31,3
E Q U
EQU
EQUANIMITY, in ethics, denotes that even and calm frameof mind and temper, under good or had fortune ; whereby a mailappears to be neither puffed up, or overjoyed with prosperity ; nord.spirited, soured, or rendered uneasy, by adversity. Set* MoralPhilosophy and Passions.
EQUANT, in the old astronomy, a circle described on thecentre of the deferent, for accounting for the eccentricity of theplanets. See Excrktric and Excentpicjty.
EQUATION, in algebra, the mutual comparing of two equalthings of different denominations, or the expression denoting thisrquaiity; which is done by setting the one in opposition to theother, with the sign of equality (—) between them : thus 3s.—3(>d.or 3 feel = 1 yard. Hence, if we put a for a foot, and b for ayard, we shall have the equation 3 a — b, in algebraical characters,bice Algebra.
Equation, Annual, of the mean motion of the sun, andmoon’s apogee and nodes. The annual equation of (he sun’smean motion depends upon the eccentricity of the earth’s orbitround him, and is l6-;-J such parts, of which the mean distance be-tween the sun and the earth is 1000 ; whence some have called itthe equation of the centre, which, when greatest, is 1° .TV 3a".The greatest equation of the moon’s mean motion is 11’ 40"; ofits apogee, 20’; and of its node, 9’ 30”. These four annual equa-tions are always mutually proportionable to each other; so thatwhen any of them is at the greatest, the three others will also begreatest; and when one diminishes, the rest diminish in the sameratio. Wherefore the annual equation of the centre of the sunbeing given, the other three corresponding equations will begiven ; so that one table of the central equations may servefor all.
Equation of Curves, an equation expressing the nature ofa curve, the relation between an absciss and a corresponding ordi-nate, or the relation of their fluxions. See Curve . When thenatures or properties of curves are expressed by algebraic equa-tions any general absciss is commonly denoted by the letter x, aiulthe corresponding ordinate by the letter y; the other, or constantlines, being represented by other lelteis. Then the equations ex-pressing tiie natures of different curves will admit of various forms,according to the situation of the point at which the absciss is sup-posed to commence, or to the relations of the co-ordinates. Thus,if the abscisses should be estimated from the vertex of the curveor from the end of the diameter then the equations are as follow:namely for the
Circle , ,2dx — x 2 = y-, where 2 d is the diameter All, (fig. 33,Plate LA II.) ; x is the absciss AP or 15P, and y is the ordi-nate PQ.
Parabola, px — ry 2 , where p is the parameter.
tY A
Ellipse,-= 2 tx — x- j
c 5 v Where 21 is the thmsverse,
TT ( and 2c the conjugate axis.
c 1 '
But if we take the abscisses from the centres, then in the
Circle, d- — x- = y 2 , where 2d is the diameter.
I 2 // 2
hyperbola,
Wi
c*
cy-
„ Where Ifore.
'■t and 2c are as be-
i u 6 ma -’ likewise make use of p the parameter in the equations0 the ellipse and hyperbola: then in the
= 2 1 — xx
x* ± 2 tx
— t 2 -
- f*
. When the abscisses are takenfrom the vertex - .
. When the abscisses are takenfrom the centre.
Equation, Construction of, in algebra, the method ofdrawing a geometrical figure whose properties shall ex’press thegiven equation, in order to demonstrate the truth of it geome-trically.
Equation, a Simple, is constructed by resolving the givensimple equation into a proportion, or finding a third or fourth pro-portional, &c. Thus, 1 . If the equation be ax — bc\ tlnnbe
a ■■ b : : c : x — —, the fourth proportional to a, b, c.a
h 2
2. If ax = b -; then a : b : : b : r = —, a third proportional to
a
a and b. _ _
3. If ax — b~ — c ! ; then, since b 2 — c 2 = b -j- c x b — c, it
(/>-(-c)X (b—c)
will be a : b -j- c : : b — c : r =-, a fourth proportional
a
to a, b -f- c, and b — c.
ab-\-mn
If the equation be x — -, first find a mean proportional
r—s y
between a and b, which suppose to be p, afro another mean pro-portional between m and n, which suppose to be q, then the equa-
tion will stand thus x — -. Which may be constructed
r—s
in the following manner. Let the base AB, (Plate LV1I. fig. 26,)of the right-angled triangle APB be made equal to q, and theperpendicular AP equal to p ; then will PB 2 be equal p 2 -J- g' 1 ,which according to the equation is to be divided by r—s.Therefore it will be as r — s : PB (—^/p 2 -j- q 2 ): : PB to a thirdproportional, which will give x required.
Equations, Quadratic, require for their construction linesof the second order; hence it is necessary to shew the nature ofthese lines. See Curve . The. general equation expressing thenature of the lines of the second order, having ail its terms and co-efficients, will be in this form :
ir 4- axy + ex*
+ b'y%dx [ = 0.
+ c J .
Where a, b, c, d, e represent any given quantities with their pro-per signs prefixed to them. If a quadratic equation is given, asi/ 2 -f- py -j- q — 0, and by comparing it with the preceding, if youtake the quantities a, b, c, d, e, and x such, that ax -|- b = p, ande.v s -J- dx -j- c = q, then will the values of y in the first equationbe equal to the values of it in the second ; and if the locus he de-scribed according to the first equation, the two values of the ordi-'nate, when ax -J- b — p, and ex 1 dx -|- e = q, will be the tworoots of the equation y 2 -\-py -f- q — 0. See Locus. And asfour of the given quantities a, b, c, d, c, may be taken at pleasure,and the fifth with the absciss x determined so, that ax -|- b may bestill equal to p, and cx 9 -J- dx -j- e — q; hence there are innumer-able ways of constructing the same equation. But these loci areto be preferred which are described most easily ; and therefore thecircle of all conic sections is to be preferred for the resolution ofquadratic equations. Let AB, fig. 27, be perpendicular to AE, andupon AB describe the semicircle BMMA. If AP be supposedequal to x, AB = a, and PM —y ; then making MllM It perpen-dicular to the diameter AB, since Alt x it 13 = BM ! , andAlt = y, RB — a — y, ltM = x, it follows that « — y X y — x~,and y 2 — ay -j- x 2 — 0. And if an equation y- — py -}- q — 0 beproposed to he resolved, its roots will be the ordinates to the cir-cle, PM and PM, to its tangent AE, if a = p, and .v 2 = q : be-cause then the equation of the circle y* — ny-j-r ! =0 will bechanged into the proposed equation — py -j- q — 0. We havetherefore this construction for finding the roots of the quadraticequation y 1 —py -f- q — 0, take AB = p, and on AB describe asemicircle: then raise AE perpendicular to AB, and on it takeAP = v / q, that is, a mean proportional between 1 and q (by 13el. 6,) then draw PM parallel to AB, meeting the semicircle inMM and the lines PM, PM, shall be the roots of the proposed
P i
equations. It appears from the construction that if q — —, or
4 -
v / ?