeks begin to use numerals
-400
Autolycus studies spheres
Hi, I'm Pythagoras, born 569BC, and I studied in Egypt and Babylon. Me and my mates set up a secretive and mysterious school in Croton… we had no possessions, were vegetarian and we called ourselves mathematikoi and studied arithmetic, music, geometry, astronomy. And we swore an oath of secrecy.
Hi, I'm Thales, born 624BC, the first of the great Greek mathematicians. I insisted on proof. And I brought Babylonian maths to Greece and used geometry to calculate the height of the pyramids and the distance of ships from the shore.
Hi, I'm Zeno of Elea, born in what is now southern Italy. I thought up paradoxes. For instance try to sum 1/2 + 1/4 + 1/8 + … backwards starting at the end!
Zero - the Babylonians use two wedge symbols to show where we would now put the zero in their numbers.
-500
I'm Hippocrates… I tried to construct a square with the same area of a circle. I failed, but discovered other stuff.
India… Panini, 520BC, works on Sanskrit grammar and opens up discussion on science, philosophy… and maths. -600
ble
Zero - the Babylonians use three hooks to show the empty space in their number system -700
C…
Hippasus proved Pythagoras wrong on one point… and got thrown in the sea!
Euclid of Alexandria, 325BC - Maths was almost all about geometry. Euclid wrote it all down in 13 books called The Elements.Euclid built up the maths, with each bit helping to prove the next bit.
Buddhism… Buddhist pilgrims carry Indian maths across Asia
Hi, I'm Eratosthenes. I was born in 276BC in Cyrene, in what is now Libya in north Africa. My nickname was Beta, because I was very, very good at lots of stuff… but never the best in anything. I measured the distance between the earth and moon, and the earth and the sun. I also measured the circumference of the earth. And I was pretty close! I also worked out how to calculate prime numbers with my sieve... which is still used in classrooms more than 2000 years later.
Autolycus studies spheres
Hi, I'm Archimedes. I estimated there were the equivalent of 1063 grains of sand in the universe. Give or take a couple, and not far out. I talked of "myriads" and was trying to fathom the scale of numbers.
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Hi, I'm Aristotle. I, and my followers gave maths logic… All men are mortal. Socrates is a man. So… Socrates must be mortal!
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Aristarchus says the earth orbits the sun
0
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China, 290BC… the classic Chou pei suan ching is written
Chinese use negative numbers
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0
China
China
Eureka! Archimedes made the first theoretical attempt to calculate pi… the ratio of the circumference to the diameter of a circle. He got 3.1418
Zero - Greek astronomers use the 0 symbol to represent the idea of nothing. It comes from the impression left in sand by counters.
I'm Zeno of Sidon born in Lebanon in 150BC. I criticised Euclid's ideas… but no one took any notice for ages. But I was right!
Chang Hong puts pi at 3.1622
The first 3x3 magic square was revealed to the Emperor Yu the Great on the back of a turtle in the Lo Shu river. The numbers in any direction add up to the magic number 15.
India
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Tsu Ch'ung Chi puts pi at 3.141592920
400
Ptolemy puts pi at 3.14166 Hi, I'm Heron of Alexandria. I wrote about the maths of things like air, steam and pressure. I also described how to make giant catapults and engines of war… and puppets!
Diaphantus of Alexandria, the father of algebra, born. He worked on equations such as ax2 + bx = c
Hi, I'm Ptolemy… I observed the sun, moon and planets and made astronomy into a mathematical science
Hi, Im Nicomarchus … I wrote the first book to treat arithmetic separate from geometry
Liu Hui in China approximates pi to 3.141014 using a polygon with 192 sides. Mayans use a place value system and a base of 20
Porphyry… biographer of Pythagoras
100
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100
China Pythagoras used for surveying and astronomy… some evidence says even earlier than Pythagoras
India - the classic era 400 to 1500
300
China - The Nine Chapters on the Mathematical Art. Common fractions, highest common denominators, lowest common multiples, areas, proportions, square and cube roots, volumes, pyramids, cylinders, cones, simultaneous linear equations, negative numbers, rightangled triangles.
I'm Heng and I was born in China in 78… among other things I invented the first machine for measuring earthquakes… a ball fell out of a dragon's mouth into a frog's
Hypatia... The first woman mathematician and philosopher to be widely recognised. Taught at Alexandria Museum, Egypt, and wrote on Euclid and geometry and Diaphantus and his equations. A pagan and follower of Plato's ideas, her flesh was scraped from her bones by militant Christians.
500 Zu Chongzhi in China approximates pi to 3.1415926 and recommends the fraction 355/113 as a close approximation and 22/7 for rough.
600 China Indian astronomer Levensita translates Indian maths into Chinese… angle measurement using 360 degrees, table of sines and explains Hindu decimal numerals.
700 Bhaskara identifies infinity
Zero - Indian mathematicians are the first to use 0 as a proper number, not just a place holder. Brahmagupta writes that the sum of 0 and a negative number is a negative, and the sum of 0 and a positive number is a positive. A number multiplied by 0 is 0.
Zero - Indian mathematician Mahavira tackles division by 0 and writes… "a number remains unchanged when divided by zero." The Indian mathematicians call zero Kha.
Bhaskara, born in what is now know as Gujarat, 600 AD. Gives formula to estimate the trigonometic sine function which has a maximum error of less than 1%. The first recorded mathematician to examine quadrilaterals where all four sides are unequal and none of the opposite sides are parallel. Bhaskara also wrote on linear equations,quadratic and cubic equations and equations with more than one unknown.
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Aryabhatiya, 476 -- arithmetic, algebra, plane geometry, trigonometry including a table of sines and the first introduction of the concept of cosine, and spherical trigonometry. Invents number system using alphabetical notation giving numbers to 10^18. Uses zero and place value system. Of all the ancients Aryabhatia has the most accurate estimate of the value for pi of 3.14159265. He estimated the circumference of the earth at 62832 miles. He said the apparent rotation of the heavens was due to axial rotation of the earth... such a shocking idea that his supporters changed his text to hide his stupidity! He correctly explained the eclipses of sun and moon, identified that the moon shines by reflecting the light of the sun, and identified the earth and the planets have elliptical orbits around the sun. He put the length of the year at 365 days 6 hours and 12 minutes and 30 seconds. Less than 25 minutes too much!
800
Abu Kamil
Mahavira uses 9 numerals
Arabia The maths of today is far closer to the developments made by the Islamic and Arabic mathematicians than the maths of the ancient Greeks. Many of the ideas previously attributed to the European mathematicians of the 16th to 18th centuries, are now known to have been developed by the Arabic mathematicians centuries earlier.
970 - Abu'l Wafa invents th
900
Jia Xian finds higher degree roots
'l Wafa invents the wall quadrant and the tangent function
Hi, I'm Omar Khayyam. I wrote books on algebra and music. I worked out the length of the year was 365.24219858156 days!
1000
Kerala, India… Taylor series developed 300 years before Taylor
1100
al-jabr… Algebra!
1200
900
Abu Kamil writes on algebra
Zero -Ibn Ezra writes The Book of Number, bringing the Indian number system and decimal fractions with place values written from left to right to the attention of European scholars. He uses the symbol 0, which he calls "galgal" meaning wheel or circle. 1100
Hi, I'm al-Biruni. I was born in 973 in what is now Uzbekistan. I did so much maths and astronomy it is hard to make a list. I lived and studied in India for a while and translated Sanskrit books on maths, astronomy, medicine, geography, grammar... 1000
Sridhara, born 870 Bengal. One of first to describe how to solve quadratic equations. Like most Indian maths… it was all written in verse!
Jabir ibn Aflah books will be translated into latin
1200
Li Yeh uses negative numbers
Zero - Islamic mathematician alSamawal writes, 'if we subtract a positive number from zero the same negative number remains. "
The origins of the Zero -Ch'in Chiu-Shao in China Yoruba people of repeats Indian number ideas and south-west Nigeria are lost in the mists of time. The Yoruba Customs link with the Abu Jafar Muhammad ibn Musa al-Khwarizmi ancient Egyptians. 825 writes Hisab al-jabr w'a'-muqabala. Recent
Beginnings of Arabic algebra, pains to link back to Babylonian and Greek but particularly Indian numeral system. Introduces Indian numerals to the west.
history dates from the early centuries of the second millenium. Numbers are based on 20 with an emphasis on subtraction. So, 19, ookandinlogun, is 20 minus 5.
The Incas
1400
1500
ed into latin
1300
The Incas
Copernicus suggests the earth circles the sun and suffers the wrath of the Catholic church
1300
1400
1500
al-Kashi finds pi to 14 places
Fibonacci describes the nine Indian number symbols and 0. I'm Nicole d'Oresme, France, and I invented coordinate geometry
Maths becomes compulsory for a degree at the University of Paris
Who discovered calculus first… Newton or Liebnitz? Bhaskaracharya in India in 1100s 1600
Early use of the +, symbols by Widman. + is a short form of the latin et, meaning 'and'
Recorde introduces the = sign Priticus uses the word trigonometry for the first time
1600
Batchet writes a book of puzzles
Van Ceulen finds pi to 35 places The Aztec Sun Stone is a calendar based on astronomy and maths. Pictographs indicate days, months and suns (cosmic cycles). The stone is 3.6 m in diameter and weighs about 24 metric tons. It took 52 years to complete, from 14271479. This calendar is 103 years older than the Gregorian calendar which is used worldwide today.
At its height during the last decades of the 15th century, the Inca empire covered much of today's Peru, Bolivia, Chile, Ecuador and Argentina. It lacked the wheel, beasts of burden and a written language. But it recorded population and statistical records on Quipus, balls of string and knots. Using relay runners a quipu could be moved some 300 miles in 24 hours. A quipu was made using different types and colours of cord and knots. According to position the knots signified units, tens... up to hundreds of thousands. The Aztecs
George Wash "The investig mathematica truths accus the mind to m and correctn reasoning…
Batchet writes a book of puzzles
1700
m in ghs ons.
I'm Leonhard Euler, born in Basel in Switzerland 1707. I was the first to use the symbol for pi, I found e, the base for natural logarithms and was the most prolific writer of maths of all time.
George Washington: "The investigation of mathematical truths accustoms the mind to method and correctness in reasoning… "
1800
1900
Rutherford finds pi to 440 places
I'm Ada Byron Lovelace. I explained how computer programmes could work… before computers were made.
I'm Srinivasa Ramanujan. I was a rail clerk in Madras. I had little formal education. I invented my own algebra. And many now say I was one of the greatest mathematicians of all time. I seemed to conjure up the most complex theorems, writing little down. 2000
1900
1800
Machin finds pi to 100 places
one
onths
I'm Charles Babbage, born in 1791 in London. I designed the first computer and changed maths forever
1700
Hi, we're the brilliant but argumentative Bernoullis, Johann, Daniel and Jacob. We helped Liebniz with calculus… but we jealously mocked each other's achievements.
I'm GH Hardy. England's top mathematician in the early 1900s. Collaborated with Ramanujan and wrote A Mathematician's Apology
I'm George Boole, born in Lincoln in 1815. I applied algebra to human reasoning. Boolean algebra is the basis of modern computer programming.
I'm Sofia Kovalevskaya, single mother, nurse to the wounded of the Paris Commune, poet and first woman professor of maths
2000
I'm Andrew Wiles and I proved Fermat's last theorem which had baffled mathematicians for more than 300 years
I'm Pierre de Fermat… I enjoyed setting maths puzzles
Fractals discovered by Mandelbrot
I'm Benjamin Banneker, son of a slave and author of astronomical almanacs.
I'm the secretive Carl Gauss… I drew a 17-gon using only ruler and compasses, plus a lot more...
I'm Marjorie Lee Browne, the first black woman in the USA to get a PhD I'm Rear Admiral Grace Hopper. I discovered the first computer bug and the first woman to be awarded the UK's and USA's highest technology awards.
1999… Kanada Takashi finds pi to 206158430000 places!
aticians e than
red
ada s pi to 000 ces!
2100
The next 30,000 years… the story continues...
2100 Produced by Mr Whitfield, Southgate School, January 2003. Main sources: The Crest of the Peacock, by Georges Gheverghese Joseph. The MacTutor History of Mathematics Archive @ http://turnbull.dcs.stand.ac.uk/~history/ Plus web sites across the world. For links see: pifactory.net Historical interpretation is down to Mr Whitfield. PS Apologies to Einstein, Russell, Descartes, Turing, Timaeus, Fourier, Nobel, Pascal, Copernicus, alKindi and all the rest missed in this time-line of the history of maths.
Mathematics… the language of all people through all time
ew Wiles oved last which
Hi, I'm… and I went to Parkrose High School in 2005
