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tequir'd but to place the longeft Branch in the Mouth of the Cannon or Mortar, and elevate or lower it, till the Thread cuts the Degree neceflary to hit a propofed Object.,
Sometimes, alfo, on one of the Surfaces of the long Branch is noted thedivifion of Diameters, and weights of Iron Bullets; as alfo the Bores of Pieces. SeeORDNANCE, Calliper, ££c.
Quadrant of Altitude, is an Appendage of the Artifi- cial Globe ; confifting of a Lamina, or Slip of Brafs, the length of a Quadrant of one of the great Circles of the Globe ; and divided into 90 Degrees.
At the end where the Divifions terminate, is a Nut ri- vetted on and furni/h'd with a Screw, by means whereof the Inftrument is fitted on to the Meridian j and moveable round upon the Rivet, to all Points of the Horizon.
Its Ufe is to ferve as a Scale in meafuring of Altitudes, Amplitudes, Azimuths, (gc. See the manner of its Ap- plication under the Ufe of the Globe.
QUADRANTAL Space, in Geometry, fee Qua- drant. ,
Q_UADRANTAL7Wa»,gfe, is a fphencal Triangle, one of whofe Sides is a Quadrant of a Circle ; and one of its Angles, a right Angle. See Spherical 'Triangle.
Quadrantal, in Antiquity, a Veffel in ufe among the Romans for themeafuring of Liquids. See Measure.
It was, at firft, call'd Amphora ; afterwards Quadrantal, from its Form, which was Square every way, like a Die. See Amphora.
Its Capacity was 80 Librie, or Pounds of Water, which made 48 Sextaries, or 6 Congii. See Con cms.
QUADRANTATATbw, in our antient Law-Books, is ufed for a quarter of an Acre ; now call'd a Rood. See Acre and Rood. See alfo Farthingdeal.
QUADRAT, call'd alfo Geo,. metrical Square and Line of Shadows, isan additional Member on the Face of the com- mon Gunter's and Sutton's Qjadrants ; of fome ufe in taking Altitudes, (3c. SeeQu adrant.
Tlie Quadrat, KLH, (Tab. Astronomy, Fig. 55.) has each of its fides divided into 100 equal Parts commen- cing from the Extremes, fo as the Number 100 falls on the' Angles; and reprefenting Tangents to the Arch of the Limb.
The Divifions are diflinguifh'd by little Lines from 5 to 5, and by Numbers from 10 to 10 ; and the Divifions being occafionally produced a-crofs, form a kind of Lattice, confiiling of 10000 little Squares.
The Proportion here, is, as Radius is to the Tangent of the Altitude at the place of Obfervation (i. e. to the parts of the Quadrat cut by the Thread) fo is the diftance be- tween the Station and foot of the Object, to its height above the Eye. See Altitude
Ufe cf the Quadrat, Geometrical Square, or Line of Shadows.
1. The Quadrat being vertically placed, and the Sights direfled to ihe Top of a Tower, or other Object, whofe Height is required 5 if the Thread cut the fide of theJ^M- drap ma&'rj right Shadows, the diftance from the Bafc of the Tower to the Point of Station is lefs than the Tower's
Height- If the Thread fall on the Diagonal of the Square,
the Diftance is juft equal to the Height If it fall on
thar fide mark'd verfed Shadows, the Diftance exceeds the Height.
Hence, meafuring the Diftance, the Height is found by the Rule of 'Three ; inafmuch as there are three Terms
giver. Indeed, their difpofition is not always the fame;
lor when the Thread cuts the fide of verfed Shadows, the firft Term in the Rule" of 'Three ought to be that part of the fide cut by the Thread, the fecond the fide of the Square,
and the third the Diftance meafur'd If the Thread cut
the other fide, the firft Term is the whole fide of the Square, the fecond the parts of the fide cut by the Thread, and rhe third the Diftance.
For an Inftance of each Suppofe, e.gr. in looking at
the Top of a Steeple the Thread cut the fide of right Sha- dows in the Point 40, and that the Diftance meafures 20 Poles ; the Cafe then will {land thus : As 40 is to 100, fo is 20 to a fourth Term, which I find to be 50 ; the Height of ihe Steeple in Poles.
Again, fuppofing the Thread to fall on the other fide, in the Point 50, and the Diftance to meafure 35 Poles ; the Terms are to be difpofed thus : As 100 is to 60 j fo is 35 toa fourth Term, viz,. 11, the Height required.
feek among the little Squares for that Perpendicular to the fide which is 20 parts from the Thread ; this Perpendicular will cut the fide ot the Square next the Centre, in the Point 5c, which is the Height requir'd in Poles.
2. If the Thread cut the fide of verfed Lines in the Point 60, and the Diftance be 3 5 Poles ; count 3 ; parrs on the fide of the Quadrat from the Centre ; counr alfo the Divi- fions of the Perpendicular from the Point 35 to the Thread, which will be2i, the Height of the Tower in Poles.
Note, In all Cafes, the Height of the Centre of the In- ftrument is to be added. See farther under Shadow.
Quadrat, in Attrology, call'd alfo Quartile, an Afpecr of the heavenly Bodies, wherein they are diftant from each other, a Quadrant, or ninety Degtees. See Aspect.
This is held a malignant Afpecr. See Quartile.
Quadrat, in Printing, is a fort of Space ; that is, a piece of Metal, caft like the Letters, to be ufed occafionally in Compofing, to make the Intervals between Words, at the ends of Lines, ($c. See Printing.
There are Quadrats of divers Sizes, as m Quadrats, 11 Quadrats, &c. which are refpect ively of the Dimenfions of fuch Letters.
QUADRAT A Legio, among the Romans, was a Legion confiding of 4000 Men. See Legion.
QUADRATIC Equation, is an Equation wherein the unknown Quantity is of two Dimenfions, i.e. is the Square of the Root or Number fought— As in x'=a -j-A*. See Equation.
Qtiadratic Equations are of two Kinds ; fimple, otfure; and adfecled.
Simple, otTure Quadratics, are thofe where the Square of the unknown Root is equal to the abfolute Num- ber given : Asin«a=3iJ; ee =146 ; ^'==133225.
The Refolution of thefe is eafy ; it being apparent that nothing more is requir'd than to extract the Square-Roor our of the Number or known Quantity. See Extraction.
Thus the Value of a in the firft Equation is equal to 6 ; in the fecond e=i2, and a little more, as being a furd Root j and h rhe third Example i y=3<>5. See Root.
jldfetted Quadratics, are rhofe which between the highell Power of the unknown Number, and the abfolute Number given, have fome intermediate Power of the un- known Number: As aa-\-i ba=tco. See Adpected.
All Equations of this Rank are in one or other of the fol- lowing Forms ; viz. aa-\-ad^=R. aa — ad=^R. ad — aa=R.
There are feveral Methods of folving adfecled Equations, or of extracting their Roots ; the moft convenient is that
of Harriot Here, x being affirmed as a part of the Root ;
a, the known Quantity of the fecond Term, will be double
the other part ; and rherefore half of a is rhe ether part
The Square, rhereof, will be compleated by adding one fourth of aa ; which done, the Root of the Square may be extract ed thus :
x* -\-ax — b
I. a a \a & add.
ee*
ax. 4 a 2 —,| a * b •
X'
,* a =z -/ ( | a " b ; )
x—la •/ C, « • l>')
Ufe of the Quadrat without Calculation.
The preceding Cafes may be pcrform'd without Calcula- tion where the Divifions of the Square are produced both ways , fo as to form the Area into little Squires.
Thus, fuppofe, 1. The Thread to fall on 40 in the fide »f right Shadows, and the Diftance be meafur'd 20 Poles 3
In lieu of rhe Characters + and — , we here ufe two Points ; to avoid the neceflity of diftinguifhing feveral Cafes. See Resolution.
ConftruBion c/Qu adrat ic Equations; fee Construc- tion.
QUADRATING of a 'Piece, among Gunners, is rhe feeing that a Piece of Ordnance be duly placed, and poiz'd in its Carriages ; that its Wheels be of an equal Height, t£c. See Carriage, Ordnance, Cannon, &c.
QU ADRATO-quadratum, or Siquadratum, rhe fourth Pov. er of Numbers : or the Product of the Cube multiply'd by the Root. See Power.
Quadrato Culms, Quadrato - Quadr ato - Cubus, and QyATiKhTO-Cubo-Cubus, are Names ufed by 2)io- phantus, Vieta,, Oughtred, and others, for the 5th, 7th, and 8th Powers of Numbers. See Powers.
QUADRATR1X, in Geometry, a mechanical Line, by means whereof we can find right Lines equal to the Circum- ference of a Circle, or orher Curve, and the feveral Parts thereof. See Circle, (£c.
Or, more accurately, the giiadratrix of a Curve, is a tranfcendental Curve, defcribed on the fame Axis, the Se- miordinates whereof being given, the Quadrature of the correfpondent Parts in the other Curve, are likewife given. See Curve.
Thus, e. gr. the Curve A N D (Tab. Analysis, Fig. 11.) may be call'd the Quadratrix of the 'Parabola A M C.fince 'tis demonftrated that A P M A is = P N', or A P M A= A P. PN, orAPMA = PNd, (Sc.
Ths.
