The commonly accepted interpretations ofFermat's method of extreme values tell us that this is a curious method, based on an approximate equality and burdened with several contradictions withinFermat's writings. In this article, both a philological approach taking into account that there is only one manuscript written inFermat's own handwriting and a mathematical approach taking into account that brilliant mathematicians usually are not so very confused when talking about their own central mathematical ideas are combined. A new hypothesis is put forward (...) which renders the mathematics clear and coherent and which does not need the assumption thatFermat was confused. Probably a number of words have been added byCarcavy in two ofFermat's papers. (shrink)