Abu Ja'far Muhammad ibn Musa Al-Khwarizmi

 Al-Mamun won the armed struggle and al-Amin was defeated and killed in 813. Following this, al-Mamun became Caliph and ruled the empire from Baghdad. He continued the patronage of learning started by his father and founded an academy called the House of Wisdom where Greek philosophical and scientific works were translated. He also built up a library of manuscripts, the first major library to be set up since that at Alexandria, collecting important works from Byzantium. In addition to the House of Wisdom, al-Mamun set up observatories in which Muslim astronomers could build on the knowledge acquired by earlier peoples.

Al-Khwarizmi and his colleagues the Banu Musa were scholars at the House of Wisdom in Baghdad. Their tasks there involved the translation of Greek scientific manuscripts and they also studied, and wrote on, algebra, geometry and astronomy. Certainly al-Khwarizmi worked under the patronage of Al-Mamun and he dedicated two of his texts to the Caliph. These were his treatise on algebra and his treatise on astronomy. The algebra treatise Hisab al-jabr w'al-muqabala was the most famous and important of all of al-Khwarizmi's works. It is the title of this text that gives us the word "algebra" and, in a sense that we shall investigate more fully below, it is the first book to be written on algebra.

Having introduced the natural numbers, al-Khwarizmi introduces the main topic of this first section of his book, namely the solution of equations. His equations are linear or quadratic and are composed of units, roots and squares. For example, to al-Khwarizmi a unit was a number, a root was x, and a square was x2. However, although we shall use the now familiar algebraic notation in this article to help the reader understand the notions, Al-Khwarizmi's mathematics is done entirely in words with no symbols being used.

He first reduces an equation (linear or quadratic) to one of six standard forms:

1. Squares equal to roots.
2. Squares equal to numbers.
3. Roots equal to numbers.
4. Squares and roots equal to numbers; e.g. x2 + 10 x = 39.
5. Squares and numbers equal to roots; e.g. x2 + 21 = 10 x.
6. Roots and numbers equal to squares; e.g. 3 x + 4 = x2.

The reduction is carried out using the two operations of al-jabr and al-muqabala. Here "al-jabr" means "completion" and is the process of removing negative terms from an equation. For example, using one of al-Khwarizmi's own examples, "al-jabr" transforms x2 = 40 x - 4 x2 into 5 x2 = 40 x. The term "al-muqabala" means "balancing" and is the process of reducing positive terms of the same power when they occur on both sides of an equation. For example, two applications of "al-muqabala" reduces 50 + 3 x + x2 = 29 + 10 x to 21 + x2 = 7 x (one application to deal with the numbers and a second to deal with the roots).

Al-Khwarizmi then shows how to solve the six standard types of equations. He uses both algebraic methods of solution and geometric methods. For example to solve the equation x2 + 10 x = 39 he writes:

... a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square, which itself, of course is 9. Nine therefore gives the square.

The geometric proof by completing the square follows. Al-Khwarizmi starts with a square of side x, which therefore represents x2 (Figure 1). To the square we must add 10x and this is


done by adding four rectangles each of breadth 10/4 and length x to the square (Figure 2). Figure 2 has area x2 + 10 x which is equal to 39. We now complete the square by adding the four little squares each of area 5/2 5/2 = 25/4. Hence the outside square in Fig 3 has area 4 25/4 + 39 = 25 + 39 = 64. The side of the square is therefore 8. But the side is of length 5/2 + x + 5/2 so x + 5 = 8, giving x = 3.

These geometrical proofs are a matter of disagreement between experts. The question, which seems not to have an easy answer, is whether al-Khwarizmi was familiar with Euclid's Elements. We know that he could have been, perhaps it is even fair to say "should have been", familiar with Euclid's work. In al-Rashid's reign, while al-Khwarizmi was still young, al-Hajjaj had translated Euclid's Elements into Arabic and al-Hajjaj was one of al-Khwarizmi's colleagues in the House of Wisdom. This would support Toomer's comments:

... in his introductory section al-Khwarizmi uses geometrical figures to explain equations, which surely argues for a familiarity with Book II of Euclid's "Elements".

Rashed writes that al-Khwarizmi's:

... treatment was very probably inspired by recent knowledge of the "Elements".

However, Gandz argues for a very different view:-

Euclid's "Elements" in their spirit and letter are entirely unknown to [al-Khwarizmi]. Al-Khwarizmi has neither definitions, nor axioms, nor postulates, nor any demonstration of the Euclidean kind.

I [EFR] think that it is clear that whether or not al-Khwarizmi had studied Euclid's Elements, he was influenced by other geometrical works. As Parshall writes:

... because his treatment of practical geometry so closely followed that of the Hebrew text, Mishnat ha Middot, which dated from around 150 AD, the evidence of Semitic ancestry exists.

Al-Khwarizmi continues his study of algebra in Hisab al-jabr w'al-muqabala by examining how the laws of arithmetic extend to an arithmetic for his algebraic objects. For example he shows how to multiply out expressions such as

(a + b x) (c + d x)




Cahit Arf

Arf contributed to the education of many of the present day mathematicians in Turkey, not only by his lectures but also through illuminating discussions in conferences and seminars. Those who had the opportunity to come into close contact with Arf, were deeply influenced by his sincere devotion to mathematics and to science in general. Especially keen to help young mathematicians, he gave them very sound advice and generous encouragement. Arf's approach to mathematics was described by M. G. Ikeda:

To every problem, he has his own idea of approach. The characteristic of his approach is thoroughness; he always seeks invariants, and prefers explicit constructions rather than combination of existing theories. Once he determines his approach, he energetically tackles the problem and never gives up until he achieves his aim. If one studies Cahit Arf's works, which are full of originality and painstaking computations, one will surely wonder where Professor Arf gets his inspirations, and how he gets insight into most complicated computations.

Much of Arf's most important work was in algebraic number theory and he invented Arf invariants which have many applications in topology. His early work was on quadratic forms in fields, particularly fields of characteristic 2. His name is not only attached to Arf invariants but he is also remembered for the Hasse-Arf Theorem which plays an important role in class field theory and in Artin's theory of L-functions. In ring theory, Arf rings are named after him.

In addition Arf worked in applied mathematics writing several papers on elastic plane bodies bounded by free boundaries and a paper on the algebraic structure of the cluster expansion in statistical mechanics. Arf described this digression into applied mathematics had been done for the wrong reasons:

I looked for applause. That's why I talked with engineers and tried to understand their work. ... Mustafa Inan was given a problem while he was making his doctorate. A bridge had collapsed in Belgium. The reason was not known. ... [Mustafa] built a model of that bridge from some material, he put loads on it and found the spots where it began to crack. It was possible to see the places where the tensions increased on that material. ...I took that problem ... I gave the formulas that would build that kind of profiles. ... I wrote five or six essays, which completed each other, about that problem. I was applauded ... but to do things for applause is not nice.

Arf presented a paper On a generalization of Green's formula and its application to the Cauchy problem for a hyperbolic equation to the volume Studies in mathematics and mechanics presented to Richard von Mises in 1954. Arf had met von Mises in 1933 in Istanbul.

An International Symposium on Algebra and Number Theory was held in Arf's honour in Silivri from 3 to 7 September 1990. At an earlier conference on Rings and Geometry held in Istanbul in 1984, Arf had presented a paper The advantage of geometric concepts in Mathematics.



Johann Bernoulli

 Johann's first publication was on the process of fermentation in 1690, certainly not a mathematical topic but in 1691 Johann went to Geneva where he lectured on the differential calculus. From Geneva, Johann made his way to Paris and there he met mathematicians in Malebranche's circle, where the focus of French mathematics was at that time. There Johann met de l'Hôpital and they engaged in deep mathematical conversations. Contrary to what is commonly said these days, de l'Hôpital was a fine mathematician, perhaps the best mathematician in Paris at that time, although he was not quite in the same class as Johann Bernoulli.

De l'Hôpital was delighted to discover that Johann Bernoulli understood the new calculus methods that Leibniz had just published and he asked Johann to teach him these methods. This Johann agreed to do and the lessons were taught both in Paris and also at de l'Hôpital's country house at Oucques. Bernoulli received generous payment from de l'Hôpital for these lessons, and indeed they were worth a lot for few other people would have been able to have given them. After Bernoulli returned to Basel he still continued his calculus lessons by correspondence, and this did not come cheap for de l'Hôpital who paid Bernoulli half a professor's salary for the instruction. However it did assure de l'Hôpital of a place in the history of mathematics since he published the first calculus book Analyse des infiniment petits pour l'intelligence des lignes courbes (1696) which was based on the lessons that Johann Bernoulli sent to him.

As one would expect, it upset Johann Bernoulli greatly that this work did not acknowledge the fact that it was based on his lectures. The preface of the book contains only the statement:-

And then I am obliged to the gentlemen Bernoulli for their many bright ideas; particularly to the younger Mr Bernoulli who is now a professor in Groningen.

The well known de l'Hôpital's rule is contained in this calculus book and it is therefore a result of Johann Bernoulli. In fact proof that the work was due to Bernoulli was not obtained until 1922 when a copy of Johann Bernoulli's course made by his nephew Nicolaus(I) Bernoulli was found in Basel. Bernoulli's course is virtually identical with de l'Hôpital's book but it is worth pointing out that de l'Hôpital had corrected a number of errors such as Bernoulli's mistaken belief that the integral of 1/x is finite. After de l'Hôpital's death in 1704 Bernoulli protested strongly that he was the author of de l'Hôpital's calculus book. It appears that the handsome payment de l'Hôpital made to Bernoulli carried with it conditions which prevented him speaking out earlier. However, few believed Johann Bernoulli until the proofs discovered in 1922.

Let us return to an account of Bernoulli's time in Paris. In 1692, while in Paris, he met Varignon and this resulted in a strong friendship and also Varignon learned much about applications of the calculus from Johann Bernoulli over the many years which they corresponded. Johann Bernoulli also began a correspondence with Leibniz which was to prove very fruitful. In fact this turned out to be the most major correspondence which Leibniz carried out. This was a period of considerable mathematical achievement for Johann Bernoulli. Although he was working on his doctoral dissertation in medicine he was producing numerous papers on mathematical topics which he was publishing and also important results which were contained in his correspondence.

Johann Bernoulli had already solved the problem of the catenary which had been posed by his brother in 1691. He had solved this in the same year that his brother posed the problem and it was his first important mathematical result produced independently of his brother, although it used ideas that Jacob had given when he posed the problem. At this stage Johann and Jacob were learning much from each other in a reasonably friendly rivalry which, a few years later, would descend into open hostility. For example they worked together on caustic curves during 1692-93 although they did not publish the work jointly. Even at this stage the rivalry was too severe to allow joint publications and they would never publish joint work at any time despite working on similar topics.

We mentioned above that Johann's doctoral dissertation was on a topic in medicine, but it was really on an application of mathematics to medicine, being on muscular movement, and it was submitted in 1694. Johann did not wish to follow a career in medicine however, but there were little prospects of a chair at Basel in mathematics since Jacob filled this post.

A stream of mathematical ideas continued to flow from Johann Bernoulli. In 1694 he considered the function y = xx and he also investigated series using the method of integration by parts. Integration to Bernoulli was simply viewed as the inverse operation to differentiation and with this approach he had great success in integrating differential equations. He summed series, and discovered addition theorems for trigonometric and hyperbolic functions using the differential equations they satisfy. This outstanding contribution to mathematics reaped its reward in 1695 when he received two offers of chairs. He was offered a chair at Halle and the chair of mathematics at Groningen. This latter chair was offered to Johann Bernoulli on the advice of Huygens and it was this post which Johann accepted with great pleasure, not least because he now had equal status to his brother Jacob who was rapidly becoming extremely jealous of Johann's progress. The fault was not all on Jacob's side however, and Johann was equally to blame for the deteriorating relations. It is interesting to note that Johann was appointed to the chair of mathematics but his letter of appointment mentions his medical skills and offered him the chance to practice medicine while in Groningen.

While he held the chair in Groningen, Johann Bernoulli competed with his brother in what was becoming an interesting mathematical tussle but an unfortunately bitter personal battle. Johann proposed the problem of the brachristochrone in June 1696 and challenged others to solve it. Leibniz persuaded him to give a longer time so that foreign mathematicians would also have a chance to solve the problem. Five solutions were obtained, Jacob Bernoulli and Leibniz both solving the problem in addition to Johann Bernoulli. The solution of the cycloid had not been found by Galileo who had earlier given an incorrect solution. Not to be outdone by his brother Jacob then proposed the isoperimetric problem, minimising the area enclosed by a curve.

Johann's solution to this problem was less satisfactory than that of Jacob but, when Johann returned to the problem in 1718 having read a work by Taylor, he produced an elegant solution which was to form a foundation for the calculus of variations.

Bernoulli also made important contributions to mechanics with his work on kinetic energy, which, not surprisingly, was another topic on which mathematicians argued over for many years. His work Hydraulica is another sign of his jealous nature. The work is dated 1732 but this is incorrect and was an attempt by Johann to obtain priority over his own son Daniel. Daniel Bernoulli completed his most important work Hydrodynamica in 1734 and published it in 1738 at about the same time as Johann published Hydraulica. This was not an isolated incident, and as he had competed with his brother, he now competed with his own son. As a study of the historical records has justified Johann's claims to be the author of de l'Hôpital's calculus book, so it has shown that his claims to have published Hydraulica before his son wrote Hydrodynamica are false.



Augustin Louis Cauchy

 In 1815 Cauchy lost out to Binet for a mechanics chair at the École Polytechnique, but then was appointed assistant professor of analysis there. He was responsible for the second year course. In 1816 he won the Grand Prix of the French Academy of Sciences for a work on waves. He achieved real fame however when he submitted a paper to the Institute solving one of Fermat's claims on polygonal numbers made to Mersenne. Politics now helped Cauchy into the Academy of Sciences when Carnot and Monge fell from political favour and were dismissed and Cauchy filled one of the two places.

In 1817 when Biot left Paris for an expedition to the Shetland Islands in Scotland Cauchy filled his post at the Collège de France. There he lectured on methods of integration which he had discovered, but not published, earlier. Cauchy was the first to make a rigorous study of the conditions for convergence of infinite series in addition to his rigorous definition of an integral. His text Cours d'analyse in 1821 was designed for students at École Polytechnique and was concerned with developing the basic theorems of the calculus as rigorously as possible. He began a study of the calculus of residues in 1826 in Sur un nouveau genre de calcul analogue au calcul infinitésimal while in 1829 in Leçons sur le Calcul Différentiel he defined for the first time a complex function of a complex variable.

Cauchy did not have particularly good relations with other scientists. His staunchly Catholic views had him involved on the side of the Jesuits against the Académie des Sciences. He would bring religion into his scientific work as for example he did on giving a report on the theory of light in 1824 when he attacked the author for his view that Newton had not believed that people had souls. He was described by a journalist who said:-

... it is certain a curious thing to see an academician who seemed to fulfil the respectable functions of a missionary preaching to the heathens.

An example of how Cauchy treated colleagues is given by Poncelet whose work on projective geometry had, in 1820, been criticised by Cauchy:-

... I managed to approach my too rigid judge at his residence ... just as he was leaving ... During this very short and very rapid walk, I quickly perceived that I had in no way earned his regards or his respect as a scientist ... without allowing me to say anything else, he abruptly walked off, referring me to the forthcoming publication of his Leçons à 'École Polytechnique where, according to him, 'the question would be very properly explored'.

Again his treatment of Galois and Abel during this period was unfortunate. Abel, who visited the Institute in 1826, wrote of him:-

Cauchy is mad and there is nothing that can be done about him, although, right now, he is the only one who knows how mathematics should be done.

Belhoste says:

When Abel's untimely death occurred on April 6, 1829, Cauchy still had not given a report on the 1826 paper, in spite of several protests from Legendre. The report he finally did give, on June 29, 1829, was hasty, nasty, and superficial, unworthy of both his own brilliance and the real importance of the study he had judged.

By 1830 the political events in Paris and the years of hard work had taken their toll and Cauchy decided to take a break. He left Paris in September 1830, after the revolution of July, and spent a short time in Switzerland. There he was an enthusiastic helper in setting up the Académie Helvétique but this project collapsed as it became caught up in political events.

Political events in France meant that Cauchy was now required to swear an oath of allegiance to the new regime and when he failed to return to Paris to do so he lost all his positions there. In 1831 Cauchy went to Turin and after some time there he accepted an offer from the King of Piedmont of a chair of theoretical physics. He taught in Turin from 1832. Menabrea attended these courses in Turin and wrote that the courses:

were very confused, skipping suddenly from one idea to another, from one formula to the next, with no attempt to give a connection between them. His presentations were obscure clouds, illuminated from time to time by flashes of pure genius. ... of the thirty who enrolled with me, I was the only one to see it through.

In 1833 Cauchy went from Turin to Prague in order to follow Charles X and to tutor his grandson. However he was not very successful in teaching the prince as this description shows:-

... exams .. were given each Saturday. ... When questioned by Cauchy on a problem in descriptive geometry, the prince was confused and hesitant. ... There was also material on physics and chemistry. As with mathematics, the prince showed very little interest in these subjects. Cauchy became annoyed and screamed and yelled. The queen sometimes said to him, soothingly, smilingly, 'too loud, not so loud'.

While in Prague Cauchy had one meeting with Bolzano, at Bolzano's request, in 1834. There are discussions on how much Cauchy's definition of continuity is due to Bolzano, Freudenthal's view that Cauchy's definition was formed before Bolzano's seems the more convincing.

Cauchy returned to Paris in 1838 and regained his position at the Academy but not his teaching positions because he had refused to take an oath of allegiance. De Prony died in 1839 and his position at the Bureau des Longitudes became vacant. Cauchy was strongly supported by Biot and Arago but Poisson strongly opposed him. Cauchy was elected but, after refusing to swear the oath, was not appointed and could not attend meetings or receive a salary.

In 1843 Lacroix died and Cauchy became a candidate for his mathematics chair at the Collège de France. Liouville and Libri were also candidates. Cauchy should have easily been appointed on his mathematical abilities but his political and religious activities, such as support for the Jesuits, became crucial factors. Libri was chosen, clearly by far the weakest of the three mathematically, and Liouville wrote the following day that he was:-

deeply humiliated as a man and as a mathematician by what took place yesterday at the Collège de France.

During this period Cauchy's mathematical output was less than in the period before his self-imposed exile. He did important work on differential equations and applications to mathematical physics. He also wrote on mathematical astronomy, mainly because of his candidacy for positions at the Bureau des Longitudes. The 4-volume text Exercices d'analyse et de physique mathématique published between 1840 and 1847 proved extremely important.

When Louis Philippe was overthrown in 1848 Cauchy regained his university positions. However he did not change his views and continued to give his colleagues problems. Libri, who had been appointed in the political way described above, resigned his chair and fled from France. Partly this must have been because he was about to be prosecuted for stealing valuable books. Liouville and Cauchy were candidates for the chair again in 1850 as they had been in 1843. After a close run election Liouville was appointed. Subsequent attempts to reverse this decision led to very bad relations between Liouville and Cauchy.



Leonhard Euler

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).

We owe to Euler the notation f (x) for a function (1734), e for the base of natural logs (1727), i for the square root of -1 (1777), ? for pi, ? for summation (1755), the notation for finite differences ?y and ?2y and many others.

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 2n + 1 were always prime if n is a power of 2. Euler verified this for n = 1, 2, 4, 8 and 16 and, by 1732 at the latest, showed that the next case 232 + 1 = 4294967297 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 ?(n), the number of integers k with 1 ? k ? n and k coprime to n. He proved another of Fermat's assertions, namely that if a and b are coprime then a2 + b2 has no divisor of the form 4n - 1, in 1749.

Perhaps the result that brought Euler the most fame in his young days was his solution of what had become known as the Basel problem. This was to find a closed form for the sum of the infinite series ?(2) = ? (1/n2), a problem which had defeated many of the top mathematicians including Jacob Bernoulli, Johann Bernoulli and Daniel Bernoulli. The problem had also been studied unsuccessfully by Leibniz, Stirling, de Moivre and others. Euler showed in 1735 that ?(2) = ?2/6 but he went on to prove much more, namely that ?(4) = ?4/90, ?(6) = ?6/945, ?(8) = ?8/9450, ?(10) = ?10/93555 and ?(12) = 691?12/638512875. In 1737 he proved the connection of the zeta function with the series of prime numbers giving the famous relation

?(s) = ? (1/ns) = ? (1 - p-s)-1

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

By 1739 Euler had found the rational coefficients C in ?(2n) = C?2n in terms of the Bernoulli numbers.

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

1/1 + 1/2 + 1/3 + ... + 1/n - logen

as n tends to infinity. He calculated the constant ? 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

?/2 - x/2 = sin x + (sin 2x)/2 + (sin 3x)/3 + ...

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 n = 3. Perhaps more significant than the result here was the fact that he introduced a proof involving numbers of the form a + b?-3 for integers a and b. 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

eix = cos x + i sin x.

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

ln(-1) = ?i

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. 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:

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 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 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.



Pierre-Simon Laplace

 He began producing a steady stream of remarkable mathematical papers, the first presented to the Académie des Sciences in Paris on 28 March 1770. This first paper, read to the Society but not published, was on maxima and minima of curves where he improved on methods given by Lagrange. His next paper for the Academy followed soon afterwards, and on 18 July 1770 he read a paper on difference equations.

Laplace's first paper which was to appear in print was one on the integral calculus which he translated into Latin and published at Leipzig in the Nova acta eruditorum in 1771. Six years later Laplace republished an improved version, apologising for the 1771 paper and blaming errors contained in it on the printer. Laplace also translated the paper on maxima and minima into Latin and published it in the Nova acta eruditorum in 1774. Also in 1771 Laplace sent another paper Recherches sur le calcul intégral aux différences infiniment petites, et aux différences finies to the Mélanges de Turin. This paper contained equations which Laplace stated were important in mechanics and physical astronomy.

The year 1771 marks Laplace's first attempt to gain election to the Académie des Sciences but Vandermonde was preferred. Laplace tried to gain admission again in 1772 but this time Cousin was elected. Despite being only 23 (and Cousin 33) Laplace felt very angry at being passed over in favour of a mathematician who was so clearly markedly inferior to him. D'Alembert also must have been disappointed for, on 1 January 1773, he wrote to Lagrange, the Director of Mathematics at the Berlin Academy of Science, asking him whether it might be possible to have Laplace elected to the Berlin Academy and for a position to be found for Laplace in Berlin.

Before Lagrange could act on d'Alembert's request, another chance for Laplace to gain admission to the Paris Académie arose. On 31 March 1773 he was elected an adjoint in the Académie des Sciences. By the time of his election he had read 13 papers to the Académie in less than three years. Condorcet, who was permanent secretary to the Académie, remarked on this great number of quality papers on a wide range of topics.

Laplace presented his famous nebular hypothesis in 1796 in Exposition du systeme du monde, which viewed the solar system as originating from the contracting and cooling of a large, flattened, and slowly rotating cloud of incandescent gas. The Exposition consisted of five books: the first was on the apparent motions of the celestial bodies, the motion of the sea, and also atmospheric refraction; the second was on the actual motion of the celestial bodies; the third was on force and momentum; the fourth was on the theory of universal gravitation and included an account of the motion of the sea and the shape of the Earth; the final book gave an historical account of astronomy and included his famous nebular hypothesis. Laplace states his philosophy of science in the Exposition as follows:-

If man were restricted to collecting facts the sciences were only a sterile nomenclature and he would never have known the great laws of nature. It is in comparing the phenomena with each other, in seeking to grasp their relationships, that he is led to discover these laws...

In view of modern theories of impacts of comets on the Earth it is particularly interesting to see Laplace's remarkably modern view of this:-

... the small probability of collision of the Earth and a comet can become very great in adding over a long sequence of centuries. It is easy to picture the effects of this impact on the Earth. The axis and the motion of rotation have changed, the seas abandoning their old position..., a large part of men and animals drowned in this universal deluge, or destroyed by the violent tremor imparted to the terrestrial globe.

Exposition du systeme du monde was written as a non-mathematical introduction to Laplace's most important work Traité de Mécanique Céleste whose first volume appeared three years later. Laplace had already discovered the invariability of planetary mean motions. In 1786 he had proved that the eccentricities and inclinations of planetary orbits to each other always remain small, constant, and self-correcting. These and many other of his earlier results formed the basis for his great work the Traité de Mécanique Céleste published in 5 volumes, the first two in 1799.

The first volume of the Mécanique Céleste is divided into two books, the first on general laws of equilibrium and motion of solids and also fluids, while the second book is on the law of universal gravitation and the motions of the centres of gravity of the bodies in the solar system. The main mathematical approach here is the setting up of differential equations and solving them to describe the resulting motions. The second volume deals with mechanics applied to a study of the planets. In it Laplace included a study of the shape of the Earth which included a discussion of data obtained from several different expeditions, and Laplace applied his theory of errors to the results. Another topic studied here by Laplace was the theory of the tides but Airy, giving his own results nearly 50 years later, wrote:-

It would be useless to offer this theory in the same shape in which Laplace has given it; for that part of the Mécanique Céleste which contains the theory of tides is perhaps on the whole more obscure than any other part...

In the Mécanique Céleste Laplace's equation appears but although we now name this equation after Laplace, it was in fact known before the time of Laplace. The Legendre functions also appear here and were known for many years as the Laplace coefficients. The Mécanique Céleste does not attribute many of the ideas to the work of others but Laplace was heavily influenced by Lagrange and by Legendre and used methods which they had developed with few references to the originators of the ideas.

Under Napoleon Laplace was a member, then chancellor, of the Senate, and received the Legion of Honour in 1805. However Napoleon, in his memoirs written on St Hélène, says he removed Laplace from the office of Minister of the Interior, which he held in 1799, after only six weeks:-

... because he brought the spirit of the infinitely small into the government.

Laplace became Count of the Empire in 1806 and he was named a marquis in 1817 after the restoration of the Bourbons.

The first edition of Laplace's Théorie Analytique des Probabilités was published in 1812. This first edition was dedicated to Napoleon-le-Grand but, for obvious reason, the dedication was removed in later editions! The work consisted of two books and a second edition two years later saw an increase in the material by about an extra 30 per cent.

The first book studies generating functions and also approximations to various expressions occurring in probability theory. The second book contains Laplace's definition of probability, Bayes's rule (so named by Poincaré many years later), and remarks on moral and mathematical expectation. The book continues with methods of finding probabilities of compound events when the probabilities of their simple components are known, then a discussion of the method of least squares, Buffon's needle problem, and inverse probability. Applications to mortality, life expectancy and the length of marriages are given and finally Laplace looks at moral expectation and probability in legal matters.

Later editions of the Théorie Analytique des Probabilités also contains supplements which consider applications of probability to: errors in observations; the determination of the masses of Jupiter, Saturn and Uranus; triangulation methods in surveying; and problems of geodesy in particular the determination of the meridian of France. Much of this work was done by Laplace between 1817 and 1819 and appears in the 1820 edition of the Théorie Analytique. A rather less impressive fourth supplement, which returns to the first topic of generating functions, appeared with the 1825 edition. This final supplement was presented to the Institute by Laplace, who was 76 years old by this time, and by his son.

We mentioned briefly above Laplace's first work on physics in 1780 which was outside the area of mechanics in which he contributed so much. Around 1804 Laplace seems to have developed an approach to physics which would be highly influential for some years. This is best explained by Laplace himself:-

... I have sought to establish that the phenomena of nature can be reduced in the last analysis to actions at a distance between molecule and molecule, and that the consideration of these actions must serve as the basis of the mathematical theory of these phenomena.

This approach to physics, attempting to explain everything from the forces acting locally between molecules, already was used by him in the fourth volume of the Mécanique Céleste which appeared in 1805. This volume contains a study of pressure and density, astronomical refraction, barometric pressure and the transmission of gravity based on this new philosophy of physics. It is worth remarking that it was a new approach, not because theories of molecules were new, but rather because it was applied to a much wider range of problems than any previous theory and, typically of Laplace, it was much more mathematical than any previous theories.



Gottfried Wilhelm von Leibniz

 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 (Dissertation on the combinatorial art). 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 (On Perplexing Cases).


... with Boineburg's encouragement, he drafted a number of monographs on religious topics, mostly to do with points at issue between the churches...

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.

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 ?  f (x) dx notation for the first time. In the same manuscript the product rule for differentiation is given. By autumn 1676 Leibniz discovered the familiar d(xn) = nxn-1dx for both integral and fractional n.

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 worker's fear that technological progress would cost them their jobs.

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 (Reflections on Knowledge, Truth, and Ideas) which clarified his theory of knowledge. In February 1686, Leibniz wrote his Discours de métaphysique (Discourse on Metaphysics).

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 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:

... 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.

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 x = 1 into 1/(1+x) = 1 - x + x2 - x3 + .... Leibniz also corresponded with Varignon on this paradox. Leibniz discussed logarithms of negative numbers with Johann Bernoulli.

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 Nicolaus(I) Bernoulli. 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.



Pafnuty Lvovich Chebyshev

 The Russian university system that Chebyshev entered had undergone considerable change. Moscow University that he entered had been founded in 1755 and modelled on the German universities. However following the Russian victory over Napoleon there was the westernising movement in the country which we mentioned above. Alexander I, the emperor of Russia, saw the universities as the breeding grounds for what he considered as dangerous doctrines coming from western Europe and the universities were put under pressure in the 1820s to dismiss staff who taught such doctrines. A new minister of education was appointed in 1833 under Nicholas I, who had become Russian emperor in 1825, and he promoted a freer intellectual atmosphere in the universities but on the other hand children of the lower classes were excluded.

At Moscow University the person who was to influence Chebyshev most was Nikolai Dmetrievich Brashman who had been professor of applied mathematics at the university since 1834. Brashman was particularly interested in mechanics but his interests were wide ranging and, in addition to courses on mechanical engineering and hydraulics, he taught his students the theory of integration of algebraic functions and the calculus of probability. Chebyshev always acknowledged the great influence Brashman had been on him while studying at university, and credited him as the main influence in directing his research interests, referring to their "precious personal talks".

The department of physics and mathematics in which Chebyshev studied announced a prize competition for the year 1840-41. Chebyshev submitted a paper on The calculation of roots of equations in which he solved the equation y = f (x) by using a series expansion for the inverse function of f. The paper was not published at the time (although it was published in the 1950s) and it was awarded only second prize in the competition rather than the Gold Medal it almost certainly deserved. Chebyshev graduated with his first degree in 1841 and continued to study for his Master's degree under Brashman's supervision.

Once, much later in his career, Chebyshev objected to being described as a "splendid Russian mathematician" and said that surely he was a "world-wide mathematician" rather than a Russian mathematician. It is very clear that right from the time he began his studies for his Master's degree that Chebyshev aimed at international recognition. His very first paper was written in French and was on multiple integrals. He submitted the paper to Liouville in late 1842 and the paper appeared in Liouville's journal in 1843. It contains a formula which is stated without proof and the following paper in the first part of volume 8 of the journal contains a proof of the formula given by Catalan. Some authors suggest that Chebyshev may have visited Paris in 1842 accompanying the Russian geographer Chikhachev who certainly met Catalan (who assisted Liouville in producing his journal) in December of that year. There is no conclusive evidence, but it must be highly likely that if Chebyshev did not personally visit Paris in 1842 then he sent his paper to Liouville via Chikhachev.

Chebyshev continued to aim at international recognition with his second paper, written again in French, appearing in 1844 published by Crelle in his journal. This paper was on the convergence of Taylor series. In the summer of 1846 Chebyshev was examined on his Master's thesis and in the same year published a paper based on that thesis, again in Crelle's journal. The thesis was on the theory of probability, and in it he developed the main results of the theory in a rigorous but elementary way. In particular the paper he published from his thesis examined Poisson's weak law of large numbers.

During 1843 Chebyshev produced a first draft of a thesis which he intended to submit to obtain his right to lecture once he found a suitable position. Times were hard and Moscow had no suitable positions available for Chebyshev but, in 1847, he was appointed to the University of St Petersburg submitting his thesis On integration by means of logarithms. In it he generalised methods of Ostrogradski to show that a conjecture which Abel made in 1826 about the integral of f (x)/?R(x), where f (x) and R(x) are polynomials, was true. In a report which he wrote about a visit to Paris in 1852, Chebyshev described how he was asked to develop the ideas further:

Liouville and Hermite suggested the idea of developing the ideas on which my thesis had been based. ... in the thesis I considered the case where the differential under the integral contains the square root of a rational function. But it was interesting in several respects to extend those principles to a root of any degree.

Although Chebyshev's thesis was not published until after his death, he published a paper containing some of its results in 1853.

Between arriving in St Petersburg and this 1853 publication Chebyshev published some of his most famous results on number theory. He wrote an important book Teoria sravneny on the theory of congruences which he submitted for his doctorate, defending it on 27 May 1849. This work also received a prize from the Academy of Sciences. He collaborated with Bunyakovsky in producing a complete edition of Euler's 99 number theory papers which they published in two volumes in 1849. Chebyshev's work on prime numbers included the determination of the number of primes not exceeding a given number, published in 1848, and a proof of Bertrand's conjecture.

In 1845 Bertrand conjectured that there was always at least one prime between n and 2n for n > 3. Chebyshev proved Bertrand's conjecture in 1850. Chebyshev also came close to proving the Prime Number Theorem, proving that if

( ?(n) log n ) / n

(with ?(n) the number of primes ? n) had a limit as n › ? then that limit is 1. He was unable to prove, however, that

lim ( ?(n) log n ) / n as n › ?

exists. The proof of this result was only completed two years after Chebyshev's death by Hadamard and (independently) de la Vallée Poussin.

We have mentioned some contributions that Chebyshev made to the theory of probability. In 1867 he published a paper On mean values which used Bienaymé's inequality to give a generalised law of large numbers. As a result of his work on this topic the inequality today is often known as the Bienaymé-Chebyshev inequality. Twenty years later Chebyshev published On two theorems concerning probability which gives the basis for applying the theory of probability to statistical data, generalising the central limit theorem of de Moivre and Laplace. Of this Kolmogorov wrote:

The principal meaning of Chebyshev's work is that through it he always aspired to estimate exactly in the form of inequalities absolutely valid under any number of tests the possible deviations from limit regularities. Further, Chebyshev was the first to estimate clearly and make use of such notions as "random quantity" and its "expectation (mean) value".

Let us mention a few further aspects of Chebyshev's work. In the theory of integrals he generalised the beta function and examined integrals of the form

?  xp (1 - x)q dx.

Other topics to which he contributed were the construction of maps, the calculation of geometric volumes, and the construction of calculating machines in the 1870s. In mechanics he studied problems involved in converting rotary motion into rectilinear motion by mechanical coupling. The Chebyshev parallel motion is three linked bars approximating rectilinear motion. He wrote many papers on his mechanical inventions; Lucas exhibited models and drawings of some of these at the Conservatoire National des Arts et Métiers in Paris. In 1893 seven of his mechanical inventions were exhibited at the World's Exposition in Chicago, organised to celebrate the 400th anniversary of Christopher Columbus' discovery of America, including his invention of a special bicycle.



Masatoshi Gündüz Ikeda

Gündüz Ikeda's mother was a school teacher and his father was a statistician. Known as Masatoshi Ikeda for the first part of his life spent in Japan, he was the youngest of his parents four children, having two elder sisters and one elder brother. He entered Osaka University to study mathematics, obtaining his Rigaku-Shi degree (equivalent to a B.Sc.) in 1948. Of course his education had taken place through the years of World War II and the extremely hard years immediately following the war. He once said of these years:-

We were out of food, at the edge of starving, yet we were reading Pontryagin's Topological Groups without skipping a word.

He remained at Osaka University, undertaking research on Frobenius and quasi-Frobenius algebras for his doctorate supervised by Kenjiro Shoda. He worked with a number of mathematicians at Osaka, joining the research group led by T Nakayama. Ikeda's first publication Supplementary remarks on Frobenius algebras was written jointly with Nakayama, and appeared in the Osaka Mathematics Journal in 1950. He was awarded his doctorate in 1953 for a thesis which included sufficient material for seven excellent published papers. In addition to the paper named above, the seven papers included two further papers on quasi-Frobenius algebras, and one on generalizing Gaschütz's results on group algebras to Frobenius rings.

Ikeda was awarded a Yukawa Scholarship to conduct research on cohomology theory of associative algebras at Nagoya University for session 1953-54. He returned to Osaka University as a Lecturer in Mathematics in 1954 where he continued working on algebras, and published three further papers on the cohomology of algebras. In 1955 he attended the Algebraic Number Theory Symposium in Tokyo and it changed the direction of his research. He decided as a result of that meeting to move towards algebraic number theory and an Alexander von Humboldt Foundation Fellowship allowed him to spend the years from 1957 to 1959 as a research fellow at the Mathematisches Seminar of Hamburg University, working with Hasse's research group. His main research during this period was on the embedding problem of Galois theory.

Another important event happened during these years for he met the Turkish research fellow Emel Ardor, also in Hamburg on an Alexander von Humboldt Fellowship. She has written [2]:-

Dr Ikeda began to show me his interests. He asked me several questions about Turkey, Turkish, and life and customs in Turkey. I soon found out the reason for this curiosity. His teacher Professor Hasse had visited Turkey a couple of times, he loved Turkey, he had friends and students in Turkey, and he was keen on learning Turkish. Further, he met Prof Orhan Içen in Hamburg, and he knew Cahit Arf by name. [Ikeda] was keen on learning new things; [and] he, too, developed a passion for Turkey and Turkish.

Ikeda remained as a Lecturer of Mathematics at Osaka University until 1960 when he first visited Izmir, Turkey, then visited Cahit Arf at Istanbul University. Arf had studied in Göttingen under Hasse, so the two had many common interests. Ikeda was appointed as Foreign Specialist in Mathematics and Statistics at the Medical School of Ege University in Izmir in 1960. After a while, however, he was appointed Associate Professor of Mathematics in the newly established Mathematics Department of Ege University. Cemal Koç writes in [3] about this period:-

In the summer of 1963 as a new graduate I paid a visit to the newly established Mathematics Department of Ege University, in Izmir, to apply for a post. At the very moment of my arrival, they introduced to me a young faculty member: Doctor Ikeda. Apparently he was Japanese, polite yet authoritative, and there was an air of general admiration about him; however the rest was a complete mystery for me.

After Koç was appointed to Ege University he assisted Ikeda with his teaching. He writes of the help that Ikeda gave to him and others at the university:-

Having been newly established, Ege University was short of facilities, resulting in, among other things, a limited number of offices in the Mathematics Department. Thus we assistants found ourselves sharing one single office. However, in time, thanks to Dr Ikeda, we transformed this office to an ongoing seminar room, for we had problems in different areas of mathematics and Dr Ikeda was the only one to lend us a hand so benevolently. His generosity and kindheartedness would touch all the faculty at Ege University, as well as all the mathematics assistants, in all of their diverse areas, supervised by himself or not. He was even often found to be helping many a faculty applicant to associate professorship or professorship in Medicine or Agriculture in their various correlation problems.

Ikeda married Emel Ardor, the Turkish research assistant he had met in Hamburg, in 1964. He also became a Turkish citizen in that year and took a more Turkish sounding name, calling himself Gündüz Ikeda. He spent time at the Middle East Technical University in Ankara, was a Visiting Professor at Hamburg during the Spring Semester of 1966, and then in the following year he was promoted to full Professor at Ege University. In May 1968 Arf and Langlands visited Ege University to deliver talks on The Cartan-Dieudonné Theorem and Automorphic Forms. At a dinner following the talks, Arf offered Ikeda a position at the Middle East Technical University in Ankara where he was now working. Ikeda accepted and took up the position of Professor there in August of 1968 and, together with Arf, worked to build up both the undergraduate and postgraduate sides of the Mathematics Department.

Although Ikeda remained in his post at the Middle East Technical University until 1976, he spent 1970-71 as Visiting Professor of Mathematics at San Diego State University, California, USA. He left the Middle East Technical University in 1976, spent the autumn term of 1976 at the Institute for Advanced Study at Princeton, then during the years 1976-78 he was Chairman of the Mathematics Department of Hacettepe University. In 1978 he returned to the Middle East Technical University where he remained until he retired in 1991.

After he retired, Ikeda was Visiting Professor of Mathematics at the Eastern Mediterranean University, Gazi Magosa, in Turkish-controlled Northern Cyprus from 1991 to 1993, then a Senior Researcher at Marmara Research Center, Gebze, Turkey from 1993 to 1995. His next appointment was as Senior Research Scientist at the National Research Institute for Electronics and Cryptology, Gebze, Turkey, between 1995 and 1997, where he headed a small research group who concentrated on cryptography. Finally he was a Senior Research Scientist at Feza Gursey Institute, Istanbul, where he returned to his main research topic of algebraic number theory.

Ikeda received many honours for his outstanding contributions. He was awarded the TUBITAK Science Prize (1979), the Mustafa Parlar Foundation Science Prize (1993), and the Marmara Research Center Merit Prize (1994). He was elected to the Turkish Academy of Sciences in 1997.

Koç, in [3], quotes some advice that Ikeda gave to young mathematicians:-

Avoid repeating yourselves in order to increase the number of your publications only. Work out real problems of mathematics no matter how big they are ...

When you have worked on a problem for a long time without any outcome, leave it alone for a while and consider also some relatively small problems to keep up your courage and productivity.

Despite his close friendship with Arf, Ikeda was critical of his contributions to Turkish mathematics. He wrote:-

To my regret, however, Cahit Arf never had pupils (in a true sense) in Turkey; it might be because he is too great, or because his works are too hard for common mathematicians. In any case, it is a pity not for Cahit Arf but for Turkish Mathematics. I really do not see why Arf rings are studied by American mathematicians yet not by Turkish mathematicians, and why his wonderful thesis is intensively re-examined by German mathematicians yet not by Turkish mathematicians. I am sure that the growth of Mathematics in a country is, as history shows, only possible if the mathematicians in that country mathematically understand and stimulate each other. So I should like to emphasize again that it is the task of young Turkish mathematicians to work in these fields, first to learn by heart what Arf did, and then to continue further study along the lines indicated by him.

Ikeda's teaching is described as follows

... he would always prefer to cover many diverse concepts and techniques with their essential features. ... His lectures included neither unnecessary material, nor unnecessary ornaments and digressions. He was meticulous about the content and rigorous in his lectures.

His character is well illustrated by the comments:-

He was so kindhearted and benevolent that probably there was very few who met him that did not receive profound suggestions and advice to improve their work and ideas. In some gatherings, participants often urged him to talk on mathematics and mathematicians in general.