be moft easily derived, or may be of the moft fimple form. The three Conic Soótions are of the second order, as their equations are u.viversally quadratic; the Cifoid of the antients is of the third order, and the forty-second species, according to Sir Isaac Newton's enumeration; this is the curve defined by the equation in page 241, when b=0. The curve delineated in page 239 is the 41ft species. When b is negative in that equation, the locus is the 43d species. The Conchoid of Nicodemes is of the fourth order ; the Cassinian curve is also of the fourth order, &c. It is to be observed, that not only the first definition of a curve may be expressed by an equation, but likewise any of those theorems called loci, in which some property is demonstrated to belong to every point of the curve. The expression of these propofitions by equations is sometimes difficult ; no general rules can be given; and it must be left to the skill and experience of the learner. 1 SCHO SCHOL I U M. This method of treating curve lines by equations, besides the uses already hinted at, has many others, which do not belong to this place ; such are, the finding the tangents of curves, their curvature, their areas and lengths, &c. The solution of these problems has been accomplished by means of the equations to curves, though by employing, concerning them, a method of reasonirig different from what has been here explained. CH A P. III. I. Construction of the Loci of Equations. "HE description of a curve, according l to the definition of it, is assumed in geometry as a Poftulate *. If * A postulate in geometry seems to be improperly called a mechanical principle. No geometrical line whatever, not even the straight line or circle, can be described mechanically according to the mathematical definition; and therefore the solutions of problems by the conic sections, or by any of the higher orders, is to be considered, in theory, as equally perfect with thofe by the circle and straight lines. It is a rule in strict geometry, not to employ a curve line in the fo. lution of a problem, if it can be performed by means of a line of an inferior order ; but, when a practical solution is required, then those lines, of whatever order, or of whatever class, and those methods of defcribing them are to be preferred, by which the con I i struction If the properties of a particular curve arc investigated, it will appear that it may be described from a variety of data different from those assumed in the postulate, by demonstrating the dependence of the former upon the latter. As the definitions of a curve may be various, so also may be the postulates, and a definition is frequently chosen from the mode of description connected with it. The particular object in view, it was formerly remarked, must determine the proper choice of struction required may be most eafily and accurately performed. Thus, even the 2d and 3d Prop. of I. B. of Euclid are constructed in practice, with much more €afe and accuracy, by transferring a distance in a pair of compasses, than by the methods there defcribed; but that principle not being assumed by Euclid as a postulate, could not be admitted in the construction of any problem in his elements. There are but few mechanical operations which adınit of tolerable accuracy, and hence the great advantage of arithmetical calculations in the practical arts founded on geometry. By these the more complicated constructions of geometry are reduced to thote simple operations which are found by experience to be capable of greatest exactness. |