Academic journal article Transactions of the American Philosophical Society

In Pursuit of the Curve

Academic journal article Transactions of the American Philosophical Society

In Pursuit of the Curve

Article excerpt


There is good evidence that work on making hyperbolic lenses by hand preceded Descartes' proof of the solution to the anaclastic. Isaac Beeckman was already a celebrated mathematician when Descartes made his acquaintance in Holland in 1618 at the age of twenty-two. The older savant and the young soldier met over a geometry problem posted at the town of Breda, where Descartes was garrisoned. His quick and elegant solution drew the attention of Beeckman, and the two men, despite several celebrated fallings-out, remained in contact until Beeckman's death in 1636.' Beeckman did not survive to see the publication of La Dioptrique in 1637 and thus did not witness the sudden popularity of the hyperbolic lens, a project he pioneered as early as 1622.

Whether Descartes played a role in Beeckman's early effort to produce an "astigmatique" lens can not be clearly determined. The entries in Beeckman's journal seem to refer only to the work of Kepler, whose earlier formulation of the anaclastic was mentioned above, but it is evident that Beeckman thought he would resolve the problem of color fringes with a new hyperbolic lens shape.2 By 1629 Beeckman was fully aware of the work being done by Ferrier under Descartes' direction, yet the work Beeckman did himself with a young artisan in the early 1620s may well have been independent of Descartes. The dispute, however, that temporarily broke the relationship between the two men, concerned, apparently, Beeckman's attempt to pass off Descartes' work as his own as early as 1618; therefore whether the hyperbolic work that led Beeckman to employ a "lunetier" was inspired by interaction with the young Descartes at Breda seems impossible to resolve.

What is certain is that Beeckman's "garçon," who was in his early teens, succeeded in making at least one approximately hyperbolic lens before 1624 by hand grinding the borders on a spherical lens in order to make it more hyperbolic in profile.3 Similar approximations were used later in the century.4 The fairly small number of references to the undertaking in Beeckman's exhaustive four-volume folio journal indicates that, though he may have had high hopes for the project, the results were less promising. Beeckman put the project aside in the late 162Os, probably after learning in 1627 about the progress of Descartes and Ferrier working in Paris.5

René Descartes returned to Paris in 1623 after a nine-year absence from France, during which he had served in the army under Prince Maurice of Nassau in Holland; Maxmillian, Duke of Bavaria; and Count Boucquoi of Hungary. He had participated in the sieges of Breda and Pressburg and watched the imperial Bavarian army decimated by Hungarian mounted swordsmen at the Battle of Neuhausel. During his travels with these armies he had won the respect and admiration of a variety of military engineers and savants, some of whom would become part of his circle on his return to Paris. In Paris, Descartes also found himself in the company of Claude Mydorge and the Père Mersenne, both of whom he seems to have known from his days as a student before 1615.6 It was during this period, before the self-imposed exile in Holland, that Descartes applied himself to the proof of the anaclastic, and during this time that he met aspiring young mathematical and mechanical practitioners like Ferrier, De Beaune, and Morin, the men who would try for the next twenty-five years to realize the hyperbolic lens making machine.

Available sources offer little on the man Ferrier: all that can be stated with certainty is that he was a promising young artisan and mechanical practitioner from Auvergne who had begun to build a reputation for himself as a craftsman and builder of mathematical instruments. Most important, he had managed to attract the attention of Descartes and had been thus introduced into his circle of savants and mathematicians in Paris. Descartes clearly thought highly of Ferrier, writing in a letter to an acquaintance that he was a man skilled in the "science of miracles" capable of realizing things thought to be the domain of "magicians aided by their demons. …

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