## The Amateur Mathematician

Pierre de Fermat

Birth: 1601, France

Death: 1665, France

Two names lead the first half of 17th-century mathematics: Rene Descartes and Pierre de Fermat. It is rather unusual since neither of them saw themselves as mathematicians as first.

Pierre de Fermat worked as a lawyer and a government official from 1631 until his death. He is regarded as the most important amateur mathematician of all time.

Entry to math circles

Fermat started working on mathematics in the late 1620s. He loves creating problems in the number theory and proving them.

Fermat became famous among mathematicians with his letter to Mersenne* dated back to April 26th, 1636. In his letter, Fermat worked for subjects such as Galileo’s free-fall experiment and Apollonius’ conics which led him to correspond with many mathematicians.

Even though his famous letter involved mathematical physics, he was interested in number theory. This is why, whatever the subject was in his letters, he somehow brought it to a problem in number theory. He wanted mathematicians to prove the problems he created and solved. But these problems were incredibly hard, and soon mathematicians started irritated by him. For instance, Frenicle de Bessy thought that Fermat was teasing him with his difficult problems.

Last Theorem

Today, Fermat is best known for his famous last theorem: Fermat’s Last Theorem. Fermat didn’t consider himself as a mathematician. Hence, he didn’t publish any of his works. In fact, sometimes he was writing his theorems and proofs in the blank parts of books.

One of those theorems was Fermat’s Last Theorem*.

* Fermat’s Last Theorem

“No three positive integers a, b, and c satisfy the equation an + bn = cn for any integer value of n greater than 2.”

Fermat also noted the following:

“I have discovered a truly remarkable proof which this margin is too small to contain.”

Fermat’s Last Theorem was finally proved by British mathematician Andrew Wiles in 1994. That means the theorem was unproven for almost 358 years!

Alienation

Between 1643 and 1654, Fermat was alienated by the science community. There were a few reasons for that such as the civil war and the plague that affected where he lived. But one of the main reasons was his dispute with Descartes.

In his time, Descartes was influential among French scientists. So, when Fermat made a negative comment about his beloved work La Geometrie, Descartes went after him. He (wrongly) criticized Fermat’s work for maxima-minima-tangent*.

Even after Fermat proved that his work was complete, Descartes continued the argument by claiming that Fermat wasn’t a sufficient mathematician.

*Maxima-Minima-Tangent

You probably heard derivatives and integral if you took a math course in university. Actually, the first developments of those two subjects belonged to a 17th-century French lawyer.

Using a simple example, I will show you how Fermat’s method worked:

Assume that we have a straight line with length a:

My ambition is to find a point on it such that it separates the line into two parts, and multiplication of the length of those two parts is maximum.

Assume that one of those parts has length x. Then the other part would have length a-x:

Then, the maximum becomes:

x . (a-x)

ax – x2

So, how can we find this length x?

At this point, Fermat discovered an ingenious method. He says that we should add length e to x:

But this length e should be so small, we might as well consider it as zero. In other words, e is infinitely small. Now we can substitute every x in our solution with x+e:

a(x+e) – (x+e)2

Use basic algebra to simplify the equation:

ax + ea – x2 – 2xe – e2

From this we can reach;

ax – x2 = ax + ea – x2 – 2xe – e2

Again simplify the equations:

ea = 2xe – e2

Now, divide everything by e:

a = 2x – e

Remember that Fermat said e is so close to zero, it is in fact zero? Then:

a = 2x

Our line a has length 2x. This means that if we split a straight line into two equal parts, their multiplication becomes maximum.

Final Years

After his dispute with Descartes ended, Fermat got in touch with mathematicians. His correspondence with Blaise Pascal led to the set up of the probability theory.

Fermat was far ahead of his time in number theory. He almost carried the whole branch by himself in the 17th-century. His findings and problems were appreciated much after his death in the modern number theory. For me, his title will always be the greatest amateur mathematician of all time.

M. Serkan Kalaycıoğlu

## A New, Strange World

Janos Bolyai

Birth: 1802 – Romania

Death: 1860 – Romania

Most people think of Count Dracula whenever Transylvania is mentioned. However, I think of another name at first: Janos Bolyai.

Failed dreams of a young mathematician

Janos Bolyai (from now on I will mention him as Bolyai) is the son of Hungarian mathematician Farkas Bolyai. He showed great potential even when he was just 5-6 years old, and mastered calculus* when he was 13.

*Calculus

The branch of mathematics that deals with the finding and properties of derivatives and integrals of functions, by methods originally based on the summation of infinitesimal differences.

In 1816, Farkas asked his old friend and mathematics teacher Gauss* to take his son as a pupil so that young Bolyai can have the best possible mathematics education. Although, Gauss rejected Farkas’ offer. This wasn’t going to be the only bad new Bolyai gets from Gauss.

*Gauss

A German mathematician who is also known as the prince of mathematics.

For young Bolyai, the best possible choice was to go to Vienna and study military engineering. He was an outstanding student there and finished 7 years of study in just 4 years and joined the army in 1823. He earned a living there until 1834.

A new geometry

Farkas Bolyai spent most of his career for finding a proof (or disproof) for Euclid’s parallel postulate* but failed in the end. It is not a surprise that young Bolyai took the matter in his own hands. He started working on the subject in the early 1820s.

Euclid’s parallel postulate

Also known as the fifth postulate that is given by the Greek mathematician Euclid in the first book of his masterpiece Elements.

Basically, it states that no two infinite parallel lines meet at a point.

Bolyai spent almost all of his spare time in his army duty for mathematics. In November 3rd, 1823 he wrote a letter to his father, mentioning his findings for the first time with these words:

“… out of nothing, I created a new and strange world.”

A year after this letter Bolyai completed his idea for non-Euclidean geometry. At first, Farkas was distant to his son’s findings, but at 1830 he realized how important his discovery was. This is why he convinced his son to write down his idea. In 1831, Bolyai wrote 24 pages-long appendix in his father’s book.

Farkas sent the appendix to his old friend Gauss to evaluate his son’s work. After reading the appendix, Gauss made two important comments to two separate people…

The idea

Bolyai’s idea at its core: Imagine a new geometry where Euclid’s fifth postulate isn’t true. In other words; a new geometry where parallel lines can meet.

In Euclidean geometry, the shortest path between two points is a straight line. However, according to Bolyai’s non-Euclidean geometry shortest path is a curve. You can read about this subject here.

One could explain Bolyai’s idea like this:

In Euclidean geometry, internal angles of any given triangle add up to 180 degrees. But if we draw a triangle on a sphere (e.g. on Earth), angles will exceed 180 degrees:

Bolyai’s geometry (today is known as hyperbolic or non-Euclidean geometry) was a brand new geometry.

The breakdown

Gauss told one of his friends “I regard this young geometer Bolyai as a genius of the first order.”. But, at the same time he wrote a letter to Farkas and showed a much different attitude:

“… to praise this work would be praising myself, as I’ve had the same ideas some 30-35 years ago.”

Today we know that Gauss, in fact, held similar ideas with Bolyai thanks to a letter of his dated back to 1824. But, this happened much after that he told. It is believed that Gauss wasn’t feeling comfortable about publishing his ideas publicly.

Gauss’ comments on his appendix hit Bolyai hard. His idea stayed unknown for the mathematics community and he was deeply affected after Gauss’ remarks. Soon after, his health had gone bad and he was forced to leave the army in 1834, and he started living an isolated life from his beloved mathematics.

Paranoid

Even after all this, Bolyai kept working on mathematics. In 1848, he received a work that is written by a Russian mathematician named Lobachevsky. Lobachevsky’s work was published in 1829 (before his appendix) and covered almost the same ideas as he had held for non-Euclidean geometry. Gauss knew about this work as well. In fact, he praised Lobachevsky for his work.

After investigating the work deeply, Bolyai believed that Lobachevsky was not a real person, and it was all Gauss behind this work. Unfortunately, Bolyai was slowly losing his mind.

Bolyai stopped his mathematics in his last years and died in poverty in 1860. It is known that he left around 20.000 pages of unpublished mathematical work behind. They can be found in the Bolyai-Teleki library located in Targu Mures.

Today, we honor him by calling non-Euclidean geometry as Bolyai-Lobachevsky geometry.

M. Serkan Kalaycıoğlu

## A Mathematician Inside Seven Sages

Thales

Birth: Around 624 BC, Miletus

Death: Around 547 BC, Miletus

Thales Theorem: The diameter of a circle always subtends a right angle to any point on the circle.

In high school math, Thales is known with this theorem even though he is one of the most important names in the history of science. The funny thing is that he most likely didn’t even find this theorem. Then, who is this Thales guy we ought to know?

Thales is known as one of the Seven Sages of Greece as well as the first known natural philosopher.

Seven Sages of Greece are the seven important figures of 7th and 6th century BC Greece including Thales, Pittacus, Bias, Solon, Cleobulus, Myson and Chilan.

Natural philosopher title comes from Thales’ diversity as it is said that he worked in mathematics (especially geometry), engineering, astronomy, and philosophy. There is not even a single written work of his left today. Everything we know about him was written by others centuries after his death. This had led many legends after his name.

Historians believe that Thales visited Egypt where he learned mathematics and engineering. He is known as the first person who introduced geometry to Greeks. It is said that while he was in Egypt, he calculated the length of pyramids by just looking at their shadow.

Whenever the sun makes 45 degrees with Earth, the shadow of a particular object becomes equal to its own length. This is known as one of the methods Thales used in his calculation.

According to another legend, he guessed the time of the solar eclipse in 585 BC. During his time, people were able to guess the lunar (moon’s) eclipse. Although, it was impossible to guess where and when the solar eclipse was gonna occur using 6th BC’s knowledge and technology. Today we believe that even if this happened, it was just an astonishing guess for Thales.

Still, some believe that Thales really guessed the solar eclipse thanks to his brilliance. After all, he was one of the Seven Sages. Actually, as Socrates stated, he was the only natural philosopher inside that prestigious group.

Thales believed that everything comes from water. According to him, the Earth was shaped like a disk and it floats on an infinite ocean. He suggested that the earthquakes were the result of the Earth’s movement on the ocean. This was a first in the history of science as Thales’ ideas were based on logic instead of supernatural phenomena.