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art of extracting, when there is occasion for it, the roots of quantities partly commensurable, and partly incommensurable: he instructed us how to form infinite series, to find by approximation the roots of numerical and literal equations of all orders: &c. most of which researches have been illustrated and commented upon in modern works.
Progress of Geometry.
FROM the commencement of the sixteenth century the ancient geometry was cultivated in Europe with rapid success. The greek geometricians, most of whom were translated into latin or italian, were taken as guides and the study of the ancient languages, being then much in vogue, multiplied both the objects and means of instruction.
Werner, who died in 1528 at the age of sixty, is mentioned as a learned geometrician. In 1522 he published some tracts at Nuremberg, almost all of which related to the theory of the conic sections.
Tartaglia and Maurolicus, who have already been mentioned, also rendered themselves useful to geometry, not only as translators of several ancient works, but as authors. The former composed a treatise in italian on Numbers and Measures; which is the first modern work, where the determination of the area of a triangle by means of it's three sides, without letting fall a perpendicular on one of the sides from the angle opposite, is to be found. The latter wrote on various subjects; and his treatise on conic sections is remarkable for perspicuity and elegance. La Hire afterward did no more than amplify the method of the sicilian geometer, and apply it to new purposes.
We ought not to forget Nonius, who was born in Portugal in 1492, and the author of several very estimable works. To him we are particularly indebted for the subdivision of the small parts of a mathematical instrument by transverse lines, called Nonius's division. He died in 1577.
Commandin, who was born in 1509, and died in 1575, was very learned in the dead languages, as well as in mathematics. He translated into latin Euclid's Elements, great part of the works of Archimedes, Ptolemy's treatises on the Planisphere and the Analemma, the book of Aristarchus of Samos on the Magnitudes and Distances of the Sun and Moon, Hero's Pneumatics, the Geodesia of the arabian geometer Mohammed of Bagdad, the Mathematical Collections of Pappus, &c. Every where Commandin displays a thorough knowledge of the subject; and he clears up the difficult passages of his authors by instructive notes, of great perspicuity and precision; a rare merit, which places Commandin far above the generality of translators and commentators.
The celebrated Ramus made no discovery in mathematics; and his Elements of Geometry and of Arithmetic are not above mediocrity: but he deserves well of the sciences for the zeal with which he defended them, and for the sacrifice he made to them of his peace, his fortune, and even his life. He was a professor in the College of France, where he founded. a chair, which still subsists; but being of the protestant religion, he fell in the horrible massacre of St Bartholomew's day by the hand of one of his fel
low professors, a zealous catholic, of the name of Charpentier.
Fernel, physician to Henry II, king of France, acquired a great name by different medical works, and some treatises and observations on the mathematics. It is said, that the great favour he enjoyed at court arose from his having taught the precious secret of removing the barrenness of Catharine de Medicis. We have a book on pure mathematics by him, entitled De Proportionibus; and two astronomical works, his Monalospherion, a kind of analemma, and his Cosmotheoria. His greatest celebrity as a mathematician is founded on his being the first of the moderns, who gave the measure of the Earth. From the number of turns made by a coach wheel on the road from Amiens to Paris, till the altitude of the pole-star was increased one degree, he estimated the length of a degree of the meridian at 56746 french toises: a conclusion pretty near the truth; but it is obvious, that it's exactness can be ascribed only to chance.
It would be as useless as tiresome, to quote here a number of geometricians, who wrote, at this period, works that deserve much praise, though they display little depth of science, and are now nearly forgotten. I shall mention, however, two german mathematicians, Peter Metius, and Hadrian Romanus, and one of Holland, Ludolph van Ceulen; all three authors of different methods for approaching much nearer to the ratio, which the circumference of the circle bears to the diameter, than had hitherto been done. Peter Metius made the remark, highly worthy our attention
and gratitude, that, if the diameter be represented by 113, the circumference will be 355; a result singu larly near the truth, considering the small number of figures by which it is expressed. Neither must I forget the celebrated Snell, another celebrated dutch mathematician, who began at the age of seventeen to write works in geometry; in which we find, among other curious things, a new determination of the ratio of the circumference of the circle to the diameter; and who afterward acquired great reputation by his inquiries concerning refraction.
The works of Regiomontanus, Tartaglia, and Bombelli, contain some geometrical problems resolved by means of algebra. But these isolated solutions, where in every particular case simple numbers were employed to express the known lines, were not founded on a regular and general method of applying algebra to geometry.
Such a method was first given by Vieta. The mutual assistance, which these two sciences lend each other, proved to him a source of many important discoveries. For instance, he observed, that every equation of the third order, containing in general either one real root only and two imaginary ones, or three real roots, the real root in the first case was to be found by the duplication of the cube, and the three real roots in the second by the trisection of an angle. It must not be forgotten, however, that he had only a confused idea of negative roots, and that they began to be made known distinctly by des Cartes.
The elements of the doctrine of angular sections are likewise, the invention of Victa, It is well known, that