QUA 
_ if. Whence y x, the fluxion of the 
a hxx 
area A B R, is, in this case, equal to j 
and consequently the fluent thereof, or the 
area itself, which, therefore, when 
a; = a, and B R coincides with H M, will 
become^ — 
the whole triangle A H M ; as is also de- 
monstrable from the principles of common 
geometry. 
Again, let the curve A R M H, (fig. 7) 
whose area you would find, be the common 
parabola ; in which case, if A B = x, and 
B R = y, and the parameter = a ; we shall 
have = ax, and y — cflx^: and therefore, 
11 2 ,1 .3 
u ( —yx ) = a^x^x ; whence « = ^ X 
X X = i y X = I X X ^ 'R- 
Hence a parabola is two-thirds of a rectan- 
gle of the same base and altitude. 
The same conclusion might have been 
found more easily in terms of y ; for a: = 
^,and.*'=^-^; and consequently m( = 
yx) = ‘^-, Whence u = ^-£=^f X 
— =~ X X =1 X A B X B R, as before. 
a ^ 
To determine the area of the hyper- 
bolic curve A M R B, (fig. 8) whose equa- 
tion is = 0 “+" ; whence we have 
m-f-n + ^ 
— = a” X ; 
2 / = ' 
and therefore 
x’‘ 
m-^n m — n 
u(=zy x) — u’‘ X x" 
X, whose fluent 
qu j_ — n m + « n^m 
i, „ X^"-; which, 
when a; =0, will also be = 0, if n be greater 
than m-, therefore the fluent requires no 
correction in this casej the area, AM RB, 
included between the asymptote, AM, and 
the ordinate B R, being truly defined by 
m+n ■n —m 
as above. But if n be less 
n — m 
than m, then the fluent, when a: = 0, will 
be infinite, because the index — - — being 
negative, 0 becomes a divisor to na® + ”; 
whence the area, AMRB,,will also be 
infinite. 
QUA 
But here, the area, B R H, compre- 
hended between the ordinate, the curve,' 
and the part, B H, of the asymptote, is 
finite, and will be truly expressed by 
? n "f* « n — fn 
^SJL. the same quantity with its 
m—n ’ ^ ■’ 
signs changed ; for the fluxion of the part 
n - m 
A M R B, being a" X x" x, that of its 
supplement B R H must consequently be 
m'^n — m 
— a" X x" X, whereof the fluent is 
m-j^n ] n n^m 
a” X x" a” X x” 
m ■ 
= the area, B R H, 
which w'ants no correction ; because when 
X is infinite and the area B R H = 0, the 
said fluent will also entirely vanish ; since 
the value of x’‘ , which is a divisor to 
■ m-j-n 
ai , is then infinite. 
For further examples see Simpson’s Flux- 
ions, vol. i. sect, vii. 
Quadrature, in astronomy, that aspect 
of the moon when she is 90 degrees distant 
from the sun; or when she is in a middle 
point of her orbit, between the points of 
conjunction and opposition, namely, in the 
first and third quarters. 
Quadrature lines, are two lines placed 
on Gunter's sector : they are marked with 
Q. and 5, 6, 7, 8, 9, 10 ; of which Q. sig- 
nifies the side of the square, and the other 
figures the sides of polygons of 5, 6, 7, &c. 
sides. S, on the same instrument, stands 
for the semi-diameter of a circle, and 90 
for a line equal to ninety degrees in circum- 
ference. 
quadrilateral, in geometry, a 
figure whose perimeter consists of four right 
lines, making four angles ; whence it is also 
called a quadrangular figure. The quadri- 
lateral figures are either a parallelogram, 
trapezium, rectangle, square, rhombus, or 
rhomboides. 
QUADRUPEDS,in zoology ,a class ofland 
animals, with hairy bodies, and four limbs or 
legs proceeding from the trunk of their bo- 
dies; add to this, that the, females of this 
class are viviparous, or bring forth their 
young alive, and nourish them with milk from 
their teats. This class, though still numerous 
enough, will be considerably lessened in 
number, by throwing out of it the frog, 
lizard, and other four-footed amphibious 
animals. See Amphibia. On the other 
hand, it will be increased by the admission 
