Jean Baptiste Joseph Fourier was a French mathematician and a scientist who engrossed himself in the applied mathematical methods of the study of vibrations and the transfer of heat. He invented what is now known as the Fourier series which uses sine and cosine waves, as a way of representing any periodic function. This led to the development of Fourier’s law of heat conduction.

During his time, he held many, varied posts, a teacher, a political prisoner, Governor of Egypt, prefect of Isere and Rhone and finally secretary of the Academie des Sciences.

Fourier was born in the year 1768, to a large and humble family in France. He was the ninth child of his father, a tailor, who died, leaving Fourier an orphan at the tender age of 9. Due to his high academic credentials in his early years he was given financial support to complete his education through the recommendation of bishop of Auxerre. He enrolled in Ecole Royale Militaire which was under Benedictine order of the covenant of St. Mark. Initially he had a great interest in literature but later discovered a greater passion for mathematics. He, therefore, studied mathematics with great enthusiasm and is reputed to have collected candle stubs so that he could study late into the night. By the year 1782, he had completed studying the six volumes of ‘Bossut’s Cours de Mathematiques.’ Moreover, in the following year he was awarded a prize for his completing a study of ‘Bossut’s mechanique en’ general.

In the year 1787 he decided to join Benedict abbey of Saint-Benoît-sur-Loire to train as a priest. He later discovered that priesthood was not his calling rather he wanted to be a mathematician and was determined to make his impact on the world through his skill in mathematics. For this reason, as well as the fact that in 1789, the newly formed Constituent Assembly in France forbade it, Fourier did not take his religious vows. Thereafter, Fourier left the Abbey and returned to the Benedictine college, Ecole Royale Militaire of Auxerre where he had earlier studied, to work on his passion in the field of mathematics as an assistant to his teacher Bonard.

Despite Fourier declining to take the priesthood vows, he was not yet fully settled on whether he should specialize in mathematical research or should concentrate on a religious life. However three years later, he found yet another calling. He became interested in politics and came into the company of the local revolutionary committee where, in 1793, he wrote: “As the natural ideas of equality developed, it was possible to conceive the sublime hope of establishing among us a free government, exempt from kings and priests, and to free from this double yoke, the long-usurped soil of Europe. I readily became enamored of this cause, in my opinion, the greatest and most beautiful one which any nation has ever undertaken.” (Grattan-Guiness, 2003)

However Fourier was disillusioned with the French revolution and he tried to resign from the committee. This was unsuccessful and he found himself trapped and unable to distance himself from it. He, therefore, maintained his position in the revolutionary committee, simultaneously continuing with his college teaching work. During the period of the Robespierre government, many of Robespierre’s opposers were sent to the guillotine. This outraged Fourier leading him to speak out against the terror he perceived in Orleans for which he was arrested. However a group of Jacobin allies arrested Robespierre and he himself was sent to the guillotine, providing Fourier a doorway to freedom.

In 1794, Fourier was granted an opportunity to further his studies at the newly founded Ecole Normale in Paris. Whilst there, he was able to interact with great mathematicians like Lagrange, Monge and Laplace. His superior ability in mathematics was highly evident amongst the other students. After completing his studies Fourier took up teaching at the College de France. Through the relationship he had cultivated with his lecturers; Lagrange, Monge, and Laplace he was able to further his mathematical career and took up a position at the Ecole Centrale des Travaux.

Unfortunately, he was arrested again over a protest speech at Orleans, but was freed during Napoleon’s reign. He returned to the Ecole polytechnique where he succeeded Lagrange in being appointed to the chair of analysis and mechanics in 1797. Throughout his teaching career he was known to be an outstanding lecturer and a gifted orator. Two years later, he joined Napoleon’s army together with other scientists. It was during the invasion of Egypt where Fourier took the role of scientific adviser. Their expedition in Egypt was, at first, a great success. They were able to take Alexandria, Malta, the Nile delta. However, the French fleet was later destroyed trapping them in Egypt. During this time Fourier organized workshops for Napoleon’s army. In addition, he worked as an administrator in the French political institutions that were being established mainly involved in archeological explorations (P, 1980)

Fourier was also involved in the establishment of the Cairo institute, specifically working in the mathematics division together with Napoleon, Monge and Malus. Fourier was appointed as secretary of the institute. He continued to hold this post for the entire time the French occupied Egypt. In addition, he was responsible for collecting and arranging all the scientific and literary discoveries that were being made during their stay in Egypt. Moreover, he had a duty of publishing large volumes of papers which were being collected in Egypt, later named Description de I’Egypte’. To this he added a detailed historical text on the ancient Egyptian civilization. All this time, he continued with his research into mathematics.

Napoleon went back to France, abandoning his army in the year 1799. Fourier followed on 2 years later together with the remainder of Napoleon’s expeditionary force. He then resumed his post as a professor of analysis in the Ecole polytechnique. Napoleon, however, had other ideas for Fourier’s career and appointed him prefect of the department of Isere, after the previous prefect had died. Fourier was unhappy with his appointment as he wanted to engage in scientific research but was unable to decline Napoleon. Therefore, he went to Grenoble where he executed many and varied duties. This made his life very busy for he was required to carry out several large projects in addition to the daily problems of the department. The previous prefect, it seems, had not left department of Isere in very good condition.

During his administrations Fourier made two great achievements; one was the supervision of construction of a new highway which was to run from Grenoble to Turin. The second was the draining the swamps of Bourgoin, which required much of his attention as there were many disagreements between the farmers and the nobility. This required him to visit all sides to seek their cooperation. Additionally, he used a great deal of his time in working on his Description de I’Egypte’. As the prefect of Isere, he held the responsibilities of a public figure. He was the one who organized the visits of pope Pius VII and the king of Spain to the region. During this time, he was under pressure to compile the results of the research work he had undertaken during the Egyptian campaign. This became too much for Fourier and he asked for leave to enable him complete the work. Upon completion of his Description de I’Egypte, in 1809 he was decorated as a baron which, at least meant he received a good wage. (Keston, 1998).

It was during Fourier’s time in Grenoble, that his mathematical and scientific work reached its peak. He had developed an intense interest in heat energy and used to keep his home unpleasantly warm and wore thick and heavy coats. This eccentricity enabled him to achieve significant developments in his theory of heat flow.

He applied mathematical techniques to the theory and wrote a memoir entitled “on the propagation of heat in solid bodies”. It was while he was working on this theory that he proposed the idea that:

"Any function of a variable, whether continuous or discontinuous, can be expanded in a series of sines of multiples of the variable"

We’ll look at what that actually means in a moment, but this is the idea which became known as the Fourier Series. He presented his memoir to the Paris institute in 1807, and also to a committee that comprised of Lagrange, Monge, Lacroix and Laplace.

Although this memoir is highly regarded now, at the time of its proposal, it was controversial. Lagrange and Laplace both challenged the idea and it wasn’t until 1829 that the German mathematician Peter Gustav Lejeune Dirichlet actually managed to demonstrate Fourier’s ideas and lay down the conditions under which the theory held. It was Dirichlet’s work that provided the foundation of what would become known as the Fourier Transform.

So what does Fourier’s theory actually mean?

"Any function of a variable, whether continuous or discontinuous, can be expanded in a series of sines of multiples of the variable"

I personally find abstract concepts like this, difficult to grasp. I like to be able to apply the things I learn to something practical. I admit that this is not always possible, however where Fourier is concerned, his theory is so useful there are many every day applications to which we can apply it.

So we need an everyday variable that we can find a function of. A function is another way of saying a signal and one of the variables we can use when looking at signals is time.

Throughout this course, we’re going to apply Fourier’s Theory using one particular type of time-dependent signal; a signal that is rather dear to my heart as a musician: Sound.

So with my apologies to Fourier, I’m going to rewrite his theory in a slightly simpler way as it applies to sound signals.

“Any sound, can be broken down into a series of sine waves at many different frequencies.”

Sounds are made when objects, such as this violin string, vibrate. The to and fro motion of the string pushes the air molecules around it together and pulls them apart again causing a wave of pressure changes in the air. These pressure changes are picked up by our ears and interpreted by our brains as sound. Many objects make can generate sound such as the vibrating cone of a speaker.

But what does sound actually look like? This might seem a bit of a strange question. Sound is something we hear, not see. In the next lecture we’re actually going to see a sound wave as we begin our journey into the Maths behind the magic of the Fourier Transform.

So let’s begin, like all good scientists, with a question:

What is sound?