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that the object of this theory is to find the general expressions of the chords or sines, for a series of arcs that are the multiples of each other; and inversely the expressions of the arcs, when the chords or sines are known. It has since received great additions from the hands of Hermann, James Bernoulli, and Euler.
Some authors have published, what others have repeated after them; and we hear every day in conversation, that des Cartes invented the application of algebra to geometry. But this is not accurate. It is ascribing more to des Cartes, than is justly his due; and forgeting the clains of his predecessors, particularly those of Vieta. The mistake is certainly pardonable, when we consider the happy, original, and extensive use, which des Cartes made of the discovery; but strict justice ought to be done, and the truth established.
By this des Cartes will lose very little: he will have the glory of being the first, who in this way completely solved the following general problem, which the ancient geometricians, Euclid, Apollonius, and Pappus, proposed to themselves, and of the solution of which they merely gave a sketch: the positions of any number of right lines, on a plane surface, being given, to find a point, from which as many other right lines may be drawn, one to each of the given lines, which shall make with them given angles; with this condition, that the product of two of the lines thus drawn shall be in a given ratio to the square of the third, if there be only three; or to the product of the other two, if there be four; or if there be five, that the product of three shall be in
a given ratio to the product of the other two and a third given line; or if there be six, &c.'
Des Cartes began by observing, that the question thus proposed is indeterminate; and that there are an infinite number of points, from which the lines required may be drawn: he conceived, that all these points might be regarded as placed in the curve described by a style, made to move on a plane according to the conditions of the problem: and he expressed this condition by an equation between the given quantities and two variable lines; so that by assuming at pleasure one of these lines, the other would be deduced from the equation; which made known at every instant the position of the describing point.
Soon after, by a fresh effort of genius, the honour of which no one shares with him, he arrived at a general method of representing the nature of curve lines by equations, and of distributing them into different classes according to the different orders of these equations: a vast and fertile field, which des Cartes laid open to the sagacity of every mathematician. By this, the law according to which a curve is to be described being given, it's course is readily traced; and it's tangents, perpendiculars, finite or infinite branches, points of inflexion or contrary flexure, and in general all the affections which characterize it, are determined. This method combines simplicity and generality in one point of view. Thus, for instance, one and the same equation of the second order between the absciss and ordinate, combined with constant quantities, may represent in general the nature of the three conic sections; then the values and
ratios of the constant quantities restrict the equation, to express, in particular cases, a parabola, an ellipsis, or an hyperbola.
We are likewise indebted to des Cartes for the manner of considering and constructing curves of double curvature by projecting them on two planes perpendicular to each other, on which they form ordinary curves, which have a common absciss and ordinate.
Of all the problems he solved in geometry no one, he said, gave him so much pleasure, as his method of drawing tangents to curve lines; by which, however, geometrical curves only are to be understood. This method gives the tangents by means of perpendiculars to the points of contact. The author assumes, that, from any point taken in the axis of a curve, a circle may be described, cutting the curve at least in two points: after which he seeks the equation that expresses the places of intersection: he then supposes, that the radius of the circle diminishes, till two adjacent intersections coincide: then the two corresponding radii form but one, which is perpendicular to the curve: and the question reduces itself to forming, from these elements, an equation, which contains two equal roots.
Des Cartes afterward proposed another method for tangents. In this he takes a point out of the curve, and on it's axis produced, on which he makes a curved line revolve, cutting the curve at least in two points and the two points of intersection he makes to coincide, by subjecting, as in the former instance, the equation of the intersections to contain two equal
Both these methods are evidently founded on the same principle, and are very ingenious, though they are much less simple, and less direct, than that of fluxions. The geometry of des Cartes appeared in 1637.
Before this period Fermat had invented his method of determining the maxima and minima, in quantities which increase at first, and then diminish; or which begin with diminishing, and afterward increase. It turns on this remark, that on each side of the point of the maximum, and also of the minimum, there are two equal magnitudes. Fermat seeks the expressions of two magnitudes at an arbitrary distance, and makes them equal to each other; then supposing the distance proposed to become infinitely small, or less than any assignable finite quantity, he obtains an equation, which gives the maximum, or the minimum. The same method serves to determine the tangents of geometrical curves, by first considering the tangent as a secant, and then making the portion of the absciss, comprised between the two ordinates answering to the two points of intersection, to vanish.
The method of fluxions rests on the same basis; yet Fermat cannot be called the inventor of fluxions. His method is not reduced to an algorithm: it is simply a general indication of the calculations to be made in each particular case; it applies only to geometrical curves; and even in this case it requires, that the radical quantities, which the equations may contain, be made to disappear; which frequently leads to calculations that are intractable, either on account
account of their length, or from the difficulty of discovering the root that satisfies the conditions of the problem.
We must refer to the head of mixed geometry several works, that appeared in the seventeenth century previous to the rise of the direct and inverse methods of fluxions: not that the methods employed in thein are founded on algebraical calculation, but because they are all more or less guided by it's spirit.
One of the most original is Cavaleri's Geometry of Indivisibles, which appeared in 1635. The method of the ancients for determining the superficies and solidities of bodies was very strict, but it had the inconvenience of requiring many collateral processes: it was necessary, that polygons should be inscribed in a figure, and circumscribed about it; or solids inscribed in a solid, and circumscribed about it; and then the limits of the ratio between the last inscribed and the last circumscribed polygon, or that of the last inscribed and the last circumscribed solid, were to be sought. Cavaleri proceeds more directly to the object: he considers plane superficies as composed of an infinite number of lines, and solids of an infinite number of planes; and he assumes as a principle, that the ratios of these infinite sums of lines, or planes, compared with the unit in each case, are the same as those of the superficies or solids, that were to be measured. The work of Cavaleri is divided into seven books. In six of these the author applies his new theory to the quadrature of the conic sections, the cubature of the solids formed by their revolution, and other questions of a like' nature on spirals: the seventh