Pierre de Carcavi received no university education. He was a counsellor to the parliament of Toulouse from 1632 until 1636. In fact he first met Fermat in 1632 when they were both members of the Parliament in Toulouse and they remained friends. In 1636 Carcavi bought an office of counsellor in the Grand Conseil in Paris.
In 1648, however, hard times struck and he was forced to sell the office to pay for the debts of his father (who had been a banker).
After this he worked for the Duke of Liancourt until 1663. In that year he was appointed Custodian of the Royal Library, a post he held for 20 years until shortly before his death.
Carcavi is best known for his correspondence with other mathematicians rather than for his own mathematics. He was friends with Huygens, Fermat (as mentioned above) and Pascal and corresponded with them.
Fermat sent many of his works to Carcavi after he moved to Paris in 1636. In 1650 Fermat sent Carcavi a treatise entitled Novus secundarum et ulterioris radicum in analyticis usus . This work contained the first known method of elimination and Fermat wanted it published. Both Pascal and Carcavi were asked to find a publisher for the work. Carcavi approached Huygens, trying to publish not only this paper of Fermat's but also other papers which Fermat had sent him. Unfortunately neither Carcavi nor Pascal succeeded and Fermat's papers were never published.
Carcavi's friendship with Pascal, like his friendship with Fermat, lasted over many years. In 1658 Pascal solved the problem of the area of any segment of the cycloid and the centre of gravity of any segment. He also solved the problems of the volume and surface area of the solid of revolution formed by rotating the cycloid about the x-axis.
Pascal published a challenge under the name of Dettonville offering two prizes for solutions to these problems. He lodged the prizes and his own solutions with Carcavi. Carcavi and Roberval were asked to judge the solutions submitted.
Pascal gave his calculating machine, the Pascaline, to Carcavi.
Carcavi also corresponded with Galileo, Mersenne, Torricelli and Descartes and he is important in transferring information from one to the other. He made many suggestions during his correspondence with other mathematicians which were later developed and incorporated in the papers of these other mathematicians.
Carcavi was elected to the Académie des Sciences in 1666. In 1668 Carcavi was appointed to a committee, along with Huygens, Roberval, Auzout, Jean Picard and Gallois, to test whether the method of determining longitude, which had been submitted to the Académie des Sciences by a German, was practical.
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