Archive for the ‘History of Algebra’ Category

I’ve been teaching an on-line History of Math course (with a HUM humanities prefix) this term.

The posts for that course are here.

The most recent post was about the French mathematicians of the 17th century – Viète, Mersenne, Fermat, Descartes and Pascal.

French Mathematics of the 17th century


Francois Viète (1540-1603)

Francois Viète was the son of a lawyer in 16th century France.  He is credited with devising a scheme* in which unknown quantities in algebra would be represented by letters that are vowels and constant quantities would be represented by letters that are consonants.  At the time, the Arabic algebra that had been transferred to Europe over the previous 500 years was based on prose writing – everything was described in words.  After Viète’s initial use of letters for unknowns and constants, René Descartes later began to use letters near the end of the alphabet for unknowns (x, y, z) and letters from the beginning of the alphabet for constants (a, b, c).  This practice continues today.

In 1593, the Dutch ambassador to France said to French King Henry IV that a well-known Dutch mathematician had posed a problem that was beyond the capabilities of ANY French mathematician.  Henry IV passed the problem along to Viète and Viète was able to solve it.

Viète began a correspondence with Roomen, the Dutch mathematician who had posed the problem originally and became one of the first internationally recognized French mathematicians.  He worked mainly in trigonometry, astronomy and the theory of equations.

*This link is a paper written by a college student at Rutgers University in New Jersey.  Papers on other subjects by other students in the same course can be found here.

Marin Mersenne (1588-1648)

Marin Mersenne was a French monk best known for his research into prime numbers.  He also did important research into the musical behavior of a vibrating string, showing that the frequency of the vibration was related to the length, tension, cross section and density of the material.

Mersenne primes are prime numbers of the form 2^p-1, where p is a prime number itself.  For example

2^2-1=3 which is prime

2^5-1=31 and so on.

Mersenne was also interested in the work that Copernicus had done on the movement of the heavenly bodies and despite the fact that, as a monk, he was closely tied to the Catholic church, he promoted the heliocentric theory in the 1600′s.

Mersenne was also known as a friend, collaborator and correspondent of many of his contemporaries.  Fermat, Pascal, Descartes, Huygens, Galileo, and Torricelli all corresponded with Mersenne and the exchange of ideas among these scientists promoted the understanding of music, weather and the solar system.

René Descartes (1596-1650)

René Descartes is probably best known for two things.  One is the conclusion “I think therefore I am” (Cogito ergo sum in Latin and Je pense donc je suis in French) and the other is the geometric coordinate system generally known as the Cartesian plane.

Descartes joined the army of Prince Maurice of Nassau in 1619 and was in Bavaria (southern Germany) and Bohemia (Czech Republic) during the beginning of the Thirty Years War.

The importance of the Cartesian Plane is difficult for us to understand today because it is a concept that we are taught at a young age.  Locating objects on a grid by their horizontal and vertical coordinates is so deeply embedded in our culture that it is difficult to imagine a time when it did not exist.

Before Descartes’ grid system took hold, there was Geometry:


and there was Algebra:


(Click on photo for larger view)

…and they were separate fields of endeavor.  The idea that a geometric shape like a parabola could be described by an algebraic formula that expressed the relationship between the curve’s horizontal and vertical components really is a ground-breaking advance.  It is so ground-breaking that once it happened, people began to forget that it hadn’t always been that way.

Once this new method for describing curves was developed, the question of finding the area under a curve was addressed.  This is the general problem of Integral Calculus.  Descartes (among others) saw that, given a polynomial curve y=x^n, the area under the curve could be found by applying the formula A=\frac{x^{n+1}}{n+1}

These were the rudimentary beginnings of the development of the Calculus that would be devised by Isaac Newton and Gottfried Leibniz in the ensuing years.

Fermat (1601-1665)

Pierre Fermat is also mostly remembered for two important ideas – Fermat’s Last Theorem and Fermat’s Little Theorem.  Fermat’s Last Theorem is a simple elegant statement – that Pythagorean Triples are the only whole number triples possible in an equation of the form a^n+b^n=c^n.

Pythagorean Triples are interesting groups of numbers that satisfy the Pythagorean relationship a^2+b^2=c^2.  Triples such as {3,4,5} {6,8,10} {8,15,17} {7, 24, 25} can be found that satisfy the equation.  But – Fermat’s Last Theorem says that if the n in the original equation is any number higher than two, then there are no whole number solutions.

It’s true – but very difficult to prove.  Mathematicians tried for 350 years or so to prove this theorem before it was finally accomplished by Andrew Wiles in 1995.

By the way, you can generate Pythagorean Triples using the following formulas:

Pick two numbers x and y, with x>y




Fermat’s Little Theorem is a useful and interesting piece of number theory that says that any prime number p divides evenly into the number a^{p-1}-1, where a is any number that doesn’t share any factors with p.

Blaise Pascal (1623-1662)

Blaise Pascal was the son of Etienne Pascal, who was a lawyer and amateur mathematician.  Etienne Pascal knew Marin Mersenne and often visited him at his Paris monastery, and when Blaise was a teenager he sometimes accompanied his father on these visits.

Pascal’s first published paper was a work on the conic sections.  He also did research on the composition of the atmosphere and noticed that the atmospheric pressure decreased as the elevation increased.  This led him to believe that beyond the atmosphere there existed a vacuum in which there was no atmospheric pressure.

René Descartes visited Pascal in 1647 and they argued about the existence of a vacuum beyond the atmosphere.  Descartes felt that this was impossible and criticized Pascal, saying that he must have a vacuum in his head.

Pascal is known for the structure of Pascal’s Triangle, which is a series of relationships that had previously been discovered by mathematicians in China and Persia.

Here is Pascal’s version:


Here is the Chinese version:


Here is a version that we often see in textbooks:


Each successive level is created by adding the two numbers above it, so in the 6th row {1,5,10,10,5,1} the 10 is created by adding the 4 and the 6 from the row above it.  These number patterns are actually quite useful in a wide variety of situations.

In raising a binomial to a power like (x+y)^5, the coefficients of each term are the same as the numbers from the 6th row:


These numbers are also related to Discrete Mathematics and Combinatorics which describes how many ways there are to choose something from a series of possibilities.

There was a lot of great mathematics happening in Italy, England, Holland and Germany during the 17th century, but this collection of French mathematicians spanning nearly 100 years produced a tremendous amount of very important mathematical ideas.

The English, Germans and Swiss would make great contributions to mathematics in the 18th century with Newton, Leibniz, the Bernoullis, Euler and others, while the French would still contribute with the works of Laplace, Lagrange and Legendre.


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What do René Descartes and Friday the 13th have to do with each other?

A few years ago, I read the book The Mystery of the Aleph by Amir Aczel and enjoyed it immensely.  In this book, he tells the story of Georg Cantor and his efforts to comprehend and mathematize infinity.

I recently picked up a copy of another one of his books,  Descartes’ Secret Notebook.

I always mention to my beginning algebra classes that Descartes’ accomplishment of combining algebra and geometry into the one discipline of analytic geometry was a masterful stroke of genius.  Even though my students often don’t appreciate the import of this accomplishment, I always make a point to mention it because I am continually amazed at this conceptual achievement.

As I read through Descartes’ Secret Notebook, what struck me the most was Descartes’ involvement in the heliocentric controversy that roiled European science and mathematics throughout the 16th and 17th centuries.  This is a well-known controversy, but I think the vehemence of opposition to the scientists and mathematicians of that era who espoused the heliocentric theory is sometimes not appreciated.

At least for me, reading the story of Descartes’ life drives home the intellectual difficulties that were created by the Catholic church’s opposition to heliocentric theory.

This is what ties Rene Descartes to Friday the 13th (at least in my mind).

I recently corresponded with Carl Bialik, who writes the Numbers Guy column at the Wall Street Journal.  His column on numerology several weeks ago considered people’s attachment of mystical properties (both good and bad) to different numbers.

In response to his question about what creates a superstition around a particular number (or type of number), my opinion was that it was due to a difference between appearance and reality.

I think that when mathematics or science arrives at a conclusion that contradicts our common sense, there is cognitive dissonance.

We can resolve this dissonance by exploring the contradiction and trying to determine its cause.

The examples I used were the ancient Greek’s fear of irrational numbers and triskaidekaphobia or fear of the number 13.

There are many reasons given for fear of the number 13, but the one that makes the most sense to me is that, although we only see 12 moon phases, the moon actually revolves around the earth 13 times each year.

This is because the earth is orbiting the sun, so that each full moon is in a slightly different position with respect to the earth (but the same in the sun-earth-moon alignment that creates the full moon).

I think that this is why the Catholic church reacted so strongly against the heliocentric theory.  By watching the sky, it seems that the sun, moon and planets move around the earth.

But, if we observe more closely and make some calculations, it turns out that this model doesn’t work.

If we think that the world is one way and mathematics and science give us different information, we can explore and analyze and try to understand the information or we can be afraid of it.

If we don’t have time to explore, analyze and understand, we may end up with fear by default.

Ben Blum-Smith gets at this same idea in his discussion of the impact of mathematical proof.  He points out that if students only ever see examples in which their intuition is correct, they don’t see the benefit of proof.

However, if they see examples (and he lists a few at the above link and a few more here) in which an initial pattern breaks down after it has been established, then students may better understand the benefits of mathematical proof and robust scientific theories.

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After the development of the Cubic and Quartic Formulas during the early to mid 1500s, mathematicians all over Europe worked to discover a formula for solving the general quintic equation, an equation of the form x5+bx4+cx3+dx2+ex+f=0.  For 250 years they failed.

During the early 1800s, two mathematicians began their brief careers and each would prove, separately, that a general equation of the 5th degree was not solvable using the four standard mathematical operations and extraction of roots.

Niels Abel was born in Norway in 1802.  Around 1824, he proved that the general quintic equation is not solvable.  He had his proof printed and sent copies to many of the leading mathematicians in Europe who, because Abel was not very well known at the time, ignored it.

Abel died in 1829 as a result of poverty and ill health.  In 1830, the French mathematician Cauchy found a copy of Abel’s paper and eventually published it in 1841.  Abel was also awarded the Grand Prix of the French Academy for his work in 1830.

Evarist Galois was born in France in 1811.  In 1830, he subbmitted a paper for consideration for the Grand Prix (which was eventually awarded to Abel).  His paper was taken home by one of the judges (Jospeh Fourier) who died shortly thereafter.  Galois’ paper was lost and not considered for the prize.

After his death in 1832, Galois’ brother and one of his friends collected Galois’ papers and delivered them to the mathematician Liouville who worked over them for for the next ten years.  Finally, by the 1840s, the work of both Abel and Galois was recognized for the genius it was.

Galois’ proof in particular used the idea of permutations, or the number of ways that objects can be combined as the basis for his proof.  This concept of permutation laid the foundation for the development what became known as Group Theory and Abstract Algebra throughout the 18-1900s.

If you’ve never heard of group theory, it can be one of the most accessible and fun areas of mathematics.  It also has important applications in physics and computer science.

As I mentioned in a previous post, there is a book called The Equation That Couldn’t Be Solved by Mario Livio that outlines the history of these ideas.

Have a great holiday break!

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Italy was a center of mathematical activity after the publication of Fibonacci’s Liber Abaci (1202) and the Treviso Arthmetic of 1478.  These books formed the foundation for European mathematics.

At some point in the early 1500’s, an Italian mathematician named Scipione del Ferro determined a general solution for what is known as the depressed cubic equation.  This is cubic equation without any x2 terms.  The general form is : x3+px=q.  As it turns out, any cubic equation of the form x3+bx2+cx+d=0 can be written as a depressed cubic, but that came later.

At the time, mathematicians didn’t publish their results, but, instead, kept them secret so that they could win the problem contests that were common at the time in Italy.  As a result, del Ferro didn’t tell anyone about his discovery until shortly before his death in 1526.  He then revealed the secret to a student of his named Antonio Maria Fior.  In 1535, Fior used this knowledge to challenge a better mathematicain named Niccolo Fontana to a problem contest.

Fontana was known as “Tartaglia,” (the stutterer) because of a speech impediment caused by an old sword wound to his jaw.  Tartaglia was a superior mathematician to Fior, but didn’t know how to solve the cubic equation yet.  So, in the time before the contest, he worked feverishly to find a solution.  Finally, he found the same solution that del Ferro had found thirty years before and was able to win the contest.

Word of Tartaglia’s victory spread among mathematicians and Giralamo Cardano decided to see if he could get Tartaglia to reveal his secret to the solution of the cubic.  At first Tartaglia refused, but then told Cardano the formula, but not how to derive it.  He also asked Cardano to promise not to reveal the result to anyone else.  Eventually, Cardano learned that del Ferro had found the solution first.  Cardano also determined the derivation of the formula that Tartaglia had shown him for himself.  So, in 1545, Cardano published the solution of the general cubic in his book Ars Magna.

It is interesting that Cardano encountered the square roots of negative numbers in working with his “cubic formula.”  These numbers, which today are called imaginary, or complex numbers were almost completely unknown at the time.  Cardano was initially flummoxed by these numbers that seemingly had no physical meaning.  However, in true trailblazing spirit, Cardano wrote in his book that although these numbers were unfamiliar, “nevertheless, we will operate,” with them using rules similar to those used for the standard number system.

Before the publication of Ars Magna one of Cardano’s students named Ludovico Ferrari found a solution for the quartic equation.  That is, an equation of the form x4+bx3+cx2+dx+e=0.  This solution was also published in Ars Magna which became known throughout Europe as the foundational text of classical European algebra.

The next step in the story took place over the ensuing 250-300 years.  Finally, in the early 1800’s it was shown that the quintic or fifth degree equation was not solvable by formula.

The stories of these mathematicians is told in An Imaginary Tale, by Paul Nahin and The Equation that Couldn’t be Solved by Mario Livio.  Next I’ll write about Galois and the evolution of Abstract Algebra from the ideas of the Classical Algebra.

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