QUA 
manac; also Robertson's Navigation, vol. i, 
pa. 340, tkc, edit. 1780. 
2. Sinicai quadrant is one of some use in 
navigation. It consists of several concentric 
quadrantal arches, divided into eight equal 
part^ by means of radii, with parallel right 
lines crossing each other at right angles. Now 
any one of the arches, as BC, tig. 8. in the Plate, 
may represent a quadrant of any great circle 
of the sphere, but is chieliy used for the ho- 
rizon or meridian. If then BC is taken for a 
quadrant of the horizon, either of the sides, 
as AB, may represent the meridian, and- the 
other side AC will represent a parallel, or 
line of east and west ; all the other lines pa- 
rallel to AB will be also meridians ; and all 
those parallel to AC, east and west lines, or 
parallels. Again, the eight species into which 
tiie arches are divided by the radii, represent 
the eight points of the compass in a quarter 
of the horizon; each containing 11" 15'. The 
arch BC is likewise divided into 90", and each 
degree subdivided into 12', diagonalwise. 
To the centre is fixed a thread, which being 
laid over any degree of the quadrant, serves 
to divide the horizon. 
It the sinicai quadrant is taken for a fourth 
part of the meridian, one side of it AB may 
be taken for the common radius of the meri- 
dian and equator ; and then the other, AC, 
will be half the axis of the world. The de- 
grees of the circumference BC will represent 
degrees of latitude; and the parallels to the 
side AB assumed from every point of latitude 
to the axis, AC, will be radii of the parallels 
of latitudes, as likewise the cosine of those 
latitudes. 
Hence, suppose it is required to find the 
degrees of longitude contained in 83 of the 
lesser leagues in the parallel of 48°; lay the 
thread over 48° of latitude on the circumfe- 
rence, and count thence the 83 leagues on 
AB, beginning at A; this will terminate in II, 
allowing every small interval four leagues. 
Then tracing out the parallel HE, from the 
point 11 to the thread; the part AE of the 
thread shews that 123 greater or equinoctial 
leagues make 6" 15'; and therefore that the 
83 lesser leagues AII, which make the dif- 
ference of longitude of the course, and are 
equal to the radius of the parallel HE, make 
6° 15' of the said parallel. 
Wlien the ship sails upon an oblique 
course, such course, beside the north and 
south greater leagues, gives lesser leagues 
easterly and westerly, to be reduced to de- 
grees of longitude of the equator. But these 
leagues being made neither on the parallel 
of departure, nor on that of arrival, but on all 
the intermediate ones, there must be found a 
mean proportional parallel between them. 
To find this, there is on the instrument a j 
scale of cross latitudes. Suppose then it were j 
required to find a mean parallel between the j 
parallels of 40° and 60°; take with the com- j 
passes the middle between the 40th and both 
degree on the scale; this middle point will 
terminate against the 31st degree, which is 
the mean parallel sought. 
The chief use of the sinicai quadrant is, to 
form upon it triangles similar to those made 
by a ship’s way with the meridians and pa- 
rallels; the sides of which triangles are mea- 
sured by the equal intervals between the con- 
centric quadrants, and the lines N and S, E 
and W; and every 5th line and arch are made 
deeper than the rest. Now suppose a ship 
Vol. 11. 
a u a a u a 53 y 
lias sailed 150 leagues north-east by north, 
or making an angle of 33" 45' with the north 
part of the meridian ; here are given the 
course and distance sailed, by which a tri- 
angle may be formed on the instrument si- 
milar to that made by the ship’s course ; and 
hence the unknown parts of the triangle may 
be found. Thus, supposing the centre A to 
represent the place of departure, count by 
means of the concentric circles along the 
point the ship sailed on, viz. A AD, 150 leagues; 
then in the triangle AED, similar to that of 
the ship’s course, find AE = difference of 
latitude, and DE — difference of longitude, 
which must be reduced according to the pa- 
rallel of latitude come to. 
Quadrant of altitude is an appendix 
to the artificial globe, consisting of a thin 
slip of brass, the length of a quarter part of 
one of the great circles of the globe, and gra- 
duated. At the end, where the division ter- 
minates, is a nut riveted on, and furnished 
with a screw, by means of which the instru- 
ment is fitted on the meridian, and moveable 
round upon the rivet to all points of the ho- 
rizon, as represented in the figure referred to. 
Its use is to serve as a scale in measuring of 
altitudes, amplitudes, azimuths, &c. 
QUADRANT AL, in Roman antiquity, a 
vessel every way square like a die, serving as 
a measure of liquids; its capacity was eighty 
librae or pounds of water, which made 48 sex- 
taries, two urine, or eight congii. 
QUADRAT, a mathematical instrument, 
called also a geometrical square, and line of 
shadows ; it is frequently an additional mem- 
ber on the face of the common quadrant, as 
also on those of Gunter’s and Sutton’s quad- 
rant; but we shall describe it by itself, as be- 
ing a distinct instrument. 
It is made of any solid matter, as brass, ' 
wood, &c. or of any four plane rules joined 
together at right angles, as represented in 
Plate Quadran's fig. 9, where A is the centre, 
from which hangs a thread with a small weight 
j at the end, serving as a plummet. Each of 
I the sides BE and DE is divided into a hun- 
dred equal parts, or if the sides are long enough 
to admit of it, into a thousand parts; C and 
F are two sights, fixed on the side AD. 
There is, moreover, an index AH, which, 
when there is occasion, is joined to the centre 
A, in such a manner that it can be moved 
freely round, and remain in any given situa- 
tion. On this instrument are two sights K, L, 
perpendicular to, the right line going from the 
centre of the instrument, i lie side DE is 
called the upright side, or the line of the di - 
rect or upright shadows; and the side BE is 
termed the reclining side, or the line of the 
versed or back shadows. 
To measure an accessible height AB, fig. 
10. by the quadrat, let the distance BD be 
measured, which suppose =96 feet, and let 
the height of the observer’s eye be six leet ; 
then holding the instrument with a steady 
hand, or rather resting it on a support, let it 
be directed towards the summit A, so that it 
may be seen clearly through both sights ; the 
perpendicular or plumb-line meanwhile hang- 
ing free, and touching the surface of the in- 
strument; let now the perpendicular be sup- 
posed to cut off on the upright side K.N 80 
equal parts; it is evident that LKN, ACIv, 
are similar triangles, and by prop. 4 lib. 6. 
of Euclid, NK \ KL I ! KC (■/'. e. BD) ", CA; 
that rs, 80 | 100' * 96 ! CA ; therefore by the 
, ,, . 96 X 100 ’ . 
rule of three, CA — — — 120 feet; and 
80 
CB, the height of the observer's eve, ~ 6 feet, 
being added, the whole height BA is 126 led. 
If the observer’s distance, as DE, is sir It 
that, when the instrument is directed as tor r 
merly towards the summit A, the perpendi- 
cular falls on the angle P, and the distance BE 
or CG is 120 feet, CA will also be 120 feet ; 
for PC: GH::GC: CA ; but PG = GB, 
therefore GC = CA ; that is, CA v ill he 1 20 
feet, and the w hole height BA =7 126 feet as 
before. 
But let the distance BF be 300 feet, and 
the perpendicular or plumb-line cut off 40 
equal parts from the reclining side. Now, in 
this case, the angles QAC, QZI, ave equal 
(29. 1. Enel.), as are also the angles QZI, 
/•IS ; therefore Z. ZIS = QAC; but ZSi = 
QCA, as being both right: hence, in the 
equiangular triangles ACQ, SZI, we have 
(by 4. 6. Eucl.) ZS : SI : : CQ : CA ; that is, 
100 : 40 : : 300 : CA, or CA = i ? . X3 °° -i2Q; 
100 
and by adding six feet, the observers height, 
the whole height BA will be 126 feet. 
To measure any distance at land or sea l)i/ 
the quadrat. In this operation the index AH, 
fig. 9, is to be applied to the instrument, as 
was shown in the description ; and by the help 
of a support, the instrument is to be placed 
horizontally at the point A, fig. 1 1 . then let it 
be turned till the remote point F, whose d : s- 
tance is to be measured, is seen through ti e 
fixed sights; and bringing the index to Le 
parallel with the other side of the instrument, 
observe through its sights any accessible mark 
B, at a distance; then carrying the i .stru- 
ment 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 cuts the right side of the square, as in 
k, the proportion will be (by 4. 6.) BE ; RK 
: : BA (the distance of the stations to be mea- 
sured with a chain) : AE, (he distance sought. 
But it (he index cuts the reclined side ot the 
square in the point L, then the proportion is, 
P.S: SB : : BA : .VC, the distance sought; 
which accordingly may be found bv the rule 
of three. 
Quadrat, in printing, a piece of metal 
cast like the letters, to till up the void spaces 
between words, &c. There are quadrats of 
different sizes, as m quadrats, n quadrats, &c. 
which are respectively ot the dimensions of 
these letters. 
QUADRATIC EQUATION, that where- 
in the unknown equality is of two dimensions, 
or raised to the second pow er. See Algebra. 
QUADRATURE, in geometry, denotes 
the squaring, or reducing a figure to a square. 
Tims the finding of a square which shall 
contain just as much surface or area as a cir- 
cle, an ellipsis, a triangle, &c. is the qua- 
drature of a circle, ellipsis, &c. 
1 he quadrature ot curvilinear spaces, as 
the circle, ellipsis, parabola, &c. is a matter 
of much deeper speculation, making a part of 
the higher geometry; wherein the doctrine of 
fluxions is of singular use. See Fluxion. 
Case I. Let ARC (Plate Quadrant, fig. 12.) be a 
curve of any kind, whose, ordinates R t, C13, are 
perpendicular to the axis AB. Imagine a right 
line 3Rg, perpendicular to AB, to move parallel 
to itself from A towards B ; and let the velocitv 
thereof, or the fluxion of the absciss A3, in any 
