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of the common parabola depends on the quadrature of the hyperbola, or on logarithms. Huygens, in the geometrical demonstrations of his Horologium oscillatorium, which appeared for the first time in 1673, rectifies curves, squares superficies, or reduces their expressions to others more simple, with an elegance and address, which the lovers of the true or linear geometry will never cease to admire.
It is commonly supposed, from the assertion of Wallis in his treatise on the Cissoid, that William Neil, his pupil, was the first who rectified the second cubic parabola. Huygens on the contrary maintains, that the theorem of Van Heuraet was circulated among geometricians, before the english had turned their thoughts to the question. As the methods are different, it may very easily have happened, that both Neil and Van Heuraet arrived at the same result, without either of them having borrowed any thing from the other. However, all these problems are but trifles, since the invention of the method of fluxions.
The cycloid began to be a little forgotten by geometricians, when Pascal brought it forward again in 1658, by proposing new problems relating to this curve, and offering prizes to those who should solve them. The total area of the cycloid, the centre of gravity of this area, the solids which the curve describes by revolving round it's basis, or the diameter of the generating circle, and the centres of gravity of these solids, had been determined: Pascal demanded what was then much more difficult, indefinite measures; that is to say, the area of any seg
ment of the cycloid whatever, the centre of gravity of this segment, the solids, and centres of gravity of the solids, which this segment describes in revolving round the ordinate, or round the absciss, whether it make a complete revolution, half a revolution, or a quarter of a revolution.
Huygens squared the segment included between the vertex and as far as a fourth of the diameter of the generating circle. Sluze measured the area of the curve by a very elegant method. The celebrated english architect sir Christopher Wren determined the length and centre of gravity of the cycloidal arc comprised between the vertex and the ordinate, and the superficies of the solids which the revolution of this arc produces. Fermat and Roberval, also from the bare enunciation of the theorems of the english geometrician, found the demonstrations of them. But all these investigations, though very profound and beautiful, did not answer the questions of the challenge, at least completely; neither were they sent in, to compete for the prize.
Wallis, and father Lallouère, a jesuit, were the only persons, who, having treated all the problems proposed, thought themselves qualified to claim the prizes. Yet Pascal demonstrated to them both, that they were mistaken in several points; and that they had advanced false conclusions, founded on errours, not in their calculations, but in their methods. He alone gave, in 1659, a true and complete solution of the problems, as well as of several others still more difficult.
In each of these investigations the common cycloid alone was the subject in question. Pascal determined in addition the dimensions of all the cycloids, curtate or prolate. He showed, that the length of these curves depends on the rectification of the ellipsis, and assigned the axes of the ellipsis in each case: when one of these axes becomes nought, the ellipsis is changed into a right line, the curve becomes the common cycloid, and Pascal concludes from his method, that the cycloidal arc is then double the corresponding chord of the generating circle; which comprises the theorem of sir C. Wren as a particular case. He likewise deduced, from his method, another very remarkable theorem, which is, that, if two cycloids, one prolate the other curtate, be such, that the base of one is equal to the circumference of the generating circle of the other, the length of these two cycloids will be equal.
In all Pascal's inventions in mathematics we discern a genius, of the most powerful kind, for the advancement of science, that the earth ever bore. Geometricians regret, that he did not dedicate to them the whole course of his short life: but then we should have lost those celebrated Provincial Letters, and those profound Thoughts, which are perhaps the masterpieces of french eloquence.
Barrow had a happy idea, which may be considered as a fresh step toward fluxions, in forming his differential triangle, for drawing tangents to curves. This triangle, as is well known, has for it's sides the element of the curve, and those of the absciss and ordinate. Barrow's method is in fact nothing more
than Fermat's abridged and simplified, as he treats the three sides of the triangle immediately as infinitely small quantities, and thus saves some length of calculation: but it still wants the essential characters of the method of fluxions, that is to say, a uniform algorithm for all cases, and the advantage of giving by one general formula the tangents of all sorts of curves, whether geometrical or mechanical. Accordingly Barrow stopped at the problem of tangents, and this even limited to the single case, where the equations are algebraical and rational, while the method of fluxions applies to an infinite number of uses besides.
The ancients set great value on simplicity and elegance in the construction of geometrical problems. And Sluze, their imitator in this respect, carried the use of geometric loci for the resolution of equations to the highest degree of perfection.
One of the greatest discoveries made in modern geometry was the theory of evolutes, invented by Huygens, which may be found in his Horologium oscillatorium already mentioned. A curve being given, Huygens constructs another curve, by drawing a series of right lines perpendicular to the former, which touch the latter: or inversely, the second curve being given, he constructs the first. From this general idea he deduces a number of remarkable propositions; such as divers theorems on the rectifications of curves; the singular property, which the cycloid has, of producing by evolution an equal and similar cycloid, placed in a reverse position; &c. The uses of this theory, in all parts of the mathematics,
are without number. Apollonius had given a general idea of it; but it remained barren, and Huygens, who, not content with clearing the ground, produced from it himself an abundant harvest, will always retain the glory of having transmitted it as a possession to future geometricians.
The english continued to enrich geometry with novelties, at that time very striking. Brouncker gave an infinite series to represent the area of the hyperbola; and Nicholas Mercator separately made the same discovery. Wallis had long taught how to square curves with monomial ordinates; and his method equally applied to curves, the ordinates of which were complex quantities raised to entire and positive powers, by developing these powers by the common principles of multiplication.
He sought to extend this theory likewise to curves with complex and radical ordinates, by endeavouring in this case to interpolate the series of the former kind with new series; but he could not succeed. Newton surmounted this difficulty: he did more; he solved the problem in a direct and much more simple manner, by means of the formula he discovered for expanding into an infinite series any power of a binonomial, whatever the exponent of the power might be, integral or fractional, positive or negative.
The infinite series thence resulting for the quadrature of the circle was found in another manner by curiJames Gregory, who formed several other very ous series. In a work never published, but of which a summary has been preserved, he gave the tangent and secant by the arc, and inversely the arc by the tangent