“Mais entre tous les écrits qu’on a recues de cette sorte, il n’y a rien de plus beau que ce qui a été envoyé par M. Wren; car outre la belle manière qu’il donne de mesurer le plan de la roulette (=cycloid), il a donné la comparaison de la ligne courbe même et ses parties, avec la ligne droit: sa proposition est que la ligne de la roulette est quadruple de son axe, dont il a envoyé l’énonciation sans démonstration. Et comme il est le premier qui l’a produite, c’est sans doute a lui que l’honneur de la première invention en appartient.”
Summing up his history of the cycloid, he concluded that the first to remark that curve in nature was P. Mersenne, that M. de Roberval first worked out some of its properties, “que le premier qui en a mesuré la ligne courbe a été M. Wren.”
The story is then taken up by a letter of Cavarci (dated December 10, 1658), the umpire, to Pascal (now masquerading under a new pseudonym—A. Dettonville = an anagram of Louis de Montalte), in which he recites the nature of the problems set—i.e., to find the dimensions and centres of gravity of the solids generated by the revolution of the cycloid. He goes on to say that there were sent solutions of the more easy problems—“savoir: le centre de gravité de la ligne courbe et la dimension des solides, lequelle M. Wren nous envoya dans ses lettres du 12 Octobre”—but concludes that of the challenge problem no solutions had been sent.
Pascal replied to this letter with a series of letters setting out a general method for dealing with such problems and the actual solutions of the problems he had proposed.
The real quarrel as to whether the problems had been solved or not was with Wallis and not with Wren. Wallis appears to have sent a solution and followed it up by various letters offering corrections. However, he was adjudged wrong in principle (see Récit de l’Examen pour les prix sur la Roulette). The prizes were not awarded. Wallis afterwards (1659) published a “Tractatus de Cycloide,” in which are included four propositions on the cycloid which Wren had given to Wallis.
Turning now to Wren’s counter problem. It is not directly connected with the cycloid, but with one of the properties of the ellipse, and it had previously been suggested by Kepler. It appears to have been confused with Pascal’s cycloid problems because Pascal showed in his general method that various cycloid problems could be referred to the ellipse. Pascal has a chapter, “L’égalité entre les lignes courbes de toutes sortes de Roulettes et les lignes elliptiques,” in the course of which he remarks, “Cette admirable égalité de la courbe de la roulette simple à une droite [=straight line] que M. Wren a trouvée, n’était, pour aussi dire, q’une égalité par accident, qui vient de ce qu’en ce cas l’ellipse se trouvé réduite à une droite.” Wren’s challenge seems to have remained unnoticed.
APPENDIX II
AN ATTEMPT AT A WREN CHRONOLOGY
The dates of Wren’s work have been set down so wildly that I prefer the omission of some buildings to the repetition of blunders. Even so I have no doubt repeated many old mistakes and made some fresh ones. An accurate chronology of Wren would be of great comfort to the student. I have checked only forty-seven of Miss Milman’s dates, but found forty-five of them wrong, by from one to twenty-five years.
The City churches are given under the year during which the first payment to the builders was made: the dates in brackets mark the last of such payments, which may well have been some years after the buildings were completed. Wren seems to have settled final accounts in batches.
L. W.