# Mathematics

Mathematics, from the point of view of science, is about developing general and systematic methods for describing and modelling aspects of nature. Mathematical structures however have often been developed purely for their own intrinsic interest without regard to applications. In some cases such abstract constructions have later proved to be important tools for science.

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## Number and Measure

A few articles, such as a bone carved with notches, survive from prehistory c.-28,000 to indicate that our ancient ancestors had some idea of number. Such tally symbols, representing numbers by sets of notches, marks, pebbles, fingers, undoubtedly preceded the invention of number words. We still use them for scoring, in darts for example, and on dominoes and playing cards.

The Egyptian civilisation c.-3000 possessed a hieroglyphic symbolism for numbers combining the tally method with special symbols for all powers of ten up to a million. This notation was purely additive, like the Roman numerals which are still used today for special purposes. The number 234 being shown by two symbols for a hundred, three symbols for ten and four unit symbols. The Ashmolean Museum in Oxford has a Egyptian royal mace attributed to King Narmer, c.-3100, with a record, possibly exaggerated, of 120,000 prisoners, 400,000 oxen and 1,422,000 goats. See also: Egyptian Mathematics and Egyptian Fractions.

The best preserved papyrus with mathematical results, our chief source of information on Egyptian mathematics, is the Ahmes or Rhind papyrus & & & copied by the scribe Ahmes (aka Ahmose) in c.-1650 in hieratic script from earlier work dating possibly to c.-2000, and collected by Rhind. Another is the Moscow papyrus c.-1700, which contains results on volume of truncated pyramid.

The parallel but less politically stable civilisations of Mesopotamia, the Akkadian, Sumerian and Babylonian, developed the cuneiform writing system. In this a positional number system based on 60 appeared, from which many tables of numerical results have survived, in particular a clay tablet of the Old Babylonian period c.-1900/-1600 known as Plimpton 322 & which shows figures relating to the right triangle rule (now known as 'Pythagorean triples'). According to MacTutor "Geometric problems relating to similar figures, area and volume were also studied and values obtained for pi."

The visual representation of men and animals in the form of cave art date from c.-30,000 in Europe and reach a high point c.-15,000 in the cave paintings at Lascaux, France. Similarly familiarity with three-dimensional shapes is evident in the craft of the potter; the earliest surviving pottery dates from c.-12,000 in Jomon, Japan, and the potter's wheel was known in mesopotamia by c.-4,500.

From c.-4,500 date crafts such as sailing, ploughing, and permanent building; megalithic structures in western Europe; towns and cities, including public buildings such as palaces and temples, develop in mesopotamia c.-4,300. Wheeled vehicles were invented in the Middle East, S. Asia and Europe c.-3,700.

The Egyptian obsession with pyramids is well known and began with Djoser c.-2,650 who was the builder of the step pyramid at Saqqara, and whose architect, physician and advisor Imhotep & is the first name other than a ruling monarch known to history. Pyramids were built by pharaohs for 1000 years, over 100 are known. The founder of the 4th dynasty Snefru & built at least four. The three great pyramids at Giza were built for other rulers of the 4th dynasty, namely: Khufu &, aka Cheops, -2,580; builder of the Great pyramid; Djedefre -2,560; who built a pyramid but on a different site, now poorly preserved; Khafre, aka Chephren -2,540, builder of the second of the Great Pyramids of Giza, and possibly of the Great Sphinx &, although this may still be uncertain; Menkaure -2,510, builder of the third and smallest of the Great Pyramids at Giza.

In Lothal, c.-2400, the ancient port city of the Harappan civilization in the Indus valley, certain shell objects have been found that are claimed to be an early form of protractor, measuring angles in multiples of 40 degrees, up to 360 degrees.

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## Greek Mathematics

Notes from MacTutor: Independent development of mathematics by the Greeks began from around -450. 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 astronomy, for example the study of trigonometry. The major Greek progress in mathematics was from -300 to +200.

Thales & & (c.-625 - c.-545), predicted an eclipse of thew sun (-586).

Anaximander (c.-610 - c.-545) &.

Anaximenes (d.-528) &.

These philosophers, all of Miletus, taught more widely than just ethics, attempting speculations, often wildly imaginative, on the nature of the cosmos.

Pythagoras (fl.c.-530), taught that "all is number", he and his followers the Pythagoreans developed propositional geometry, investigated proportions in music of lyre strings.

Zeno of Elea (c.-490 - c.-430) - originator of the paradoxes of infinity.

Plato & & & & (-427 - -347) describes the trial of Socrates and features him as a speaker in his Dialogues.

Archytas (c.-428 - c.-350) mathematician interested in the problem of duplication of the cube.

Thaetetus (c.-417 - c.-369), mathematician, discoverer of the icosahedron (completing the five 'Platonic' solids).

Eudoxus (-408 - -355), believed to have originated the ideas in Euclid Book V for dealing with incommensurable magnitudes (like the side and diagonal of a square), used the method of exhaustion for areas.

Heraclides (-387 - -312), postulated a rotating earth.

Aristotle & & & & & & (-384 - -322), his works include Physics, Metaphysics, Nichomachean Ethics, Politics, Poetics, De Anima. His groundbreaking work on Logic, known as the Organon, consists of six titles: Prior Analytics, Posterior Analytics, On Interpretation, Topics, Sophistical Refutations, and Categories.

Euclid & & (c.-325 - -265), compiled all the Elements & of arithmetic and geometry known at his time into a systematic logical development. Other works: Catoptrics included the law of reflection.

Archimedes & (-287 - -212), mathematician and engineer, The Sandreckoner deals with very large numbers, On Spirals, On plane equilibria principle of the lever and centres of gravity, Quadrature of the parabola area by method of exhaustion, On the sphere and cylinder determines volumes and areas, On conoids and spheroids, On floating bodies Archimedes' principle of buoyancy, Measurement of a circle approximation to pi, The Method.

Aristarchus of Samos & & & & & (c.-310 - c.-230), On the Sizes and Distances of the Sun and Moon, he also according to Archimedes proposed a sun-centred universe, though not in this book.

Eratosthenes (-276 - -194), made a determination of the radius of the Earth.

Apollonius of Perga (c.-262 - c.-190), Conics names the ellipse, parabola, and hyperbola.

Hipparchus & (-190 - -120), astronomer: -150 used parallax to determine the Moon is roughly 380,000 km away, -134 makes a detailed star map, using a 'magnitude' scale of luminosity, and discovers the precession of the equinoxes from knowledge of earlier observations of Babylonian astronomers.

The Antikythera Mechanism & & &, recovered in 1902 from an ancient Greek shipwreck near the island of Antikythera, is a clockwork type mechanism, dated to 100 to 150 BC, apparently including sufficient data to predict an eclipse, and possibly part of a more elaborate model of the planetary motions. "The mechanism is the oldest known complex scientific instrument. It has several accurate scales, and is essentially an analog computer made with gears. It is based on theories of astronomy and mathematics developed by Greek astronomers."

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## Early Mediaeval Mathematics

Heron, aka Hero of Alexandria (1st century) mathematician and inventor, Metrica, Mechanics, and Pneumatics.

Claudius Ptolemy & & (c.85 - 165). made astronomical observations 127-141, compiled and systematised the knowledge of his day: Harmonics on music, Optics on light, including angles of refraction for several media, Geographia including map projections, Tetrabiblos on astrology, Planetary Hypothesis on cosmology, Syntaxis or 'Almagest' on astronomy.

The Nine Chapters c.210, a Chinese mathematical work.

Diophantus (fl.3rd century) Alexandrian mathematician, Arithmetica introduced 'diophantine equations' in theory of numbers.

450 Tsu Ch'ung-Chih and Tsu K^eng-Chih compute pi to six decimal places

Aryabhata the Elder (476 - 550) Aryabhatiya & Indian mathematician and astronomer, uses Sanskrit syllabic numeration.

550 Hindu mathematicians give zero a numeral representation in a positional notation system

Boethius & & & & (480 - 525).

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.

Alcuin (735 - 804) educationalist, calligrapher.

"A Persian astronomer Mohammad al-Fazari &, and his father Ibrahim, c.750, investigated the plane astrolabe, and built an example, an astronomical instrument and analog computer that was important in locating and predicting the positions of the Sun, Moon, Planets and stars." [But see Antikythera Mechanism -100].

Al Khwarizmi (c.780 - c.850) mathematician, from whose name we derive the term "algorithm", and from the title of whose main work we derive the word "algebra".

al Kindi & & & mathematician, maintained that space and time must be finite, since infinity leads to paradoxes.

al-Battani & aka Albategnius (853 - 929), astronomer who calculated accurate values for the length of the year, precession of equinoxes, and inclination of Earth's axis, his work is cited by Copernicus.

as Suli (880 - 946) shatranj (chess) expert, solved the knight's tour problem.

Ibn Yunus (c.950 - 1009) astronomer al-Zij al-Hakimi al-Kabir (Astronomical Handbook) and Kitab ghayat al-intifa' (Book of Useful Tables).

al Haitham & & & & & & aka Alhazen (965 - 1040), physicist who studied light and vision Kitab-al-Manadhir (Book of Optics).

Ibn Sina aka Avicenna (980 - 1037) physician and philosopher.

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## Later Mediaeval 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.

Omar Khayyam & & (c.1048 - c.1122) Persian mathematician, astronomer and poet, The Rubaiyat &.

Adelard of Bath & (1075 - 1160), traveller and scholar.

Gerard of Cremona (c.1114 - 1187), translated 87 works from Arabic to Latin, including editions of Euclid's Elements, Ptolemy's Almagest and Al Khwarizmi's Algebra.

Robert Grosseteste (1175 - 1253).

Leonardo of Pisa, aka Fibonacci & & & (1170 - 1250), Liber Abbaci 1202 and 1228 introduced Hindu decimal numerals, for use in conjunction with the Abacus, in place of the Roman numerals then in use in Europe. The important series now known by his name appears in a problem about breeding rabbits, Practica Geometriae 1220, Liber Quadratorum and Flos 1225 solved some quadratic and cubic equations.

Johannes de Sacrobosco aka John of Halifax or Holywood (d.c.1256), author of widely used school texts on arithmetic Algorismus and astronomy De Sphaera Mundi.

William of Ockham & & (1287 - 1347) commentaries with extracts and links.

Jean Buridan & (1300 - 1358). Secular philosopher and logician, Summulae de dialectica [Compendium of Dialectic]. He developed the concept of impetus, the first step toward the modern concept of inertia. His name is most familiar through the thought experiment known as Buridan's ass.

Nicolas Oresme & (1323 - 1382). Anticipated Descartes in use of coordinates in geometry. He proved the divergence of the harmonic series.

Filippo Brunelleschi (1377 - 1446) architect of the Dome of Santa Maria del Fiore in Florence.

Ghiyath al-Kashi (c.1380 - 1429), Treatise on the circumference 1424 computes 2pi to sixteen decimal places using inscribed and cirumscribed polygons.

Leone Battista Alberti (1404 - 1472), De Pictura 1435 on laws of perspective.

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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. The progress in algebra had a major effect and enthusiasm for mathematical research, spread from Italy to Stevin in Belgium and Viète in France. The equals sign "=" was introduced by Recorde in 1557. Negative and imaginary numbers began to be introduced, in particular by Bombelli 1572.

Luca Pacioli (1445 - 1517), Summa de arithmetica, geometria, proportioni et proportionalita 1494, encyclopedic textbook explaining double-entry book-keeping and summarising the mathematics of Euclid, Boethius, Sacrobosco and Fibonacci, Divina proportione 1509, illustrated by da Vinci.

Leonardo da Vinci & & & (1452 - 1519), artist and scientist.

Scipione del Ferro (1465 - 1526), first to solve cubic equations 1520.

Nicolo Fontana aka Tartaglia (1500 - 1557), solved cubic equations (independent of the earlier work of del Ferro), Quesiti et Inventioni 1546.

Girolamo Cardano (1501 - 1576), physician, mathematician, gambler, Ars Magna 1545 solution of cubic and quartic equations.

Georg Joachim Rheticus (1514 - 1574), Narratio Prima 1540 on Copernican system, Opus Palatinum de triangulis trigonometric tables, completed by Valentine Otho 1596.

Lodovico Ferrari (1522 - 1565), solves the quartic equation 1540.

Rafael Bombelli (1526 - 1572), Algebra 1572, uses negative and complex numbers.

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An important device, the vernier, which allows the accurate mechanical measurement of angles and distances was invented by Pierre Vernier in 1631. Also important for the advancement of natural philosophy were the mathematical techniques of logarithms and slide rules. The 17th Century saw Napier, Briggs and others greatly extend the power of mathematics as a calculatory science with the discovery of logarithms. Laboriously compiled logarithmic tables were used for 350 years as the main tool in arithmetical calculations. An amazing amount of effort was saved using logarithms, how could the heavy calculations necessary in the sciences ever have taken place without logs. The logarithm remains an important mathematical function but its use in calculating has gone for ever with the advent of electronic calculators and computers.

Ludolf van Ceulen (1540 - 1610), computed pi to 35 decimal places On The Circle 1596.

John Napier (1550 - 1617), natural logarithms Mirifici Logarithmorum Canonis Descriptio 1614.

Thomas Harriot (1560 - 1621).

Henry Briggs (1561 - 1630), decimal logarithms Logarithmorum Chilias Prima 1617.

Johannes Kepler & & & & & & (1571 - 1630), Mysterium Cosmographicum 1596, Astronomia Pars Optica 1604, De Stella Nova 1606, Astronomia Nova gave the first two laws of planetary motion 1609, Dioptrice 1611, Nova Stereometria Dolorium 1615 uses primitive calculus to estimate volumes of revolution of wine barrels, Harmonice Mundi includes work on polyhedra, and the third law of planetary motion relating size and period of orbits 1619, Epitome Astronomiae 1621 influential textbook, Rudolphine Tables accurate astronomical data 1627.

William Oughtred (1574 - 1660), Clavis Mathematicae 1631, slide rule.

Claude Gaspar Bachet Sieur de Meziriac (1581 - 1638), Problèmes plaisans et delectables 1612.

Descartes added the power of algebraic methods to geometry. The convention of using letters near the end of the alphabet representing unknowns was introduced by Descartes in 1637.

Progress towards the calculus continued with Cavalieri who used "indivisibles" and Fermat who developed a theory of maxima and minima. Fermat, together with Pascal, began the mathematical study of probability.

Rene Descartes & & & & & (1596 - 1650), Discourse on Method & &.

Bonaventura Cavalieri (1598 - 1647), Geometria indivisibilibus 1635.

Pierre de Fermat (1601 - 1665), number theorist.

Gilles Roberval & (1602 - 1675) mathematician, founding member of the Academie Royale des Sciences, Paris 1666, made contributiuons to development of calculus Traité des indivisibles, but not published until 1693.

John Wallis (1616 - 1703), Arithmetica Infinitorum 1656.

Blaise Pascal & & & (1623 - 1662), mathematician and theologian Pensees.

Christian Huygens & & (1629 - 1695), Horologium Oscillatorium 1673, Treatise on Light 1690.

Marin Mersenne & & & & & (1588 - 1648) Minim Friar, mathematician and organiser of scientific meetings and correspondence with many scholars, L'harmonie universelle on music and physics of sound 1627, studied prime Mersenne numbers of the form 2^n - 1 with n prime.

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## Calculus

The methods of calculus and their applications developed significabtly from 1660 onwards. Newton, building on the work of many earlier mathematicians such as his teacher Barrow, developed the method he called "fluxions" to deal with the varying quantities, found in the study of mechanics and astronomy. Leibniz at the same time developed an approach based on "infinitesimals", and introduced the notations for integration and differentiation now most commonly used. Leibniz's notation for calculus led more easily to extending the ideas of the calculus to functions of two or more variables. His work was taken up by the brothers Jacob and Johann Bernoulli and later by Johann's son Daniel and student Euler among many others.

Isaac Newton & & & & & & & (1642 - 1727), Mathematical Principles of Natural Philosophy & 1687.

Gottfried Wilhelm von Leibniz & & & (1646 - 1716) philosopher and mathematician, calculus.

Jacob Bernoulli (1654 - 1705) applied calculus to non-algebraic curves such as the logarithmic spiral and catenary, Ars Conjectandi 1713, tried to provide a rigorous proof of the "law of large numbers" to justify probability theory.

Johann Bernoulli (1667 - 1748) cooperated, and competed, with his older brother Jacob, to develop the theory of differential equations, father of Daniel.

Brook Taylor & (1685 - 1731), Methodus Incrementorum 1715, Taylor's series for f(x + h).

Colin Maclaurin (1698 - 1746), Treatise of Fluxions 1742, Maclaurin's series.

Daniel Bernoulli & (1700 - 1782) applied calculus to fluid mechanics (Hydrodynamica 1738), applied probability to calculation of risk, and statistics to smallpox data.

Leonhard Euler & & (1707 - 1783). The most prolific mathematician of the 18th Century. His works standardised much of the notation and terminology of mathematics. He developed two new branches of calculus; the calculus of variations and differential geometry. Euler was also important in pushing forward with research in number theory begun so effectively by Fermat.

Joseph-Louis Lagrange (1736 - 1813), Euler's successor in Berlin, Lagrange began a rigorous theory of functions and of mechanics.

Pierre-Simon Laplace & & & (1749 - 1827), Mecanique Celeste consolidated all the work of the previous century.

Major progress in synthetic geometry by Monge and Carnot.

Fourier's work on heat and analysis of functions.

Plücker produced fundamental work on analytic geometry and Steiner in synthetic geometry.

Non-euclidean geometry developed by Lobachevsky and Bolyai led to characterisation of geometry by Riemann.

Gauss 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.

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. Analysis was driven by the requirements of mathematical physics and astronomy.

Algebraic geometry was carried forward by Cayley whose work on matrices and linear algebra complemented that by Hamilton and Grassmann.

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.

## Discrete Logic and Electronic Computers

The concepts necessary for the development of programmable computers were pioneered in the work of Babbage and Boole. At the end of the 19th century Frege and Cantor invent set theory and analyze the concept of number.

Charles Babbage (1791 - 1871), inventor of a difference engine, precursor of programmable computers.

George Boole (1815 - 1864), mathematical logic.

Bertrand Russell (1872 - 1970), Russell's Paradox (1901), Principles of Mathematics (1903).

Kurt Godel

Further understanding of the physics of metals, semiconductors and insulators led a team of three men at Bell labs, William Shockley, Walter Brattain and John Bardeen in 1947 to the first transistor and then to many important variations, especially the bipolar junction transistor. Further developments in the fabrication and miniaturization of integrated circuits in the years to come produced fast, compact computers that came to revolutionize the way physics was done--simulations and complex mathematical calculations became possible that were undreamed of even a few decades previous.

Alan Turing & & & & & & & & (1912 - 1954), Computing Machinery and Intelligence 1950.

Gregory Chaitin & (1947 - ), Godel's Theorem and Information 1982.

Stephen Wolfram & & (1959 - ), A New Kind of Science, Reviews by S. Weinberg and D. Naiditch.

## Websites on History of Mathematics

D. E. Joyce, Clark University
Math Archives, University of Tennessee
History of Computers
History of Computers 1820 -
Wikipedia on History of Mathematics
Think Quest development of mathematics (translation from Korean)
Who was Fibonacci?
Ancient Geometry
The MacTutor History of Mathematics Archive.
Sheffield College
British Society for the History of Mathematics
MacTutor Overview
Chronology of Mathematics &.
MacTutor History of Mathematics
Intute
D. R. Wilkins, Trinity College Dublin
W. W. Rouse Ball biographies of mathematicians of 17th and 18th centuries.