Life and work PIERRE DE FERMAT

Carlos Eduardo de Oliveira

The Calculus Creator: Newton or Leibniz?

Fermat in Truth

Jurandi Leão Santos

1601 (?) - 1665

PIERRE DE FERMAT

PIERRE DE FERMAT

- Son of Dominique de Fermat (merchant of skins);

- He studied at the Franciscan Monastery Grandselve and later at the University of Toulouse;

- Toulouse Parliament Counsellor;

- He is interested in mathematics in his leisure time;

- Influenced by the Arithmetic of Diophantus;

- Contributions in Number Theory, Probability Theory, Analytic Geometry.

PIERRE DE FERMAT

“Here, on January 13, 1665, was buried Pierre de Fermat, the Parliament counsellor and

mathematician of great renown, famous for his theorem,

for n>2”.

FERMAT TOMBSTONE

n n na b c

PIERRE DE FERMAT

FERMAT LETTER

PIERRE DE FERMAT

OEUVRES DE FERMAT

Excerpt from the letter from Descartes to Fermat :

“I know that my approval is not necessary for you about the opinion you have of yourself, but if it can contribute in something, then you give me the honor to answer. I think I have to confess to you frankly

that I have never met anyone who made me try to understand geometry as you.”

FERMAT WORKS

* Maximum and minimum method

Let an AB segment, measuring a. Find the point C on AB so that the product AC.CB is maximum:

2

1( )P b a b ab b Max

Consider the segment of measure b with a (very small) certain increase e in such a way that the product remains maximum:

2( )( )P b e a b e

2 2

2P ab b be ae be e Max

Equaling P1 and P2 because the two products represent the maximum - note that e should be zero for this to happen, so we will take e = 0 at the appropriate time:

ab 2b ab 2b 2 22 0be ae be e ae be e

Dividing the expression by e: 2 0a b e

As we know that e=0: 2 0 22

aa b a b b

FERMAT WORKS

This was one problem that Fermat presented as exemplification of his method: Divide the line AC at point E, such that the product between AE square and EC is maximum"

2 2 3

1( )P b a b ab b

2

2( ) ( )P b e a b e

2 3 2 2 2 3

23 2 3P ab b b e abe be ae e

Equaling : 1 2

P e P2ab 3b 2ab 3b 2 2 2 33 2 3b e abe be ae e

2 2 2 30 3 2 3b e abe be ae e

Dividing the expression by e: 2 23 2 3 0b ab be ae e

Taking e = 0: 23 2 0b ab

Dividing the expression by b: 2

3 2 03

ab a b

CONCLUSION

Let's look at this method in a more current language. Consider a segment with measure a and let f(b) the product of the two parts of this segment when the first of them measures b, or f(b) = b(a – b)

Adopting an increase e to b, we are calculating f(b+e) = (b + e)(a – b – e)

Equating these two functions: ( ) ( ) ( ) ( ) 0f b f b e f b f b e

Dividing the expression by e: ( ) ( )0

f b f b e

e

At the appropriate time, making e = 0: 0

( ) ( )lim 0 '( ) 0e

f b f b ef b

e

extreme or critical point

THANK YOU

Bibliography

TANNERY, Paul. HENRY, Charles. Oeuvres de Fermat.

EVES, Howard. Introdução à história da matemática.

HAROLD M. Edwards. Fermat’s Last Theorem.

LAI, Giampaolo. Alle origini del calcolo infinitesimale: il metodo dei

massimi e minimi di Fermat.

MAHONEY, Michael S. The mathematical carrer of Pierre de Fermat.

PITOMBEIRA, João B. ROQUE, Tatiana. Tópicos de história da matemática.

BOYER, Carl B. História da Matemática.

PARADÍS, Jaume. PLA, Josep. VIADER, Pelegrí. Fermat’s treatise on quadrature: a new reading.

DESCARTES. René. Discurso do Método.

ACZEL, Amir D.. O Caderno Secreto de Descartes