PARABaLAi 
ether point, neither in the curve nor in the 
axis produced, through which the tangent 
is to pass : draw DEG perpendicular to 
the axis, and take D H a mean proportional 
between D E and DG, and draw H G 
parallel to the axis, so shall C be the point 
of contact through which, and the given 
point D, the tangent D C T is to be drawn. 
When the tangent is to make a given 
angle with the ordinate at the point of con- 
tact : take the absciss A I equal to half tire 
parameter, or to double the focal distanee, 
and draw the ordinate I E : also draw A H 
to make with A I the angle H A I equal 
to the given angle ; then draw H C parallel 
to the axis, and it will cut the curve in C 
the point of contact, where a line drawn to 
make the given angle with G B will be the 
tangent required. 
“ To find the Area of a Parabola.” Mul- 
tiply the base E G by the perpendicular 
height A I, and | of the product will be the 
area of the space A E G A ; because the 
parabolic space is | of its cii'cinnscribing 
parallelogram. 
“ To find the Length of the Curve AG,” 
commencing at the vertex. Let y = the 
ordinate B G, p =: 
the parameter, 5 =± 
and s = \/ 1 + 5 ^ ; then shall Xiqs-\- 
hyp. log. of 5 -}- s) be the length of the 
curve A G. 
Parabola, Cartesian is a curve of the 
second order, expressed by the equation 
xy =-ax^ ~\-b 3s^ cx -j- d, containing 
four infinite legs, viz. two hyperbolic ones, 
M M, B m, (Plate Parabola, fig. 8 ), (A E 
being the asymptote)tending contrary ways, 
and two parabolic legs BN, MN joining 
them, being the sixty-sixth species of lines 
of the third order, according to Sir Isaac 
Newton, called by him a trident: it is 
made use of by Des Gartes, in the third 
book of his Geometry, for finding the roots 
of equations of six dimensions by its inter- 
sections with a circle. Its most simple 
equation is xy xx x^ -j- a\ and tlie points 
through which it is to pass, may be easily 
found by means of a common parabola, 
whose absciss is « -j- A .r -f" 
hyperbola whose absciss is — ; for y wilt be 
equal to the sum or difference of the corres- 
pondent ordinates of this parabola and hy- 
perbola. 
Parabola, diverging, a name given by 
Sir Isaac Newton to five different lines 
of the third order, expressed by the equa- 
tion yyzxax’-^bx^-j-cx-l-d. 
PARABOLIC asymptote, in geometry, 
is used for a parabolic line approaching to 
a curve, so that they never meet; yet, by 
producing both indefinitely, their distance 
from each other becomes less than any 
given line. Maclaurin observes, that there 
may be as many different kinds, of these 
asymptotes as there are parabolas of difter- 
ent orders. 
When a curve has a common parabola 
for its asymptote, the ratio of the subtan- 
gent to the absciss approaches continually 
to the ratio of two to one, when the axis of 
the parabola coincides with the base ; but 
this ratio of the subtangent to the absciss 
approaches to that of one to two, when the 
axis is perpendicular to the base. And by 
observing the limit to which the ratio of 
the subtangeut and absciss approaches, 
parabolic asymptotes of various kinds may 
be discovered. 
Parabolic conoid, in geometry, a solid 
generated by the rotation of a parabola 
about its axis : its solidity is = i of that 
of its circumscribing cylinder. The circles, 
conceived to be t,he elements of this figure, 
are in arithmbtical proportion, decreasing 
towards the vertex. A parabolic conoid 
is to a cylinder of the same base and height, 
as t to 2 , and to a cone of the same, base 
and height, as li to 1 . See the article 
Gaogino. 
Parabolic citnevs, a solid figure formed 
by multiplying all the DB's (Plate Para- 
bola, fig. 9) into' the D S’s ; or, which 
amounts to the same, on the base APB 
erect a prism, whose altitude is A S ; this 
will be a parabolical cuneus,. which of neces- 
sity will be equal to the parabolical pyrami- 
doid, as the component rectangles in one 
are severally equal to all the component 
squares in the other. 
Parabolic pyramidoid, a solid figure 
generated by supposing all the squares of 
the ordinate applicates in the parabola so 
placed, as that the axis shall pass through 
all the centres at right angles ; in which 
case, the aggregate of the planes will form 
the parabolic pyramidoid. 
The solidity hereof is had by multiplying 
the base by half the altitude, the reason of 
which is obvious ; for the component planes 
being a series of arithmetical proportionals 
beginning from 0 , their sum will be equal 
to the extremes multiplied by half the num- 
ber of terms. 
Parabolic space, the area contained be- 
tween any entire ordinate as W (Plate 
