Quotations in Context: Fermat

“And perhaps, posterity will thank me for having shown it
that the ancients did not know everything.”

The French scholar and lawyer Pierre de Carcavi held a number of different jobs during his career, which was capped off by maintaining the Royal Library in Paris for two decades. He corresponded with many of the greatest mathematicians and scientists in Europe, including Descartes, Pascal, Torricelli, Roberval, Huygens, Leibniz, and Galileo. Early in his career, Carcavi joined the Parliament of Toulouse in 1632, where he met Pierre de Fermat. Even after Carcavi left Toulouse for Paris in 1636, they both continued to correspond, and Carcavi made many (unsuccessful) attempts to publish the mathematical manuscripts that Fermat shared with him. 

Engraving of Carcavi by Gérard Edelinck, 1675.
Public domain, Los Angeles County Museum of Art.

In August 1659, Fermat sent Carcavi a short letter entitled Relation des nouvelles découvertes en la science des nombres, which can be found on pages 213–216 of Recherches sur les Manuscrits de Pierre de Fermat by Charles Henry, published in 1880. In this letter, Fermat described his method of infinite descent, stating that at first he was able to use the method only to show negative results, such as the fact that if a right triangle has sides with whole number lengths, then the area of the triangle cannot be a square number. Fermat claimed that if such a triangle exists, then there would have to be a smaller triangle with the same property, and then an even smaller triangle, and so on, leading to a contradiction since whole numbers cannot be decreased indefinitely. 

Engraving of Fermat by François de Poilly. 
Public domain, Wikimedia Commons.

Fermat intentionally did not justify his claim that if a right triangle with such a property existed, there would have to be a smaller triangle with the same property, stating both that the explanation would be too long, and that he wanted to keep mathematicians such as Blaise Pascal and Gilles de Roberval guessing: 

Je n’adjouste pas la raison d’ou j’infere que s’il y auoit un triangle rectangle de cette nature, il y en auroit un autre de mesme nature moindre que le premier, parce que le discours en seroit trop long, et que c’est la tout le mystere de ma method. Je seray bien aise que les Pascals et les Roberuals et tant d’autres scavants la cherchent sur mon indication [Henry 1880, p. 214]. 

Fermat stated that eventually he saw how to apply the method to positive assertions, and he gave the example of the claim that any prime number which is one more than a multiple of four can be written as a sum of squares. Fermat again provided an outline of the argument, stating that if the prime could not be written as a sum of squares, there would have to be a smaller prime with the same property, and then a smaller such number, and so on until finally the prime was decreased to five, the smallest possible prime that is one more than a multiple of four. But \(5=1^2+2^2\), which would be a contradiction. As before, Fermat did not provide the details for how one finds the smaller prime in each case of the descent. 

Fermat included several more examples of problems that can be resolved using infinite descent, and noted in conclusion that he wrote this short summary because he feared he would never have the time to write a detailed account of the method. He expressed a hope that Carcavi would share the letter with other mathematicians, who might be inspired to master the method and publish a complete account themselves. It is at this point that the subject quotation of this column appears: 

Et peut estre la posterite me scaura gré de luy avoir fait connoistre que les anciens n'ont pas tout sceu [Henry 1880, p. 216].

References:

Henry, Charles. 1880. Recherches sur les Manuscrits de Pierre de Fermat.  Rome: Imprimerie des Sciences Mathématiques et Physiques.

“Quotations in Context” is a regular column written by Michael Molinsky that has appeared in the CSHPM/SCHPM Bulletin of the Canadian Society for History and Philosophy of Mathematics since 2006 (this installment was first published in May 2020). In the modern world, quotations by mathematicians or about mathematics frequently appear in works written for a general audience, but often these quotations are provided without listing a primary source or providing any information about the surrounding context in which the quotation appeared. These columns provide interesting information on selected statements related to mathematics, but more importantly, the columns highlight the fact that students today can do the same legwork, using online databases of original sources to track down and examine quotations in their original context.