We now enter upon the 17th century, the most fertile of any in mathematical discoveries; in fact, the progress since made in the science is little more than their expan sion ; and whatever perfection it may attain in future ages, a great share of the glory will belong to the period at which we are now arrived.
One of the earliest geometers of the 17th century was Lucas Valerius, an Italian, and Professor of Mathematics at Home. He determined the centre of gravity in complete colloids and spheroids, as well as in their segments cut off by planes parallel to the base. Archimedes had resolved this problem only in the ease of the parabolic conoid ; and Commandinus had extended the subject a little farther, to the easiest cases; but Valerius went beyond them both.
Marinus Ghetaldus, a native of Ragusa, was well ac quainted with the ancient geometry. Guided by the indi cations of Pappus, he attempted to restore the lost book of Apollonius on Int/Mations, in a work called 4pollonius Redivivus. He also wrote a supplement to the ?lliollonius Gallus of Vieta. He died on a mission to Turkey in 1609.
Alexander Anderson was one of the earliest of the Scot tish geometers. He appears to have been a friend or scho lar of Vieta, some of whose posthumous works he publish ed. He was well acquainted with the geometrical analysis; and of this he has given proof in his Suliplementum nii Redivivi, where he endeavours to supply what Ghetal dus has left incomplete in his work. See ANDERSON.
The Low Countries produced in this period several ma thematicians, whose labours were conducive to the progress of geometry. Ludolph Van Ceulen claims attention, on account of the immensely long calculation by which he de .termined that the diameter of a circle being supposed 1, the circumference will be between the number 3.14159,26535,89793,23846,26433,83279,50288, and the same number increased by unity in the last figure. It must be acknowledged that there was more patience than genius displayed in this effort ; for he proceeded simply af ter the manner of Archimedes, inscribing polygons in a circle, and describing others of an equal number of sides about it, until he found an inscribed and circumscribing po lygon to agree in 56 figures. After the example of Ar chimedes, he desired that this, his greatest discovery, should be inscribed on his tomb. Geometry, however, de rived more real service from his other labours.
Willebrod Snellius was another of the Dutch mathema ticians : At the age of seventeen, he undertook to restore Apollonius's book of Determinate Sections, and he pub lished his divination with the title .9pollonius Batavus. He
also treated of the approximate value of the circumference of a circle in his Cyclometria. He here shewed how Van Ceulen might have greatly shortened his labour, by two li mits nearer to the circumference than the circumscribing and inscribed polygons; and he verified the calculation, by computing the perimeter of a polygon of 1073741824 sides, which, according to the other method, would have given only 20 figures of the number.
Albert Girard, another Fleming, was highly estimable as a geometer. Ile was the first that found the surface of a spherical triangle, or of a polygon bounded by great cir cles on the sphere. lie deserves still more honour, how ever, for his divination of the Porisms of Euclid, if, as he asserted, he really had succeeded in restoring the work of the ancient geometer. Unfortunately his lahours on this subject have never been published.
Want of room obliges us to pass over several whose re putation as geometers is excelled by that which they ac quired in other branches of mathematics : we must not, however, omit the celebrated Kepler ; he was the first that bad the boldness to introduce the name and the idea of infinity into the language of geometry. The circle lie considered as composed of an infinite number of triangles, having their vertices at the centre, and of which the bases form the circumference ; and the cone, as made up of an infinite number of pyramids, whose bases formed its base, and which had with it a common vertex.
By the aid of these, and similar views, Kepler, in his Nova Stereometria, a work on gauging, demonstrated, in a direct manner, and with great brevity, those truths, which the ancients had established by tedious and very peculiar modes of reasoning. Kepler opened in this book a vast field for speculation ; for, passing beyond the views of Archimedes, he formed a multitude of new bodies, and he investigated their solidities. Archimedes limited his en quires to those generated by the rotation of conic sections about an axis, but Kepler treated of solids generated by the rotation of these curves about any line whatever in their plane. Ile thus considered ninety solids besides those handled by the Sicilian geometer. Upon the whole, this book contained views, which appear to have had great influence on the improvements that soon afterwards took place in geometry.