of a definition; the fimplicity of it, the ease with which the other properties of the figure may be derived from it, and fometimes even the eafe with which it can be executed mechanically, may be confidered as important circumftances.

In the ftraight line, the circle, the conic fections, and a few curves of the higher orders, the moft convenient definitions, and the poftulates connected with them, are generally known and received. An equation to a curve may also be affumed as a definition of it, and the defcription of it, according to that definition, may be confidercd as a poftulate; but, if the geometrical conftruction of problems is to be investigated by means of algebra, it is often useful to deduce from the equation to a curve those data, which, from the geometrical theory of the curve, are known to be neceffary to its description in the original poftulate, or in any problem founded upon it. This is called conftructing the locus of an equation, and from this method are generally derived the most elegant constructions which can be obtained

obtained by the use of algebra. In the following section, there is an example of a problem refolved by fuch conftructions.

Sometimes a mechanical defcription of a curve line defined by an equation is useful; and, as the exhibition of it by fuch a motion as is supposed in that definition, is rarely practicable, it generally becomes neceffary to contrive fome more fimple motion which may in effect correspond with the other, and may defcribe the curve with the degree of accuracy which is wanted. Frequently, indeed, the only method which can be conveniently practised, is the finding a number of points in the curve by the resolution of numeral equations, in the manner mentioned in Sect. 1. of this Chapter, and then joining these points by the hand; and, though this operation is manifeftly imperfect, it is on fome occafions useful.

II. Solution of Problems.

The folution of geometrical problems by algebra is much promoted, by defcribing

the

the loci of the equations arifing from these problems.

1

For this purpose, equations are to be derived according to the methods formerly defcribed, and then to be reduced to two,, containing each the fame two unknown quantities. The loci of these equations are to be described, the two unknown quantities being confidered as the co-ordinates, and placed at the fame angle in both. The co-ordinates at an interfection of the loci will be common to both, and give a folution of the problem.

The fimplicity of a conftruction obtained by this method will depend upon a proper notation, and the choice of the equations of which the loci are to be defcribed. These will frequently be different from what would be proper in a different method of folution.

PROB.

To find a point F in the bafe of the given triangle ABC, fo that the fum of the Squareš of FE, FD drawn from it perpendicular upon the two fides, may be equal to a gi ven space.

A

Draw BA, CG perpendicular on the two fides, and let FD= x, FE=y, BF=%, BC=6,BH=p,CG =r, and the given fpace FD2+FE2: m2.

From fimilar triangles, z:x::b:r, D and x=bx.

r

Alfo b-x:y::b:p, and=b2; there-B

bx

by

P

G

H

E

fore=b; ; that is, y=p—x, an equa

tion to a straight line.

But

But x2+y2=m2, of which the locus is a circle, having m for the radius. By conftructing these loci, their interfection will give a folution of the problem.

N

Let KL=CG (=r) be at right angles to LM=BH (=p), join KM, to which let LN be parallel; LN is the locus of the equation px

y=p ; for, let any line OPQ be drawn parallel to LM, K if KP=x, then PQ= px, and QO=LM=p, therefore PO=y=p—

px

About the center K, with a distance

P

equal to the line m, let

a circle be described; that circle will be the locus of the equation m2x2+y2; for it is plain that if OP be any perpendicular from the circumference upon KL, KP being x, OP will be y. Either of the points therefore in which these two loci interfect each other, as O, will give OP an ordinate in both equa

tions,

M