Fermat / Analytical Geometry

Fermat (1601-1665) was not an expert in mathematics because he first was a lawyer and then a councilor by profession. Even so, he had brought about some significant results in the field of number theory, and among them, Fermat's last (or great) theorem is particularly well known. Fermat had made several other discoveries that became part of the basis of mathematics.

[Coordinates]

Fermat was the inventor of coordinates. However, since the theory was constituted with only one coordinate axis, it could be referred to as a uniaxial analysis. Moreover, the term coordinates was formed by Leibniz and the orthogonal coordinate axis came into widespread use in the 19th Century. Today, the orthogonal coordinate is referred to as "Cartesian coordinates" because when Descartes published his book "Discours de la methode", he defined coordinates for the first time in the world in this book and it is believed that this first definition had influenced the use of the term "Cartesian coordinates" in after ages.

[Differentiation]
In his treatise "Method to find the maximal and minimal values", Fermat described the method of finding out the maximal and minimal values of the curved line of a given polynomial equation.

Suppose a given point is the extremal value, and when the value of and the value of passing through the nearby point are comparedAthe difference would be nearly unnoticeable although even though the values may be different.

The values of point and can be calculated by the formulation and by dividing the numerical result by and assuming that .

Fermat also found the method to calculate tangent.

In the calculation of tangent at point , point is assumed to be on the curved line as well as on the tangent.

and are similar.

The value of can be calculated by dividing the denominator and numerator by and when is assumed.

This method is similar to the "Method to find maximal and minimal values".

Descartes disqualified this method when he learned of it in 1638 and said; "This method is not correct."

[Integration]

Besides calculating the tangent of , Fermat also calculated the downside area of the curved line.

He broke down the distance into using , which is close enough to 1.

He approximated the downside area by drawing vertical lines against the curved line from each point. Fermat used the term "squaring" to mean "integration".

The sum of the areas of given rectangles

Given that Athen Area

The above relation could be held good even in the case where being a rational number. Fermat calculated in a similar way for the cases where being negative numbers but the calculation did not work out only in the case where .

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