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.