The Great Mathematicians
Rene Descartes (1596-1650)
Born near Tours into a good family, Descartes entered the Jesuit College at La Fleche when he was eight to receive his basic education. While there, he developed his lifelong habit of lying in bed until late in the morning, at first due to ill health, and later regarded these hours as his most productive. He took his degree at Poitier and went to Paris to study mathematics with Mersenne. In 1617, he began a number of years of travelling, mostly involved in various military campaigns, initially with Prince Maurice of Orange. However, he always sought out distinguished mathematicians with whom to converse. Eventually returning to Paris, he continued with his mathematics and his scientific philosophy, and then chose to move to Holland, where he stayed for twenty years. In 1649, he reluctantly accepted and invitation from Queen Christina of Sweden to instruct her in philosophy, and to establish an Academy of Sciences in Stockholm. The Scandinavian winter however was too much for him and he died early in 1650. Descartes' most successful period of mathematical output was during his stay in Holland. In particular, "La Geometrie" of 1637, a 3-part appendix to one of his philosophical discourses, introduces his arithmetisation of geometry, which we recognise as analytic, or Cartesian in his honour.
Pierre de Fermat (1601-1665)
Fermat was born near Toulouse, where he studied law, and served in local parliament, first as a lawyer, and then as councillor. Thus he was a busy man, but nevertheless found ample time to indulge in his hobbies, one of which was mathematics. Although Fermat published little in his lifetime, his correspondences with many of Europe's leading mathematicians of the time considerably influenced his contemporaries. Much of the credit here must go to Father Marin Mersenne, a friar and great instigator of research and correspondence, without whom mathematics would have suffered. Fermat was one of a number of mathematicians who paved the way for Newton and Leibnitz' invention of the calculus, with his studies of analytic geometry, and his techniques for dealing with tangents and areas (differentiation and integration respectively). He also co-founded the modern approach to probability, in correspondence with Pascal, but his personal delight and greatest legacy was in the theory of numbers, where for long he was the lone voice, and left a multitude of original results, although rarely providing any proofs, being content to state them. In particular, his "Last Theorem" took three hundred years and many advances in mathematics to prove. He died at Castres in 1665.
Blaise Pascal (1623-1662)
Born in Auvergne, Pascal was a child prodigy amd mathematical genius. At the early age of fourteen, he joined the informal Mersenne academy of meetings, and at this time was already producing results in geometry. In the next few years, he added to the study of conics and applied himself also to physics. In 1654, a question on chance by a gambling friend prompted him, in correspondence with Fermat, to provide the effective beginnings of modern probability. In fact, he had already started on a book about the arithmetical triangle, which he so developed in importance that it is now named after him, although it had been extant for hundreds of years before him. Late in 1654 however, Pascal abandoned mathematics for religion after what he took to be a divine revelation, and from then on devoted himself to religious philosophy. He did on occasion return to mathematics for ease of mind when he thought that divine will allowed and contributed a fairly full account of the geometry of the cycloid. However, it is to be regretted that he did not produce more in his short life, which ended in Paris in 1662.
Christiaan Huygens (1629-1695)
Born in The Hague, Huygens studied at Leyden under van Schooten. In 1651 he published his first article, and this was followed by a number of papers on conics and curves. He discovered the isochronous property of the cycloid and also was an important observational astronomer, constructing high-quality lenses for the purpose. In 1657 he wrote the first formal account of probability, based on the Pascal-Fermat correspondence. Before this, in 1656, he had invented the pendulum clock, and this instigated his work on dynamics, which resulted in the major text "Horologium Oscilatorium", which deals with most of the elementary dynamics of particles in linear or circular motion, and appeared in 1673 in Paris, where he had moved in 1665 to benefit from a pension awarded by Louis XIV. He returned to Holland in 1681, and in 1689 visited Newton in London. Returning home again he published a treatise expounding his wave theory of light, in contrast to Newton's emission theory. He died in The Hague in 1695.
John Wallis (1616-1703)
The most billiant student of William Oughtred, and the most able and influential British mathematician before Newton, Wallis was educated at Cambridge, but became Savilian professor at Oxford, a position he held for 54 years until his duties took lesser precedence than mathematics. Wallis was an able writer of textbooks, including the first history of mathematics in England, but it was in his research work that he brought forward the ideas of the infinitesimal so far that the final step made by Newton was made considerably easier than at first appears for the inventor of the calculus, anticipating many results on definite integrals. Among Wallis' best known results is the infinite product representation of pi, obtained by applying his techniques to the curve sqrt(1-x^2), although the process is more complicated than we would use now. Wallis was also a founder member of the Royal Society, and assisted the government as a cryptologist.
James Gregory (1638-1675)
Born in Scotland, where he later came to be professor of mathematics at both St Andrews and Edinburgh, Gregory's chief work dealt with infinite series. Coming from a mathematical family, Gregory early found a patron who enabled him to study and travel to meet many mathematicians on the continent. Indeed, Gregory studied in Italy for several years where, in 1667, he published works containing significant results in finitesimal analysis. Gregory was something of a prophet in his anticipation of the need for understanding of convergence of series, not realised for many years afterwards. Unfortunately most of his work was couched in a geometrical context rather than analytic, and so in the main he was superseded by Newton. He did however independently find the binomial theorem for fractional powers, was the discoverer of Taylor series, and his name is given to the arctanx expansion in infinite series.
Isaac Newton (1642-1727)
Born on Christmas Day in Woolsthorpe after the death of his father, Newton was brought up by his grandmother when his mother remarried, and after showing skill in mathematics, entered Cambridge at the insistence of an uncle who recognised his ability. Newton enrolled at Trinity College in 1661, and it was not long before his genius flowered. In 1665 he discovered the general binomial theorem, and during the months of 1665-1666 when the college was closed due to the plague, Newton went home to indulge in the most productive period in mathematical history, laying the foundations of the calculus, the laws of gravity, and the physics of light and colour. In 1669 he became Lucasian professor at Cambridge when Barrow resigned in his favour. During these years, he chiefly studied optics, and in 1679 verified his laws of gravitation by astronomical study. Many of the most distinguished British mathematicians of the time encouraged his work, and were responsible for its general acceptance. In 1684 the astronomer Halley visited Newton and prompted the normally reluctant publisher to collect together his researches in celestial mechanics. Halley himself accounted for the publication expenses of the monumental "Principia", the first complete version appearing in 1687 and immediately hailed throughout Europe. In his later years Newton was feted with honours and positions including Master of the Mint, President of the Royal Society and a knighthood, but these were slightly tarnished by the unfortunate controversy with Leibnitz over precedence of discovering the calculus. He was buried in Westminster Abbey. It is important to note that although Newton is famous for a certain few results, his contributions to all parts of pure and applied mathematics was immense.
Gottfried Wilhelm Leibnitz (1646-1716)
Born in Leipzig, where he attended university, earning his degree at 17 after studying law, theology, philosophy and mathematics, Leibnitz was refused a doctorate of laws, ostensibly because of his youth, whereupon he took his doctorate at Altdorf in Nuremberg. He then entered the diplomatic service, first for the Elector of Mainz, and then from about 1676 to his death, for the Duke of Brunswick at Hanover. As an influential government official, Leibnitz travelled widely, visiting Huygens in Paris and the Royal Society in London in 1672 and 1673 respectively. The invention of the calculus seems to originate somewhere between these visits and 1676 when he again visited London. Most of his mathematical papers were published in the "Acta Eruditorum" which he co-founded, his written output chiefly between 1682 and 1692, and his work had a wide circulation in Europe. His calculus actually appeared in 1684, before Newton's, but generally accepted to be later in conception. The discrepancy of dates resulted in the controversy between Newton's and Leibnitz' supporters, although the two men were mostly indifferent to the fact themselves and continued to correspond. In 1700, Leibnitz founded the Berlin Academy, and was responsible for the creation of others. In 1714 his employer became king of Britain, however, and Leibnitz was left behind and died neglected. It is recognisd that Leibnitz was the greatest all-round genius of his century, being a philosopher, linguist and inventor as well as a mathematician.
Jacques Bernoulli (1654-1705)
The most senior and one of the most gifted in a celebrated family of Swiss mathematicians, Jacques Bernoulli was born in Basle, his ancestors having come from the Spanish Netherlands. He became interested in the infinitesimals via the work of Wallis and Barrow, and giving up his previous calling of the ministry, quickly mastered the new methods of Leibnitz, realising the power of the new mathematics and contributing important results of his own soon after. From 1687 until his death he occupied the chair of mathematics at Basle. Jacques was interested in infinite series, raising the question of convergence. He was also a prolific studier of higher curves, rediscovering many results independently, as well as contributing new ones, especially in respect to the logarithmic spiral where he expressed an arc length as an integral, the first instance of what we now know as an elliptic integral. His best known publication, however, is the "Ars Conjectandi", the earliest substantial volume in probability theory, incorporating Huygens monograph, and giving many results in elementary analysis and number theory, introducing the Bernoulli numbers in his formula for the sums of powers of positive integers, and containing the celebrated law of large numbers.
Jean Bernoulli (1667-1748)
As his brother Jacques, born in Basle, Jean was to have been a merchant or physician, but he became so involved in the new calculus that in 1691-92 he wrote two books on differentiation and integration, though these didn't appear in print until long after. A jealous and argumentative man, Jean was forever igniting conflict, not the least with his brother as to who was the better mathematician. While in Paris in 1692 he instructed the French Marquis de l'Hopital in the new discipline and made a pact under which in return for a salary he sent his mathematical discoveries to l'Hopital for that man to do with them as he wished. Thus l'Hopital's Rule is in fact one of Jean's results. After l'Hopital's death, however, Jean virtually accused him of plagiarism. Bernoulli was a prolific publisher on analysis, as was his brother, using the calculus to good effect in studying curves, and in 1705 took over the chair in Basle left clear by his brother's death. It was Jean who was chief in supporting Leibnitz against Newton, attacking with unwarranted aggression. He even drove his son Daniel from his home for winning a prize for which he had also competed. However, Jean was an outstanding teacher and researcher, responsible for the quick dissemination of the calculus, and is frequently regarded as the inventor of the calculus of variations by his work on the brachistochrone, though many before him had studied similar problems without the aid of the calculus.
Abraham de Moivre (1667-1754)
Born a French Huguenot, but emigrating to England after the revocation of the Edict of Nantes, where he became friendly with Newton and Halley, de Moivre became a private teacher of mathematics and was elected to the Royal Society as well as the Academies of Paris and Berlin. In 1711 he published a memoir in the Philosophical Transactionson the laws of chance, which he later expanded into the celebrated "Doctrine of Chances" which first appeared in 1718, building on the work of the Bernoullis, and continued to produce important results on probability. De Moivre also studied complex numbers and circular functions, and his "Miscellanea Analytica" is important in the analytic side of trigonometry to which he contributed the theorem named after him. In view of the depth of his results he was regarded by Newton as a mathematician of great power.
Colin Maclaurin (1698-1746)
Born in Scotland and educated in Glasgow where he entered university at the age of 11, Maclaurin was perhaps the outstanding British mathematician of his day. He became professor of mathematics at Aberdeen while still only 19, and later also held that position in Edinburgh. His early works were published in the Philosophical Transactions on conics, and on continuing the Newton and Stirling on plane curves, as well as higher algebraic curves. His geometrical work is of particular importance, but Maclaurin mastered most areas of mathematics, writing his "Treatise on Fluxions" in defence of the Newtonian view of calculus, which contains a number of new results, and his "Treatise on Algebra" which contains the rule for solving simultaneous equations by determinants, now called Cramer's rule. When Bonnie Prince Charlie marched against Edinburgh in 1745, Maclaurin escaped the city, being an active opponent of the Young Pretender, but the flight to York was too much for him and he died in 1746. With he death, and that of de Moivre eight years later, British mathematics went into eclipse, insulating itself from the rest of Europe, only in part because of the Newton-Leibnitz controversy, and not gaining recognition again for a hundred years.
Leonhard Euler (1707-1783)
Born in Basle, Euler was heavily influenced in his early years by the Bernoullis, Jean as a teacher and Jean's sons Daniel and Nicolaus as friends, and although Euler's father initially wanted his son to enter the ministry, he helped to instruct his son in mathematics, as he himself was a capable mathematician. Thus Euler had a broad education, which enabled him to take up a position in medicine at the St Petersburg Academy in 1727, where the younger Bernoullis had gone as professors of mathematics, and who recommended him for the post. However, the patron of the academy, Catherine I, died on the day Euler arrived in Russia, and after a period when the academy nearly succumbed to government pressure, Euler found himself in the chair of physics rather than medicine. Thus when Daniel Bernoulli left in 1733 to return to Basle, Nicolaus having died earlier, Euler became the academy's foremost mathematician. Indeed, from the outset he had started to contribute a spate of mathematical articles to the academy's research journal, and he continued this output for the rest of his life, eventually publishing more than 500 books and articles, averaging 800 pages of mathematics a year, the most prolific mathematician of all time, even the loss of sight in one eye in 1735 not hindering him in any way. It was almost impossible for him not to write about mathematics, so easily did the ideas flow, and his proficiency in language made his work acceptable in many countries. In fact, soon after arriving at the academy, Euler achieved an international reputation, and in 1741 accepted an invitation from Frederick the Great to join the Berlin Academy, and he spent 25 years there, although still receiving a pension from Russia and submitting papers to both academies. However the stay in Berlin gradually grew intolerable as the monarch preferred scholars of a more philosophical bent and ready wit, and in 1766 Euler returned to St Petersburg at the suggestion of Catherine the Great. During these years his remaining eye gradually lost its sight by cataract and, after the temporary reprieve of an operation in 1771, he spent the last 12 years of his life totally blind, even this not diminishing the flood of research and publication. The advances made by Euler in every branch of mathematics are legion. He instituted the treatment of complex numbers as an analytical tool, introducing the Euler identities connecting the transcendental functions exp, sin and cos, and giving many other results; he studied infinite series and products, giving many results including the sums of reciprocals of even powers from 2 to 26; he contributed fundamental results to the calculus, especially in his best-known textbook "Introductio in Analysin infinitorum", differential equations, differential geometry and the calculus of variations and introduced the Beta and Gamma integrals; he provided the relation v-e+f=2 connecting the number of vertices, edges and faces of a simple polyhedron; he instigated combinatorics and graph theory with publications on recreational mathematics; he considerably enriched number theory, being the first to prove many of Fermat's results including the special case of Fermat's Last Theorem for n=3, introducing the phi-function, and provided striking results connecting number theory with analysis, in particular the representation of the zeta-function as an infinite product involving primes only; he introduced much of our modern notation, as in the letter e for exp(1), of which he proved irrationality; he also contributed greatly to applied mathematics. He died in 1783, sipping tea in the company of one of his grandchildren.
Jean le Rond d'Alembert (1717-1783)
The leading French mathematician of the mid-18th century, d'Alembert was discovered as an infant abandoned on the steps of the church of St Jean Baptiste le Rond in Paris, from which part of his name derives, the rest taken at a later date. However, it was soon discovered that he was the son of the autocratic sister of a cardinal, Madame de Tencin, and the Chevalier Destouches, a general. The child was brought up by a glazier's wife and in later years he refused to accept his real mother and considered himself the son of his foster parents. D'Alembert was broadly educated, his tuition paid for privately by his real father, which served him well when from 1751 to 1772 he collaborated with Diderot in the celebrated "Encyclopedie". At the age of 24 he was elected to the Paris Academy and later became its permanent secretary. Towards the end of Euler's residency in Berlin, Frederick the Great of Prussia invited d'Alembert to head the academy there, but he declined as he thought it inappropriate to hold a superior position over the great Euler. D'Alembert spent much of him time attempting a proof of what is today known as the fundamental theorem of algebra, and the theorem is named after him to this day in France, although its first full proof was by Gauss much later. Connected with this work, d'Alembert studied the idea of applying calculus to complex variables, and in 1752 published a paper containing the now-famous Cauchy-Riemann equations. Even though d'Alembert's political and philosophical outlook helped pave the way for the French revolution, in mathematics his conservative views sometimes clashed with those of Euler, whose ideas became more widely accepted. However, d'Alembert has left considerable contributions to applied mathematics and partial differentials, in particular being the first man to arrive at the general solution for the second-order partial differential wave equation, via the study of vibrating strings, as well as the mechanical principle that bears his name. He died in Paris in 1783.
Johann Heinrich Lambert (1728-1777)
Born in Mulhouse, the son of a poor tailor, and largely self-taught, Lambert was a man of exceptional ability in a great variety of fields. In mathematics however, he is known as one of the forerunners of non-Euclidean geometry. He questioned the belief that the sum of angles of a plane triangle must be two right angles, and in studying Euclid's parallel postulate was led to consider the quadrilateral now known by his name. The Lambert quadrilateral has three right angles, and he considered the possibilities that the fourth angle was acute, right or obtuse. Since plane and spherical geometry cover the right and obtuse cases respectively, Lambert proposed the existence of a surface for which the fourth angle is acute, and although he could not provide one, he was vindicated in 1868 when Beltrami discovered the pseudosphere. Lambert also proved the irrationality of pi and was the first to give a systematic development of the theory of hyperbolic functions.
Joseph-Louis Lagrange (1736-1813)
The youngest of 11 children, of whom only he survived to adulthood, Lagrange was born in Turin of mixed French and Italian parentage, where he received his education, and as a young man became professor of mathematics at the military academy. Here his reputation was established by publications in the academy's Miscellanea between 1759 and 1761 of results in the calculus of variations, Euler encouraging the work over his own, which he held back in favour of Lagrange. In 1766, Euler and d'Alembert advised Frederick the Great of Prussia on Euler's successor at Berlin, and Lagrange accepted the rather flamboyant offer, leaving only on Frederick's death twenty years later. While in Berlin, Lagrange published important works in mechanics and in the theories of functions and equations. In 1770 he considered the solubility of equations in terms of permutations, which work led eventually to the theory of groups and the insolubility of the quintic by Galois and Abel. He also proved important results in number theory, in particular proving a result of Fermat that every positive integer can be expressed as the sum of at most four squares. After Berlin, Lagrange found the patronage of Louis XVI in France, where in 1788 he published the "Mechanique analytique" which raised the physics of mechanics to the level of pure mathematics in a 'scientific poem' of great achievement. Thinking seriously of leaving France at the fall of the Bastille, Lagrange was invited to teach at the newly formed Ecole's Normale and Polytechnique, out of which grew his famous textbooks, in paticular the "Theorie des fonctions anaytiques" which emphasised the importance of rigour in mathematics. A prominent member of the Committee on Weights and Measures formed to reform the old system, Lagrange survived the ban on foreigners imposed by the authorities by specific exemption, and although revolted by the cruelties of the revolution, he always remained politically neutral. Lagrange is regarded as the greatest mathematician of his time, and the pioneer of modern mathematics in his instigation of rigour.
Gaspard Monge (1746-1818)
Born at Beaune, the son of a tradesman, and educated both there and at Lyons, Monge was permitted to attend courses at the Ecole Militaire at Mezieres through the influence of an army officer who recognised the boy's ability. While there, Monge so impressed the governors that he was soon a member of staff, becoming the professor of mathematics in 1768, and later of physics also. After the revolution, of which he was an ardent supporter, Monge was placed on the Committee for Weights and Measures, though he did not take up the position immediately, having previously become an examiner for the Navy. On his return to Paris in 1792, he was also named Minister of the Navy, in which capacity it fell to him to sign the official record of the execution of the king. Though Monge stepped down from the Ministry in disgust at its disorganisation, he remained active in politics and governmental work, and was instrumental in establishing, in 1795, the Ecole Polytechnique for advanced engineering, where he became both administrator and teacher. Throughout his time at the Polytechnique, Monge was forced to write textbooks on a variety of subjects, albeit with great reluctance, and it fell to many of his most eminent students to compile those he should have written. However Monge did publish a few tracts, in the "Feuille d'analyse" systematically organising three-dimensional analytic geometry, and in 1807 co-writing a memoir on solid analytical geometry with Hachette. He also lectured at the short-lived Ecole Normale, which led to the publication of his "Geometrie descriptive" whose ideas revolutionised engineering design. Primarily a geometer, Monge is recognised as the founder of modern pure geometry, and as one of the most gifted and influential mathematics teachers of all time, inspired many of his pupils to follow his work. A loyal supporter of Napolean, even to accompanying him on the ill-fated Egyptian expedition, Monge was banished and stripped of honours at the restoration of the monarchy and died shortly after.
Pierre-Simon Laplace (1749-1827)
Born of poor parents in Paris, Laplace's mathematical ability earned him good teaching posts soon after leaving school, and he was highly regarded as a mathematician at the time of the revolution. However, Laplace took virtually no part in revolutionary activities, preferring to accept each new political authority with equanimity. This did not mean that he disassociated himself from his more active colleagues, and he did sit on the Committee for Weights and Measures as well as being a respected member of staff at the Normale and the Polytechnique. At one point in fact, he held the position of Minister of the Interior but proved ill-equipped for the job. Perhaps another reason for taking such a small part in the organisation of his country was that his particular interests in mathematics were of no use to the new regime, and it is to later generations that his work is of major importance, in probability and celestial mechanics. The theory of probability owes more to Laplace than to any other mathematician. From 1774 he wrote many papers on the subject, eventually embodied in the "Theorie analytique des probabilites" of 1812, which involves extensive use of advanced calculus, a tool which Laplace wielded powerfully, introducing Bayes' ideas on inverse probability, giving the first formal proof of Legendre's least squares method and introducing the Laplace transform. But it is in the "Mecanique celeste" that Laplace's calculus is at its most impressive. Here is the culmination of the Newtonian view of gravitation, explaining the origin of the solar system, and introducing the Laplacian idea of a potential, and the Laplace equation. Here, if nowhere else, Laplace rivals Newton.
Adrien-Marie Legendre (1752-1833)
The youngest of the thee L's, and similar in that he remained politically aloof from the revolution, Legendre experienced no difficulty in obtaining an education, and early achieved fame for his triangulation of France, though at first he was excluded from a place on the Committee for Weights and Measures due to a foolish oversight. However, the Committee was so impressed with his measurements of the terrestrial meridian that the metre was eventually defined as one ten millionth of the distance from equator to pole. Subsequently Legendre did serve on the Committee during its phase of control by the Institut National. In 1794 appeared Legendre's "Elements de geometrie" which arose from an attempt to put geometry in a more rigorous frame. This became one of the most successful textbooks of the day and influenced teaching in many countries, though its author was primarily an analyst. Legendre produced many books and papers on the calculus, in particular providing the Legendre polynomials so useful in physics, studying differential equations and the calculus of variations. However Legendre was most satisfied with his work on elliptical integrals and indefinite forms, and on the theory of numbers, writing the first treatise wholly devoted to the subject, the "Essai sut la theorie des nombres" of 1797-98. In particular he proved the insolubility of Fermat's Last Theorem for n=5, rediscovered and proved the quadratic reciprocity law and studied the distribution of primes. Late in life he was deprived of his pension for resisting governmental control of the reformed Academy of Sciences, but continued with his mathematics until his death.
Lazare Carnot (1753-1823)
The family of Carnot is perhaps second only to that of the Bernoulli in producing capable mathematicians, the first of any repute, and perhaps the most talented, being Lazare. Born into this rich Parisian family, he attended the Ecole Militaire where Monge was one of his teachers, and then entered the army, though lacking a title he was forbidden to rise any further up the ranks than captain. However, with the onset of the revolution, Carnot threw himself into politics as an ardent republican, and won the admiration of his countrymen for his military successes, acclaimed the "Organiser of Victory". He was also active in the formation of the Polytechnique, bearing a charmed life against all threats until 1797 when, having been a member of most of the powerful assemblies and committees of the new republic, he refused to support Napolean in his coup d'etat and was deported. His exile in Geneva gave him an opportunity to complete much mathematical work, including his "Reflexions", a philosophically inclined work on infinitesmal calculus which sought for greater rigour. However, Carnot's fame rests on geometry, and in this area he published in 1801 and 1803 two works, "De la correlation des figures de geometrie" and "Geometrie de position", which place him with Monge as the co-founder of pure geometry, that is, using no analytic techniques. He returned to France but was exiled again in 1804, when, in his position of Tribunem he voted against naming Napolean as emperor, but he later served willingly for the welfare of France both as soldier and politician. Following the restoration of the monarchy, Carnot sought exile once more, this time in Magdeburg, accepting final defeat and continuing his his scholarly pursuits.
Joseph Fourier (1768-1830)
Born in Auxerre, a tailor's son, Fourier received his education through the Benedictine Order, and at one point intended to become a priest. Instead he became a teacher of mathematics, first at the local military school and then at the Normale and Polytechnique. He was a great supporter of Napolean, and accompanied him, together with Monge, to Egypt. Returning to France he held administrative posts but continued with his research. Fourier's great contribution to mathematics was the "Theorie analytique de la chaleur", published in 1822, a mathematical study of heat conduction, developed by him over a period of ten years, in which he showed that any function f(x) can be represented by an infinite series now known as the Fourier series for f(x), which still exists if the function is discontinuous or having points with no derivative. Thus functions need not be well-behaved to be understood. This major step also went a long way into bringing the function concept into general use. On the Bourbon restoration, Fourier fell from grace but his work has ever since been fundamental in both mathematics and physics.
Carl Friedrich Gauss (1777-1855)
Perhaps the greatest and most complete mathematician of all time, and so German that he never left Germany, Gauss was born in Brunswick, his father a hard-working labourer with stubborn views on education. However, Gauss' mother, though uneducated herself, encouraged her son in his studies and was greatly proud of his later fame. An infant prodigy, the young Gauss came to the attention of the Duke of Brunswick, who ensured that the boy entered college at Brunswick when he was 15, and in 1795 at Gottingen. While in Gottingen, Gauss paid frequent visits to the University of Helmstadt, from where he received his doctorate in 1798, his thesis proving the fundamental theorem of algebra. Indeed, by this time he had already produced several results of major importance, and it was this early success that led him to become a mathematician rather than a philologist. Only two years later, Gauss published his single most important work, the "Disquisitiones arithmeticae", of fundamental importance to the modern theory of numbers, which Gauss always thought as the pinnacle of mathematics, and which includes the construction of regular polygons of his youth, the congruence notation and a proof of the quadratic reciprocity law. He was also interested in the study of primes, in particular giving the conjecture that pi(x) approximates to x/log(x), which was proved 100 years later. Much of Gauss' number theory intersects with and supersedes that of Legendre, who had given partial proofs of some results, and the latter came to be jealous of the younger man. However, in the 19th century, Gauss virtually abandoned his "Queen of Mathematics", his attention distracted by too many other subjects. In 1801 he calculated the orbit of the planetoid Ceres from scant data by his own method, which met with great success, and in 1807 he became professor of mathematics and director of the observatory at Gottingen, posts he held for the rest of his life. He also carried out numerous researches and experiments into electricity and magnetism, mostly with his colleague Weber, and the two invented the telegraph. In pure mathematics, Gauss was reluctant to publish, and it frequently happened that results discovered later by others had been proved previously by him but never released. Such is the case with the ideas of elliptic integrals and non-Euclidean geometry, and mathematicians lived in dread that what they had just produced would already have been done by him. Gauss died at home in Gottingen in 1855at the peak of his fame.
Simeon-Denis Poisson (1781-1840)
Born in Pithiviers of a local administrator who had taken charge of local affairs at the revolution, Poisson was brought up as a republican but later became a legitimist and was honoured in 1825 by a baronage. At first it was hoped that he would become a physician, but his interest in mathematics led him in 1798 to the Ecole Polytechnique where he eventually became professor and examiner. His great love of mathematics prompted Poisson to publish almost 400 works during his life, and he had a reputation as a good teacher. In research, Poisson followed the work done by Cauchy and Gauss, and much of his mathematics is devoted to various fields of physics such as electromagnetism, heat and celestial mechanics, and his name is attached to several occurrences in these subjects, his work being primarily analytic in nature. He also introduced the familiar Poisson distribution into probability in the publication "Recherches sur le probabilite des jugements" of 1837.
Augustin Louis Cauchy (1789-1857)
Born in Paris, just after the fall of the Bastille, Cauchy received his early education from his father, who was personally acquainted with both Laplace and Lagrange. Cauchy then attended the Ecole Centrale du Pantheon, where he excelled in classical studies, before entering the Ecole Polytechnique in 1805, studying under aforementioned Laplace and Lagrange. After two years there he moved once more, this time to the Ecole des Ponts et Chausees, before taking a position in Cherbourg as an engineer in 1810. However, due to ill-health he returned to Paris after three years, and being persuaded to concentrate on pure science, held a professorship at the Polytechnique. On the Bourbon restoration in 1816, the French Academy was purged and Cauchy accepted a seat made available by the expulsion of Monge, continuing also with his previous professorship, lecturing on many topics. It was during this tenure, in 1821, that he was encouraged by Laplace and Poisson to publish the "Cours d'Analyse de l'Ecole Royale Polytechnique", a work of great importance to modern mathematics in its greatly improved treatment of the fundamental principles of calculus, and the introduction of rigour in the study of series. This first book was followed by two others, in 1823 and 1829, which continued and extended the treatment of calculus given in the first. Cauchy's lectures on the subject so impressed Laplace that the latter, convinced of the danger of using series without investigating convergence, promptly checked his "Mecanique celeste", fortunately finding no such incorrect assumptions. On the expulsion of Charles X, and the accession of Louis Philippe in 1830, Cauchy went into voluntary exile, as a devout catholic finding himself unable to take the oath of allegiance demanded of him. After a brief stay in Switzerland, he took a post in Turin that was especially created for him. In 1833 he took on the responsibility of educating the grandson of his exiled king Charles X, and travelled extensively during this time. Cauchy finally returned to Paris in 1838, but it was not until 1848 that by special dispensation of the emperor he was allowed to occupy the chair of mathematics at the Polytechnique without taking the oath. Cauchy was a prodigious mathematician, publishing between 1830 and 1859 over 600 articles, of varying quality, and second only to Euler in extent of output. His work covers a wide range of subjects and his prompt publishing led to him being one the most influential mathematicians of the 19th century. As well as encouraging rigour in all his work, and especially in those mentioned previously, he published in 1812 and 84-page memoir on determinants, bringing together and extending ideas that previously existed in a slightly different form via Gauss and Laplace. He is also almost single-handedly responsible for the development of the calculus of complex variables. He was also an influential teacher, enjoying this task as much as Gauss disliked it, and prepared numerous textbooks.
Jacob Steiner (1796-1863)
Born at Utzendorf in Switzerland, his father a peasant, Steiner had no opportunity to learn reading and writing until he was 14. However, at 18 he became a pupil of Pestalozzi's famous school in Yverdon. In 1818 he went to Heidelberg, and three years later to Berlin, supporting himself by giving lessons. His "Systematische Entwickelungen" was published in 1832 and made his name. Impressed by his work, Crelle and Jacobi obtained for him a chair of geometry in Berlin, which he occupied until his death, after many years of ill-health. Most of Steiner's researches appeared in Crelle's Journal, covering various purely geometrical topics such as algebraic curves, maxima and minima, giving the first synthetic proofs of many properties of surfaces previously obtained only by analytical methods. His studies are totally confined to synthetic, or pure, geometry, pioneered previously by Monge and Carnot, of which he is considered to be the foremost exponent, and he professed an intense dislike for the analytical approach to the subject.
Pierre Gustav Lejeune Dirichlet (1805-1859)
Dirichlet holds a pivotal position in the development of mathematics in the 19th century. Born at Duren, he was educated at the gymnasium in Bonn and subsequently the Jesuit gymnasium in Cologne. In 1822, he was attracted to Paris by the leading names of Laplace, Legendre, Fourier, Poisson and Cauchy. At that time, the teaching facilities in Paris were far better than in Germany, where Gauss was the only great figure. However, Dirichlet'sinterests in mathematics all rested on Gauss' "Disquisitiones Arithmeticae", which he read in Paris, and henceforth he never ceased to study and admire it. Indeed, much of its contents were simplified by Dirichlet, and therefore made more palatable to the rest of the mathematical community. Dirichlet's first memoir was presented to the French Academy in 1825, on the non-existence of solutions to Fermat's equation for n=5, the proof of which was taken partly from work done by Legendre. In 1827, Dirichlet obtained a professorship at Breslau and returned to Germany. Before long, he accepted a position in Berlin, and remained there until in 1855 he finally succeeded Gauss at Gottingen. On this move, Dirichlet had hoped to concentrate on completing Gauss' unfinished works, but unfortunately died himself soon after. Dirichlet was principally analyst, doing extensive research on Fourier series, but his love of Gauss led him continually to apply analysis to number theory. Apart from the above-mentioned result on Fermat's conjecture, he provided the first proof of Bertrand's Postulate, and his most well known result, which carries his name, shows the infinitude of primes in certain arithmetic progressions. Dirichlet showed in these results that the domain of number theory could not be studied in total isolation from the rest of mathematics, particularly analysis, which indeed Gauss had used on occasion while working on binary forms. Dirichlet also followed Cauchy's lead in bringing in more rigour to mathematics.
Niels Henrik Abel (1802-1829)
Born into a large family at Findo in Norway, Abel was the son of the village pastor. He had a normal childhood, attending the cathedral school in Christiania, and it was not until he was 16, when his schoolteacher Holmhoe encouraged him to read the "Disquitiones" of Gauss, that the young Abel first showed any interest in mathematics. However, he quickly displayed great aptitude for the subject, and one of his first results was to show that the binomial theorem is true for all real exponents. When Abel was 18, his father died, and responsilibity for looking after the family fell on him. It was during this period of difficulty that Abel succeeded in proving the insolubility of the general quintic, a results he eventually published in 1824. In 1821 he went to the University of Christiania, where he studied the works of the leading contemporary mathematicians. While here he began his work on elliptic integrals, where his main claim to fame lies. His aptitude for mathematics led to the offer of a stipend from the government that he might continue his work in Germany and France, and so he left Norway in 1825. He spent the first 6 months in Berlin, where he met Crelle and Steiner, and was instrumental in starting Crelle's Journal in 1826, to which he contributed several important papers. Abel then travelled via Freiberg to Paris, where he arrived in July 1826, remaining for almost a year and meeting Dirichlet, Legendre and Cauchy among other prominent mathematicians, but he was little appreciated among these, since Crelle's Journal was little known at the time. Finally, short of funds, he returned home via Berlin, giving private lessons to support himself. In 1829, Crelle managed at last to secure an appointment for Abel in Berlin, but the news did not reach Norway until after Abel's death of tuberculosis at Arendal. Abel studied many areas of analysis, including infinite series, the theory of functions, and integration, always trying to introduce rigour to the loose methods then prevailing, but his most important work was on the discovery of elliptic higher transcendental functions. A memoir containing the important result known as Abel's theorem was sent to the French Academy in 1826, a few months after Abel had arrived in Paris. Cauchy and Legndre were appointed to examine it, but nothing was said about it until after Abel's death, when Jacobi, following up on a reference to it, asked Legendre what had happened. Legendre related that the manuscript had been so badly written as to be illegible and Abel had failed to produce a better version when prompted, the original remaining with Cauchy. Cauchy eventually unearthed the manuscript in 1830, and when it had been properly studied, was hailed as an outstanding piece of work by all concerned. Indeed Jacobi, who admired the work greatly, hailed Abel as the greatest researcher into integral calculus of the 19th century. It finally saw print in 1841.
Carl Gustav Jacob Jacobi (1804-1851)
Born of Jewish parents in Potsdam, his father being a prosperous banker, Jacobi, the second son, had a comfortable childhood, and entered the University of Berlin on leaving school, to study phililogy and mathematics, having been introduced to the latter on reading the works of Euler. He took his doctoral thesis in 1825, and gave lectures in Berlin for two years before becoming extraordinary professor of mathematics at Konigsberg, where he was soon promoted to full professor. After the publication of his classic treatise on elliptic functions, the "Fundamenta nova theoriae functionem ellipticanum", in 1829, Jacobi spent much of his time travelling, meeting Gauss at Gottingen and Legendre, Fourier, Poisson and Cauchy in Paris. It was while here that he forced Abel's ignored memoir on the integral calculus into light. In 1842, the year that he visited London with his colleague Bessel, Jacobi relinquished his chair at Konigsberg to return to Berlin with a pension given to him by the Prussian government, and remained there until his death. Jacobi was introduced to elliptic integrals by Legendre's "Exercises", and his independent discovery of double periodicity, shared with Gauss and Abel, would have earned his reputation, but he developed both his own results and those of Abel to a considerable extent, expanding the idea of the Abelian function that Abel only hinted at, his results often appearing in Crelle's Journal. Jacobi also published important articles on functional determinants bringing them into general use, and they now called Jacobians in his honour. He was also one of the most gifted and influential teachers of mathematics, delighting in instructing others, and revolutionising the presentation of lectures.
Karl Weierstrass (1815-1897)
Born in Westphalia, Weierstrass was the eldest of four children, none of whom ever married. He was brought up as a devout, but liberal, catholic, his father having been converted from Protestantism, His early school work was good enough for his father to send him tothe University of Bonn to study law, but his time there was misspent in drinking and fencing, and he left without a degree. He then prepared to become a schoolteacher at Munster, where his instructor Gundermann became his mentor. Gundermann spent a great deal of time with Weierstrass, encouraging the younger man in the pursuit of mathematics. Gundermann, as will happen, also transferred his own interest in higher function theory to Weierstrass, and many of Weierstrass' early papers were on hyperelliptic and Abelian functions, where he made significant advances. Gundermann also persuaded Weierstrass of the usefulness of power series representations of functions, where he was to create his best work. Weierstrass obtained his teaching certificate at the comparatively late age of 26, and for more than 12 years taught at various secondary schools before a paper of his on Abelian functions, which appeared in Crelle's Journal in 1854, brought him some recognition, and he was soon offered a professorship at Berlin. For the next 40 years, during which he became known as one of the greatest teachers of advanced mathematics in the world, he continued to use and encourage rigour in analysis, seeking to remove any geometric connotation. This process became known as the Arithmetisation of Analysis. He invented the concept of uniform convergence of sequences and series and founded the study of power series representations of complex functions, providing the theory with the idea of analytic continuation, a method for enlarging the domains of definition of complex functions. He also independently obtained the results now known as the Bolzano-Weierstrass Theorem, that a bounded infinite set contains a limit point, and the first widely-known example of a continuous nowhere-differentiable functions, although Bolzano had discovered one earlier. Despite his late start as a professionsl mathematician, Weierstrass is regarded as the leading analyst of the 19th century.
Georg Friedrich Bernhard Riemann (1826-1866)
One of the most original mathematicians of the 19th century, Riemann was born at Breselenz in Hanover, the son of the village pastor. Though his upbringing was modest, and he always remained frail and shy, Riemann showed his natural gifts from an early age and had little trouble in obtaining a good education. He entered the University of Gottingen, with his father being keen to channel him into theology and philology. While there, however, Riemann attended some lectures on mathematics and was so taken by it that he abandoned his previous calling. After studying under Gauss for some time, he moved to Berlin, where the then strngth of mathematics lay in Jacobi, Dirichlet, Steiner and Eisenstein. In 1850, he returned to Gottingen in order to study for his doctorate with Weber, his thesis arousing the interest of Gauss, dealing as it did with the theory of functions of a complex variable, including the famous Cauchy-Riemann equations, and also the basis which led to the concept of a Riemann surface. In 1854, Riemann became Privatdozent at Gottingen, and his inaugural lecture to the faculty, the contents of which were not published until 1867, consisted of the foundations of Riemannian geometry, and entirely revolutionised, unified and generalised the study of non-Euclidean geometry to arbitrary dimensions and introduced the concept of the Riemannian metric, which eventually led to general relativity. This, and further contributions to theoretical physics, resulted in Riemann being appointed as successor to Dirichlet's chair at Gottingen, which Gauss had earlier held. Greater recognition followed, but from 1862 his health deteriorated rapidly, and he was persuaded to visit Italy regularly to convalesce, where he died during his third visit, at Selasca, in 1866. During his later years as a professor, Riemann continued to study, and though he published little, that which was produced is outstanding in conciseness and originality. As well as his work on analytic geometry, he was active in the fields of number theory and analysis, on many occasions combining the two to provide striking results. In particular, his generalisation of the zeta function of Euler and Dirichlet to complex variables, now named after him, leads to the famous Riemann Hypothesis, the conjecture that all the complex roots of the zeta function have real part equal to 1/2. This has never been proved, but many interesting statistical theorems now exist, all based on Riemann's initial work. He also worked on the distribution of primes, and elliptic and Abelain functions. Modern calculus is also indebted to Riemann for his definition of the integral and the necessary and sufficient conditions for a function to be integrable.