be moft eafily derived, or may be of the moft fimple form.

The three Conic Sections are of the fecond order, as their equations are univerfally quadratic; the Ciffoid of the antients is of the third order, and the forty-fecond fpecies, according to Sir Ifaac Newton's enumeration; this is the curve defined by the equa tion in page 241, when bo. The curve delineated in page 239 is the 4ift fpecies. When bis negative in that equation, the locus is the 43d fpecies. The Conchoid of Nicodemes is of the fourth order; the Caffinian 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 fome property is demonftrated to belong to every point of the curve. The expreffion of these propofitions by equations is fometimes difficult; no general rules can be given; and it must be left to the fkill and experience of the learner.

SCHO

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SCHOLIUM.

This method of treating curve lines by equations, befides the ufes already hinted at, has many others, which do not belong to this place; fuch are, the finding the tangents of curves, their curvature, their areas and lengths, &c. The folution of these problems has been accomplished by means of the equations to curves, though by employing, concerning them, a method of reafoning different from what has been here explained.

С НА Р.

CHA P. III.

1. Conftruction of the Loci of Equations.

HE defcription of a curve, according to the definition of it, is affumed in geometry as á Poftulate *:

If

* A poftulate in geometry feems to be improperly called a mechanical principle. No geometrical line whatever, not even the straight line or circle, can be defcribed mechanically according to the mathematical definition; and therefore the folutions of problems by the conic fections, or by any of the higher orders, is to be confidered, in theory, as equally perfect with those by the circle and ftraight lines. It is a rule in ftrict geometry, not to employ a curve line in the folution of a problem, if it can be performed by means of a line of an inferior order; but, when a practical folution is required, then those lines, of whatever order, or of whatever clafs, and thofe methods of defcribing them are to be preferred, by which the con

1

be

If the properties of a particular curve are investigated, it will appear that it may described from a variety of data different from those affumed in the poftulate, by demonftrating the dependence of the former upon the latter.

As the definitions of a curve may be various, so also may be the poftulates, and a definition is frequently chofen from the mode of description connected with it. The particular object in view, it was formerly remarked, muft 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 conftructed in practice, with much more eafe and accuracy, by transferring a diftance in a pair of compaffes, than by the methods there defcribed; but that principle not being affumed by Euclid as a poftulate, could not be admitted in the construction of any problem in his elements. There are but few mechanical operations which admit of tolerable accuracy, and hence the great advantage of arithmetical calculations in the practical arts founded on geometry. By thefe the more complicated conftructions of geometry are reduced to thofe fimple operations which are found by experience to be capable of greatest exactness.