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seventh is employed in demonstrating the same things by principles independent of indivisibles, and to establish the perfect accuracy of the new method by the coincidence of their results.
The french geometricians, on their part, resolved similar but more difficult problems. For instance, Fermat, Roberval, and des Cartes squared parabolas of the higher orders, and determined the solids which all these curves form by revolving round the absciss or ordinate, as likewise the centres of gravity of these solids; thus completing the theory, which Archimedes had given for the common parabola.
The method of Roberval, like that of Cavaleri, was founded on the principle of indivisibles; but it was exhibited in a point of view more conformable to geometrical strictness, because Roberval considered the planes, or solids, as having for their elements rectangles of infinitely small altitudes, or sections of infinitely little thickness, and not simple lines or planes. There are proofs, that he employed this method as early as 1634, and consequently he borrowed nothing from Cavaleri.
About the same time Roberval applied his methods to the cycloid, a curve that had become celebrated by it's numerous and singular properties. He determined the area of this curve, and the solids it generated by turning round it's base or it's axis: he likewise found the centre of gravity of the area of the same curve, and those of it's parts on each side
of the axis.
These new problems having been proposed to Fermat and des Cartes, they both resolved them. They likewise taught how to draw tangents to the cycloid, which, being a mechanical curve, required methods different from those already employed for drawing tangents to geometrical curves.
Roberval had framed a general method for tangents, which was applicable to geometrical and mechanical curves indifferently, and by this he determined the tangents of the cycloid. This method deserves notice for it's analogy, in respect to the metaphysical principle, with that of fluxions, which Newton produced long after. A curve being supposed to be described by the motion of a point, Roberval considers this point as acted upon at every instant by two velocities given by the nature of the curve: he constructs a parallelogram, the sides of which are proportional to these velocities: and he assumes as a principle, that the direction of the element, or of the tangent, must fall on the diagonal; so that the position of this diagonal being known, we have that of the tangent, Thus, for example, in the ellipsis, where the sum of the two lines drawn from the foci to one and the same point in the curve is always the same, if one of the lines be diminished in any degree, the other will be increased by an equal quantity: then the parallelogram becomes a rhomboid; and consequently the tangent must divide into two equal parts the angle formed by the prolongations of the two lines proposed. But the method does not apply to all cases with the same facility; and frequently it even becomes impracticable by the diffi
culty of determining the two velocities of the describing point: while in the method of fluxions, the metaphysical principle being reduced to an algorithm of calculation, disencumbered of all superfluous operations, one and the same general formula gives, without the least difficulty, the tangents for all curves, of which we have the equation.
Unhappily, Roberval united with great geometrical talents a vain and peevish disposition. He was continually at war with des Cartes and other french geometricians, and very often he was in the wrong.
On the subject of the problems of the cycloid he mortally offended Torricelli. This illustrious italian geometer having published solutions of these problems, as of his own invention, in 1644, Roberval claimed them; maintaining that they were fundamentally the same with his own, which one Beaugrand had communicated to Galileo, on whose death they fell into the hands of Torricelli, his pupil, and the inheritor of his papers. Torricelli was so much affected by this charge of plagiarism, that it brought him to his grave in the flower of his age. If any one follow Torricelli attentively in his demonstrations, he will be fully convinced, that they are his own; and that probably he had never read either the pretended copies of Roberval's solutions, sent to Galileo, or the Universal Harmony of father Mersenne, published in 1637, where these same solutions are given.
The jesuit Gregory St. Vincent, who was born in the Netherlands in 1584, and died in 1667, acquired considerable reputation in the mathematics by a work, in which he attempted to square the circle, and failed,
but which notwithstanding abounded in accurate and profound theories on the measure of the ungulæ of different bodies formed by the revolution of the conic
Herigon, also, deserves to be mentioned here, not as a mathematician of the first order, but for having collected into one course all the branches of the mathematical sciences, in the state in which they were at that period. His book, published in 1644 in french and latin, was widely circulated, and was of great utility. Beside the general knowledge of arithmetic, algebra, geometry, mechanics, astronomy, geography, &c., Herigon has introduced into his collection several works of the ancient geometricians; such as Euclid's Elements, his Data, his Optics and Catoptrics, the Geometry of Tactions of Apollonius, the Spherics of Theodosius, &c. In all of which his neat, clear, and strict method of demonstration deserves praise.
The celebrated inverse method of Tangents originated from a problem, which Beaune proposed to his friend des Cartes in 1647. This was, to find a curve, such that the ordinate shall be to the subtangent, as a given line is to the part of the ordinate comprised between the curve and a line inclined at a given angle.' Des Cartes pointed out the construction of the curve, and several properties of it; but he could not completely accomplish the solution, which was reserved for the method of fluxions.
While Roberval, and some others of the french geometricians, endeavoured to depreciate the geometry of des Cartes, it found a crowd of admirers of the greatest merit in foreign countries. Of these the
chief was Schooten, professor of mathematics at Leyden, who displayed and extended it in an excellent commentary, published for the first time in 1649, and afterward reprinted with considerable additions. He had already distinguished himself by his Geometrical Exercitations, published in 1646.
In England geometry acquired new treasures of a different kind. By his Arithmetic of Infinites Wallis resolved a great number of elegant problems relating to the quadrature of curves, the cubature of solids, the determination of centres of gravity, &c.
When parabolas of every order had been squared, it was natural to think of determining their curvatures, or generally to find a right line, which should be equal in length to the perimeter of a given curve. This new problem was then attended with the greatest difficulty. As early as the year 1657 Huygens gave, by letters, some openings for solving it. His countryman Van Heuraet reduced the question to geometrical constructions, which were a little embarrassing, but which at length led him to a very beautiful discovery. He found, that the second cubic parabola, where the squares of the ordinates are as the cubes of the abscisses, is equal to a right line, which he assigns. This discovery was published in 1659, at the end of the second edition of Schooten's Commentary on the Geometry of des Cartes.
The other parabolas are not susceptible of rectification algebraically: but they may be measured at least by methods of approximation, by employing series, or quadratures, of certain curvilinear spaces easy to be calculated: for example, the rectification