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II. The ftraight line PM moving along the other, is called an Ordinate, and is ufually denoted by y.
III. The fegment of the base AP between a given point in it A, and an ordinate PM, is called an Abfcifs with refpect to that ordinate, and is denoted by x. The ordinate and abfcifs together are called Co-ordinates.
IV. If the relation of the variable abfcifs and ordinate, AP and PM, be expreffed by an equation, which befides x and y contains only known quantities, the curve MO defcribed by the extremity of the ordinate, moving along the bafe, is called the Locus of that equation.
V. If the equation is finite, the curve is called Algebraical*. It is this class only which is here confidered.
* The terms Geometrical and Algebraical, as applied to curve lines, are used in different fenfes, by different writers; there are feveral other claffes of curves befides what is here called algebraical, which can be treated of mathematically, and even by means of algebra. See Scholium at the end.
VI. The dimenfions of fuch equations are eftimated from the highest fum of the exponents of x and y in any term.-According to this definition, the terms x, x3y, x2y2, xy, y, are all of the fame dimension.
VII. Curve lines are divided into orders from the dimenfions of their equations, when freed from fractions and furds.
In these general definitions, the straight line is fuppofed to be comprehended, as it is the locus of fimple equations. The loci of quadratic equations are fhewn to be the conic fections, which are hence called lines of the fecond order, &c.
It is fufficiently plain from the nature of an equation, containing two variable quantities, that it must determine the position of
every point of the curve, defined by it in the manner now defcribed: for if any particular known value of one of the variable quantities, as of x, be affumed, the equation will then have one unknown quantity only, and being refolved, will give a precife num
ber of correfponding values of y, which determine fo many points of the curve.
As every point of the locus of an equation has the fame general property, it must be one curve only, and from this equation all its properties may be derived. It is plain also, that any curve line defined from the motion of a point, according to a fixeď rule, muft either return into itself, or be extended ad infinitum with a continued cur
The equation, however, is fuppofed to be irreducible; because, if it is not, the locus will be a combination of inferior lines; but this combination will poffefs the general properties of the lines of the order of the given equation.
It is to be observed all along, that the pofitive values of the ordinate, as PM, being taken upwards, the negative Pm will be placed downwards, on the oppofite fide of the base: And if positive values of the abfcifs, as AP, be affumed to the right from its beginning, the negative values AP will
the left, and from these the points
of the curve M, m, on that fide are to be determined.
In the general definition of curves it is ufual to fuppofe the co-ordinates to be at right angles. If the locus of any equation be defcribed, and if the abfcifs be affumed on another base, and the ordinate be placed at a different angle, the new equation expreffing their relation, though of a different form, will be of the fame order as the ori ginal equation; and likewife will have, in common with it, thofe properties which diftinguish the equations of that particular
This method of defining curves by equations may not be the fitteft for a full investigation of the properties of a particular curve; but, as their number is without limit, fuch a minute inquiry concerning all, would be not only useless, but impoffible. It has this great advantage, however, that many of the general affections of all curves, and of the distinct orders, and also some of the
the most useful properties of particular curves, may be easily derived from it.
I. The Determination of the Figure of a Curve from its Equation.
The general figure of the curve may be found by substituting fucceffively particular values of x the abfcifs, and finding by the refolution of thefe equations the correfponding values of y the ordinate, and of confequence fo many points of the curve. If numeral values be substituted for x, and alfo certain numbers for the known letters, the refolution of the equation gives numeral expreffions of the ordinates; and from thefe, by means of fcales, a mechanical description of the curve will be obtained, which may often be useful, both in pointing out the general difpofition of the figure, and alfo in the practical applications of geometry.
Some more general fuppofitions may be of ufe in determining the figure; but these