READING NOTES FOR USE IN PRESENTATIONS AND CURRICULUM DEVELOPMENT

This is split into four sections of notes from my sabbatical readings:

A. Mathematicians

B. Historical Sweep

C. Other

D. Notation

E. Education

F. Locations

G. Basic Skills Topics

H. Applications

I. Research Leads

J. Algebra Thread 101

K. Logic Thread 101

L. Notation Thread 101

M. Geometry Thread 101

N. Quotes

O. Disambiguation

P. Womens Issues

Q. Bibliography

MATHEMATICIANS

Ahmes (c. 1650BC) Ahmes is the first person whose name we know who had some definite connection with mathematics. He is the scribe of the Rhind Papyrus, which is also called the Ahmes Papyrus. It is not known whether he was actually a mathematician himself, but there are so few mathematical errors in the papyrus, that it seems he knew what he was doing. He was copying over a document written in the twelfth dynasty (c. 1990-1780 BC), which may have been one of the text preservation projects initiated by the Hyksos rulers, who were respectful of the then ancient Egyptian civilization. (Derbyshire [5] 29)

Thales (c.624 c. 546BC) Thales, the teacher of Pythagoras, is credited with being the first man in history to insist on proof rather than intuition in geometry.

Pythagoras (570-495 BC) Pythagoras perpetuated the notion of his teacher, Thales, in insisting on proof rather than intuition in geometry.

Hippocrates of Chios (470-410 BC) live a generation before Plato, so Plato had his work available. His work was somewhat like that of Euclid, with propositions and proofs. It is no longer extant, having been so eclipsed by Euclids elements. This Hippocrates was a contemporary of the famous physician, but they are not the same person. (Grabiner 1st elements lecture)

Plato (427BC-423BC) Let no one ignorant of geometry enter here Socrates taught him, he taught Aristotle, Aristotle taught Alexander the Great, Alexander conquered the world and founded Alexandria. According to the earliest source of information (a letter from about 150 BC) the library was initially organized by Demetrius of Phaleron, a student of Aristotle under the reign of Ptolemy I Soter though it may have been Ptolemy II (Grabiner states Ptolemy I).

---- Socractes (470-399 BC) ---- Plato (428-347 BC) ---- Aristotle (384-322) ----

Euclid (c. 300BC) Euclid lived, taught, and founded a school in the city of Alexandria when it, and the rest of Egypt, was ruled by Alexanders general Ptolemy. Euclid was the first head of the mathematics department (really?) in Alexandria (Singh 46) Euclid himself is thought to have gotten his mathematical training in Athens at the school of Plato.

Diophantus (200-285 AD) actually, were not even sure about the century in which he lived! (Derbyshire [5] 35). His arrival in Alexandria could have been anytime within a five-century window. In his writings Diophantus quotes Hypsicles and therefore he must have lived after 150 BC; on the other hand, his own work is quoted by Theon of Alexandria and therefore he must have lived before AD 364. A date around AD 250 is generall accepted as being a reasonable estimate. (Singh 51) Its also not clear how much of his notation is original with him, but he is the first one we can put a name with. A document that predates him is the so-called Michigan Papyrus (see Derbyshire [5] 36). His work, Arithmetica, gave rise to Fermats Last Theorem, and the phrase Diophantine analysis means to seek whole number solutions to polynomial equations. (see Derbyshire [5] 39)

Hypatia (360-415 AD) Hypatia is the first woman in mathematics of whom we have considerable knowledge, but she is not the first female mathematician (see introduction to Osen for role of women in Mesopotamia, especially under Hammurabi) (Osen 22, Derbyshire [5] 44) Enthusiastic students came from Europe, Asia, and Africa to hear her lecture on the Arithmetica of Diophantus. (Osen 26) When asked why she never married, she said she was wedded to truth (Singh 99) It seems she was the last to teach at the Museion of Alexandria and that her appalling death marked the end of mathematics in the ancient European world. (Derbyshire [5] 45)

al-Khwarizmi (780-850 AD) main algebraic achievement was to bring forward the idea of equations as objects of interest, classifying all equations of the first and second degrees in one unknown and giving rules for manipulating them (Derbyshire [5] 50)

Omar Khayyam (1048-1131) seems to have been the first to recognize cubic equations as a distinct class or problems (Derbyshire [5] 55)

NB There was some strong activity in medieval Europe including Oresme and Regiomontanus Black Death hit 134653 Thomas Bradwardine, mathematician and Archbishop of Canterbury (mentioned in Chaucers Tales) died in 1349, possibly of plague a third to a half of Europes population died which may be why math stalled for a couple centuries see Boyers History for this and also for mentions of fall of Constantinople and if that really did bring more documents in to Europe (or the opposite)

Fra Luca Bartolomeo de Pacioli(c.14471517) First person to publish a book on double-entry book-keeping. He is also the first man who wrote about the problem of the unfinished game. There was a flaw in his reasoning; this problem was later picked up by Cardano and then, most famously by Pascal and Fermat. (Devlin 16-17) He is remembered not so much for original mathematical contributions as for his book (Summa . . . published in 1494), which catalogued systematically all of what was know mathematically at that time. The book draws heavily on both Elements and Liber Abaci but also contains much that had been discovered sincet hos two earlier works appeared, particularly in algebra where great strides had been made in solving polynomial equations. The Summa includes the problem of the points, which provided a solid platform for Cardano who set the stage for the Pascal-Fermat probability correspondence. He was good friends with Leonardo da Vinci, and da Vinci illustrated Paciolis book Divina Proportione, published in 1509. (Devlin 34-35)

Girolamo Cardano (1501-1576) Cardanos father was a friend of Leonardo da Vinci (with whom Pacioli was a collaborator). Cardanos lasting contribution to the creation of probability theory came in a manuscript he wrote in 1564, titled The Book of Games of Chance and published in 1663, almost a century after his death. He wrote it not as a mathematics treatise but as a practical guide for gamblers. It is the first scientific study of dice rolling, based on the premise that there are fundamental principles governing the likelihood of particular outcomes. The book is partly observational and partly a theoretical analysis of how chance events, such as particular outcomes of rolls of dice, aggregate when repeated many times. In other words, it was the first study of frequentist probability (Devlin 43-44) The issue of the unfinished problem was first published by Luca Pacioli. His reasoning was flawed, and the problem was later taken up by Cardano and, in the next century, most famously by Pascal and Fermat. Though the reasoning of Pacioli and Cardano was flawed, Cardano did make several key observations that established the beginning of what, following Pascal and Fermats work, became probability theory. (Devlin 16-18) Cardano was the second most famous physician in Europe, after Andreas Vesalius. The society of his time clamored for his services. He seems to have been averse to travel, however, venturing far afield only once to Scotland in 1552, to cure the asthma of John Hamilton, the last Roman Catholic archbishop of that country. Cardanos fee was 2000 gold crowns. The cure seems to have been completely successful. Hamilton lived until 1571 when he was hanged, in full pontificals, on the public gibbet at Stirling for complicity in the murder of Lord Darnely, husband of Mary Qyeen of Scots. Passing through London on his way home from Scotland, Cardano cast horoscopes for the boy kind Edward VI. (Derbyshire [5] 73) According to Hellman (p. 24) Tartaglia handed Cardano over to the Inquisition. The most damning piece of evidence was his horoscope of Jesus. Though Cardano was not tortured or killed he was thrown into jail. He sought desperately for help and was able to reach out to an official in the church, Archbishop Hamilton, who had in the past asked to be called upon if need be. The archbishop came through for Cardano, who was released a few months later. It was just in time, for not long after, the archbishops own fortunes changed; he was captured by the forces of Mary, Queen of Scots, and beheaded. (Hellman 24) Cardano wrote a book of advice for the sorrowing titled Consolation. It was translated into English in 1573 and was read by William Shakespeare. The sentiments in Hamlets soliloquy closely resemble some remarks about sleep in Consolation, and this may be the book that Hamlet is traditionally carrying when he comes on stage to deliver that soliloquy. Cardano was the first mathematician to take the strange creatures that were complex numbers even half seriously. (Derbyshire [5] 82) (reference to Cardan universal joint pp. 72 ff. also his prediction of his own death and its fulfillment) He was the first to describe typhoid fever (1553) and invented the combination lock. (Burger lecture 17). He also invented the gimbal consisting of three concentric rings allowing a supported compass or gyroscope to rotate freely.

Rafael Bombelli (1526-1572) The first mathematician to tackle complex numbers with any confidence was Rafael Bombelli. Around 1560, Bombelli was in Rome. There he met and talked math with Antonio Maria Pazzi, who taught the subject at the university in that city. Pazzi mentioned that he had found in the Vatican library a manuscript on arithmetic and algebra by a certain Diophantus, a Greek author of the ancient world. The two men examine the text and decided to make a translation of it. The translation was never finished, but there is no doubt that Bombelli got much inspiration from studying Diopahntus. He included 143 of Diophantuss problems in lAlgebra, and it was through his book that Diophantuss work first became known to European mathematicians of the time. (Derbyshire [5] 82-83)

John Dee (1527-1608) John Dee provided a widely respected Mathematical Preface along with copious notes and supplementary material, to the first English edition of Euclids Elements by Henry Billingsley (the first translator of Euclid into English). In his early twenties he had been invited to lecture on Euclid in Paris. He straddled the worlds of science and magic just as they were becoming distinguishable. He was an advisor to Queen Elizabeth I. He was born in the Tower Ward of London. He studied with Mercator and did work on navigation. Before it was vandalized his personal library was the largest in England. Dee was also a friend of Tycho Brahe and was familiar with the works of Copernicus. Dee applied Copernican theory to the problem of calendar reform. In 1583, he was asked to advise the Queen about the new Gregorian calendar that had been promulgated by Pope Gregory XIII from October 1582. (Wikipedia)

Francois Viete (1540-1603) was likely the first to use the term polynomial, which comes from Greek and Latin and means having many names, where names is understood to mean named parts. It showed up in English about 100 years after Vietes use of the term. (early in Derbyshire?) Viete was born into a Huguenot family yet grew up to serve Charles IX of France who authorized the general massacre of Huguenots on St. Bartholomews Eve (August 23, 1572); yet the following year, Viete, a Huguenot, was appointed by the king to a government position in Brittany. When Charles died in 1574 and was succeeded by his younger brother Henry III, Viete returned to Paris to take up a position as advisor to this king. After this king was assassinated and replaced by Henry of Navarre (Henry IV), first king of the Bourbon Dynasty, Viete served him as well this kind being a protestant. Philip of Spain desired his daughter to have the throne and intrigued with actions at the French court on her behalf. These intrigues relied on letters written in code. Finding he had a mathematician at hand, Henry set Viete the task of cracking the Spanish code. Viete did so, and when it dawned on Philip that his unbreakable code had been broken, he complained to the Pope that Henry was using witchcraft. Vietes book, the Isaoge represents a great step forward in algebra and a small step backward. He used letters for known and unknown quantities capital vowels (unknowns) and capital consonants (knowns). This was the step forward. The step back was that he was strongly influenced by the geometry of the ancient and wanted to base his algebra rigorously on geometrical concepts and therefore there were issues with polynomials of degree higher than three, and yet he was able to solve a 45th degree polynomial! It was Viete who opened up the line of inquiry that led to the study of symmetries of an equations solutions, and there from to Galois theory, the theory of groups and all of modern algebra. (Derbyshire [5] 85-89) [With this, Vieta marked the end of medieval algebra (from Al-Khwarizmi to Stevin) and opened the modern period. Wikipedia]

John Napier (1550-1617) worshipped at St. Cuthberts Church and was an elder there. He was buried there, and a plaque says, Near this spot was laid the body of . . . Napier was the first to use the constant e. (Later exploration was done by Jakob Bernoulli and then by Euler, of course). (Burger, lecture 14)

William Oughtred (1574-1660) gave us the times sign (x)

Father Marin Mersenne (1588-1648) - He has often been referred to as Frances walking scientific journal, and it was by his effort that Galileos work became known outside his own country. (Hellman 30)

Rene Descartes (1596 -1650) Descartes developed a vortex theory that postulated a matter-filled universe in which the motion of any body can be caused only by contact with another. It had enormous influence for a while. Its advantage was that it provided a mechanical explanation for many previously puzzling phenomena, as well as for explanations that had previously depended on spirits and ghosts. (Hellman 30) The Fundamental Theorem of Algebra was first stated by him in his La Geometrie (1637) though in a tentative form, as he was not at ease with complex numbers.

Pierre de Fermat (1601-1665) was born to a family of wealthy merchants. He was a man in the Renaissance tradition, who was at the center of the rediscovery of ancient Greek knowledge (Singh ix) He studied law at Toulouse and Orleans, and mathematics at Bordeaux. His law studies prepared him for his career as a lawyer and jurist, and his mathematical training prepared him for his lifelong interest in the subject. He is nearly always described as an amateur mathematician, but he actually did study it at Bordeaux; he devoted immense amounts of time to it, and he was famed in his day as one of the greatest mathematicians in Europe. He is perhaps called an amateur because he did not get paid for his research, because he published almost nothing during his life, and because he worked in another profession. (Singh) Judges in seventeenth-century France were discouraged from socializing on the grounds that friends and acquaintances might one day be called before the court. Fraternizing with the locals could only lead to favoritism. Isolated from the rest of Toulouses high society, Fermat could concentrate on his hobby. (Singh 56) Fermat developed algebraic coordinate geometry independently of Descartes; he also made important contributions to the early development of calculus, and Newton wrote that he developed his calculus based on Monsieur Fermats method of drawing tangents. (Devlin 22, Singh 44). Fermats approach to analytic geometry is closer to ours, but Fermat championed Vietes cumbersome notation, while Descartess symbolic notation is quite modern. (Hellman 50). He also invented modern number theory essentially single-handedly (Devlin 22) Along with Pascal he is a father of probability theory (Singh 43) Despite encouragements of Father Mersenne, Fermat steadfastly refused to reveal his proofs. Publication and recognition meant nothing to him, and he was satisfied with the simple pleasure of being able to create new theorems undisturbed. Though he was shy and retiring, he did have a mischievous streak and would tease other mathematicians by writing letters stating his most recent theorem and challenging them to find a proof. Descartes referred to him as a braggart, and Wallis referred to him as that damned Frenchman. (Singh 39-40) Isolated from the French school of mathematics and not necessarily fondly remembered by his frustrated correspondents, Fermats discoveries were at risk of being lost forever. (??) Fortunately his eldest son, Clement-Samuel, was determined that his discoveries should not be lost to the world. It is thanks to his efforts that we know anything at all about Fermats remarkable breatkthroughs in number theory, and, in particular, if it were not for Clement Samuel, the enigma known as FLT would have died with its creator. (Singh 62)

Blaise Pascal (1623-1662) was a child prodigy. Strangely, his father. A wealthy tax collector and keen amateur mathematician, decided that his son should not study mathematics before the age of fifteen and, accordingly, removed all mathematics texts from their house. This prohibition only raised you Blaises curiosity about the banned subject, and he started to work on geometry in secret at the age of twelve. He discovered on his own that the sum of the angles of a triangle is two right angles; when his father found out, he was so impressed that he removed the ban and allowed his son to read mathematics texts, starting with the Elements. He also started to take the obviously gifted Blaise to meetings of Mersennes Academy. When Pascal was sixteen, he wrote his first paper, on conic sections, and presented it to Mersennes Academy. To assist his fathers tax-collecting work, the teenage Pascal also invented the calculation machine the Pascaline and oversaw its manufacture and sale. The device sold only in small numbers and eventually went out of production. The adult Pascal devoted his life to the study of mathematics, natural science, and religion, all privately supported by his familys fortune (he never took a university position). Pascal became ill in his twenties and never fully regained his health (died at 39); he lived most of his adult life in great pain. In his later years, he lost interest in mathematics and focused his attention on writing religious treatises. At a time when his father was injured and Jansenist brothers came to care for him, Pascal came under their influence, which caused him to feel that mathematics was a distraction from concern about his eternal soul that should be avoided, but the terror of eternal damnation eventually became too great, and his doctor recommended he go back to normal activities for a young man, at which point he took up mathematics again. Pascals results re: binomial theorem led Isaac Newton to his discovery of the general binomial theorem (which includes fractional and negative powers). (Devlin 16-22,54-56)

Sir Isaac Newton (1643(2)-1727) When Newton spoke of having stood on the shoulders of giants he was specifically referring to the work in physics and astronomy of Galileo and Kepler (he wasnt the first person to have used this quote). (Devlin 32) Pascals results re: binomial theorem led Isaac Newton to his discovery of the general binomial theorem (which includes fractional and negative powers). (Devlin 56)

Leibniz (1646-1714) He had a desire to unify the Catholic and Protestant churches. (Hellman 71). According to Hellman pp70ff, Leibniz had a hard time of things towards the end of his life, but in terms of calculus, though he had lost the battle in his own age, he eventually won the war.

Jakob Bernoulli (1655-1705) Jakob took up the baton of Fermat, Pascal and Huygens in terms of probability theory in his Ars conjectandi (The Art of Conjecture). One of his most significant contributions to mathematics was his law of large numbers. (Devlin 106-107)

Johann Bernoulli (1667-1748) Johann, along with his brother Jakob were the first mathematicians to recognize the importance of the calculus, to put it to use, and to spread the word of its significance. Two of Johanns pupils were Leonhard Euler and the Marquis de lHospital. (Note to self: lHospital wrote the first systematic textbook on calculus and is a much better mathematician than I was led to believe through rumors that he paid Bernoulli to let him put his name on lHospitals rule!!). (Hellman 74) Johann Bernoulli espoused the use of experiments to learn more about nature and came under attack because of it (Hellman 89).

Thomas Bayes (1702-1761) studied logic and theology at the University of Edinburgh and is known for his work in statistics and his statistical law. His father was a non-conformist minister (one of the first to be ordained in England), so Bayes was not allowed to study at Oxford or Cambridge.

Leonhard Euler (1707-1783) - in 1742, almost a century after Fermats death, Leonhard Euler asked a friend of his to search Fermats house in case some vital scrap of paper [relating to a proof] still remained. (Singh 32) Euler was the first mathematician to make a breakthrough toward proving FLT the case of n=3. (Singh 73,81) Euler showed that by incorporating imaginary numbers into his proof he could plug holes in the proof and force the method of infinite descent to work for the case n=3 (Singh 87) In his twenties, Euler lost the sight in one eye; this was something he saw as only a minor handicap, claiming, now I will have less distraction. Forty years later, at the age of sixty, his situation worsened when a cataract in his good eye meant he was destined to become completely blind. He was determined not to give in and began to practice writing with his fading eye closed in order to perfect his technique before the onset of darkness. Within weeks he was blind. He continued to produce mathematics for the next seventeen years, and, if anything, he was more productive than ever. It was during his blindness that he computed the lunar positions, which for the emperors of Europe was the most prized of mathematical achievements, a problem that had confounded the greatest mathematicians in Europe, including Newton. (Singh 87-88)

Sophie Germain (1776-1831) - She was 13 years old when the Bastille fell, an event that turned Paris into Bedlam. The streets were filled with discontented Parisians demonstrating their revolutionary sentiments, foraging for food, and reveling in the general anarchy. It was no place for a young girl. Sophies family was relatively wealthy and able to shield her, but the cost of this for her was long hours of solitude. She spent these hours in her fathers ample library. It was here that she same across the legend of Archimedes death as recorded by J. E. Montuclas History of Mathematics. She felt that if geometry was so engaging it must hold wonders worth exploring. [As Singh puts it on page 103: Germain concluded that if someone could be so consumed by a geometric problem that it could lead to their death, then mathematics must be the most captivating subject in the world.] Sophies family firmly and stubbornly opposed to her decision because they were concerned for her health, based on the customary wild stories of what happened to young girls who were too studious. They took desperate measures, denying her light and heat for her bedroom and confiscating her clothing once she retired for the night. After her parents were in bed, she would wrap herself in quilts, take out a store of hidden candles, and work at her books all night. After finding her asleep at her desk in the morning with the ink frozen in the ink horn and her slate covered with calculations, he parents finally hid the wisdom and grace to relent. Alone and untutored she went through every mathematics book her fathers library had as well as teaching herself Greek and Latin so that she could read Newton and Euler. She spent the years of the Reign of Terror studying Differential Calculus. In 1794 the Ecole Polytechnique opened, and she diligently set about collecting lectures of various professors and found the analysis of Lagrange particularly interesting a correspondence began with her using the pseudonym M. le Blanc, a former student at the academy. Her excellent work caught his attention, and Lagrange requested a meeting with her/him, and she was forced to reveal her identity. He was astonished to meet the young woman and became her mentor and friend. (Singh 104) Gausss Disquisitiones came out in 1801, and by 1804 Sophie had become so entranced that she sent him some results of her own investigations again using a pseudonym. [She came across Fermats Last Theorem and worked on the problem for several years, and, 75 years after Eulers breakthrough with n=3, worked towards a general approach using a particular type of prime that is now named for her 2p+1, where 2p+1 is prime and p is the Sophie Germain prime that generates it. Dirichlet and Legendre, both using her method, showed FLT for n=5. Also using Germains method, Lame was able to prove n=7. (Singh 104-106)] The two had an extensive correspondence. In 1807 her identity became known to him when she was concerned for his safety when French troops were besieging Breslau near Gausss home. (Osen 83-86) She sent a message to her friend General Joseph-Marie Pernety, who was in charge of the advancing forces. She asked him to guarantee the safety of Guass. As a result the general took special care of the German mathematician, explaining to him that he owed his life to Mademoiselle Sophie Germain. Gauss was grateful but surprised, as he had never heard of such a lady. In her next letter to him she reluctantly revealed her identity (Singh 107) Both Gauss and Lagrange were gracious to Sophie, and encouraging, after finding out her true identity. Gauss once wrote a sympathetic letter on this [see above] prejudice to Sophie Germain, the great French mathematician. In Gausss words, But when a person of the sex, which according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarize herself with these thorny researches, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, then without doubt she must have the noblest courage, quite extraordinary talents and a superior genius (Osen 4 from Bell also see Singh p108) Most of Sophies early work was in number theory, but at around the turn of the century mathematical interest in Paris began to focus on the work of Ernst Chladni who had studied the vibration of elastic surfaces by sprinkling them with a fine powder, strumming the edges with a bow, then noting the figures formed by the nodal lines. (Osen 88) She is considered one of the founders of mathematical physics. (Osen 83) Her father, though he opposed her studies during childhood, funded her research in adulthood; she never married (Singh 103) Her most important contribution to the subject was Memoirs on the vibrations of elastic plates, a brilliant and insightful paper that laid the foundation for the modern theory of elasticity. (Singh 108-109) Sadly, though her work applied to the building of the Eiffel tower, her name is not to be found among the 72 names of French scientists, mathematicians and engineers engraved thereon. Im finding conflicting accounts of the recommendation of a doctoral degree pushed for (or suggested) by Gauss. Either way, she died before she could receive it (or before it was considered not sure). She died of breast cancer at the age of 55. (Osen 91) When the Eiffel Tower was erected, special attention to the elasticity of the materials used had to be paid by the engineers. The names of 72 savants who had contributed to this field were listed, but Sophie Germain was not listed even though her researches contributed so much. (Osen 92)

Carl Friedrick Gauss (1777-1855) - Gauss once wrote a sympathetic letter on this [see above] prejudice to Sophie Germain, the great French mathematician. In Gausss words, But when a person of the sex, which according to our customs and prejudices, must encounter infinitely more difficulties than men to familiarize herself with these thorny researches, succeeds nevertheless in surmounting these obstacles and penetrating the most obscure parts of them, then without doubt she must have the noblest courage, quite extraordinary talents and a superior genius (Osen 4 from Bell). Gauss also wrote graciously to Caroline Herschel who presented her arrangement and complication of the work of John Flamsteed at Gottingen: These works so precious by themselves but much more so by the numerous enrichments from your own hand, shall always be considered as the greatest ornaments of the library of our Observatory. (Osen 79) Gauss had so many great ideas that he could often do no more than record them in his journal. Before he left the University of Gottingen, at the age of 21, he had virtually completed one of the masterpieces of number theory and of mathematics, the Disquisitiones Arithmeticae. (Ried 47)

Mary Fairfax Somerville (1780-1872) Marys mother taught her to read the Bible and say her prayers, but otherwise she was allowed to grow up a wild creature. Once she completed her domestic duties (caring for the poultry and the dairy) there was nothing to occupy her time. She had no playmates and abhorred childhood pastimes such as playing with dolls. Many of her lonely hours were spent exploring along the Scottish seacoast and through the dark moors that bordered her home. These explorations were to have a lasting effect on her life and interests in nature. Her early education was haphazard, scant, and self-directed. By age 10 she could hardly read. After her father returned from a long sea voyage he was shocked to find her a savage as a result of her carefree life, and he sent her away to a fashionable girls school in Musselburgh. Mary was utterly wretched there, and even in her old age her writings reflect the horror of the experience. After a year of study she was allowed to return home where she mostly studied birds and flowers and read the small collection of books available to her. This was fine with her family, but an aunt was appalled that Mary could not sew more than if she were man, so she was enrolled in sewing school to learn the tiresome art of stitchery. During this time of her life she was bored and her life didnt seem to hold out much promise. There was an attic window that faced north that afforded her a cozy place to muse about her problems and to study the stars by night, a pastime that also had a strong influence on her development. For a while the family took an apartment in Edinburgh, during which time she was able to learn piano, arithmetic, writing, and Latin, but these studies were discontinued when her family returned to Burntisland. She was feeling an acute need for education, but her family was opposed. It was during this time that she was at a tea party where she came across a fashion magazine where she chanced upon some algebraic symbols. As she puts it, At the end of the magazine I read what appeared to me to be simply an arithmetical question, but on turning the page I was surprised to see strange-looking lines mixed with letters, chiefly Xs and Ys, and asked, What is that? Oh, said the friend, its a kind of arithmetic; they call it Algebra; but I can tell you nothing about it. And we talked about other things; but on going home I thought I would look if any of our books could tell me what was meant by Algebra. Sadly, she could find no books in her home that addressed this topic. But chance favored her once again in an unlikely time and place. She had been sent to learn painting and dancing at Naysmiths Academy, and it was here in a discussion of perspective that Mary overheard the master of the school advice a male student to study Euclids elements, a book the master considered the foundation of perspective and mechanical science. She still faced the difficulty of obtaining a copy. It was considered unacceptable for a young girl to walk into a bookseller and ask for a copy of Euclid. She eventually got her copy through her youngest brothers tutor, Mr. Gaw. Mary happened to be sewing one day in the room where her brother was doing his lessons, and she involuntarily prompted him when he stumbled over an answer, which astonished the tutor. (Osen 96-102) As with Sophie Germain, when Marys parents caught her studying they confiscated her supply of candles, but she had covered most of the first 6 books of Euclid by then and worked from memory. [Singh also compares her and Germain on page 103 saying that Marys father stated, We must put a stop to this, or we shall have Mary in a straitjacket one of these days.} At age 33 she bought an excellent little library. She writes, I could hardly believe that I possessed such a treasure when I looked back on the day that I first saw the mysterious word, `Algebra and the long years in which I persevered almost without hope. It taught me never to despair. She also writes, I was considered eccentric and foolish, and my conduct was highly disapproved by many, especially by some members of my own family. They expected me to enteratin and keep a gay house for them, and in that they were disappointed. As I was quite independent. I did not care fo their criticism. A great part of the day I was osccupied with my children: in the evening I worked. (Osen 104) After her second marriage (each marriage had been to a cousin), William Somerville, a surgeon who was head of the Army medical department, her circle of friends grew. She got to know Laplace and discussed calculus and astronomy with him. She also came to know the Herschels and Dr. Whewell. (Osen 104-105) Marys work in translating Laplaces Celestial Mechanics was part of what re-opened British scholarly work to the continent after the passing of Sir Isaac Newton (Lord Brougham had written her husband to enlist his support in encouraging her to do the translation). (Osen 106-107) Her work inspired John Couch Adams to discover Neptune a sentence in her book inspired him to look for it based on unexplained movements of Uranus. Her translation of Laplace became a required textbook for the honor students at Cambridge. She was preached against at York Cathedral for her work Physical Geography. (Osen 110) Her book, The Connections of the Physical Sciences was called by some critics of the time the best general survey of physical sciences published in Britain up to that time. (Osen 113) She lived to be 92 and remained strong and active until the end, with several projects yet ongoing when she died. It is amazing to think what she could have accomplished had she been provided with education and support at a younger age. (Osen 114-115)

Augustin-Louis Cauchy (1789-1857) More concepts and theorems have been named for Cauchy than for any other mathematician. (Derbyshire [5] 126) Cauchy established a good, coherent theory of determinants. In a long paper that he read to the French Institute in 1812 he gave a full and systematic description of determinants, their symmetries, and the rules for manipulating them. This paper is generally considered the starting point of modern matrix algebra. It took 46 years, however, to get from the manipulation of determinants to a true abstract algebra of matrices (the first person to use the word matrix in this algebraic context was Sylvester in 1850).

Niels Henrik Abel (1802-1829) Abel was the first of the great trio of 19th century Norwegian algebraists. Though his life was very pitiable he seems to have been a cheerful and sociable man despite it all. He is generally credited with proving the algebraic unsolvability of the general quintic. At about the time Abel arrived in Berlin August Crelle had decided to found his mathematics journal. He published Abels unsolvability proof and many others of Abels papers. The unsolvability of the quintic was merely one minor aspect of Abels wide-ranging mathematical interests. His major work was in analysis, the theory of functions. Abels proof closed out the first great epoch in the history of algebra, but that wasnt recognized at the time, and it took a long time for his proof to become widely known. His proof also didnt put an end to the general theory of polynomial equations in a single unknown. Even though there is no algebraic solution to the general quintic, we know particular quintics have solutions in roots, so the question that arises is which quintic equations can be solved algebraically. A complete answer to this was given by Galois. (Derbyshire [5] 127-131)

William Rowan Hamilton (1805-1865) The germ of matrix theory, which is needed in quantum mechanics, existed in the quaternions developed by William Rowan Hamilton more than three quarters of a century before. (Ried 180-181)

Augustus De Morgan (1806-1871) He earned his bachelors degree at Trinity College, Cambridge; George Peacock had been one of his teachers. He was deeply religious but it was personal to him and he was no friend of any organized church, so he declined to take the required religious test upon graduation, went home to London and resigned himself to becoming a lawyer. He had barely registered at Lincolns Inn, however when the University of London was founded in Gower Street a university open to students of any sex, religion or political opinion. He was offered the chair of mathematics and he took it. De Morgans wife ran an intellectual salon, in the old French style, from their home at 30 Cheyne Walk, Chelsea. De Morgan had a puzzlers mind, with a great love of verbal and mathematical curiosities, some of which he collected in his popular boo A Budget of Paradoxes. (He was pleased to know that he was s years old in the year x2. De Morgans importance for the history of algebra is in his attempt to overhaul logic an improve its notations. Logic had undergone very little development since its origins under Aristotle. (Derbyshire [5] 182-183)

Ernst Kummer (1810-1893) spurred on by FLT was led to the introduction of ideal numbers and to the discover of the law of the unique decomposition of the numbers of a cyclotomic field into ideal prime factors. (Ried 75)

Evariste Galois (1811-1832) [For a great site that weights all the sources on Galois, read Tony Rothmans according to Derbyshire [5] 206] Abels proof closed out the first great epoch in the history of algebra, but that wasnt recognized at the time, and it took a long time for his proof to become widely known. His proof also didnt put an end to the general theory of polynomial equations in a single unknown. Even though there is no algebraic solution to the general quintic, we know particular quintics have solutions in roots, so the question that arises is which quintic equations can be solved algebraically. A complete answer to this was given by Galois. (Derbyshire [5] 130-131) The idea of a group as applied to permutations or substitutions is due to Galois, and the introduction of it may be considered as marking an epoch in the progress of the theory of algebraic equations (quote from Cayley Derbyshire [5] 191) The issue Galois was concerned with, the issue of what form the solutions of our equation will take, depends on the relationship between these two fields, the coefficient field and the solution field. THAT was Galoiss great insight. His discovery was that this relationship can be expressed in the language of group theory, which, in 1830, means the language of permutations. Among all useful permutations of the solution field there is a subfamily of permutations that leave the coefficient field unchanged. That subfamily forms a group, which we call the Galois group of the equation. (Derbyshire [5] 202-203) The group of even permutations of n objects is simple when n is 5 or greater that, fundamentally, is why the general quintic equation has no algebraic solution. (Derbyshire [5] 222) There is much to be said about Galoiss death in a duel perhaps best to go back to readings when the time comes see the Pistols at Dawn chapter of Derbyshire [5] noting especially the time-line given on page 210 - among other readings). His death took place at the same time in which the story of Les Miserables takes place (Derbyshire [5] 207), and had General Lemarque not have died it may have been the plan to use Galoiss death as the reason for the barricades to go up; Galois died May 30, 1832, and Lemarque died June 1, 1832). Galoiss brother and friends copied out his papers and circulated them to the big name mathematicians of the day, including Gauss, but there was no immediate result. Ten years later the French mathematician Joseph Liouville took an interest in the papers and published all of Galoiss papers three years later in a math journal he had founded himself. At that time Galoiss name became known to the larger mathematical community. (Derbyshire [5] 211)

James Joseph Sylvester (1814-1897) was the first mathematician to use the term matrix in an algebraic setting. This was 46 years after Cauchys definitive paper on determinants. Sylvester at this time (1850) was still thinking of matrices in terms of determinants. The first formal recognition of a matrix as a mathematical object in its own right was in a paper by Cayley in 1858. (Derbyshire [5] 174)

Ada Byron Lovelace (1815-1852) Ada had a very privileged background and was almost one of Queen Victorias ladies in waiting. Dickens, Faraday and Babbage were just some of the close acquaintances of Ada and her husband. Adas mother saw her father as having volatile, poetic, insanity, so she went about purging her childhood of any element of her fathers personality force-feeding her mathematics and science instead although they were seen at the time as the preserve of the male mind. Augustus de Morgan (who founded the math department at University College London) was Adas main tutor. Even de Morgan saw mathematics as a male domain, writing her mother why women should avoid doing hard math, the reason is obvious the very great tension of mind which mathematics requires is beyond the strength of a womans physical power of application although he write in the same letter that, Lady L. has unquestionably as much power as would require all the strength of a mans constitution. Ada threw herself into understanding the Analytical Engine, and saw, even more than Babbage himself, the full extent of what the analytical engine could actually think about. The hardware was only half the story the computer needed software if it were to be versatile enough to calculate any type of equation, and it was here that Ada would show her genius. During her twenties Ada said to her mother that she wanted to compensate for Byrons misguided genius if he has transmitted to me any portion of his genius then I will use it to bring out great truths and principles. Ada is certainly the worlds first published computer programmer. She saw the possibility of interchanging numbers and symbols (not just number crunching), one example would be that the engine might compose elaborate and scientific music, but Babbage just saw it as a number cruncher . Turing, who had built his machine, The Bombe, which he had search for patterns and which could exchange numbers and symbols, came across Adas work in the 1940s (after he had created his machine), and it is thanks to him that her work has come to light. A new programming language developed in 1986 was named ADA and was adopted by the UK air traffic control; Ada finally got to fly (she had invented a mechanical bird when she was young). (Calculating Ada: The Countess of Computers BBC)

George Boole (1815-1864) - De Morgans notational system for logic is now regarded as a historical curiosity, since De Morgan merely improved the traditional way of writing out logical formulas. What was really needed for progress in logic was a fully modern algebraic symbolism. That was supplied by George Boole (Derbyshire [5] 184) Boole was one of the new men of the early 19th century Britain, from humble origins and self-taught, financed by no patron and with nothing but his own merit and energy to help him rise. At the age of 16 his fathers affairs had collapsed, and George had to take a job as a schoolmaster to support the family (hes had such learning as his parents could afford, supplementing it with intensive studies of his own and at age 14 was producing translations of Greek verse). He continued schoolmastering for 18 years, running his own school for the most part. He opened his first when he was just 19. He took up a serious study of mathematics at about 17 and quickly taught himself calculus. By his mid-20s he was publishing regularly in the Cambridge Mathematical Journal. De Morgan helped Boole to get a paper on differential equations published by the Royal Society. In 1846, the British government announced an expansion of higher education in Ireland, Booles admirers (De Morgan, Cayley, and Lord Kelvin), agitated for Boole to be given a professorship at one of the new colleges. In 1849 he had become professor of mathematics at Queens College, Cork. He served in that position for 15 years until a November day in 1864 when he walked the two miles from his house to the college in pouring rain, lectured in wet clothes, and caught a chill. His wife believed that a disease should be treated by methods resembling the cause, so she put George in bed and threw buckets of icy water over him he died. His wife was the niece of Sir Geoerge Everest, the man the mountain was named after. They had five daughters, the middle one of whom, Alicia Boole Stott, became a self-taught mathematician and did important work in multidimensional geometry. Booles great achievement was the alebraization of logic the elevation of logic into a branch of mathematics by the use of algebraic symbols.

Arthur Cayley (1821-1895) - The first formal recognition of a matrix as a mathematical object in its own right was in a paper by Cayley in 1858. James Joseph Sylvester was the first mathematician to use the term matrix in an algebraic setting. This was 46 years after Cauchys definitive paper on determinants. Sylvester at this time (1850) was still thinking of matrices in terms of determinants. (Derbyshire [5] 174) Cayley has a fair claim to having been the founder of modern abstract group theory (H: Cayley Tables). (Derbyshire [5] 189)

Leopold Kronecker (1823-1891) Contrary to what I had thought previously, Kronecker was not a traditionalist but rather a maverick, as was Cantor. Because of the perception of Kroneckers persecution of Cantor, he has been unfairly stigmatized. His work foreshadowed the 20th century intuitionists as Cantors work foreshadowed the 20th century formalists. Cantor is not the only one with whom Kronecker fell out. (Hellman 141). Kronecker objected bitterly to Cantors introduction of completed infinities into set theory. Kronecker argued that uncountable sets like the real numbers do not belong in math that math can be developed without them, that they bring unwanted and unnecessary metaphysical baggage into the subject and that mathematics should be rooted in counting, algorithms and computation (note it is this school of thought that Brouwer brought forward into the 20th century. Brouwers version was called intuitionism, and the view of a mathematician who picked up his baton (Bishop) was called constructivism) (Derbyshire [5] 287) Kronecker was a tiny man, scarcely five feet tall, who had so successfully managed his familys business and agricultural affairs that he had been able to retire at the age of 30 and devote the rest of his life to his hobby, mathematics. As a member of the Berlin Academy, he had regularly taken advantage of his prerogative to deliver lectures at the University. He was now 63 and only recently, since the retirement of Kummer, had he become an official professor. Kronecker had made very important contributions, especially to the higher algebra. (Ried 25-26)

Bernhard Riemann (1826-1866) Riemanns dissertation is seen as one of the most important events of modern mathematical history. (Ried 68) Derbyshire refers to Riemann as perhaps the most imaginative mathematician that ever lived. Riemann created all of modern differential geometry, laying out the mathematics that Albert Einstein would pick up, 60 years later, to use as a framework for his general theory of relativity (Derbyshire [5] 265 & 268)

James Clerk Maxwell (1831-1879) In the 1840s two entirely new mathematical objects came forth, even if they werent entirely understood by their creators: vector space and algebra. Both ideas, even in their primitive states, created wide new opportunities for mathematical investigation and physical investigation as well. Faraday had come up with the concept of an electromagnetic field, which he saw all in terms of lines of force. His math, however, was insufficient to make his ideas rigorous. It was James Clerk Maxwell who filled out the math and found vectors to be exactly what they needed to express these new understandings. (Derbyshire [5] 156-157)

Peter Guthrie Tait (1831 1901) - a mathematical physicist best known for an energy physics textbook co-written with Lord Kelvin. He is also well known for his early investigations into knot theory, which contributed to the eventual formation of topology as a mathematical discipline. (Wiki)

Richard Dedekind (1831-1916) According to Derbyshire ([5] 230-231) Richard Dedekind was the best in the midcentury crop of German mathematicians who came in the wake of the Napoleonic Wars and the overhaul of German educational systems. This crop includes Dirichlet, Kummer, Helmholtz, Kronecker, Eisenstein, Riemann and Clebsch. Dedekind contributed to algebra in three ways first he gave us the concept of an ideal second he (with Heinrich Weber) opened up the theory of function fields third he began the process of axiomatization of algebra (the definition of algebraic objects as pure abstractions in the language of set theory). This axiomatic approach, when it reached full maturity half-century later became the foundation of the modern algebraic point of view. (Derbyshire [5] 231).

Lewis Carroll (1832-1898) Charles Lutwidge Dodgson (Dodson) his father was one of a long line of clergy stretching back for generations. He was awarded a Studentship (i.e. Fellowship) at Christ Church, Oxford, which entitled him to live in college for the rest of his life as long as he remained unmarried and prepared for holy orders. He enjoyed entertaining his many siblings, love the animal world (including snails and toads), and inherited a deep religious conviction that governed him. (Wilson 21-24) One day when he was a very small boy he came across a book of logarithms, brought it to his father (who had won a double first at Christ Church in Classics and Mathematics but forfeited his Studentship by marrying) and asked for an explanation. His father said he was much to young to understand such a difficult subject, but Charles insisted, But, Please explain! (Wilson 24). In the Wilson book many excerpts from Carrolls diary are included which really resonated with me. There is much about his approach to his teaching, his feelings about his teaching and himself as a teacher and also about concerns about his use of time (Wilson 51-66). He may have been the first to use recreational topics as a vehicle for conveying more serious mathematical topics, and many of his puzzles are identical to what Arthur Benjamin uses today! (Wilson 63-65) When he began publishing recreations in a magazine it was then that his publisher suggested he use a pseudonym. Dodgson sent in four possibilities, all plays on his name, and Lewis Carroll is the one selected by his publisher. Were he not so well known as a writer, he would have been remembered as the second greatest British photographer of the Victorian era, and the foremost photographer of children. He photographed Tennyson, Faraday, Ruskin and Dante Gabriel Rossetti. Sadly much nonsense has been written about Dodgsons friendships with children . . . Subjecting him to modern analysis rather than judging him in the context of his time is bad history and bad psychology and says more about the writer than it does about Carroll (Wilson 108-109). A boating trip in which he made up a story for Alice Liddell is the origin of Alice in Wonderland (1863); she asked him to write it down afterwards. (Wilson 109) In a passage of the Alice books Carroll is represented by a Dodo bird; he and the Liddell children had seen a stuffed one I their regular visits to the Oxford University Museum. (Wilson 111) Dodgson created a method for finding determinants of matrices called the Method of Condensation (which shortens the process, especially for matrices 4x4 and larger). (Wilson 118-120) In the early 1870s activity at the college caused Dodgson to become involved in election processes. He did much study here and this may be the most important of all his mathematical works. They have been called the most important in this field after Condorcets. (Wilson 132) He had some impact on voting in Britain and could have had more had he lived longer or been able to get around to publishing a second book on the topic (Wilson 148). Many (most) people enjoyed and continue to enjoy Carrolls mathematical riddles and puzzles, but others were (and probably still are) put off by them. Some interesting stories of this are shared in Wilsons book (163, 184). There is discussion of Carrolls Game of Logic in Wilsons book a full chapter is given to it and it seems it might be an improvement in some ways on Venn Diagrams, at least for working with syllogisms.

Edwin Abbott Abbott (1838-1926) Abbott was a boys school headmaster of a reforming and progressive cast of mind, inspired by a personal view of Christianity and skeptical of many of the conventions of Victorian society.

Sophus Lie (1842-1899) very interesting character see Derbyshire [5] 269-271

Georg Cantor (1845-1918) Cantor found comfort in interacting with Roman Catholic clergy and felt that he was a reporter of Gods revelation, claiming, From me, Christian Philosophy will be offered for the first time a true theory of the infinite. (Hellman 122, Dauben 144-147) In the late 1890s, with encouragement from Hilbert, Minkowski took advantage of the fact that he was a full professor to deliver a course of lectures on Cantors theory of th infinite. It was a time when, according to Hilbert, the work of Cantor was still actually taboo in German mathematical circles. (Ried 50) Cantor is the one who developed the concept of well-ordering; one outcome of this is that Z and Z^+ would have the same cardinality but a different order type. (Zermelo supplied the critical item needed for proving the well-ordering principle, the Axiom of Choice). (Hellman 145)

Felix Klein - (1849-1925) Known for his Erlangen Program, Klein was full professor at Gottingen when Hilbert replaced Weber. Kleins reputation drew students to Gottingen from all over the world, but particularly from the US. Klein had established a reading room (Lesezimmer) on the third floor of the classroom building (Auditorienaus), which became the center of mathematical life in Gottingen. The Lesezimmer was entirely different from any other mathematical library in existence at that time. Books were on open shelves and the students could go directly to them. Klein had also established on the third floor a collection of mathematical models. Klein prepared so thoroughly for his lectures that they were recognized as classics. Before he began he mapped out in his mind an arrangement of formulas, diagrams and citations. Nothing put on the blackboard during the lecture ever had to be erased. At the conclusion the board contained a perfect summary of the presentation, every square inch being appropriately filled and logically ordered. It was Kleins theory that students should work out proofs for themselves. He gave them only a general sketch of the method. The result was that a student had to spend at least four hours outside the class for every hour spent in class if he wished to master the material. (Ried 48) Klein was the force behind the desire for a separate building for mathematics at Gottingen, a Mathematical Institute (Ried 136)

Sonya Kovalevsky (1850-1891) Kronecker said of her, The history of mathematics will speak of her as one of the rarest investigators. (Osen 1) When she was a child she had long philosophical discussions with her Uncle Piotr, who, though he was not a trained mathematician, cherished a profound respect for mathematics and was able to transmit this reverence to Sonya. (Osen 119) (Her story of this reminds me of an essay by Borges who speaks of poetry speaking to him as a child before he was able to understand it. Sonya writes that he spoke to her of many mathematical ideas the sense of which I could not of course understand as yet; but which acted on my inspiration Osen 120 from Leffler) Another factor that may have attracted her to mathematics was the singular wallpaper that had been in one of the childrens rooms at Palibino. Not enough wallpaper had been sent out from St. Petersburg to repaper all of the rooms at the old estate. While waiting for more wallpaper to arrive the room had been partially covered with sheets of lithographed lectures by Ostrogradsky on differential and integral calculus that her father had bought in his youth. When, many years later as a girl of 15, she took her first lesson in differential calculus, her teacher was astonished at the quickness with which she grasped the material as if she had known them before. As soon as he began to explain the concepts she immediately and vividly recalled that all of this had stood on the pages that papered the wall. (Osen 120-121) Sonya was also gifted in a literary sense, as was her sister Anuita, who, while still in her teens published a short story in a popular magazine edited by Dostoevsky, thus beginning a close friendship with an elite circle of Eyropean intellectural living in Moscow. (Osen 121) Sonyas father had reluctantly allowed her to study mathematics at the naval school in St. Petersburg, he was unsettled at the prospect of her following this uncommon pursuit as a serious career. Another serious impediment to her study was the fact that Russian universities were closed to women students. (Osen 121) Sonya found a convenient means of escape from these restrictions by contracting a nominal, platonic marriage for the sole purpose of gaining the freedom to travel. She married Vladimir Kovalevsky over her parents objections. He was a student of paleontology and was quite amenable to the plans. They married in 1868 and set out for Heidelberg. She later studied with Weierstrass at Berlin (Osen 122-123) in this case private lessons as the university doors remained closed to her no matter how strong the recommendations in her favor. Weierstrass was by far the most influential teacher in her life. She was granted a doctorate from the University of Gottingen. Weierstrass attempted to find a position for her that was worthy of her talents, but he was rebuffed, leaving him disgusted with the bigoted and orthodox mentality of academic cliques. (Osen 127) During a stay in St. Petersburg in 1876, she met Mittag-Leffler who was able to persuade the Swedish authorities to let her lecture in Stockholm. Mittag-Leffler raised private funds to give her an official appointment at the university. Several donors pledged a small sum that the university matched so that she was able to earn a scant living for herself and her daughter Foufie. There continued to be discussion about the propriety of a woman teaching at the university, but 5 years after she began, Leffler was able to secure for her a tenured appointment. (Osen 131-132) She died unexpectedly at the age of 41. She was overwhelmed, was traveling back and forth from Stockhold to Moscow where her sister was dying a slow, painful death. A love affair had just ended. She was in a dispirited and hopeless mood. Her last journey between the two cities was a particularly difficult one. Her preoccupation with her worry and discouragement kept her from thinking about the details of travel, and she found herself caught in the middle of the night at a cold, deserted station where she was alone to struggle with heavy luggage. Bone tired and frozen, she became feverish and succumbed to influenza, which was rampant at the time.

Paul Wolfskehl (1856-1906) - Paul Wolfskehl an industrialist with an interest in mathematics - was obsessed with a beautiful woman, whose identify has never been established. Depressingly for Wolfskehl the mysterious woman rejected him and he was left in such a state of utter despair that he decided to commit suicide. He was a passionate man, but not impetuous, and he planned his death with meticulous detail. He set a date for his suicide and would shoot himself through the head at the stroke of midnight. In the days that remained he settled all his outstanding business affairs, and on the final day he wrote his will and completed letters to all his close friends and family. Wolfskehl had been so efficient that everything was completed slightly ahead of his midnight deadline, so to while away the hours he went to the library and began browsing through the mathematical publications. It was not long before he found himself staring at Kummers classic paper explaining the failure of Cauchy and Lame to solve Fermats Last Theorem. It was one of the great calculations of the age and suitable reading for the final moments of a suicidal mathematician. Wolfskehl worked through the calculation line by line. Suddenly he was startled wt what appeared to be a gap in the logicKummer had made an assumption and failed to justify a step in his argument. Wolfskehl wondered whether he had uncovered a serious flaw or whether Kummers assumption was justified. If the former were true, then there was chance that proving FLT might be a good deal easier than many had presumed. He sat down, explored the inadequate segment of the proof, and became engrossed in developing a mini-proof that would either consolidate Kummers work or prove that his assumption was wrong in which case all Kummers work would be invalidated. By dawn his work was complete. The bad news, as far as mathematics was concerned, was that Kummers proof has been remedied and the last Theorem remained in the realm of the unattainable. The good news was that the appointed time of the suicide had passed, and Wolfskehl was so proud that he had discovered and corrected a gap in the work of the great Ernst Kummer that this despair and sorrow evaporated. Mathematics had renewed his desire for life. Wolfskehl tore up his farewell letters and rewrote his will in the light of what had happened that night. Upon his death in 1908 the new will was read out, and the Wolfskehl family were shocked to discover that Paul had bequeathed a large proportion of his fortune as a prize to be awarded to whomsoever could prove Fermats Last Theorem. The reward of 100,000 Marks, apprxomiately the equivalent of a million dollars by todays standards, was his way of repaying a debt to the conundrum that had saved his life. The money was put into the charge of Gottingen University. (Singh 122-123) The interest on which was used to bring visiting lecturers to the university (which caused Hilbert to hope FLT would not be solved, because that would kill the goose the laid the golden egg!)

David Hilbert (1862-1943) Hilberts parents selected the renowned Friedrichskolleg (1872) for him a gymnasium that was reputed to be the best in Konigsberg, and where Kant had gone. It was an unfortunate choice for two reasons one was its traditional and rigid curriculum, and the other was that there were a plethora of boys at that time in Konigsberg who were strong in the sciences, but they all attended Altstadt Gymnasium, so Hilbert did not become acquainted with any of them. Hilberts passion was mathematics, but language classes formed by far the largest part of the curriculum. There was little opportunity for independent thinking or expression. Science was not offered. (Ried 3) David was considered to be a bit off his head by his family. His mother wrote his school essays for him, but he could explain mathematics problems to his teachers. No one really understood him at home. At a young age he had found the school subject that was perfectly suited to his mind and a source of inexhaustible delight. He said later that mathematics first appealed to him because it was bequem easy, effortless. It required no memorization. He could always figure it out again for himself. He knew he could not study it at university (i.e. go on to university) unless he obtained a diploma from gymnasium, so he slighted his favorite subject and concentrated on getting by in Greek and Latin. (Ried 6) Once at university (1880), the faculty members chose what they wanted to teach and the students what they wished to learn. There were no specified requirements, no minimum number of units, no roll call, no examinations until the taking of a degree, so many students responded to this sudden freedom by spending their first university years drinking and dueling, but for the 18-year-old Hilbert the university offered something more alluring, the freedom to concentrate, at least, on mathematics. (Ried 9-10) The days at university were a time of triumphant pessimism, a reaction against the almost religious belief in the power of science, which had flourished in the previous century. Emil duBois-Reymond, a physiologist turned philosopher, were widely read and much quoted. DuBois-Reymond concerned himself with the limits of the knowledge of nature. He maintained that certain problems were unsolvable even in principle. His gloomy concession, Ignoramus et ignorabimus we are ignorant and we shall remain ignorant was the catchword of many of the scientific philosophical discussions at the university. But to Hilbert, and to his friend Minkowski, such a concession was thoroughly abhorrent. (Ried 13) Once he became docent, Hilbert was resolved that he should educate himself as well as his students through his choice of subjects and that he would not repeat lectures (as many docents did). (Ried 28) What Hilbert considered to be a great and fruitful problem had all of the three characteristics: It was clear and easy to comprehend; it was difficult yet not completely inaccessible, and it was significant (a guidepost on the tortuous paths to hidden truths. (Ried 31) While a Privatdozent at Knigsberg, Hilbert published a proof of Gordans problem; Klein was so impressed by this proof that he decided then and there to get Hilbert on the staff. (Derbyshire {5] 282) Hilbert was a happy young man with a reputation as a snappy dancer and a charmer; he flirted outrageously. (Ried 36) Hilbert developed a series of lectures titled Foundations of Geometry, which, when they appeared in print, attracted attention all over the mathematical world. Unlike Euclid, Hilbert required that his axioms satisfy certain logical demands: that they were complete (so that all theorems could be derived from them), that they were independent (so that the removal of any one axiom from the set would make it impossible to prove at least some of the theorems) and that they were consistent (so that no contradictory theorems could be established by reasoning with them. (Ried 62-63) Hilbert would become totally taken up with one area of mathematics, and it was the only thing he would talk about, but then he could enter a new area of mathematics and produce in it, immediately, great mature work. (Ried 64) In 1899 and 1900 Hilberts mathematical interests were more varied than they had previously been. It was in the midst of this uncharacteristically diversified activity that an invitation arrived for him to make one of the major addresses at the second International Congress of Mathematics in Paris in the summer of 1900He had frequently reflected upon the importance of individual problems in the development of mathematics . . . he concluded his extensive introduction with a stirring re-iteration of his conviction that every definite mathematical problem must necessarily be susceptible of an exact settlement, either in the form of an actual answer to the question asked, or by the proof of the impossibility of its solution and therewith the necessary failure of all attempts. He took the opportunity to deny publicly and emphatically Ignoaramus et ignorabimus of duBois-Reymond, which had been so popular in the century that was passing. In mathematics there is no ignorabimus. (Ried 69-70) Hilbert, beginning in the late 19teens felt prompted to popularize mathematics: For it is true, he conceded, that mathematics is not, generally speaking, a popular subject. He saw the reason for the lack of popularity in the common superstition that mathematics is . . . a further development of the fine art of arithmetic, of juggling with numbers . . . . He thought he could bring about a greater enjoyment of the subject he himself enjoyed so thoroughly if he could make it possible for his listeners to penetrate to the essence of mathematics without having to weight themselves down under a laborious course of studies. He planned instead a leisurely walk in the big garden that is geometry so that each may pick for himself a bouquet to his liking. (Ried 154) Though, seemingly contradictory to this is Hilberts comment about a mathematician who had become a novelist. When the people in Gottingen marveled at this and wondered how he could do so, Hilbert replied, But that is completely simple. He did not have enough imagination for mathematics, but he had enough for novels. (Ried 175) He expressed that it was harder for mathematicians to popularize their subject than for those in the other sciences, especially biology. He said it still must happen, if we do it right that we find a beautiful dessin. During 1929 and 1930 he gave Sunday morning lectures in a series that was held a Gottingen. (Ried 192-193)

Hermann Minkowski (1864-1909) In the late 1890s, with encouragement from Hilbert, Minkowski too advantage of the fact that he was a full professor to deliver a course of lectures on Cantors theory of th infinite. It was a time when, according to Hilbert, the work of Cantor was still actually taboo in German mathematical circles. (Ried 50) In front of a group, Minkowski suffered from stagefright. Even after he became a teacher he was still embarrassed by attention even from younger people. In Zurich, his shy, stammering delivery of lectures completely put off a student named Albert Einstein. (Ried 92) Later, after 1905, when Minkowski was concentrating almost entirely upon electrodynamics, the work of the Bern patent clerk Einstein became known in Gottingen, and Minkowski recalled his former student, Ach, der Einstein, he said ruefully, Oh, that Einstein, always missing lectures I really would not have believed him capable of it! (Ried 105)

Bertrand Russel (1872-1970) notation from Peano

G. H. Hardy (1877-1947) a frequent visitor to Gottingen. When he left here to make his voyage home on the choppy North Sea is when he wrote his postcard to Harald Bohr that he had solved the Riemann Hypothesis as part of his personal war with God, knowing God would not allow him to die with such glory (Ried 163)

Emmy Noether (1882-1935) See Hilbert bio, esp pp. 142-143 also Derbyshire algebra Emmy grew up the daughter of math professor Max Noether of the University of Erlangen. Her home life was filled with provocative discussions that sparked an interest in her that was overpowering. (Osen 142) In 1900 when she decided to study mathematics at Erlangen, women were allowed to sit in on university classes, but only as auditors and only with the professors permission. By 1907 there had been some modest reforms, and Noether was awarded a doctorate by Erlangen, only the second doctorate in mathematics given to a woman by a German university. The habilitation degree, however, the second doctorate that would have allowed her to teach at university level, was still not open to women. For eight years she worked at Erlangen as an unpaid supervisor of doctoral students and occasional lecturer. (Derbyshire {5] 236) After her father retired and her mother passed away and her brother entered the army she was persuaded to move to Gottingen. Her own interests were close to those of David Hilbert, and on one of her visits he persuaded her to remain. Her work was useful to Hilbert and Klein who were then working on the general theory of relativity. (Osen 143-144) though German universities had granted doctorates to women, it was still hard to get habilitation permission for them -Hilbert fought hard for her - ended up declaring lecture in his own name but then she gave them big fight with senate on this though Emmy was accepted by fellow mathematicians, the era was not at all politically correct! (see Derbyshire [5] 235: Wilhelmine Germany was an exceptionally misogynist society, even by late 19th century standards. The German expression Kinder, Kirche, Kche (chidren, church, kitchen), supposedly identifying a womans proper place in society, is I think known even to people who dont speak German.) It was during the time of WWI that she arrived at Gottingen. She had been born and raised in Erlangen, and her father Max was a math professor there. From Ried: Emmy Noether had little in common with the legendary female mathematician Sonya Kovalevsky, who had bewitched even Weierstrass with her young charms as well as her mind. She was not even feminine in her appearance of manner. This is the first thing, eve today, that the men who knew her recall. She had a loud and disagreeable voice She looked like an energetic and very nearsighted washerwoman. Her clothes were always baggy. And they still quote with delight the gentle remark of Hermann Weyl that the graces did not preside at her cradle. But she was to be much more important to mathematics than the bewitching Sonya. (Ried 142-143) Also on page 143 of Ried is a description of Hilberts efforts to obtain a habilitation for her the argument of the philosophical faculty of what the returning soldiers would think about learning at the feet of a woman, and Hilberts retort about the senate not being a bathing establishment. When Hilbert still could not obtain her habilitation he would announce lectures in his own name but have them delivered by her. (Ried 143 & Derbyshire {5] 238) One of the most fertile circles of research in post WWI Germany revolved around her. The desired position of Privatdozent had at last been obtained for her in 1919 after the general liberalization of German society following defeat in WWI (Derbyshire [5] 238). This was still the lowest possible rank on the university scale, not a job, but a privilege. But she was delighted with the appointment. It had been 13 years since she defended her doctoral dissertation before Gordan. Once Dedekinds approach to the concept of ideal became familiar, it was clear that rings could have interesting internal structures like groups. People who were using rings at that time always had particular applications in mind, so there was no ring theory as such until Emmy Noether (The Lady of the Rings) came along after World War I that a coherent theory emerged, embracing all these areas and setting them on a firm axiomatic footing. (Derbyshire [5]233-234) In 1922 she became a nicht beamteter ausserordentlicher Professor an unofficial extraordinary, or associate, professor. There were no obligations connected with this new title and no salary. By this time inflation had so reduced the students ability to pay fees that if the Privatdozents were not to starve away they had to be given some small sums by the University for delivering lectures in their specialties. Such a Lehrauftrag for algebra was no awarded to Emmy Noether, the first and only salary she was ever to be paid in Gottingen. (Ried) By 1930 she had established herself as the most vigorous focus for the proud mathematical tradition at Gottingen; her personal life was a quiet one, and her days were spent at work in the beautiful new Mathematical Institute. (Osen 148-149) Early in 1933, the National Socialists rose to power, and after Hitler came to power Emmy had to leave Gottingen, having three strikes against her that of being an educated woman, that of being a Jew, and that of being liberal. A place was found for her at Bryn Mawr in the US. She returned to Gottingen in the summer of 1934 her heart knew no malice; she did not believe in evil indeed it never entered her mind that it could play a role among men. (Ried 207) After a year and a half at Bryn Mawr and Princeton she died very suddenly on April 14, 1935 following an apparently successful operation. She was only 53 years old and at the apex of her productive power and technical skill. Albert Einstein said of her In the judgment of the most competent living mathematicians Fraulein Noether was the most significant creative mathematical genius thus far produced since the higher education for women began. In the realm of algebra in which the most gifted mathematicians have been busy for centuries she discovered methods which have problems of enormous importance in the development of the present day younger generation of mathematicians. (Osen) Hermann Weyl, who cared for her very much, also wrote a touching tribute to her, in which he mentions her nickname der Noether as signaling how she had broken through barriers restricting her sex. (Osen)

E. T. Bell (1883-1960) Bell spent the last year of his life in Watsonville hospital. He had a room there in which he had his books, reprints, novels and his popular mathematics books. However, he spent most of the days reading poetry. Often he would have a kitten on his bed. At first he smoked cigars but after he set the bed on fire he was told that he could only smoke if someone else was in the room. He died in the hospital fifteen days after signing the contract to haveThe Last Problempublished by Simon and Schuster. He was a rock-solid mathematician with many publications to his name, but when it came to writing the history of mathematics he allowed himself to romanticize it and not to discriminate between anecdote and fact. Some of his popular work was more accurate than other and the reason I mention his work The Last Problem is that this is the book that inspired 10-year-old Andrew Wiles. Julia Robinson felt that this work was quite accurate (as opposed to his Men of Mathematics, though his Men of Mathematics did inspire many people to go into mathematics). I find the timing both interesting and touching. Wiles was born in 1953. Bell died in 1960. Bells book was published in 1961, and Wiles found it in Milton Road Library, Cambridge in 1963.

Hermann Weyl (1885-1955) It was not just mathematics that interested Weyl. There was philosophy. Art. Literature. He was convinced that the problems of science could not be separated from the problems of philosophy; also convinced that mathematics like art, music and literature was a creative activity of mankind. He loved to write, and wrote well. (Ried) He took Hilberts place at Gottingen after Hilberts retirement (Ried 191)

Andrew Wiles (b. 1953) Quote: It looked so simple, and yet all the great mathematicians in history couldnt solve it. Here was a problem that I, a ten-year-old, could understand, and I knew from the moment that I would never let it go. I had to solve it. (Singh 6) Andrew Wiles had come across the book The Last Problem by Eric Temple Bell at the Milton Road Library in Cambridge. (Singh 5)

HISTORICAL SWEEP

Ancient

Some of the oldest written texts known to us that contain any mathematics at all contain material that can fairly be called algebraic. Those texts date from the first half of the second millennium BC, from 37 or 38 centuries ago, and were written by people living in Mesopotamia and Egypt. To a person of our time, that world seems inconceivably remote. The year 1800 BC was almost as far back in Julius Caesars past as Caesar is in ours. (Derbyshire [5] 19)

In about 1650 BC the scribe Ahmes copied over what would come to be called the Rhind Papyrus. This was during the Hyksos Dynasty (first invasion of Egypt by outsiders no such thing had happened in its previous millennia and a half existence). What Ahmes was copying was a document written in the Twelfth Dynasty (c. 1990-1780 BC). This may have been one of the restoration projects by the Hyksos who were respectful of the then-ancient Egyptian civilization. (Derbyshire [5] 29)

Through all the changes and conquests in Mesopotamia the people went on writing in cuneiform all the way until the conquest by the Parthians in 141 BC. There are mathematical texts in cuneiform through this time, and it is an astonishing thing, testified to by everyone who has studied this subject that there was almost no progress in mathematical symbolism, technique, or understanding during the millennium and a half that separates Hammurabis empire from the Parthian conquest. The mathematician John Conway, who has made a study of cuneiform tablets, says that the ONLY DIFFERENCE that presents itself to the eye is a positional zero . . . As with Mesopotamia, so with Egypt: We have no ground for thinking that Egyptian mathematics made any notable progress from the 16th to the 4th century BC . . . Though it seems mathematicians from Mesopotamia and Egypt made no mathematical progress over this scope of time, their brilliant early discoveries had spread throughout the ancient West, and possibly beyond. From this point on in fact, from the 6th century BC the story of algebra in the world is a Greek story. (Derbyshire [5] 32)

That there are other kinds of numbers, neither whole nor rational, was discovered by the Greeks about 500 BC. The discovery made a profound impression on Greek thought and raised questions that even today have not been answered to the satisfaction of all mathematicians and philosophers. (Derbyshire [5] 10)

The ancient Greeks inherited a legacy of accumulated knowledge and ideas from older Babylonian and Egyptian cultures a legacy that the Greeks used well . . . One mathematical contribution that is distinctly Greek, however, is the whole notion of proof, deduction and abstraction. (Osen 14) There are some interesting comments about this by Grabiner in her first lecture on the Elements noting that the Greeks were the first and the only in the world to have developed the axiomatic method and why this may have come about. She gave 3 theories (one having to do with differences in the values of pi given by the Babylonians and the Mesopotamians and wanting to be able to have a solid, agreed-upon foundation other possibilities have to do with the environment of argumentation and philosophy, mathematic s and logic having become linked another having to do with the Greeks having a desire to explain everything based on a single principle (all is water, all is air, all is number, the atoms of Democritus, etc.)

Hippocrates of Chios (470-410 BC) lived a generation before Plato, so Plato had his work available. His work was somewhat like that of Euclid, with propositions and proofs. It is no longer extant, having been so eclipsed by Euclids elements. (Grabiner 1st elements lecture) Plato posted over his academy, Let no one ignorant of geometry enter here Socrates had taught him, he taught Aristotle, Aristotle taught Alexander the Great, Alexander conquered the world and founded Alexandria. According to the earliest source of information (a letter from about 150 BC) the library was initially organized by Demetrius of Phaleron, a student of Aristotle under the reign of Ptolemy I Soter though it may have been Ptolemy II (Grabiner states Ptolemy I).

The peculiarity of Greek mathematics is that prior to Diophantus it was mainly geometrical. The usual reason given for this . . . is that the school of Pythagoras (late 6th century BC) had the idea to found all mathematics and music and astronomy on number but that the discovery of irrational numbers so disturbed the Pythagoreans, they turned away from arithmetic, which seemed to contain numbers that could not be written, to geometry, where such numbers could be represented infallibly by the lengths of line segments. (Derbyshire [5] 32)

Alexandrian mathematics went into something of a decline after the glory days of the 3rd century BC, and in the disorderly first century BC (think of Anthony and Cleopatra) seems to have died out altogether. With the more settled conditions of the early Roman imperial era, there was a revival. There was also a turn of thought away from purely geometrical thinking, and it was in this new era that Diophantus (c.200 c. 285) lived and worked. [The term era is used broadly here, as we are not even sure of the century in which he lived the 3rd century is the most popular guess] (Derbyshire [5] 34)

After his time the literal notation of Diophantus, which was SUCH a huge leap forward, was for some reason either forgotten or ignored, which seems surprising given that it was taught at Alexandria by Hypatia (360-415 AD) and that Enthusiastic students came from Europe, Asia, and Africa to hear her lecture on the Arithmetica of Diophantus. (Osen 26)

Hypatia (360-415 AD) wrote On the Conics of Apollonius, popularizing his text. It is interesting to note that with the close of the Greek period, interest in conic sections waned, and after Hypatia, these curves were largely neglected by mathematicians until the first half of the seventeenth century. (Osen 27) It seems she was the last to teach at the Museion of Alexandria and that her appalling death marked the end of mathematics in the ancient European world. (Derbyshire [5] 45)

700-800

Baghdad under the Abbasid Caliphs (5th, 6th and 7th) was a great cultural center. All that was needed to make it an ideal center for the preservation and enrichment of knowledge was an academy, a place where written documents could be consulted and lectures and scholarly conferences held. Such an academy soon appeared. It was called Dar al-Hikma, the House of Wisdom. This was the time when al-Khwarizmi lived and worked.

900s

The growth of any discipline depends on the ability to communicate and develop ideas, and this in turn relies on a language that is sufficiently detailed and flexible. The ideas of Pythagoras and Euclid were no less elegant for their awkward expression, but translated into the symbols of Arabia they would blossom and give fruit to newer and richer concepts. In the tenth century the French scholar Gerbert of Aurillac learned the new counting system from the Moors of Spain and through his teaching positions at churches and school throughout Europe he was able to introduce the new system into the (academic) West. In 999 he was elected Pope Sylvester II, an appointment that allowed him to further encourage the use of Indo-Arabic numerals. Although the efficiency of the system revolutionized accounting and was rapidly adopted by merchants (?), , it did little to inspire a revival in European mathematics. (Singh 55)

1200s Liber Abbaci was, by the standards of its time, wonderfully innovative and very influential. For 300 years it was the best math textbook available that had been written since the end of the ancient world. It is often credited with having introduced Arabic numerals, including zero, to the West.

1200s-1300s+

NB There was some strong activity in medieval Europe including Fibonacci, Oresme and Regiomontanus Black Death hit 134653 Thomas Bradwardine, mathematician and Archbishop of Canterbury (mentioned in Chaucers Tales) died in 1349, possibly of plague a third to a half of Europes population died which may be why math stalled for a couple centuries see Boyers History for this and also for mentions of fall of Constantinople and if that really did bring more documents in to Europe (or the opposite)

1400s

A portion of Hypatias original treatise On the Astronomical Canon of Diophantus was found during the fi