« PreviousContinue »
II. The straight line PM moving along the other, is called an Ordinate, and is usually denoted by y.
III. The segment of the base AP between à given point in it A, and an ordinate PM, is called an Abscifs with respect to that ordinate, and is denoted by %. The ordinate and absciss together are called Co-ordinates,
IV. If the relation of the variable absciss and ordinate, AP and PM, be expressed by an equation, which besides x and y contains only known quantities, the curve MO described by the extremity of the ordinate, moving along the base, 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 considered.
* The terms Geometrical and Algebraical, as applied to curve lines, are ufed in different senses, by different writers; there are several other classes of curves besides what is here called algebraical, which can be treated of mathematically, and even by means of algebra. See Scholium at the end.
VI. The dimensions of such equations are estimated from the highest sum of the exponents of x and y in any term.-According to this definition, the terms x4, x3y, x?y?, xy, y, are all of the fame dimension.
VII. Curve lines are divided into orders from the dimensions of their equations; when freed from fractions and surds.
În these general definitions, the straight line is supposed to be comprehended, as it is the locus of simple equations. The loci of quadratic equations are shewn to be the conic sections, which are hence called lines of the second order, &c.
It is sufficiently 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 described : for if any particular known value of one of the variable quantities, as of x, be assumed, the equation will then have one unknown quantity only, and being resolved, will give a precise num-.
ber of corresponding values of y, which determine so many points of the curve.
As every point of the locus of an equation has the same 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 fixed rule, must either return into itself, or be extended ad infinitum with a continued curvature,
The equation, however, is supposed to be irreducible; because, if it is not, the locus will be a combination of inferior lines; but this combination will possess the general properties of the lines of the order of the given equation.
It is to be observed all along, that the positive values of the ordinate, as PM, being taken upwards, the negative Pm will be placed downwards, on the opposite side of the base: And if positive values of the absciss, as AP, be assuined to the right from its beginning, the negative values AP will
be upon 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 usual to suppose the co-ordinates to be at right angles. If the locus of any equation be described, and if the absciss be affumed on another base, and the ordinate be placed at a different angle, the new equation expressing their relation, though of a different form, will be of the same order as the original equation ; and likewise will have, in common with it, those properties which distinguish the equations of that particular
This method of defining curves by equations may not be the fittest for a full investigation of the properties of a particular curve; but, as their number is without limit, such a minute inquiry concerning all, would be not only useless, but impossible. 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 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 successively particular values of x the absciss, and finding by the resolution of these equations the corresponding values of y the ordinate, and of consequence so many points of the curve. If numeral values be substituted for x, and also certain numbers for the known letters, the resolution of the equation gives numeral expressions of the ordinates ; and from these, by means of scales, a mechanical description of the curve will be obtained, which
often be useful, both in pointing out the general disposition of the figure, and also in the practical applications of geometry.
Some more general suppositions may be of use in determining the figure; but these