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Quick Info

Born

17 September 1826
Breselenz, Hanover (now Germany)

Died

20 July 1866
Selasca, Italy

Summary

Bernhard Riemann's ideas concerning geometry of space had a profound effect on the development of modern theoretical physics. He clarified the notion of integral by defining what we now call the Riemann integral.

Biography

Bernhard Riemann's father, Friedrich Bernhard Riemann, was a Lutheran minister. Friedrich Riemann married Charlotte Ebell when he was in his middle age. Bernhard was the second of their six children, two boys and four girls. Friedrich Riemann acted as teacher to his children and he taught Bernhard until he was ten years old. At this time a teacher from a local school named Schulz assisted in Bernhard's education.

In 1840 Bernhard entered directly into the third class at the Lyceum in Hannover. While at the Lyceum he lived with his grandmother but, in 1842, his grandmother died and Bernhard moved to the Johanneum Gymnasium in Lüneburg. Bernhard seems to have been a good, but not outstanding, pupil who worked hard at the classical subjects such as Hebrew and theology. He showed a particular interest in mathematics and the director of the Gymnasium allowed Bernhard to study mathematics texts from his own library. On one occasion he lent Bernhard Legendre's book on the theory of numbers and Bernhard read the 900 page book in six days.

In the spring of 1846 Riemann enrolled at the University of Göttingen. His father had encouraged him to study theology and so he entered the theology faculty. However he attended some mathematics lectures and asked his father if he could transfer to the faculty of philosophy so that he could study mathematics. Riemann was always very close to his family and he would never have changed courses without his father's permission. This was granted, however, and Riemann then took courses in mathematics from Moritz Stern and Gauss.

It may be thought that Riemann was in just the right place to study mathematics at Göttingen, but at this time the University of Göttingen was a rather poor place for mathematics. Gauss did lecture to Riemann but he was only giving elementary courses and there is no evidence that at this time he recognised Riemann's genius. Stern, however, certainly did realise that he had a remarkable student and later described Riemann at this time saying that he:-

... already sang like a canary.
Riemann moved from Göttingen to Berlin University in the spring of 1847 to study under Steiner, Jacobi, Dirichlet and Eisenstein. This was an important time for Riemann. He learnt much from Eisenstein and discussed using complex variables in elliptic function theory. The main person to influence Riemann at this time, however, was Dirichlet. Klein writes in :-
Riemann was bound to Dirichlet by the strong inner sympathy of a like mode of thought. Dirichlet loved to make things clear to himself in an intuitive substrate; along with this he would give acute, logical analyses of foundational questions and would avoid long computations as much as possible. His manner suited Riemann, who adopted it and worked according to Dirichlet's methods.
Riemann's work always was based on intuitive reasoning which fell a little below the rigour required to make the conclusions watertight. However, the brilliant ideas which his works contain are so much clearer because his work is not overly filled with lengthy computations. It was during his time at the University of Berlin that Riemann worked out his general theory of complex variables that formed the basis of some of his most important work.

In 1849 he returned to Göttingen and his Ph.D. thesis, supervised by Gauss, was submitted in 1851. However it was not only Gauss who strongly influenced Riemann at this time. Weber had returned to a chair of physics at Göttingen from Leipzig during the time that Riemann was in Berlin, and Riemann was his assistant for 18 months. Also Listing had been appointed as a professor of physics in Göttingen in 1849. Through Weber and Listing, Riemann gained a strong background in theoretical physics and, from Listing, important ideas in topology which were to influence his ground breaking research.

Riemann's thesis studied the theory of complex variables and, in particular, what we now call Riemann surfaces. It therefore introduced topological methods into complex function theory. The work builds on Cauchy's foundations of the theory of complex variables built up over many years and also on Puiseux's ideas of branch points. However, Riemann's thesis is a strikingly original piece of work which examined geometric properties of analytic functions, conformal mappings and the connectivity of surfaces.

In proving some of the results in his thesis Riemann used a variational principle which he was later to call the Dirichlet Principle since he had learnt it from Dirichlet's lectures in Berlin. The Dirichlet Principle did not originate with Dirichlet, however, as Gauss, Green and Thomson had all made use if it. Riemann's thesis, one of the most remarkable pieces of original work to appear in a doctoral thesis, was examined on 16 December 1851. In his report on the thesis Gauss described Riemann as having:-
... a gloriously fertile originality.
On Gauss's recommendation Riemann was appointed to a post in Göttingen and he worked for his Habilitation, the degree which would allow him to become a lecturer. He spent thirty months working on his Habilitation dissertation which was on the representability of functions by trigonometric series. He gave the conditions of a function to have an integral, what we now call the condition of Riemann integrability. In the second part of the dissertation he examined the problem which he described in these words:-
While preceding papers have shown that if a function possesses such and such a property, then it can be represented by a Fourier series, we pose the reverse question: if a function can be represented by a trigonometric series, what can one say about its behaviour.
To complete his Habilitation Riemann had to give a lecture. He prepared three lectures, two on electricity and one on geometry. Gauss had to choose one of the three for Riemann to deliver and, against Riemann's expectations, Gauss chose the lecture on geometry. Riemann's lecture Über die Hypothesen welche der Geometrie zu Grunde liegen 10 June 1854, became a classic of mathematics.

There were two parts to Riemann's lecture. In the first part he posed the problem of how to define an -dimensional space and ended up giving a definition of what today we call a Riemannian space. Freudenthal writes in :-
It possesses shortest lines, now called geodesics, which resemble ordinary straight lines. In fact, at first approximation in a geodesic coordinate system such a metric is flat Euclidean, in the same way that a curved surface up to higher-order terms looks like its tangent plane. Beings living on the surface may discover the curvature of their world and compute it at any point as a consequence of observed deviations from Pythagoras's theorem.
In fact the main point of this part of Riemann's lecture was the definition of the curvature tensor. The second part of Riemann's lecture posed deep questions about the relationship of geometry to the world we live in. He asked what the dimension of real space was and what geometry described real space. The lecture was too far ahead of its time to be appreciated by most scientists of that time. Monastyrsky writes in :-
Among Riemann's audience, only Gauss was able to appreciate the depth of Riemann's thoughts. ... The lecture exceeded all his expectations and greatly surprised him. Returning to the faculty meeting, he spoke with the greatest praise and rare enthusiasm to Wilhelm Weber about the depth of the thoughts that Riemann had presented.
It was not fully understood until sixty years later. Freudenthal writes in :-
The general theory of relativity splendidly justified his work. In the mathematical apparatus developed from Riemann's address, Einstein found the frame to fit his physical ideas, his cosmology, and cosmogony: and the spirit of Riemann's address was just what physics needed: the metric structure determined by data.
So this brilliant work entitled Riemann to begin to lecture. However :- Ⓣ( On the hypotheses at the foundations of geometry ) , delivered on
Not long before, in September, he read a report "On the Laws of the Distribution of Static Electricity" at a session of the Göttingen Society of Scientific researchers and Physicians. In a letter to his father, Riemann recalled, among other things, "the fact that I spoke at a scientific meeting was useful for my lectures". In October he set to work on his lectures on partial differential equations. Riemann's letters to his dearly-loved father were full of recollections about the difficulties he encountered. Although only eight students attended the lectures, Riemann was completely happy. Gradually he overcame his natural shyness and established a rapport with his audience.
Gauss's chair at Göttingen was filled by Dirichlet in 1855. At this time there was an attempt to get Riemann a personal chair but this failed. Two years later, however, he was appointed as professor and in the same year, 1857, another of his masterpieces was published. The paper Theory of abelian functions was the result of work carried out over several years and contained in a lecture course he gave to three people in 1855-56. One of the three was Dedekind who was able to make the beauty of Riemann's lectures available by publishing the material after Riemann's early death.

The abelian functions paper continued where his doctoral dissertation had left off and developed further the idea of Riemann surfaces and their topological properties. He examined multi-valued functions as single valued over a special Riemann surface and solved general inversion problems which had been solved for elliptic integrals by Abel and Jacobi. However Riemann was not the only mathematician working on such ideas. Klein writes in :-
... when Weierstrass submitted a first treatment of general abelian functions to the Berlin Academy in 1857, Riemann's paper on the same theme appeared in Crelle's Journal, Volume 54. It contained so many unexpected, new concepts that Weierstrass withdrew his paper and in fact published no more.
The Dirichlet Principle which Riemann had used in his doctoral thesis was used by him again for the results of this 1857 paper. Weierstrass, however, showed that there was a problem with the Dirichlet Principle. Klein writes :-
The majority of mathematicians turned away from Riemann ... Riemann had quite a different opinion. He fully recognised the justice and correctness of Weierstrass's critique, but he said, as Weierstrass once told me, that he appealed to Dirichlet's Principle only as a convenient tool that was right at hand, and that his existence theorems are still correct.
We return at the end of this article to indicate how the problem of the use of Dirichlet's Principle in Riemann's work was sorted out.

In 1858 Betti, Casorati and Brioschi visited Göttingen and Riemann discussed with them his ideas in topology. This gave Riemann particular pleasure and perhaps Betti in particular profited from his contacts with Riemann. These contacts were renewed when Riemann visited Betti in Italy in 1863. In two letter from Betti, showing the topological ideas that he learnt from Riemann, are reproduced.

In 1859 Dirichlet died and Riemann was appointed to the chair of mathematics at Göttingen on 30 July. A few days later he was elected to the Berlin Academy of Sciences. He had been proposed by three of the Berlin mathematicians, Kummer, Borchardt and Weierstrass. Their proposal read :-
Prior to the appearance of his most recent work [Theory of abelian functions], Riemann was almost unknown to mathematicians. This circumstance excuses somewhat the necessity of a more detailed examination of his works as a basis of our presentation. We considered it our duty to turn the attention of the Academy to our colleague whom we recommend not as a young talent which gives great hope, but rather as a fully mature and independent investigator in our area of science, whose progress he in significant measure has promoted.
A newly elected member of the Berlin Academy of Sciences had to report on their most recent research and Riemann sent a report on On the number of primes less than a given magnitude another of his great masterpieces which were to change the direction of mathematical research in a most significant way. In it Riemann examined the zeta function which had already been considered by Euler. Here the sum is over all natural numbers while the product is over all prime numbers. Riemann considered a very different question to the one Euler had considered, for he looked at the zeta function as a complex function rather than a real one. Except for a few trivial exceptions, the roots of all lie between 0 and 1. In the paper he stated that the zeta function had infinitely many nontrivial roots and that it seemed probable that they all have real part . This is the famous Riemann hypothesis which remains today one of the most important of the unsolved problems of mathematics.

Riemann studied the convergence of the series representation of the zeta function and found a functional equation for the zeta function. The main purpose of the paper was to give estimates for the number of primes less than a given number. Many of the results which Riemann obtained were later proved by Hadamard and de la Vallée Poussin.

In June 1862 Riemann married Elise Koch who was a friend of his sister. They had one daughter. In the autumn of the year of his marriage Riemann caught a heavy cold which turned to tuberculosis. He had never had good health all his life and in fact his serious heath problems probably go back much further than this cold he caught. In fact his mother had died when Riemann was 20 while his brother and three sisters all died young. Riemann tried to fight the illness by going to the warmer climate of Italy.

The winter of 1862-63 was spent in Sicily and he then travelled through Italy, spending time with Betti and other Italian mathematicians who had visited Göttingen. He returned to Göttingen in June 1863 but his health soon deteriorated and once again he returned to Italy. Having spent from August 1864 to October 1865 in northern Italy, Riemann returned to Göttingen for the winter of 1865-66, then returned to Selasca on the shores of Lake Maggiore on 16 June 1866. Dedekind writes in :-
His strength declined rapidly, and he himself felt that his end was near. But still, the day before his death, resting under a fig tree, his soul filled with joy at the glorious landscape, he worked on his final work which unfortunately, was left unfinished.
Finally let us return to Weierstrass's criticism of Riemann's use of the Dirichlet's Principle. Weierstrass had shown that a minimising function was not guaranteed by the Dirichlet Principle. This had the effect of making people doubt Riemann's methods. Freudenthal writes in :-
All used Riemann's material but his method was entirely neglected. ... During the rest of the century Riemann's results exerted a tremendous influence: his way of thinking but little.
Weierstrass firmly believed Riemann's results, despite his own discovery of the problem with the Dirichlet Principle. He asked his student Hermann Schwarz to try to find other proofs of Riemann's existence theorems which did not use the Dirichlet Principle. He managed to do this during 1869-70. Klein, however, was fascinated by Riemann's geometric approach and he wrote a book in 1892 giving his version of Riemann's work yet written very much in the spirit of Riemann. Freudenthal writes in :-
It is a beautiful book, and it would be interesting to know how it was received. Probably many took offence at its lack of rigour: Klein was too much in Riemann's image to be convincing to people who would not believe the latter.
In 1901 Hilbert mended Riemann's approach by giving the correct form of Dirichlet's Principle needed to make Riemann's proofs rigorous. The search for a rigorous proof had not been a waste of time, however, since many important algebraic ideas were discovered by Clebsch, Gordan, Brill and Max Noether while they tried to prove Riemann's results. Monastyrsky writes in :-
It is difficult to recall another example in the history of nineteenth-century mathematics when a struggle for a rigorous proof led to such productive results.

1. H Freudenthal, Biography in Dictionary of Scientific Biography (New York 1970-1990). See THIS LINK.

2. Biography in Encyclopaedia Britannica. http://www.britannica.com/biography/Bernhard-Riemann

3. R Dedekind, Biography of Riemann, in H Weber and R Dedekind (eds.), The Collected Works of Riemann (New York, 1953).

4. F Klein, Development of mathematics in the 19th century (Brookline, Mass., 1979).

5. M Monastyrsky, Rieman, Topology and Physics (Boston-Basel, 1987).

6. G Schulz, Riemann, in H Wussing and W Arnold, Biographien bedeutender Mathematiker (Berlin, 1983).

7. H Grauert, Bernhard Riemann and his ideas in philosophy of nature, in Analysis, geometry and groups: a Riemann legacy volume (Palm Harbor, FL, 1993), 124-132.

8. F Klein, Riemann und seine Bedeutung für die Entwicklung der modernen Mathematik, Ges. Math. Abh. (1923), 482-497.

9. E Scholz, Riemann's vision of a new approach to geometry, in 1830-1930: a century of geometry (Berlin, 1992), 22-34.

10. A P Yushkevich and S S Demidov, Bernhard Riemann : on the 150th anniversary of his birth (Russian), Mat. v Skole (4) (1977), 76-80.

Written by J J O'Connor and E F Robertson
Last Update September 1998

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Born 17 September 1826 Breselenz, Hanover (now Germany) Died 20 July 1866 Selasca, Italy Summary Bernhard Riemann's ideas concerning geometry of space had a profound effect on the development of modern theoretical physics. He clarified the notion of integral by defining what we now call the Riemann integral.

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Quick Info

Born

Died

2 February 1704
Paris, France

Summary

De L'Hôpital was a French mathematician who wrote the first textbook on calculus, which consisted of the lectures of his teacher Johann Bernoulli.

Biography

To give Guillaume de l'Hôpital's full name would take a whole paragraph so we give just a much shortened version: Guillaume-François-Antoine Marquis de l'Hôpital, Marquis de Sainte-Mesme, Comte d'Entremont and Seigneur d'Ouques-la-Chaise. The family had been a prominent one in France over many generations going back to around the 12 th century. There are various spellings of the name Hôpital, the earlier versions being l'Hospital or Lhospital with l'Hôpital being a relatively modern form of the name. His father was Anne-Alexandre de l'Hôpital, a Lieutenant-general in the King's Army; he was Comte de Sainte-Mesme and Duc d'Orléans. Guillaume's mother was Elisabeth Gobelin, the daughter of Claude Gobelin who was an Intendant in the King's Army and a Councillor of State. Sturdy writes :-

The prominent association which Anne-Alexandre de l'Hôpital had with the house of Orléans and the trust which Gaston d'Orléans placed in him conferred a brilliant lustre on the reputation of the l'Hôpitals; it also afforded them a protection second only to that of the king himself.
As a child, l'Hôpital had no talent for subjects like Latin, but he developed strong mathematical abilities and a real passion for the subject. Bernard de Fontenelle, in , recounts that when l'Hôpital was fifteen years old he was, on one occasion, discussing mathematics with the Duke of Roannès and a Mr Arnaud. They told him about a very difficult problem on the cycloid that Blaise Pascal had proposed. A few days later l'Hôpital had solved the problem. Given his family background it is not surprising that Guillaume de l'Hôpital followed a military career and served as a captain in a cavalry regiment. He certainly did not give up his interest in mathematics, as Fontenelle explains :-
He entered into the service, but without giving up his dearest passion. He studied geometry even in his tent. It is not just that he retired there to study, it was also to hide his application to study. For it must be admitted that the French nation, although as well mannered as any other, is still in that sort of barbarism by which it wonders whether the sciences, taken to a certain point, are incompatible with nobility, and whether it is not more noble to know nothing. ... I have personally seen some of those who served at the same time, greatly astonished that a man who lived like them was one of the leading mathematicians in Europe.
However he resigned from the army because of nearsightedness being unable to see beyond ten paces. Julian Coolidge writes :-
One is inclined to believe that it was l'Hôpital's love of mathematics rather than the imperfection of his vision that led him to abandon a military career in favour of a scientific one.
Certainly from that time on he directed his attention to mathematics. It is almost certain that l'Hôpital would be totally unknown in the world of mathematics today but for a chance meeting between him and Johann Bernoulli towards the end of 1691. Bernoulli at this time was 24 years old and he had just arrived in Paris after giving lectures on the latest development in mathematics, namely Leibniz's differential calculus. l'Hôpital was at the time a member of Nicolas Malebranche's circle at the Congregation of the Oratory which contained the leading mathematicians and scientists of Paris. It was an obvious place for Bernoulli to go to meet the leading French mathematicians and he soon discovered that l'Hôpital was the most enthusiastic. l'Hôpital, for his part, was intrigued to meet Bernoulli, for it quickly became clear to him that he was much more knowledgeable about the new developments in infinitesimal methods than anyone else in Paris. Bernoulli told l'Hôpital, and others in Malebranche's circle, that he knew the general formula for the radius of curvature of a curve. Although others such as Huygens, Leibniz and Newton knew this, it was thought in Paris to be an important open question so l'Hôpital, although probably one of the best mathematicians in France, realised he could learn much from Bernoulli. Bernoulli agreed to give four lectures a week to Malebranche's circle and he did so for the next six months. l'Hôpital attended these lectures but then moved from Paris to his estate at Ouques where he employed Bernoulli to give him private lessons. By November 1692 Bernoulli had left Ouques and returned to Basel from where he carried out a correspondence with l'Hôpital.

L'Hôpital had published a few brief mathematical notes, but in 1692, while Bernoulli was giving him lessons at Ouques, l'Hôpital sent a solution of de Beaune's problem to Huygens. Florimond de Beaune had asked for a curve for which the subtangent had a fixed length and Bernoulli had included the solution in the course he had given l'Hôpital. L'Hôpital did not claim that the solution he sent Huygens was his own but Huygens made the reasonable assumption that it was. Shortly after this l'Hôpital published the solution under a pseudonym. By the time Bernoulli saw the published solution he was back in Basel and, naturally enough, he was highly displeased. For about six months the correspondence between l'Hôpital and Bernoulli ceased then, on 17 March 1694, l'Hôpital sent a letter to Bernoulli with a remarkable proposition :-
I will be happy to give you a retainer of 300 pounds, beginning with the first of January of this year. ... I promise shortly to increase this retainer, which I know is very modest, as soon as my affairs are somewhat straightened out. ... I am not so unreasonable as to demand in return all of your time, but I will ask you to give me at intervals some hours of your time to work on what I request and also to communicate to me your discoveries, at the same time asking you not to disclose any of them to others. I ask you even not to send here to Mr Varignon or to others any copies of the writings you have left with me; if they are published, I will not be at all pleased. Answer me regarding all this ...
Although no copy of Johann Bernoulli's reply has been found, we know from l'Hôpital's next letter that Bernoulli rapidly accepted the proposition. The correspondence continued and in one of the letters Bernoulli sent to l'Hôpital in 1695 he made the following remarkable promise :-
You have only to let me know your definite wishes, if I am to publish nothing more in my life, for I will follow them precisely and nothing more by me will be seen.
On 1 September 1695 Johann Bernoulli left Basel to take up a new appointment as professor of mathematics at Groningen. In 1696 L'Hôpital's famous book Analyse des infiniment petits pour l'intelligence des lignes courbes was published; it was the first text-book to be written on the differential calculus. In the introduction l'Hôpital acknowledges his indebtedness to Leibniz, Jacob Bernoulli and Johann Bernoulli but l'Hôpital regarded the foundations provided by him as his own ideas. He wrote:-
I must own myself very much obliged to the labours of Messieurs Bernoulli, but particularly to those of the present Professor at Groningen. I have made free use of their discoveries, as well as those of Mr Leibniz, so that I frankly return to them whatever they please to claim as their own.
He was also aware of Newton's contributions, writing:-
I must here in justice own (as Mr Leibniz himself has done in 'Journal des Sçavans' for August 1694) that the learned Sir Isaac Newton likewise discovered something like the Calculus Differentialis ... Bit the method of Mr Leibniz is much more easy and expeditious, on account of the notation he uses, not to mention the wonderful assistance it affords on many occasions.
The book begins with two definitions:

Definition 1. Variable quantities are those that increase or decrease continuously while a constant quantity remains the same while other vary.

Definition 2. The infinitely small part by which a variable quantity increases or decreases continuously is called the differential of that quantity.

There follow two axioms:

Axiom 1. Grant that two quantities whose difference is an infinitely small quantity may be taken (or used) indifferently for each other; or (what is the same thing) that a quantity which is increased or decreased only by an infinitesimally small quantity may be considered as remaining the same.

Axiom 2. Grant that a curved line may be considered as the assemblage of an infinite number of infinitely small straight lines; or (what is the same thing) as a polygon with an infinite number of sides, each of infinitely small length such that the angle between adjacent lines determines the curvature of the curve.

In the second chapter of the work L'Hôpital went on to determine tangents to a curve. Given his definition of a curve as a polygon with an infinite number of sides each of infinitely small length, he can define the tangent at a point on the curve as being the straight line produced from the infinitely small straight line at that point. He gives many examples of computing tangents as well as a general method. In the third chapter he considers maximum and minimum problems giving examples from mechanics and geography. In later chapters he goes on to consider points of inflection, cusps, curvature, evolutes, involutes, and higher order derivatives. In Chapter 9 is found the rule, now known as L'Hôpital's rule, for finding the limit of a function giiven as a fraction whose numerator and denominator tend to zero at a point.

This book was an extremely important contribution. It was used for a long time, with new editions produced until 1781, and it was also a model for the next generation of calculus books. But, of course, we have to examine the question of how dependent the work was on Johann Bernoulli. Well, he did not complain too vigorously when the book appeared and only after L'Hôpital's death did he become more forceful in saying that the book was essentially his. Johann Bernoulli, although having an outstanding reputation as a mathematician, had been involved in various priority disputes, particularly with his brother Jacob Bernoulli, and his claims were not taken too seriously. On the other hand, L'Hôpital's personality and deep understanding of the concepts led colleagues to take his side. For example, Robinson writes in :-
L'Hôpital possessed a very attractive personality, being, among other things, modest and generous, two qualities which were not widespread among mathematicians of his time.
Only in 1921 did a manuscript copy of the course given by Johann Bernoulli to L'Hôpital come to light and it was seen how closely the book followed the course notes. Also when the agreement between the two men was seen, more understanding of the events became possible. In fact Bernoulli had not been in a position to complain when L'Hôpital's book was published because of the agreement between them. In Truesdell tries to put the arrangement into perspective:-
We should not judge L'Hôpital's procedure too harshly. While perhaps financial necessity compelled Bernoulli to accept the arrangement initially, it continued after he had settled in his professorship at Groningen in 1695. L'Hôpital, being a noblemen, was accustomed to pay for the services of others, and what he did would not have been considered wrong had Bernoulli been a politician, a lawyer, perhaps even an architect. Certainly it was nothing for L'Hôpital to be proud of. Careful examination of the letters in which L'Hôpital reported his mathematical progress to Leibniz and Huygens shows that with one or two possible exceptions L'Hôpital did not lie, but rather referred to Bernoulli in a condescending tone without acknowledging any debt whatever to him and in matters of provenance wrote in such a way as to suggest without actually asserting.
So why did Johann Bernoulli agree so quickly to L'Hôpital's proposal? Although as Truesdell suggests, money may have played a part, it does not seem to be the whole reason. Certainly social status was far more significant at this time than in the modern world and Bernoulli would, to quite a large extent, feel obliged to be subservient to a nobleman. Towards the end of his life Bernoulli boasted of the money he had received from L'Hôpital, exaggerating the amount he had received. But it is also possible that it may have been more important to the youthful Bernoulli that his mathematical discoveries became well known than that the world knew him to be the author.

But we have to give L'Hôpital credit for understanding quickly the novel mathematics that was being presented to him. He was a very competent mathematician and certainly was able to write with great clarity. It does appear, however, that he made few if any mathematical discoveries of his own and his solution of the brachistochrone problem was probably not his own. The fact that this problem was solved independently by Newton, Leibniz and Jacob Bernoulli would put l'Hôpital in very good company indeed if the solution was indeed due to him. As well as the problems a mathematician solves, it is often the quality of the questions they ask which single them out for greatness. In this respect it is worth noting the fact that l'Hôpital was wondering whether the notion of a fractional derivative makes sense. There is a letter from him to Leibniz dated 1695 in which he asks whether has a meaning for .

It might be thought that being from the nobility, L'Hôpital would have found it easy to be elected to the Académie des Sciences. However, his nobility made election under the normal processes impossible but after the reorganisation of the Academy in 1699 he was given honorary status.

L'Hôpital considered publishing a work on integration but, on learning that Leibniz was going to publish a work on the topic, he dropped his plans. A manuscript of a book was discovered following his death and this was published under the title Traité anlytique des sections coniques in 1707. A second edition of this text appeared in 1720.

L'Hôpital married Marie-Charlotte de Romilley de La Chesnelaye; they had one son and three daughters.

Let us end by giving Coolidge's assessment of L'Hôpital :-
He was a shining example of a man of the highest social distinction whose love of learning drove him to devote much of his short life to scientific writing.

Written by J J O'Connor and E F Robertson
Last Update December 2008

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Born Died 2 February 1704 Paris, France Summary De L'Hôpital was a French mathematician who wrote the first textbook on calculus, which consisted of the lectures of his teacher Johann Bernoulli. To give Guillaume de l'Hôpital's full name would take a whole paragraph so we give just a much shortened version: Guillaume-François-Antoine Marquis de l'Hôpital, Marquis de Sainte-Mesme, Comte d'Entremont and Seigneur d'Ouques-la-Chaise.

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Biography

Leonhard Euler's father was Paul Euler. Paul Euler had studied theology at the University of Basel and had attended Jacob Bernoulli's lectures there. In fact Paul Euler and Johann Bernoulli had both lived in Jacob Bernoulli's house while undergraduates at Basel. Paul Euler became a Protestant minister and married Margaret Brucker, the daughter of another Protestant minister. Their son Leonhard Euler was born in Basel, but the family moved to Riehen when he was one year old and it was in Riehen, not far from Basel, that Leonard was brought up. Paul Euler had, as we have mentioned, some mathematical training and he was able to teach his son elementary mathematics along with other subjects.

Leonhard was sent to school in Basel and during this time he lived with his grandmother on his mother's side. This school was a rather poor one, by all accounts, and Euler learnt no mathematics at all from the school. However his interest in mathematics had certainly been sparked by his father's teaching, and he read mathematics texts on his own and took some private lessons. Euler's father wanted his son to follow him into the church and sent him to the University of Basel to prepare for the ministry. He entered the University in 1720, at the age of 14, first to obtain a general education before going on to more advanced studies. Johann Bernoulli soon discovered Euler's great potential for mathematics in private tuition that Euler himself engineered. Euler's own account given in his unpublished autobiographical writings, see , is as follows:-

... I soon found an opportunity to be introduced to a famous professor Johann Bernoulli. ... True, he was very busy and so refused flatly to give me private lessons; but he gave me much more valuable advice to start reading more difficult mathematical books on my own and to study them as diligently as I could; if I came across some obstacle or difficulty, I was given permission to visit him freely every Sunday afternoon and he kindly explained to me everything I could not understand ...

In 1723 Euler completed his Master's degree in philosophy having compared and contrasted the philosophical ideas of Descartes and Newton. He began his study of theology in the autumn of 1723, following his father's wishes, but, although he was to be a devout Christian all his life, he could not find the enthusiasm for the study of theology, Greek and Hebrew that he found in mathematics. Euler obtained his father's consent to change to mathematics after Johann Bernoulli had used his persuasion. The fact that Euler's father had been a friend of Johann Bernoulli's in their undergraduate days undoubtedly made the task of persuasion much easier.

Euler completed his studies at the University of Basel in 1726. He had studied many mathematical works during his time in Basel, and Calinger has reconstructed many of the works that Euler read with the advice of Johann Bernoulli. They include works by Varignon, Descartes, Newton, Galileo, van Schooten, Jacob Bernoulli, Hermann, Taylor and Wallis. By 1726 Euler had already a paper in print, a short article on isochronous curves in a resisting medium. In 1727 he published another article on reciprocal trajectories and submitted an entry for the 1727 Grand Prize of the Paris Academy on the best arrangement of masts on a ship.

The Prize of 1727 went to Bouguer, an expert on mathematics relating to ships, but Euler's essay won him second place which was a fine achievement for the young graduate. However, Euler now had to find himself an academic appointment and when died in St Petersburg in July 1726 creating a vacancy there, Euler was offered the post which would involve him in teaching applications of mathematics and mechanics to physiology. He accepted the post in November 1726 but stated that he did not want to travel to Russia until the spring of the following year. He had two reasons to delay. He wanted time to study the topics relating to his new post but also he had a chance of a post at the University of Basel since the professor of physics there had died. Euler wrote an article on acoustics, which went on to become a classic, in his bid for selection to the post but he was not chosen to go forward to the stage where lots were drawn to make the final decision on who would fill the chair. Almost certainly his youth (he was 19 at the time) was against him. However Calinger suggests:-

This decision ultimately benefited Euler, because it forced him to move from a small republic into a setting more adequate for his brilliant research and technological work.

As soon as he knew he would not be appointed to the chair of physics, Euler left Basel on 5 April 1727. He travelled down the Rhine by boat, crossed the German states by post wagon, then by boat from Lübeck arriving in St Petersburg on 17 May 1727. He had joined the St Petersburg Academy of Sciences two years after it had been founded by Catherine I the wife of Peter the Great. Through the requests of Daniel Bernoulli and Jakob Hermann, Euler was appointed to the mathematical-physical division of the Academy rather than to the physiology post he had originally been offered. At St Petersburg Euler had many colleagues who would provide an exceptional environment for him :-

Nowhere else could he have been surrounded by such a group of eminent scientists, including the analyst, geometer Jakob Hermann, a relative; Daniel Bernoulli, with whom Euler was connected not only by personal friendship but also by common interests in the field of applied mathematics; the versatile scholar Christian Goldbach, with whom Euler discussed numerous problems of analysis and the theory of numbers; F Maier, working in trigonometry; and the astronomer and geographer J-N Delisle.

Euler served as a medical lieutenant in the Russian navy from 1727 to 1730. In St Petersburg he lived with Daniel Bernoulli who, already unhappy in Russia, had requested that Euler bring him tea, coffee, brandy and other delicacies from Switzerland. Euler became professor of physics at the Academy in 1730 and, since this allowed him to become a full member of the Academy, he was able to give up his Russian navy post.

Daniel Bernoulli held the senior chair in mathematics at the Academy but when he left St Petersburg to return to Basel in 1733 it was Euler who was appointed to this senior chair of mathematics. The financial improvement which came from this appointment allowed Euler to marry which he did on 7 January 1734, marrying Katharina Gsell, the daughter of a painter from the St Petersburg Gymnasium. Katharina, like Euler, was from a Swiss family. They had 13 children altogether although only five survived their infancy. Euler claimed that he made some of his greatest mathematical discoveries while holding a baby in his arms with other children playing round his feet.

We will examine Euler's mathematical achievements later in this article but at this stage it is worth summarising Euler's work in this period of his career. This is done in as follows:-

... after 1730 he carried out state projects dealing with cartography, science education, magnetism, fire engines, machines, and ship building. ... The core of his research program was now set in place: number theory; infinitary analysis including its emerging branches, differential equations and the calculus of variations; and rational mechanics. He viewed these three fields as intimately interconnected. Studies of number theory were vital to the foundations of calculus, and special functions and differential equations were essential to rational mechanics, which supplied concrete problems.

The publication of many articles and his book Mechanica (1736-37), which extensively presented Newtonian dynamics in the form of mathematical analysis for the first time, started Euler on the way to major mathematical work.

Euler's health problems began in 1735 when he had a severe fever and almost lost his life. However, he kept this news from his parents and members of the Bernoulli family back in Basel until he had recovered. In his autobiographical writings Euler says that his eyesight problems began in 1738 with overstrain due to his cartographic work and that by 1740 he had :-

... lost an eye and [the other] currently may be in the same danger.

However, Calinger in argues that Euler's eyesight problems almost certainly started earlier and that the severe fever of 1735 was a symptom of the eyestrain. He also argues that a portrait of Euler from 1753 suggests that by that stage the sight of his left eye was still good while that of his right eye was poor but not completely blind. Calinger suggests that Euler's left eye became blind from a later cataract rather than eyestrain.

By 1740 Euler had a very high reputation, having won the Grand Prize of the Paris Academy in 1738 and 1740. On both occasions he shared the first prize with others. Euler's reputation was to bring an offer to go to Berlin, but at first he preferred to remain in St Petersburg. However political turmoil in Russia made the position of foreigners particularly difficult and contributed to Euler changing his mind. Accepting an improved offer Euler, at the invitation of Frederick the Great, went to Berlin where an Academy of Science was planned to replace the Society of Sciences. He left St Petersburg on 19 June 1741, arriving in Berlin on 25 July. In a letter to a friend Euler wrote:-

I can do just what I wish [in my research] ... The king calls me his professor, and I think I am the happiest man in the world.

Even while in Berlin Euler continued to receive part of his salary from Russia. For this remuneration he bought books and instruments for the St Petersburg Academy, he continued to write scientific reports for them, and he educated young Russians.

Maupertuis was the president of the Berlin Academy when it was founded in 1744 with Euler as director of mathematics. He deputised for Maupertuis in his absence and the two became great friends. Euler undertook an unbelievable amount of work for the Academy :-

... he supervised the observatory and the botanical gardens; selected the personnel; oversaw various financial matters; and, in particular, managed the publication of various calendars and geographical maps, the sale of which was a source of income for the Academy. The king also charged Euler with practical problems, such as the project in 1749 of correcting the level of the Finow Canal ... At that time he also supervised the work on pumps and pipes of the hydraulic system at Sans Souci, the royal summer residence.

This was not the limit of his duties by any means. He served on the committee of the Academy dealing with the library and of scientific publications. He served as an advisor to the government on state lotteries, insurance, annuities and pensions and artillery. On top of this his scientific output during this period was phenomenal.

During the twenty-five years spent in Berlin, Euler wrote around 380 articles. He wrote books on the calculus of variations; on the calculation of planetary orbits; on artillery and ballistics (extending the book by Robins); on analysis; on shipbuilding and navigation; on the motion of the moon; lectures on the differential calculus; and a popular scientific publication Letters to a Princess of Germany (3 vols., 1768-72).

In 1759 Maupertuis died and Euler assumed the leadership of the Berlin Academy, although not the title of President. The king was in overall charge and Euler was not now on good terms with Frederick despite the early good favour. Euler, who had argued with d'Alembert on scientific matters, was disturbed when Frederick offered d'Alembert the presidency of the Academy in 1763. However d'Alembert refused to move to Berlin but Frederick's continued interference with the running of the Academy made Euler decide that the time had come to leave.

In 1766 Euler returned to St Petersburg and Frederick was greatly angered at his departure. Soon after his return to Russia, Euler became almost entirely blind after an illness. In 1771 his home was destroyed by fire and he was able to save only himself and his mathematical manuscripts. A cataract operation shortly after the fire, still in 1771, restored his sight for a few days but Euler seems to have failed to take the necessary care of himself and he became totally blind. Because of his remarkable memory he was able to continue with his work on optics, algebra, and lunar motion. Amazingly after his return to St Petersburg (when Euler was 59) he produced almost half his total works despite the total blindness.

Euler of course did not achieve this remarkable level of output without help. He was helped by his sons, Johann Albrecht Euler who was appointed to the chair of physics at the Academy in St Petersburg in 1766 (becoming its secretary in 1769) and Christoph Euler who had a military career. Euler was also helped by two other members of the Academy, W L Krafft and A J Lexell, and the young mathematician N Fuss who was invited to the Academy from Switzerland in 1772. Fuss, who was Euler's grandson-in-law, became his assistant in 1776. Yushkevich writes in :-

.. the scientists assisting Euler were not mere secretaries; he discussed the general scheme of the works with them, and they developed his ideas, calculating tables, and sometimes compiled examples.

For example Euler credits Albrecht, Krafft and Lexell for their help with his 775 page work on the motion of the moon, published in 1772. Fuss helped Euler prepare over 250 articles for publication over a period on about seven years in which he acted as Euler's assistant, including an important work on insurance which was published in 1776.

He also wrote a eulogy of Euler, which you can see at THIS LINK.

Yushkevich describes the day of Euler's death in :-

On 18 September 1783 Euler spent the first half of the day as usual. He gave a mathematics lesson to one of his grandchildren, did some calculations with chalk on two boards on the motion of balloons; then discussed with Lexell and Fuss the recently discovered planet Uranus. About five o'clock in the afternoon he suffered a brain haemorrhage and uttered only "I am dying" before he lost consciousness. He died about eleven o'clock in the evening.

After his death in 1783 the St Petersburg Academy continued to publish Euler's unpublished work for nearly 50 more years.

Euler's work in mathematics is so vast that an article of this nature cannot but give a very superficial account of it. He was the most prolific writer of mathematics of all time. He made large bounds forward in the study of modern analytic geometry and trigonometry where he was the first to consider sin, cos etc. as functions rather than as chords as Ptolemy had done.

He made decisive and formative contributions to geometry, calculus and number theory. He integrated Leibniz's differential calculus and Newton's method of fluxions into mathematical analysis. He introduced beta and gamma functions, and integrating factors for differential equations. He studied continuum mechanics, lunar theory with Clairaut, the three body problem, elasticity, acoustics, the wave theory of light, hydraulics, and music. He laid the foundation of analytical mechanics, especially in his Theory of the Motions of Rigid Bodies (1765).

Let us examine in a little more detail some of Euler's work. Firstly his work in number theory seems to have been stimulated by Goldbach but probably originally came from the interest that the Bernoullis had in that topic. Goldbach asked Euler, in 1729, if he knew of Fermat's conjecture that the numbers were always prime if is a power of 2. Euler verified this for = 1, 2, 4, 8 and 16 and, by 1732 at the latest, showed that the next case is divisible by 641 and so is not prime. Euler also studied other unproved results of Fermat and in so doing introduced the Euler phi function , the number of integers with and coprime to . He proved another of Fermat's assertions, namely that if and are coprime then has no divisor of the form , in 1749.

Here the sum is over all natural numbers while the product is over all prime numbers.

By 1739 Euler had found the rational coefficients in in terms of the Bernoulli numbers.

Other work done by Euler on infinite series included the introduction of his famous Euler's constant γ, in 1735, which he showed to be the limit of

as tends to infinity. He calculated the constant γ to 16 decimal places. Euler also studied Fourier series and in 1744 he was the first to express an algebraic function by such a series when he gave the result

in a letter to Goldbach. Like most of Euler's work there was a fair time delay before the results were published; this result was not published until 1755.

Euler wrote to James Stirling on 8 June 1736 telling him about his results on summing reciprocals of powers, the harmonic series and Euler's constant and other results on series. In particular he wrote :-

Concerning the summation of very slowly converging series, in the past year I have lectured to our Academy on a special method of which I have given the sums of very many series sufficiently accurately and with very little effort.

He then goes on to describe what is now called the Euler- Maclaurin summation formula. Two years later Stirling replied telling Euler that Maclaurin:-

... will be publishing a book on fluxions. ... he has two theorems for summing series by means of derivatives of the terms, one of which is the self-same result that you sent me.

Euler replied:-

... I have very little desire for anything to be detracted from the fame of the celebrated Mr Maclaurin since he probably came upon the same theorem for summing series before me, and consequently deserves to be named as its first discoverer. For I found that theorem about four years ago, at which time I also described its proof and application in greater detail to our Academy.

Some of Euler's number theory results have been mentioned above. Further important results in number theory by Euler included his proof of Fermat's Last Theorem for the case of . Perhaps more significant than the result here was the fact that he introduced a proof involving numbers of the form for integers and . Although there were problems with his approach this eventually led to Kummer's major work on Fermats Last Theorem and to the introduction of the concept of a ring.

One could claim that mathematical analysis began with Euler. In 1748 in Introductio in analysin infinitorum Euler made ideas of Johann Bernoulli more precise in defining a function, and he stated that mathematical analysis was the study of functions. This work bases the calculus on the theory of elementary functions rather than on geometric curves, as had been done previously. Also in this work Euler gave the formula

In Introductio in analysin infinitorum Euler dealt with logarithms of a variable taking only positive values although he had discovered the formula

in 1727. He published his full theory of logarithms of complex numbers in 1751.

Analytic functions of a complex variable were investigated by Euler in a number of different contexts, including the study of orthogonal trajectories and cartography. He discovered the Cauchy- Riemann equations in 1777, although d'Alembert had discovered them in 1752 while investigating hydrodynamics.

In 1755 Euler published Institutiones calculi differentialis which begins with a study of the calculus of finite differences. The work makes a thorough investigation of how differentiation behaves under substitutions.

In Institutiones calculi integralis (1768-70) Euler made a thorough investigation of integrals which can be expressed in terms of elementary functions. He also studied beta and gamma functions, which he had introduced first in 1729. Legendre called these 'Eulerian integrals of the first and second kind' respectively while they were given the names beta function and gamma function by Binet and Gauss respectively. As well as investigating double integrals, Euler considered ordinary and partial differential equations in this work.

The calculus of variations is another area in which Euler made fundamental discoveries. His work Methodus inveniendi lineas curvas ... published in 1740 began the proper study of the calculus of variations. In it is noted that Carathéodory considered this as:-

... one of the most beautiful mathematical works ever written.

Problems in mathematical physics had led Euler to a wide study of differential equations. He considered linear equations with constant coefficients, second order differential equations with variable coefficients, power series solutions of differential equations, a method of variation of constants, integrating factors, a method of approximating solutions, and many others. When considering vibrating membranes, Euler was led to the Bessel equation which he solved by introducing Bessel functions.

Euler made substantial contributions to differential geometry, investigating the theory of surfaces and curvature of surfaces. Many unpublished results by Euler in this area were rediscovered by Gauss. Other geometric investigations led him to fundamental ideas in topology such as the Euler characteristic of a polyhedron.

In 1736 Euler published Mechanica which provided a major advance in mechanics. As Yushkevich writes in :-

The distinguishing feature of Euler's investigations in mechanics as compared to those of his predecessors is the systematic and successful application of analysis. Previously the methods of mechanics had been mostly synthetic and geometrical; they demanded too individual an approach to separate problems. Euler was the first to appreciate the importance of introducing uniform analytic methods into mechanics, thus enabling its problems to be solved in a clear and direct way.

In Mechanica Euler considered the motion of a point mass both in a vacuum and in a resisting medium. He analysed the motion of a point mass under a central force and also considered the motion of a point mass on a surface. In this latter topic he had to solve various problems of differential geometry and geodesics.

Mechanica was followed by another important work in rational mechanics, this time Euler's two volume work on naval science. It is described in as:-

Outstanding in both theoretical and applied mechanics, it addresses Euler's intense occupation with the problem of ship propulsion. It applies variational principles to determine the optimal ship design and first established the principles of hydrostatics ... Euler here also begins developing the kinematics and dynamics of rigid bodies, introducing in part the differential equations for their motion.

Of course hydrostatics had been studied since Archimedes, but Euler gave a definitive version.

In 1765 Euler published another major work on mechanics Theoria motus corporum solidorum Ⓣ(Theory of the motion of solid bodies) in which he decomposed the motion of a solid into a rectilinear motion and a rotational motion. He considered the Euler angles and studied rotational problems which were motivated by the problem of the precession of the equinoxes.

Euler's work on fluid mechanics is also quite remarkable. He published a number of major pieces of work through the 1750s setting up the main formulae for the topic, the continuity equation, the Laplace velocity potential equation, and the Euler equations for the motion of an inviscid incompressible fluid. In 1752 he wrote:-

However sublime are the researches on fluids which we owe to Messrs Bernoulli, Clairaut and d'Alembert, they flow so naturally from my two general formulae that one cannot sufficiently admire this accord of their profound meditations with the simplicity of the principles from which I have drawn my two equations ...

Euler contributed to knowledge in many other areas, and in all of them he employed his mathematical knowledge and skill. He did important work in astronomy including :-

... determination of the orbits of comets and planets by a few observations, methods of calculation of the parallax of the sun, the theory of refraction, consideration of the physical nature of comets, .... His most outstanding works, for which he won many prizes from the Paris Académie des Sciences, are concerned with celestial mechanics, which especially attracted scientists at that time.

In fact Euler's lunar theory was used by Tobias Mayer in constructing his tables of the moon. In 1765 Mayer's widow received £3000 from Britain for the contribution the tables made to the problem of the determination of the longitude, while Euler received £300 from the British government for his theoretical contribution to the work.

Euler also published on the theory of music, in particular he published Tentamen novae theoriae musicae in 1739 in which he tried to make music:-

... part of mathematics and deduce in an orderly manner, from correct principles, everything which can make a fitting together and mingling of tones pleasing.

However, according to the work was:-

... for musicians too advanced in its mathematics and for mathematicians too musical.

Cartography was another area that Euler became involved in when he was appointed director of the St Petersburg Academy's geography section in 1735. He had the specific task of helping Delisle prepare a map of the whole of the Russian Empire. The Russian Atlas was the result of this collaboration and it appeared in 1745, consisting of 20 maps. Euler, in Berlin by the time of its publication, proudly remarked that this work put the Russians well ahead of the Germans in the art of cartography.

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Leonhard Euler's father was Paul Euler. Paul Euler had studied theology at the University of Basel and had attended Jacob Bernoulli's lectures there. In fact Paul Euler and Johann Bernoulli had both lived in Jacob Bernoulli's house while undergraduates at Basel. Paul Euler became a Protestant minister and married Margaret Brucker, the daughter of another Protestant minister.

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Quick Info

Born

4 January 1643
Woolsthorpe, Lincolnshire, England

Died

31 March 1727
London, England

Summary

Isaac Newton was the greatest English mathematician of his generation. He laid the foundation for differential and integral calculus. His work on optics and gravitation make him one of the greatest scientists the world has known.

Biography

Isaac Newton's life can be divided into three quite distinct periods. The first is his boyhood days from 1643 up to his appointment to a chair in 1669. The second period from 1669 to 1687 was the highly productive period in which he was Lucasian professor at Cambridge. The third period (nearly as long as the other two combined) saw Newton as a highly paid government official in London with little further interest in mathematical research.

Isaac Newton was born in the manor house of Woolsthorpe, near Grantham in Lincolnshire. Although by the calendar in use at the time of his birth he was born on Christmas Day 1642, we give the date of 4 January 1643 in this biography which is the "corrected" Gregorian calendar date bringing it into line with our present calendar. (The Gregorian calendar was not adopted in England until 1752.) Isaac Newton came from a family of farmers but never knew his father, also named Isaac Newton, who died in October 1642, three months before his son was born. Although Isaac's father owned property and animals which made him quite a wealthy man, he was completely uneducated and could not sign his own name.

You can see a picture of Woolsthorpe Manor as it is now at THIS LINK.

Isaac's mother Hannah Ayscough remarried Barnabas Smith the minister of the church at North Witham, a nearby village, when Isaac was two years old. The young child was then left in the care of his grandmother Margery Ayscough at Woolsthorpe. Basically treated as an orphan, Isaac did not have a happy childhood. His grandfather James Ayscough was never mentioned by Isaac in later life and the fact that James left nothing to Isaac in his will, made when the boy was ten years old, suggests that there was no love lost between the two. There is no doubt that Isaac felt very bitter towards his mother and his step-father Barnabas Smith. When examining his sins at age nineteen, Isaac listed:-

Threatening my father and mother Smith to burn them and the house over them.
Upon the death of his stepfather in 1653, Newton lived in an extended family consisting of his mother, his grandmother, one half-brother, and two half-sisters. From shortly after this time Isaac began attending the Free Grammar School in Grantham. Although this was only five miles from his home, Isaac lodged with the Clark family at Grantham. However he seems to have shown little promise in academic work. His school reports described him as 'idle' and 'inattentive'. His mother, by now a lady of reasonable wealth and property, thought that her eldest son was the right person to manage her affairs and her estate. Isaac was taken away from school but soon showed that he had no talent, or interest, in managing an estate.

An uncle, William Ayscough, decided that Isaac should prepare for entering university and, having persuaded his mother that this was the right thing to do, Isaac was allowed to return to the Free Grammar School in Grantham in 1660 to complete his school education. This time he lodged with Stokes, who was the headmaster of the school, and it would appear that, despite suggestions that he had previously shown no academic promise, Isaac must have convinced some of those around him that he had academic promise. Some evidence points to Stokes also persuading Isaac's mother to let him enter university, so it is likely that Isaac had shown more promise in his first spell at the school than the school reports suggest. Another piece of evidence comes from Isaac's list of sins referred to above. He lists one of his sins as:-
... setting my heart on money, learning, and pleasure more than Thee ...
which tells us that Isaac must have had a passion for learning.

We know nothing about what Isaac learnt in preparation for university, but Stokes was an able man and almost certainly gave Isaac private coaching and a good grounding. There is no evidence that he learnt any mathematics, but we cannot rule out Stokes introducing him to Euclid's Elements which he was well capable of teaching (although there is evidence mentioned below that Newton did not read Euclid before 1663). Anecdotes abound about a mechanical ability which Isaac displayed at the school and stories are told of his skill in making models of machines, in particular of clocks and windmills. However, when biographers seek information about famous people there is always a tendency for people to report what they think is expected of them, and these anecdotes may simply be made up later by those who felt that the most famous scientist in the world ought to have had these skills at school.

Newton entered his uncle's old College, Trinity College Cambridge, on 5 June 1661. He was older than most of his fellow students but, despite the fact that his mother was financially well off, he entered as a sizar. A sizar at Cambridge was a student who received an allowance toward college expenses in exchange for acting as a servant to other students. There is certainly some ambiguity in his position as a sizar, for he seems to have associated with "better class" students rather than other sizars. Westfall (see or ) has suggested that Newton may have had Humphrey Babington, a distant relative who was a Fellow of Trinity, as his patron. This reasonable explanation would fit well with what is known and mean that his mother did not subject him unnecessarily to hardship as some of his biographers claim.

Newton's aim at Cambridge was a law degree. Instruction at Cambridge was dominated by the philosophy of Aristotle but some freedom of study was allowed in the third year of the course. Newton studied the philosophy of Descartes, Gassendi, Hobbes, and in particular Boyle. The mechanics of the Copernican astronomy of Galileo attracted him and he also studied Kepler's Optics. He recorded his thoughts in a book which he entitled Quaestiones Quaedam Philosophicae . It is a fascinating account of how Newton's ideas were already forming around 1664. He headed the text with a Latin statement meaning " Plato is my friend, Aristotle is my friend, but my best friend is truth" showing himself a free thinker from an early stage.

How Newton was introduced to the most advanced mathematical texts of his day is slightly less clear. According to de Moivre, Newton's interest in mathematics began in the autumn of 1663 when he bought an astrology book at a fair in Cambridge and found that he could not understand the mathematics in it. Attempting to read a trigonometry book, he found that he lacked knowledge of geometry and so decided to read Barrow's edition of Euclid's Elements. The first few results were so easy that he almost gave up but he:-
... changed his mind when he read that parallelograms upon the same base and between the same parallels are equal.
Returning to the beginning, Newton read the whole book with a new respect. He then turned to Oughtred's Clavis Mathematica and Descartes' La Géométrie . The new algebra and analytical geometry of Viète was read by Newton from Frans van Schooten's edition of Viète's collected works published in 1646. Other major works of mathematics which he studied around this time was the newly published major work by van Schooten Geometria a Renato Des Cartes which appeared in two volumes in 1659-1661. The book contained important appendices by three of van Schooten's disciples, Jan de Witt, Johan Hudde, and Hendrick van Heuraet. Newton also studied Wallis's Algebra and it appears that his first original mathematical work came from his study of this text. He read Wallis's method for finding a square of equal area to a parabola and a hyperbola which used indivisibles. Newton made notes on Wallis's treatment of series but also devised his own proofs of the theorems writing:-
Thus Wallis doth it, but it may be done thus ...
It would be easy to think that Newton's talent began to emerge on the arrival of Barrow to the Lucasian chair at Cambridge in 1663 when he became a Fellow at Trinity College. Certainly the date matches the beginnings of Newton's deep mathematical studies. However, it would appear that the 1663 date is merely a coincidence and that it was only some years later that Barrow recognised the mathematical genius among his students.

Despite some evidence that his progress had not been particularly good, Newton was elected a scholar on 28 April 1664 and received his bachelor's degree in April 1665. It would appear that his scientific genius had still not emerged, but it did so suddenly when the plague closed the University in the summer of 1665 and he had to return to Lincolnshire. There, in a period of less than two years, while Newton was still under 25 years old, he began revolutionary advances in mathematics, optics, physics, and astronomy.

While Newton remained at home he laid the foundations for differential and integral calculus, several years before its independent discovery by Leibniz. The 'method of fluxions', as he termed it, was based on his crucial insight that the integration of a function is merely the inverse procedure to differentiating it. Taking differentiation as the basic operation, Newton produced simple analytical methods that unified many separate techniques previously developed to solve apparently unrelated problems such as finding areas, tangents, the lengths of curves and the maxima and minima of functions. Newton's De Methodis Serierum et Fluxionum was written in 1671 but Newton failed to get it published and it did not appear in print until John Colson produced an English translation in 1736.

When the University of Cambridge reopened after the plague in 1667, Newton put himself forward as a candidate for a fellowship. In October he was elected to a minor fellowship at Trinity College but, after being awarded his Master's Degree, he was elected to a major fellowship in July 1668 which allowed him to dine at the Fellows' Table. In July 1669 Barrow tried to ensure that Newton's mathematical achievements became known to the world. He sent Newton's text De Analysi to Collins in London writing:-
[ Newton] brought me the other day some papers, wherein he set down methods of calculating the dimensions of magnitudes like that of Mr Mercator concerning the hyperbola, but very general; as also of resolving equations; which I suppose will please you; and I shall send you them by the next.
Collins corresponded with all the leading mathematicians of the day so Barrow's action should have led to quick recognition. Collins showed Brouncker, the President of the Royal Society, Newton's results (with the author's permission) but after this Newton requested that his manuscript be returned. Collins could not give a detailed account but de Sluze and Gregory learnt something of Newton's work through Collins. Barrow resigned the Lucasian chair in 1669 to devote himself to divinity, recommending that Newton (still only 27 years old) be appointed in his place. Shortly after this Newton visited London and twice met with Collins but, as he wrote to Gregory:-
... having no more acquaintance with him I did not think it becoming to urge him to communicate anything.
Newton's first work as Lucasian Professor was on optics and this was the topic of his first lecture course begun in January 1670. He had reached the conclusion during the two plague years that white light is not a simple entity. Every scientist since Aristotle had believed that white light was a basic single entity, but the chromatic aberration in a telescope lens convinced Newton otherwise. When he passed a thin beam of sunlight through a glass prism Newton noted the spectrum of colours that was formed.

He argued that white light is really a mixture of many different types of rays which are refracted at slightly different angles, and that each different type of ray produces a different spectral colour. Newton was led by this reasoning to the erroneous conclusion that telescopes using refracting lenses would always suffer chromatic aberration. He therefore proposed and constructed a reflecting telescope.

In 1672 Newton was elected a fellow of the Royal Society after donating a reflecting telescope. Also in 1672 Newton published his first scientific paper on light and colour in the Philosophical Transactions of the Royal Society. The paper was generally well received but Hooke and Huygens objected to Newton's attempt to prove, by experiment alone, that light consists of the motion of small particles rather than waves. The reception that his publication received did nothing to improve Newton's attitude to making his results known to the world. He was always pulled in two directions, there was something in his nature which wanted fame and recognition yet another side of him feared criticism and the easiest way to avoid being criticised was to publish nothing. Certainly one could say that his reaction to criticism was irrational, and certainly his aim to humiliate Hooke in public because of his opinions was abnormal. However, perhaps because of Newton's already high reputation, his corpuscular theory reigned until the wave theory was revived in the 19 th century.

Newton's relations with Hooke deteriorated further when, in 1675, Hooke claimed that Newton had stolen some of his optical results. Although the two men made their peace with an exchange of polite letters, Newton turned in on himself and away from the Royal Society which he associated with Hooke as one of its leaders. He delayed the publication of a full account of his optical researches until after the death of Hooke in 1703. Newton's Opticks appeared in 1704. It dealt with the theory of light and colour and with

1. investigations of the colours of thin sheets

2. 'Newton's rings' and

3. diffraction of light.

To explain some of his observations he had to use a wave theory of light in conjunction with his corpuscular theory.

Another argument, this time with the English Jesuits in Liège over his theory of colour, led to a violent exchange of letters, then in 1678 Newton appears to have suffered a nervous breakdown. His mother died in the following year and he withdrew further into his shell, mixing as little as possible with people for a number of years.

Newton's greatest achievement was his work in physics and celestial mechanics, which culminated in the theory of universal gravitation. By 1666 Newton had early versions of his three laws of motion. He had also discovered the law giving the centrifugal force on a body moving uniformly in a circular path. However he did not have a correct understanding of the mechanics of circular motion.

Newton's novel idea of 1666 was to imagine that the Earth's gravity influenced the Moon, counter- balancing its centrifugal force. From his law of centrifugal force and Kepler's third law of planetary motion, Newton deduced the inverse-square law.

In 1679 Newton corresponded with Hooke who had written to Newton claiming:-
... that the Attraction always is in a duplicate proportion to the Distance from the Center Reciprocall ...
M Nauenberg writes an account of the next events:-
After his 1679 correspondence with Hooke, Newton, by his own account, found a proof that Kepler's areal law was a consequence of centripetal forces, and he also showed that if the orbital curve is an ellipse under the action of central forces then the radial dependence of the force is inverse square with the distance from the centre.
This discovery showed the physical significance of Kepler's second law.

In 1684 Halley, tired of Hooke's boasting [M Nauenberg]:-
... asked Newton what orbit a body followed under an inverse square force, and Newton replied immediately that it would be an ellipse. However in 'De Motu..' he only gave a proof of the converse theorem that if the orbit is an ellipse the force is inverse square. The proof that inverse square forces imply conic section orbits is sketched in Cor. 1 to Prop. 13 in Book 1 of the second and third editions of the 'Principia', but not in the first edition.
Halley persuaded Newton to write a full treatment of his new physics and its application to astronomy. Over a year later (1687) Newton published the Philosophiae naturalis principia mathematica or Principia as it is always known.

The Principia is recognised as the greatest scientific book ever written. Newton analysed the motion of bodies in resisting and non-resisting media under the action of centripetal forces. The results were applied to orbiting bodies, projectiles, pendulums, and free-fall near the Earth. He further demonstrated that the planets were attracted toward the Sun by a force varying as the inverse square of the distance and generalised that all heavenly bodies mutually attract one another.

Further generalisation led Newton to the law of universal gravitation:-
... all matter attracts all other matter with a force proportional to the product of their masses and inversely proportional to the square of the distance between them.
Newton explained a wide range of previously unrelated phenomena: the eccentric orbits of comets, the tides and their variations, the precession of the Earth's axis, and motion of the Moon as perturbed by the gravity of the Sun. This work made Newton an international leader in scientific research. The Continental scientists certainly did not accept the idea of action at a distance and continued to believe in Descartes' vortex theory where forces work through contact. However this did not stop the universal admiration for Newton's technical expertise.

James II became king of Great Britain on 6 February 1685. He had become a convert to the Roman Catholic church in 1669 but when he came to the throne he had strong support from Anglicans as well as Catholics. However rebellions arose, which James put down but he began to distrust Protestants and began to appoint Roman Catholic officers to the army. He then went further, appointing only Catholics as judges and officers of state. Whenever a position at Oxford or Cambridge became vacant, the king appointed a Roman Catholic to fill it. Newton was a staunch Protestant and strongly opposed to what he saw as an attack on the University of Cambridge.

When the King tried to insist that a Benedictine monk be given a degree without taking any examinations or swearing the required oaths, Newton wrote to the Vice-Chancellor:-
Be courageous and steady to the Laws and you cannot fail.
The Vice-Chancellor took Newton's advice and was dismissed from his post. However Newton continued to argue the case strongly preparing documents to be used by the University in its defence. However William of Orange had been invited by many leaders to bring an army to England to defeat James. William landed in November 1688 and James, finding that Protestants had left his army, fled to France. The University of Cambridge elected Newton, now famous for his strong defence of the university, as one of their two members to the Convention Parliament on 15 January 1689. This Parliament declared that James had abdicated and in February 1689 offered the crown to William and Mary. Newton was at the height of his standing - seen as a leader of the university and one of the most eminent mathematicians in the world. However, his election to Parliament may have been the event which let him see that there was a life in London which might appeal to him more than the academic world in Cambridge.

After suffering a second nervous breakdown in 1693, Newton retired from research. The reasons for this breakdown have been discussed by his biographers and many theories have been proposed: chemical poisoning as a result of his alchemy experiments; frustration with his researches; the ending of a personal friendship with Fatio de Duillier, a Swiss-born mathematician resident in London; and problems resulting from his religious beliefs. Newton himself blamed lack of sleep but this was almost certainly a symptom of the illness rather than the cause of it. There seems little reason to suppose that the illness was anything other than depression, a mental illness he must have suffered from throughout most of his life, perhaps made worse by some of the events we have just listed.

Newton decided to leave Cambridge to take up a government position in London becoming Warden of the Royal Mint in 1696 and Master in 1699. However, he did not resign his positions at Cambridge until 1701. As Master of the Mint, adding the income from his estates, we see that Newton became a very rich man. For many people a position such as Master of the Mint would have been treated as simply a reward for their scientific achievements. Newton did not treat it as such and he made a strong contribution to the work of the Mint. He led it through the difficult period of recoinage and he was particularly active in measures to prevent counterfeiting of the coinage.

In 1703 he was elected president of the Royal Society and was re-elected each year until his death. He was knighted in 1705 by Queen Anne, the first scientist to be so honoured for his work. However the last portion of his life was not an easy one, dominated in many ways with the controversy with Leibniz over which of them had invented the calculus.

Given the rage that Newton had shown throughout his life when criticised, it is not surprising that he flew into an irrational temper directed against Leibniz. We have given details of this controversy in Leibniz's biography and refer the reader to that article for details. Perhaps all that is worth relating here is how Newton used his position as President of the Royal Society. In this capacity he appointed an "impartial" committee to decide whether he or Leibniz was the inventor of the calculus. He wrote the official report of the committee (although of course it did not appear under his name) which was published by the Royal Society, and he then wrote a review (again anonymously) which appeared in the Philosophical Transactions of the Royal Society.

Newton's assistant Whiston had seen his rage at first hand. He wrote:-
Newton was of the most fearful, cautious and suspicious temper that I ever knew.

1. I B Cohen, Biography in Dictionary of Scientific Biography (New York 1970-1990).
SeeTHIS LINK.

2. Biography in Encyclopaedia Britannica.
http://www.britannica.com/biography/Isaac-Newton

3. G E Christianson, In the Presence of the Creator: Isaac Newton and His Times (1984).

4. R S Westfall, Never at Rest: A Biography of Isaac Newton (1990).

5. M Bartolozzi and R Franci, A fragment of the history of algebra : Newton's rule on the number of imaginary roots in an algebraic equation (Italian), Rend. Accad. Naz. Sci. XL Mem. Sci. Fis. Natur. (5) (2, II) (1990), 69-85.

6. S B Engelsman, Orthogonaltrajektorien im Prioritätsstreit zwischen Leibniz und Newton, in 300 Jahre 'Nova methodus' von G W Leibniz (1684-1984) (Wiesbaden, 1986), 144-156.

7. C Ferrini, On Newton's demonstration of Kepler's second law in Hegel's 'De orbitis planetarum' (1801), Philos. Natur. (1) (1994), 150-170.

8. A T Grigor'yan, Isaac Newton's work of genius (Russian), Studies in the history of physics and mechanics 1987 'Nauka' (Moscow, 1987), 177-191; 245-246.

9. U Grigull, Das Newtonsche Abkühlungsgesetz : Bemerkungen zu einer Arbeit von Isaac Newton aus dem Jahre 1701, Physis - Riv. Internaz. Storia Sci. (1-4) (1978), 213-235.

10. J W Herivel, Newtonian studies. III : The originals of the two propositions discovered by Newton in December 1679?, Arch. Internat. Histoire Sci. (1961), 23-33.

11. R C Hovis, What can the history of mathematics learn from philosophy? A case study in Newton's presentation of the calculus, Philos. Math. (2) (1) (1989), 35-57.

12. V S Kirsanov, The correspondence between Isaac Newton and Robert Hooke : 1679-80 (Russian), Voprosy Istor. Estestvoznan. i Tekhn. (4) (1996), 3-39; 173.

13. F D Kramar, Questions of the foundations of analysis in the works of Wallis and Newton (Russian), Trudy Sem. MGU Istor. Mat. Istor.-Mat. Issledov. 1950 (3) (1950), 486-508.

14. T M Kuk, On the question of Newton's physical concept of force (Russian), Sketches on the history of mathematical physics 'Naukova Dumka' (Kiev, 1985), 80-83; 185.

15. L L Kul'vetsas, The content of the concept of force in Newton's mechanics (Russian), in Studies in the history of physics and mechanics, 1990 'Nauka' (Moscow, 1990), 131-149.

16. V P Lishevskii, The genius of the natural sciences (on the occasion of the 350th anniversary of the birth of Isaac Newton) (Russian), Vestnik Ross. Akad. Nauk (1) (1993), 33-37.

17. J A Lohne, Newton's table of refractive powers. Origins, accuracy, and influence, Sudhoffs Arch. 61 (3) (1977), 229-247.

18. H Nakajima, Two kinds of modification theory of light : some new observations on the Newton-Hooke controversy of 1672 concerning the nature of light, Ann. of Sci. (3) (1984), 261-278.

19. A Pérez de Laborda, Newtons Fluxionsrechnung im Vergleich zu Leibniz' Infinitesimalkalkül, in 300 Jahre 'Nova methodus' von G W Leibniz (1684-1984) (Wiesbaden, 1986), 239-257.

20. I A Tyulina, On the foundations of Newtonian mechanics (on the 300th anniversary of Newton's 'Principia') (Russian), Istor. Metodol. Estestv. Nauk (1989), 184-196.

21. I A Tyulina, On the foundations of Newtonian mechanics (on the 300th anniversary of Newton's 'Principia') (Russian), Istor. Metodol. Estestv. Nauk (1989), 184-196.

22. A P Youschkevitch, Newton and the mathematical natural sciences (on the 300th anniversary of the 'Principia') (Russian), Mat. v Shkole (1) (1988), 64-67.

23. A P Youschkevitch, Comparaison des conceptions de Leibniz et de Newton sur le calcul infinitésimal, in Leibniz in Paris (1672-1676) (Wiesbaden, 1978), 69-80.

Written by J J O'Connor and E F Robertson
Last Update January 2000

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Born 4 January 1643 Woolsthorpe, Lincolnshire, England Died 31 March 1727 London, England Summary Isaac Newton was the greatest English mathematician of his generation. He laid the foundation for differential and integral calculus. His work on optics and gravitation make him one of the greatest scientists the world has known.

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Quick Info

Born

1 July 1646
Leipzig, Saxony (now Germany)

Died

14 November 1716
Hannover, Hanover (now Germany)

Summary

Gottfried Leibniz was a German mathematician who developed the present day notation for the differential and integral calculus though he never thought of the derivative as a limit. His philosophy is also important and he invented an early calculating machine.

Biography

Gottfried Leibniz was the son of Friedrich Leibniz, a professor of moral philosophy at Leipzig. Friedrich Leibniz :-

...was evidently a competent though not original scholar, who devoted his time to his offices and to his family as a pious, Christian father.
Leibniz's mother was Catharina Schmuck, the daughter of a lawyer and Friedrich Leibniz's third wife. However, Friedrich Leibniz died when Leibniz was only six years old and he was brought up by his mother. Certainly Leibniz learnt his moral and religious values from her which would play an important role in his life and philosophy.

At the age of seven, Leibniz entered the Nicolai School in Leipzig. Although he was taught Latin at school, Leibniz had taught himself far more advanced Latin and some Greek by the age of 12. He seems to have been motivated by wanting to read his father's books. As he progressed through school he was taught Aristotle's logic and theory of categorising knowledge. Leibniz was clearly not satisfied with Aristotle's system and began to develop his own ideas on how to improve on it. In later life Leibniz recalled that at this time he was trying to find orderings on logical truths which, although he did not know it at the time, were the ideas behind rigorous mathematical proofs. As well as his school work, Leibniz studied his father's books. In particular he read metaphysics books and theology books from both Catholic and Protestant writers.

In 1661, at the age of fourteen, Leibniz entered the University of Leipzig. It may sound today as if this were a truly exceptionally early age for anyone to enter university, but it is fair to say that by the standards of the time he was quite young but there would be others of a similar age. He studied philosophy, which was well taught at the University of Leipzig, and mathematics which was very poorly taught. Among the other topics which were included in this two year general degree course were rhetoric, Latin, Greek and Hebrew. He graduated with a bachelors degree in 1663 with a thesis De Principio Individui which:-
... emphasised the existential value of the individual, who is not to be explained either by matter alone or by form alone but rather by his whole being.
In this there is the beginning of his notion of "monad". Leibniz then went to Jena to spend the summer term of 1663.

At Jena the professor of mathematics was Erhard Weigel but Weigel was also a philosopher and through him Leibniz began to understand the importance of the method of mathematical proof for subjects such as logic and philosophy. Weigel believed that number was the fundamental concept of the universe and his ideas were to have considerable influence of Leibniz. By October 1663 Leibniz was back in Leipzig starting his studies towards a doctorate in law. He was awarded his Master's Degree in philosophy for a dissertation which combined aspects of philosophy and law studying relations in these subjects with mathematical ideas that he had learnt from Weigel. A few days after Leibniz presented his dissertation, his mother died.

After being awarded a bachelor's degree in law, Leibniz worked on his habilitation in philosophy. His work was to be published in 1666 as Dissertatio de arte combinatoria . In this work Leibniz aimed to reduce all reasoning and discovery to a combination of basic elements such as numbers, letters, sounds and colours.

Despite his growing reputation and acknowledged scholarship, Leibniz was refused the doctorate in law at Leipzig. It is a little unclear why this happened. It is likely that, as one of the younger candidates and there only being twelve law tutorships available, he would be expected to wait another year. However, there is also a story that the Dean's wife persuaded the Dean to argue against Leibniz, for some unexplained reason. Leibniz was not prepared to accept any delay and he went immediately to the University of Altdorf where he received a doctorate in law in February 1667 for his dissertation De Casibus Perplexis .

Leibniz declined the promise of a chair at Altdorf because he had very different things in view. He served as secretary to the Nuremberg alchemical society for a while (see ) then he met Baron Johann Christian von Boineburg. By November 1667 Leibniz was living in Frankfurt, employed by Boineburg. During the next few years Leibniz undertook a variety of different projects, scientific, literary and political. He also continued his law career taking up residence at the courts of Mainz before 1670. One of his tasks there, undertaken for the Elector of Mainz, was to improve the Roman civil law code for Mainz but :-
Leibniz was also occupied by turns as Boineburg's secretary, assistant, librarian, lawyer and advisor, while at the same time a personal friend of the Baron and his family.
Boineburg was a Catholic while Leibniz was a Lutheran but Leibniz had as one of his lifelong aims the reunification of the Christian Churches and :-
... with Boineburg's encouragement, he drafted a number of monographs on religious topics, mostly to do with points at issue between the churches...
Another of Leibniz's lifelong aims was to collate all human knowledge. Certainly he saw his work on Roman civil law as part of this scheme and as another part of this scheme, Leibniz tried to bring the work of the learned societies together to coordinate research. Leibniz began to study motion, and although he had in mind the problem of explaining the results of Wren and Huygens on elastic collisions, he began with abstract ideas of motion. In 1671 he published Hypothesis Physica Nova . In this work he claimed, as had Kepler, that movement depends on the action of a spirit. He communicated with Oldenburg, the secretary of the Royal Society of London, and dedicated some of his scientific works to the Royal Society and the Paris Academy. Leibniz was also in contact with Carcavi, the Royal Librarian in Paris. As Ross explains in :-
Although Leibniz's interests were clearly developing in a scientific direction, he still hankered after a literary career. All his life he prided himself on his poetry (mostly Latin), and boasted that he could recite the bulk of Virgil's "Aeneid" by heart. During this time with Boineburg he would have passed for a typical late Renaissance humanist.
Leibniz wished to visit Paris to make more scientific contacts. He had begun construction of a calculating machine which he hoped would be of interest. He formed a political plan to try to persuade the French to attack Egypt and this proved the means of his visiting Paris. In 1672 Leibniz went to Paris on behalf of Boineburg to try to use his plan to divert Louis XIV from attacking German areas. His first object in Paris was to make contact with the French government but, while waiting for such an opportunity, Leibniz made contact with mathematicians and philosophers there, in particular Arnauld and Malebranche, discussing with Arnauld a variety of topics but particularly church reunification.

In Paris Leibniz studied mathematics and physics under Christiaan Huygens beginning in the autumn of 1672. On Huygens' advice, Leibniz read Saint-Vincent's work on summing series and made some discoveries of his own in this area. Also in the autumn of 1672, Boineburg's son was sent to Paris to study under Leibniz which meant that his financial support was secure. Accompanying Boineburg's son was Boineburg's nephew on a diplomatic mission to try to persuade Louis XIV to set up a peace congress. Boineburg died on 15 December but Leibniz continued to be supported by the Boineburg family.

In January 1673 Leibniz and Boineburg's nephew went to England to try the same peace mission, the French one having failed. Leibniz visited the Royal Society, and demonstrated his incomplete calculating machine. He also talked with Hooke, Boyle and Pell. While explaining his results on series to Pell, he was told that these were to be found in a book by Mouton. The next day he consulted Mouton's book and found that Pell was correct. At the meeting of the Royal Society on 15 February, which Leibniz did not attend, Hooke made some unfavourable comments on Leibniz's calculating machine. Leibniz returned to Paris on hearing that the Elector of Mainz had died. Leibniz realised that his knowledge of mathematics was less than he would have liked so he redoubled his efforts on the subject.

The Royal Society of London elected Leibniz a fellow on 19 April 1673. Leibniz met Ozanam and solved one of his problems. He also met again with Huygens who gave him a reading list including works by Pascal, Fabri, Gregory, Saint-Vincent, Descartes and Sluze. He began to study the geometry of infinitesimals and wrote to Oldenburg at the Royal Society in 1674. Oldenburg replied that Newton and Gregory had found general methods. Leibniz was, however, not in the best of favours with the Royal Society since he had not kept his promise of finishing his mechanical calculating machine. Nor was Oldenburg to know that Leibniz had changed from the rather ordinary mathematician who visited London, into a creative mathematical genius. In August 1675 Tschirnhaus arrived in Paris and he formed a close friendship with Leibniz which proved very mathematically profitable to both.

It was during this period in Paris that Leibniz developed the basic features of his version of the calculus. In 1673 he was still struggling to develop a good notation for his calculus and his first calculations were clumsy. On 21 November 1675 he wrote a manuscript using the notation for the first time. In the same manuscript the product rule for differentiation is given. By autumn 1676 Leibniz discovered the familiar for both integral and fractional .

Newton wrote a letter to Leibniz, through Oldenburg, which took some time to reach him. The letter listed many of Newton's results but it did not describe his methods. Leibniz replied immediately but Newton, not realising that his letter had taken a long time to reach Leibniz, thought he had had six weeks to work on his reply. Certainly one of the consequences of Newton's letter was that Leibniz realised he must quickly publish a fuller account of his own methods.

Newton wrote a second letter to Leibniz on 24 October 1676 which did not reach Leibniz until June 1677 by which time Leibniz was in Hanover. This second letter, although polite in tone, was clearly written by Newton believing that Leibniz had stolen his methods. In his reply Leibniz gave some details of the principles of his differential calculus including the rule for differentiating a function of a function.

Newton was to claim, with justification, that
... not a single previously unsolved problem was solved ...
by Leibniz's approach but the formalism was to prove vital in the latter development of the calculus. Leibniz never thought of the derivative as a limit. This does not appear until the work of d'Alembert.

Leibniz would have liked to have remained in Paris in the Academy of Sciences, but it was considered that there were already enough foreigners there and so no invitation came. Reluctantly Leibniz accepted a position from the Duke of Hanover, Johann Friedrich, of librarian and of Court Councillor at Hanover. He left Paris in October 1676 making the journey to Hanover via London and Holland. The rest of Leibniz's life, from December 1676 until his death, was spent at Hanover except for the many travels that he made.

His duties at Hanover :-
... as librarian were onerous, but fairly mundane: general administration, purchase of new books and second-hand libraries, and conventional cataloguing.
He undertook a whole collection of other projects however. For example one major project begun in 1678-79 involved draining water from the mines in the Harz mountains. His idea was to use wind power and water power to operate pumps. He designed many different types of windmills, pumps, gears but :-
... every one of these projects ended in failure. Leibniz himself believed that this was because of deliberate obstruction by administrators and technicians, and the workers' fear that technological progress would cost them their jobs.
In 1680 Duke Johann Friedrich died and his brother Ernst August became the new Duke. The Harz project had always been difficult and it failed by 1684. However Leibniz had achieved important scientific results becoming one of the first people to study geology through the observations he compiled for the Harz project. During this work he formed the hypothesis that the Earth was at first molten.

Another of Leibniz's great achievements in mathematics was his development of the binary system of arithmetic. He perfected his system by 1679 but he did not publish anything until 1701 when he sent the paper Essay d'une nouvelle science des nombres to the Paris Academy to mark his election to the Academy. Another major mathematical work by Leibniz was his work on determinants which arose from his developing methods to solve systems of linear equations. Although he never published this work in his lifetime, he developed many different approaches to the topic with many different notations being tried out to find the one which was most useful. An unpublished paper dated 22 January 1684 contains very satisfactory notation and results.

Leibniz continued to perfect his metaphysical system in the 1680s attempting to reduce reasoning to an algebra of thought. Leibniz published Meditationes de Cognitione, Veritate et Ideis which clarified his theory of knowledge. In February 1686, Leibniz wrote his Discours de métaphysique .

Another major project which Leibniz undertook, this time for Duke Ernst August, was writing the history of the Guelf family, of which the House of Brunswick was a part. He made a lengthy trip to search archives for material on which to base this history, visiting Bavaria, Austria and Italy between November 1687 and June 1690. As always Leibniz took the opportunity to meet with scholars of many different subjects on these journeys. In Florence, for example, he discussed mathematics with Viviani who had been Galileo's last pupil. Although Leibniz published nine large volumes of archival material on the history of the Guelf family, he never wrote the work that was commissioned.

In 1684 Leibniz published details of his differential calculus in Nova Methodus pro Maximis et Minimis, itemque Tangentibus... in Acta Eruditorum, a journal established in Leipzig two years earlier. The paper contained the familiar d notation, the rules for computing the derivatives of powers, products and quotients. However it contained no proofs and Jacob Bernoulli called it an enigma rather than an explanation.

In 1686 Leibniz published, in Acta Eruditorum, a paper dealing with the integral calculus with the first appearance in print of the ∫ notation.

Newton's Principia appeared the following year. Newton's 'method of fluxions' was written in 1671 but Newton failed to get it published and it did not appear in print until John Colson produced an English translation in 1736. This time delay in the publication of Newton's work resulted in a dispute with Leibniz.

Another important piece of mathematical work undertaken by Leibniz was his work on dynamics. He criticised Descartes' ideas of mechanics and examined what are effectively kinetic energy, potential energy and momentum. This work was begun in 1676 but he returned to it at various times, in particular while he was in Rome in 1689. It is clear that while he was in Rome, in addition to working in the Vatican library, Leibniz worked with members of the Accademia. He was elected a member of the Accademia at this time. Also while in Rome he read Newton's Principia. His two part treatise Dynamica studied abstract dynamics and concrete dynamics and is written in a somewhat similar style to Newton's Principia. Ross writes in :-
... although Leibniz was ahead of his time in aiming at a genuine dynamics, it was this very ambition that prevented him from matching the achievement of his rival Newton. ... It was only by simplifying the issues... that Newton succeeded in reducing them to manageable proportions.
Leibniz put much energy into promoting scientific societies. He was involved in moves to set up academies in Berlin, Dresden, Vienna, and St Petersburg. He began a campaign for an academy in Berlin in 1695, he visited Berlin in 1698 as part of his efforts and on another visit in 1700 he finally persuaded Friedrich to found the Brandenburg Society of Sciences on 11 July. Leibniz was appointed its first president, this being an appointment for life. However, the Academy was not particularly successful and only one volume of the proceedings were ever published. It did lead to the creation of the Berlin Academy some years later.

Other attempts by Leibniz to found academies were less successful. He was appointed as Director of a proposed Vienna Academy in 1712 but Leibniz died before the Academy was created. Similarly he did much of the work to prompt the setting up of the St Petersburg Academy, but again it did not come into existence until after his death.

It is no exaggeration to say that Leibniz corresponded with most of the scholars in Europe. He had over 600 correspondents. Among the mathematicians with whom he corresponded was Grandi. The correspondence started in 1703, and later concerned the results obtained by putting into . Leibniz also corresponded with Varignon on this paradox. Leibniz discussed logarithms of negative numbers with Johann Bernoulli, see .

In 1710 Leibniz published Théodicée a philosophical work intended to tackle the problem of evil in a world created by a good God. Leibniz claims that the universe had to be imperfect, otherwise it would not be distinct from God. He then claims that the universe is the best possible without being perfect. Leibniz is aware that this argument looks unlikely - surely a universe in which nobody is killed by floods is better than the present one, but still not perfect. His argument here is that the elimination of natural disasters, for example, would involve such changes to the laws of science that the world would be worse. In 1714 Leibniz wrote Monadologia which synthesised the philosophy of his earlier work, the Théodicée .

Much of the mathematical activity of Leibniz's last years involved the priority dispute over the invention of the calculus. In 1711 he read the paper by Keill in the Transactions of the Royal Society of London which accused Leibniz of plagiarism. Leibniz demanded a retraction saying that he had never heard of the calculus of fluxions until he had read the works of Wallis. Keill replied to Leibniz saying that the two letters from Newton, sent through Oldenburg, had given:-
... pretty plain indications... whence Leibniz derived the principles of that calculus or at least could have derived them.
Leibniz wrote again to the Royal Society asking them to correct the wrong done to him by Keill's claims. In response to this letter the Royal Society set up a committee to pronounce on the priority dispute. It was totally biased, not asking Leibniz to give his version of the events. The report of the committee, finding in favour of Newton, was written by Newton himself and published as Commercium epistolicum near the beginning of 1713 but not seen by Leibniz until the autumn of 1714. He learnt of its contents in 1713 in a letter from Johann Bernoulli, reporting on the copy of the work brought from Paris by his nephew . Leibniz published an anonymous pamphlet Charta volans setting out his side in which a mistake by Newton in his understanding of second and higher derivatives, spotted by Johann Bernoulli, is used as evidence of Leibniz's case.

The argument continued with Keill who published a reply to Charta volans. Leibniz refused to carry on the argument with Keill, saying that he could not reply to an idiot. However, when Newton wrote to him directly, Leibniz did reply and gave a detailed description of his discovery of the differential calculus. From 1715 up until his death Leibniz corresponded with Samuel Clarke, a supporter of Newton, on time, space, freewill, gravitational attraction across a void and other topics, see , , and .

In Leibniz is described as follows:-
Leibniz was a man of medium height with a stoop, broad-shouldered but bandy-legged, as capable of thinking for several days sitting in the same chair as of travelling the roads of Europe summer and winter. He was an indefatigable worker, a universal letter writer (he had more than 600 correspondents), a patriot and cosmopolitan, a great scientist, and one of the most powerful spirits of Western civilisation.
Ross, in , points out that Leibniz's legacy may have not been quite what he had hoped for:-
It is ironical that one so devoted to the cause of mutual understanding should have succeeded only in adding to intellectual chauvinism and dogmatism. There is a similar irony in the fact that he was one of the last great polymaths - not in the frivolous sense of having a wide general knowledge, but in the deeper sense of one who is a citizen of the whole world of intellectual inquiry. He deliberately ignored boundaries between disciplines, and lack of qualifications never deterred him from contributing fresh insights to established specialisms. Indeed, one of the reasons why he was so hostile to universities as institutions was because their faculty structure prevented the cross-fertilisation of ideas which he saw as essential to the advance of knowledge and of wisdom. The irony is that he was himself instrumental in bringing about an era of far greater intellectual and scientific specialism, as technical advances pushed more and more disciplines out of the reach of the intelligent layman and amateur.

Written by J J O'Connor and E F Robertson
Last Update October 1998

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Gottfried Leibniz was a German mathematician who developed the present day notation for the differential and integral calculus though he never thought of the derivative as a limit. His philosophy is also important and he invented an early calculating machine.

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J. L. Coolidge, The Number e, The American Mathematical Monthly, Vol. 57, No. 9 (Nov., 1950), pp. 591-602