CONIC SECTIONS. 
firft or major axis, and ab for their fecond or minor axis; 
and if dae.,fbg, be two other oppofite hyperbolas, hav¬ 
ing the fame axis, but in the contrary order, viz. ab 
their firft axis, and A B their fecond ; then thefe two 
latter curves, dae, fbg, are called the conjugate hyper¬ 
bolas to the two former, D A E, F B G ; and each pair of 
oppofite curves mutually conjugate to the other. And 
if tangents be drawn to the four vertices of the curves, 
or extremities of the axis, forming the inferibed rebtangie 
HIKE; the diagonals HC K and I C L, of this rebtan- 
crle, are called the afymptqt.es ot the curves. 
° Scholium .—The redlangle inferibed between the four 
conjugate hyperbolas, is fimilar to a rebtangle circum- 
feribed about an ellipfe, by drawing tangents, in like 
manner to the four extremities ot the two axes ; and the 
afymptotes or diagonals in the hyperbola, are analogous 
to thofe in the ellipfe, cutting this curve in fimilar points, 
and making the pair of equal conjugate diameters. More¬ 
over, the whole figure, formed by the four hyperbolas, 
is, as it were, an ellipfe turned iniide out, cut open at 
the extremities D, E, F, G, of the laid equal conjugate 
diameters, and thofe four points drawn out to an infinite 
diftance, the curvature being turned the contrary way, 
but the axis, and the reftangle palling through their ex¬ 
tremities, continuing fixed. From the foregoing defini¬ 
tions are derived the following general corollaries to the 
fedtions : 
Ellipfe. Hyperbola. ' Parabola. 
Cor. i. In the ellipfe, the femiconjugate axis, C D or 
1 CE, is a mean proportional between CO and CP, the 
parts of the diameter O P of a circular fedtionof the cone, 
drawn through the center C of the ellipfe, and parallel 
to the bafe of the cone. For D E is a double ordinate 
in this circle, being perpendicular to O P as well as to 
AB. In like manner, ii> the hyperbola, the length of 
the femiconjugate axis, CD or CE, is a mean propor¬ 
tional between C O and C P, drawn parallel to the bafe, 
and meeting the Tides of the cone in O and P. Or, if 
A O' be drawn parallel to the fide V B, and meet PC 
produced in O', making CO' = CO ; and on this diame¬ 
ter O'P a circle be drawn parallel to the bafe ; then the 
femiconjugate C D or C E will be an ordinate of this cir¬ 
cle, being perpendicular to O'P as well as to AB. Or, 
in-both figures, the whole conjugate axis DE is a mean 
proportional between QA and BR, parallel to the bafe 
.of the cone. In the parabola, both the tranfverfe and 
conjugate are infinite; for A B and B R are both infinite. 
Cor. 2. In all the ledlions, AG will be equal to the 
parameter of the axis, if QG be drawn making the angle 
AQG equal to the angle BAR. In like manner B^ 
will be equal to the fame parameter, if R" be drawn to 
make the angle B Rg — the angle A B Q. 
Cor. 3. Flence the upper hyperbolic feclion, or fedtion 
of the oppofite cone, is equal and fimilar to the lower one. 
For the two fedtions have the fame tranfverfe or major axis 
A B, and the fame conjugate or minor axis D E ; which is 
the mean proportional between AQand RB; and they 
have aifo equal parameters AG, Bg\ So that the two 
oppofite fedtions make, as it were, but the two oppofite 
ends of one entire ledtion or hyperbola, the two being every 
where mutually equal and fimilar. Like the two halves 
of an ellipfe, with their ends turned the contrary way. 
Cor. 4. And lienee, although both the tranfverfe and 
conjugate axis in the parabola be infinite, yet the former 
is infinitely greater than the hitter, or has an infinite ra¬ 
tio to it. F'or the tranfverfe has the fame ratio to the 
conjugate, as the conjugate has to the parameter, that 
is, as an infinite to a finite quantity, which conllitutes an 
infinite ratio.. 
F'rom the foregoing definitions it appears, that the co¬ 
nic fedtions are in themfelves a fyltem of regular curves, 
naturally allied to each other ; and that one is changed 
into another perpetually, when it is either increafed, or 
diminilhed, in infinitum. Thus, the curvature of a circle 
being ever fo little increafed or diminilhed, paffes into an 
ellipfe ; and again, the center of the ellipfe going ofi' in¬ 
finitely, and the curvature being thereby diminilhed, is 
changed into a parabola; and, laftly, the curvature of 
a parabola, being ever fo little changed, there arifeth the 
firfl of the hyperbolas ; the innumerable lpecies of which 
will all of them arife orderly by a gradual diminution of 
the curvature ; till this quite vanifhing, the laft hyper¬ 
bola ends in a right line. From whence it is manifeft, 
that every regular curvature, like that of a circle, from 
the circle itfelf to a right line, is a conical curvature, 
and is diftinguifhed with its peculiar name, according to 
the divers degrees of that curvature. That all diame¬ 
ters in a circle and ellipfe interfedt one another in the 
center of the figure within the fedtion ; that in the pan+- 
bofa they are all parallel among themfelves, and to the 
axis ; but, in the hyperbola, they interfedt one another, 
without tire figure, in the common center of the oppo¬ 
fite and conjugate ledlions. 
In the circle, the latus reElurh, or parameter, is double 
the diftance from the vertex to the focus, which is alfo 
the center. But, in ellipfes, the parameters are in all 
proportions to that diftance, between the double and 
quadruple, according to their different fpecies. While, 
in the parabola, the parameter is juft quadruple that dif¬ 
tance. And, laftly, in hyperbolas, the parameters are in 
all proportions beyond the quadruple, according to their 
various kinds. 
ON THE PARABOLA. 
Definitions.— 1. Let a point S> be affumed in the 
line LZ, fig. 1. and ELD be drawn perpendicular to itj 
then if a line MW move parallel to LZ, interfedling 
another line S H revolving about S, fo that M P may be 
always equal to S P, the curve palling through all the 
interfedtions P, will be a parabola. 
Cor ..—Hence, the curve cuts L Z in a point A which 
bifedts LS, fo that A is the vertex. Alfo, the two parts 
of the curve on each fide of A Z are fimilar and equal. 
2. The point S is the focus. 3. The line AZ is the axis ; 
and any line P W parallel to A Z, is the diameter. 4. The 
line E D is called the direElrix. 5. If PN be drawn per¬ 
pendicular to AZ, AN is an abfcijfa, and PN an ordinate 
to the axis. 6. If QV be drawn parallel to a tangent 
