# Brief History Of Mathematics

Introduction

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There are many excellent reasons to study the history of mathematics. It helps students develop a deeper understanding of the mathematics they have already studied by seeing how it was developed over time and in various places. It encourages creative and flexible thinking by allowing students to see historical evidence that there are different and perfectly valid ways to view concepts and to carry out computations. Ideally, a History of Mathematics course should be a part of every mathematics major program.

The nature of mathematics can be traced from ancient history of mathematics to contemporary one. The word” mathematics” comes from the Greek  “máthema” which means

Science, knowledge, or learning; and “mathematikós” means “fond of learning”. The invention of printing has largely solved the problem of obtaining secure texts and has allowed historians of mathematics to concentrate their editorial efforts on the correspondence or the unpublished works of mathematicians. The achievements of prehistoric mathematics and the flowering of Pythagorean have significant evidences to trace the nature of mathematics. While the challenge of non-Euclidean geometry to Euclidean geometry has some impacts to the development of contemporary mathematics.

## BRIEF HISTORY OF MATHEMATICS

Mathematics starts with counting. It is not reasonable, however, to suggest that early counting was mathematics. Only when some record of the counting was kept and, therefore, some representation of numbers occurred can mathematics be said to have started.

In Babylonia mathematics developed from 2000 BC. Earlier a place value notation number system had evolved over a lengthy period with a number base of 60. It allowed arbitrarily large numbers and fractions to be represented and so proved to be the foundation of more high powered mathematical development.

Number problems such as that of the Pythagorean triples (a,b,c) with a2+b2 = c2 were studied from at least 1700 BC. Systems of linear equations were studied in the context of solving number problems. Quadratic equations were also studied and these examples led to a type of numerical algebra.

Geometric problems relating to similar figures, area and volume were also studied and values obtained for π.

The Babylonian basis of mathematics was inherited by the Greeks and independent development by the Greeks began from around 450 BC. Zeno of Elea’s paradoxes led to the atomic theory of Democritus. A more precise formulation of concepts led to the realisation that the rational numbers did not suffice to measure all lengths. A geometric formulation of irrational numbers arose. Studies of area led to a form of integration.

The theory of conic sections shows a high point in pure mathematical study by Apollonius. Further mathematical discoveries were driven by the astronomy, for example the study of trigonometry.

The major Greek progress in mathematics was from 300 BC to 200 AD. After this time progress continued in Islamic countries. Mathematics flourished in particular in Iran, Syria and India. This work did not match the progress made by the Greeks but in addition to the Islamic progress, it did preserve Greek mathematics. From about the 11th Century Adelard of Bath, then later Fibonacci, brought this Islamic mathematics and its knowledge of Greek mathematics back into Europe.

Major progress in mathematics in Europe began again at the beginning of the 16th Century with Pacioli, then Cardan, Tartaglia and Ferrari with the algebraic solution of cubic and quartic equations. Copernicus and Galileo revolutionised the applications of mathematics to the study of the universe.

The progress in algebra had a major psychological effect and enthusiasm for mathematical research, in particular research in algebra, spread from Italy to Stevin in Belgium and Viète in France.

The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculatory science with his discovery of logarithms. Cavalieri made progress towards the calculus with his infinitesimal methods and Descartes added the power of algebraic methods to geometry.

Progress towards the calculus continued with Fermat, who, together with Pascal, began the mathematical study of probability. However the calculus was to be the topic of most significance to evolve in the 17th Century.

Newton, building on the work of many earlier mathematicians such as his teacher Barrow, developed the calculus into a tool to push forward the study of nature. His work contained a wealth of new discoveries showing the interaction between mathematics, physics and astronomy. Newton’s theory of gravitation and his theory of light take us into the 18th Century.

However we must also mention Leibniz, whose much more rigorous approach to the calculus (although still unsatisfactory) was to set the scene for the mathematical work of the 18th Century rather than that of Newton. Leibniz’s influence on the various members of the Bernoulli family was important in seeing the calculus grow in power and variety of application.

The most important mathematician of the 18th Century was Euler who, in addition to work in a wide range of mathematical areas, was to invent two new branches, namely the calculus of variations and differential geometry. Euler was also important in pushing forward with research in number theory begun so effectively by Fermat.

Toward the end of the 18th Century, Lagrange was to begin a rigorous theory of functions and of mechanics. The period around the turn of the century saw Laplace’s great work on celestial mechanics as well as major progress in synthetic geometry by Monge and Carnot.

The 19th Century saw rapid progress. Fourier’s work on heat was of fundamental importance. In geometry Plücker produced fundamental work on analytic geometry and Steiner in synthetic geometry.

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Non-euclidean geometry developed by Lobachevsky and Bolyai led to characterisation of geometry by Riemann. Gauss, thought by some to be the greatest mathematician of all time, studied quadratic reciprocity and integer congruences. His work in differential geometry was to revolutionise the topic. He also contributed in a major way to astronomy and magnetism.

The 19th Century saw the work of Galois on equations and his insight into the path that mathematics would follow in studying fundamental operations. Galois’ introduction of the group concept was to herald in a new direction for mathematical research which has continued through the 20th Century.

Cauchy, building on the work of Lagrange on functions, began rigorous analysis and began the study of the theory of functions of a complex variable. This work would continue through Weierstrass and Riemann.

Algebraic geometry was carried forward by Cayley whose work on matrices and linear algebra complemented that by Hamilton and Grassmann. The end of the 19th Century saw Cantor invent set theory almost single handedly while his analysis of the concept of number added to the major work of Dedekind and Weierstrass on irrational numbers

Analysis was driven by the requirements of mathematical physics and astronomy. Lie’s work on differential equations led to the study of topological groups and differential topology. Maxwell was to revolutionise the application of analysis to mathematical physics. Statistical mechanics was developed by Maxwell, Boltzmann and Gibbs. It led to ergodic theory.

The study of integral equations was driven by the study of electrostatics and potential theory. Fredholm’s work led to Hilbert and the development of functional analysis.

## Notation and communication

There are many major mathematical discoveries but only those which can be understood by others lead to progress. However, the easy use and understanding of mathematical concepts depends on their notation.

For example, work with numbers is clearly hindered by poor notation. Try multiplying two numbers together in Roman numerals. What is MLXXXIV times MMLLLXIX? Addition of course is a different matter and in this case Roman numerals come into their own, merchants who did most of their arithmetic adding figures were reluctant to give up using Roman numerals.

What are other examples of notational problems. The best known is probably the notation for the calculus used by Leibniz and Newton. Leibniz’s notation lead more easily to extending the ideas of the calculus, while Newton’s notation although good to describe velocity and acceleration had much less potential when functions of two variables were considered. British mathematicians who patriotically used Newton’s notation put themselves at a disadvantage compared with the continental mathematicians who followed Leibniz.

Let us think for a moment how dependent we all are on mathematical notation and convention. Ask any mathematician to solve ax = b and you will be given the answer x = b/a. I would be very surprised if you were given the answer a = b/x, but why not. We are, often without realising it, using a convention that letters near the end of the alphabet represent unknowns while those near the beginning represent known quantities.

It was not always like this: Harriot used a as his unknown as did others at this time. The convention we use (letters near the end of the alphabet representing unknowns) was introduced by Descartes in 1637. Other conventions have fallen out of favour, such as that due to Viète who used vowels for unknowns and consonants for knowns.

Of course ax = b contains other conventions of notation which we use without noticing them. For example the sign “=” was introduced by Recorde in 1557. Also ax is used to denote the product of a and x, the most efficient notation of all since nothing has to be written!

## How we view history

We view the history of mathematics from our own position of understanding and sophistication. There can be no other way but nevertheless we have to try to appreciate the difference between our viewpoint and that of mathematicians centuries ago. Often the way mathematics is taught today makes it harder to understand the difficulties of the past.

There is no reason why anyone should introduce negative numbers just to be solutions of equations such as x + 3 = 0. In fact there is no real reason why negative numbers should be introduced at all. Nobody owned -2 books. We can think of 2 as being some abstract property which every set of 2 objects possesses. This in itself is a deep idea. Adding 2 apples to 3 apples is one matter. Realising that there are abstract properties 2 and 3 which apply to every sets with 2 and 3 elements and that 2 + 3 = 5 is a general theorem which applies whether they are sets of apples, books or trees moves from counting into the realm of mathematics.

Negative numbers do not have this type of concrete representation on which to build the abstraction. It is not surprising that their introduction came only after a long struggle. An understanding of these difficulties would benefit any teacher trying to teach primary school children. Even the integers, which we take as the most basic concept, have a sophistication which can only be properly understood by examining the historical setting.

## Mathematics in the Ancient World

Prehistoric people must have used simple arithmetic. However when people became civilized mathematics became far more important. Proper record keeping was essential. In Iraq a people called the Sumerians counted in sets of 60. We still divide hours into 60 minutes and minutes into 60 seconds. We also divide circles into 360 degrees.

The Egyptians had some knowledge of practical geometry which they used to build the pyramids. However the Greeks were interested in ideas for their own sake. Around 600 BC a Greek called Thales calculated the height of a pyramid by measuring its statue. But the most famous Greek mathematician was Pythagoras. (c. 570-495 BC). Pythagoras is famous for his theorem The square on the hypotenuse is equal to the sum of the squares on the other two sides.

Theano of Crotona was a great woman mathematician. Euclid (325-265 BC) is most famous for his book about geometry Elements. A man named Eratosthenes (c.276-194 BC) calculated the circumference of the Earth. Archimedes (287-212 BC) worked out formulas for the area of shapes and the volumes of solids. The last great mathematician of the Ancient World was a woman named Hypatia (died 415 AD).

Roman numerals consisted of I meaning one, X meaning ten, L meaning fifty and C meaning 100. They had no symbol meaning zero. However the Indians invented a symbol for zero and the numerals we now use were invented by them. They were later used by the Arabs and were first used in Europe in the Middle Ages.

## Mathematics 500-1800

There were a number of great Indian mathematicians during this era. Among them were Aryabhata (c. 476-550) and Brahmagupta (c 598-670). A Persian named Al-Khwarizmi was also a famous mathematician. He lived in the early 9th century. He wrote about Indian numerals and algebra.

In Europe An Italian called Fibonacci (c 1175-1250) was a great mathematician of the Middle Ages. He discovered the Fibonacci series of numbers. (Each number is equal to the sum of the previous two numbers 1, 1, 3,, 5, 8, 13 etc.) In 1489 a German named Johannes Widmann invented the + sign for plus and the – sign for minus. The = sign for equals was invented by a Welshman called Robert Recorde in 1557.

During the 17th century mathematics made rapid progress. A Scot named John Napier (1550-1617) invented logarithms. Englishman William Oughtred (1575-1660) invented the slide rule. He also began using the symbol X for multiplication. John Graunt (1620-1674) was the first man to study statistics. Meanwhile a Frenchman named Blaise Pascal (1623-1662) studied probability. Renes Descartes (1596-1650) invented the Cartesian coordinate system with x and y axes. Gottfried Leibniz (1646-1716) invented calculus. One of the greatest mathematicians of the 18th century was Leonhard Euler (1707-1783). Euler made many discoveries and he wrote hundreds of books on mathematics. Another great mathematician was Maria Agnesi.

## Modern Mathematics

In the 19th century Carl Friedrich Gauss (1777-1855) made contributions to algebra, geometry and probability. Charles Babbage (1791-1871) is called the father of the computer because he designed a mechanical calculating machine he called an analytical engine (although it wasn’t actually built in his lifetime). Babbage was assisted by another great mathematician called Ada Lovelace (1815-1852). George Boole (1815-1864) created Boolean algebra. Meanwhile in 1801 William Playfair (1759-1823) invented the pie chart. (Florence Nightingale did not invent the pie chart although she did use them). John Venn (1834-1923) invented the venn diagram.

One of the most famous mathematicians of the 20th century was Alan Turing (1912-1954). He is famous for the Turing test, which states that a computer can be considered intelligent if a human being communicating with it cannot tell it is a computer. In the late 20th century computers became very useful to mathematicians.

Nature’s abacus Soon after language develops, it is safe to assume that humans begin counting – and that fingers and thumbs provide nature’s abacus. The decimal system is no accident. Ten has been the basis of most counting systems in history.

When any sort of record is needed, notches in a stick or a stone are the natural solution. In the earliest surviving traces of a counting system, numbers are built up with a repeated sign for each group of 10 followed by another repeated sign for 1.

Arithmetic cannot easily develop until an efficient numerical system is in place. This is a late arrival in the story of mathematics, requiring both the concept of place value and the idea of zero.

As a result, the early history of mathematics is that of geometry and algebra. At their elementary levels the two are mirror images of each other. A number expressed as two squared can also be described as the area of a square with 2 as the length of each side. Equally 2 cubed is the volume of a cube with 2 as the length of each dimension.

Babylon and Egypt: from 1750 BC The first surviving examples of geometrical and algebraic calculations derive from Babylon and Egypt in about 1750 BC.

Of the two Babylon is far more advanced, with quite complex algebraic problems featuring on cuneiform tablets. A typical Babylonian maths question will be expressed in geometrical terms, but the nature of its solution is essentially algebraic (see a Babylonian maths question). Since the numerical system is unwieldy, with a base of 60, calculation depends largely on tables (sums already worked out, with the answer given for future use), and many such tables survive on the tablets.

Egyptian mathematics is less sophisticated than that of Babylon; but an entire papyrus on the subject survives. Known as the Rhind papyrus, it was copied from earlier sources by the scribe Ahmes in about 1550 BC. It contains brainteasers such as problem 24: – What is the size of the heap if the heap and one seventh of the heap amount to 19?

The papyrus does introduce one essential element of algebra, in the use of a standard algebraic symbol – in this case h or aha, meaning ‘quantity’ – for an unknown number.

Pythagoras: 6th century BC Ancient mathematics has reached the modern world largely through the work of Greeks in the classical period, building on the Babylonian tradition. A leading figure among the early Greek mathematicians is Pythagoras.

In about 529 BC Pythagoras moves from Greece to a Greek colony at Crotona, in the heel of Italy. There he establishes a philosophical sect based on the belief that numbers are the underlying and unchangeable truth of the universe. He and his followers soon make precisely the sort of discoveries to reinforce this numerical faith.

The Pythagoreans can show, for example, that musical notes vary in accordance with the length of a vibrating string; whatever length of string a flute player starts with, if it is doubled the note always falls by exactly an octave (still the basis of the scale in music today).

The followers of Pythagoras are also able to prove that whatever the shape of a triangle, its three angles always add up to the sum of two right angles (180 degrees).

The most famous equation in classical mathematics is known still as the Pythagorean theorem: in any right-angle triangle the square of the longest side (the hypotenuse) is equal to the sum of the squares of the two other sides. It is unlikely that the proof of this goes back to Pythagoras himself. But the theorem is typical of the achievements of Greek mathematicians, with their primary interest in geometry.

This interest reaches its peak in the work compiled by Euclid in about 300 BC.

Euclid and Archimedes: 3rd century BC Euclid teaches in Alexandria during the reign of Ptolemy. No details of his life are known, but his brilliance as a teacher is demonstrated in the Elements, his thirteen books of geometrical theorems. Many of the theorems derive from Euclid’s predecessors (in particular Eudoxus), but Euclid presents them with a clarity which ensures the success of his work. It becomes Europe’s standard textbook in geometry, retaining that position until the 19th century.

Archimedes is a student at Alexandria, possibly within the lifetime of Euclid. He returns to his native Syracuse, in Sicily, where he far exceeds the teacher in the originality of his geometrical researches.

The fame of Archimedes in history and legend derives largely from his practical inventions and discoveries, but he himself regards these as trivial compared to his work in pure geometry. He is most proud of his calculations of surface area and of volume in spheres and cylinders. He leaves the wish that his tomb be marked by a device of a sphere within a cylinder.

A selection of titles from his surviving treatises suggests well his range of interests: On the Sphere and the Cylinder; On Conoids and Spheroids; On Spirals; The Quadrature of the Parabola; or, closer to one of his practical discoveries, On Floating Bodies.

The circumference of the earth: calculated c. 220 BC Eratosthenes, the librarian of the museum at Alexandria, has more on his mind than just looking after the scrolls. He is making a map of the stars (he will eventually catalogue nearly 700), and he is busy with his search for prime numbers; he does this by an infinitely laborious process now known as the Sieve of Eratosthenes.

But his most significant project is working out the circumference of the earth.

Eratosthenes hears that in noon at midsummer the sun shines straight down a well at Aswan, in the south of Egypt. He finds that on the same day of the year in Alexandria it casts a shadow 7.2 degrees from the vertical. If he can calculate the distance between Aswan and Alexandria, he will know the circumference of the earth (360 degrees instead of 7.2 degrees, or 50 times greater).

He discovers that camels take 50 days to make the journey from Aswan, and he measures an average day’s walk by this fairly predictable beast of burden. It gives him a figure of about 46,000 km for the circumference of the earth. This is, amazingly, only 15% out (40,000 km is closer to the truth).

Algebra: from the 2nd century AD The tradition of Babylonian algebra is revived by the Greeks in Alexandria, where Diophantus writes a treatise called Arithmetica in about AD 200; he uses a special sign for minus, and adopts the letter s for the unknown quantity. Greek algebra in its turn spreads to India, China and Japan. But it achieves its widest influence through the Arabic transmission of Greek culture.

In this the most significant event is a book written in Baghdad in about AD 825 by al-Khwarizmi. Its title is Kitab al jabr w’al-muqabala (‘Book of Restoration and Reduction’). The success of this work in Europe provides, from part of the title (al jabr), the word ‘algebra’.

The most important Renaissance work on algebra, written by Girolamo Cardano and published in 1545, expresses in its title the status of the art; it is called Ars Magna, the ‘great art’ as opposed to the lesser art of arithmetic. But there are still no standard symbols.

These emerge during the next century. Both plus (+) and minus (-) derive from abbreviations used in Latin manuscripts. The square root sign √ is perhaps a version of r for radix (‘root’ in Latin). The equal sign (=) is attributed to an English author, Robert Record, in a book of 1556. In the 17th century Descartes introduces the use of x, y and z for unknown quantities, and the convention for writing squared and cubed numbers.

### Brief History Of Mathematics

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