MATHEMATICS, HISTORY OF

history of the field from ancient times to the present. As a consequence of the exponential growth of science most mathematics has developed since the 15th century AD, and it is a historical fact that from the 15th century to the late 20th century new developments in mathematics have been largely concentrated in Europe and North America. For these reasons the bulk of this article is devoted to European developments since 1500. This does not mean, however, that developments elsewhere have been unimportant. Indeed, to understand the history of mathematics in Europe it is necessary to know its history at least in Mesopotamia and Egypt, in ancient Greece, and in Islamic civilization from the 9th to the 15th centuries. The way in which these civilizations influenced one another, and the important direct contributions Greece and Islam made to later developments, are discussed in the first parts of this article. India's contributions to the development of contemporary mathematics were made through the considerable influence of Indian achievements on Isl amic mathematics during its formative years. In order to provide a portrait of the mathematical achievements of one major Asian civilization, the article contains an overview of some of the principal periods and achievements of mathematics in China. It is important to be aware of the character of the sources for the study of the history of mathematics. The history of Mesopotamian and Egyptian mathematics is based on the many extant original documents written by scribes. Although in the case of Egypt these documents are few, they are all of a type and leave little doubt that Egyptian mathematics was, on the whole, elementary and profoundly practical in its orientation. For Mesopotamian mathematics, on the other hand, there are a large number of clay tablets, which reveal mathematical achievements of a much higher order than those of the Egyptians. The tablets indicate that the Mesopotamians had a great deal of remarkable mathematical knowledge, although they offer no evidence that this knowledge was organized into a deductive system. Future research may reveal more about the early development of mathematics in Mesopotamia or about its influence on Greek mathematics, but it seems likely that this picture of Mesopotamian mathematics will stand. From the period before Alexander the Great no Greek mathematical documents have been preserved except for fragmentary paraphrases, and even for the subsequent period it is well to remember that the oldest copies of Euclid's Elements are in Byzantine manuscripts dating from the 10th century AD. This stands in complete contrast to the situation described above for Egyptian and Babylonian documents. Although in general outline the present account of Greek mathematics is secure, in such important matters as the origin of the axiomatic method, the pre-Euclidean theory of ratios, and the discovery of the conic sections, historians have given competing accounts based on fragmentary texts, quotations of early writings culled from nonmathematical sources, and a considerable amount of conjecture. Many important treatises from the early period of Islamic mathematics have not survived or have survived only in Latin translations, so that there are still many unanswered questions about the relationship between early Islamic mathematics and the mathematics of Greece and India. In addition, the amount of surviving material from later centuries is so large in comparison with that which has been studied that it is not yet possible to offer any sure judgment of what medieval Islamic mathematics did not contain, and this means that it is not yet possible to evaluate with any assurance what was original in European mathematics from the 11th to the 15th century. In modern times 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. However, the exponential growth of mathematics means that, for the period from the 19th century on, historians are able to treat only the major figures in any detail. In addition there is, as the period gets nearer the present, the problem of perspective. Mathematics, like any other human activity, has its fashions, and the nearer one is to a given period, the more likely these fashions are to look like the wave of the future. For this reason, the present article makes no attempt to assess the most recent developments in the subject. John L. Berggren Additional reading General sources Two standard texts are Carl B. Boyer, A History of Mathematics, 2nd ed. edited by Uta C. Merzbach (1989); and, on a more elementary level, Howard Eves, An Introduction to the History of Mathematics, 5th ed. (1983). Discussions of the mathematics of various periods may be found in O. Neugebauer, The Exact Sciences in Antiquity, 2nd ed. (1957, reissued 1969); Morris Kline, Mathematical Thought from Ancient to Modern Times (1972); and Bartel L. van der Waerden, Science Awakening, 4th ed. (1975; originally published in Dutch, 1950). See also Kenneth O. May, Bibliography and Research Manual of the History of Mathematics (1973); and Joseph W. Dauben, The History of Mathematics from Antiquity to the Present: A Selective Bibliography (1985). A good source for biographies of mathematicians is Charles Coulston Gillispie (ed.), Dictionary of Scientific Biography, 16 vols. (197080, reissued 16 vol. in 8, 1981). Those wanting to study the writings of the mathematicians themselves will find the following source books useful: Henrietta O. Midonick (ed.), The Treasury of Mathematics: A Collection of Source Material in Mathematics (1965); John Fauvel and Jeremy Gray (eds.), The History of Mathematics: A Reader (1987); D.J. Struik (ed.), A Source Book in Mathematics, 12001800 (1969); and David Eugene Smith, A Source Book in Mathematics (1929; reissued in 2 vol., 1959). A study of the development of numeric notation can be found in Georges Ifrah, From One to Zero (1985; originally published in French, 1981). Mathematics in ancient Mesopotamia Editions of mathematical tablets include O. Neugebauer (ed. and trans.), Mathematische Keilschrift-Texte, 3 vol. (193537, reprinted 3 vol. in 2, 1973); and F. Thureau-Dangin (ed. and trans.), Textes mathmatiques babyloniens (1938). O. Neugebauer and A. Sachs, Mathematical Cuneiform Texts (1945), is the principal English edition of mathematical tablets. A brief look at Babylonian mathematics is contained in the first chapter of Asger Aaboe, Episodes from the Early History of Mathematics (1964), pp. 531. Mathematics in ancient Egypt Editions of the basic texts are T. Eric Peet (ed. and trans.), The Rhind Mathematical Papyrus (1923, reprinted 1970); A.B. Chace et al. (eds. and trans.), The Rhind Mathematical Papyrus, 2 vol. (192729); and W.W. Struve (V.V. Struve) (ed.), Mathematischer papyrus des staatlichen Museums der schnen Knste in Moskau (1930). A brief but useful summary appears in G.J. Toomer, Mathematics and Astronomy, ch. 2 in J.R. Harris (ed.), The Legacy of Egypt, 2nd ed. (1971), pp. 2754. For an extended account of Egyptian mathematics, see Richard J. Gillings, Mathematics in the Time of the Pharaohs (1972, reprinted 1982). Greek mathematics Critical editions of Greek mathematical texts include The Thirteen Books of Euclid's Elements, trans. by Thomas L. Heath, 2nd ed. rev., 3 vol. (1926, reprinted 1956); The Works of Archimedes, trans. by Thomas L. Heath (1897, reprinted 1953); E.J. Dijksterhuis, Archimedes, trans. from Dutch (1956, reprinted 1987); Thomas L. Heath, Apollonius of Perga: Treatise on Conic Sections (1896, reissued 1961), and Diophantus of Alexandria: A Study in the History of Greek Algebra, 2nd ed. (1910, reprinted 1964); Roshdi Rashed (trans.), Les Arithmtiques (1984 ), of which vol. 3 and 4 contain Books IVVII of Diophantus; and Jacques Sesiano, Books IV to VII of Diophantus' Arithmetica in the Arabic Translation Attributed to Qusta ibn Luq (1982). General surveys are Thomas L. Heath, A History of Greek Mathematics, 2 vol. (1921, reprinted 1981); Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1968; originally published in German, 1934); and Wilbur Richard Knorr, The Ancient Tradition of Geometric Problems (1986). Special topics are examined in O.A.W. Dilke, Mathematics and Measurement (1987); rpd Szab, The Beginnings of Greek Mathematics (1978; originally published in German, 1969); and Wilbur Richard Knorr, The Evolution of the Euclidean Elements: A Study of the Theory of Incommensurable Magnitudes and Its Significance for Early Greek Geometry (1975). Mathematics in medieval Islam Sources for Arabic mathematics include J.P. Hogendijk (ed. and trans.), Ibn Al-Haytham's Completion of the Conics, trans. from Arabic (1985); Martin Levey and Marvin Petruck (eds. and trans.), Principles of Hindu Reckoning, trans. from Arabic (1965), the only extant text of Kushyar ibn Labban's work; Martin Levey (ed. and trans.), The Algebra of Abu Kamil, trans. from Arabic and Hebrew (1966), with a 13th-century Hebrew commentary by Mordecai Finzi; Daoud S. Kasir (ed. and trans.), The Algebra of Omar Khayyam, trans. from Arabic (1931, reprinted 1972); Frederic Rosen (ed. and trans.), The Algebra of Mohammed ben Musa, trans. from Arabic (1831, reprinted 1986); and A.S. Saidan (ed. and trans.), The Arithmetic of al-Uqlidisi, trans. from Arabic (1978). Islamic mathematics is examined in J.L. Berggren, Episodes in the Mathematics of Medieval Islam (1986); E.S. Kennedy et al., Studies in the Islamic Exact Sciences (1983); and Roshdi Rashed, Entre arithmtique et algbre: recherches sur l'histoire des mathmatiques arabes (1984). European mathematics during the Middle Ages and Renaissance An overview is provided by Michael S. Mahoney, Mathematics, in David C. Lindberg (ed.), Science in the Middle Ages (1978), pp. 145178. A.P. Juschkewitsch, Geschichte der Mathematik im Mittelalter (1964; originally published in Russian, 1961), pp. 326434, is the definitive modern work. Other sources include Alexander Murray, Reason and Society in the Middle Ages (1978, reprinted 1985), ch. 68; George Sarton, Introduction to the History of Science, vol. 2, From Rabbi Ben Ezra to Roger Bacon, 2 parts (1931, reprinted 1975), and vol. 3, Science and Learning in the Fourteenth Century, 2 parts (194748, reprinted 1975); and, on a more advanced level, Edward Grant and John E. Murdoch (eds.), Mathematics and Its Applications to Science and Natural Philosophy in the Middle Ages (1987). For the Renaissance, see Paul Lawrence Rose, The Italian Renaissance of Mathematics: Studies on Humanists and Mathematicians from Petrarch to Galileo (1975). Mathematics in the 17th and 18th centuries An overview of this period is contained in Derek Thomas Whiteside, Patterns of Mathematical Thought in the Later Seventeenth Century, Archive for History of Exact Sciences, 1(3):179388 (1961). Specific topics are examined in Margaret E. Baron, The Origins of the Infinitesimal Calculus (1969, reprinted 1987); Roberto Bonola, Non-Euclidean Geometry: A Critical and Historical Study of Its Development (1955; originally published in Italian, 1912); Carl B. Boyer, The Concepts of the Calculus: A Critical and Historical Discussion of the Derivative and the Integral (1930, reissued with the title The History of the Calculus and Its Conceptual Development, 1949, reprinted 1959); Herman Goldstine, A History of Numerical Analysis from the 16th Through the 19th Century (1977); Judith V. Grabiner, The Origins of Cauchy's Rigorous Calculus (1981); I. Grattan-Guinness, The Development of the Foundations of Mathematical Analysis from Euler to Riemann (1970); Roger Hahn, The Anatomy of a Scientific Institution: The Paris Academy of Sciences, 16661803 (1971); and Lubo Nov, Origins of Modern Algebra, trans. from Czech (1973). Mathematics in the 19th and 20th centuries Surveys include Herbert Mehrtens, Henk Bos, and Ivo Schneider (eds.), Social History of Nineteenth Century Mathematics (1981); William Aspray and Philip Kitcher (eds.), History and Philosophy of Modern Mathematics (1988); and Keith Devlin, Mathematics: The New Golden Age (1988). Special topics are examined in Umberto Bottazzini, The Higher Calculus: A History of Real and Complex Analysis from Euler to Weierstrass (1986; originally published in Italian, 1981); Julian Lowell Coolidge, A History of Geometrical Methods (1940, reissued 1963); Joseph Warren Dauben, Georg Cantor: His Mathematics and Philosophy of the Infinite (1979); Harold M. Edwards, Fermat's Last Theorem: A Genetic Introduction to Algebraic Number Theory (1977); I. Grattan-Guinness (ed.), From the Calculus to Set Theory, 16301910: An Introductory History (1980); Jeremy Gray, Ideas of Space: Euclidian, Non-Euclidean, and Relativistic (1979); Thomas Hawkins, Lebesgue's Theory of Integration: Its Origins and Development, 3rd ed. (1979); Jesper Ltzen, The Prehistory of the Theory of Distributions (1982); and Michael Monastyrsky, Riemann, Topology, and Physics, trans. from Russian (1987). Mathematics in China and Japan Chinese mathematics is discussed in Joseph Needham, Science and Civilisation in China, vol. 3, Mathematics and the Sciences of the Heavens and the Earth (1959, reprinted 1970), pp. 1168; Ancient China's Technology and Science (1983), a group of papers prepared by the Institute of the History of Natural Sciences, Chinese Academy of Science, in Peking; Yan Li (Yen Li) and Shiran Du (Shih-jan Tu), Chinese Mathematics: A Concise History, trans. from Chinese (1987); Ulrich Libbrecht, Chinese Mathematics in the Thirteenth Century (1973); and Lay Yong Lam, A Critical Study of the Yang Hui Suan Fa: A Thirteenth-Century Chinese Mathematical Treatise, trans. from Chinese (1977). Useful journal articles include Donald Blackmore Wagner, An Early Chinese Derivation of the Volume of a Pyramid: Liu Hui, Third Century A.D., Historia Mathematica, 6(2):164188 (May 1979), and Liu Hui and Tsu Keng-Chih on the Volume of a Sphere, Chinese Science, 3:5979 (1978). Besides Historia Mathematica (quarterly) and Chinese Science (irregular), many papers on Chinese mathematics may be found in Archive for History of Exact Sciences (8/yr.).Overviews of Japanese mathematics include David Eugene Smith and Yoshio Mikami, A History of Japanese Mathematics (1914); Yoshio Mikami, The Development of Mathematics in China and Japan, 2nd ed. (1974); and Shigeru Nakayama, Japanese Scientific Thought, in Charles Coulston Gillispie (ed.), Dictionary of Scientific Biography, vol. 15 (1978), pp. 728758. Historia Scientiarum: International Journal of the History of Science Society of Japan (annual), contains many papers on Japanese mathematics. John L. Berggren Wilbur R. Knorr Menso Folkerts Craig G. Fraser Jeremy John Gray Karine Carole Chemla European mathematics during the Middle Ages and Renaissance Until the 11th century only a small part of the Greek mathematical corpus was known in the West. Because almost no one could read Greek, what little was available came from the poor texts written in Latin in the Roman Empire, together with the very few Latin translations of Greek works. Of these the most important were the treatises by Boethius, who in about AD 500 made Latin redactions of a number of Greek scientific and logical writings. His Arithmetic, which was based on Nicomachus, was well known and was the means by which medieval scholars learned of Pythagorean number theory. Boethius and Cassiodorus provided the material for the part of the monastic education called the quadrivium: arithmetic, geometry, astronomy, and music theory. Together with the trivium (grammar, logic, rhetoric), these subjects formed the seven liberal arts, which were taught in the monasteries, cathedral schools, and, from the 12th century on, in the universities and which constituted the principal university instruction until modern times. For monastic life it sufficed to know how to calculate with Roman numerals. The principal application of arithmetic was a method for determining the date of Easter, the computus, that was based on the lunar cycle of 19 solar years (i.e., 235 lunar revolutions) and the 28-year solar cycle. Between the time of Bede (d. 735), when the system was fully developed, and about 1500, the computus was reduced to a series of verses that were learned by rote. Until the 12th century, geometry was largely concerned with approximate formulas for measuring areas and volumes in the tradition of the Roman surveyors. About AD 1000 the French scholar Gerbert of Aurillac, later Pope Sylvester II, introduced a type of abacus, in which numbers were represented by stones bearing Arabic numerals. Such novelties were known to very few. The transmission of Greek and Arabic learning In the 11th century a new phase of mathematics began with the translations from Arabic. Scholars throughout Europe went to Toledo, Crdoba, and elsewhere in Spain to translate into Latin the accumulated learning of the Muslims. Along with philosophy, astronomy, astrology, and medicine, important mathematical achievements of the Greek, Indian, and Islamic civilizations became available in the West. Particularly important were Euclid's Elements, the works of Archimedes, and al-Khwarizmi's treatises on arithmetic and algebra. Western texts called algorismus (a Latin form of the name al-Khwarizmi), introduced the Hindu-Arabic numerals and applied them in calculations. Thus modern numerals first came into use in universities and then became common among merchants and other laymen. It should be noted that, up to the 15th century, calculations were often performed with board and counters. Reckoning with Hindu-Arabic numerals was used by merchants at least from the time of Leonardo of Pisa (beginning of the 13th century) first in Italy, then in the trading cities of southern Germany and France, where maestri d'abbaco or Rechenmeister taught commercial arithmetic in the various vernaculars. Some schools were private, while others were run by the community. Mathematics in China and Japan When speaking of mathematics in East Asia, it is necessary to take into account its development in China, Korea, and Japan as a whole. Mathematicians from these countries can be considered as part of the same working community. Moreover, these scholars usually wrote with Chinese characters and thus could read one another's texts. The common basis for the development of mathematics in East Asia was books that were written in China from the 1st through the 13th century, to which most of the subsequent works refer. For this period, the two main sources, the references found in the texts and those given in the bibliographies compiled for the dynastic annals, indicate that there are many lacunae in the books that have survived. The oldest works extant survived because they became official books. In the 17th century few ancient Chinese mathematical works were known, and those that were known were not fully understood. Thereafter, as Chinese mathematicians became aware of European achievements, they began to look for such works throughout the country. Editions of the texts they found began to appear at the end of the 18th century and have become the main sources for the history of Chinese mathematics. Discoveries of new sources are now rare, contrary to the situation for Arabic mathematics, for example. Nevertheless, in the 20th century a mathematical book was discovered in a grave dating from the 2nd or 3rd century, which pushed back by more than three centuries what was previously known about the subject. Chinese mathematics to the 13th century Outline of the history The Nine Chapters on the Mathematical Procedures (or the Nine Chapters) was probably compiled in the 1st century AD. This book gathered and organized many mathematical achievements from preceding periods. It played an important part in the development of mathematics in China, for all Chinese mathematicians refer to it and most of the subjects they have worked on stem from it. Its format, which was adopted by most subsequent authors, consists of problems for which a numerical answer and a procedure for solution are given. These problems are arranged in classes that come under the heading of a general method. Many scholars wrote commentaries on the Nine Chapters, adding explanations and proofs, rewriting procedures, and suggesting new formulas. The most important of the commentaries to survive, attributed to Liu Hui (3rd century), contains the richest set of proofs within this tradition. Some of the books written subsequently are known because, gathered together with the Nine Chapters and the astronomical treatise Mathematical Classic of the Gnomon of Chou and commented on in the 7th century by a group under the leadership of Li Ch'un-feng, they became the Ten Classics of Mathematics, the manual for officials trained in the then newly established office of mathematics. Although some people were thus officially trained as mathematicians, no major breakthrough seems to have been achieved until the 11th century. At that time (1084) the Ten Classics of Mathematics was edited and printed, and this seems to have been related to renewed activity in mathematics during the 11th and 12th centuries, known today only through later quotations. This period probably paved the way for the major achievements of Chinese tradition, as they are known today only through the few books that have come down from the second half of the 13th century. China was then divided into two countries, North and South, and achievements by mathematicians of both sides are known: in the South, those of Ch'in Chiu-shao and Yang Hui (who were minor officials), and in the North, those of Li Yeh (a recluse scholar) and Chu Shih-chieh (a wandering teacher). Their contributions seem to have been arrived at independently but they attest to a common basis. From the period after this, no valuable works survive, and there is no evidence that any important mathematical works were written. Still, some major works of the 13th century are recorded in the encyclopaedia compiled under the Yung-lo emperor (140224), but commentaries on these books written by the end of the 15th century show that by this time they were no longer understood. It was only in the 16th century that the abacus appears to have come into widespread use, and most of the books of the period discuss it. But it is not possible to date the moment when it actually appeared in China. The following discussion of the evolution of mathematical subjects within the Chinese tradition emphasizes several common characteristics of most of the achievements: a specific use of algorithms and the importance given to position, to configurations of numbers, and to parallelisms between procedures. Mathematics in medieval Islam Origins In Hellenistic times and in late antiquity, scientific learning in the eastern part of the Roman world was spread over a variety of centres, and Justinian's closing of the pagan academies in Athens in 529 gave further impetus to this diffusion. An additional factor was the translation and study of Greek scientific and philosophical texts sponsored both by monastic centres of the various Christian churches in the Levant, Egypt, and Mesopotamia and by enlightened rulers of the Sasanian dynasty in places like the medical school at Gondeshapur. Also important were developments in India in the first few centuries AD. Although the decimal system for whole numbers was apparently not known to the Indian astronomer Aryabhata I (b. 476), it was used by his pupil Bhaskara I in 620, and by 670 the system had reached northern Mesopotamia, where the Nestorian bishop Severus Sebokht praised its Hindu inventors as discoverers of things more ingenious than those of the Greeks. Earlier, in the late 4th or early 5th century, the anonymous Hindu author of an astronomical handbook, the Surya Siddhanta, had tabulated the sine function (unknown in Greece) for every 33/4 of arc from 33/4 to 90. Within this intellectual context the rapid expansion of Islam took place between the time of Muhammad's return to Mecca from his exile in Medina in 630 and the Muslim conquest of lands extending from Spain to the borders of China by 715. Not long afterward, Muslims began the acquisition of foreign learning, and, by the time of the caliph al-Mansur (d. 775), such Indian and Persian astronomical material as the Brahma-sphuta-siddhanta and the Shah's Tables had been translated into Arabic. The subsequent acquisition of Greek material was greatly advanced when the caliph al-Ma'mun constructed a translation and research centre, the House of Wisdom, in Baghdad during his reign (813833). Most of the translations were done from Greek and Syriac by Christian scholars, but the impetus and support for this activity came from Muslim patrons. These included not only the caliph but also wealthy individuals such as the three brothers known as the Banu Musa, whose treatises on geometry and mechanics formed an important part of the works studied in the Islamic world. Of Euclid's works, the Elements, the Data, the Optics, the Phaenomena, and On Divisions were translated. Of Archimedes' works only twoSphere and Cylinder and Measurement of the Circleare known to have been translated, but these were sufficient to stimulate independent researches from the 9th to the 15th century. On the other hand, virtually all of Apollonius' works were translated, and of Diophantus and Menelaus one book each, the Arithmetica and the Sphaerica, respectively, were translated into Arabic. Finally, the translation of Ptolemy's Almagest furnished important astronomical material. Of the minor writings, Diocles' treatise on mirrors, Theodosius' Spherics, Pappus' work on mechanics, Ptolemy's Planisphaerium, and Hypsicles' treatises on regular polyhedra (the so-called Books XIV and XV of Euclid's Elements) were among those translated. Mathematics in the 9th century Thabit ibn Qurrah (836901), a Sabian from Harran in northern Mesopotamia, was an important translator and reviser of these Greek works. In addition to translating works of the major Greek mathematicians (for the Banu Musa, among others), he was a court physician. He also translated Nicomachus of Gerasa's Arithmetic and discovered a beautiful rule for finding amicable numbers, a pair of numbers such that each number is the sum of the set of proper divisors of the other number. The investigation of such numbers formed a continuing tradition in Islam. Kamal ad-Din al-Farisi (d. c. 1320) gave the pair 17,926 and 18,416 as an example of Thabit's rule, and in the 17th century Muhammad Baqir Yazdi gave the pair 9,363,584 and 9,437,056. One scientist typical of the 9th century was Muhammad ibn Musa al-Khwarizmi. Working in the House of Wisdom, he introduced Indian material in his astronomical works and also wrote an early book explaining Hindu arithmetic, the Book of Addition and Subtraction According to the Hindu Calculation. In another work, the Book of Restoring and Balancing, he provided a systematic introduction to algebra, including a theory of quadratic equations. Both works had important consequences for Islamic mathematics. Hindu Calculation began a tradition of arithmetic books that, by the middle of the next century, led to the invention of decimal fractions (complete with a decimal point), and his Restoring and Balancing became the point of departure and model for later writers such as the Egyptian Abu Kamil. Both books were translated into Latin, and Restoring and Balancing was the origin of the word algebra, from the Arabic word for restoring in its title (al-jabr). The Hindu Calculation, from a Latin form of the author's name, algorismi, yielded the word algorithm. Al-Khwarizmi's algebra also served as a model for later writers in its application of arithmetic and algebra to the distribution of inheritances according to the complex requirements of Muslim religious law. This tradition of service to the Islamic faith was an enduring feature of mathematical work in Islam and one which, in the eyes of many, justified the study of secular learning. In the same category are al-Khwarizmi's method of calculating the time of visibility of the new moon (which signals the beginning of the Muslim month) and the expositions by astronomers of methods for finding the direction to Mecca for the five daily prayers. Mathematics in the 17th and 18th centuries The 17th century The 17th century, the period of the scientific revolution, witnessed the consolidation of Copernican heliocentric astronomy and the establishment of inertial physics in the work of Kepler, Galileo, Descartes, and Newton. This period was also one of intense activity and innovation in mathematics. Advances in numerical calculation, the development of symbolic algebra and analytic geometry, and the invention of the differential and integral calculus resulted in a major expansion of the subject areas of mathematics. By the end of the 17th century a program of research based in analysis had replaced classical Greek geometry at the centre of advanced mathematics. In the next century this program would continue to develop in close association with physics, more particularly mechanics and theoretical astronomy. The extensive use of analytic methods, the incorporation of applied subjects, and the adoption of a pragmatic attitude to questions of logical rigour distinguished the new mathematics from traditional geometry. Institutional background Until the middle of the 17th century, mathematicians worked alone or in small groups, publishing their work in books or communicating with other researchers by letter. At a time when people were often slow to publish, invisible colleges, networks of scientists who corresponded privately, played an important role in coordinating and stimulating mathematical research. Marin Mersenne in Paris acted as a clearinghouse for new results, informing his many correspondentsincluding Fermat, Descartes, Blaise Pascal, Gilles Personne de Roberval, and Galileoof challenge problems and novel solutions. Later in the century John Collins, librarian of London's Royal Society, performed a similar function among British mathematicians. In 1660 the Royal Society of London was founded, to be followed in 1666 by the Academy of Sciences in France, in 1700 by the Berlin Academy, and in 1724 by the St. Petersburg Academy. The official publications sponsored by the academies, as well as independent journals such as the Acta Eruditorum (founded in 1682), made possible the open and prompt communication of research findings. Although universities in the 17th century provided some support for mathematics, they became increasingly ineffective as state-supported academies assumed direction of advanced research. Mathematics in the 19th and 20th centuries Most of the powerful abstract mathematical theories in use today originated in the 19th century, so that any historical account of the period should be supplemented by reference to detailed treatments of these topics. Moreover, mathematics grew so much during this period that any account must necessarily be selective. Nonetheless, some broad features stand out. The growth of mathematics as a profession was accompanied by a sharpening division between mathematics and the physical sciences, and contact between the two subjects takes place today across a clear professional boundary. One result of this separation has been that mathematics, no longer able to rely on its scientific import for its validity, developed markedly higher standards of rigour. It was also freed to develop in directions that had little to do with applicability. Some of these pure creations have turned out to be surprisingly applicable, while the attention to rigour has led to a wholly novel conception of the nature of mathematics and logic. Moreover, many outstanding questions in mathematics yielded to the more conceptual approaches that came into vogue. Projective geometry The French Revolution provoked a radical rethinking of education in France, and mathematics was given a prominent role. The cole Polytechnique was established in 1794 with the ambitious task of preparing all candidates for the specialist civil and military engineering schools of the republic. Mathematicians of the highest calibre were involved; the result was a rapid and sustained development of the subject. The inspiration for the cole was that of Gaspard Monge, who believed strongly that mathematics should serve the scientific and technical needs of the state. To that end he devised a syllabus that promoted his own descriptive geometry, which was useful in the design of forts, gun emplacements, and machines and which was employed to great effect in the Napoleonic survey of Egyptian historical sites. In Monge's descriptive geometry, three-dimensional objects are described by their orthogonal projections onto a horizontal and a vertical plane, the plan and elevation of the object. A pupil of Monge, Jean-Victor Poncelet, was taken prisoner in Napoleon's retreat from Moscow and sought to keep up his spirits while in jail in Saratov by thinking over the geometry he had learned. He dispensed with the restriction to orthogonal projections and decided to investigate what properties figures have in common with their shadows. There are several of these properties: a straight line casts a straight shadow, and a tangent to a curve casts a shadow that is tangent to the shadow of the curve. But some properties are lost: the lengths and angles of a figure bear no relation to the lengths and angles of its shadow. Poncelet felt that the properties that survive are worthy of study, and, by considering only those properties that a figure shares with all its shadows, Poncelet hoped to put truly geometric reasoning on a par with algebraic geometry. In 1822 Poncelet published the Trait des proprits projectives des figures (Treatise on the Projective Properties of Figures). From his standpoint, every conic section is equivalent to a circle, so his treatise contained a unified treatment of the theory of conic sections. It also established several new results. Geometers who took up his work divided into two groups: those who accepted his terms and those who, finding them obscure, reformulated his ideas in the spirit of algebraic geometry. On the algebraic side it was taken up in Germany by August Ferdinand Mbius, who seems to have come to his ideas independently of Poncelet, and then by Julius Plcker. They showed how rich was the projective geometry of curves defined by algebraic equations and thereby gave an enormous boost to the algebraic study of curves, comparable to the original impetus provided by Descartes. Germany also produced synthetic projective geometers, notably Jakob Steiner (born in Switzerland but educated in Germany) and Karl George Christian von Staudt, who emphasized what can be understood about a figure from a careful consideration of all its transformations. Figure 3: Duality associates with the point P the line RS, and vice versa. Within the debates about projective geometry emerged one of the few synthetic ideas to be discovered since the days of Euclid, that of duality. This associates with each point a line and with each line a point, in such a way that (1) three points lying in a line give rise to three lines meeting in a point, and, conversely, three lines meeting in a point give rise to three points lying on a line, and (2) if one starts with a point (or a line) and passes to the associated line (point) and then repeats the process, one returns to the original point (line). One way of doing this (presented by Poncelet) is to pick an arbitrary conic and then to associate with a point P lying outside the conic the line that joins the points R and S at which the tangents through P to the conic touch the conic (see Figure 3). A second method is needed for points on or inside the conic. The feature of duality that makes it so exciting is that one can apply it mechanically to every proof in geometry, interchanging point and line and collinear and concurrent throughout, and so obtain a new result. Sometimes a result turns out to be equivalent to the original, sometimes to its converse, but at a single stroke the number of theorems was more or less doubled.