biografías de matemáticos.docx

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George Francis FitzGerald

Born: 3 Aug 1851 in Kill-o'-the Grange, Monkstown, Co. Dublin, IrelandDied: 21 Feb 1901 in Dublin, Ireland

George FitzGerald was a brilliant mathematical physicist who today is known by mostscientists as one of the proposers of the FitzGerald-Lorentz contraction in the theory ofrelativity. However, this suggestion by FitzGerald, as we shall see below, was not in thearea in which he undertook most of his research, and he would certainly not have ratedthis his greatest contribution.

George FitzGerald's parents were William FitzGerald and Anne Frances Stoney. Hisfather William was a minister in the Irish Protestant Church and rector of St Ann'sDublin at the time of George's birth. William, although having no scientific interestshimself, was an intellectual who went on to become Bishop of Cork and later Bishop ofKillaloe. It seems that George's later interest in metaphysics came from his father's sideof the family. George's mother was the daughter of George Stoney from Birr in King'sCounty and she was also from an intellectual family. George Johnstone Stoney, whowas Anne's brother, was elected a Fellow of the Royal Society of London and GeorgeFitzGerald's liking for mathematics and physics seems to have come mainly from hismother's side of the family.

William and Anne had three sons, George being the middle of the three. MauriceFitzGerald, one of George's two brothers, also went on to achieve academic success inthe sciences, becoming Professor of Engineering at Queen's College Belfast. George'sschooling was at home where, together with his brothers and sisters, he was tutored byM A Boole, who was George Boole's sister. It is doubtful whether Miss Boole realisedwhat enormous potential her pupil George had, for although he showed himself to be anexcellent student of arithmetic and algebra, he was no better than an average pupil atlanguages and had rather a poor verbal memory. However, when the tutoring progressedto a study of Euclid'sElements then George showed himself very able indeed, and healso exhibited a great inventiveness for mechanical constructions, having great

dexterity. He was also an athletic boy yet he had no great liking for games.

Miss Boole prepared her pupils very well for their university studies. She noticed oneremarkable talent in her pupil George, that was his skill as an observer. Many years laterFitzGerald, clearly thinking of his own youth, wrote:-

The cultivation and training of the practical ability to do things and to learn fromobservation, experiment and measurement, is a part of education which the clergymanand the lawyer can maybe neglect, because they have to deal with emotions and words,but which the doctor and the engineer can only neglect at their own peril and that ofthose who employ them. These habits should be carefully cultivated from the earliest

years while a child's character is being developed. As the twig is bent so the treeinclines.


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FitzGerald certainly showed that he had acquired the ability to learn from observation,experiment and measurement. He entered Trinity College Dublin at the young age of 16to study his two best subjects which were mathematics and experimental science, and hewas soon putting the training he had received at home to good use. At Trinity College,FitzGerald [8]:-

... attained all the distinctions that lay in his path with an ease, and wore them with agrace, that endeared him to his rivals and contemporaries.

It was not an undergraduate career devoted entirely to study, however, for FitzGeraldplayed a full part in literary clubs and social clubs. He also continued his athleticinterests, taking to gymnastics and to racquet sports. In 1871 he graduated as the beststudent in both mathematics and experimental science. He won a University Studentshipand two First Senior Moderatorships in his chosen topics.

The aim of FitzGerald was now to win a Trinity College Fellowship but at this time

these were few and far between. He was to spend six years studying before he obtainedthe Fellowship he wanted, but during these years he laid the foundation of his researchcareer. He studied the works of Lagrange, Laplace, Franz Neumann, and those of hisown countrymen Hamilton and MacCullagh. In addition he absorbed the theories putforward by Cauchy and Green. Then, in 1873, a publication appeared which would playa major role in his future. This wasElectricity and Magnetism by Maxwell which, forthe first time, contained the four partial differential equations, now known as Maxwell'sequations. FitzGerald immediately saw Maxwell's work as providing the framework forfurther development and he began to work on pushing forward the theory.

It is worth noting that FitzGerald's reaction to Maxwell's fundamental paper was notthat of most scientists. Very few seemed to see the theory as a starting point, rather mostsaw it only as a means to produce Maxwell's own results. It is a tribute to FitzGerald'sinsight as a scientist that he saw clearly from the beginning the importance ofElectricityand Magnetism. Maxwell's theory was for many years, in the words of Heaviside,"considerably underdeveloped and little understood" but a few others were to see it inthe same light as FitzGerald including Heaviside, Hertz and Lorentz. FitzGerald wouldexchange ideas over the following years with all three of these scientists.

During the six years he spent working for the Fellowship, FitzGerald also studiedmetaphysics, a topic which he had not formally studied as an undergraduate, and he was

particularly attracted to Berkeley's philosophy. His liking for metaphysics and his deepunderstanding of the topic combined with his other great talents in his future career. Hewon his Fellowship and became a tutor at Trinity College Dublin in 1877. This was nothis first attempt at winning a Fellowship, rather it was his second since he failed to wina Fellowship at his first attempt. At Trinity College he was attached to the Departmentof Experimental Physics and soon he was exerting the greatest influence on the teachingof the physical sciences in the College.

In 1881 John R Leslie, the professor of natural philosophy at Dublin, died andFitzGerald succeeded him to the Erasmus Smith Chair of Natural and ExperimentalPhilosophy. At the time of his appointment he gave up his duties as College tutor, a role

in which he had been extremely successful, to concentrate on his duties as a professor.One of FitzGerald's long running battles at Trinity College Dublin was to increase the


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amount of teaching of experimental physics. He soon set up classes in an old chemicallaboratory that he was able to obtain for his use, and he gathered round him colleagueswho would help in the practical aspects of the subject. As is so often the case inuniversities, however, he was restricted in the progress he could make from a lack offunds.

In a lecture which he gave to the Irish Industrial League in 1896 FitzGerald emphasisedhis lifelong belief in practical studies:-

The fault of our present system is in supposing that learning to use words teaches us touse things. This is at its best. It really does not even teach children to use words, it onlyteaches them to learn words, to stuff their memories with phrases, to be a pack of

parrots, to suffocate thought with indigestible verbiage. Take the case of experimenting.How can you teach children to make careful experiments with words? Yet it is greatimportance that they should be able to learn from experiments.

However, practical applications are built on theoretical foundations and FitzGerald fullyunderstood this. In his inaugural lecture on 22 February 1900 as President of the DublinSection of the Institution of Electrical Engineers, he spoke of how electricity had beenapplied to the benefit of mankind during the nineteenth century. Behind a practicalinvention such as telegraphy there was a wealth of theoretical work:-

... telegraphy owes a great deal to Euclid and other pure geometers, to the Greek andArabian mathematicians who invented our scale of numeration and algebra, to Galileoand Newton who founded dynamics, to Newton and Leibniz who invented the calculus,to Volta who discovered the galvanic coil, to Oersted who discovered the magneticactions of currents, to Ampre who found out the laws of their action, to Ohm whodiscovered the law of resistance of wires, to Wheatstone, to Faraday, to Lord Kelvin, toClerk Maxwell, to Hertz. Without the discoveries, inventions, and theories of theseabstract scientific men telegraphy, as it now is, would be impossible.

We should also look at FitzGerald's idea of the purpose of a university since it was, likehis other educational beliefs, the driving force in how he carried out his professorialduties. He believed that the primary purpose of a university was not to teach the fewstudents who attended but, through research, to teach everyone. He wrote in 1892:-

The function of the University is primarily to teach mankind. .. at all times the greatest

men have always held that their primary duty was the discovery of new knowledge, thecreation of new ideas for all mankind, and not the instruction of the few who found itconvenient to reside in their immediate neighbourhood. ... Are the Universities to devotethe energies of the most advanced intellects of the age to the instruction of the wholenation, or to the instruction of the few whose parents can afford them an - in some

places fancy - education that can in the nature of things be only attainable by the rich?

As can be seen from the quotations we have given from FitzGerald's writing, his interestin education went well beyond the narrow confines of his own department. It was notmerely a theoretical interest for, true to his own beliefs, he took a very practical role ineducation. He was an examiner in physics at the University of London beginning in

1888 and he served as a Commissioner of National Education in Ireland in 1898 beingconcerned with reforming primary education in Ireland. As part of this task he travelled


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to the United States on a fact finding tour in the autumn of 1898. As one might haveexpected, his aim was to bring far more practical topics into the syllabus of primaryschools. At the time of his death he was involved in the reform of intermediateeducation in Ireland and he also served on the Board which was considering technicaleducation.

In 1883 FitzGerald married Harriette Mary Jellett. She was the daughter of the Rev J HJellett, the Provost of Trinity College and an outstanding scientist who had beenawarded the Royal Medal by the Royal Society. It was through his personal friendshipwith Jellett, and also their joint scientific studies, that FitzGerald got to know Harriette.Although the couple had been married just under eight years at the time of FitzGerald'sdeath, they had eight children during this time; three sons and five daughters. FitzGeraldwas elected a Fellow of the Royal Society in 1883 and, like his father-in-law, he was toreceive its Royal medal. This was in 1899 when the prestigious award was made toFitzGerald for his contributions to theoretical physics, especially to optics andelectrodynamics. Lord Lister, presenting the medal, said [3]:-

His critical activity pervades an unbounded field, enlivened and enriched throughout bythe fruits of a luxuriant imagination.

We should now examine the research for which FitzGerald received these honours.

Beginning in 1876, before he obtained his Fellowship, FitzGerald began to publish theresults of his research. His first work On the equations of equilibrium of an elasticsurface filled in cases of a problem studied by Lagrange. His second paper in the sameyear was on magnetism and he then, still in the year 1876, published On the rotation ofthe plane of polarisation of light by reflection from the pole of a magnetin theProceeding of the Royal Society. He had already begun to contribute to Maxwell'stheory and, as well as theoretical contributions, he was conducting experiments inelectromagnetic theory. His first major theoretical contribution was On theelectromagnetic theory of the reflection and refraction of lightwhich he sent to theRoyal Society in October 1878. Maxwell, in reviewing the paper, noted that FitzGeraldwas developing his ideas in much the same general direction as was Lorentz.

At a meeting of the British Association in Southport in 1883, FitzGerald gave a lecturediscussing electromagnetic theory. He suggested a method of producing electromagneticdisturbances of comparatively short wavelengths:-

... by utilising the alternating currents produced when an accumulator is dischargedthrough a small resistance. It would be possible to produce waves of as little as 10metres wavelength or less.

So FitzGerald, using his own studies of electrodynamics, suggested in 1883 that anoscillating electric current would produce electromagnetic waves. However, as he laterwrote:-

... I did not see any feasible way of detecting the induced resonance.

In 1888 FitzGerald addressed the Mathematical and Physical Section of the BritishAssociation in Bath as its President. He was able to report to British Association that


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Heinrich Hertz had, earlier that year, verified this experimentally. Hertz had verifiedthat the vibration, reflection and refraction of electromagnetic waves were the same asthose of light. In this brilliant lecture, given to a general audience, FitzGerald describedhow Hertz:-

... has observed the interference of electromagnetic waves quite analogous to those oflight.

After his appointment to the chair, FitzGerald had continued to produce manyinnovative ideas but no major theories. For example despite his ideas onelectromagnetic waves he had not followed through the research and the finalexperimental verification had been achieved by Hertz. The reason for this is perhapsbest understood with a quotation from a letter which FitzGerald sent to Heaviside on 4February 1889 (see for example [1]):-

I admire from a distance those who contain themselves till they worked to the bottom of

their results but as I am not in the very least sensitive to having made mistakes I rushout with all sorts of crude notions in hope that they may set others thinking and lead tosome advance.

Although FitzGerald is modestly talking down his contributions in this quotation, thecomment he made about himself is essentially correct. O J Lodge [9] gives a similar, butfairer, analysis of FitzGerald's work:-

... the leisure of long patient analysis was not his, nor did his genius altogether lie inthis direction: he was at his best when, under the stimulus of discussion, his mindteemed with brilliant suggestions, some of which he at once proceeded to test by roughquantitative calculation, for which he was an adept in discerning the necessary data.The power of grasping instantly all the bearings of a difficult problem was his to anextraordinary degree ...

Again Heaviside wrote (see for example [8]):-

He had, undoubtedly, the quickest and most original brain of anybody. That was a greatdistinction; but it was, I think, a misfortune as regards his scientific fame. He saw toomany openings. His brain was too fertile and inventive. I think it would have been better

for him if he had been a little stupid -- I mean not so quick and versatile, but more

plodding. He would have been better appreciated, save by a few.Finally we should examine the contribution for which FitzGerald is universally knowntoday. There had been many attempts to detect the motion of the Earth relative to theaether, a medium in space postulated to carry light waves. A A Michelson and E WMorley conducted an accurate experiment to compare the speed of light in the directionof the Earth's motion and the speed of light at right angles to the Earth's motion. Despitethe difference in relative motion to the aether, the velocity of light was found to be thesame. In 1889, two years after the Michelson-Morley experiment, FitzGerald suggestedthat the shrinking of a body due to motion at speeds close to that of light would accountfor the result of that experiment. Lodge [9] writes that the idea:-


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... flashed on him in the writer's study at Liverpool as he was discussing the meaning ofthe Michelson-Morley experiment.

Lorentz, independently in 1895, gave a much more detailed description of the samekind. It was typical of these two great men that both were more than ready to

acknowledge the contribution of the other, but there is little doubt that each had the ideaindependently of the other. The FitzGerald-Lorentz contraction now plays an importantrole in relativity.

Sadly FitzGerald died at the age of only 49 years. Maxwell, whose work had proved sofundamental for FitzGerald, had died at the age of 48 while Hertz died at the age of 36.In fact in 1896 FitzGerald had reviewed the publication of Hertz'sMiscellaneousPapers forNature after Hertz's death. Four years later, in September 1900, FitzGeraldbegan to complain of indigestion and began to have to be careful what he ate. A fewweeks later he complained that he was finding it difficult to concentrate on a problem.His health rapidly deteriorated and despite having an operation the end came quickly.

W Ramsay, on hearing of FitzGerald's death wrote (see [8]):-

... to me, as to many others, FitzGerald was the truest of true friends; always interested,always sympathetic, always encouraging, whether the matter discussed was a personalone, or one connected with science or with education. And yet I doubt if it were thesequalities alone which made his presence so attractive and so inspiring. I think it was the

feeling that one was able to converse on equal terms with a man who was so muchabove the level of one's self, not merely in intellectual qualities of mind, but in everyrespect. ... he had no trace of intellectual pride; he never put himself forward, and hadno desire for fame; he was content to do his duty. And he took this to be the task ofhelping others to do theirs.

FitzGerald was described by Lord Kelvin (William Thomson) as (see [10]):-

... living in an atmosphere of the highest scientific and intellectual quality, but always acomrade with every fellow-worker of however humble quality.... My scientific sympathyand alliance with him have greatly ripened during the last six or seven years over theundulatory theory of light and the aether theory of electricity and magnetism.

On his death the Faculty of Science of the University of London adopted the resolution

[3]:-That this meeting ... having heard with profound sorrow of the premature death of thelate Professor George Francis FitzGerald, desires to place on record its highappreciation of his brilliant qualities as a man, as a teacher, as an investigator, and asa leader of scientific thought ...

Article by:J J O'ConnorandE F Robertson

October 2003


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MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/Biographies/FitzGerald.html]

Niels Henrik Abel

Born: 5 Aug 1802 in Frinde (near Stavanger), NorwayDied: 6 April 1829 in Froland, Norway

Niels Abel's life was dominated by poverty and we begin by putting this in context bylooking briefly at the political problems which led to economic problems in Norway. Atthe end of the 18th century Norway was part of Denmark and the Danish tried to remainneutral through the Napoleonic wars. However a neutrality treaty in 1794 wasconsidered a aggressive act by Britain and, in 1801, the British fleet destroyed most ofthe Danish fleet in a battle in the harbour at Copenhagen. Despite this Denmark-Norwayavoided wars until 1807 when Britain feared that the Danish fleet might be used by theFrench to invade. Using the philosophy that attack is the best form of defence, theEnglish attacked and captured the whole Danish fleet in October 1807.

Denmark then joined the alliance Britain Britain. The continental powers blockadedBritain, and as a counter to this Britain blockaded Norway. The twin blockade was acatastrophe to Norway preventing their timber exports, which had been largely toBritain, and preventing their grain imports from Denmark. An economic crisis inNorway followed with the people suffering hunger and extreme poverty. In 1813Sweden attacked Denmark from the south and, at the treaty of Kiel in January 1814,Denmark handed over Norway to Sweden. An attempt at independence by Norway afew months later led to Sweden attacking Norway in July 1814. Sweden gained controlof Norway, setting up a complete internal self-government for Norway with agovernment in Christiania (which is called Oslo today). In this difficult time Abel wasgrowing up in Gjerstad in south-east Norway.

Abel's father, Sren Georg Abel, had a degree in theology and philology and his father(Niels Abel's grandfather) was a Protestant minister at Gjerstad near Risor. Sren Abelwas a Norwegian nationalist who was active politically in the movement to makeNorway independent. Sren Abel married Ane Marie Simonson, the daughter of amerchant and ship owner, and was appointed as minister at Finnoy. Niels Abel, thesecond of seven children, was one year old when his grandfather died and his father wasappointed to succeed him as the minister at Gjerstad. It was in that town that Abel wasbrought up, taught by his father in the vicarage until he reached 13 years of age.However, these were the 13 years of economic crisis for Norway described above andAbel's parents would have not been able to feed their family that well. The problemswere not entirely political either for [14]:-


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[Abel's] father was probably a drunkard and his mother was accused of having laxmorals.

Abel's father was, however, important in the politics of Norway and, after Swedengained control of Norway in 1814, he was involved in writing a new constitution for

Norway as a member of the Storting, the Norwegian legislative body. In 1815 Abel andhis older brother were sent to the Cathedral School in Christiania. The founding of theUniversity of Christiania had taken away the good teachers from the Cathedral Schoolto staff the University when it opened for teaching in 1813. What had been a goodschool was in a bad state when Abel arrived. Uninspired by the poor school, he proved arather ordinary pupil with some talent for mathematics and physics.

When a new mathematics teacher Bernt Holmbo joined the school in 1817 thingschanged markedly for Abel. The previous mathematics teacher had been dismissed forpunishing a boy so severely that he had died. Abel began to study university levelmathematics texts and, within a year of Holmbo's arrival, Abel was reading the works

of Euler, Newton, Lalande and d'Alembert. Holmbo was convinced that Abel had greattalent and encouraged him greatly taking him on to study the works of Lagrange andLaplace. However, in 1820 tragedy struck Abel's family when his father died.

Abel's father had ended his political career in disgrace by making false charges againsthis colleagues in the Storting after he was elected to the body again in 1818. His habitsof drinking to excess also contributed to his dismissal and the family was therefore inthe deepest trouble when he died. There was now no money to allow Abel to completehis school education, nor money to allow him to study at university and, in addition,Abel had the responsibility of supporting his mother and family.

Holmbo was able to help Abel gain a scholarship to remain at school and Abel wasable to enter the University of Christiania in 1821, ten years after the university wasfounded. Holmbo had raised money from his colleagues to enable Abel to study at theuniversity and he graduated in 1822. While in his final year at school, however, Abelhad begun working on the solution of quintic equations by radicals. He believed that hehad solved the quintic in 1821 and submitted a paper to the Danish mathematicianFerdinand Degen, for publication by the Royal Society of Copenhagen. Degen askedAbel to give a numerical example of his method and, while trying to provide anexample, Abel discovered the mistake in his paper. Degen had given Abel someimportant advice that was to set him working on an area of mathematics (see [2]):-

... whose development would have the greatest consequences for analysis andmechanics. I refer to elliptic integrals. A serious investigator with suitablequalifications for research of this kind would by no means be restricted to the manybeautiful properties of these most remarkable functions, but could discover a Strait of

Magellan leading into wide expanses of a tremendous analytic ocean.

At the University of Christiania Abel found a supporter in the professor of astronomyChristopher Hansteen, who provided both financial support and encouragement.Hansteen's wife began to care for Abel as if he was her own son. In 1823 Abelpublished papers on functional equations and integrals in a new scientific journal started

up by Hansteen. In Abel's third paper, Solutions of some problems by means of definiteintegrals he gave the first solution of an integral equation.


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Abel was given a small grant to visit Degen and other mathematicians in Copenhagen.While there he met Christine Kemp who shortly afterwards became his fiance.Returning to Christiania, Abel tried to get the University of Christiania to give him alarger grant to enable him to visit the top mathematicians in Germany and France. Hedid not speak French of German so, partly to save money, he was given funds to remain

in Christiania for two years to give him the chance to become fluent in these languagesbefore travelling. Abel began working again on quintic equations and, in 1824, heproved the impossibility of solving the general equation of the fifth degree in radicals.He published the work in French and at his own expense since he wanted an impressivepiece of work to take with him when he was on his travels. As Ayoub writes in [6]:-

He chose a pamphlet as the quickest way to get it into print, and in order to save on theprinting costs, he reduced the proof to fit on half a folio sheet[six pages].

By this time Abel seems to have known something of Ruffini's work for he had studiedCauchy's work of 1815 while he was an undergraduate and in this paper there is a

reference to Ruffini's work. Abel's 1824 paper begins ([6]):-

Geometers have occupied themselves a great deal with the general solution of algebraicequations and several among them have sought to prove the impossibility. But, if I amnot mistaken, they have not succeeded up to the present.

Abel sent this pamphlet to several mathematicians including Gauss, who he intended tovisit in Gttingen while on his travels. In August 1825 Abel was given a scholarshipfrom the Norwegian government to allow him to travel abroad and, after taking a monthto settle his affairs, he set out for the Continent with four friends, first visitingmathematicians in Norway and Denmark. On reaching Copenhagen, Abel found thatDegen had died and he changed his mind about taking Hansteen's advice to go directlyto Paris, preferring not to travel alone and stay with his friends who were going toBerlin. As he wrote in a later letter ([7]):-

Now I am so constituted that I cannot endure solitude. Alone, I am depressed, I getcantankerous, and I have little inclination to work.

In Copenhagen Abel was given a letter of introduction to Crelle by one of themathematicians there. Abel met Crelle in Berlin and the two became firm friends. Thisproved the most useful part of Abel's whole trip, particularly as Crelle was about to

begin publishing a journal devoted to mathematical research. Abel was encouraged byCrelle to write a clearer version of his work on the insolubility of the quintic and thisresulted inRecherches sur les fonctions elliptiques which was published in 1827 in thefirst volume ofCrelle's Journal, along with six other papers by Abel. While in Berlin,Abel learnt that the position of professor of mathematics at the University ofChristiania, the only university in Norway, had been given to Holmbo. With noprospects of a university post in Norway, Abel began to worry about his future.

Crelle's Journal continued to be a source for Abel's papers and Abel began to work toestablish mathematical analysis on a rigorous basis. He wrote to Holmbo from Berlin[2]:-


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My eyes have been opened in the most surprising manner. If you disregard the verysimplest cases, there is in all of mathematics not a single infinite series whose sum hadbeen rigorously determined. In other words, the most important parts of mathematicsstand without foundation. It is true that most of it is valid, but that is very surprising. Istruggle to find a reason for it, an exceedingly interesting problem.

It had been Abel's intention to travel with Crelle to Paris and to visit Gauss in Gttingenon the way. However, news got back to Abel that Gauss was not pleased to receive hiswork on the insolubility of the quintic, so Abel decided that he would be better not to goto Gttingen. It is uncertain why Gauss took this attitude towards Abel's work since hecertainly never read it - the paper was found unopened after Gauss's death. Ayoub givestwo possible reasons [6]:-

... the first possibility is that Gauss had proved the result himself and was willing to letAbel take the credit. ... The other explanation is that he did not attach very muchimportance to solvability by radicals...

The second of these explanations does seem the more likely, especially since Gauss hadwritten in his thesis of 1801 that the algebraic solution of an equation was no better thandevising a symbol for the root of the equation and then saying that the equation had aroot equal to the symbol.

Crelle was detained in Berlin and could not travel with Abel to Paris. Abel therefore didnot go directly to Paris, but chose to travel again with his Norwegian friends to northernItaly before crossing the Alps to France. In Paris Abel was disappointed to find therewas little interest in his work. He wrote back to Holmbo ([7]):-

The French are much more reserved with strangers than the Germans. It is extremelydifficult to gain their intimacy, and I do not dare to urge my pretensions as far as that;

finally every beginner had a great deal of difficulty getting noticed here. I have justfinished an extensive treatise on a certain class of transcendental functions to present itto the Institute which will be done next Monday. I showed it to Mr Cauchy, but hescarcely deigned to glance at it.

The contents and importance of this treatise by Abel is described in [2]:-

It dealt with the sum of integrals of a given algebraic function. Abel's theorem states

that any such sum can be expressed as a fixed number p of these integrals, withintegration arguments that are algebraic functions of the original arguments. Theminimal number p is the genus of the algebraic function, and this is the first occurrenceof this fundamental quantity. Abel's theorem is a vast generalisation of Euler's relation

for elliptic integrals.

Two referees, Cauchy and Legendre, were appointed to referee the paper and Abelremained in Paris for a few months [14]:-

... emaciated, gloomy, weary and constantly worried. He ... could only afford to eat onemeal a day.


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He published some articles, mainly on the results he had already written for Crelle'sJournal, then with no money left and his health in a very poor state, he returned toBerlin at the end of 1826. In Berlin, Abel borrowed some money and continued workingon elliptic functions. He wrote a paper in which [2]:-

... he radically transformed the theory of elliptic integrals to the theory of ellipticfunctions by using their inverse functions ...

Crelle tried to persuade Abel to remain in Berlin until he could find an academic postfor him and he even offered Abel the editorship ofCrelle's Journal. However, Abelwanted to get home and by this time he was heavily in debt. He reached Christiania inMay 1827 and was awarded a small amount of money by the university although theymade sure they had the right to deduct a corresponding amount from any future salaryhe earned. To make a little more money Abel tutored schoolchildren and his fiance wasemployed as a governess to friends of Abel's family in Froland.

Hansteen received a major grant to investigate the Earth's magnetic field in Siberia anda replacement was needed to teach for him at the University and also at the MilitaryAcademy. Abel was appointed to this post which improved his position a little.

In 1828 Abel was shown a paper by Jacobi on transformations of elliptic integrals. Abelquickly showed that Jacobi's results were consequences of his own and added a note tothis effect to the second part of his major work on elliptic functions. He had beenworking again on the algebraic solution of equations, with the aim of solving theproblem of which equations were soluble by radicals (the problem which Galois solveda few years later). He put this to one side to compete with Jacobi in the theory of ellipticfunctions, quickly writing several papers on the topic.

Legendre saw the new ideas in the papers which Abel and Jacobi were writing and said([2]):-

Through these works you two will be placed in the class of the foremost analysts of ourtimes.

Abel continued to pour out high quality mathematics as his health continued todeteriorate. He spent the summer vacation of 1828 with his fiance in Froland. Themasterpiece which he had submitted to the Paris Academy seemed to have been lost and

so he wrote the main result down again [3]:-The paper was only two brief pages, but of all his many works perhaps the most

poignant. He called it only "A theorem": it had no introduction, contained nosuperfluous remarks, no applications. It was a monument resplendent in its simple lines- the main theorem from his Paris memoir, formulated in few words.

Abel travelled by sled to visit his fiance again in Froland for Christmas 1828. Hebecame seriously ill on the sled journey and despite an improvement which allowedthem to enjoy Christmas, he soon became very seriously ill again. Crelle was told andhe redoubled his efforts to obtain an appointment for Abel in Berlin. He succeeded and

wrote to Abel on the 8 April 1829 to tell him the good news. It was too late, Abel hadalready died. Ore [3] describes his last few days:-


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... the weakness and cough increased and he could remain out of bed only the fewminutes while it was being made. Occasionally he would attempt to work on hismathematics, but he could no longer write. Sometimes he lived in the past, talking abouthis poverty and about Fru Hansteen's goodness. Always he was kind and patient. ...

He endured his worst agony during the night of April 5. Towards morning he becamemore quiet and in the forenoon, at eleven o'clock, he expired his last sigh.

After Abel's death his Paris memoir was found by Cauchy in 1830 after much searching.It was printed in 1841 but rather remarkably vanished again and was not found until1952 when it turned up in Florence. Also after Abel's death unpublished work on thealgebraic solution of equations was found. In fact in a letter Abel had written to Crelleon 18 October 1828 he gave the theorem [13]:-

If every three roots of an irreducible equation of prime degree are related to oneanother in such a way that one of them may be expressed rationally in terms of the other

two, then the equation is soluble in radicals.

This result is essentially identical to one given by Galois in his famous memoir of 1830.In this same year 1830 the Paris Academy awarded Abel and Jacobi the Grand Prix fortheir outstanding work.


June 1998

MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/Biographies/Abel.html]

Leone Battista Alberti

Born: 18 Feb 1404 in Genoa, French Empire (now Italy)Died: 3 April 1472 in Rome, Papal States (now Italy)

Leone Battista Alberti's father was Lorenzo Alberti. We do not know who his motherwas, and there is reason to believe that he was an illegitimate child. His father's familywere wealthy and had been involved in banking and commercial business in Florenceduring the 14th century. In fact the success of the city of Florence during this period isto a large extent a consequence of the success of the Alberti family, whose firm hadbranches spread widely through north Italy. Not content with their major financialachievements, however, members of the family became involved in politics. This turnedout to be a disaster and the family was driven out of Florence after decrees were passed


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to exile them. It was for this reason that Lorenzo Alberti came to be living in Genoa atthe time his son was born, for there he was safe yet still able to continue his wealthy lifestyle within a local branch of the family firm.

As a child Leone Battista received his mathematical education from his father Lorenzo.

However, when the plague struck Genoa, Lorenzo rapidly went with his children toVenice where the firm also had a major branch run by members of the Alberti family.However, Lorenzo died shortly after arriving in Venice and Leone Battista began livingwith one of his uncles. This arrangement was short-lived for the uncle soon vanished. Itis likely that by this time members of the family were attempting by unscrupulousmeans to gain access to the family fortune. Leone Battista attended a school in Paduathen, from 1421, he attended the University of Bologna where he studied law but didnot enjoy this topic. He became ill through overwork but still managed to gain a degreein canon law. It was around this time that he became interested in pursuing hismathematical studies, rather as a way to relax when stressed out by his law studieswhich he found made far too large demands on memory. Also around this time he wrote

a comedy Philodoxius (Lover of Glory, 1424), composed in Latin verse.

By this time the decrees which had forced his family to flee from Florence had beenrevoked and Alberti went to live in the city where he met Brunelleschi and the twobecame good friends. They shared an interest in mathematics and, through Brunelleschi,Alberti became interested in architecture. At this stage, however, his interest was purelytheoretical and he did not put his theories into practice. In 1430 Alberti began workingfor a cardinal of the Roman Catholic Church. This post meant that he travelled a lot, inparticular to France, Belgium and Germany. In 1432 he began following a literarycareer as a secretary in the Papal Chancery in Rome writing biographies of the saints inelegant Latin. Going to Rome was highly significant for Alberti, for there he fell in lovewith the ancient classical architecture which he saw all around. This led him to studynot only classical architecture but also painting and sculpture. Alberti served PopeEugene IV but this was a period of considerable weakness for the Papacy and militaryaction against the Pope forced Eugene IV out of Rome on several occasions. Alberti leftRome with the Pope at such times and spent time at the court in Rimini. Nicholas V,who was Pope from 1447 to 1455, was an enthusiast for classical studies and producedan environment much suited to Alberti who presented him with his book on architecture

De re aedificatoria in 1452. Alberti modelled the book on the classical work byVitruvius and copied his format by dividing his text into ten books. Vitruvius (1stcentury BC) was the author of the famous treatiseDe architectura (On Architecture).

The methods of fortification which Alberti set out in the text were highly influential andwere used in the fortification of towns for several hundred years. In 1447, the yearNicholas V became Pope, Alberti became a canon of the Metropolitan Church ofFlorence and Abbot of Sant' Eremita of Pisa. Pope Nicholas V employed him on anumber of major architectural projects and we describe below some of his remarkablebuildings.

Alberti studied the representation of 3-dimensional objects and, in 1435, wrote the firstgeneral treatiseDe Pictura on the laws of perspective. This was first published in Latinbut in the following year Alberti published an Italian version under the titleDella

pittura. The book was dedicated to Brunelleschi who had indeed been a great

inspiration to him. It was printed in 1511. Simon writes in [13]:-


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Alberti explained and justified his method of perspective construction by using themetaphor of a window opening onto the world. The picture surface is conceived asintersecting the pyramid of vision without altering it.

Alberti wrote about how he enjoyed applying mathematics to artistic undertakings:-

Nothing pleases me so much as mathematical investigations and demonstrations,especially when I can turn them to some useful practice drawing from mathematics the

principles of painting perspective and some amazing propositions on the moving ofweights .

Field [9] also comments on how mathematics influenced the arts through thecontributons of Alberti and others around the same period:-

What we seem to be seeing in this progress of perspective towards the applied arts inthe sixteenth century is the progress of mathematics as an increasingly important

component in the training and practice of craftsmen in general, and of architects inparticular.

Alberti also worked on maps (again involving his skill at geometrical mappings) and hecollaborated with Paolo Toscanelli who supplied Columbus with the maps for his firstvoyage. He also wrote the first book on cryptography which contains the first exampleof a frequency table. In this area he introduced polyalphabetic substitution [10]. This isthe method of cipher in which the kth letter of a text which is the ith letter in thealphabet is replaced by thejth letter of the alphabet wherej =f(i, k) for some functionf.Polyalphabetic substitution was introduced into diplomatic practice by Alberti, who alsoinvented a simple mechanical device to speed up coding and decoding, consisting of afixed and a movable ring.

Alberti is best known, however, as an architect. We mentioned above that Alberti spenttime in Rimini and it was there that he designed the facade of the Tempio Malatestiano,his first attempt to put his theoretical ideas about architecture into practice. It wasdesigned in the style of the Arch of Augustus in Rimini and is the first example in thehistory of art of a classical building becoming the model for a Renaissance one.

Gombrich writes [4]:-

Brunelleschi's idea had been to introduce the forms of classical buildings, the columns,pediments and cornices which he had copied from Roman ruins. He had used theseforms in his churches. His successors were eager to emulate him in this. [The Church ofS Andrea, Mantua, is] a church planned by the Florentine architect Leone Battista

Alberti, who conceived its facade as a gigantic triumphal arch in the Roman manner.But how was this new programme to be applied to an ordinary dwelling-house in a citystreet? No private houses had survived from Roman times, and even if they had, needsand customs had changed so much that they might have offered little guidance. The

problem, then, was to find a compromise between the traditional house, with walls andwindows, and the classical forms which Brunelleschi had taught the architects to use. Itwas again Alberti who found the solution that remained influential up to our own days.

When he built a palace for the rich Florentine merchant family Rucellai, he designed anordinary three-storeyed building. There is little similarity between this facade and a


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classical ruin. And yet Alberti stuck to Brunelleschi's programme and used classicalforms for the decoration of the facade. Instead of building columns or half-columns, hecovered the house with a network of flat pilasters and entablatures which suggest aclassical order without changing the structure of the building. ... Alberti ... merely'translated' a Gothic design into classical forms by smoothing out the 'barbaric' pointed

arch and using the elements of the classical order in a traditional context.

The Church of S Andrea, Mantua, which Gombrich comments on in the above quote,was designed by Alberti in 1470 and work on it began two years later. Alberti did notlive to see his design take shape for he died in the year in which building started and bythe time the facade and portico were in position he had been dead for 18 years. Thechurch is discussed in [5] where the author writes that Alberti's:-

... avowed architectural aim, to schematise in the spatial form of the church theimmanent, harmonious order of the world, found majestic realization in [his] ownchurch of Sant' Andrea in Mantua. This was his final architectural work ... and it

carries out these theoretical ideas with perfect artistic clarity.

Alberti made numerous innovations in his design with the traditional division into naveand aisles discarded in favour of providing a continuous space. There is certainly amathematical flavour to the way that Alberti has sequences of small and large chapelsalternating along the sides of the main space.

In addition to the Church of S Andrea, Mantua, Alberti had earlier articulated the facadeof the Santa Maria Novella in Florence, which he began work on in 1447, and thePalazzo Rucellai, mentioned in the quote of Gombrich above. Both works wereundertaken for the Florence merchant Giovanni di Paolo Rucellai. The Palazzo Rucellaiwas designed between 1446 and 1451 and stands in the Via della Vigna. In the square,on the right, is the Loggia dei Rucellai built by Alberti in 1460 as a formal hall for theRucellai family.

As to Alberti's character and appearance, Gille writes in [1]:-

Alberti was, we are told, amiable, very handsome, and witty. He was adept at directingdiscussions and took pleasure in organising small conversational groups.

In fact Alberti wrote some autobiographical notes which survive in which he boasts of

his physical abilities. He claimed he was capable of:-... standing with his feet together, and springing over a man's head.

In a similar vein he also claimed that he:-

... excelled in all bodily exercises; could, with feet tied, leap over a standing man; couldin the great cathedral, throw a coin far up to ring against the vault; amused himself bytaming wild horses and climbing mountains.

Even if untrue, these delightful quotes tell us much of Alberti's personality.



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August 2006

MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/Biographies/Alberti.html]

Jean Robert Argand

Born: 18 July 1768 in Geneva, SwitzerlandDied: 13 Aug 1822 in Paris, France

Jean-Robert Argand was an accountant and bookkeeper in Paris who was only anamateur mathematician. Little is known of his background and education. We do knowthat his father was Jacques Argand and his mother Eves Canac. In addition to his date ofbirth, the date on which he was baptized is known - 22 July 1768.

Among the few other facts known of his life is a little information about his children.His son was born in Paris and continued to live there, while his daughter, Jeanne-Franoise-Dorothe- Marie-Elizabeth Argand, married Flix Bousquet and they lived inStuttgart.

Argand is famed for his geometrical interpretation of the complex numbers where i isinterpreted as a rotation through 90. The concept of the modulus of a complex numberis also due to Argand but Cauchy, who used the term later, is usually credited as theoriginator this concept. The Argand diagram is taught to most school children who arestudying mathematics and Argand's name will live on in the history of mathematicsthrough this important concept. However, the fact that his name is associated with thisgeometrical interpretation of complex numbers is only as a result of a rather strangesequence of events.

The first to publish this geometrical interpretation of complex numbers was CasparWessel. The idea appears in Wessel's work in 1787 but it was not published until

Wessel submitted a paper to a meeting of the Royal Danish Academy of Sciences on 10March 1797. The paper was published in 1799 but not noticed by the mathematicalcommunity. Wessel's paper was rediscovered in 1895 when Juel draw attention to itand, in the same year, Sophus Lie republished Wessel's paper.

This is not as surprising as it might seem at first glance since Wessel was a surveyor.However, Argand was not a professional mathematician either, so when he publishedhis geometrical interpretation of complex numbers in 1806 it was in a book which hepublished privately at his own expense. His knowledge of the book trade allowed him toput out this small edition but one would have expected it to be in a less noticable placethan Wessel's work which after all was published by the Royal Danish Academy.Perhaps even more surprisingly, Argand's name did not even appear on the book so itwas impossible to identify the author.


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The way that Argand's work became known is rather complicated. Legendre was sent acopy of the work and he sent it to Franois Franais although neither knew the identityof the author. After Franois Franais's death in 1810 his brother Jacques Franaisworked on his papers and he discovered Argand's little book among them. In September1813 Jacques Franais published a work in which he gave a geometric representation of

complex numbers, with interesting applications, based on Argand's ideas. JacquesFranais might easily have claimed these ideas for himself, but he did quite the reverse.He ended his paper by saying that the idea was based on the work of an unknownmathematician and he asked that the mathematician should make himself known so thathe might receive the credit for his ideas.

The article by Jacques Franais appeared in Gergonne's journalAnnales demathmatiques and Argand responded to Jacques Franais's request by acknowledgingthat he was the author and submitting a slightly modified version of his original workwith some new applications to theAnnales de mathmatiques. There is nothing like anargument to bring something to the attention of the world and this is exactly what

happened next. A vigorous discussion between Jacques Franais, Argand and Servoistook place in the pages ofGergonne's Journal. In this correspondence Jacques Franaisand Argand argued in favour of the validity of the geometric representation, whileServois argued that complex numbers must be handled using pure algebra.

One might have expected that Argand would have made no other contributions tomathematics. However this is not so and, although he will always be remembered forthe Argand diagram, his best work is on the fundamental theorem of algebra and for thishe has received little credit. He gave a beautiful proof (with small gaps) of thefundamental theorem of algebra in his work of 1806, and again when he published hisresults in Gergonne's Journal in 1813. Certainly Argand was the first to state thetheorem in the case where the coefficients were complex numbers. Petrova, in [6],discusses the early proofs of the fundamental theorem and remarks that Argand gave analmost modern form of the proof which was forgotten after its second publication in1813.

After 1813 Argand did achieve a higher profile in the mathematical world. He publishedeight further articles, all in Gergonne's Journal, between 1813 and 1816. Most of theseare based on either his original book, or they comment on papers published by othermathematicians. His final publication was on combinations where he used the notation(m, n) for the combinations ofn objects selected from m objects.

In [1] Jones sums up Argand's work as follows:-

Argand was a man with an unknown background, a nonmathematical occupation, andan uncertain contact with the literature of his time who intuitively developed a criticalidea for which the time was right. He exploited it himself. The quality and significanceof his work were recognised by some of the geniuses of his time, but breakdowns incommunication and the approximate simultaneity of similar developments by otherworkers force a historian to deny him full credit for the fruits of the concept on which helaboured.


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July 2000

MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/Biographies/Argand.html]

Charles Babbage

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

Both the date and place ofCharles Babbage's birth were uncertain but have now beenfirmly established. In [1] and [12], for example, his date of birth is given as 26December 1792 and both give the place of his birth as near Teignmouth. Also in [18] itis stated:-

Little is known of Mr Babbage's parentage and early youth except that he was born on26 December1792.

However, a nephew wrote to The Times a week after the obituary [18] appeared, sayingthat Babbage was born on 26 December 1791. There was little evidence to prove whichwas right until Hyman (see [8]) in 1975 found that Babbage's birth had been registeredin St Mary's Newington, London on 6 January 1792. Babbage's father was BenjaminBabbage, a banker, and his mother was Betsy Plumleigh Babbage. Given the place thathis birth was registered Hyman says in [8] that it is almost certain that Babbage wasborn in the family home of 44 Crosby Row, Walworth Road, London.

Babbage suffered ill health as a child, as he relates in [4]:-

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 placedunder the care of a clergyman (who kept a school at Alphington, near Exeter), withinstructions to attend to my health; but, not to press too much knowledge upon me: amission which he faithfully accomplished.

Since his father was fairly wealthy, he could afford to have Babbage educated at privateschools. 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 formathematics but a dislike for the classics. On leaving the academy, he continued tostudy at home, having an Oxford tutor to bring him up to university level. Babbage in[4] lists the mathematics books he studied in this period with the tutor:-


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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 'Thorie des Fonctions'. I possessed also the 'Fluxions' ofMaclaurin and of Simson.

Babbage entered Trinity College, Cambridge in October 1810. However the groundinghe had acquired from the books he had studied made him dissatisfied with the teachingat Cambridge. He wrote [4]:-

Thus it happened that when I went to Cambridge I could work out such questions as thevery 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 thusacquired a distaste for the routine of the studies of the place, and devoured the papersof Euler and other mathematicians scattered through innumerable volumes of theacademies 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 bepenetrated with the superior power of the notation of Leibniz.

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

Babbage tried to buy Lacroix's book on the differential and integral calculus but this didnot prove easy in this period of war with Napoleon. When he did find a copy of thework 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 [4]:-

I then drew up the sketch of a society to be instituted for translating the small work ofLacroix on the Differential and Integral Calculus. It proposed that we should haveperiodical meetings for the propagation of d's; and consigned to perdition all whosupported 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'sfriend some years later- see the article on Green) who encouraged him to set up hisSociety. The Analytical Society was set up in 1812 and its members were all Cambridgeundergraduates. Nine mathematicians attended the first meeting but the two mostprominent members, in addition to Babbage, were John Herschel and George Peacock.

Babbage and Herschel produced the first of the publications of the Analytical Societywhen they publishedMemoirs of the Analytical Society in 1813. This is a remarkablydeep 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:-


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It is a lamentable consideration, that that discovery which has most of any done honourto the genius of man, should nevertheless bring with it a train of reflections so little tothe 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 calculdiffrentiel et intgral published in 1816 and a book of examples on the calculus whichthey published in 1820.

Babbage had moved from Trinity College to Peterhouse and it was from that Collegethat he graduated with a B.A. in 1814. However, Babbage realised that Herschel was amuch more powerful mathematician than he was so [12]:-

He did not compete for honours, believing Herschel sure of first place and not caring tocome 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 functionalequations in 1815 and 1816. Also in 1816, at the early age of 24, he was elected afellow of the Royal Society of London. He wrote papers on several differentmathematical topics over the next few years but none are particularly important andsome, 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 hisfeelings on how the Royal Society was run:-

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

However in 1820 he was elected a fellow of the Royal Society of Edinburgh, and in thesame year he was a major influence in founding the Royal Astronomical Society. Heserved as secretary to the Royal Astronomical Society for the first four years of itsexistence 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 Professorof 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 wouldhave expected of the holder, was that by this time he had become engrossed in what wasto became the main passion of his life, namely the development of mechanicalcomputers.

Babbage is without doubt the originator of the concepts behind the present daycomputer. The computation of logarithms had made him aware of the inaccuracy ofhuman calculation around 1812. He wrote in [4]:-

... I was sitting in the rooms of the Analytical Society, at Cambridge, my head leaningforward on the table in a kind of dreamy mood, with a table of logarithms lying open


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before me. Another member, coming into the room, and seeing me half asleep, calledout, Well, Babbage, what are you dreaming about?" to which I replied "I am thinkingthat 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 preciseand he formulated a plan to construct tables using the method of differences bymechanical means. Such a machine would be able to carry out complex operations usingonly the mechanism for addition. Babbage began to construct a small difference enginein 1819 and had completed it by 1822. He announced his invention in a paperNote onthe application of machinery to the computation of astronomical and mathematicaltables 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 theresults 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 termsare 2, 4, 6, 8, .. and the second differences are 2, 2, 2, ..... The difference engine is giventhe 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 membersof 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 theFrench Government of producing logarithmic and trigonometric tables with teams ofpeople to carry out the calculations. He argued that a large difference engine could dothe 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 hisdevelopment of the difference engine. He then met the Chancellor of the Exchequer toseek public funds for the construction of a large difference engine. The Royal Societyhad already given positive advice to the government:-

Mr Babbage has displayed great talent and ingenuity in the construction of his machinefor computation, which the committee thanks fully adequate to the attainment of the

objects proposed by the inventory; and they consider Mr Babbage as highly deservingof 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 hebelieved he could complete in three years. He set out to produce an engine with [3]:-

... six orders of differences, each of twenty places of figures, whilst the first threecolumns 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 hadbeen calculating, and it was intended to have a printer to print out the results

automatically. However the construction proceeded slower than had been expected. By1827 the expenses were getting out of hand.


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The year 1827 was a year of tragedy for Babbage; his father, his wife and two of hischildren all died that year. He own health gave way and he was advised to travel on theContinent. After his travels he returned near the end of 1828. Further attempts to obtaingovernment support eventually resulted in the Duke of Wellington, the Chancellor ofthe Exchequer and other members of the government visiting Babbage and inspecting

the work for themselves. By February 1830 the government had paid, or promised topay, 9000 towards the project.

In 1830 Babbage publishedReflections on the Decline of Science in England, acontroversial work that resulted in the formation, one year later, of the BritishAssociation for the Advancement of Science. In 1834 Babbage published his mostinfluential work On the Economy of Machinery and Manufactures, in which heproposed 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 thattime 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 decisionas to whether to continue support. In 1842 the decision not to proceed was taken byRobert Peel's government. Dubbey in [6] writes:-

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

By 1834 Babbage had completed the first drawings of the analytical engine, theforerunner of the modern electronic computer. His work on the difference engine hadled him to a much more sophisticated idea. Although the analytic engine neverprogressed beyond detailed drawings, it is remarkably similar in logical components toa present day computer. Babbage describes five logical components, the store, the mill,the control, the input and the output. The store contains [4]:-

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

The mill is the analogue of the cpu in a modern computer and it is the place [4]:-... 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 typedevice. It was operated by punched cards and the punched cards contained the programfor the particular task [4]:-

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

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


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

Babbage visited Turin in 1840 and discussed his ideas with mathematicians thereincluding Menabrea. During Babbage's visit, Menabrea collected all the material neededto describe the analytical engine and he published this in October 1842. Lady AdaLovelace translated Menabrea's article into English and added notes considerably moreextensive than the original memoir. This was published in 1843 and included [7]:-

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

Although Babbage never built an operational, mechanical computer, his design conceptshave been proved correct and recently such a computer has been built followingBabbage's own design criteria. He wrote in 1851 (see [7]):-

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

Despite this last statement, Babbage never did quite give up hope that the analyticalengine would be built writing in 1864 in [4]:-

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

After Babbage's death a committee,whose members included Cayley and Clifford, wasappointed by the British Association [12]:-

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

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


October 1998

MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/Biographies/Babbage.html]


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Thomas Bayes

Born: 1702 in London, EnglandDied: 17 April 1761 in Tunbridge Wells, Kent, England

Thomas Bayes' father, Joshua Bayes, was one of the first six Nonconformist ministersto be ordained in England. He was ordained in 1694 and moved to Box Lane Chapel,Bovington, about 25 miles from London. Thomas's mother was Anne Carpenter. Thefamily moved to Southwark, London, when Thomas was young and there Joshuabecame an assistant at St Thomas's and also an assistant at the Chapel in Leather Lane,Holborn. Thomas was the eldest of his parents seven children, four boys and three girls.

In [1] Hacking claims that Thomas was educated privately, something he says appearsnecessary for the son of a Nonconformist minister at that time. If this is the case, thennothing is known of his tutors but Barnard in [6] points out the intriguing possibilitythat he could have been tutored by de Moivre who was certainly giving private tuition inLondon at the time. However other historians (see for example [21]) suggest thatThomas received a liberal education for the ministry. In that case it is likely that heattended Fund Academy in Tenter Alley which was the only school with the rightreligious connections near where Bayes lived.

We do know that in 1719 Bayes matriculated at the University of Edinburgh where hestudied logic and theology. He had to choose a Scottish university if he was to obtainhis education without going overseas since, at this time, Nonconformists were notallowed to matriculate at Oxford or Cambridge. Records which still survive at theUniversity of Edinburgh record that he gave the homily on 14 January 1721 with thetext being "Matthew Chapter 7 verses 24-27", and again on 20 January 1722 with thetext being "Matthew Chapter 11 verses 29-30". He left the University as a probationer,but he was not ordained at this stage. At some time Bayes must have studiedmathematics but there is no evidence that he did so at Edinburgh University. However,he certainly had the opportunity to study mathematics at Edinburgh and when he wroteat age 34:-

I have long ago thought that the first principles and rules of the method of Fluxionsstood in need of more full and distinct explanation and proof ...

he certainly suggests that his interest went back to his student days or perhaps shortlyafterwards.

Thomas Bayes was ordained, a Nonconformist minister like his father, and at firstassisted his father in Holborn. In about 1733 he became minister of the PresbyterianChapel in Tunbridge Wells, 35 miles southeast of London, on the death of the previousminister John Archer. There is, however, evidence that he was associated with

Tunbridge Wells before 1733. One report states (see for example [4] where this isquoted):-


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John Archer died in 1733 but was succeeded by Thomas Bayes in 1730, who was a manof considerable attainment. It appears that Thomas Bayes left Tunbridge Wells in 1728and returned in 1731. During this time he was at Leather Lane Presbyterian Church,

London, where his father was pastor.

On 24 August 1746 William Whiston describes having breakfast with Bayes who hesays is:-

... a dissenting Minister at Tunbridge Wells, and a Successor, though not immediate, toMr Humphrey Ditton, and like him a very good mathematician.

Bayes apparently tried to retire from the ministry in 1749 but remained minister atTunbridge Wells until 1752 when he did retire, but continued to live in TunbridgeWells.

Bayes set out his theory of probability inEssay towards solving a problem in the

doctrine of chances published in the Philosophical Transactions of the Royal Society ofLondon in 1764. The paper was sent to the Royal Society by Richard Price, a friend ofBayes', who wrote:-

I now send you an essay which I have found among the papers of our deceased friendMr Bayes, and which, in my opinion, has great merit... In an introduction which he haswrit to this Essay, he says, that his design at first in thinking on the subject of it was, to

find out a method by which we might judge concerning the probability that an event hasto happen, in given circumstances, upon supposition that we know nothing concerning itbut that, under the same circumstances, it has happened a certain number of times, and

failed a certain other number of times.

Dale writes in [4]:-

What may the reader expect to find in this Essay? As regards probability, he will expect,of course, some or other version of what has become known as 'Bayes's theorem': andsuch expectation will indeed be met. In addition he will find a clear discussion of thebinomial distribution and if he should probe even deeper he will find ... the firstoccurrence of a probability logic result involving conditional probability. The Essayshould be of interest to mathematicians for the evaluation of the incomplete beta-

function. We note too the use of approximations to various integrals made here and in

the Supplement by both Bayes and Price, and the attention paid to the question of theerror incurred in the making of such approximation. The Essay, then, mainly, andperhaps justly, remembered for the solution of the problem posed by Bayes, should alsobe remembered for its contribution to pure mathematics.

Bayes's conclusions were accepted by Laplace in a 1781 memoir, rediscovered byCondorcet (as Laplace mentions), and remained unchallenged until Boole questionedthem in theLaws of Thought. Since then Bayes' techniques have been subject tocontroversy.

Bayes also wrote an articleAn Introduction to the Doctrine of Fluxions, and a Defence

of the Mathematicians Against the Objections of the Author of The Analyst(1736)


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attacking Berkeley for his attack on the logical foundations of the calculus. In thePreface Bayes gives his reasons for writing the text:-

I have long ago thought that the first principles and rules of the method of Fluxionsstood in need of more full and distinct explanation and proof, than what they had

received either from their first incomparable author, or any of his followers; andtherefore was not at all displeased to find the method itself opposed with so muchwarmth by the ingenious author of the Analyst; and had it been his only design to bringthis point to a fair issue, whether a demonstration by the method of Fluxions be trulyscientific or not, I should have heartily applauded his conduct, and have thought hedeserved the thanks even of the Mathematicians themselves. But the invidious light inwhich he has put this debate, by representing it as of consequence to the interests ofreligion, is, I think, truly unjustifiable, as well as highly imprudent.

Bayes writes that Berkeley:-

...represents the disputes and controversies among mathematicians as disparaging theevidence of their methods: and ... he represents Logics and Metaphysics as proper toopen their eyes, and extricate them from their difficulties. ... If the disputes of the

professors of any science disparage the science itself, Logics and Metaphysics are muchmore disparaged than Mathematics, why, therefore, if I am half blind, must I take formy guide one that can't see at all?

Bayes was elected a Fellow of the Royal Society in 1742 despite the fact that at thattime he had no published works on mathematics, indeed none were published in hislifetime under his own name, the article on fluxions referred to above was publishedanonymously. Another mathematical publication on asymptotic series appeared after hisdeath where he showed that the series for logz! given by Stirling and de Moivre, wasnot valid since it diverged.

There are a few other pieces of mathematics which have come down to us from Bayesand we now look at some of these. The first we mention is a letter which he wrote,probably around 1756. Bayes wrote:-

You may remember a few days ago we were speaking of Mr Simpson's attempt to showthe great advantage of taking the mean between several astronomical observationsrather than trusting to a single observation carefully made, in order to diminish the

errors arising from the imperfection of instrument and the organs of sense.In fact Simpson had made the same error that the French would make nearly fifty yearslater with Borda's repeating circle, in believing that one could make the error in theobservation as small as one desired by making multiple observations. However, Bayesrealised that this was not so and wrote in his letter:-

Now that the errors arising from the imperfection of the instrument and the organs ofsense should be thus reduced to nothing or next to nothing only by multiplying thenumber of observations seems to me extremely incredible. On the contrary the moreobservations you make with an imperfect instrument the more it seems to be that the

error in your conclusion will be proportional to the imperfection of the instrument madeuse of ...


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In [4] a notebook which was almost certainly written by Bayes is examined in detail.This notebook contains a considerable amount of mathematical work, includingdiscussions of probability, trigonometry, geometry, solution of equations, series, anddifferential calculus. There are also sections on natural philosophy in which Bayes looksat topics which include electricity, optics and celestial mechanics.


June 2004

MacTutor History of Mathematics[http://www-history.mcs.st-andrews.ac.uk/Biographies/Bayes.html]

George Boole

Born: 2 Nov 1815 in Lincoln, Lincolnshire, EnglandDied: 8 Dec 1864 in Ballintemple, County Cork, Ireland

George Boole's parents were Mary Ann Joyce and John Boole. John made shoes but hewas interested in science and in particular the application of mathematics to scientificinstruments. Mary Ann was a lady's maid and she married John on 14 September 1806.They moved to Lincoln where John opened a cobbler's shop at 34 Silver Street. Thefamily were not well off, partly because John's love of science and mathematics meantthat he did not devote the energy to developing his business in the way he might havedone. George, their first child, was born after Mary Ann and John had been married fornine years. They had almost given up hope of having children after this time so it wasan occasion for great rejoicing. George was christened the day after he was born, anindication that he was a weak child that his parents feared might not live. He was namedafter John's father who had died in April 1815. Over the next five years Mary Ann and

John had three further children, Mary Ann, William and Charles.

If George was a weak child after his birth, he certainly soon became strong and healthy.George first attended a school in Lincoln for children of tradesmen run by two MissesClarke when he was less than two years old. After a year he went to a commercialschool run by Mr Gibson, a friend of John Boole, where he remained until he was sevenyears old. His early instruction in mathematics, however, was from his father who alsogave George a liking for constructing optical instruments. When he was seven Georgeattended a primary school where he was taught by Mr Reeves. His interests turned tolanguages and his father arranged that he receive instruction in Latin from a localbookseller.


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Having learnt Latin from a tutor, George went on to teach himself Greek. By the age of14 he had become so skilled in Greek that it provoked an argument. He translated apoem by the Greek poet Meleager which his father was so proud of that he had itpublished. However the talent was such that a local schoolmaster disputed that any 14year old could have written with such depth. By this time George was attending

Bainbridge's Commercial Academy in Lincoln which he had entered on 10 September1828. This school did not provide the type of education he would have wished but itwas all his parents could afford. However he was able to teach himself French andGerman studying for himself academic subjects that a commercial school did not cover.

Boole did not study for an academic degree, but from the age of 16 he was an assistantschool teacher at Heigham's School in Doncaster. This was rather forced on him sincehis father's business collapsed and he found himself having to support financially hisparents, brothers and sister. He maintained his interest in languages, began to studymathematics seriously, and gave up ideas which he had to enter the Church. The firstadvanced mathematics book he read was Lacroix'sDifferential and integral calculus.

He was later to realise that he had almost wasted five years in trying to teach himself thesubject instead of having a skilled teacher. In 1833 he moved to a new teaching positionin Liverpool but he only remained there for six months before moving to Hall'sAcademy in Waddington, four miles from Lincoln. In 1834 he opened his own school inLincoln although he was only 19 years old.

In 1838 Robert Hall, who had run Hall's Academy in Waddington, died and Boole wasinvited to take over the school which he did. His parents, brothers and sister moved toWaddington and together they ran the school which had both boarding and day pupils.At this time Boole was studying the works of Laplace and Lagrange, making noteswhich would later be the basis for his first mathematics paper. However he did receiveencouragement from Duncan Gregory who at this time was in Cambridge and the editorof the recently founded Cambridge Mathematical Journal. Boole was unable to takeDuncan Gregory's advice and study courses at Cambridge as he required the incomefrom his school to look after his parents. In the summer of 1840 he had opened aboarding school in Lincoln and again the whole family had moved with him. He beganpublishing regularly in the Cambridge Mathematical Journal and his interests wereinfluenced by Duncan Gregory as he began to study algebra.

Boole had begun to correspond with De Morgan in 1842 and when in the following yearhe wrote a paper On a general method of analysis applying algebraic methods to the

solution of differential equations he sent it to De Morgan for comments. It waspublished by Boole in the Transactions of the Royal Society in 1844 and for this workhe received the Society's Royal Medal in November 1844. His mathematical work wasbeginning to bring him fame.

Boole was appointed to the chair of mathematics at Queens College, Cork in 1849. Infact he made an application for a chair in any of the new Queen's Colleges of Ireland in1846 and in September of that year De Morgan, Kelland, Cayley, and Thomson wereamong those writing testimonials in support. De Morgan wrote (see for example [7]):-

I can speak confidently to the fact of his being not only well-versed in the highest

branches of mathematics, but possessed of original power for their extension whichgives him a very respectable rank among their English cultivators of this day.


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Kelland wrote:-

From the originality of his conceptions and the extent and accuracy of his knowledge, Iconceive he has few superiors in Europe ...

Boole's father died in December 1848 before the decision had been made concerning theIrish chairs but an announcement came in August 1849 that Boole was to become thefirst Professor of Mathematics at Queen's College, Cork, and he took up the position inNovember. He taught there for the rest of his life, gaining a reputation as an outstandingand dedicated teacher. However the position was not without difficulty as the Collegebecame embroiled in religious disputes. Boole wrote to De Morgan on 17 October 1850(see for example [7]):-

... if you should hear of any situation in England that would be likely to suit me ... let meknow of it. I am not terrified by the storm of religious bigotry which is at this momentraging round us here. I am not dissatisfied with my duties and I may venture to say that

I am on good terms with my colleagues and with my pupils. But I cannot helpentertaining a feeling ... that recent events in this College have laid the foundation of alack of mutual trust and confidence among us ...

In May 1851 Boole was elected as Dean of Science, a role he carried outconscientiously. By this time he had already met Mary Everest (a niece of Sir GeorgeEverest, after whom the mountain is named) whose uncle was the professor of Greek atCork and a friend of Boole. They met first in 1850 when Mary visited her uncle in Corkand again in July 1852 when Boole visited the Everest family in Wickwar,Gloucestershire, England. Boole began to give Mary informal mathematics lessons onthe differential calculus. At this time he was 37 years old while Mary was only 20. In1855 Mary's father died leaving her without means of support and Boole proposedmarriage. They married on 11 September 1855 at a small ceremony in Wickwar. Itproved a very happy marriage with five daughters: Mary Ellen born in 1856, Margaretborn in 1858, Alicia (later Alicia Stott) born in 1860, Lucy Everest born in 1862, andEthel Lilian born in 1864. MacHale writes [7]:-

The large gap in their ages seemed to count for nothing because they were kindredspirits with an almost complete unity of purpose.

Let us now look at Boole's most important work. In 1854 he publishedAn investigation

into the Laws of Thought, on Which are founded the Mathematical Theories of Logicand Probabilities. Boole approached logic in a new way reducing it to a simple algebra,incorporating logic into mathematics. He pointed out the analogy between algebra

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