LOCUS GEOMETRICUS. 
M thereof, parallel to A Q, then we shall 
always have PM=:FM — PF, that is 
h X 
Hence it appears, that all the loci of the 
first degree are straight lines ; which may 
be easily found, because all their equations 
may be reduced to some one of the forego- 
ing formulas. 
All loci of the second degree are conic 
sections, viz. either the parabola, the circle, 
ellipsis, or hyperbola ; if an equation there- 
fore be given, whose locus is of the second 
degree, and it be required to draw the conic 
> section, which is the locus thereof; first 
draw a parabola, ellipsis, or hyperbola ; so 
as that the equations expressing the natures 
thereof may be as compound as possible. 
In order to get general equations, or formu- 
las, by examining the peculiar properties 
whereof we may know which of these for- 
mulas the given equation ought to have re- 
gard to ; that is, which of the conic sec- 
tions will be the locus of the proposed 
equation. This known, compare all the 
terms of the proposed equation with the 
terms of the general formula of that conic 
section, which you have found will be the 
locus of the given equation ; by which 
means you will find how to draw the sec- 
tion, which is the locus of the equation 
given. 
For example ; let A P = x, P M == y, be 
unknown, and variable straight lines (fig. 
9) ; and let m, n, p, r, s, be given right 
lines : in the line A P take A B = ?», and 
draw BE = n, AD = r and parallel to 
P M ; and through the point A draw A E 
= e, and through tlie point D the indefinite 
right line D G parallel to A E. In D G 
take D C = s, and with C G, as a diameter, 
having its ordinates parallel to P M, and the 
line C H = p, as the parameter, describe a 
parabola C M : then the portion thereof, in- 
cluded in the angle PAD, will be the locus 
yy — 
2 K X y . nnx X 
2ry + 
2 n r a? 
m 
ep X , 
h 2 ) s. = 0 . 
m ' 
-f r ■ 
For if from any point M of that portion 
there be drawn the right line M P, making 
any angle A P M with M P ; the triangles 
A B E, A P F, shall be similar ; therefore, 
AB:AE::AP: AF or DG; that is. 
m:e::x:—. And A B : B E :: A P. : P F ; 
m ’ 
that is, m: n :: X : 
m 
And consequently. 
GMorPM — PF — FG = y — __.r. 
m 
AndCGorDG— DC= — — s. But 
m 
from the nature of the parabola G = 
C G X C H ; which equation will become 
that of the general formula, by putting the 
literal values of those lines. 
Again, if through the fixed point A you 
draw the indefinite right line A Q (fig. 10), 
parallel to P M, and you take A B = m, 
and draw B E = « and parallel to A P, and 
through the determinate points A E, the 
line A E = e ; and if in A P you take A D 
= r : and draw the indefinite straight line 
DG parallel to AE, and take DC = s: 
this being done, if with the diameter C G, 
whose ordinates are parallel to A P, and 
parameter the line CH=p, you describe 
a parabola C M ; the portion of this para- 
bola contained in the angle BAP shall be 
the locus of this second equation, or for- 
mula : 
yx 
XX — 
m 
nnyy 
mm, 
2 r X -|- 
S 71 r 1 / 
m 
I epy , 
j' r BS. — 0. 
m‘ 
For, if the line M Q be drawn from any 
point M, therein, parallel to AP; then 
will, AB : AE::,AQorPM: AFor DG; 
that is, and A B ; B E :: A Q 
' m 
; Q F ; that is, m-.n:\y And there- 
711 
fore G M orQM — QF — FG = x — 
— r; and C G or D G — D C = 
m nt 
— s. 
And so by the common property of the 
parabola, you will have the foregoing se- 
cond equation, or formula. So likewise 
may be found general equations for the 
other conic sections. 
Now if it be required to draw the para- 
bola, which we find to be the locus of this 
proposed equation yy — Say — ix-f-cc 
= o; compare 'every terra of the first for- 
mula with the terms of the equation, 
because yy in both is without fractions; 
and then will — = o, because the rectan- 
m 
gle xy not being in the proposed equation, 
the said rectangle may be esteemed as multi- 
plied by 0 ; whence w = o, and m=zc; be- 
cause the line A E falling in A B, that is, in 
A P in the construction of the formula, the 
points B E, do coincide. Therefore de- 
stroying all the terms adfected with — 
m 
