E QU

316

EQIT

»/q = ^p, then AP = JAB, and the ordinate FN touches the Icurve in N, the two loots PM, PM, in that case becoming equal jto one another, and to PN. If AP lie taken greater than -JAB,that is, when the */q is greater than \p, or q greater than J p, theordinates do not meet the circle, and the roots of the equation be-come imaginary. The roots of the same equation may be other-wise thus determined. 1. [f it be a simple quadratic, it maybereduced to this form x a = ab ; and hence a: x : : x: b, or x =\/<ib, a mean proportional between a and b. Therefore upon astraight line (tig. 28,) take AB=r«, and BC = b; then upon thediameter AC describe a semicircle, and raise the perpendicularBD to meet it in T> ; so shall Bl) be = x, the mean proportionalsought between AB and BC, or between a and b.

2. If the quadratic be affected, let it first be x 2 -|- 2 ax ~ b 2 ;then form the right-angled triangle whose base AB (fig. 2c),) is a,and perpendicular 15C is b\ and with the centre A and radius AC,describe the semicircle DCE ; so shall DBand BE be the tworoots of the given quadratic equation x- -}-2ax = IP.

3. If the quadratic he x 2 — 2ax = b-, then the construction willbe the verv same as of the preceding one x 2 -f- 2 ax = IP.

4. But if the form be ‘2ax — x 2 = b -; form a right-angled tri-angle whose hypothenuse I'G is a, (fig. 30,) and perpendicularGII is b ; then with the radius FG and centre F describe a semi-circle IGK ; so shall HI and Ilk be the two roots of the givenequation 2 ax — x 2 = 6 2 , or x 2 — 2ax == — b' ! .

Equations, Cubic and Biquadratic, may be constructedby means of the conic sections. The roots of any equation maybe determined by the intersections of a straight line with a curveof the same dimensions as the equation, or by the intersections ofany two curves whose indices multiplied by each other, give aproduct equal to the index of the proposed equation. Thus theroots of a biquadratic equation inay be determined by the inter-sections of two conic sections ; for the equation by which the or-dinates from the four points in which these conic sections may cutone another can be determined, will arise to four dimensions: andthe conic sections may be assumed in such a manner, as to makethis equation coincide with any proposed biquadratic: so that theordinates from these four intersections, will be equal to the roots jof the proposed biquadratic. If one of the intersections of theconic section falls upon the axis, then one of the ordinates va-nishes, and the equation by which these ofdinates are determinedwill then be of three dimensions only, or a cubic, to which anyproposed cubic equation may be accommodated. So that thethree remaining ordinates will "be the three roots of that proposedcubic. Those conic sections ought to be preferred for this pur-pose that .are most easily described. They must not, however,be both circles ; for their intersections are only two, and can serveor.ly for the resolution of quadratic equations. Yet the circleought to be one, as being most easily described, and the parabolais commonly assumed for the other. Their intersections are de-termined in the following manner. Let APE he the commonApollonian parabola, (fig. 31.) "Fake on its axis the line A B =half of its parameter. Let C be any point in the plane of the na-rabola, and from it, as a centre, describe with any radius CP acircle meeting the parabola in P. Let PM, GO, be perpendicu-lars on the axis in M and D, and let CM parallel (o life axis meetPM in N. Then will always CP 2 = CN S + NP 2 (47. 1. Euc.)Put CP = a, the parameter of the parabola — h, AD = c,

DC = d, AM = x, PM =y. Then CN 2 = x + c NF- =

y -f d u ; and x -f c * + y -f d = (P. That is .■> 2 ± 2c.r -j- r 2+ ir±2‘ty + ^ = « J . ^ llt Irom nature the parabola,y*

iflzz.br. and x 2 = —; substituting therefore these values for x 2y 4 2c;/ 2

and x, it will be —i-hi/ 2 i 2dy -f- c- + d : — a- = 0.

IP _b _ _

Or multiplying by i 2 , if 4; 'Ibc -(- IPX ‘2cUPy -f- C'-^cP—cPx IP j=zf). Which mav r present any biquadratic equation that |wants the second term ; since such value; may be found for «, b,c, and d, by comparing this with any proposed biquadratic as to I

make them coincide. And then the ordinates from the points P,P, P, P, on the axis, will be equal to the roots of (hat proposedbiquadratic ; and this may he done though the parameter of theparabola, viz. b, be given : that is, if you have a parabola alreadymade or given, bv it alone you may resolve all biquadratic equa-tions, and you will only need to vary the centre of your circle andits radius. If the circle described from the centre (lig. 32,) passthrough the vertex A, then CP 2 — CA-’ = C I) 2 -{- AD 2 , that is<P zz a 2 + c 2 ; and the last term of the biquadratic (c 2 -j~ d 1 — « 2 )will vanish ; therefore dividing the vest by y, there arises the cubicy 3 -)' i 26c -j- b• x y i 2db- = 0. Let the cubic equation pro-posed to be resolved be ;/ 3 4: py 4; r = 0. Compare the terms

of these two equations and you will have i 2 be -f- b- =z 4; p, andbp r

4: 2 db- = 4; r, or 4- c = — + —, and d = 4:-• From which

2 2 b 2 b 2

you have this construction of the cubic ?/ 3 4; py ±r = 0, by

means of any given parabola APE. From the point B, take inthe axis (forward if the equation has — p, but backward if p be

P

positive) the line BD =: — ; then raise the perpendicular DC2br

-, and from C describe »a circle passing through the vertex A,

2 IP

meeting the parabola in P ; so shall the ordinate PM be one of theroots ot the cubic i py f r = 0. The ordinates that standon the same side of the axis with the centre C, are negative oraffirmative, according as the last term r is negative or atlirmative ;and those ordinates have always contrary signs that stand on dif-ferent sides of the axis. See Maclaurin’s Algebra, part iii . cap. 2;and Simpson’s Algebra , p. 2f>7. \

Equation of Timf., in astronomy and chronology, the re-duction of the apparent time or motion of the sun, to equable,mean, or true time. The difference between true and apparenttime arises from two causes, the eccentricity of the earth’s orbit,and the obliquity of the ecliptic. See Astronomy.

EQUATOR. See Astronomy and Geography.

EQUATORIAL Sector. See Astronomy.

EQUERRY, or EQUERY , popularly called Querry, anofficer of the king’s stables, under the master of the horse. Ofthese there are seven, who, when his majesty goes abroad, ride inthe leading coach, are in waiting one at a time monthly, and havea table with the gentlemen ushers during the time, ami a salary of3001. a-year each.

Equerry of the Crown Stable has that appellation, asbeing employed in managing and breaking the saddle horses, andpreparing them for the king’s riding.

EQUERY, or ECURY, [Ewurie, French , or cqnile, Lat. astable, from equus, a horse,} a grand stable for horses, furnishedwith stalls, manger, rack, &c. Some think the word stable pro-perly relates only to bullocks, cows, sheep, hogs, &c. and cquery,to horses, mules, &c. A simple cquery is that provided for onerow of horses; a double cquery that provided fur two, witli a pas-sage in the middle, or two passages; the horses being placed headto head, as in the little equery at Versailles . Under cquery aresometimes also comprehended "the lodgings and apartments ot theequerries, grooms, pages, tkc.

Equf.ry, [esnuyt i',1 is also an officer who has the care of thehorses of a king or prince.

EQUES, [from equu.i, a horse,] in antiquity, a night, or horse-man. Sec Equitfs.

Kquks, Auratus, a knight bachelor; called auratus, i. e. gilt,because anciently none but knights might gild or beautify theirarmour with gold. In law this term is not used, but instead of itmiles, and sometimes chevalier.

EQUF.STKIA, 'tt Roman antiquity, a place in the theatre where

the eqnites or knights sat.

EQUES ilUAX GAMES, among the ancient Romans, horse-race--, of which there were live kinds, the prodromtis or plainhorse-race, the chariot-race, the decursory race about funeralpiles, the ludi sevirales, and the ludi neptunales.

Equestrian Order, among the Romans, signified their equi-lesor knights; also their horsemen in the field ; the first of which

orders