QUA 
€ A ; but P G = G H, therefore G C = 
C A j that is, C A will be 120 feet, anti the 
whole height B A = 126 feet, as before. 
But let the distance B F {ibid.) be 300 
feet, and the perpendicular or plum line 
cut off 40 equal parts from the reclining 
side. Now, in this case, the angles Q A C, 
Q Z I, are equal (29. 1. End.) as are also 
the angles Q Z I, Z I S ; therefore the angle 
ZIS = QAC; but ZSI = QCA, as 
being both right ; hence, in the equiangular 
triangles A C Q, S Z I, we have (by 4. 6. 
Eucl.) ZS;SI::CQ:CA; that is, 100 : 
40 : : 300 : C A, or C A = ^;^^ .-=:120 ; 
and by adding 6 feet, the observer’s height, 
the whole height B A will be 126 feet. 
To measure any distance, at land or sea, 
by the quadrat. In this operation the 
index, A H, is to be applied to the instru- 
ment, as was shown in the description ; and, 
by the help ofa support, the instrument is to 
be placed hoi izontally at the point A (fig. 4) 
then let it be turned till the remote point 
P, whose distance is to be measured, be 
seen throbgh the fixed sights ; and bringing 
the index to be parallel with the other side 
of the instrument, observe through its sights 
any accessible mark, B, at a distance ; then 
carrying the instrument to the. point B, 
let the immoveable sights be directed to 
the first station A, and the sights of the 
index to the point F. If the index cut the 
right side of the square, as in K, the pro- 
portion will be (by 4. 6.) B R: RK : : BA 
(the distance of the stations to be measured 
with a chain) : A F, the distance sought. 
But if the index cut the reclined side of flie 
square, in the point L ; then the proportion 
is L S : S B : ; B A ; A G, the distance 
sought ; which,.accordingly, may be found 
by the rule of three. 
The quadrat may be used without calcu- 
lation, where the divisions of the square are 
produced both ways so as to form the area 
into little squares. Ex. Suppose the thread 
to fall on 40 in the side of right shadows, and 
the distance to be measured 20 poles ; seek 
among the litlle squares for that perpendi- 
cular to the side of which is 20 parts from 
the thread, this perpendicular will cut the 
side of the square next the centre, in the 
point 50, which is the height of the required 
poles. If the thread cut the side of the 
versed shadow's in the point 60, and the dis- 
tance be 35 poles, count 35 parts on the 
side of the quadrat from the centre, count 
also the divisions of the perpendieular from 
QUA 
the point 35 to the thread, which will be 
21, the height of the tower in poles. 
Quadrat, in printing, a piece of metal 
cast like the letters, to fill up the void spaces 
betw'een words, &c. There are quadrats 
of different sizes, as m quadrats, n quadrats, 
&c. which are, respectively, of the dimen- 
sions of these letters. 
QUADRATIC equation, in algebra, that 
wherein the unknown equality is of two 
dimensions, or raised to the second power. 
See Algebra. 
QUADRATURE, in geometry, denotes 
the squaring, or reducing a figure to a square. 
Thus, the finding of a square, which shall 
contain just as much surface, or area, as a 
circle, an ellipsis, a triangle, &c. is the qua- 
drature of a circle, ellipsis, &c. The qua- 
drature of rectilinear figures, or method of 
finding their areas, has been already deli- 
vered. See Mensuration. 
But the quadrature of curvilinear spaces, 
as the circle, ellipsis, parabola, &c. is a mat- 
ter of much deeper speculation, making a 
part of the higher geometry ; wherein the 
doctrine of fluxions is of singular use. We 
shall give an example or two. 
Let A R C (Plate XIII. Miscell. fig. 5) 
be a curve of any kind, whose ordinates 
R b, C B, are perpendicular to the axis 
A B. Imagine a right line, 5 R g, perpendi- 
cular to A B, to move parallel to itself from 
A towards B ; and let the velocity thereof, 
or the fluyion of the absciss, A h, in any pro- 
posed position of that line, be denoted by 
bd, then will, bn, the rectangle under bd 
and the ordinate, b R, express the corres- 
ponding fluxion of {he generating area, 
A 6 R ; which fluxion, if A 6 = .r, and 
6 R = y, will be y x. From whence, by 
substituting for y or x, according to the 
equation of the curve, and taking the fluent, 
the area itseltj A 6 R, will become known. 
But in order to render this still more 
plain, we shall give some examples, where- 
in X, y, z, and u are all along put to de- 
note the absciss, ordinate, curve-line, and 
the area respectively, unless where the 
contrary is expressly specified. Thus, if 
the area of a right angled triangle be re- 
quired; put the base A H (fig 6) = a, the 
perpendicular H M = 6, and let A B = a;, , 
be any portion of the base, considered as 
a flowing quantity ; and let B R = y be 
the ordinate, or perpendicular correspond- 
ing. Then because of the similar triangles, 
A H M and A B R, we shall have a:by.x:y 
