LOCUS GEOMETRICUS. 



which a local or indeterminate problem is 

 solved. See LOCAL PHOHLEM. 



A locus is a line, any point of which 

 may equally solve an indeterminate pro- 

 blem. Thus, if a I'ight line suffice for 

 the construction of the equation, it is call- 

 ed focus ad rectum; if a circle, locus cud 

 circulum ; if a parabola, locus ad parabo. 

 lam ; if an ellipsis, locus ad ellipsin , and 

 so of the rest of the conic sections. 



The loci of such equations as are right 

 lines, or circles, the ancients called plain 

 loci,- and of those that are parabolas, hy- 

 perbolas, &c. solid loci. But VVolfius, and 

 others, among the moderns, divide the 

 loci more commodiously into orders, ac- 

 cording 1 to the numbers of dimensions to 

 which the indeterminate quantities rise. 

 Thus, it will be a locus of the first order, 



if the equation is x = ; a locus of the 



second or quadratic order, if y i =ax, or 

 y-=a* a? a ; a locus of the third or cu- 

 bic order, if y3 ==i a 1 x ) or yi=.ax* x?, 

 &c. 



The better to conceive the nature of 

 the locus, suppose two unknown and va- 

 riable right lines A P, P M (Plate VIII. 

 Mi seel. tig. 4 and 5) making any given 

 angle A P M with each other; the one 

 whereof, as A P, we call x, having a fixed 

 origin in ihe point A, and extending it- 

 self indefinitely along a right line given 

 in position ; the other P M, which we call 

 y, continually changing its position, but 

 always parallel to itself. An equation on- 

 ly containing these two unknown quanti- 

 ties, x and y, mixed with known ones, 

 which expresses the relation of every va- 

 riable quantity A P, (x}, to its correspon- 

 dent variable quantity P M, (y) : the line 

 passing through the extremities of all the 

 values of y, i. e. through all the points M, 

 is called a geometrical locus, in general, 

 and the locus of that equation in particu- 

 lar. 



All equations, whose loci are of the 

 first order, may be reduced to some one 

 of the four following formulas : 1. y = 

 bx bx , b x 



7T- 2 ^=T+ C ' 3.2 = --c. 4. 



y = c . Where the unknown quan- 

 tity, z/, is supposed always to be freed 

 from fractions, and the fraction that mul- 

 tiplies the other unknown quantity, x, to 



be reduced to this expression -, and all 

 the known terms to c. 



The locus of the first formula being al- 

 ready determined : to find that of the se- 



VOL. IV. 



cond, y = -- \-c ; in the line A P, fig. 6, 



take A B = a, and draw B E = b, A D= 

 c, and parallel to P M. On the same side 

 A P, draw the line AE of an indefinite 

 length towards E, and the indefinite 

 straight line D M parallel to A E. Then 

 the line I) M is the locus of the aforesaid 

 equation, or formula ; for if the line M P 

 be drawn from any point M thereof paral- 

 lel to A Q, the triangles A B E, and A PF, 

 will be similar : and therefore A B (a) : 



B E (b) :: A P O) P F =1 ; and con- 



sequently PM (y) = P 

 F M (c). 



To find the locus of the third form, #== 

 b x 

 -- c, proceed thus : assume A B =. a 



(fig. 7) ; and draw the right lines B E = 

 b, A I) = c and parallel to P M, the one 

 on one side A P, and the other on the 

 other side : and through the points A E, 

 draw the line AE of an indefinite length 

 towards E, and through the point D, the 

 line D M parallel to A E : then the inde- 

 finite right line GM shall be the locus 

 sought ; for we shall have always P M = 



Lastly, to find the locus of the fourth 

 formula, y = c -- - ; inAP (fig. 8) : 



take A B = , and draw B E = b, A D= 

 c, and parallel to P M, the one on one side 

 A P, and the other on the other side ; and 

 through the points A and E, draw the 

 line A E indefinitely towards E, and 

 through the point D draw the line D M 

 parallel to A E. Then D G shall be the 

 locus sought; for if the line M P be 

 drawn from any point M thereof parallel 

 to A Q, then we shall always have P M 



FM PF, thatis, y = c 

 a 



Hence it appears, that all the loci of the 

 first degree are straight lines ; which may 

 be easily found, because all their equa- 

 tions may be reduced to some one of the 

 foregoing formulas. 



All loci of the second degree are conic 

 sections, liz. either the parabola, the 

 circle, ellipsis, or hyperbola : n an equa- 

 tion therefore be given, whose locus is of 

 the second degree, audit be required to 

 draw the conic section, which is the locus 

 thereof; first draw a parabola, ellipsis, or 

 hyperbola ; so as that the equations ex- 

 pressing the natures thereof may be as 



