Monday 28 November 2011

Edward Waring


Edward Waring

Edward Waring, despite not being very well known even today, was one of the most talented mathematicians of his time. He was cursed by the inability to properly communicate his ideas coupled with a fascination for esoteric mathematical topics. One of his biographers wrote:

Waring was one of the profoundest mathematicians of the eighteenth century; but the inelegance and obscurity of his writings prevented him from obtaining that reputation to which he was entitled.
Waring was born in Shropshire, England in 1736. His father was a successful farmer and he attended school in Shrewsbury. In 1753, he entered Magdalene College at Cambridge. Originally, he entered as a sizar which meant that he paid a reduced admission fee but had to take on extra duties at the school.

His mathematical abilities soon drew the attention of his teachers. He graduated with top honors in 1757. One year after graduation, he was elected as a fellow to Magdalene College. In 1759, his name was put forward as the Cambridge Lucasian Chair of mathematics even though he was only two years past graduation.

William Powell, one of the professors at St. John's College challenged this nomination. Powell wrote a pamphlet questioning Waring's mathematical knowledge. Waring responded to this with his own pamphlet and Powell wrote a rebuttal. The debate finally ended when a famous mathematician of this time, John Wilson, intervened on Waring's behalf. In 1760, Edward Waring became Lucasian Professor of Mathematics. He had not yet turned 24.

In 1762, Waring published his most famous work: Meditationes Algebraicae. The work shows his thoughts on topics in equations, number theory, and geometry. The work was well received and he was elected to the Royal Society in 1763. He would later extend this work into three separate volumes.

Despite the book's high praise by many top mathematicians, the book was not widely read among mathematicians. In 1764, an influential math book claimed that there were no first rate mathematicians in England. Waring was alarmed at being overlooked but admitted:

... never could hear of any reader in England, out of Cambridge, who took pains to read and understand it ...
Even though he was the Lucasian Professor of Mathematics, he decided to also study medicine. He received a medical degree in 1767. His medical career did not go as well as he had hoped and he gave up medicine by 1770. In 1776, he married Mary Oswell.

After quitting medicine, he expanded his original mathematical work: releasing a volume on geometry in 1770 and a volume on number theory and equations in 1772.. In these works, he did significant work with symmetric functions and also the cyclotomic equation. His ideas were a precursor to what later became group theory. In number theory, he presented a problem that was later solved by David Hilbert in 1909.

Despite being the Lucasian Professor, he did not lecture much. In fact, he did not correspond very often with the mathematicians of his day. His works were not systematic and most of his ideas were hundreds of years ahead of their time.

As he grew older, he struggled with poverty. In 1795, three years before his death, he resigned from the Royal Society because he could not afford its dues. He died on August 15, 1798.

François Viète


François Viète

François Viète was born in 1540 in France; he was a successful politician who also made very important contributions to algebra. He is credited for example with introducing letters to represent known and unknown values in mathematical equations. His writings were also important in establishing the symbol (+) as representing addition and (-) as representing subtraction.

His family was well connected; his mother, for example, was the first cousin of the president of Parliament in Paris. Viète was able to take advantage of these connections through out his life. He attended the University of Poitiers and in 1560, he graduated with a law degree.

Although a lawyer by profession, he took a strong interest in math and science and published his first mathematical paper in 1571. He was especially interested in the works of Pappus and Diophantus.

Viète was a Huguenot and he lived in Paris during the St. Bartholomew's Day Massacre where many thousands of protestant Hugeuenots were killed.

In 1573, he became a councillor at Rennes where he remained until 1580. At that time, he was appointed by King Henry III to be a royal privy councillor in Paris. In 1584, King Henry III's younger brother became ill and died; this meant that Henry of Navarre, a protestant, became heir to the throne. In the struggle that erupted, Viète, a protestant, was kicked out of office.

Viète left Paris and headed to the small town of Beauvoir-sur-Mer. He spent five years there where he was able to devote time to studying mathematics. It was during this time that he did most of his most imporant work relating to cubic equations and mathematical notation.

Viète believed that the Greeks had not revealed all their mathematical insights. He believed that they had secret methods which he hoped to rediscover. In this way, he introduced what would later become variables and coefficients. In Viète's view, he had restored the hidden mathematical methods of the Greeks.

In 1589, he was called back to parliament in Tours. This happened just before Henry III was assassinated on August 1, 1589. He stayed on as the protestant Henry of Navarre became King Henry IV.

In 1590, a coded letter to King Philip II of Spain was intercepted by the French. Viète, at this time, had a very strong reputation in mathematics and he was given the task of deciphering the note. This he accomplished in March of 1590. The MacTutor web site quotes a historical text:

... when Philip, assuming that the cipher could not be broken, discovered that the French were aware of his military plans, he complained to the Pope that black magic was being employed against his country.
When in 1592, King Henry IV converted to Catholicism. Viète did the same.

In 1593, there was a challenge made to all mathematicians by Adriaan van Roumen, a professor of math at Louvain. Roumen asked for a solution to an equation which had 45 terms. Viète was able to solve this problem. In fact, Roumen asked for one solution. Viète was able to demonstrate that there were 23 positive solutions.

Viète was very wealthy. He published all of his writings himself and sent them out to scholars throughout Europe. In all of his writings, he only focuses on positive values. Like many of the mathematicians before him, he did not include consideration of negative numbers in his writings.

He was a member of the Royal Privy Council until his death in February, 1603.

Pierre Laurent Wantzel


Pierre Laurent Wantzel

Pierre Larent Wantzel was born on June 5, 1814 in Paris, France. His father was a professor of applied mathematics at the Ecole speciale du Commerce. From an early age, Wantzel showed tremendous ability in mathematics. Ademar Jean Claude Barre de Saint-Venantrelates an anecdote that when Wantzel was 9 years old, his teacher would send for him to help with certain difficult surveying problems.

He entered the Ecole des Arts et Metiers when he was 12. After a year, Wantzel decided that the school was not academic enough and switched to the College Charlesmagne in 1828. He would later marry the daughter of his language coach at the College Charlesmagne.

At 15, Wantzel edited a famous mathbook, Antoine-Andre-Louis Reynaud's Treatise on Arithmetic and added a proof for a method of finding square roots that had previously not had a proof. Later, in 1831, he he was awarded first prize in Latin disseration. When he applied in 1832 for the prestigious Ecole Polytechnique and the Ecole Normale, he placed first in both examinations. By 1838, he had become a lecturer in mathematics at the Ecole Polytechnique and in 1841, he also became a professor of applied mathematics at Ecole de Ponts et Chaussees.

In 1837, Wantzel became the first to prove the impossibility of duplicating the cube and trisecting an angle using only ruler and compass. The great Carl Friedrich Gauss had stated that both of these methods were impossible but had never provided proof.

In 1845, Wantzel gave a revised proof of Abel's Theorem on the impossibility of solving all equations of n ≥ 5 by radicals. In this presentation, Wantzel gave a revised proof of the one done by Paolo Ruffini. In all, he wrote over 20 works on a wide range of subjects.

By May 12, 1848, Wantzel's never-ending pattern of very hard work without break and opium-use caught up with him and he died at the very early age of 33.

Saint-Venant writes:
... one could reproach him for having been too rebellious against those counselling prudence. He usually worked during the evening, not going to bed until late in the night, then reading, and got but a few hours of agitated sleep, alternatively abusing coffee and opium, taking his meals, until his marriage, at odd and irregular hours.
One might wonder how why Wantzel doesn't rank with the greatest mathematicians. Without a doubt, he showed tremendous promise as a child and with all of his hard work, one wonders why he did not reach the highest heights in mathematics. Saint-Venant writes:
...I believe that this is mostly due ot the irregular manner in which he worked, to the excessive number of occupations in which he was engaged, to the continual movement and feverishness of his thoughts, and even to the abuse of his own facilities. Wantzel improvised more than he elaborated, he probably did not give himself the leisure nor the calm necessary to linger long on the same subject.

Nicolo Fontana Tartaglia


Nicolo Fontana Tartaglia

Nicolo Fontana "Tartaglia" was born around 1500 in Venice. His father was a "mail rider", a person who would ride from town to town, to make deliveries. When Tartaglia was six years old, his father was murdered during one of these trips.

When Tartaglia was 12, the French took over his town of Brescia. There had been fierce fighting and in retaliation, there was a great slaughter of the Brescian townfolks. It is said that 46,000 inhabitants were killed in the fighting and the revenge killings. Tartaglia, his mother, and his sister hid away in a cathedral. In the course of the fighting, Tartaglia was attacked and left for dead.

Somehow, he survived but for the rest of his life, he bore terrible scars near his jaw and had difficulty speaking. It was in this way that he got his nickname "Tartaglia" which means "the stammerer". As an adult, he would always wear a beard in order to hide the scars.

Tartaglia took an interest in mathematics and demonstrated strong ability which earned him the patronage of Ludovico Balbisonio who paid for him to attend the University of Padua. By 1516, he had moved to Verona where he made his living by lecturing on mathematics.

In 1534, Tartaglia was working as a math teacher in Venice. While this was still a low paying position, he began to develop a reputation by participating in public mathematical challenges.

In 1535, there was a famous math contest between Tartaglia and Fior. Fior was a student of Scipione del Ferro who had already solved the depressed cubic equation and had given the solution to Fior before he died. Tartaglia had by this time found a solution to one form of the cubic equation which was not directly solvable by the depressed cubic equation. Both Tartaglia and Fior submitted 30 problems that the other had to solve. Tartaglia could tell that Fior had a solution to the depressed cubic so he worked hard to find it. On February 13, 1535, Tartaglia solved the depressed cubic. Tartaglia easily solved all 30 of Fior's problems and won the contest.

Girolamo Cardano heard about Tartaglia's solution of the depressed cubic. Cardano, by this time, had a reputation as a brilliant and well-connected mathematician and physician. He approached Tartaglia with the hope of including Tartaglia's result in a book he was working on. Tartaglia declined.

Cardano promised to provide a letter of introduction to a very influential patron in Milan and invited Tartaglia to his home. Tartaglia, who still held his teacher's job, accepted Cardano's invitation to visit. While there, Cardano was able to convince Tartaglia to reveal his method for solving the depressed cubic. Cardano promised that he would not publish the solution without getting permission from Tartaglia.

Using Tartaglia's method, Cardano was able to solve the general cubic equation and his assistant Lodovico Ferrari was able to solve the general quartic equation. Tartaglia refused to give Cardano permission to publish the method. It is probable that Tartaglia believed that he could use his method to continue to win math challenges.

When Cardano learned that Scipione del Ferro was the first to solve the depressed cubic, he decided to go ahead and publish his book Ars Magna even without Tartaglia's permission. In the book, he gives Tartaglia and del Ferro credit for the solving the depressed cubic. He then shows how the general cubic equation can be reduced to a form of the depressed cubic.

This publishing of Cardano's book infuriated Tartaglia who proceeded to publish his own book called New Problems and Inventions. In it, he attacked Cardano. Despite this, Cardano's book was a bestseller and established Cardano as the leading mathematician of his age.

Tartaglia challenged Cardano to a public math debate but Cardano refused. Cardano's assistant Ferrari offered to take up the challenge in Cardano's place. At this time, Ferrari was a relatively unknown mathematician and Tartaglia had little to gain from the challenge. On the other hand, if he had succeeded in challenging Cardano, because of Cardano's reputation, the debate could have helped his mathematical standing.

Tartaglia was offered a very lucrative post that he could get if he defeated Ferrari in the challenge. The math debate between Tartaglia and Ferrari occurred on August 10, 1548. Ferrari had clearly mastered the general cubic and quartic equations and it was very clear that Tartaglia would not be able to answer all of Ferrari's problems. Rather than admit this, Tartaglia left before the challenge completed.

Having lost the challenge, Tartaglia found that his existing position was threatened. In this way, Tartaglia saw his reputation collapse. It is said that for the rest of his life, he harbored resentment against Cardano. It is also said that he died in poverty. Tartaglia died on December 13, 1557.

Alexandre-Theophile Vandermonde


Alexandre-Theophile Vandermonde

Alexandre-Theophile Vandermonde was born on Feb 28, 1735 in Paris, France. From a young age, he showed a strong interest in music and this interest was encouraged by his father. Vandermonde played the violin and for these early years did not show any special interest in mathematics. Today, Vandermonde is remembered primarily for his mathematical achievements.

Vandermonde did not take up math until he was already 35 years old. At 35, he wrote a paper that created enough of a stir in the scientific community of France to get him selected to the Academie des Sciences in 1771. In total, he wrote only four mathematical papers.

In 1778, he chose to work on paper regarding musical theory. Many might have expected a mathematical analysis that extended the work of the Ancient Greeks. Instead, Vandermonde argued that there should be no theory of music as mathematical theory, but, instead, musicians should judge music solely based on their trained ear.

Throughout his life, Vandermonde collaborated with the greatest talents of his time. In 1777, he collaborated with Etienne Bezout and Antoine-Laurent Lavoisier to study the effects of the severe frost of 1776. In 1787, he worked with Gaspard Monge and Claude Louis Berthollet to research the manufacturing of steel with the aim of improving bayonets. In this effort, he investigated different combinations of iron and carbon.

In Vandermonde's first paper on mathematics, he presented a solution to the eleventh root of unity using ideas of symmetrical polynomials and permutations. These ideas would predate both the work of Carl Friedrich Gauss and Evariste Galois.

Henri Lebesgue wrote (quoted from Jean Tignol, see below):
Surely, any man who discovers something truly important is left behind by his own discovery; he himself hardly understands it, and only by pondering it for a long time. Vandermonde never came back to his algebraic investigations because he did not realize their importance in the first place, and if he did not understand them afterwards, it is precisely because he did not reflect deeply on them: he was interested in everything, he was busy with everything; he was not able to go slowly to the bottom of anything...
Today, Vandermonde is best known for something that may have been accidentally named after him: the Vandermonde determinant. The idea is not found in any of his papers. Still, the last of his four math papers did focus on determinants in which for the first time, a determinant is presented as a mathematical function and Vandermonde proceeds to investigate its mathematical properties. One mathematician, Thomas Muir, has gone so far to say that Vandermonde is "the only one fit to be viewed the founder of the theory of determinants."

Paolo Ruffini


Paolo Ruffini

Paolo Ruffini was born on September 22, 1765 in what is today Italy. In 1783, he entered the University of Modena where he studied geometry, calculus and medicine. In 1787, he was asked to teach the course on calculus even though he was still a student. Later, in 1788, he became a professor at the university.

Even as he taught mathematics, he continued to study medicine and in 1791, he received his license to practice medicine.

1796 was the time of Napoleon Bonaparte who took over Modena. Napoleon set up what became the Cisalpine Republic that included Modena. All faculty of the university were required to swear an oath of allegiance to the new republic.

Ruffini refused to swear this oath and so lost his job and was no longer allowed to teach. This change enabled him to focus on medicine and to actively study his own projects in mathematics.

One of his math projects was an effort to solve the quintic equation in terms of radicals. Thequadratic equation was long known to be solvable. The cubic equation had been solved byGiralomo Cardano and the quartic equation had been solved by Lodovico Ferrari. No one had yet solved the quintic.

It was at this point, that Paolo Ruffini approached the problem differently than others had before him. He was the first person to propose that the problem was not solvable in terms of radicals. In 1799, Ruffini published his math treatise which he titled: General theory of equations in which it is shown that the algebraic solution of the general equation of degree greater than four is impossible.

In the introduction, he writes:
The algebraic solution of general equations of degree greater than four is always impossible. Behold a very important theorem which I believe I am able to assert (if I do not err): to present the proof of it is the main reason for publishing this volume. The immortal Lagrange, with his sublime reflections, has provided the basis of my proof.
In writing this work, he made arguments in terms of group theory. Since group theory had not yet been invented, Ruffini had to invent concepts which made his argument even hard to follow. In this work, we find such concepts as order, cycles, and the notion of primitive elements. Joseph-Louis Lagrange had previously written about permutations but Ruffini applied these points to his argument in ways that Lagrange had not. In this work, he successfully proved some very important theorems of group theory.

Unfortunately, he did not succeed in proving his claim about the quintic equation. He came very close. So close, in fact, that it was later accepted by Augustin Louis Cauchy as valid. In retrospect, it is clear that his argument had a significant gap which would later be resolved by Niels Abel. Ruffini assumed but failed to demonstrate that:
"...if an expression by radicals is a root of the general equation of some degree, then every function of it is composed is a rational expression of the roots." (Jean-Pierre Tignol, p211).
In fairness to Ruffini, he received no response from the mathematical community on this point. He sent a copy to Lagrange in 1801. He got no response. He sent out a second copy of his book with the following message:
Because of the uncertainty that you may have received my book, I send you another copy. If I have erred in any proof, or if I have said something which I believed new, and which is in reality not new, finally if I have written a useless book, I pray you point it out to me sincerely.
He still got no response. In 1802, he wrote a follow up letter. Again, no response.

A mathematician named Gian Malfatti did respond with objections that today are viewed as a misunderstanding of Ruffini's ideas. Ruffini responded with additional proofs that are today accepted as answering Malfatti's objections.

Why didn't anyone respond? The answer seems to be that Ruffini was tackling a question in a way that no one wanted to be right. The mathematical community was not yet ready to concede that quintic equations were not solvable by radicals. In addition, Ruffini's proof was over 516 pages and the mathematical argument was difficult to follow.

In 1810, Ruffini asked the Institute of Science in Paris to make a formal statement on the validity of his proof. After over a year, Adrien-Marie Legendre, Lagrange, and Sylvestre Francois de Lacroix concluded that there was nothing of importance here and that it was not "worthy of attention."

Ruffini made the same request of the Royal Society. The response was polite but again reflected disinterest in the topic area.

In 1814, after the fall of Napoleon, Ruffini was reinstated back at the University of Modena. All this time, Ruffini had a reputation for honesty and integrity. As a doctor, he treated patients from the richest to the poorest backgrounds. In 1817, there was a typhus epidemic. Ruffini proceeded to continue to treat patients and caught the disease himself. Although, he made a recovery, he never regained his health and had to resign from his university position in 1819.

The year before his death, he received the following correspondence from Cauchy which would represent the only recognition he received while he was alive from a major mathematician:
... your memoir on the general resolution of equations is a work which has always seemed to me worthy of the attention of mathematicians and which, in my judgement, proves completely the impossibility of solving algebraically equations of higher than the fourth degree.
Even so, Ruffini does not get credit for proof that the quintic equation is not solvable by radicals. That honor goes to Niels Abel. On this point, R. G. Ayoub writes:
... the mathematical community was not ready to accept so revolutionary an idea: that a polynomial could not be solved in radicals. Then, too, the method of permutations was too exotic and, it must be conceeded, Ruffini's early account is not easy to follow. ... between 1800 and 1820 say, the mood of the mathematical community ... changed from one attempting to solve the quintic to one proving its impossibility...
Paolo Ruffini died on May 10, 1822 in Modena.

Jacques Charles Francois Sturm


Jacques Charles Francois Sturm

Jacques Charles Francois Sturm was born on September 29, 1803 in Geneva, Switzerland. His father was a math teacher. When Sturm was 16, his father died and his family fell into a difficult financial situation.

At the Geneva Academy, Sturm's strong mathematical ability was recognized by his instructors. One of his teachers, Jean-Jacques Schaub, arranged financial support for young Sturm so he could attend school full time. At the Geneva Academy, Sturm met Daniel Colladon whose friendship and collaboration was an important part of his early work in mathematics.

When Sturm graduated from the Academy, he accepted a position as the tutor to Madame de Stael's youngest son in 1823. Madam de Stael had been a very successful and famous French writer who had died in 1817.

The family spent six months each year in Paris and Sturm was able to join them. Through the family, he was able to meet many of the intellectual luminaries of French society including Dominique Francois Jean Arago, Pierre-Simon Laplace, Simeon Denis Poisson, Jean Baptiste Joseph Fourier, Joseph Louis Gay-Lussac, and Andre Marie Ampere among others.

In 1824, Sturm and Colladon attempted to win a prize offered by the Paris Academy on the compressibility of water. The results were not as expected and Colladon severely injured his hand. They tried again in 1825. This time Sturm got a job tutoring Arago's son and was able to use Ampere's laboratory and received support and advice from Fourier. With all this new help, even if they did not win, they had made significant improvement from the previous year.

The next year, both Sturm and Colladon worked as assistants to Fourier. Additionally, they continued their experiments on the compressability of water and this time, they won the Grand Prix of the Academies de Sciences. The prize money was enough that they could stay in Paris and devote themselves to their research.

In 1829, Sturm published what would become one of his most famous papers: Mémoire sur la résolution des équations numériques. In it, he presented a major simplification of a method discovered by Cauchy to identify the number of real roots that an equation had over a specified interval. His method was largely based on methods from Fourier but the result was undeniably impressive. Her is Hermite's response:
Sturm's theorem had the good fortune of immediately becoming a classic and of finding a place in teaching that it will hold forever. His demonstration, which utilises only the most elementary considerations, is a rare example of simplicity and elegance.
Despite the well-received paper, Sturm had trouble finding work until the revolution of 1830. With the help of Arago, Sturm became professor of mathematics at the College Rollin. Three years later, he became a French citizen and three years after that he was admitted to the Academie des Sciences.

He would make significant contributions to differential equations relating to Poisson's theory of heat. Today, this work along with with the work done by Liouville form what is known as Sturm-Liouville Theory. Later in his career, he was professor at the Ecole Polytechnique. He made contributions to infinitesimal geometry, projective geometry, differential geometry, and geometric optics.

He died on December 18, 1855 in Paris.

Claudius Ptolemy


Claudius Ptolemy

Claudius Ptolemy was born around 85AD in Egypt. He lived during a time of Hellenized Egypt but very little is known of his personal life.

His reputation rests on his unprecedented contributions to astronomy and geography and to the controversy that these works later caused as astronomers reacted to the famous work byCopernicus. In Ptolemy's universe, the earth sits at the center.

Ptolemy lived in Alexandria, Egypt. We know very little of his education. He mentions the astronomic data of Theon the mathematician who was probably Theon of Smyrna, one of his teachers. Many of his early works are dedicated to Syrus who may have also been one of his teachers.

Many of Ptolemy's major works still exist. His magnus opus is the Almagest which is Arabic for "the greatest". Its original name was the Mathematical Compilations. The Greeks soon started calling it The Greatest Compilations. Later on, the Arabic version became the basis of the Latin translation which is why Ptolemy's work is largely known as the Almagest.

The Almagest was one of Ptolemy's earliest works. It presents the Greek theory about the motions of the Sun, the moon, and the planets. This stood as the standard text on astronomy until Copernicus released his heliocentric theory in 1543.

It is a very ambitious work. Ptolemy writes (taken from MacTutor Biography):

We shall try to note down everything which we think we have discovered up to the present time; we shall do this as concisely as possible and in a manner which can be followed by those who have already made some progress in the field. For the sake of completeness in our treatment we shall set out everything useful for the theory of the heavens in the proper order, but to avoid undue length we shall merely recount what has been adequately established by the ancients. However, those topics which have not been dealt with by our predecessors at all, or not as usefully as they might have been, will be discussed at length to the best of our ability.
His geocentric astronomic model was based on Aristotle even though Aristarchus of Samoshad argued for a heliocentric model. After presenting this model, he then introduces trigonometric models to predict the motions of the orbits. He presents tables of chords (which are similar to the sine function) and uses the concept of "epicycles" (circles-within-circles) to model the planetary movements. It is his theory of planets which is his greatest contribution. In his day, there were only 5 planets recognized and he presented detailed mathematical models for explaining their motions.

Although Hipparchus gets credit for much of the theory that Ptolemy presents, it is Ptolemy who was able to apply that theory to the planetary data.

The historian Toomer wrote (from MacTutor web site):

As a didactic work the "Almagest" is a masterpiece of clarity and method, superior to any ancient scientific textbook and with few peers from any period. But it is much more than that. Far from being a mere 'systemisation' of earlier Greek astronomy, as it is sometimes described, it is in many respects an original work.
There have been many who questioned whether Ptolemy deserves his reputation for being the greatest astronomer of his day. Tycho Brahe noticed that all of Ptolemy's star catalogue data was consistently off by 1 degree longitude. Although Ptolemy claimed that he had gathered the data. Brahe believed that Ptolemy had copied the data from another source. Some scholars have argued that Hipparchus is the true genius and Ptolemy merely copied Hipparchus's work. It is very hard to assess the truth of any of these claims since almost of all of Hipparchus's work is lost.

Regardless of these criticisms, it is undeniable that Ptolemy played the lead role in the geocentric theory of the solar system that ruled the day until the publication of Copernicus's famous work. Hipparchus may be the father of trigonometry but it is Ptolemy who became the lead voice in the application of trigonometry to the motions of the planets, sun, and moon.

Bernhard Riemann


Bernhard Riemann

Bernhard Riemann was born in the Kingdom of Hanover (now, Germany) on September 17, 1826. He was the second of six children. He was home schooled by his father, Lutheran minister, until he was 10. Through out his life, he remained very close to his family and very religious.

When he was 14, Riemann moved in with his grandmother and attended school in Lyceum. He entered the gymnasium in Luneburg in 1842. At the gymnasium, Riemann showed strong interest in mathematics. It is said that he read a 900 page book by Adrien Legendre on number theory in 6 days.

In 1846, Riemann enrolled in theology at the University of Gottingen. He continued to take classes in mathematics and later, after consulting with his father, changes his focus from theology to mathematics. At Gottingen, he was able to takes courses from the legendary mathematician Carl Friedrich Gauss.

Despite having Gauss on its faculty, the University of Gottingen was, at this time, secondary in mathematics to the University of Berlin. Riemann transferred there in 1847 and was able to attend courses in advanced mathematics given by Jakob Steiner, Carl Jacobi, Johann Dirichlet, and Ferdinand Eisenstein. These were exciting times for Riemann and he became particularly influenced by the theories of Dirichlet. It is said that at this time, Riemann built up what would become his general theory of complex variables.

Riemann returned to the University of Gottingein in 1849 to work on his Ph.D. thesis under the guidance of Gauss. Riemann was also greatly influenced by Wilhem Weber in theoretical physics and Johann Listing in topology.

Riemann's thesis was on what are today known as Riemann surfaces. In this groundbreaking work, Riemann used topology to analyze complex variables. The MacTutor biography describes this work as: "a strikingly original piece of work which examined the geometric properties of analytic functions, conformal mappings, and the connectivity of surfaces."

Today, Riemann's Ph.D. thesis is considered to be one of the most impressive that has ever been produced. Based on Gauss's recommendation, Riemann was offered a post at the University of Gottingen as a lecturer. To become a lecturer, he needed to achieve a post-doctoral degree called a Habilitation. To complete this degree, he needed to make a presentation on an advanced topic. He proposed three topics to Gauss and to his surprise, Gauss selected geometry. So, on June 10, 1854, Riemann presented a lecture on what is today known as Riemannian geometry which would later be the basis of Einstein's general theory of relativity. The lecture is today considered a classic.

The story goes that Riemann's lecture was so advanced for its time that only Gauss appreciated the depths of the ideas. While the other technical members of the audience listened politely, Gauss was greatly excited. The MacTutor biography quotes an eyewitness: "The lecture exceeded all his [Gauss's] expectations and greatly surprised him. Returning to the faculty meet, he [Gauss] spoke with the greatest phrase and rare enthusiasm to Wilhelm Weber about the depth of the thoughts that Riemann had presented."

When Gauss died in 1855, his replacement was Johann Dirichlet. In 1857, Riemann became a professor of mathematics. At this time, he wrote a paper on the theory of abelian functions. At the same time, Karl Weierstrass was working on the same topic. Riemann's paper was very advanced. Felix Klein writes: "It [Riemann's paper] contained so many unexpected, new concepts that Weierstrass withdrew his paper and in fact published no more."

In 1859, Dirichlet died and Riemann replaced him as the chair of mathematics. He was also admitted to the the Berlin Academy of Sciences. His nomination read (from MacTutor):
Prior to the appearance of his most recent work [Theory of abelian functions], Riemann was almost unknown to mathematicians. This circumstance excuses somewhat the necessity of a more detailed examination of his works as a basis of our presentation. We considered it our duty to turn the attention of the Academy to our colleague whom we recommend not as a young talent which gives great hope, but rather as a fully mature and independent investigator in our area of science, whose progress he in significant measure has promoted.
As a newly elected member of the Academy of Sciences, Riemann was expected to make a techincal presentation. Riemann's presentation unleashed what is today known as the Riemann Hypothesis. This is the most famous and most important open problem in number theory. It was identified by David Hilbert among his famous collection of 23 mathematical problems. The purpose of the paper was to outline a method for determining the number primes less than a given number.

At 36, he married Elise Koch and it seemed like his mathematical impact was only just beginning. Unfortunately, around this time, he got sick with what would later turn out to be tuberculosis. He continued to travel to Italy where he hoped a warmer environment would help his health and then returned to Gottingen. He died on July 20, 1866 in Italy.

Over all, Riemann's output was small but the influence of that output makes him one of the most influential mathematicians of the nineteenth century. He masterfully combined topics of topology, geometry, analysis, and number theory to show how each complemented the other areas. His impact on such a wide range of topics makes him one of the most important mathematicians of all time.

Sir Isaac Newton


Sir Isaac Newton

Sir Isaac Newton was born in Lincolnshire, England in 1643, the same year that Galileo died. He was born premature and was not expected to live. His father, a farmer also named Isaac, had died three months before his birth. His father had owned animals and property. By the standards of the day, he was born into a wealthy family. Still, his father had been uneducated and unable to sign his name. It was the intention of his mother, Hanna, that young Isaac would become a farmer.

When Isaac was two years old, his mother married Barnabas Smith, a church minister from a neighboring town. From this point on, Isaac lived with his grandparents. He was raised as an orphan and it is said that it was not a happy childhood.

In 1653, Barnabas Smith died and Isaac moved back with his mother. Isaac began attending a grammar school in Grantham. As a child, he did not do well in school. His teachers considered him to be 'idle' and 'inattentive.' Rather than socialize with other students, he kept to himself.

When he was 17, his mother called him home in order to run the farm. Newton seemed to show little interest in learning the farm responsibilities. He prefered to experiment and build gadgets rather than watching the sheep. Eventually, it was decided that farming was not the career for young Newton. Instead, in 1661, he was sent to Cambridge.

Despite the property owned by his mother, Newton was sent to Cambridge as a subsizar, that is, a student who had to perform labor for other students in order to go to Cambridge. His mother refused to pay any money for Newton's schooling.

At Cambridge, Newton began to study law. In addition, he took time to study mathematics and philsophy. At the time, philosophy was dominated by Aristotle. Despite this, Newton took strong interest in the works of Gassendi, Boyle, and Kepler, and Descartes.

Newton's interests in mathematics supposedly began in 1663, when he purchased a book on astrology and was unable to understand its mathematical details. He started on Euclid's Elements but found the initial proofs too simple. It was only when he came upon the principle that parallelograms that have the same base and the same parallels are equal (for details on this proof, see Euclid's Book I, Proposition 35). This gave Newton a new respect for the concept of proof and he read Euclid's Elements very thoroughly from beginning to end.

In the summer of 1665, an outbreak of the Bubonic Plague forced the closure of Cambridge. For the next two years, Newton returned to his mother's house where he worked on his independent projects which included ideas that would later become his theory of calculus and his laws of motion. It is often said that 1666 was the year that Newton came up with his most important ideas.

Cambridge reopened in 1667 and Newton returned. In 1666, Newton wrote three very important papers on calculus. He was 24. Before these papers, no one had known who Newton was. These papers covered slopes on curves (differentiation) and areas under curves (integration) and the relationship between these two methods (fundamental theorem of calculus). These results had been applied to very specific cases before Newton but nothing before matched Newton's generalized methods and nothing compared to the depth of Newton's understanding of the two methods and their relationship. In 1667, Newton became a minor fellow of Trinity College. One might wonder with his accomplishments with calculus why he would receive only a minor fellowship. The reason was probably because Newton kept to himself for the most part. He did not talk during dinner and did not join the other scholars in social activities.

In 1668, Newton built the first ever reflecting telescope. Previous to this, all the telescopes were refracting telescopes, that is, a person looked directly through the telescopic lense. In a reflecting telescope, a person looked through a mirror that reflected the lense. This resulted in an unprecedented clarity of image. This enabled telescopes to be made smaller with significantly greater power of magnification.

Newton was so proud of his telescope which he had built himself that he started to demonstrate it to others. Word spread and it created a sensation. Newton was then became a member of the Royal Society of Science. The most prestigious group in science at the time.

In 1669, Newton became the Cambridge Lucasian Professor of mathematics. He received this office primarily from the stong support of Dr. Isaac Barrow who had held the post previously and decided to step down. At the time, all professors were expected to be ordained as ministers. Newton asked to be free from this requirement so that he could spend more time studying mathematics. His request was approved by King Charles II.

From 1670 to 1672, Newton focused his time on optics. It was a this time that he did his very famous experiment with the prism which demonstrated that white light was composed of a spectrum of colors. It was also at this time that he presented his theory of light as a particle and the concept of the ether.

In 1671, Newton presented his telescope to the Royal Society. He submitted a paper on optics detailing the ideas that led to the reflecting telescope. At the time, Robert Hooke was seen as the leading expert on optics. He aggressively attacked many of the ideas of in Newton's paper.

In 1678, Newton experienced a mental breakdown. He was accused of plagiarism and later on, his mother died. Newton withdrew from his colleagues and became focused on alchemy. It is believed by scholars that it was Newton's deep interest in alchemy that helped him with his ideas about universal gravitation. Gravity, after all, is a mysterious force at a distance that is not explained so much as it is described mathematically.

In 1684, the problem of Kepler's planetary motions had become a very important topic of discussion for the Royal Society. The astronomer Edmund Halley had begun discussions with many of the leading scientists about their ideas of whether Kepler's laws were correct and whether they implied an inverse square force between planets. It was at this time that Halley spoke about this with Newton. He was very surprised to hear that Newton had already worked out the details and could demonstrate Kepler's principles with a proof. This work would become Newton's most famous work, the Principia. Interestingly, Newton did not have the funds to publish this work and it was mostly through the contributions of Halley that the Principia was completed and released to the world.

In 1689, Newton was elected to Parliament to represent Cambridge. Thus, began his career in London. In 1699, he became Master of the Mint. With this position, he became a very wealthy man. He took the position very seriously and was responsible for a significant "recoining" that occured.

In 1703, he was elected President of the Royal Society. Each year, he was reelected until his death in 1727. In 1705, he was knighted by Queen Anne.

Sir Isaac Newton is perhaps the most important scientist and mathematician who has ever lived.

Blaise Pascal


Blaise Pascal

Blaise Pascal was born in Clement, France in 1623. From an early age, he was identified as a prodigy. When he was eleven, he wrote an essay on the sounds of vibrating bodies. His father did not take well to this and forbid him from studying mathematics. At 12, he came up with his own proof that the sum of the angles of a triangle add up to sum of two right angles. His was father was impressed with this and decided that it was ok if Pascal read Euclid's Elements. At the age of 16, Pascal wrote an essay on conic sections which included a result known today as Pascal's Theorem. At age 18, he created a mechanical calculator capable of addition and subtraction (he was the second person to do this; Wilhelm Schickard had built the first one in 1624).

Pascal's father, Etienne, was a local magistrate and was an acquaintance of many of the most famous intellectuals of his day including Rene DescartesPierre Gassendi, and Girard Desargues. It is said that when Descartes saw Pascal's writing on conic sections, he refused to believe that the boy had written it. Pascal's mother died when he was three years old so he was raised and educated by his father.

In 1653, Pascal wrote his famous treatise on what is today known as Pascal's Triangle. This is a method for determining the binomial coefficients for a given value of (a + b)n. Interestingly, Pascal was not the first to come up with this method. It had been discovered 400 years earlier by the Chinese mathematician Yang Hui.

In 1654, Pascal became interested in figuring out the odds associated with gambling. He wrote some letters to Pierre de Fermat seeking his opinion and their correspondences became the foundation for Probability Theory.

Pascal made numerous contributions to science. He studied the effects of air pressure on fluids and correctly proposed that a vacuum explained the the dynamics of mercury in a barometer (at the time, it was not clear how mercury-based barometers worked). His experiments with barometers attracted attention across Europe. At the time, Descartes argued that a vacuum could not exist.

In 1654, Pascal's carriage almost fell off a bridge. He was crossing the bridge when his horses lost their step and went over. The carriage would have followed but the reins broke and the carriage balanced over the edge of the bridge. Pascal and his passenger managed to get back to the bridge but fifteen days later, Pascal had a religious experience. After this, he began writing works of a philosophical and religious nature. His Provincial Letters and Penseesbecame very famous. The Pensees was released after his death.

In 1659, Pascal grew seriously ill; through out his life, his health had been bad but this time, he did not recover. In 1662, when he was 39, he died.

Today Pascal stands as one of the giants of his age. The pascal is the SI unit of pressure and his literary works are considered among the greatest of his time. He is seen as one of the fathers of the computer and his work with Fermat on probability theory is at the foundation of economics and the social sciences.

Gottfried Wilhelm Leibniz


Gottfried Wilhelm Leibniz

Gottfried Wilhelm Leibniz was born on July 1, 1646 in Leipzig, Saxony which, today, is part of Germany. His father was a professor of moral philosophy at the University of Leipzig which had opened in Saxony in 1409. His father died when Gottfried was six years old. Young Leibniz inherited his father's library.

Leibniz was raised by his mother. At age seven, he entered the Nicolai School in Leibzig. Leibniz immersed himself in self-study in an effort to be able to read his father's books. By the age of 12 he had grown very advanced in Latin and had begun to study Greek. He would later write about his dissatisfaction with the logic of Aristotle.

At 14, he entered the University of Leipzig where he most likely studied philosophy, mathematics, rhetoric, Latin, Greek, and Hebrew. During the summer of 1663, he visited the University of Jena where he gained his first exposure to fundamental mathematical ideas such as proofs. Leipzig at the time was not very strong in mathematics so it is believed that Jena played a very important role in the development of his understanding of mathematics. Leibniz was greatly influenced by the ideas of Erhard Weigel who believed that all the universe could be viewed in terms of numbers.

Leibniz received his bachelors degree in law and a masters in philosophy from Leipzig. Despite this great progress, when he presented his thesis for his doctorate, his advancement was denied. The details for why this occurred are unclear. Normally, this meant that Leibniz would need to wait a year before resubmitting his doctoral thesis. Instead, Leibniz presented his doctorate thesis to the University of Altdorf where he gained his doctorate.

Leibniz was offered a position at the University of Altdorf which he decided not to accept. Later, he made the acquaintance of Baron Johann Christian von Boineburg. He soon had become "secretary, assistant, librarian, lawyer, and advisor the Baron and his family." (E J Aiton, Leibniz: A biography, Bristol-Boston, 1984) At this point, Leibniz's interests and works rested primarily in literary ambitions. One writer noted that during this period of Leibniz's life, he would have passed as "a typical late renaissance humanist." (G M Ross,Leibniz, Oxford, 1984)

In 1672, Leibniz was sent by Boineburg to meet with the French in an effort to dissuade Louis XIV from invading the German regions. Leibniz put forward a plan of invading Egypt that was very similar to the plan the Napolean would later carry out. At this point, Leibniz met the mathematicians and scientists of Paris. In particular, he studied mathematics and science under Christian Huygens from the Netherlands. His ventures in math and science in Paris were more successful than his political efforts.

Baron Boineburg died on December 15 of 1672. The Baron's family continued to sponsor Leibniz. In January of 1673, Leibniz gave up on his efforts at peace in Paris and went now to England to convince the British of peace. There, he met with Hooke, Boyle, and Pell. Leibniz presented his ideas for an automatic counting machine. On April 19, 1673, Leibniz was elected as a member to the Royal Society of London. Once again, his scientific pursuits were more successful than the political ones.

In 1674, Leibniz began to take interest in the problem of infinitesimals. He corresponded with Oldenburg from the Royal Society who let him know that Newton and Gregory had found very general methods to the problem. By autumn of 1676, Leibniz had worked out much of his notation for calculus. At this time, he received a letter from Newton. In 1677, he received a second letter from Newton. In this second letter, Newton questions whether Leibniz stole Newton's method. Newton pointed out that not a "single previously unsolved problem was solved." (Quoted in the MacTutor article) Later, Leibniz's notation would prove very important in the advancement of calculus.

Leibniz had hoped to join the Academy of Sciences in Paris but no opportunity to join came his way. In October of 1676, he accepted a position as librarian and Court Councillor to the Duke of Hanover, Johann Friedrich.

During this time, Leibniz worked on many outside projects. He worked unsuccessfully on developing wind-powered pumps to drain water from the Harz mountain mines. From these efforts, he developed his knowledge of geology and proposed a theory that the earth was at one time molten lava.

By 1679, he had developed a "binary system of arithmetic." He also worked on the problem of determinants which had written around 1684.

In 1680, the Duke of Hanover died and Leibniz began working for his brother Ernst August. Leibniz began to work on the family tree which included the House of Brunswick. As part of this effort, he traveled to Bavaria, Austria, and Italy. In each of these places, he met with scholars and other famous writers. He would publish the results of his research in nine large volumes. Still, he never completed the work that Ernst August had requested.

In 1684 and 1686, Leibniz began to publish his theory of calculus. The next year, in 1687, Newton published his Principia.

In 1710, Leibniz published Theodicee in which he argued that even if the world is not perfect, it is the best possible world. In 1714, he published his famous work Monadologia.

Unfortunately, it was the dispute with Newton that filled his last years. The main argument against Leibniz were the two letters that he had received from Oldenburg. Leibniz claimed that there was not enough information presented in the letters to give him the methods he found. In 1711, a paper by Keill was read to the Royal Society which accused Leibniz of plagiarism. In 1713, the Royal Society investigated the issue and ruled against Leibniz. Leibniz was never asked to give his version of events and Newton himelf wrote the final report.

In 1714, George Ludwig from the House of Hanover became King of England. Leibniz was not invited to join him.

Leibniz died on November 14, 1716. In his life, he had corresponded with over 600 figures of his time. The Berlin Academy of Science emerged as the result of Leibniz's work. While it is clear that Newton invented calculus first, it is also clear that Leibniz extended Newton's work in very important ways that Newton did not appreciate.

Joseph Liouville


Joseph Liouville

Joseph Liouville was born on March 24, 1809 in Saint-Omer, France. His father was an officer in Napoleon's army so for his earliest years, he lived with his uncle. Only when Napoleon was defeated did his father return and the family live together in Toul, France.

In Paris, Liouville studied mathematics at the College St. Louis. Already at this time, he showed interest and talent in advanced mathematical topics.

In 1825, when he was 16, he entered the Ecole Polytechnique. There, he attended lectures by Andre Marie Ampere and Dominique Francois Jean Arago. While he did not attend any lectures by Augustin Louis Cauchy, he was greatly influenced by him. Among his examiners when he graduated were Gaspard de Pony and Simeon Denis Poisson.

He graduated in 1827 and entered the Ecole des Ponts et Chaussees with the intention of becoming an engineer. Engineering projects in those days were physically demanding and Liouville found his health severely affected. He took some time off by returning to Toul. He got married to Marie-Louise Balland and decided to resign from the Ecole des Ponts et Chaussees which he did in 1830.

In 1831, he accepted an academic position at the Ecole Polytechnique. He was assistant to Claude Louis Mathieu and the role carried with it responsibilities of 35-40 hours of lectures per week. In doing this, Liouville developed a reputation for focusing on advanced topics and for being difficult to follow.

Many times, Liouville attempted to improve his position at the Ecole Polytechnique without success. He was also frustrated by the quality of math journals in France at the time. In 1836, he started his own math journal, Journal de Mathematiques Pures et Appliqueeswhich was also known as the Journal de Liouville. By this time, he had developed an international reputation based on his papers which he published in Crelle's Journal. The journal was a success and published many significant papers by French mathematicians.

By 1838, he became Professor of Analysis and Mechanics at the Ecole Polytechnique. Many honors followed. In 1840, he was elected to the Academie des Sciences in Astronomy and he was also elected to the Bureau des Longitudes.

Liouville had become close friends with Arago who become the head of the Republican Party in France. Liouville was encouraged to run for office. In 1848, Liouville was elected to the Constituting Assembly. Unfortunately, his political career did not last long for he was voted out of office the following year. This had a big impact on his spirits as noted by one of his biographers:
The political defeat changed Liouville's personality. In earlier letters, he was often depressed because of illness, and could vent his anger towards his enemies such as Libri, but he always fought for what he believed was right. After the election in 1849, he resigned and became bitter, even towards his old friends. When he sat down at his desk, he did not only work, ... he also pondered his ill fate. ... his mathematical notes were interrupted with quotes from poets and philosophers...
In 1850, the mathematical chair at the College de France opened. Up for consideration were Liouville and Cauchy. After a heated contest, the position went to Liouville in 1851.

Liouville's mathematical output was phenomenal writing over 400 mathematical papers. Over 200 of papers were on number theory. Other papers covered a range of topics including mathematical physics, astronomy, as well as pure mathematics.

He introduced the fractional calculus as part of his analysis of electromagnetism. He was also to the first to prove the existence of transcendental numbers, numbers that is not algebraic (that is, it cannot be a solution to an equation of a nonconstant polynomial with rational coefficients). He did very significant work on the boundary value problems with differential equations in what is today called Sturm-Liouville Theory. He did important work in statistical mechanics and measure theory.

Perhaps, his most important impact in mathematics was his discovery of the memoir byEvariste Galois. In 1843, he announced to the Paris Academy that he had discovered very brilliant insights by Galois. Galois's memoir was then published in 1846 which would introduce group theory and place Galois among the most celebrated mathematicians in the history of the subject.

Liouville died on September 8, 1882. Many historians consider him the greatest mathematician of his day. The Liouville Crater on the Moon is named in his honor.

Gabriel Lamé


Gabriel Lamé

Gabriel Lamé was born in Tours, France on July 22, 1795. He entered the Ecole Polytechnic in 1813 and while there, published his first mathematical paper in 1816. He graduated Ecole Polytechnic in 1817 and attended the Ecole de Mines from 1817 to 1820.

In 1820, Lamé and his colleague Emile Clapeyron were invited to come to Russia to teach mathematics. Lamé was appointed professor at the Institut et Corps du Genie des Voies de Communication located in St. Petersburg. He stayed at this position for 12 years and during this time he lectured and wrote papers on a range of subjects in mathematics, physics, and chemistry. It was while in Russia that he would develop his great interest in railroad development.

Lamé returned to Paris in 1832. That year, he became the Chair of Physics at the Ecole Polytechnique. In 1836, he became the Chief Engineer of Mines. He was also extensively involved in the building of railroads from Paris to Versailles and Paris to St. Germain.

He was elected to the Academie de Sciences in 1843 and that same year, he left Ecole Polytechnique to join Sorbonne where he was a professor of mathematics and probability. He became the chair of mathematical physics and probability at Sorbonne in 1851.

In his lifetime, Lamé worked on a range of very advanced topics including: the stability of vaults, design of suspension bridges, elasticity theory, conduction of heat, general theory of curvilinear coordinates, Fermat's Last Theorem for n=7, differential geometry, and number theory.

Outside of France, he was considered the leading French mathematician of his time by many of his contemporaries such as Carl Friedrich Gauss. Within France, his reputation was less bright. Lamé considered his general theory of curvilinear coordinates to be his most important achievement. Still, this contribution was made obsolete by the discovery of more modern methods.

Adrien-Marie Legendre


Adrien-Marie Legendre


Adrien-Marie Legendre did not like to talk about his personal life. His colleague Simeon Poisson wrote:
Our colleague has often expressed the desire that, in speaking of him, it would only be the matter of his works, which are, in fact, his entire life. (FromMacTutor Biography)
He was born September 15, 1752 in France. He may have been born in Toulouse but from a very early time, his family lived in Paris. His family was wealthy and he attended the College Mazarin in Paris. His area of focus was mathematics and physics.

In 1775, he became a lecturer at the Ecole Militaire based on the recommendation of the well known mathematician Jean D'Alembert. One of his fellow lecturers at the Ecole Militaire was Pierre-Simon Laplace who would later become very well known himself.

Legendre's reputation was made when he was able to win the Berlin Academy Prize in 1782. The Berlin Academy had proposed a very difficult problem that involved calculating the projectory path of a cannon ball undergoing air resistance. He later wrote an influential paper on the force of attraction between two confocal ellipsoids. In January of 1783, he was appointed as an associate to the French Academy of Sciences.

Legendre proceeded to do influential investigations of celestrial mechanics, number theory, and elliptic functions. In number theory, his work includes an important result on the quadratic reciprocity of residues and arithmetic progressions of prime numbers. Both of these results were later superceded by Carl Friedrich Gauss's work on quadratic reciprocity of residues and Johann Dirichlet's work on arithmetic progressions of primes. In 1787, he was elected to the Royal Society of London.

After the French Revolution of 1793, Legendre lost most of his fortune. In 1794, he published his Elements of Geometry which became the leading elementary text on geometry at the time. In this work, he attempted to make the proofs behind Euclid's Elements clearer and better organized.

In 1801, Gauss criticized Legendre's results on quadratic reciprocity and claimed that he himself had invented the correct method. It is clear that Legendre was not happy with Gauss's words but in 1808, when Legendre came out with the next version of his textbook on number theory, he included Gauss's proof instead of his own.

In 1824, Legendre refused to support the government's candidate for the National Institute. As a result of his refusal, he lost his pension and lived his remaining years close to poverty.

In 1830, he offered his proof for Fermat's Last Theorem n = 5 which was based on the work done by Dirichlet and also the proof by Sophie Germain.

Legendre was fascinated by Euclid's parallel postulate and for many years attempted to provide a proof. He refused to believe in the existence of non-Euclidean geometries which were first proposed by Nikolai Ivanovich Lobechevsky in 1829.

Legendre died on January 10, 1833. He had made significant achievements in geometry, differential equations, calculus, function theory, number theory, and statistics.