EQUATION. 
struction of right lines or curves, or the re- 
ducing given equations into geometrical 
figures. And this is effected by lines or 
curves, according to the order or rank of 
the equation. The foots of any equation 
may be determined, that is, the equation 
may be constructed, by the intersections of 
a straight line with another line or curve of 
tiie same dimensions as the equation to be 
constructed : for the roots of the equation 
are the ordinates of the curve at the points 
of intersection with the right line ; and it is 
well known that a curve may be cut by a 
right line in as many points as its dimen- 
sions amount to. Thus, then, a simple 
equation will be constructed by the inter- 
section of one right line with another ; a 
quadratic equation, or an affected equation 
of the second rank, by the intersections of a 
right line with a circle, or any of the conic 
sections, which are all lines of the second 
order ; and which may he cut by the right 
line in two points, thereby giving the two 
roots of the quadratic equation. A cubic 
equation may be constructed by the inter- 
section of the right line with a line of the 
third order, and so on. But if, instead of 
the right line, some other line of a higher 
order be used, then the second line, whose 
intersections witli the former are to deter- 
mine the roots of the equation, may be 
taken as many dimensions lower as the for- 
mer is taken higher. And, in general, an 
equation of any height will be constructed 
by the intersection of two lines, whose di- 
mensions multiplied together produce the 
dimension of the given equation. Thus, the 
intersections of a circle with tire conic sec- 
tions, or of these with each other, will con- 
struct the biquadratic equations, or those of 
the fourth power, because 2 X 2 = 4; and 
the intersections of the circle, or conic sec- 
tions, with a line of the third order, will 
construct the equations of the fifth and sixth 
power, and so on. — For example : 
To construct a simple equation . This is 
done by resolving the given simple equa- 
tion into a proportion, or finding a third or 
fourth proportional, Sic. Thus, 1, If the 
equation be a x = b c ; then a : b :: c : x 
J) c 
“, the fourth proportional to a, b, c. 2 . If 
h 2 
ax =b 2 ; then a : h :: b : x , a third 
a 
proportional to a and b. S. If ax~b 2 — 
f ; then, since b 2 — c 2 = b X 6 — c, it 
Will bea:b + c::b — c:x = b ^ CX ' , ~ c 
I ■ ' « ’ 
a fourth proportional to a, fc c, and b — c. 
4. If a x = b 2 -j- c 2 ; then construct the 
right-angled triangle ABC (Plate V. Miscel. 
fig. 5.) whose base is b, and perpendicular 
is c, so shall the square of the hypothenuse 
be 6 2 -}- c 2 , which call h 2 ; then the equation 
b 2 
is (ii= h 2 , and x = — a third proportional 
to a and h. 
To construct a quadratic equation. 1. If it 
be a simple quadratic, it may be reduced to 
this form x 2 — ab; and hence a ; x :: x : b, 
or x = a b, a mean proportional between 
a and b. Therefore upon a straight line 
take A B = a, and B C = 6 ; then upon the 
diameter A C describe a semicircle, and 
raise the perpeudicular B D to meet it in 
D ; so shall B D be = x, the mean propor- 
tional sought between A B and B C, or be- 
tween a and b. 2. If the quadratic be af- 
fected, let it first be x 2 -|- 2 a x = b 2 ; then 
form the right-angled triangle whose base 
A B is a, and perpendicular B C is 6; and 
with the centre A and radius AC describe 
the semicircle DCE;so shall D B and B E 
be the two roots of the given quadratic 
equation x 2 2 a x = b 2 . 3. If the qua- 
dratic .be x 2 — 2 a x = b 2 , then the con- 
struction will be the very same as of the 
preceding one x 2 -}- 2 a x = 4. But if 
the form be 2 ax — x 2 = b 2 , form a right- 
angled triangle (fig. .) whose hypothenuse 
FG is a, and perpendicular GH is 6; then 
with the radius F G and centre F describe 
a semi-circle I G K : so shall I H and H R 
be the two roots of the given equation 
2 ax — x 2 = h 1 , or x 2 — 2 a x = — b ? \ 
To construct cubic and biquadratic equa- 
tions. These are constructed by the inter- 
sections of two conic sections ; for the equa- 
tion will rise to four dimensions, by which 
are determined the ordinates from the four 
points in which these conic sections may cut 
one another ; and the conic sections mav be 
assumed in such a manner as to make" this 
equation coincide with any proposed biqua- 
dratic ; so that the ordinates from these four 
intersections will be equal to the roots of 
the proposed biquadratic. When one of 
the intersections of the conic section falls 
upon the axis, then one of the ordinates 
vanishes and the equation by which these or- 
dinates are determined, will then be of three 
dimensions only, or a cubic to which any pro- 
posed cubic equation may be accommodat- 
ed ; so that the three remaining ordinates 
will be the roots of that proposed cubic. 
The conic sections for this purpose should 
be such as are most easily described ; the 
