Nicolaus(II) Bernoulli

Nicolaus(II) Bernoulli was the favourite of three sons of Johann Bernoulli. He entered the University of Basel when only 13 years of age and, like many other members of his family, studied both mathematics and law. In 1715 he became a licentiate in jurisprudence.

Born: 6 Feb 1695 in Basel, Switzerland
Died: 31 July 1726 in St Petersburg, Russia

Nicolaus worked as his father's assistant helping him with correspondence. In particular he was involved with writing letters concerning the famous priority dispute between Newton and Leibniz. He not only replied to Taylor regarding the dispute but he also made important mathematical contributions to the problem of trajectories while working on the mathematical arguments behind the dispute.

Nicolaus worked on curves, differential equations and probability. He died only 8 months after taking up an appointment in St Petersburg at a young age when his talents promised so much for the future. Fleckstein writes in [1]:-

With his brother Daniel he travelled in France and Italy, where both received appointments to the St Petersburg Academy. Within a year, however, he contracted and died of a hectic fever.

Nicolaus(I) Bernoulli

Born: 21 Oct 1687 in Basel, Switzerland
Died: 29 Nov 1759 in Basel, Switzerland



Nicolaus(I) Bernoulli was a nephew of Jacob Bernoulli and Johann Bernoulli. His early education involved studying mathematics with his uncles. In fact it was Jacob Bernoulli who supervised Nicolaus's Master's degree at the University of Basel which he was awarded in 1704. Five years later he was received a doctorate for a dissertation which studied the application of probability theory to certain legal questions.

In 1712 Nicolaus Bernoulli toured Europe visiting Holland, England and France. It was in France that he met Montmort and the two mathematicians became close friends and collaborated on mathematical questions in a long correspondence.

Nicolaus Bernoulli was appointed to Galileo's chair at Padua in 1716 which Hermann had filled immediately prior to Nicolaus's appointment. There he worked on geometry and differential equations. In 1722 he left Italy and returned to his home town to take up the chair of logic at the University of Basel. After nine years, remaining at the University of Basel, he was appointed to the chair of law. In addition to these academic appointments, he did four periods as rector of the university.

J O Fleckenstein, writing in [1], describes Nicolaus Bernoulli's contribution to mathematics:-

Nicolaus was a gifted but not very productive mathematician. As a result, his most important achievements are hidden throughout his correspondence, which comprises about 560 items. The most important part of his correspondence with Montmort (1710-1712) was published in the latter's "Essai d'analyse sur les jeux de hazard" (Paris, 1713).

From Montmort's work we can see that Nicolaus formulated certain problems in the theory of probability, in particular the problem which today is known as the St Petersburg problem. Nicolaus also corresponded with Leibniz during the years 1712 to 1716. In these letters Nicolaus discussed questions of convergence, and showed that (1+x)n diverges for x > 0.

Nicolaus also corresponded with Euler. Again quoting [1]:-

In his letters to Euler (1742-43) he criticises Euler's indiscriminate use of divergent series. In this correspondence he also solved the problem of the sum of the reciprocal squares (1/n2) = π2/6, which had confounded Leibniz and Jacob Bernoulli.

Nicolaus Bernoulli assisted in the publication of Jacob Bernoulli's Ars conjectandi. Later Nicolaus edited Jacob Bernoulli's complete works and supplemented it with results taken from Jacob's diary. Other problems he worked on involved differential equations. He studied the problem of orthogonal trajectories, making important contributions by the construction of orthogonal trajectories to families of curves, and he proved the equality of mixed second-order partial derivatives. He also made significant contributions in studying the Riccati equation.

One of the great controversies of the time was the Newton Leibniz argument. As might be expected Nicolaus supported Leibniz but he did produce some good arguments in his favour such as observing that Newton failed to understand higher derivatives properly which had led him into errors in the problem of inverse central force in a resisting medium.

Nicolaus(I) Bernoulli received many honours for his work. For example he was elected a member of the Berlin Academy in 1713, a Fellow of the Royal Society of London in 1714, and a member of the Academy of Bologna in 1724.

Johann Bernoulli

Born: 27 July 1667 in Basel, Switzerland
Died: 1 Jan 1748 in Basel, Switzerland


Johann Bernoulli was the tenth child of Nicolaus and Margaretha Bernoulli. He was the brother of Jacob Bernoulli but Johann was twelve years younger than his brother Jacob which meant that Jacob was already a young man while Johann was still a child. The two brothers were to have an important influence on each others mathematical development and it was particularly true that in his early years Johann must have been greatly influenced by seeing Jacob head towards a mathematical career despite the objections of his parents. As to his education as a child, Johann wrote in his autobiography that his parents:-

... spared no trouble or expense to give me a proper education in both morals and religion.

This religion was the Calvinist faith which had forced his grandparents to flee from Antwerp to avoid religious persecution.

Nicolaus and Margaretha Bernoulli tried to set Johann on the road to a business career but, despite his father's strong pushing, Johann seemed to be totally unsuited to a future in business. Johann's father had intended him to take over the family spice business and in 1682, when he was 15 years old, Johann worked in the spice trade for a year but, not liking the work, he did not do well. It was with great reluctance that Johann's father agreed in 1683 to Johann entering the University of Basel. The subject that Johann Bernoulli was to study at university was medicine, a topic that many members of the Bernoulli family ended up studying despite their liking for mathematics and mathematical physics.

At Basel University Johann took courses in medicine but he studied mathematics with his brother Jacob. Jacob was lecturing on experimental physics at the University of Basel when Johann entered the university and it soon became clear that Johann's time was mostly devoted to studying Leibniz's papers on the calculus with his brother Jacob. After two years of studying together Johann became the equal of his brother in mathematical skill.

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

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

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

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

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

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

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

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

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

Johann Bernoulli had married Drothea Falkner and their first child was seven months old when the family departed for Holland on 1 September 1695. This first child was Nicolaus(II) Bernoulli who also went on to become a mathematician. Perhaps this is a good time to note that two other of Johann's children went on to become mathematicians, Daniel Bernoulli, who was born while the family was in Groningen, and Johann(II) Bernoulli.

Neither Bernoulli's wife nor his father-in-law had been happy about the move to Groningen especially since the journey was such a difficult one with a young baby. After setting out on 1 September they had to cross a region where armies were fighting, then travel down the Rhine by boat, finally taking a carriage and another boat to their destination. They arrived on 22 October to begin ten years in Groningen which were to be filled with difficulties. Johann was involved in a number of religious disputes, his second child was a daughter who was born in 1697 and only lived for six weeks, and he suffered so severe an illness that he was reported to have died.

In one dispute he was accused of denying the resurrection of the body, a charge based on medical opinions he held. In a second dispute in 1702 Bernoulli was accused by a student at the University of Groningen, Petrus Venhuysen, who published a pamphlet which basically accused Bernoulli of following Descartes' philosophy. The pamphlet also accused him of opposing the Calvinist faith and depriving believers of their comfort in Christ's passion. Bernoulli wrote a long twelve page reply to the Governors of the University, which still exists [16]:-

... I would not have minded so much if [Venhuysen] had not been one of the worst students, an utter ignoramus, not known, respected, or believed by any man of learning, and he is certainly not in a position to blacken an honest man's name, let alone a professor known throughout the learned world...

... all my life I have professed my Reformed Christian belief, which I still do... he would have me pass for an unorthodox believer, a very heretic; indeed very wickedly he seeks to make me an abomination to the world, and to expose me to the vengeance of both the powers that be and the common people...

This was not Johann's only dispute while in Groningen. He introduced physics experiments in his teaching, but Sierksma writes in [16] that these:-

... were objectionable to scientists of the Cartesian persuasion and Calvinists alike. The Cartesians naturally highlighted 'reason' and held the view that... the world of sensory perception is of minor importance; the Calvinists attempted to fathom God's underlying plan by scrupulously analysing natural phenomenon. Interpretations of these natural phenomenon alone would be incompatible with either.

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

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

In 1705 the Bernoulli family in Groningen received a letter saying that Johann's father-in-law was pining for his daughter and grandchildren and did not have long to live. They decided to return to Basel along with Nicolaus(I) Bernoulli, his nephew, who had been studying mathematics in Groningen with his uncle. They left Groningen two days after Jacob's death but, of course, they were not aware that he had died of tuberculosis then, and they only learnt of his death while they were on their journey. Hence Johann was not returning to Basel expecting the chair of mathematics, rather he was returning to fill the chair of Greek. Of course the death of his brother was to lead to a change of plan.

Before reaching Basel, however, Johann was tempted by an offer of a chair at the University of Utrecht. The head of the University of Utrecht was so keen to have Bernoulli come there that he set out after the Bernoulli's catching up with them in Frankfurt. He tried to persuade Johann to go to Utrecht but Bernoulli was set on returning to Basel.

On his return to Basel Johann worked hard to ensure that he succeeded to his brother's chair and soon he was appointed to Jacob's chair of mathematics. It is worth remarking that Bernoulli's father-in-law lived for three years in which he greatly enjoyed having his daughter and grandchildren back in Basel. There were other offers that Johann turned down, such as Leiden, a second offer from Utrecht and a generous offer for him to return to Groningen in 1717.

In 1713 Johann became involved in the Newton-Leibniz controversy. He strongly supported Leibniz and added weight to the argument by showing the power of his calculus in solving certain problems which Newton had failed to solve with his methods. Although Bernoulli was essentially correct in his support of the superior calculus methods of Leibniz, he also supported Descartes' vortex theory over Newton's theory of gravitation and here he was certainly incorrect. His support in fact delayed acceptance of Newton's physics on the Continent.

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

Johann Bernoulli attained great fame in his lifetime. He was elected a fellow of the academies of Paris, Berlin, London, St Petersburg and Bologna. He was known as the "Archimedes of his age" and this is indeed inscribed on his tombstone.

Johann(III) Bernoulli

Born: 4 Nov 1744 in Basel, Switzerland
Died: 13 July 1807 in Berlin, Germany


Johann(III) Bernoulli was a son of Johann(II) Bernoulli. He was certainly considered a prodigy when a child with an encyclopedic knowledge and, like many other members of his extraordinarily talented family, he studied law and took an interest in mathematics.

At the early age of fourteen he graduated with the degree of master of law. He was appointed to a chair at Berlin Academy at the age of only 19. Frederick II asked him to revive the astronomical observatory of the Academy but this was not a task for which Johann(III) was particularly well suited. His health had never been particularly good and his qualities as an astronomical observer were relatively poor.

Johann(III) Bernoulli wrote a number of works on astronomy, reporting on astronomical observations and calculations, but these are of little importance. Strangely his most important contributions were the accounts of his travels in Germany which were to have a historical impact.

In the field of mathematics he worked on probability, recurring decimals and the theory of equations. As in his astronomical work there was little of lasting importance. He did, however, publish the Leipzig Journal for Pure and Applied Mathematics between 1776 and 1789.

He was well aware of the famous mathematical line from which he was descended and he looked after the wealth of mathematical writings that had passed between members of the family. He sold the letters to the Stockholm Academy where they remained forgotten about until 1877. At that time when these treasures were examined, 2800 letters written by Johann(III) Bernoulli himself were found in the collection.

Johann(II) Bernoulli

Born: 28 May 1710 in Basel, Switzerland
Died: 17 July 1790 in Basel, Switzerland



Johann(II) Bernoulli was one of three sons of Johann Bernoulli. In fact he was the most successful of the three. He originally studied law and in 1727 he obtained the degree of doctor of jurisprudence.

He worked on mathematics both with his father and as an independent worker. He had the remarkable distinction of winning the Prize of the Paris Academy on no less than four separate occasions. On the strength of this he was appointed to his father's chair in Basel when Johann Bernoulli died.

However, quoting [1]:-

... thereafter his mathematical production dwindled to occasional academic papers and a treatise, although he lived to almost as old as his father. His shyness and frail constitution did not, however, prevent him from engaging in extensive scientific correspondence (about 900 items) and from furthering the publication, in four volumes, of his father's Opera Omnia. He personified the mathematical genius of his native city in the second half of the eighteenth century.

Johann(II) Bernoulli worked mainly on heat and light.

Maupertuis, who was President of the Berlin Academy, was accused by Samuel König of plagiarising Leibniz's work. Voltaire was so critical of Maupertuis' work that eventually he left Berlin and, in 1756, travelled to Basel where he took refuge in Johann(II) Bernoulli home. Maupertuis remained in Johann's home for the last three years of his life.

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' 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 was told by his father to study medicine but while he was studying that topic he asked his brother Jacob 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 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 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 [1]:-

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 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 [1]:-

... 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 [1] 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".

Jacob(II) (Jacques(II)) Bernoulli

Born: 17 Oct 1759 in Basel, Switzerland
Died: 15 Aug 1789 in St Petersburg, Russia



Jacob(II) Bernoulli was one of the sons of Johann(II) Bernoulli. Following the family tradition he took a degree in law but his interests were in mathematics and mathematical physics.

In 1782 Jacob(II) Bernoulli's uncle Daniel Bernoulli died and his chair of physics in Basel became vacant. Jacob(II) applied for the chair and presented a work on mathematical physics to support his application. The decison as to who should fill the vacant chair was not made on academic grounds but was made by drawing lots. Jacob(II) Bernoulli was unlucky and he was not offered this position he would really have liked.

He was then appointed as secretary to the Imperial Envoy to Turin and Venice. However, he was soon given the chance of another academic post when he received an offer from St Petersburg. He went to St Petersburg and began to write important works on mathematical physics which he presented to the St Petersburg Academy of Sciences. These treatises were on elasticity, hydrostatics and ballistics.

Despite the rather harsh climate, the city of St Petersburg had great attractions for Jacob(II) Bernoulli since his uncle Daniel Bernoulli had worked there with Euler. In fact Jacob(II) married a granddaughter of Euler in St Petersburg but, tragically, the city was to lead to his death.

St Petersburg is located on the delta of the Neva River, at the head of the Gulf of Finland. St Petersburg, built on 42 islands in the Neva River, is a city of waterways and bridges and because of this it is called the "Venice of the North." This has great attraction but Jacob(II) Bernoulli drowned, while still only 29 years of age, in the Neva River while he was swimming.

Daniel Bernoulli

Born: 8 Feb 1700 in Groningen, Netherlands
Died: 17 March 1782 in Basel, Switzerland


Daniel Bernoulli was the son of Johann Bernoulli. He was born in Groningen while his father held the chair of mathematics there. His older brother was Nicolaus(II) Bernoulli and his uncle was Jacob Bernoulli so he was born into a family of leading mathematicians but also into a family where there was unfortunate rivalry, jealousy and bitterness.

When Daniel was five years old the family returned to their native city of Basel where Daniel's father filled the chair of mathematics left vacant on the death of his uncle Jacob Bernoulli. When Daniel was five years old his younger brother Johann(II) Bernoulli was born. All three sons would go on to study mathematics but this was not the course that Johann Bernoulli planned for Daniel.

Johann Bernoulli's father had tried to force Johann into a business career and he had resisted strongly. Rather strangely Johann Bernoulli now tried exactly the same with his own son Daniel. First however Daniel was sent to Basel University at the age of 13 to study philosophy and logic. He obtained his baccalaureate examinations in 1715 and went on to obtain his master's degree in 1716. Daniel, like his father, really wanted to study mathematics and during the time he studied philosophy at Basel, he was learning the methods of the calculus from his father and his older brother Nicolaus(II) Bernoulli.

Johann was determined that Daniel should become a merchant and he tried to place him in an apprenticeship. However Daniel was as strongly opposed to this as his own father had been and soon Johann relented but certainly not as far as to let Daniel study mathematics. Johann declared that there was no money in mathematics and so he sent Daniel back to Basel University to study medicine. This Daniel did spending time studying medicine at Heidelberg in 1718 and Strasbourg in 1719. He returned to Basel in 1720 to complete his doctorate in medicine.

By this stage Johann Bernoulli was prepared to teach his son more mathematics while he studied medicine and Daniel studied his father's theories of kinetic energy. What he learned on the conservation of energy from his father he applied to his medical studies and Daniel wrote his doctoral dissertation on the mechanics of breathing. So like his father Daniel had applied mathematical physics to medicine in order to obtain his medical doctorate.

Daniel wanted to embark on an academic career like his father so he applied for two chairs at Basel. His application for the chair of anatomy and botany was decided by drawing of lots and he was unlucky in this game of chance. The next chair to fall vacant at Basel that Daniel applied for was the chair of logic, but again the game of chance of the final selection by drawing of lots went against him. Having failed to obtain an academic post, Daniel went to Venice to study practical medicine.

In Venice Daniel was severely ill and so was unable to carry out his intention of travelling to Padua to further his medical studies. However, while in Venice he worked on mathematics and his first mathematical work was published in 1724 when, with Goldbach's assistance, Mathematical exercises was published. This consisted of four separate parts being four topics that had attracted his interest while in Venice.

The first part described the game of faro and is of little importance other than showing that Daniel was learning about probability at this time. The second part was on the flow of water from a hole in a container and discussed Newton's theories (which were incorrect). Daniel had not solved the problem of pressure by this time but again the work shows that his interest was moving in this direction. His medical work on the flow of blood and blood pressure also gave him an interest in fluid flow. The third part of Mathematical exercises was on the Riccati differential equation while the final part was on a geometry question concerning figures bounded by two arcs of a circle.

While in Venice, Daniel had also designed an hour glass to be used at sea so that the trickle of sand was constant even when the ship was rolling in heavy seas. He submitted his work on this to the Paris Academy and in 1725, the year he returned from Italy to Basel, he learnt that he had won the prize of the Paris Academy. Daniel had also attained fame through his work Mathematical exercises and on the strength of this he was invited to take up the chair of mathematics at St Petersburg. His brother Nicolaus(II) Bernoulli was also offered a chair of mathematics at St Petersburg so in late 1725 the two brothers travelled to St Petersburg.

Within eight months of their taking up the appointments in St Petersburg Daniel's brother died of fever. Daniel was left, greatly saddened at the loss of his brother and also very unhappy with the harsh climate. He thought of returning to Basel and wrote to his father telling him how unhappy he was in St Petersburg. Johann Bernoulli was able to arrange for one of his best pupils, Leonard Euler, to go to St Petersburg to work with Daniel. Euler arrived in 1727 and this period in St Petersburg, which Daniel left in 1733, was to be his most productive time.

One of the topics which Daniel studied in St Petersburg was that of vibrating systems. As Straub writes in [1]:-

From 1728, Bernoulli and Euler dominated the mechanics of flexible and elastic bodies, in that year deriving the equilibrium curves for these bodies. ... Bernoulli determined the shape that a perfectly flexible thread assumes when acted upon by forces of which one component is vertical to the curve and the other is parallel to a given direction. Thus, in one stroke he derived the entire series of such curves as the velaria, lintearia, catenaria...

While in St Petersburg he made one of his most famous discoveries when he defined the simple nodes and the frequencies of oscillation of a system. He showed that the movements of strings of musical instruments are composed of an infinite number of harmonic vibrations all superimposed on the string.

A second important work which Daniel produced while in St Petersburg was one on probability and political economy. Daniel makes the assumption that the moral value of the increase in a person's wealth is inversely proportional to the amount of that wealth. He then assigns probabilities to the various means that a person has to make money and deduces an expectation of increase in moral expectation. Daniel applied some of his deductions to insurance.

Undoubtedly the most important work which Daniel Bernoulli did while in St Petersburg was his work on hydrodynamics. Even the term itself is based on the title of the work which he produced called Hydrodynamica and, before he left St Petersburg, Daniel left a draft copy of the book with a printer. However the work was not published until 1738 and although he revised it considerably between 1734 and 1738, it is more the presentation that he changed rather then the substance.

This work contains for the first time the correct analysis of water flowing from a hole in a container. This was based on the principle of conservation of energy which he had studied with his father in 1720. Daniel also discussed pumps and other machines to raise water. One remarkable discovery appears in Chapter 10 of Hydrodynamica where Daniel discussed the basis for the kinetic theory of gases. He was able to give the basic laws for the theory of gases and gave, although not in full detail, the equation of state discovered by Van der Waals a century later.

Daniel Bernoulli was not happy in St Petersburg, despite the obvious scientific advantage of working with Euler. By 1731 he was applying for posts in Basel but probability seemed to work against him and he would lose out in the ballot for the post. The post was neither one in mathematics nor physics but Daniel preferred to return to Basel and give lectures on botany rather than remain in St Petersburg. By this time his younger brother Johann(II) Bernoulli was also with him in St Petersburg and they left St Petersburg in 1733, making visits to Danzig, Hamburg, Holland and Paris before returning to Basel in 1734.

Daniel Bernoulli submitted an entry for the Grand Prize of the Paris Academy for 1734 giving an application of his ideas to astronomy. This had unfortunate consequences since Daniel's father, Johann Bernoulli, also entered for the prize and their entries were declared joint winners of the Grand Prize. The result of this episode of the prize of the Paris Academy had unhappy consequences for Daniel. His father was furious to think that his son had been rated as his equal and this resulted in a breakdown in relationships between the two. The outcome was that Daniel found himself back in Basel but banned from his father's house. Whether this caused Daniel to become less interested in mathematics or whether it was the fact that his academic position was a non mathematical one, certainly Daniel never regained the vigour for mathematical research that he showed in St Petersburg.

Although Daniel had left St Petersburg, he began an immediate correspondence with Euler and the two exchanged many ideas on vibrating systems. Euler used his great analytic skills to put many of Daniel's physical insights into a rigorous mathematical form. Daniel continued to work on polishing his masterpiece Hydrodynamica for publication and added a chapter on the force of reaction of a jet of fluid and the force of a jet of water on an inclined plane. In this chapter, Chapter 13, he also discussed applications to the propulsion of ships.

The 1737 prize of the Paris Academy also had a nautical theme, the best shape for a ship's anchor, and Daniel Bernoulli was again the joint winner of this prize, this time jointly with Poleni. Hydrodynamica was published in 1738 but, in the following year Johann Bernoulli published Hydraulica which is largely based on his son's work but Johann tried to make it look as if Daniel had based Hydrodynamica on Hydraulica by predating the date of publication on his book to 1732 instead of its real date which is probably 1739. This was a disgraceful attempt by Johann to gain credit for work which was not his and at the same time to discredit his own son and shows the depths to which the bad feeling between them had reached.

It is fair to say that there is no evidence that Daniel was in any way to blame for the breakdown of relationships with his father. Rather the reverse since there is evidence that he tried to mend the relationship with such acts as describing himself on the frontispiece of Hydrodynamica as 'Daniel Bernoulli, son of Johann'. Another sign that Daniel was not jealous of members of his own family in the way the Johann Bernoulli and Jacob Bernoulli had been is the fact that he did produce joint work with his younger brother Johann(II) Bernoulli.

Botany lectures were not what Daniel wanted and things became better for him in 1743 when he was able to exchange these for physiology lectures. In 1750, however, he was appointed to the chair of physics and taught physics at Basel for 26 years until 1776. He gave some remarkable physics lectures with experiments performed during the lectures. Based on experimental evidence he was able to conjecture certain laws which were not verified until many years later. Among these was Coulomb's law in electrostatics.

Daniel Bernoulli did produce other excellent scientific work during these many years back in Basel. In total he won the Grand Prize of the Paris Academy 10 times, for topics in astronomy and nautical topics. He won in 1740 (jointly with Euler) for work on Newton's theory of the tides; in 1743 and 1746 for essays on magnetism; in 1747 for a method to determine time at sea; in 1751 for an essay on ocean currents; in 1753 for the effects of forces on ships; and in 1757 for proposals to reduce the pitching and tossing of a ship in high seas.

Another important aspect of Daniel Bernoulli's work that proved important in the development of mathematical physics was his acceptance of many of Newton's theories and his use of these together with the tolls coming from the more powerful calculus of Leibniz. Daniel worked on mechanics and again used the principle of conservation of energy which gave an integral of Newton's basic equations. He also studied the movement of bodies in a resisting medium using Newton's methods.

He also continued to produce good work on the theory of oscillations and in a paper he gave a beautiful account of the oscillation of air in organ pipes. His strengths and weaknesses are summed up by Straub in [1]:-

Bernoulli's active and imaginative mind dealt with the most varied scientific areas. Such wide interests, however, often prevented him from carrying some of his projects to completion. It is especially unfortunate that he could not follow the rapid growth of mathematics that began with the introduction of partial differential equations into mathematical physics. Nevertheless he assured himself a permanent place in the history of science through his work and discoveries in hydrodynamics, his anticipation of the kinetic theory of gases, a novel method for calculating the value of an increase in assets, and the demonstration that the most common movement of a string in a musical instrument is composed of the superposition of an infinite number of harmonic vibrations...

Daniel Bernoulli was much honoured in his own lifetime. He was elected to most of the leading scientific societies of his day including those in Bologna, St Petersburg, Berlin, Paris, London, Bern, Turin, Zurich and Mannheim.

Baudhayana

To write a biography of Baudhayana is essentially impossible since nothing is known of him except that he was the author of one of the earliest Sulbasutras. We do not know his dates accurately enough to even guess at a life span for him, which is why we have given the same approximate birth year as death year.

He was neither a mathematician in the sense that we would understand it today, nor a scribe who simply copied manuscripts like Ahmes. He would certainly have been a man of very considerable learning but probably not interested in mathematics for its own sake, merely interested in using it for religious purposes. Undoubtedly he wrote the Sulbasutra to provide rules for religious rites and it would appear an almost certainty that Baudhayana himself would be a Vedic priest.

The mathematics given in the Sulbasutras is there to enable the accurate construction of altars needed for sacrifices. It is clear from the writing that Baudhayana, as well as being a priest, must have been a skilled craftsman. He must have been himself skilled in the practical use of the mathematics he described as a craftsman who himself constructed sacrificial altars of the highest quality.

The Sulbasutras are discussed in detail in the article Indian Sulbasutras. Below we give one or two details of Baudhayana's Sulbasutra, which contained three chapters, which is the oldest which we possess and, it would be fair to say, one of the two most important.

The Sulbasutra of Baudhayana contains geometric solutions (but not algebraic ones) of a linear equation in a single unknown. Quadratic equations of the forms ax2 = c and ax2 + bx = c appear.

Several values of π occur in Baudhayana's Sulbasutra since when giving different constructions Baudhayana uses different approximations for constructing circular shapes. Constructions are given which are equivalent to taking π equal to 676/225 (where 676/225 = 3.004), 900/289 (where 900/289 = 3.114) and to 1156/361 (where 1156/361 = 3.202). None of these is particularly accurate but, in the context of constructing altars they would not lead to noticeable errors.

An interesting, and quite accurate, approximate value for √2 is given in Chapter 1 verse 61 of Baudhayana's Sulbasutra. The Sanskrit text gives in words what we would write in symbols as

√2 = 1 + 1/3 + 1/(34) - 1/(3434)= 577/408

which is, to nine places, 1.414215686. This gives √2 correct to five decimal places. This is surprising since, as we mentioned above, great mathematical accuracy did not seem necessary for the building work described. If the approximation was given as

√2 = 1 + 1/3 + 1/(34)

then the error is of the order of 0.002 which is still more accurate than any of the values of π. Why then did Baudhayana feel that he had to go for a better approximation?

Johann Jakob Balmer

Born: 1 May 1825 in Lausen, Basel-Land, Switzerland
Died: 12 March 1898 in Basel, Switzerland



Johann Balmer's father was also named Johann Jakob Balmer and he was a Chief Justice. Johann's mother was Elizabeth Rolle Balmer. Johann was the eldest of his parents sons. He attended his first school in Liestal, a town which had just become the capital of the half canton of Basel-Landschaft. Then, for his secondary education, he studied at a school in Basel where he excelled in mathematics and decided to study that topic at university.

For his university studies in mathematics Balmer attended the University of Karlsruhe and the University of Berlin. His course of studies led to a doctorate which he received from the University of Basel in 1849 for a dissertation on the cycloid.
Balmer taught in Basel all his life. From 1859 until his death in 1898 he was a school teacher of mathematics at a secondary school for girls in the city. From 1865 until 1890 he was also a university lecturer in mathematics at the University of Basel where his main field of interest was geometry. He married Christine Pauline Rinck in 1868 when he was 43 years old and they had six children.

However, despite being a mathematics teacher and lecturer all his life, Balmer is best remembered for his work on spectral series and his formula, given in 1885, for the wavelengths of the spectral lines of the hydrogen atom. This was set out in one of only two papers which he wrote on spectra of the elements, the second being in 1897. It is surprising to realise that Balmer was sixty years old when he wrote the paper for which he is famous on the spectral lines of the hydrogen atom and that he was seventy-two when he wrote his only other work on this topic. But for his work on what amounts to a problem in physics, Balmer would be unknown today within the history of mathematics since he made no contribution to geometry of special significance despite it being the topic of interest throughout his life.

The major contribution which Balmer made, however, depended much more on his mathematical skills than on his understanding of physics, for he produced a formula which gave the wavelengths of the observed lines produced by the hydrogen atom without giving any physical explanation. Balmer's famous formula is

= hm2/(m2 - n2).

Putting n = 2 and h = 3654.6 10-8 cm, the wavelengths given by the formula for m = 3, 4, 5, 6 were correct to a high degree of accuracy. Previous attempts had looked for formulae of quite different types and had failed to come up with anything which matched the experimental evidence. Putting m = 7 gave Balmer a predicted value for the next line and indeed a colleague at the University of Basel was able to tell Balmer that this line had been observed and the wavelength agreed with a high level of accuracy with the one Balmer's formula predicted.

In his paper of 1885 Balmer suggested that giving n other small integer values would give the wavelengths of other series produced by the hydrogen atom. Indeed this prediction turned out to be correct and these series of lines were later observed. The reason why the formula holds was not understood in Balmer's lifetime and had to wait until the theoretical work of Niels Bohr in 1913.

Balmer's formula led to more general formulae for the spectral lines of other atoms. Others who, basing their ideas on those of Balmer, were able to achieve such results included Rydberg, Kayser and Runge.

Charles Babbage

Born: 26 Dec 1791 in London, England
Died: 18 Oct 1871 in London, England



Both the date and place of Charles Babbage's birth were uncertain but have now been firmly established. In  and , for example, his date of birth is given as 26 December 1792 and both give the place of his birth as near Teignmouth. Also in it is stated:-

Little is known of Mr Babbage's parentage and early youth except that he was born on 26 December 1792.

However, a nephew wrote to The Times a week after the obituary  appeared, saying that Babbage was born on 26 December 1791. There was little evidence to prove which was right until Hyman  in 1975 found that Babbage's birth had been registered in St Mary's Newington, London on 6 January 1792. Babbage's father was Benjamin Babbage, a banker, and his mother was Betsy Plumleigh Babbage. Given the place that his birth was registered Hyman says in  that it is almost certain that Babbage was born in the family home of 44 Crosby Row, Walworth Road, London.

Babbage suffered ill health as a child, as he relates in :-

Having suffered in health at the age of five years, and again at that of ten by violent fevers, from which I was with difficulty saved, I was sent into Devonshire and placed under the care of a clergyman (who kept a school at Alphington, near Exeter), with instructions to attend to my health; but, not to press too much knowledge upon me: a mission which he faithfully accomplished.

Since his father was fairly wealthy, he could afford to have Babbage educated at private schools. After the school at Alphington he was sent to an academy at Forty Hill, Enfield, Middlesex where his education properly began. He began to show a passion for mathematics but a dislike for the classics. On leaving the academy, he continued to study at home, having an Oxford tutor to bring him up to university level. Babbage in lists the mathematics books he studied in this period with the tutor:-

Amongst these were Humphry Ditton's 'Fluxions', of which I could make nothing; Madame Agnesi's 'Analytical Instructions' from which I acquired some knowledge; Woodhouse's 'Principles of Analytic Calculation', from which I learned the notation of Leibniz; and Lagrange's 'Théorie des Fonctions'. I possessed also the 'Fluxions' of Maclaurin and of Simson.

Babbage entered Trinity College, Cambridge in October 1810. However the grounding he had acquired from the books he had studied made him dissatisfied with the teaching at Cambridge. He wrote :-


Thus it happened that when I went to Cambridge I could work out such questions as the very moderate amount of mathematics which I then possessed admitted, with equal facility, in the dots of Newton, the d's of Leibniz, or the dashes of Lagrange. I thus acquired a distaste for the routine of the studies of the place, and devoured the papers of Euler and other mathematicians scattered through innumerable volumes of the academies of St Petersburg, Berlin, and Paris, which the libraries I had recourse to contained.

Under these circumstances it was not surprising that I should perceive and be penetrated with the superior power of the notation of Leibniz.

It is a little difficult to understand how Woodhouse's Principles of Analytic Calculation was such an excellent book from which to learn the methods of Leibniz, yet Woodhouse was teaching Newton's calculus at Cambridge without any reference to Leibniz's methods. Woodhouse was one of Babbage's teachers at Cambridge yet he seems to have taken no part in the Society that Babbage was to set up to try to bring the modern continental mathematics to Cambridge.

Babbage tried to buy Lacroix's book on the differential and integral calculus but this did not prove easy in this period of war with Napoleon. When he did find a copy of the work he had to pay seven guineas for it - an incredible amount of money in those days. Babbage then thought of setting up a Society to translate the work :-

I then drew up the sketch of a society to be instituted for translating the small work of Lacroix on the Differential and Integral Calculus. It proposed that we should have periodical meetings for the propagation of d's; and consigned to perdition all who supported the heresy of dots. It maintained that the work of Lacroix was so perfect that any comment was unnecessary.

Babbage talked with his friend Edward Bromhead (who would become George Green's friend some years later- see the article on Green) who encouraged him to set up his Society. The Analytical Society was set up in 1812 and its members were all Cambridge undergraduates. Nine mathematicians attended the first meeting but the two most prominent members, in addition to Babbage, were John Herschel and George Peacock.

Babbage and Herschel produced the first of the publications of the Analytical Society when they published Memoirs of the Analytical Society in 1813. This is a remarkably deep work when one realises that it was written by two undergraduates. They gave a history of the calculus, and of the Newton, Leibniz controversy they wrote:-

It is a lamentable consideration, that that discovery which has most of any done honour to the genius of man, should nevertheless bring with it a train of reflections so little to the credit of his heart.

Two further publications of the Analytical Society were the joint work of Babbage, Herschel and Peacock. These are the English translation of Lacroix's Sur le calcul différentiel et intégral published in 1816 and a book of examples on the calculus which they published in 1820.

Babbage had moved from Trinity College to Peterhouse and it was from that College that he graduated with a B.A. in 1814. However, Babbage realised that Herschel was a much more powerful mathematician than he was so :-

He did not compete for honours, believing Herschel sure of first place and not caring to come out second.

Indeed Herschel was first Wrangler, Peacock coming second. Babbage married in 1814, then left Cambridge in 1815 to live in London. He wrote two major papers on functional equations in 1815 and 1816. Also in 1816, at the early age of 24, he was elected a fellow of the Royal Society of London. He wrote papers on several different mathematical topics over the next few years but none are particularly important and some, such as his work on infinite series, are clearly incorrect.

Babbage was unhappy with the way that the learned societies of that time were run. Although elected to the Royal Society, he was unhappy with it. He was to write of his feelings on how the Royal Society was run:-

The Council of the Royal Society is a collection of men who elect each other to office and then dine together at the expense of this society to praise each other over wine and give each other medals.

However in 1820 he was elected a fellow of the Royal Society of Edinburgh, and in the same year he was a major influence in founding the Royal Astronomical Society. He served as secretary to the Royal Astronomical Society for the first four years of its existence and later he served as vice-president of the Society.

Babbage, together with Herschel, conducted some experiments on magnetism in 1825, developing methods introduced by Arago. In 1827 Babbage became Lucasian Professor of Mathematics at Cambridge, a position he held for 12 years although he never taught. The reason why he held this prestigious post yet failed to carry out the duties one would have expected of the holder, was that by this time he had become engrossed in what was to became the main passion of his life, namely the development of mechanical computers.

Babbage is without doubt the originator of the concepts behind the present day computer. The computation of logarithms had made him aware of the inaccuracy of human calculation around 1812. He wrote in :-

... I was sitting in the rooms of the Analytical Society, at Cambridge, my head leaning forward on the table in a kind of dreamy mood, with a table of logarithms lying open before me. Another member, coming into the room, and seeing me half asleep, called out, Well, Babbage, what are you dreaming about?" to which I replied "I am thinking that all these tables" (pointing to the logarithms) "might be calculated by machinery."

Certainly Babbage did not follow up this idea at that time but in 1819, when his interests were turning towards astronomical instruments, his ideas became more precise and he formulated a plan to construct tables using the method of differences by mechanical means. Such a machine would be able to carry out complex operations using only the mechanism for addition. Babbage began to construct a small difference engine in 1819 and had completed it by 1822. He announced his invention in a paper Note on the application of machinery to the computation of astronomical and mathematical tables read to the Royal Astronomical Society on 14 June 1822.

Although Babbage envisaged a machine capable of printing out the results it obtained, this was not done by the time the paper was written. An assistant had to write down the results obtained. Babbage illustrated what his small engine was capable of doing by calculating successive terms of the sequence n2 + n + 41.

The terms of this sequence are 41, 43, 47, 53, 61, ... while the differences of the terms are 2, 4, 6, 8, .. and the second differences are 2, 2, 2, ..... The difference engine is given the initial data 2, 0, 41; it constructs the next row 2, (0 + 2), [41 + (0 + 2)], that is 2, 2, 43; then the row 2, (2 + 2), [43 + (2 + 2)], that is 2, 4, 47; then 2, 6, 53; then 2, 8, 61; ... Babbage reports that his small difference engine was capable of producing the members of the sequence n2 + n + 41 at the rate of about 60 every 5 minutes.

Babbage was clearly strongly influenced by de Prony's major undertaking for the French Government of producing logarithmic and trigonometric tables with teams of people to carry out the calculations. He argued that a large difference engine could do the work undertaken by teams of people saving cost and being totally accurate.

On 13 July 1823 Babbage received a gold medal from the Astronomical Society for his development of the difference engine. He then met the Chancellor of the Exchequer to seek public funds for the construction of a large difference engine. The Royal Society had already given positive advice to the government:-

Mr Babbage has displayed great talent and ingenuity in the construction of his machine for computation, which the committee thanks fully adequate to the attainment of the objects proposed by the inventory; and they consider Mr Babbage as highly deserving of public encouragement, in the prosecution of his arduous undertaking.

His initial grant was for 1500 and he began work on a large difference engine which he believed he could complete in three years. He set out to produce an engine with :-

... six orders of differences, each of twenty places of figures, whilst the first three columns would each have had half a dozen additional figures.

Such an engine would easily have been able to compute all the tables that de Prony had been calculating, and it was intended to have a printer to print out the results automatically. However the construction proceeded slower than had been expected. By 1827 the expenses were getting out of hand.

The year 1827 was a year of tragedy for Babbage; his father, his wife and two of his children all died that year. He own health gave way and he was advised to travel on the Continent. After his travels he returned near the end of 1828. Further attempts to obtain government support eventually resulted in the Duke of Wellington, the Chancellor of the Exchequer and other members of the government visiting Babbage and inspecting the work for themselves. By February 1830 the government had paid, or promised to pay, 9000 towards the project.

In 1830 Babbage published Reflections on the Decline of Science in England, a controversial work that resulted in the formation, one year later, of the British Association for the Advancement of Science. In 1834 Babbage published his most influential work On the Economy of Machinery and Manufactures, in which he proposed an early form of what today we call operational research.

The year 1834 was the one in which work stopped on the difference engine. By that time the government had put 17000 into the project and Babbage had put 6000 of his own money. For eight years from 1834 to 1842 the government would make no decision as to whether to continue support. In 1842 the decision not to proceed was taken by Robert Peel's government. Dubbey in writes:-

Babbage had every reason to feel aggrieved about his treatment by successive governments. They had failed to understand the immense possibilities of his work, ignored the advice of the most reputable scientists and engineers, procrastinated for eight years before reaching a decision about the difference engine, misunderstood his motives and the sacrifices he had made, and ... failed to protect him from public slander and ridicule.

By 1834 Babbage had completed the first drawings of the analytical engine, the forerunner of the modern electronic computer. His work on the difference engine had led him to a much more sophisticated idea. Although the analytic engine never progressed beyond detailed drawings, it is remarkably similar in logical components to a present day computer. Babbage describes five logical components, the store, the mill, the control, the input and the output. The store contains:-

... all the variables to be operated upon, as well as all those quantities which had arisen from the results of other operations.

The mill is the analogue of the cpu in a modern computer and it is the place :-

... into which the quantities about to be operated upon are always bought.

The control on the sequence of operations to be carried out was by a Jacquard loom type device. It was operated by punched cards and the punched cards contained the program for the particular task :-

Every set of cards made for any formula will at any future time recalculate the formula with whatever constants may be required.

Thus the Analytical Engine will possess a library of its own. Every set of cards once made will at any time reproduce the calculations for which it was first arranged.

The store was to hold 1000 numbers each of 50 digits, but Babbage designed the analytic engine to effectively have infinite storage. This was done by outputting data to punched cards which could be read in again at a later stage when needed. Babbage decided, however, not to seek government support after his experiences with the difference engine.

Babbage visited Turin in 1840 and discussed his ideas with mathematicians there including Menabrea. During Babbage's visit, Menabrea collected all the material needed to describe the analytical engine and he published this in October 1842. Lady Ada Lovelace translated Menabrea's article into English and added notes considerably more extensive than the original memoir. This was published in 1843 and included :-

... elaborations on the points made by Menabrea, together with some complicated programs of her own, the most complex of these being one to calculate the sequence of Bernoulli numbers.

Although Babbage never built an operational, mechanical computer, his design concepts have been proved correct and recently such a computer has been built following Babbage's own design criteria. He wrote in 1851 :-

The drawings of the Analytical Engine have been made entirely at my own cost: I instituted a long series of experiments for the purpose of reducing the expense of its construction to limits which might be within the means I could myself afford to supply. I am now resigned to the necessity of abstaining from its construction...

Despite this last statement, Babbage never did quite give up hope that the analytical engine would be built writing in 1864 in :-

... if I survive some few years longer, the Analytical Engine will exist...

After Babbage's death a committee,whose members included Cayley and Clifford, was appointed by the British Association :-

... to report upon the feasibility of the design, recorded their opinion that its successful realisation might mark an epoch in the history of computation equally memorable with that of the introduction of logarithms...

This was an underestimate. The construction of modern computers, logically similar to Babbage's design, have changed the whole of mathematics and it is even not an exaggeration to say that they have changed the whole world.

Abraham bar Hiyya Ha-Nasi

Abraham bar Hiyya was a Spanish Jewish mathematician and astronomer. In the Hebrew of his time 'Ha-Nasi' meant 'the leader' but he is also known by the Latin name Savasorda which comes from his 'job description' showing that he held an official position in the administration in Barcelona.

Abraham bar Hiyya is famed for his book Hibbur ha-Meshihah ve-ha-Tishboret (Treatise on Measurement and Calculation), translated into Latin by Plato of Tivoli as Liber embadorum in 1145. This book is the earliest Arab algebra written in Europe. It contains the complete solution of the general quadratic and is the first text in Europe to give such a solution. Rather strangely, however, 1145 was also the year that al-Khwarizmi's algebra book was translated by Robert of Chester so Abraham bar Hiyya's work was rapidly joined by a second text giving the complete solution to the general quadratic equation.

It is interesting to see the areas of mathematics and the mathematicians with which Abraham was familiar. Of course he knew geometry through the works of Euclid, but he also knew the contributions to geometry from other Greek texts such as Theodosius's Sphaerics in three books, On the Moving Sphere which is a work on the geometry of the sphere by Autolycus, Apollonius's Conics, and the later contributions by Heron of Alexandria and Menelaus of Alexandria. Abraham had also studied some of the important works on algebra by Arab mathematicians, in particular al-Khwarizmi and al-Karaji.

Among other texts written by Abraham bar Hiyya was Yesod ha-Tebunah u-Migdal ha-Emunah (The Foundation of Understanding and the Tower of Faith). This work is an encyclopaedia of mathematics, astronomy, optics and music. It is the first encyclopaedia in the Hebrew language.

Abraham also wrote a number of texts on astronomy; in particular he wrote on the form of the Earth and the calculation of the paths of the stars on the celestial sphere. His book Tables of the Prince refers to the tables of al-Battani while Abraham's treatise Sefer ha-Ibbur (Book of Intercalation), written in 1122-23, is the first Hebrew work devoted exclusively to a study of the calendar.

In the philosophical treatise Hegyon ha-Nefesh ha-Azuva (Meditation of the Sad Soul) Abraham deals with the nature of good and evil and ethics. Megillat ha-Megalleh (Scroll of the Revealer) outlines Abraham's view of history based on astrology. It claims to forecast the messianic future.

Perhaps one of the most important features of Abraham bar Hiyya's work is the fact that it appears to have stimulated an interest in Arabic mathematics and, together with the work of Abraham ibn Ezra, marks the beginning of Hebrew scholarly study of mathematics. As the author of [5] writes:-

The major part of the mathematical 'classics' in Hebrew were translated from Arabic between the second third of the thirteenth century and the first third of the fourteenth century, within the northern littoral of the western Mediterranean. This movement occurred after the original works by Abraham bar Hiyya and Abraham ibn Ezra became available to a wide readership.

It is rather difficult to place Abraham bar Hiyya in the development of mathematics since in most respects he did not fit nicely into one culture but spanned several. It may indeed be for just that reason that he is important since he produced a cross-fertilisation of ideas between these cultures. As Levey (the author of [6]) writes in [1], Abraham:-

... did not definitely belong definitely to one mathematical group. He spent most of his life in Barcelona, an area of both Arab and Christian learning, and was active in translating the masterpieces of Arab science. ... he deplored the lack of knowledge of Arab science and language among the people of Provence. He wrote his own works in Hebrew, but he helped translate ... works into Latin....

Edwin Abbott Abbott

Born: 20 Dec 1838 in Marylebone, Middlesex, England
Died: 12 Oct 1926 in Hampstead, London, England


Edwin Abbott Abbott's parents were Jane Abbott and Edwin Abbott. His mother Jane was a first cousin of his father, so both had the name of Abbott which explains Edwin Abbott Abbott having 'Abbott' as both a surname and a middle name. Edwin Abbott was headmaster of the Philological School at Marylebone.

Abbott was educated at the City of London School which had gained a fine reputation under Dr G F W Mortimer who was headmaster throughout the years during which he studied there. Following a fine school education, Abbott entered St John's College, Cambridge, in 1857. After an outstanding academic career as an undergraduate he was the Senior Classics medallist in 1861 and was elected to a fellowship at his college in the following year. In the same year he was ordained a deacon and in 1863 he became a priest. At this time College fellows were not allowed to marry so, when Abbott wished to marry Mary Elizabeth Rangeley from Unstone, Derbyshire, in 1863, he had to resign the fellowship. Edwin and Mary had one son and one daughter.

After leaving Cambridge, Abbott taught at King Edward's School, Birmingham, and then at Clifton College. In 1865 he was appointed as headmaster of the City of London School on the retirement of his former headmaster Dr Mortimer. It was a post which Abbott held for 24 years until he retired in 1889. Of course Abbott was relatively young when he retired being only 50 years old. He did not retire to give up work, rather he enjoyed writing and retired so that he could devote more time to his literary efforts. Before looking at some of the books which he published, we should first say a little about his fine qualities as a teacher and as a headmaster which saw the already excellent City of London School reach even higher standards during his years in charge:-

His greatness as an educator derived partly from his organization of new methods of instruction, partly from his initiation of many innovations in the school curriculum, and partly from what can only be called his genius for teaching. Having a reverence for physical science not often found among the classical scholars of his day, he made an elementary knowledge of chemistry compulsory throughout the upper school.

He made many innovations to the curriculum taught at the school in addition to the sciences referred to in the above quote, and he transmitted his own enthusiasm for literature, both English literature and classical literature, to pupils at the school.

As a scholar, Abbott was very broad writing excellent works on a wide variety of topics. He published Shakespearean Grammar (1870), English Lessons for English People (1871) and How to Write Clearly (1872). He was a leading expert on Francis Bacon, published Bacon and Essex (1877) and wrote an introduction to Bacon's Essays (1886). Some of his works on textual criticism contain excellent statistical analyses, for example Johannine Vocabulary (1905) and Johannine Grammar (1906). Among numerous religious writings we mention Philochristus (1878), Onesimus: Memoirs of a Disciple of Paul (1882), and Silanus the Christian (1906).

Of course we have not yet mentioned his most famous work, and certainly the one which merits his inclusion in this archive. This was Flatland: a romance of many dimensions (1884) which Abbott wrote under the pseudonym of A Square. The book has seen many editions, the sixth edition of 1953 being reprinted by Princeton University Press in 1991 with an introduction by Thomas Banchoff. Flatland is an account of the adventures of A Square in Lineland and Spaceland. In it Abbott tries to popularise the notion of multidimensional geometry but the book is also a clever satire on the social, moral, and religious values of the period. Here is Abbott's introduction to Flatland:-

I call our world Flatland, not because we call it so, but to make its nature clearer to you, my happy readers, who are privileged to live in Space. Imagine a vast sheet of paper on which straight Lines, Triangles, Squares, Pentagons, Hexagons, and other figures, instead of remaining fixed in their places, move freely about, on or in the surface, but without the power of rising above or sinking below it, very much like shadows - only hard with luminous edges - and you will then have a pretty correct notion of my country and countrymen. Alas, a few years ago, I should have said "my universe": but now my mind has been opened to higher views of things.

In such a country, you will perceive at once that it is impossible that there should be anything of what you call a "solid" kind; but I dare say you will suppose that we could at least distinguish by sight the Triangles, Squares, and other figures, moving about as I have described them. On the contrary, we could see nothing of the kind, not at least so as to distinguish one figure from another. Nothing was visible, nor could be visible, to us, except Straight Lines; and the necessity of this I will speedily demonstrate. Place a penny on the middle of one of your tables in Space; and leaning over it, look down upon it. It will appear a circle. But now, drawing back to the edge of the table, gradually lower your eye (thus bringing yourself more and more into the condition of the inhabitants of Flatland), and you will find the penny becoming more and more oval to your view, and at last when you have placed your eye exactly on the edge of the table (so that you are, as it were, actually a Flatlander) the penny will then have ceased to appear oval at all, and will have become, so far as you can see, a straight line.

In Abbott's Flatland, the more sides you have then the higher is your class. Workers are equilateral triangles, the author himself is A Square, a person of middle class, while the highest classes are the circles who are priests. The Flatland world is visited by a sphere which A Square sees at first as a dot which grows into a disk, then shrinks again to a dot and vanishes. The sphere opens his eyes to the possibility of a third dimension, and he suggests to the sphere that he might live in a world with four or more dimensions, but the sphere makes fun of this suggestion. When A Square tells his fellow Flatlanders about the third dimension they ridicule him, and eventually he is put in prision where he writes the book.

It is worth noting that this remarkable piece of writing by Abbott predated by many years Einstein's four dimensional world of relativity. Abbott wrote a Preface which contains the following:-

It is true that we have really in Flatland a Third unrecognised Dimension called 'height,' just as it also is true that you have really in Spaceland a Fourth unrecognised Dimension, called by no name at present, but which I will call 'extra-height.' But we can no more take cognisance of our 'height' than you can of your 'extra-height.' ...

Suppose a person of the Fourth Dimension, condescending to visit you, were to say, 'Whenever you open your eyes, you see a Plane (which is of Two Dimensions) and you infer a Solid (which is of Three); but in reality you also see (though you do not recognise) a Fourth Dimension, which is not colour nor brightness nor anything of the kind, but a true Dimension, although I cannot point out to you its direction, nor can you possibly measure it.' What would you say to such a visitor? Would not you have him locked up? Well, that is my fate: and it is as natural for us Flatlanders to lock up a Square for preaching the Third Dimension, as it is for you Spacelanders to lock up a Cube for preaching the Fourth. Alas, how strong a family likeness runs through blind and persecuting humanity in all Dimensions! Points, Lines, Squares, Cubes, Extra-Cubes - we are all liable to the same errors, all alike the Slavers of our respective Dimensional prejudices ...

More recently, in 2002, an annotated version of Flatland has been produced with an introduction and notes by Ian Stewart who gives extensive discussion of mathematical topics related to passages in Abbott's text.

Abbott is described in  as follows:-

In spite of a frail and delicate physique, Abbott could keep discipline without effort. He was an impressive preacher: in the pulpit he was a bold and original exponent of advanced broad church doctrines.

Abbott died of influenza at his home, Wellside, Well Walk in Hampstead, and was buried in Hampstead cemetery.

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Jagadish Chandra Bose

Physicist turned plant biologist Jagadish Chandra Bose was born on 30 November 1858 in Mymensingh, India (now in Bangladesh) to a well-to-do family. His father Bhagabanchandra Bose was a Deputy Magistrate. A distinguished student, he began attending St. Xavier’s College, Calcutta in 1875 and received the BA degree from Calcutta University in 1877. In 1880 the twenty-two-year old Bose left India for England. For a year he studied medicine at London University, England, but had to give it up because of his own ill health. Within a year he moved to Cambridge to take up a scholarship to study Natural Science at Christ's College Cambridge. One of his lecturers at Cambridge was Professor Rayleigh, who clearly had a profound influence on his later work. He graduated from there in 1884 with a Natural Science Tripos (a special course of study at Cambridge). That same year Bose also received the BS degree from London University. Just one year later Bose became a Professor of Physical Science at Presidency College of Calcutta, where for the next 30 years he taught and conducted research. As a teacher Bose was very popular and engaged the interest of his students by making extensive use of scientific demonstrations. Many of his students at the Presidency College were destined to become famous in their own right - for example S.N. Bose, later to become well known for the Bose-Einstein statistics.

In 1894, J.C. Bose converted a small enclosure adjoining a bathroom in the Presidency College into a laboratory. He carried out experiments involving refraction, diffraction and polarization. To receive the radiation, he used a variety of different junctions connected to a highly sensitive galvanometer. He plotted in detail the voltage-current characteristics of his junctions, noting their non-linear characteristics. He developed the use of galena crystals for making receivers, both for short wavelength radio waves and for white and ultraviolet light. Patent rights for their use in detecting electromagnetic radiation were granted to him in 1904. In 1954 Pearson and Brattain gave priority to Bose for the use of a semi-conducting crystal as a detector of radio waves. Sir Neville Mott, Nobel Laureate in 1977 for his own contributions to solid-state electronics, remarked that "J.C. Bose was at least 60 years ahead of his time" and "In fact, he had anticipated the existence of P-type and N-type semiconductors." In 1895 Bose gave his first public demonstration of electromagnetic waves, using them to ring a bell remotely and to explode some gunpowder. In 1896 the Daily Chronicle of England reported: "The inventor (J.C. Bose) has transmitted signals to a distance of nearly a mile and herein lies the first and obvious and exceedingly valuable application of this new theoretical marvel."

During the years 1894-1900, Bose performed pioneering research on radio waves and created waves as short as 5 mm. Bose’s work actually predates that of Guglielmo Marconi who is most often associated with the development of radio. Unlike Marconi who sought to commercialize his work with radio waves, Bose was purely interested in radio waves as a scientific endeavor. Bose also developed equipment for generating, transmitting, and receiving radio waves and used it to demonstrate conclusively the waves’ properties such as reflection, total reflection, refraction, double refraction, and polarization. Bose also experimented with galena to form an early type of semiconductor diode, which may be used as a detector of electromagnetic waves. Bose's demonstration of remote wireless signalling has priority over Marconi; he was the first to use a semiconductor junction to detect radio waves, and he invented various now commonplace microwave components. Outside of India he is rarely given the deserved recognition. Further work at millimeter wavelengths was almost nonexistent for nearly 50 years. J.C. Bose was at least this much ahead of his time. Research into the generation and detection of millimeter waves, and the properties of substances at these wavelengths, was being undertaken in some detail one hundred years ago, by J.C. Bose in Calcutta.


After about 1900, Bose began pursuing another longtime interest—animal and plant physiology. This included studies of the effects of electromagnetic radiation on plants, a topical field today. His contributions to this field were pioneering. He introduced many delicate and sensitive instruments, such as the Chrestograph, which was used for recording plant growth. It could magnify a small movement as much as a million times. Another device he developed demonstrated the effects of electromagnetic waves on living and nonliving matter.

Bose retired in 1915 and was appointed Emeritus Professor, Presidency College, Calcutta, for a period of 5 years. In 1917 he founded the Bose Research Institute in Calcutta which was the first scientific research institute in India. That same year a knighthood was conferred on Bose. In 1920 he became the first Indian scientist to be elected to Great Britain’s prestigious Royal Society.

Bose traveled frequently to Europe and the United States on various scientific missions and gave lectures on electromagnetic waves, the effects of electromagnetic waves on living and nonliving matter, and plant physiology. On a personal level, Bose believed in the free exchange of scientific knowledge and strongly believed that knowledge grows by sharing it with fellow scientists. Bose died on 23 November 1937 at the age of 78.