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Joseph-Louis Lagrange

Born: 25 Jan 1736 in Turin, Sardinia-Piedmont (now Italy)

Died: 10 April 1813 in Paris, France

Joseph-Louis Lagrange is usually considered to be a French mathematician, but the Italian Encyclopaedia [40] refers to him as an Italian mathematician. They certainly have some justification in this claim since Lagrange was born in Turin and baptised in the name of Giuseppe Lodovico Lagrangia. Lagrange’s father was Giuseppe Francesco Lodovico Lagrangia who was Treasurer of the Office of Public Works and Fortifications in Turin, while his mother Teresa Grosso was the only daughter of a medical doctor from Cambiano near Turin. Lagrange was the eldest of their 11 children but one of only two to live to adulthood.

Turin had been the capital of the duchy of Savoy, but become the capital of the kingdom of Sardinia in 1720, sixteen years before Lagrange’s birth. Lagrange’s family had French connections on his father’s side, his great-grandfather being a French cavalry captain who left France to work for the Duke of Savoy. Lagrange always leant towards his French ancestry, for as a youth he would sign himself Lodovico LaGrange or Luigi Lagrange, using the French form of his family name.

Despite the fact that Lagrange’s father held a position of some importance in the service of the king of Sardinia, the family were not wealthy since Lagrange’s father had lost large sums of money in unsuccessful financial speculation. A career as a lawyer was planned out for Lagrange by his father, and certainly Lagrange seems to have accepted this willingly. He studied at the College of Turin and his favourite subject was classical Latin. At first he had no great enthusiasm for mathematics, finding Greek geometry rather dull.

Lagrange’s interest in mathematics began when he read a copy of  Halley’s 1693 work on the use of algebra in optics. He was also attracted to physics by the excellent teaching of Beccaria at the College of Turin and he decided to make a career for himself in mathematics. Perhaps the world of mathematics has to thank Lagrange’s father for his unsound financial speculation, for Lagrange later claimed:-

If I had been rich, I probably would not have devoted myself to mathematics.

He certainly did devote himself to mathematics, but largely he was self taught and did not have the benefit of studying with leading mathematicians. On 23 July 1754 he published his first mathematical work which took the form of a letter written in Italian to  Giulio Fagnano. Perhaps most surprising was the name under which Lagrange wrote this paper, namely Luigi De la Grange Tournier. This work was no masterpiece and showed to some extent the fact that Lagrange was working alone without the advice of a mathematical supervisor. The paper draws an analogy between the  binomial theorem and the successive derivatives of the product of functions.

Before writing the paper in Italian for publication, Lagrange had sent the results to  Euler, who at this time was working in Berlin, in a letter written in Latin. The month after the paper was published, however, Lagrange found that the results appeared in correspondence between  Johann Bernoulli and  Leibniz. Lagrange was greatly upset by this discovery since he feared being branded a cheat who copied the results of others. However this less than outstanding beginning did nothing more than make Lagrange redouble his efforts to produce results of real merit in mathematics. He began working on the  tautochrone, the curve on which a weighted particle will always arrive at a fixed point in the same time independent of its initial  position. By the end of 1754 he had made some important discoveries on the tautochrone which would contribute substantially to the new subject of the  calculus of variations (which mathematicians were beginning to study but which did not receive the name ‘calculus of variations’ before  Euler called it that in 1766).

Lagrange sent  Euler his results on the tautochrone containing his method of maxima and minima. His letter was written on 12 August 1755 and  Euler replied on 6 September saying how impressed he was with Lagrange’s new ideas. Although he was still only 19 years old, Lagrange was appointed professor of mathematics at the Royal Artillery School in Turin on 28 September 1755. It was well deserved for the young man had already shown the world of mathematics the originality of his thinking and the depth of his great talents.

In 1756 Lagrange sent  Euler results that he had obtained on applying the calculus of variations to mechanics. These results generalised results which  Euler had himself obtained and  Euler consulted  Maupertuis, the president of the Academy, about this remarkable young mathematician. Not only was Lagrange an outstanding mathematician but he was also a strong advocate for the principle of least action so  Maupertuis had no hesitation but to try to entice Lagrange to a position in Prussia. He arranged with  Euler that he would let Lagrange know that the new position would be considerably more prestigious than the one he held in Turin. However, Lagrange did not seek greatness, he only wanted to be able to devote his time to mathematics, and so he shyly but politely refused the position.

 Euler also proposed Lagrange for election to the Berlin Academy and he was duly elected on 2 September 1756. The following year Lagrange was a founding member of a scientific society in Turin, which was to become the Royal Academy of Science of Turin. One of the major roles of this new Society was to publish a scientific journal the Mélanges de Turin which published articles in French or Latin. Lagrange was a major contributor to the first volumes of the Mélanges de Turin volume 1 of which appeared in 1759, volume 2 in 1762 and volume 3 in 1766.

The papers by Lagrange which appear in these transactions cover a variety of topics. He published his beautiful results on the calculus of variations, and a short work on the  calculus of probabilities. In a work on the foundations of dynamics, Lagrange based his development on the principle of least action and on kinetic energy.

In the Mélanges de Turin Lagrange also made a major study on the propagation of sound, making important contributions to the theory of vibrating strings. He had read extensively on this topic and he clearly had thought deeply on the works of  Newton,  Daniel Bernoulli,  Taylor,  Euler and  d’Alembert. Lagrange used a discrete mass model for his vibrating string, which he took to consist of n masses joined by weightless strings. He solved the resulting system of n+1  differential equations, then let n tend to infinity to obtain the same functional solution as  Euler had done. His different route to the solution, however, shows that he was looking for different methods than those of  Euler, for whom Lagrange had the greatest respect.

In papers which were published in the third volume, Lagrange studied the integration of differential equations and made various applications to topics such as fluid mechanics (where he introduced the Lagrangian function). Also contained are methods to solve systems of linear differential equations which used the characteristic value of a linear substitution for the first time. Another problem to which he applied his methods was the study the orbits of Jupiter and Saturn.

The Académie des Sciences in Paris announced its prize competition for 1764 in 1762. The topic was on the libration of the Moon, that is the motion of the Moon which causes the face that it presents to the Earth to oscillate causing small changes in the position of the lunar features. Lagrange entered the competition, sending his entry to Paris in 1763 which arrived there not long before Lagrange himself. In November of that year he left Turin to make his first long journey, accompanying the Marquis Caraccioli, an ambassador from Naples who was moving from a post in Turin to one in London. Lagrange arrived in Paris shortly after his entry had been received but took ill while there and did not proceed to London with the ambassador.  D’Alembert was upset that a mathematician as fine as Lagrange did not receive more honour. He wrote on his behalf:-

Monsieur de la Grange, a young geometer from Turin, has been here for six weeks. He has become quite seriously ill and he needs, not financial aid, for the Marquis de Caraccioli directed upon leaving for England that he should not lack for anything, but rather some signs of interest on the part of his native country … In him Turin possesses a treasure whose worth it perhaps does not know.

Returning to Turin in early 1765, Lagrange entered, later that year, for the Académie des Sciences prize of 1766 on the orbits of the moons of Jupiter.  D’Alembert, who had visited the Berlin Academy and was friendly with Frederick II of Prussia, arranged for Lagrange to be offered a position in the Berlin Academy. Despite no improvement in Lagrange’s position in Turin, he again turned the offer down writing:-

It seems to me that Berlin would not be at all suitable for me while M  Euler is there.

By March 1766  d’Alembert knew that  Euler was returning to St Petersburg and wrote again to Lagrange to encourage him to accept a post in Berlin. Full details of the generous offer were sent to him by Frederick II in April, and Lagrange finally accepted. Leaving Turin in August, he visited  d’Alembert in Paris, then Caraccioli in London before arriving in Berlin in October. Lagrange succeeded  Euler as Director of Mathematics at the Berlin Academy of Science on 6 November 1766.

Lagrange was greeted warmly by most members of the Academy and he soon became close friends with  Lambert and  Johann(III) Bernoulli. However, not everyone was pleased to see this young man in such a prestigious position, particularly  Castillon who was 32 years older than Lagrange and considered that he should have been appointed as Director of Mathematics. Just under a year from the time he arrived in Berlin, Lagrange married his cousin Vittoria Conti. He wrote to  d’Alembert:-

My wife, who is one of my cousins and who even lived for a long time with my family, is a very good housewife and has no pretensions at all.

They had no children, in fact Lagrange had told  d’Alembert in this letter that he did not wish to have children.

Turin always regretted losing Lagrange and from time to time his return there was suggested, for example in 1774. However, for 20 years Lagrange worked at Berlin, producing a steady stream of top quality papers and regularly winning the prize from the Académie des Sciences of Paris. He shared the 1772 prize on the  three body problem with  Euler, won the prize for 1774, another one on the motion of the moon, and he won the 1780 prize on perturbations of the orbits of comets by the planets.

His work in Berlin covered many topics: astronomy, the stability of the  solar system, mechanics, dynamics, fluid mechanics, probability, and the foundations of the calculus. He also worked on  number theory proving in 1770 that every positive integer is the sum of four squares. In 1771 he proved  Wilson’s theorem (first stated without proof by  Waring) that n is  prime if and only if (n -1)! + 1 is divisible by n. In 1770 he also presented his important work Réflexions sur la résolution algébrique des équations which made a fundamental investigation of why equations of degrees up to 4 could be solved by  radicals. The paper is the first to consider the roots of a equation as abstract quantities rather than having numerical values. He studied  permutations of the roots and, although he does not compose permutations in the paper, it can be considered as a first step in the development of  group theory continued by  Ruffini,  Galois and  Cauchy.

Although Lagrange had made numerous major contributions to mechanics, he had not produced a comprehensive work. He decided to write a definitive work incorporating his contributions and wrote to  Laplace on 15 September 1782:-

I have almost completed a Traité de mécanique analytique, based uniquely on the principle of virtual velocities; but, as I do not yet know when or where I shall be able to have it printed, I am not rushing to put the finishing touches to it.

Caraccioli, who was by now in Sicily, would have liked to see Lagrange return to Italy and he arranged for an offer to be made to him by the court of Naples in 1781. Offered the post of Director of Philosophy of the Naples Academy, Lagrange turned it down for he only wanted peace to do mathematics and the position in Berlin offered him the ideal conditions. During his years in Berlin his health was rather poor on many occasions, and that of his wife was even worse. She died in 1783 after years of illness and Lagrange was very depressed. Three years later Frederick II died and Lagrange’s position in Berlin became a less happy one. Many Italian States saw their chance and attempts were made to entice him back to Italy.

The offer which was most attractive to Lagrange, however, came not from Italy but from Paris and included a clause which meant that Lagrange had no teaching. On 18 May 1787 he left Berlin to become a member of the Académie des Sciences in Paris, where he remained for the rest of his career. Lagrange survived the French Revolution while others did not and this may to some extent be due to his attitude which he had expressed many years before when he wrote:-

I believe that, in general, one of the first principles of every wise man is to conform strictly to the laws of the country in which he is living, even when they are unreasonable.

The Mécanique analytique which Lagrange had written in Berlin, was published in 1788. It had been approved for publication by a committee of the Académie des Sciences comprising of  Laplace, Cousin,  Legendre and  Condorcet.  Legendre acted as an editor for the work doing proof reading and other tasks. The Mécanique analytique summarised all the work done in the field of mechanics since the time of  Newton and is notable for its use of the theory of differential equations. With this work Lagrange transformed mechanics into a branch of mathematical analysis. He wrote in the Preface:-

One will not find figures in this work. The methods that I expound require neither constructions, nor geometrical or mechanical arguments, but only algebraic operations, subject to a regular and uniform course.

Lagrange was made a member of the committee of the Académie des Sciences to standardise weights and measures in May 1790. They worked on the metric system and advocated a decimal base. Lagrange married for a second time in 1792, his wife being Renée-Françoise-Adélaide Le Monnier the daughter of one of his astronomer colleagues at the Académie des Sciences. He was certainly not unaffected by the political events. In 1793 the Reign of Terror commenced and the Académie des Sciences, along with the other learned societies, was suppressed on 8 August. The weights and measures commission was the only one allowed to continue and Lagrange became its chairman when others such as the chemist Lavoisier,  Borda,  Laplace,  Coulomb,  Brisson and  Delambre were thrown off the commission.

In September 1793 a law was passed ordering the arrest of all foreigners born in enemy countries and all their property to be confiscated. Lavoisier intervened on behalf of Lagrange, who certainly fell under the terms of the law, and he was granted an exception. On 8 May 1794, after a trial that lasted less than a day, a revolutionary tribunal condemned Lavoisier, who had saved Lagrange from arrest, and 27 others to death. Lagrange said on the death of Lavoisier, who was guillotined on the afternoon of the day of his trial:-

It took only a moment to cause this head to fall and a hundred years will not suffice to produce its like.

The École Polytechnique was founded on 11 March 1794 and opened in December 1794 (although it was called the École Centrale des Travaux Publics for the first year of its existence). Lagrange was its first professor of analysis, appointed for the opening in 1794. In 1795 the École Normale was founded with the aim of training school teachers. Lagrange taught courses on elementary mathematics there. We mentioned above that Lagrange had a ‘no teaching’ clause written into his contract but the Revolution changed things and Lagrange was required to teach. However, he was not a good lecturer as  Fourier, who attended his lectures at the École Normale in 1795 wrote:-

His voice is very feeble, at least in that he does not become heated; he has a very pronounced Italian accent and pronounces the s like z … The students, of whom the majority are incapable of appreciating him, give him little welcome, but the professors make amends for it.

Similarly Bugge who attended his lectures at the École Polytechnique in 1799 wrote:-

… whatever this great man says, deserves the highest degree of consideration, but he is too abstract for youth.

Lagrange published two volumes of his calculus lectures. In 1797 he published the first theory of functions of a real variable with Théorie des fonctions analytique although he failed to give enough attention to matters of convergence. He states that the aim of the work is to give:-

… the principles of the differential calculus, freed from all consideration of the infinitely small or vanishing quantities, of limits or fluxions, and reduced to the algebraic analysis of finite quantities.

Also he states:-

The ordinary operations of algebra suffice to resolve problems in the theory of curves.

Not everyone found Lagrange’s approach to the calculus the best however, for example  de Prony wrote in 1835:-

Lagrange’s foundations of the calculus is assuredly a very interesting part of what one might call purely philosophical study: but when it is a case of making transcendental analysis an instrument of exploration for questions presented by astronomy, marine, geodesy, and the different branches of science of the engineer, the consideration of the infinitely small leads to the aim in a manner which is more felicitous, more prompt, and more immediately adapted to the nature of the questions, and that is why the Leibnizian method has, in general, prevailed in French schools.

The second work of Lagrange on this topic Leçons sur le calcul des fonctions appeared in 1800.

Napoleon named Lagrange to the Legion of Honour and Count of the Empire in 1808. On 3 April 1813 he was named grand croix of the Ordre Impérial de la Réunion. He died a week later.

J J O’Connor and E F Robertson

Список литературы

Для подготовки данной работы были использованы материалы с сайта http://www-history.mcs.st-andrews.ac.uk/

Aristotle

Born: 384 BC in Stagirus, Macedonia, Greece

Died: 322 BC in Chalcis, Euboea, Greece

Aristotle was not primarily a mathematician but made important contributions by systematising deductive logic. He wrote on physical subjects: some parts of his Analytica posteriora show an unusual grasp of the mathematical method. Primarily, however, he is important in the development of all knowledge for, as the authors of write:-

Aristotle, more than any other thinker, determined the orientation and the content of Western intellectual history. He was the author of a philosophical and scientific system that through the centuries became the support and vehicle for both medieval Christian and Islamic scholastic thought: until the end of the 17th century, Western culture was Aristotelian. And, even after the intellectual revolutions of centuries to follow, Aristotelian concepts and ideas remained embedded in Western thinking.

Aristotle was born in Stagirus, or Stagira, or Stageirus, on the Chalcidic peninsula of northern Greece. His father was Nicomachus, a medical doctor, while his mother was named Phaestis. Nicomachus was certainly living in Chalcidice when Aristotle was born and he had probably been born in that region. Aristotle’s mother, Phaestis, came from Chalcis in Euboea and her family owned property there.

There is little doubt that Nicomachus would have intended Aristotle to become a doctor, for the tradition was that medical skills were kept secret and handed down from father to son. It was not a society where people visited a doctor but rather it was the doctors who travelled round the country tending to the sick. Although we know nothing of Aristotle’s early years it is highly likely that he would have accompanied his father in his travels. We do know that Nicomachus found the conditions in Chalcidice less satisfactory than in the neighbouring state of Macedonia and he began to work there with so much success that he was soon appointed as the personal physician to Amyntas III, king of Macedonia.

There is no record to indicate whether Aristotle lived with his father in Pella, the capital of Macedonia, while Nicomachus attended to king Amyntas at the court there. However, Aristotle was certainly friendly with Philip, king Amyntas’s son, some years later and it seems reasonable to assume that the two, who were almost exactly the same age, had become friendly in Pella as young children.

When Aristotle was about ten years old his father died. This certainly meant that Aristotle could not now follow in his father’s profession of doctor and, since his mother seems also to have died young, Aristotle was brought up by a guardian, Proxenus of Atarneus, who was his uncle (or possibly a family friend as is suggested by some authors). Proxenus taught Aristotle Greek, rhetoric, and poetry which complemented the biological teachings that Nicomachus had given Aristotle as part of training his son in medicine. Since in latter life Aristotle wrote fine Greek prose, this too must have been part of his early education.

In 367 BC Aristotle, at the age of seventeen, became a student at Plato’s Academy in Athens. At the time that Aristotle joined the Academy it had been operating for twenty years. Plato was not in Athens, but rather he was on his first visit to Syracuse. We should not think of Plato’s Academy as a non-political organisation only interested in abstract ideas. The Academy was highly involved in the politics of the time, in fact Plato’s visit to Sicily was for political reasons, and the politics of the Academy and of the whole region would play a major role in influencing the course of Aristotle’s life.

When Aristotle arrived in Athens, the Academy was being run by Eudoxus of Cnidos in Plato’s absence. Speusippus, Plato’s nephew, was also teaching at the Academy as was Xenocrates of Chalcedon. After being a student, Aristotle soon became a teacher at the Academy and he was to remain there for twenty years. We know little regarding what Aristotle taught at the Academy. In [10] Diogenes Laertius, writing in the second century AD, says that Aristotle taught rhetoric and dialectic. Certainly Aristotle wrote on rhetoric at this time, issuing Gryllus which attacked the views on rhetoric of Isocrates, who ran another major educational establishment in Athens. All Aristotle’s writings of this time strongly support Plato’s views and those of the Academy.

Towards the end of Aristotle’s twenty years at the Academy his position became difficult due to the political events of the time. Amyntas, the king of Macedonia, died around 369 BC, a couple of years before Aristotle went to Athens to join the Academy. Two of Amyntas’s sons, Alexander II and Perdiccas III, each reigned Macedonia for a time but the kingdom suffered from both internal disputes and external wars. In 359 BC Amyntas’s third son, Philip II came to the throne when Perdiccas was killed fighting off an Illyrian invasion. Philip used skilful tactics, both military and political, to allow Macedonia a period of internal peace in which they expanded by victories over the surrounding areas.

Philip captured Olynthus and annexed Chalcidice in 348 BC. Stagirus, the town of Aristotle’s birth, held out for a while but was also defeated by Philip. Athens worried about the powerful threatening forces of Macedonia, and yet Aristotle had been brought up at the Court of Macedonia and had probably retained his friendship with Philip. The actual order of events is now a little uncertain. Plato died in 347 BC and Speusippus assumed the leadership of the Academy. Aristotle was certainly opposed to the views of Speusippus and he may have left the Academy following Plato’s death for academic reasons or because he failed to be named head of the Academy himself. Some sources, however, suggest that he may have left for political reasons before Plato died because of his unpopularity due to his Macedonian links.

Aristotle travelled from Athens to Assos which faces the island of Lesbos. He was not alone in leaving the Academy for Xenocrates of Chalcedon left with him. In Assos Aristotle was received by the ruler Hermias of Atarneus with much acclaim. It is likely that Aristotle was acting as an ambassador for Philip and he certainly was treated as such by Hermias. Aristotle married Pythias, the niece and adopted daughter of Hermias, and they had one child, a daughter also called Pythias. However, Aristotle’s wife died about 10 years after their marriage. It is thought that she was much younger than Atistotle, being probably of age of about 18 when they married.

On Assos, Aristotle became the leader of the group of philosophers which Hermias had gathered there. It is possible that Xenocrates was also a member of the group for a time. Aristotle had a strong interest in anatomy and the structure of living things in general, an interest which his father had fostered in him in his early years, that helped him to develop a remarkable talent for observation. Aristotle and the members of his group began to collect observations while in Assos, in particular in zoology and biology. Barnes writes in that Aristotle’s:-

… studies on animals laid the foundations of the biological sciences; and they were not superseded until more than two thousand years after his death. The enquiries upon which those great works were based were probably carried out largely in Assos and Lesbos.

Aristotle probably begun his work Politics on Assos as well as On Kingship which is now lost. He began to develop a philosophy distinct from that of Plato who had said the kings should be philosophers and philosophers kings. In On Kingship Aristotle wrote that it is:-

… not merely unnecessary for a king to be a philosopher, but even a disadvantage. Rather a king should take the advice of true philosophers. Then he would fill his reign with good deeds, not with good words.

However, Aristotle’s time in Assos was ended by political events. The Persians attacked the town and Hermias was captured and executed. Aristotle escaped and stopped on the island of Lesbos on his way to Macedonia. It was more than a passing visit for he remained there for about a year and must have had the group of scientists from Assos with him for they continued their biological researches there.

Macedonia was now at peace with Athens, for Philip had made a treaty in 346 BC. In 343 BC Aristotle reached the Court of Macedonia and he was to remain there for seven years. The often quoted story that he became tutor to the young Alexander the Great, the son of Philip, is almost certainly a later invention as was pointed out by Jaeger, see . Grayeff in  suggests that Philip saw in Aristotle a future head of the Academy in Athens. Certainly this would have suited Philip well for Speusippus, the then head of the Academy, was strongly opposed to Philip and strongly encouraging Athens to oppose the rise of Macedonia.

The treaty between Athens and Macedonia began to fall apart in 340 BC and preparations for war began. The following year Speusippus died but Aristotle, although proposed as head of the Academy, was not elected. The position went to Xenocrates and Philip lost interest in his support for Aristotle. He moved back to his home in Stagirus and took with him to Stagirus his circle of philosophers and scientists.

Aristotle did not marry again after the death of his wife but he did form a relationship with Herpyllis, who came from his home town of Stagirus. It is not clear when they first met but together they had a son, Nicomachus, named after Aristotle’s father.

Philip was now at the height of his power but, as so often happens, that proved the time for internal disputes. Aristotle supported Alexander, Philip’s son who soon became king. Alexander decided on a policy similar to his father in regard to Athens and sought to assert his power by peaceful means. Alexander protected the Academy and encouraged it to continue with its work. At the same time, however, he sent Aristotle to Athens to found a rival establishment.

In 335 BC Aristotle founded his own school the Lyceum in Athens. He arrived in the city with assistants to staff the school and a large range of teaching materials he had gathered while in Macedonia; books, maps, and other teaching material which may well have been intended at one stage to support Aristotle in his bid to become head of the Academy. The Academy had always been narrow in its interests but the Lyceum under Aristotle pursued a broader range of subjects. Prominence was given by Aristotle to the detailed study of nature and in this and all the other subjects he studied:-

His own researches were carried out in company, and he communicated his thoughts to his friends and pupils, never thinking to retain them as a private treasure-store. He thought, indeed, that a man could not claim to know a subject unless he was capable of transmitting his knowledge to others, and he regarded teaching as the proper manifestation of knowledge.

Whether the works that come down to us are due to Aristotle or to later members of his school was questioned by a number of scholars towards the end of the 19th century. The reasons are discussed by Jaeger , but in this work Jaeger argues that the apparent differences in the approach by Aristotle in different works can be explained by his ideas developing over a number of years. Grayeff [6] examines certain texts in detail and again claims that they represent developments in the ideas of Aristotle’s school long after his death. He writes:-

According to a tradition which arose about two hundred and fifty years after his death, which then became dominant and even today is hardly disputed, Aristotle in these same years lecturer — not once, but two or three times, in almost every subject — on logic, physics, astronomy, meteorology, zoology, metaphysics, theology, psychology, politics, economics, ethics, rhetoric, poetics; and that he wrote down these lectures, expanding them and amending them several times, until they reached the stage in which we read them. However, still more astounding is the fact that the majority of these subjects did not exist as such before him, so that he would have been the first to conceive of and establish them, as systematic disciplines.

After the death of Alexander the Great in 323 BC, anti-Macedonian feeling in Athens made Aristotle retire to Chalcis where he lived in the house which had once belonged to his mother and was still retained by the family. He died the following year from a stomach complaint at the age of 62.

It is virtually impossible to give an impression of Aristotle’s personality with any certainty but the authors of write:-

The anecdotes related of him reveal him as a kindly, affectionate character, and they show barely any trace of the self-importance that some scholars think they can detect in his works. His will, which has been preserved, exhibits the same kindly traits; he makes references to his happy family life and takes solicitous care of his children, as well as his servants.

Barnes writes:-

He was a bit of a dandy, wearing rings on his fingers and cutting his hair fashionably short. He suffered from poor digestion, and is said to have been spindle-shanked. He was a good speaker, lucid in his lectures, persuasive in conversation; and he had a mordant wit. His enemies, who were numerous, made him out to be arrogant and overbearing. … As a man he was, I suspect, admirable rather than amiable.

We have commented above on the disputes among modern scholars as to whether Aristotle wrote the treatises now assigned to him. We do know that his work falls into two distinct parts, namely works which he published during his lifetime and are now lost (although some fragments survive in quotations in works by others), and the collection of writings which have come down to us and were not published by Aristotle in his lifetime. We can say with certainty that Aristotle never intended these 30 works which fill over 2000 printed pages to be published. They are certainly lecture notes from the courses given at the Lyceum either being, as most scholars believe, the work of Aristotle, or of later lecturers. Of course it is distinctly possible that they are notes of courses originally given by Aristotle but later added to by other lecturers after Aristotle’s death.

The works were first published in about 60 BC by Andronicus of Rhodes, the last head of the Lyceum. Certainly:-

The form, titles, and order of Aristotle’s texts that are studied today were given to them by Andronicus almost three centuries after the philosopher’s death, and the long history of commentary upon them began at this stage.

What do these works contain? There are important works on logic. Aristotle believed that logic was not a science but rather had to be treated before the study of every branch of knowledge. Aristotle’s name for logic was «analytics», the term logic being introduced by Xenocrates working at the Academy. Aristotle believed that logic must be applied to the sciences:-

The sciences — at any rate the theoretical sciences — are to be axiomatised. What, then, are their axioms to be? What conditions must a proposition satisfy to count as an axiom? again, what form will the derivations within each science take? By what rules will theorems be deduced from axioms? Those are among the questions which Aristotle poses in his logical writings, and in particular in the works known as Prior and Posterior Analytics.

In fact in Prior Analytics Aristotle proposed the now famous Aristotelian syllogistic, a form of argument consisting of two premises and a conclusion. His example is:-

(i) Every Greek is a person.

(ii) Every person is mortal.

(iii) Every Greek is mortal.

Aristotle was not the first to suggest axiom systems. Plato had made the bold suggestion that there might be a single axiom system to embrace all knowledge. Aristotle went for the somewhat more possible suggestion of an axiom system for each science. Notice that Euclid and his axiom system for geometry came after Aristotle.

Another topic to which Aristotle made major contributions was natural philosophy or rather physics by today’s terminology. (I [EFR] show my age and the traditional nature of St Andrews University if I remark that in the 1960s a pass in ‘General Natural Philosophy’ formed part of my degree.) Aristotle looks at matter, change, movement, space, position, and time. He also made contributions to the study of astronomy where in particular he studied comets, geography with an examination of features such as rivers), chemistry where he was interested in processes such as burning, as well as meteorology and the study of rainbows.

As well as important works on zoology and psychology, Aristotle wrote his famous work on metaphysics. This, according to Aristotle, studies:-

… the most general or abstract features of reality and the principles that have universal validity. … metaphysics studies whatever must be true of all existent things just insofar as they exist, [and] it studies the general conditions which any existing thing must satisfy.

Although Aristotle does not appear to have made any new discoveries in mathematics, he is important in the development of mathematics. As Heath explains in :-

The importance of a proper understanding of the mathematics in Aristotle lies principally in the fact that most of his illustrations of scientific method are taken from mathematics.

Clearly Aristotle had a thorough grasp of elementary mathematics and believed mathematics to have great importance as one of three theoretical sciences. However, it is fair to say that he did not agree with Plato, who elevated mathematics to such a prominent place of study that there was little room for the range of sciences studied by Aristotle. The other two theoretical sciences, Aristotle claimed, were (using modern terminology) philosophy and theoretical physics.

 Heath notes in the introduction to  some of the mathematics referred to by Aristotle in his works:-

… Aristotle was aware of the important discoveries of Eudoxus which affected profoundly the exposition of the Elements by Euclid. One allusion clearly shows that Aristotle knew of Eudoxus’s great Theory of Proportion which was expounded by Euclid in his Book V, and recognised the importance of it. Another passage recalls the fundamental assumption on which Eudoxus based his ‘ method of exhaustion’ for measuring areas and volumes; and, of course, Aristotle was familiar with the system of concentric spheres by which Eudoxus and Callippus accounted theoretically for the independent motions of the sun, moon, and planets. …

The incommensurable is mentioned over and over again, but the case mentioned is that of the diagonal of a square in relation to its side; there is no allusion to the extension of the theory to other cases by Theodorus and Theaetetus…

 Heath  also mentions the mathematics which Aristotle, perhaps surprising, does not refer to. There is:-

… no allusion to conic sections, to the doubling of the cube, or to the trisection of an angle. The problem of squaring the circle is mentioned in connection with the attempts of Antiphon, Bryson, and Hippocrates to solve it; but there is nothing about the curve of Hippias …

While Heath  discusses the many mathematical references in Aristotle, the book attempts to construct (or reconstruct) a work on Aristotle’s view of the philosophy of mathematics. As Apostle writes in:-

… numerous passages on mathematics are distributed throughout the works we possess and indicate a definite philosophy of mathematics, so that an attempt to construct or reconstruct that philosophy with a fairly high degree of accuracy is possible.

We end our discussion with an illustration of Aristotle’s ideas of ‘continuous’ and ‘infinite’ in mathematics. Heath  explains Aristotle’s idea that ‘continuous’:-

… could not be made up of indivisible parts; the continuous is that in which the boundary or limit between two consecutive parts, where they touch, is one and the same…

As to the infinite Aristotle believed that it did not actually exist but only potentially exists. Aristotle writes in Physics (see for example ):-

But my argument does not anyhow rob mathematicians of their study, although it denies the existence of the infinite in the sense of actual existence as something increased to such an extent that it cannot be gone through; for, as it is, they do not need the infinite or use it, but only require that the finite straight line shall be as long as they please. … Hence it will make no difference to them for the purpose of proofs.

J J O’Connor and E F Robertson

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Galileo Galilei

Born: 15 Feb 1564 in Pisa (now in Italy)

Died: 8 Jan 1642 in Arcetri (near Florence) (now in Italy)

Galileo Galilei’s parents were Vincenzo Galilei and Guilia Ammannati. Vincenzo, who was born in Florence in 1520, was a teacher of music and a fine lute player. After studying music in Venice he carried out experiments on strings to support his musical theories. Guilia, who was born in Pescia, married Vincenzo in 1563 and they made their home in the countryside near Pisa. Galileo was their first child and spent his early years with his family in Pisa.

In 1572, when Galileo was eight years old, his family returned to Florence, his father’s home town. However, Galileo remained in Pisa and lived for two years with Muzio Tedaldi who was related to Galileo’s mother by marriage. When he reached the age of ten, Galileo left Pisa to join his family in Florence and there he was tutored by Jacopo Borghini. Once he was old enough to be educated in a monastery, his parents sent him to the Camaldolese Monastery at Vallombrosa which is situated on a magnificent forested hillside 33 km southeast of Florence. The Camaldolese Order was independent of the Benedictine Order, splitting from it in about 1012. The Order combined the solitary life of the hermit with the strict life of the monk and soon the young Galileo found this life an attractive one. He became a novice, intending to join the Order, but this did not please his father who had already decided that his eldest son should become a medical doctor.

Vincenzo had Galileo return from Vallombrosa to Florence and give up the idea of joining the Camaldolese order. He did continue his schooling in Florence, however, in a school run by the Camaldolese monks. In 1581 Vincenzo sent Galileo back to Pisa to live again with Muzio Tedaldi and now to enrol for a medical degree at the University of Pisa. Although the idea of a medical career never seems to have appealed to Galileo, his father’s wish was a fairly natural one since there had been a distinguished physician in his family in the previous century. Galileo never seems to have taken medical studies seriously, attending courses on his real interests which were in mathematics and natural philosophy. His mathematics teacher at Pisa was Filippo Fantoni, who held the chair of mathematics. Galileo returned to Florence for the summer vacations and there continued to study mathematics.

In the year 1582-83 Ostilio Ricci, who was the mathematician of the Tuscan Court and a former pupil of  Tartaglia, taught a course on  Euclid’s Elements at the University of Pisa which Galileo attended. During the summer of 1583 Galileo was back in Florence with his family and Vincenzo encouraged him to read Galen to further his medical studies. However Galileo, still reluctant to study medicine, invited Ricci (also in Florence where the Tuscan court spent the summer and autumn) to his home to meet his father. Ricci tried to persuade Vincenzo to allow his son to study mathematics since this was where his interests lay. Certainly Vincenzo did not like the idea and resisted strongly but eventually he gave way a little and Galileo was able to study the works of  Euclid and  Archimedes from the Italian translations which  Tartaglia had made. Of course he was still officially enrolled as a medical student at Pisa but eventually, by 1585, he gave up this course and left without completing his degree.

Galileo began teaching mathematics, first privately in Florence and then during 1585-86 at Sienna where he held a public appointment. During the summer of 1586 he taught at Vallombrosa, and in this year he wrote his first scientific book The little balance [La Balancitta] which described  Archimedes’ method of finding the specific gravities (that is the relative densities) of substances using a balance. In the following year he travelled to Rome to visit  Clavius who was professor of mathematics at the Jesuit Collegio Romano there. A topic which was very popular with the Jesuit mathematicians at this time was centres of gravity and Galileo brought with him some results which he had discovered on this topic. Despite making a very favourable impression on  Clavius, Galileo failed to gain an appointment to teach mathematics at the University of Bologna.

After leaving Rome Galileo remained in contact with  Clavius by correspondence and Guidobaldo del  Monte was also a regular correspondent. Certainly the theorems which Galileo had proved on the centres of gravity of solids, and left in Rome, were discussed in this correspondence. It is also likely that Galileo received lecture notes from courses which had been given at the Collegio Romano, for he made copies of such material which still survive today. The correspondence began around 1588 and continued for many years. Also in 1588 Galileo received a prestigious invitation to lecture on the dimensions and location of hell in Dante’s Inferno at the Academy in Florence.

Fantoni left the chair of mathematics at the University of Pisa in 1589 and Galileo was appointed to fill the post (although this was only a nominal position to provide financial support for Galileo). Not only did he receive strong recommendations from  Clavius, but he also had acquired an excellent reputation through his lectures at the Florence Academy in the previous year. The young mathematician had rapidly acquired the reputation that was necessary to gain such a position, but there were still higher positions at which he might aim. Galileo spent three years holding this post at the university of Pisa and during this time he wrote De Motu a series of essays on the theory of motion which he never published. It is likely that he never published this material because he was less than satisfied with it, and this is fair for despite containing some important steps forward, it also contained some incorrect ideas. Perhaps the most important new ideas which De Motu contains is that one can test theories by conducting experiments. In particular the work contains his important idea that one could test theories about falling bodies using an inclined plane to slow down the rate of descent.

In 1591 Vincenzo Galilei, Galileo’s father, died and since Galileo was the eldest son he had to provide financial support for the rest of the family and in particular have the necessary financial means to provide dowries for his two younger sisters. Being professor of mathematics at Pisa was not well paid, so Galileo looked for a more lucrative post. With strong recommendations from  Guidobaldo del Monte, Galileo was appointed professor of mathematics at the University of Padua (the university of the Republic of Venice) in 1592 at a salary of three times what he had received at Pisa. On 7 December 1592 he gave his inaugural lecture and began a period of eighteen years at the university, years which he later described as the happiest of his life. At Padua his duties were mainly to teach  Euclid’s geometry and standard (geocentric) astronomy to medical students, who would need to know some astronomy in order to make use of astrology in their medical practice. However, Galileo argued against  Aristotle’s view of astronomy and natural philosophy in three public lectures he gave in connection with the appearance of a New Star (now known as ‘ Kepler’s supernova’) in 1604. The belief at this time was that of  Aristotle, namely that all changes in the heavens had to occur in the lunar region close to the Earth, the realm of the fixed stars being permanent. Galileo used parallax arguments to prove that the New Star could not be close to the Earth. In a personal letter written to  Kepler in 1598, Galileo had stated that he was a Copernican (believer in the theories of  Copernicus). However, no public sign of this belief was to appear until many years later.

At Padua, Galileo began a long term relationship with Maria Gamba, who was from Venice, but they did not marry perhaps because Galileo felt his financial situation was not good enough. In 1600 their first child Virginia was born, followed by a second daughter Livia in the following year. In 1606 their son Vincenzo was born.

We mentioned above an error in Galileo’s theory of motion as he set it out in De Motu around 1590. He was quite mistaken in his belief that the force acting on a body was the relative difference between its specific gravity and that of the substance through which it moved. Galileo wrote to his friend Paolo Sarpi, a fine mathematician who was consultor to the Venetian government, in 1604 and it is clear from his letter that by this time he had realised his mistake. In fact he had returned to work on the theory of motion in 1602 and over the following two years, through his study of inclined planes and the pendulum, he had formulated the correct law of falling bodies and had worked out that a projectile follows a parabolic path. However, these famous results would not be published for another 35 years.

In May 1609, Galileo received a letter from Paolo Sarpi telling him about a spyglass that a Dutchman had shown in Venice. Galileo wrote in the Starry Messenger (Sidereus Nuncius) in April 1610:-

About ten months ago a report reached my ears that a certain Fleming had constructed a spyglass by means of which visible objects, though very distant from the eye of the observer, were distinctly seen as if nearby. Of this truly remarkable effect several experiences were related, to which some persons believed while other denied them. A few days later the report was confirmed by a letter I received from a Frenchman in Paris, Jacques Badovere, which caused me to apply myself wholeheartedly to investigate means by which I might arrive at the invention of a similar instrument. This I did soon afterwards, my basis being the doctrine of refraction.

From these reports, and using his own technical skills as a mathematician and as a craftsman, Galileo began to make a series of telescopes whose optical performance was much better than that of the Dutch instrument. His first telescope was made from available lenses and gave a magnification of about four times. To improve on this Galileo learned how to grind and polish his own lenses and by August 1609 he had an instrument with a magnification of around eight or nine. Galileo immediately saw the commercial and military applications of his telescope (which he called a perspicillum) for ships at sea. He kept Sarpi informed of his progress and Sarpi arranged a demonstration for the Venetian Senate. They were very impressed and, in return for a large increase in his salary, Galileo gave the sole rights for the manufacture of telescopes to the Venetian Senate. It seems a particularly good move on his part since he must have known that such rights were meaningless, particularly since he always acknowledged that the telescope was not his invention!

By the end of 1609 Galileo had turned his telescope on the night sky and began to make remarkable discoveries. Swerdlow writes (see [16]):-

In about two months, December and January, he made more discoveries that changed the world than anyone has ever made before or since.

The astronomical discoveries he made with his telescopes were described in a short book called the Starry Messenger published in Venice in May 1610. This work caused a sensation. Galileo claimed to have seen mountains on the Moon, to have proved the  Milky Way was made up of tiny stars, and to have seen four small bodies orbiting Jupiter. These last, with an eye to getting a position in Florence, he quickly named ‘the Medicean stars’. He had also sent Cosimo de Medici, the Grand Duke of Tuscany, an excellent telescope for himself.

The Venetian Senate, perhaps realising that the rights to manufacture telescopes that Galileo had given them were worthless, froze his salary. However he had succeeded in impressing Cosimo and, in June 1610, only a month after his famous little book was published, Galileo resigned his post at Padua and became Chief Mathematician at the University of Pisa (without any teaching duties) and ‘Mathematician and Philosopher’ to the Grand Duke of Tuscany. In 1611 he visited Rome where he was treated as a leading celebrity; the Collegio Romano put on a grand dinner with speeches to honour Galileo’s remarkable discoveries. He was also made a member of the Accademia dei Lincei (in fact the sixth member) and this was an honour which was especially important to Galileo who signed himself ‘Galileo Galilei Linceo’ from this time on.

While in Rome, and after his return to Florence, Galileo continued to make observations with his telescope. Already in the Starry Messenger he had given rough periods of the four moons of Jupiter, but more precise calculations were certainly not easy since it was difficult to identify from an observation which moon was I, which was II, which III, and which IV. He made a long series of observations and was able to give accurate periods by 1612. At one stage in the calculations he became very puzzled since the data he had recorded seemed inconsistent, but he had forgotten to take into account the motion of the Earth round the sun.

Galileo first turned his telescope on Saturn on 25 July 1610 and it appeared as three bodies (his telescope was not good enough to show the rings but made them appear as lobes on either side of the planet). Continued observations were puzzling indeed to Galileo as the bodies on either side of Saturn vanished when the ring system was edge on. Also in 1610 he discovered that, when seen in the telescope, the planet Venus showed phases like those of the Moon, and therefore must orbit the Sun not the Earth. This did not enable one to decide between the Copernican system, in which everything goes round the Sun, and that proposed by Tycho  Brahe in which everything but the Earth (and Moon) goes round the Sun which in turn goes round the Earth. Most astronomers of the time in fact favoured  Brahe’s system and indeed distinguishing between the two by experiment was beyond the instruments of the day. However, Galileo knew that all his discoveries were evidence for Copernicanism, although not a proof. In fact it was his theory of falling bodies which was the most significant in this respect, for opponents of a moving Earth argued that if the Earth rotated and a body was dropped from a tower it should fall behind the tower as the Earth rotated while it fell. Since this was not observed in practice this was taken as strong evidence that the Earth was stationary. However Galileo already knew that a body would fall in the observed manner on a rotating Earth.

Other observations made by Galileo included the observation of sunspots. He reported these in Discourse on floating bodies which he published in 1612 and more fully in Letters on the sunspots which appeared in 1613. In the following year his two daughters entered the Franciscan Convent of St Matthew outside Florence, Virginia taking the name Sister Maria Celeste and Livia the name Sister Arcangela. Since they had been born outside of marriage, Galileo believed that they themselves should never marry. Although Galileo put forward many revolutionary correct theories, he was not correct in all cases. In particular when three comets appeared in 1618 he became involved in a controversy regarding the nature of comets. He argued that they were close to the Earth and caused by optical refraction. A serious consequence of this unfortunate argument was that the Jesuits began to see Galileo as a dangerous opponent.

Despite his private support for Copernicanism, Galileo tried to avoid controversy by not making public statements on the issue. However he was drawn into the controversy through Castelli who had been appointed to the chair of mathematics in Pisa in 1613. Castelli had been a student of Galileo’s and he was also a supporter of  Copernicus. At a meeting in the Medici palace in Florence in December 1613 with the Grand Duke Cosimo II and his mother the Grand Duchess Christina of Lorraine, Castelli was asked to explain the apparent contradictions between the Copernican theory and Holy Scripture. Castelli defended the Copernican position vigorously and wrote to Galileo afterwards telling him how successful he had been in putting the arguments. Galileo, less convinced that Castelli had won the argument, wrote Letter to Castelli to him arguing that the Bible had to be interpreted in the light of what science had shown to be true. Galileo had several opponents in Florence and they made sure that a copy of the Letter to Castelli was sent to the Inquisition in Rome. However, after examining its contents they found little to which they could object.

The Catholic Church’s most important figure at this time in dealing with interpretations of the Holy Scripture was Cardinal Robert Bellarmine. He seems at this time to have seen little reason for the Church to be concerned regarding the Copernican theory. The point at issue was whether  Copernicus had simply put forward a mathematical theory which enabled the calculation of the positions of the heavenly bodies to be made more simply or whether he was proposing a physical reality. At this time Bellarmine viewed the theory as an elegant mathematical one which did not threaten the established Christian belief regarding the structure of the universe.

In 1616 Galileo wrote the Letter to the Grand Duchess which vigorously attacked the followers of  Aristotle. In this work, which he addressed to the Grand Duchess Christina of Lorraine, he argued strongly for a non-literal interpretation of Holy Scripture when the literal interpretation would contradict facts about the physical world proved by mathematical science. In this Galileo stated quite clearly that for him the Copernican theory is not just a mathematical calculating tool, but is a physical reality:-

I hold that the Sun is located at the centre of the revolutions of the heavenly orbs and does not change place, and that the Earth rotates on itself and moves around it. Moreover … I confirm this view not only by refuting  Ptolemy’s and  Aristotle’s arguments, but also by producing many for the other side, especially some pertaining to physical effects whose causes perhaps cannot be determined in any other way, and other astronomical discoveries; these discoveries clearly confute the Ptolemaic system, and they agree admirably with this other position and confirm it.

Pope Paul V ordered Bellarmine to have the Sacred Congregation of the Index decide on the Copernican theory. The cardinals of the Inquisition met on 24 February 1616 and took evidence from theological experts. They condemned the teachings of  Copernicus, and Bellarmine conveyed their decision to Galileo who had not been personally involved in the trial. Galileo was forbidden to hold Copernican views but later events made him less concerned about this decision of the Inquisition. Most importantly Maffeo Barberini, who was an admirer of Galileo, was elected as Pope Urban VIII. This happened just as Galileo’s book Il saggiatore (The Assayer) was about to be published by the Accademia dei Lincei in 1623 and Galileo was quick to dedicate this work to the new Pope. The work described Galileo’s new scientific method and contains a famous quote regarding mathematics:-

Philosophy is written in this grand book, the universe, which stands continually open to our gaze. But the book cannot be understood unless one first learns to comprehend the language and read the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other geometric figures without which it is humanly impossible to understand a single word of it; without these one is wandering in a dark labyrinth.

Pope Urban VIII invited Galileo to papal audiences on six occasions and led Galileo to believe that the Catholic Church would not make an issue of the Copernican theory. Galileo, therefore, decided to publish his views believing that he could do so without serious consequences from the Church. However by this stage in his life Galileo’s health was poor with frequent bouts of severe illness and so even though he began to write his famous Dialogue in 1624 it took him six years to complete the work.

Galileo attempted to obtain permission from Rome to publish the Dialogue in 1630 but this did not prove easy. Eventually he received permission from Florence, and not Rome. In February 1632 Galileo published Dialogue Concerning the Two Chief Systems of the World — Ptolemaic and Copernican. It takes the form of a dialogue between Salviati, who argues for the Copernican system, and Simplicio who is an Aristotelian philosopher. The climax of the book is an argument by Salviati that the Earth moves which was based on Galileo’s theory of the tides. Galileo’s theory of the tides was entirely false despite being postulated after  Kepler had already put forward the correct explanation. It was unfortunate, given the remarkable truths the Dialogue supported, that the argument which Galileo thought to give the strongest proof of  Copernicus’s theory should be incorrect.

Shortly after publication of Dialogue Concerning the Two Chief Systems of the World — Ptolemaic and Copernican the Inquisition banned its sale and ordered Galileo to appear in Rome before them. Illness prevented him from travelling to Rome until 1633. Galileo’s accusation at the trial which followed was that he had breached the conditions laid down by the Inquisition in 1616. However a different version of this decision was produced at the trial rather than the one Galileo had been given at the time. The truth of the Copernican theory was not an issue therefore; it was taken as a fact at the trial that this theory was false. This was logical, of course, since the judgement of 1616 had declared it totally false.

Found guilty, Galileo was condemned to lifelong imprisonment, but the sentence was carried out somewhat sympathetically and it amounted to house arrest rather than a prison sentence. He was able to live first with the Archbishop of Siena, then later to return to his home in Arcetri, near Florence, but had to spend the rest of his life watched over by officers from the Inquisition. In 1634 he suffered a severe blow when his daughter Virginia, Sister Maria Celeste, died. She had been a great support to her father through his illnesses and Galileo was shattered and could not work for many months. When he did manage to restart work, he began to write Discourses and mathematical demonstrations concerning the two new sciences.

After Galileo had completed work on the Discourses it was smuggled out of Italy, and taken to Leyden in Holland where it was published. It was his most rigorous mathematical work which treated problems on impetus, moments, and centres of gravity. Much of this work went back to the unpublished ideas in De Motu from around 1590 and the improvements which he had worked out during 1602-1604. In the Discourses he developed his ideas of the inclined plane writing:-

I assume that the speed acquired by the same movable object over different inclinations of the plane are equal whenever the heights of those planes are equal.

He then described an experiment using a pendulum to verify his property of inclined planes and used these ideas to give a theorem on acceleration of bodies in free fall:-

The time in which a certain distance is traversed by an object moving under uniform acceleration from rest is equal to the time in which the same distance would be traversed by the same movable object moving at a uniform speed of one half the maximum and final speed of the previous uniformly accelerated motion.

After giving further results of this type he gives his famous result that the distance that a body moves from rest under uniform acceleration is proportional to the square of the time taken.

One would expect that Galileo’s understanding of the pendulum, which he had since he was a young man, would have led him to design a pendulum clock. In fact he only seems to have thought of this possibility near the end of his life and around 1640 he did design the first pendulum clock. Galileo died in early 1642 but the significance of his clock design was certainly realised by his son Vincenzo who tried to make a clock to Galileo’s plan, but failed.

It was a sad end for so great a man to die condemned of heresy. His will indicated that he wished to be buried beside his father in the family tomb in the Basilica of Santa Croce but his relatives feared, quite rightly, that this would provoke opposition from the Church. His body was concealed and only placed in a fine tomb in the church in 1737 by the civil authorities against the wishes of many in the Church. On 31 October 1992, 350 years after Galileo’s death, Pope John Paul II gave an address on behalf of the Catholic Church in which he admitted that errors had been made by the theological advisors in the case of Galileo. He declared the Galileo case closed, but he did not admit that the Church was wrong to convict Galileo on a charge of heresy because of his belief that the Earth rotates round the sun.

J J O’Connor and E F Robertson

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Mathematics as a science

Mathematics as a science, viewed as a whole, is a collection of branches.

The largest branch is that which builds on the ordinary whole numbers, fractions, and irrational numbers, or what, collectively, is called the real number system.

Arithmetic, algebra, the study of functions, the calculus, differential equations, and various other subjects which follow the calculus in logical order are all developments of the real number system.

This part of mathematics is termed the mathematics of number.

A second branch is geometry consisting of several geometries.

Mathematics contains many more divisions.

Each branch has the same logical structure: it begins with certain concepts, such as the whole numbers or integers in the mathematics of number, and such as point, line and triangle in geometry.

These concepts must verify explicitly stated axioms.

Some of the axioms of the mathematics of number are the associative, commutative, and distributive properties and the axioms about equalities.

Some of the axioms of geometry are that two points determine a line, all right angles are equal, etc.

From the concepts and axioms theorems are deduced.

Hence, from the standpoint of structure, the concepts, axioms and theorems are the essential components of any compartment of mathematics.

We must break down mathematics into separately taught subjects, but this compartmentalization taken as a necessity, must be compensated for as much as possible.

Students must see the interrelationships of the various areas and the importance of mathematics for other domains.

Knowledge is not additive but an organic whole and mathematics is an inseparable part of that whole.

The full significance of mathematics can be seen and taught only in terms of its intimate relationships to other fields of knowledge.

If mathematics is isolated from other provinces, it loses importance.

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Jacob (Jacques) Bernoulli

Born: 27 Dec 1654 in Basel, Switzerland

Died: 16 Aug 1705 in Basel, Switzerland

Jacob Bernoulli’s father, Nicolaus Bernoulli (1623-1708) inherited the spice business in Basel that had been set up by his own father, first in Amsterdam and then in Basel. The family, of Belgium origin, were refugees fleeing from persecution by the Spanish rulers of the Netherlands. Philip, the King of Spain, had sent the Duke of Alba to the Netherlands in 1567 with a large army to punish those opposed to Spanish rule, to enforce adherence to Roman Catholicism, and to re-establish Philip’s authority. Alba set up the Council of Troubles which was a court that condemned over 12000 people but most, like the Bernoulli family who were of the Protestant faith, fled the country.

Nicolaus Bernoulli was an important citizen of Basel, being a member of the town council and a magistrate. Jacob Bernoulli’s mother also came from an important Basel family of bankers and local councillors. Jacob Bernoulli was the brother of  Johann Bernoulli and the uncle of  Daniel Bernoulli. He was compelled to study philosophy and theology by his parents, which he greatly resented, and he graduated from the University of Basel with a master’s degree in philosophy in 1671 and a licentiate in theology in 1676.

During the time that Jacob Bernoulli was taking his university degrees he was studying mathematics and astronomy against the wishes of his parents. It is worth remarking that this was a typical pattern for many of the Bernoulli family who made a study of mathematics despite pressure to make a career in other areas. However Jacob Bernoulli was the first to go down this road so for him it was rather different in that there was no tradition of mathematics in the family before Jacob Bernoulli. Later members of the family must have been much influenced by the tradition of studying mathematics and mathematical physics.

In 1676, after taking his theology degree, Bernoulli moved to Geneva where he worked as a tutor. He then travelled to France spending two years studying with the followers of  Descartes who were led at this time by  Malebranche. In 1681 Bernoulli travelled to the Netherlands where he met many mathematicians including  Hudde. Continuing his studies with the leading mathematicians and scientists of Europe he went to England where, among others, he met  Boyle and  Hooke. At this time he was deeply interested in astronomy and produced a work giving an incorrect theory of comets. As a result of his travels, Bernoulli began a correspondence with many mathematicians which he carried on over many years.

Jacob Bernoulli returned to Switzerland and taught mechanics at the University in Basel from 1683, giving a series of important lectures on the mechanics of solids and liquids. Since his degree was in theology it would have been natural for him to turn to the Church, but although he was offered an appointment in the Church he turned it down. Bernoulli’s real love was for mathematics and theoretical physics and it was in these topics that he taught and researched. During this period he studied the leading mathematical works of his time including Descartes’s Géométrie and van  Schooten’s additional material in the Latin edition. Jacob Bernoulli also studied the work of  Wallis and  Barrow and through these he became interested in  infinitesimal geometry. Jacob began publishing in the journal Acta Eruditorum which was established in Leipzig in 1682.

In 1684 Jacob Bernoulli married Judith Stupanus. They were to have two children, a son who was given his grandfather’s name of Nicolaus and a daughter. These children, unlike many members of the Bernoulli family, did not go on to become mathematicians or physicists.

One of the most significant events concerning the mathematical studies of Jacob Bernoulli occurred when his younger brother,  Johann Bernoulli, began to work on mathematical topics.  Johann Bernoulli was told by his father to study medicine but while he was studying that topic he asked his brother Jacob Bernoulli to teach him mathematics. Jacob Bernoulli was appointed professor of mathematics in Basel in 1687 and the two brothers began to study the calculus as presented by  Leibniz in his 1684 paper on the differential calculus in Nova Methodus pro Maximis et Minimis, itemque Tangentibus… published in Acta Eruditorum. They also studied the publications of von  Tschirnhaus. It must be understood that  Leibniz’s publications on the calculus were very obscure to mathematicians of that time and the Bernoullis were the first to try to understand and apply  Leibniz’s theories.

Although Jacob and  Johann Bernoulli both worked on similar problems their relationship was soon to change from one of collaborators to one of rivals.  Johann Bernoulli’s boasts were the first cause of Jacob’s attacks on him and Jacob wrote that Johann was his pupil whose only achievements were to repeat what his teacher had taught him. Of course this was a grossly unfair statement. Jacob Bernoulli continued to attack his brother in print in a disgraceful and unnecessary fashion, particularly after 1697. However he did not reserve public criticism for his brother. He was critical of the university authorities at Basel and again he was very public in making critical statements that, as one would expect, left him in a difficult situation at the university. Jacob probably felt that Johann was the more powerful mathematician of the two and, this hurt since Jacob’s nature meant that he always had to feel that he was winning praise from all sides. Hofmann writes in:-

Sensitivity, irritability, a mutual passion for criticism, and an exaggerated need for recognition alienated the brothers, of whom Jacob had the slower but deeper intellect.

As suggested by this quote the brothers were equally at fault in their quarrel.  Johann Bernoulli would have liked the chair of mathematics at Basel which Jacob held and he certainly resented having to move to Holland in 1695. This was another factor in the complete breakdown of relations in 1697.

Of course the dispute between the brothers over who could obtain the greatest recognition was a particularly stupid one in the sense that both made contributions to mathematics of the very greatest importance. Whether the rivalry spurred them on to greater things or whether they might have achieved more had they continued their initial collaboration, it is impossible to say. We shall now examine some of the major contributions made by Jacob Bernoulli at an important stage in the development of mathematics following  Leibniz’s work on the calculus.

Jacob Bernoulli’s first important contributions were a pamphlet on the parallels of logic and algebra published in 1685, work on  probability in 1685 and geometry in 1687. His geometry result gave a construction to divide any triangle into four equal parts with two perpendicular lines.

By 1689 he had published important work on infinite series and published his law of large numbers in probability theory. The interpretation of probability as relative-frequency says that if an experiment is repeated a large number of times then the relative frequency with which an event occurs equals the probability of the event. The law of large numbers is a mathematical interpretation of this result. Jacob Bernoulli published five treatises on infinite series between 1682 and 1704. The first two of these contained many results, such as fundamental result that (1/n) diverges, which Bernoulli believed were new but they had actually been proved by  Mengoli 40 years earlier. Bernoulli could not find a closed form for (1/n2) but he did show that it converged to a finite limit less than 2.  Euler was the first to find the sum of this series in 1737. Bernoulli also studied the exponential series which came out of examining compound interest.

In May 1690 in a paper published in Acta Eruditorum, Jacob Bernoulli showed that the problem of determining the isochrone is equivalent to solving a first-order nonlinear  differential equation. The isochrone, or curve of constant descent, is the curve along which a particle will descend under gravity from any point to the bottom in exactly the same time, no matter what the starting point. It had been studied by  Huygens in 1687 and  Leibniz in 1689. After finding the differential equation, Bernoulli then solved it by what we now call separation of variables. Jacob Bernoulli’s paper of 1690 is important for the history of calculus, since the term integral appears for the first time with its integration meaning. In 1696 Bernoulli solved the equation, now called «the Bernoulli equation»,

y’ = p(x)y + q(x)yn

and Hofmann describes this part of his work as:-

… proof of Bernoulli’s careful and critical work on older as well as on contemporary contributions to infinitesimal mathematics and of his perseverance and analytical ability in dealing with special pertinent problems, even those of a mechanical-dynamic nature.

Jacob Bernoulli also discovered a general method to determine  evolutes of a curve as the envelope of its circles of curvature. He also investigated caustic curves and in particular he studied these associated curves of the  parabola, the logarithmic spiral and epicycloids around 1692. The lemniscate of Bernoulli was first conceived by Jacob Bernoulli in 1694. In 1695 he investigated the drawbridge problem which seeks the curve required so that a weight sliding along the cable always keeps the drawbridge balanced.

Jacob Bernoulli’s most original work was Ars Conjectandi published in Basel in 1713, eight years after his death. The work was incomplete at the time of his death but it is still a work of the greatest significance in the theory of probability. In the book Bernoulli reviewed work of others on probability, in particular work by van  Schooten,  Leibniz, and Prestet. The  Bernoulli numbers appear in the book in a discussion of the exponential series. Many examples are given on how much one would expect to win playing various game of chance. There are interesting thoughts on what probability really is:-

… probability as a measurable degree of certainty; necessity and chance; moral versus mathematical expectation; a priori an a posteriori probability; expectation of winning when players are divided according to dexterity; regard of all available arguments, their valuation, and their calculable evaluation; law of large numbers …

In Hofmann sums up Jacob Bernoulli’s contributions as follows:-

Bernoulli greatly advanced algebra, the infinitesimal calculus, the  calculus of variations, mechanics, the theory of series, and the theory of probability. He was self-willed, obstinate, aggressive, vindictive, beset by feelings of inferiority, and yet firmly convinced of his own abilities. With these characteristics, he necessarily had to collide with his similarly disposed brother. He nevertheless exerted the most lasting influence on the latter.

Bernoulli was one of the most significant promoters of the formal methods of higher analysis. Astuteness and elegance are seldom found in his method of presentation and expression, but there is a maximum of integrity.

Jacob Bernoulli continued to hold the chair of mathematics at Basel until his death in 1705 when the chair was filled by his brother Johann. Jacob had always found the properties of the logarithmic spiral to be almost magical and he had requested that it be carved on his tombstone with the Latin inscription Eadem Mutata Resurgo meaning «I shall arise the same though changed».

J J O’Connor and E F Robertson

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Luca Pacioli

Born: 1445 in Sansepolcro, Italy

Died: 1517 in Sansepolcro, Italy

Luca Pacioli’s father was Bartolomeo Pacioli, but Pacioli does not appear to have been brought up in his parents house. He lived as a child with the Befolci family in Sansepolcro which was the town of his birth. This town is very much in the centre of Italy about 60 km north of the city of Perugia. As far as Pacioli was concerned, perhaps the most important feature of this small commercial town was the fact that Piero della  Francesca had a studio and workshop in there and della  Francesca spent quite some time there despite frequent commissions in other towns

Although we know little of Pacioli’s early life, the conjecture that he may have received at least a part of his education in the studio of della  Francesca in Sansepolcro must at least have a strong chance of being correct. One reason that this seems likely to be true is the extensive knowledge that Pacioli had of the work of  Piero della Francesca and Pacioli’s writings were very strongly influenced by those of  Piero.

Pacioli moved away from Sansepolcro while he was still a young lad. He moved to Venice to enter the service of the wealthy merchant Antonio Rompiansi whose house was in the highly desirable Giudecca district of that city. One has to assume that Pacioli was already well educated in basic mathematics from studies in Sansepolcro and he certainly must have been well educated generally to have been chosen as a tutor to Rompiansi’s three sons. However, Pacioli took the opportunity to continue his mathematical studies at a higher level while in Venice, studying mathematics under Domenico Bragadino. During this time Pacioli gained experience both in teaching, from his role as tutor, and also in business from his role helping with Rompiansi’s affairs.

It was during his time in Venice that Pacioli wrote his first work, a book on arithmetic which he dedicated to his employer. This was completed in 1470 probably in the year that Rompiansi died. Pacioli certainly seemed to know all the right people for he left Venice and travelled to Rome where he spent several months living in the house of Leone Battista  Alberti who was secretary in the Papal Chancery. As well as being an excellent scholar and mathematician,  Alberti was able to provide Pacioli with good religious connections. At this time Pacioli then studied theology and, at some time during the next few years, he became a friar in the Franciscan Order.

In 1477 Pacioli began a life of travelling, spending time at various universities teaching mathematics, particularly arithmetic. He taught at the University of Perugia from 1477 to 1480 and while there he wrote a second work on arithmetic designed for the classes that he was teaching. He taught at Zara (now called Zadar or Jadera in Croatia but at that time in the Venetian Empire) and there wrote a third book on arithmetic. None of the three arithmetic texts were published, and only the one written for the students in Perugia has survived. After Zara, Pacioli taught again at the University of Perugia, then at the University of Naples, then at the University of Rome. Certainly Pacioli become acquainted with the duke of Urbino at some time during this period. Pope Sixtus IV had made Federico da Montefeltro the duke of Urbino in 1474 and Pacioli seems to have spent some time as a tutor to Federico’s son Guidobaldo who was to become the last ruling Montefeltro when his father died in 1482. The court at Urbino was a notable centre of culture and Pacioli must have had close contact with it over a number of years.

In 1489, after two years in Rome, Pacioli returned to his home town of Sansepolcro. Not all went smoothly for Pacioli in his home town, however. He had been granted some privileges by the Pope and there was a degree of jealousy among the men from the religious orders in Sansepolcro. In fact Pacioli was banned from teaching there in 1491 but the jealousy seemed to be mixed with a respect for his learning and scholarship for in 1493 he was invited to preach the Lent sermons.

During this time in Sansepolcro, Pacioli worked on one of his most famous books the Summa de arithmetica, geometria, proportioni et proportionalita which he dedicated to Guidobaldo, the duke of Urbino. Pacioli travelled to Venice in 1494 to publish the Summa. The work gives a summary of the mathematics known at that time although it shows little in the way of original ideas. The work studies arithmetic, algebra, geometry and trigonometry and, despite the lack of originality, was to provide a basis for the major progress in mathematics which took place in Europe shortly after this time. As stated in the Summa was:-

… not addressed to a particular section of the community. An encyclopaedic work (600 pages of close print, in folio) written in Italian, it contains a general treatise on theoretical and practical arithmetic; the elements of algebra; a table of moneys, weights and measures used in the various Italian states; a treatise on double-entry bookkeeping; and a summary of  Euclid’s geometry. He admitted to having borrowed freely from  Euclid,  Boethius,  Sacrobosco,  Fibonacci, …

The geometrical part of Pacioli’s Summa is discussed in detail in [6]. The authors write:-

The geometrical part of L Pacioli’s Summa [Venice, 1494] in Italian is one of the earliest printed mathematical books. Pacioli broadly used  Euclid’s Elements, retelling some parts of it. He referred also to Leonardo of Pisa ( Fibonacci).

Other interesting aspects of the Summa was the fact that it studied games of chance. Pacioli studied the problem of points, see, although the solution he gave is incorrect.

Ludovico Sforza was the second son of Francesco Sforza, who had made himself duke of Milan. When Francesco died in 1466, Ludovico’s elder brother Galeazzo Sforza became duke of Milan. However, Galeazzo was murdered in 1476 and his seven year old son became duke of Milan. Ludovico, after some political intrigue, became regent to the young man in 1480. With very generous patronage of artists and scholars, Ludovico Sforza set about making his court in Milan the finest in the whole of Europe. In 1482  Leonardo da Vinci entered Ludovico’s service as a court painter and engineer. In 1494 Ludovico became the duke of Milan and, around 1496, Pacioli was invited by Ludovico to go to Milan to teach mathematics at Ludovico Sforza’s court. This invitation may have been made at the prompting of  Leonardo da Vinci who had an enthusiastic interest in mathematics.

At Milan Pacioli and  Leonardo quickly became close friends. Mathematics and art were topics which they discussed at length, both gaining greatly from the other. At this time Pacioli began work on the second of his two famous works, Divina proportione and the figures for the text were drawn by  Leonardo. Few mathematicians can have had a more talented illustrator for their book! The book which Pacioli worked on during 1497 would eventually form the first of three books which he published in 1509 under the title Divina proportione (see for example). This was the first of the three books which finally made up this treatise, and it studied the ‘Divine Proportion’ or ‘ golden ratio’ which is the ratio a : b = b : (a + b). It contains the theorems of  Euclid which relate to this ratio, and it also studies regular and semiregular polygons (see in particular for a discussion of Pacioli’s work on regular polygons). Clearly the interest of  Leonardo in this aesthetically satisfying ratio both from a mathematical and artistic point of view was an important influence on the work. The golden ratio was also of importance in architectural design and this topic was to form the second part of the treatise which Pacioli wrote later. The third book in the treatise was a translation into Italian of one of della  Francesca’s works.

Louis XII became king of France in 1498 and, being a descendant of the first duke of Milan, he claimed the duchy. Venice supported Louis against Milan and in 1499 the French armies entered Milan In the following year Ludovico Sforza was captured when he attempted to retake the city. Pacioli and  Leonardo fled together in December 1499, three months after the French captured Milan. They stopped first at Mantua, where they were the guests of Marchioness Isabella d’Este, and then in March 1500 they continued to Venice. From Venice they returned to Florence, where Pacioli and  Leonardo shared a house.

The University of Pisa had suffered a revolt in 1494 and had moved to Florence. Pacioli was appointed to teach geometry at the University of Pisa in Florence in 1500. He remained in Florence, teaching geometry at the university, until 1506.  Leonardo, although spending ten months away working for Cesare Borgia, also remained in Florence until 1506. Pacioli, like  Leonardo, had a spell away from Florence when he taught at the University of Bologna during 1501-02. During this time Pacioli worked with Scipione del  Ferro and there has been much conjecture as to whether the two discussed the algebraic solution of  cubic equations. Certainly Pacioli discussed this topic in the Summa and some time after Pacioli’s visit to Bologna, del  Ferro solved one of the two cases of this classic problem.

During his time in Florence Pacioli was involved with Church affairs as well as with mathematics. He was elected the superior of his Order in Romagna and then, in 1506, he entered the monastery of Santa Croce in Florence. After leaving Florence, Pacioli went to Venice where he was given the sole rights to publish his works there for the following fifteen years. In 1509 he published the three volume work Divina proportione and also a Latin translation of  Euclid’s Elements. The first printed edition of  Euclid’s Elements was the thirteenth century translation by  Campanus which had been published in printed form in Venice in 1482. Pacioli’s edition was based on that of  Campanus but it contained much in the way of annotation by Pacioli himself.

In 1510 Pacioli returned to Perugia to lecture there again. He also lectured again in Rome in 1514 but by this time Pacioli was 70 years of age and nearing the end of his active life of scholarship and teaching. He returned to Sansepolcro where he died in 1517 leaving unpublished a major work De viribus amanuensis on recreational problems, geometrical problems and proverbs. This work makes frequent reference to  Leonardo da Vinci who worked with him on the project, and many of the problems in this treatise are also in  Leonardo’s notebooks. Again it is a work for which Pacioli claimed no originality, describing it as a compendium.

Despite the lack of originality in Pacioli’s work, his contributions to mathematics are important, particularly because of the influence which his book were to have over a long period. In  the importance of Pacioli’s work is discussed, in particular his computation of approximate values of a square root (using a special case of  Newton’s method), his incorrect analysis of certain games of chance (similar to those studied by  Pascal which gave rise to the  theory of probability), his problems involving  number theory (similar problems appeared in  Bachet’s compilation), and his collection of many  magic squares.

In 1550 there appeared a biography of Piero della  Francesca written by Giorgio Vasari. This biography accused Pacioli of plagiarism and claimed that he stole della  Francesca’s work on perspective, on arithmetic and on geometry. This is an unfair accusation, for although there is truth that Pacioli relied heavily on the work of others, and certainly on that of della  Francesca in particular, he never attempted to claim the work as his own but acknowledged the sources which he used.

J J O’Connor and E F Robertson

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Albrecht Dürer

Born: 21 May 1471 in Imperial Free City of Nürnberg (now in Germany)

Died: 6 April 1528 in Imperial Free City of Nürnberg (now in Germany)

Albrecht Dürer was the third son of Albrecht Dürer and Barbara Holfer. He was one of their eighteen children. The Dürer family came from Hungary, Albrecht Dürer senior being born there, and at this time the family name was Ajtos. The name Ajtos means «door» in Hungarian and when Dürer senior and his brothers came to Germany they chose the name Türer which sounds like the German «Tür» meaning door. The name changed to Dürer but Albrecht Dürer senior always signed himself Türer rather than Dürer.

Here are  portraits of his father and mother.

Albrecht Dürer senior was a jeweller who had served his apprenticeship with Hieronymus Holfer, and then married Holfer’s daughter. Albrecht Dürer junior wrote about his father and his upbringing (see for example):-

My father suffered much and toiled painfully all his life, for he had no resources other than the proceeds of his trade from which to support himself and his wife and family. He led an honest, God-fearing life. His character was gentle and patient. He was friendly towards all and full of gratitude to his Maker. He cared little for society and nothing for worldly amusements. A man of very few words and deeply pious, he paid great attention to the religious education of his children. His most earnest hope was that the high principles he instilled into their minds would render them ever more worthy of divine protection and the sympathy of mankind. He told us every day that we must love God and be honourable in our dealings with our neighbours.

As a young boy Dürer was educated at the Lateinschule in St Lorenz and he also worked in his father’s workshop learning the trade of a goldsmith and jeweller. By the age of 13 he was already a skilled painter as seen from a  self portrait which he painted at that time. This was the first of many self-portraits which Dürer painted and they provide a wonderful record. Here is our  collection of such self-portraits.

In 1486 Dürer became an apprentice painter and woodcut designer to Michael Wolgemut, the leading producer of altarpieces. After an apprenticeship of four years, Dürer had learnt all he could from Wolgemut and had reached a level of artistic quality exceeding that of his famous teacher. Wolgemut advised Dürer to travel to widen his experience and meet other artists. Following Wolgemut’s advice, Dürer delayed visiting Italy (which Wolgemut himself never visited), where there were very different artistic styles, until he had fully developed his own style and learnt more techniques from other German artists.

Here is a  portrait of Wolgemut.

Dürer travelled first to Nördlingen, where he met artists of the Swabian school. The Swabian style had been influenced by Dutch artistic design which Dürer had not met before. His next visit was to Ulm where he met more artists of the Swabian school. Dürer:-

… participated with keen enjoyment in the discussions among artists of his own age, in the low-ceilinged taverns, over foaming mugs of beer. These youthful enthusiasts, in common with those of all nations throughout history, were bent on rejuvenation of the art of the world. They were delighted with Dürer’s drawings, with his first engravings and the small pictures he had already painted, independently of Wolgemut’s directions or opinions.

Leaving Ulm, Dürer made his way to Constance which charmed him with its fairyland appearance. Basel was the next town which Dürer visited, and he found it quite similar to his home town of Nürnberg. Finally Dürer returned home, making visits to Colmar and Strasbourg on the way.

It had been a long journey of great importance to Dürer which had taken nearly four years, but after he returned to Nürnberg in 1494 he felt disappointed that he had not visited Italy. He had also become convinced that:-

… the new art must be based upon science — in particular, upon mathematics, as the most exact, logical, and graphically constructive of the sciences.

Italy was not only a country with new ideas to offer Dürer in art, but it was also leading the world at this time in the revival of mathematics. Before setting out for Italy, however, Dürer married Agnes Frey, the daughter of a learned man Hans Frey who had made quite a lot of money through making jewellery, musical instruments, and mechanical devices.

Here are  portraits of Agnes.

The marriage seems to have been more the idea of the parents of Agnes and Albrecht, and the pair were married on 7 July 1494. It was a marriage which helped raise Dürer’s status in Nürnberg, as well as provide him with money which helped him set up his own studio.

Before the end of 1494, Dürer was on his travels again, leaving Agnes behind in Nürnberg. First he visited Augsburg where he met strong Italian artistic influences for the first time. Travelling through the Tyrol, he reached Trento and his first view of Italy.

Here is  one of his paintings of Trento.

He travelled on to Verona before reaching Venice which was his main objective. In Venice, Dürer, as he had done throughout his journeys, sketched scenes, visited galleries and churches, and met with the local artists. One of the artists that he met in Venice, Giovanni Bellini, had an important influence on Dürer for:-

… everything that [Venice] could teach him was to be found in Giovanni’s paintings. He cultivated the artist’s society, therefore, with a devotion both impassioned and deferential, retaining throughout his life, with his whole heart and soul, unbounded feelings of gratitude to the man whose pictures had unveiled so wonderful a world to him.

Dürer returned to Nürnberg in 1495, and although he does not seem to have met with any of the major Italian mathematicians on his journeys, he did meet Jacopo de Barbari who told him of the mathematical work of  Pacioli and its importance to the theory of beauty and art. Nor did Dürer meet with  Leonardo da Vinci while in Italy, but he learnt of the importance which that artist placed in mathematics. Back in Nürnberg, Dürer began a serious study of mathematics. He read  Euclid’s Elements and the important treatise De architectura (On Architecture) by  Vitruvius (1st century BC), the famous Roman architect and engineer. He also became familiar with the work of  Alberti and  Pacioli on mathematics and art, in particular work on proportion.

It was not only this scientific approach to art that influenced Dürer as he began his artistic career in Nürnberg, but he also benefited from seeing different artistic styles and the different scenery which he had viewed:-

The variety of regions through which Dürer had passed in the course of his travels and the care he had taken with the drawings and water-colours he had made of the most attractive or unfamiliar of them had provided him with a great range of pictorial motives emanating from the most diverse sources.

In 1495 Dürer was still not well known as an artist in the highest circles but news of his skill reached Frederick the Wise, Elector of Saxony, and Dürer was commissioned to paint his portrait. Frederick liked  his portrait which Dürer painted in April 1496 when Frederick had visited Nürnberg. Despite Frederick’s attempts to persuade Dürer to move to Weimar and become Court painter, the artist did not wish to leave Nürnberg. He was deeply attached to Nürnberg, painting  these views of the city in 1497.

From about 1500 Dürer’s art showed the influence of the mathematical theory of proportion which he continued to spend so much time studying. It is claimed that his  self-portrait in a wig made in 1500 has the dimensions of the head constructed proportionally. For the engraving  Adam and Eve made in 1504, Dürer described the intricate  ruler and compass constructions which he made to construct the figures. It was not only the mathematical theory of proportion which influenced Dürer’s art at this period, but also his mastery of perspective through his study of geometry. This is most clearly seen in his woodcuts  Life of the Virgin made between 1502 and 1505.

During the ten years after 1496 Dürer went from a relatively unknown artist to someone with a wide reputation as both an artist and a mathematician. His personal circumstances had changed greatly. His father had died in 1502 and Dürer was left to care for his invalid, and nearly blind, mother. He had set up his own printing press while he, or often his wife, sold his works to buyers at local fairs. It was a difficult life and one in which Dürer’s health began to suffer. In fact he would never regain full health during the rest of his life.

From 1505 to 1507 Dürer made a second visit to Italy, spending much time again in Venice. It was a very different visit from his first, with Dürer now more interested in his international fame than in learning about art. He was so conscious of his fame, and the threat he perceived that he might hold to the local artists, that:-

… he refused invitations to dinner in case someone should try to poison him.

It was not about art that Dürer now wished to learn from the Italians, but rather about mathematics. He visited Bologna to meet with  Pacioli whom he considered held the mathematical secrets of art. He also visited Jacopo de Barbari and the great efforts which Dürer made to meet de Barbari shows the importance which Dürer more and more attached to mathematical knowledge. Dürer returned to Nürnberg from this second visit to Italy feeling that he must delve yet more deeply into the study of mathematics.

In about 1508 Dürer began to collect material for a major work on mathematics and its applications to the arts. This work would never be finished but Dürer did use parts of the material in later published work. He continued to produce art of outstanding quality, and he produced one of his most famous engravings  Melancholia in 1514. It contains the first  magic square to be seen in Europe, cleverly including the date 1514 as two entries in the middle of the bottom row. Also of mathematical interest in Melancholia is the polyhedron in the picture. The faces of the polyhedron appear to consist of two equilateral triangles and six somewhat irregular pentagons. An interesting reconstruction of the polyhedron is given in , see also  for further details.

Dürer worked for Maximilian I, the Holy Roman emperor, from about 1512. Maximilian, however, had little in the way of wealth to pay for Dürer’s work and he asked the councillors of Nürnberg to exempt Dürer from taxes as compensation. He then asked the councillors to pay Dürer a pension on his behalf, which certainly did not please them. From about 1515 the councillors tried to avoid paying this pension. Dürer met Maximilian personally for the first time in 1518 and, probably from one sitting in Augsburg, painted  Maximilian’s portrait. The following year Maximilian died and this was the final excuse for the councillors to refuse to make any further payment, saying that the new emperor Charles would have to agree to the pension.

Although Dürer was fairly well off by this time and the pension was not necessary for him, it was more a matter of prestige to have his pension restored. He set off for Antwerp on 15 July 1520 with his wife and their maid to visit the Emperor Charles V. Passing through Aachen, Dürer sketched  the cathedral at Aachen.

Dürer had a second reason for this visit to the Netherlands, for he believed that Maximilian’s daughter had a book by Jacopo de Barbari on applications of mathematics to art, and Dürer had long sought the truths which he believed this work contained. On meeting Maximilian’s daughter he offered her the portrait of her father which he had painted, but was distressed to find that she did not want the portrait. She had already given the book by Jacopo de Barbari to another artist so Dürer’s quest was in vain. He did persuade Charles V to restore his pension, however, which was formally agreed on 12 November 1520.

After returning to Nürnberg, Dürer’s health became still worse. He did not slacken his work on either mathematics or painting but most of his effort went into his work Treatise on proportion. Although it was completed in 1523, Dürer realised that it required mathematical knowledge which went well beyond what any reader could be expected to have, so he decided to write a more elementary text. He published this more elementary treatise, in four books, in 1525 publishing the work through his own publishing company.

This treatise, Unterweisung der Messung mit dem Zirkel und Richtscheit, is the first mathematics book published in German (if one discounts an earlier commercial arithmetic book) and places Dürer as one of the most important of the Renaissance mathematicians. Dürer’s sources for this work are discussed in [21] where three main sources are suggested (i) the practical recipes of craftsmen, (ii) classical mathematics from printed works and manuscripts, and (iii) the manuals of Italian artists. The article [16] gives many details of the mathematics contained in the treatise.

The first of the four books describes the construction of a large number of curves, including the  Spiral of Archimedes, the  Equiangular or Logarithmic Spiral, the  Conchoid,  Dürer’s Shell Curves, the  Epicycloid, the  Epitrochoid, the  Hypocycloid, the  Hypotrochoid, and the  Limaçon of Pascal (although of course Dürer did not use that name!). Details about Dürer’s descriptions of the curves, in particular one he calls a «muschellini», is given in.

In the second book he gave exact and approximate methods to construct regular polygons. Dürer’s constructions of regular polygons with 5, 7, 9, 11 and 13 sides is discussed in [12]. Dürer also gave approximate methods to  square the circle using ruler and compass constructions in this book. A method to obtain a good approximation to the  trisector of an angle by Euclidean construction is also given.

Book three considers pyramids, cylinders and other solid bodies. The second part of this book studies sundials and other astronomical instruments. The final book studies the five Platonic solids as well as the semi-regular Archimedean solids. Also in this book is Dürer’s theory of shadows and an introduction to the theory of perspective.

In 1527 Dürer published another work, this time on fortifications. There were strong reasons why he produced a work on fortifications at this time, for the people of Germany were in fear of an invasion by the Turks. Many cities, including Nürnberg, would improve their fortifications using the methods set out by Dürer in this book. Dürer’s final masterpiece was his Treatise on proportion which was at the proof stage at the time of his death.

Descriptive geometry originated with Dürer in this work although it was only put on a sound mathematical basis in later work of  Monge. One of the methods of overcoming the problems of projection, and describing the movement of bodies in space, is descriptive geometry. Dürer’s remarkable achievement was through applying mathematics to art, he developed such fundamentally new and important ideas within mathematics itself.

J J O’Connor and E F Robertson

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About Lobachevsky

Science is a very important and interesting field of activity of man. Mathematics as a science is a good example of such activity.

Russian and Soviet mathematicians made a great contribution to the development of mathematics. World known are the name of S. Kovalevskaya, N. Lobachevsky and others. There are scientists who seem to be born scientists. And their life is not always a bed of roses, sometimes their life turns really tragic. Lobachevsky’s life is a good example and a proof of this statement.

Lobachevsky was born in 1792 near Nizhni Novgorod. When he was a child, his family moved to Kazan, where Lobachevsky entered the gymnasium. His progress in mathematics was very rapid there.

At the age of 15 he entered the University of Kazan. At 19 he was awarded a Master’s degree. At the age of 24 Lobachevsky became a Professor of mathematics at Kazan University. When he was 35, Lobachevsky was appointed rector of the University. Lobachevsky was 34 years old, when he founded non-Euclidean geometry of two-dimensions. His geometry is valid for pseudosfherical surfaces. It is based on the assumption that the axiom on parallel to a given straight line.

It was one of the masterpieces of mathematics. Strange as it may seem, scientists of old Russia and Europe did not support his geometry. But Lobachevsky was not discouraged. In 1846 he was dismissed from his office. And still Lobachevsky continued his research work, now at home. When he got blind, he dictated his new masterpiece-pan-geometry — to his pupils, Lobachevsky died in 1856 at the age of 63.

So, we see that revolutionary idens in science, they struggle their way through misunderstanding. And only a really talented and persistent personality can win.

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Leonhard Euler

Born: 15 April 1707 in Basel, Switzerland

Died: 18 Sept 1783 in St Petersburg, Russia

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, von 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 Nicolaus(II) Bernoulli 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 nor 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 Science 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 became 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 [24] 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 [24] 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 [1]:-

… 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 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.

 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).

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), p 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) = p2/6 but he went on to prove much more, namely that (4) = p4/90, (6) = p6/945, (8) = p8/9450, (10) = p10/93555 and (12) = 691p12/638512875. 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) = Cp2n 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

p/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 [60]:-

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 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) = pi

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 establish 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 formulas 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 Tobias 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.

J J O’Connor and E F Robertson

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Rapid growth in the nature and applications of mathematics means that the Newtonian core — calculus, analysis, and differential equations — is now just one part of a more diverse mathematical landscape. Yet most scientists have explored only this original territory, because that is all that was included in their curriculum in high school, college, and graduate school. With the exception of statistics, an old science widely used across all disciplines that has become largely mathematical during the 20th century, the narrow Newtonian legacy of analysis is the principal connection between practicing scientists and broad mathematical foundations of their disciplines.

The dramatic changes in the mathematical sciences of the last quarter century are largely invisible to those outside the small community of research mathematicians. Today’s mathematical sciences, like yesterday’s Gaul, can be divided into three parts of roughly comparable size: statistical science, core mathematics, and applied mathematics. Each of these three major areas is led (in the United States) by a few thousand active researches and receives approximately $50 million in federal research support annually. Although the boundaries between these parts overlap considerably, each province has an identifiable character paradigm established by Newton: data, deduction, and observation.

Core mathematics investigates properties of number and space, ideas rooted in antiquity. Its tools are abstraction and deduction; its edifices include functions, equations, operators, and infinite-dimensional space. Within core mathematics are found the traditional subjects of number theory, algebra, geometry, analysis, and topology. After a half-century of explosive specialized growth, core mathematics is experiencing a renaissance of renewed integrity based on the unexpected but welcome discovery of deep links among its various components.

The faculty of mathematics of Irkutsk State University

1. The faculty of mathematics was set up a year after the establishment of Irkutsk State University, in 1919.

2. It occupied the building of the East Siberian Institute for girls of noble origin.

3. First, it was the faculty of physics and mathematics.

4. Later it was decided to have two faculties, the faculty of physics and the faculty of mathematics.

5. It happened in 1965.

6. The first dean of the mathematical faculty was Professor Vladimir Vasilyev.

7. In 1959 a computer center was set up at the University, and University students began studying computers and programming.

8. In 1997 the Internet Center was set up on the basis of our University Computer Center.

9. And Now University students have a free access to Internet and World Wide Web.

10. In 1998 our faculty was reformed, and the Institute of mathematics and Economics was set up.

11. Its director is Professor Oleg Vasilyev, a corresponding member of the Russian Academy of Sciences.

12. Students are trained in three specialities; they are mathematics, applied mathematics, mathematical methods and operations research in economy.

13. Our students study many special subjects and humanities, too.

14. They are mathematical analysis, algebra, geometry and topology, differential equations, the theory of probability, mathematical modeling, numerical methods, operations research, and system analyses, optimization and optimum management, mathematical cybernetics, mathematical software, and others.

15. As for humanities, they are the Russian and the English languages, History and philosophy.

16. There are 13 departments at the Institute.

17. They are headed by Professors and assistant professors, such as Vasilyev, Bludov, Zaharova, Mantsivoda and others.

18. I specialize in mathematics.

19. When at school I got interested in mathematical sciences and after finishing school I entered the institute of mathematics and economics.

20. After graduating from the University I’ll be a teacher of mathematics.

21. Students of mathematics have their school practice every year.

22. I like my future profession and I do my best to get profound (deep) knowledge in it.

23. Our students also go in for sports, attend different students clubs and take an active part in the life of the University.

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