[228] If the hyperbola is referred to the asymptotes as axes, the area between two ordinates (x = a, x = b) is the difference of the logarithms of a and b to the base e. E.g., in the case of the hyperbola xy = 1, the area between x = a and x = 1 is log a.
[229] "On ne peut lui refuser la justice de remarquer que personne avant lui ne s'est porté dans cette recherche avec autant de génie, & même, si nous en exceptons son objet principal, avec autant de succès." Quadrature du Cercle, p. 66.
[230] The title proceeds: Seu duae mediae proportionales inter extremas datas per circulum et per infinitas hyperbolas, vel ellipses et per quamlibet exhibitae.... René Francois, Baron de Sluse (1622-1685) was canon and chancellor of Liège, and a member of the Royal Society. He also published a work on tangents (1672). The word mesolabium is from the Greek μεσολάβιον or μεσόλαβον, an instrument invented by Eratosthenes for finding two mean proportionals.
[231] The full title has some interest: Vera circuli et hyperbolae quadratura cui accedit geometriae pars universalis inserviens quantitatum curvarum transmutationi et mensurae. Authore Jacobo Gregorio Abredonensi Scoto ... Patavii, 1667. That is, James Gregory (1638-1675) of Aberdeen (he was really born near but not in the city), a good Scot, was publishing his work down in Padua. The reason was that he had been studying in Italy, and that this was a product of his youth. He had already (1663) published his Optica promota, and it is not remarkable that his brilliancy brought him a wide circle of friends on the continent and the offer of a pension from Louis XIV. He became professor of mathematics at St Andrews and later at Edinburgh, and invented the first successful reflecting telescope. The distinctive feature of his Vera quadratura is his use of an infinite converging series, a plan that Archimedes used with the parabola.
[232] Jean de Beaulieu wrote several works on mathematics, including La lumière de l'arithmétique (n.d.), La lumière des mathématiques (1673), Nouvelle invention d'arithmétique (1677), and some mathematical tables.
[233] A just estimate. There were several works published by Gérard Desargues (1593-1661), of which the greatest was the Brouillon Proiect (Paris, 1639). There is an excellent edition of the Œuvres de Desargues by M. Poudra, Paris, 1864.
[234] "A certain M. de Beaugrand, a mathematician, very badly treated by Descartes, and, as it appears, rightly so."
[235] This is a very old approximation for π. One of the latest pretended geometric proofs resulting in this value appeared in New York in 1910, entitled Quadrimetry (privately printed).
[236] "Copernicus, a German, made himself no less illustrious by his learned writings; and we might say of him that he stood alone and unique in the strength of his problems, if his excessive presumption had not led him to set forth in this science a proposition so absurd that it is contrary to faith and reason, namely that the circumference of a circle is fixed and immovable while the center is movable: on which geometrical principle he has declared in his astrological treatise that the sun is fixed and the earth is in motion."
[237] So in the original.