V A II 
V A R 
V A R 
545 
VARIABLE, in geometry and analytics, 
is a term applied by mathematicians to such 
quantities as are considered in a variable or 
changeable state, either increasing or de- 
creasing. Thus the abscisses and ordinates 
of an ellipsis, or other curve line, are variable 
quantities ; because they vary or change 
their magnitude together, the one at the same 
time with the other. But some quantities 
may be variable by themselves alone, or 
while those connected with them are con- 
stant; as the abscisses of a parallelogram, 
whose ordinates may be considered as all 
equal, and therefore constant. Also the di- 
ameter of a circle, and the parameter of a 
conic section, are constant, while their ab- 
scisses are variable. 
Variable quantities are usually denoted by 
the last letters of the alphabet, z, y, x, See. ; 
while the constant ones are denoted by the 
leading letters, a, b, c, &c. 
Some authors, instead of variable and con- 
stant quantities, use the terms fluent and 
stable quantities. 
The indefinitely small quantity by which a 
variable quantity is continually increased or 
decreased in very small portions of time, is 
called the differential, or increment or decre- 
ment. And the rate of its increase or de- 
crease at any point, is called i<s fluxion ; 
while the variable quantity itself is called the 
fluent. And the calculation of these, is the 
subject of the new methodus differentialis, 
or doctrine of fluxions. 
VARIANCE, in law, signifies any alter- 
ation of a thing formerly laid in a plea; or 
where the declaration in a cause differs from 
the writ, or from the deed upon which it is 
grounded. 2 Lit. Abr. 629. 
If there is a variance between the decla- 
ration and the writ, it is error, and the writ 
shall abate. And if there appears to "be a 
material variance between the matter pleaded, 
and the manner of pleading it, this is not a 
good plea ; for the manner and matter of 
pleading ought to agree in substance, or 
there will be no certainty in it. Cro. Jac. 
479. 
VARIATION, in geography and navi- 
gation, is the deviation of the magnetical 
needle, in the mariner’s compass, from the 
true north point, towards either the east or 
west ; or it is an arch of the horizon, inter- 
cepted between the meridian of the place of 
observation and the magnetic meridian. See 
Magnetism. 
Variation, in astronomy. The variation 
of the moan, called by Bulliald the reflection 
of her light, is the third inequality observed 
in the moon’s motion ; by which, when out 
of the quadratures, her true place differs from 
her place twice equated. See Astron om y. 
Newton makes the moon’s variation to 
arise partly from the form of her orbit, which 
is an ellipsis ; and partly from the inequality 
of the spaces which the moon describes in 
equal times, by a radius drawn to the earth. 
To find the greatest variation. Observe 
the moon’s longitude in the octants ; and to 
the time of observation compute the moon’s 
place twice equated ; then the difference be- 
tween the computed and observed place, is 
the greatest variation. 
Tycho makes the greatest variation 40' 
20" ; and Kepler makes it 5 1' 49". But New- 
9 
ton makes the greatest variation, at a mean 
distance between the sun and the earth, to be 
35' 10"; at the other distances, the greatest 
variation is in a ratio compounded of the 
duplicate ratio of the times of the moon’s sy- 
nodical revolution directly, and the triplicate 
ratio of the distance! of the sun from the earth 
inversely. And therefore in the sun’s apo 
e, the greatest variation is 33' 14", and in 
is perigee 37' 11"; provided that the eccen- 
tricity of the sun is to the transverse semidi- 
ameter of the orbis magnus, as 16 i~4 to 
1000. Or, taking the mean motions ot the 
moon from the sun, as they are stated in Dr. 
Halley’s tables, then the greatest variation 
at the mean distance of the earth from the 
sun will be 35' 7", in the apogee of the sun 
33' 27", and in his perigee 36' 51". 
Variation of curvature, in. geometry, 
is used for that inequality or change which 
takes place in the curvature of all curves ex- 
cept the circle, by which their curvature 
is more or less in different parts of them. 
And tins variation constitutes the quality of 
the curvature of any line. 
Sir Isaac Newton makes the index of the 
inequality, or variation of curvature, to be 
the ratio of the fluxion of the radius of cur- 
vature to the fluxion of the curve itself: and 
Maclanrin, to avoid the perplexity that diffe- 
rent notions, connected with the same terms, 
occasions to learners, has adopted the same 
definition ; but he suggests, that this ratio 
gives rather the variation of the ray of cur- 
vature, and that it might have been proper 
to have measured the variation of curvature 
rather by the ratio of the fluxion of the cur- 
vature itself to the fluxion of the curve ; so 
that, the curvature being inversely as the 
radius of curvature, and consequently it 
fluxion as the fluxion of the radius itself di 
rectly, and the square of the radius inversely, 
its variation would have been directly as the 
measure of it' according to Newton’s defi- 
nition, and inversely as the square of the 
radius of curvature 
According to this notion, it would have 
been measured by the angle of contact con- 
tained by the curve and circle of curvature, 
in the same manner as the curvature itself 
is measured by the angle of contact contained 
by the curve and tangent. The reason of 
this remark may appear from this example : 
The variation of curvature, according to 
Newton's explication, is uniform in the loga 
rithmic spiral, the fluxion of the radius of 
curvature in this figure being always in the 
same ratio to the fluxion of the curve; and 
yet, while the spiral is produced, though its 
curvature decreases, it never vanishes : which 
must appear a strange paradox to those who 
do not attend to the import of sir Isaac New- 
ton’s definition. 
The variation of curvature at any point of 
a conic section, is always as the tangent of 
the angle contained by the diameter that 
passes through the point of contact, and the 
perpendicular to the curve at the same point 
or to the angle formed bv the diameter of the 
section, and of the circle of curvature. Hence 
the variation- of curvature vanishes at the ex- 
tremities of either axis, and is greatest when 
the acute angle, contained by the diameter 
passing through the point of contact and the 
tangent, is least. 
When the conic section is a parabola, the 
variation is as the tangent of the angle, con- 
tained by the right line drawn from the point 
of contact to the focus, and the perpendicu- 
lar to the curve. See Curvature. 
From sir Isaac Newton’s definition may be 
derived practical rules for the variation of cur- 
vature, as follows: 
1 Find the radius of curvature, or rather its 
fluxion then divide this fluxion by the fluxion 
of the curve, and the quotient will give the va- 
riation of curvature; exterminating the fluxions 
when necessary, by the equation of the curve, 
or perhaps by expressing their ratio by help of 
the tangent, or ordinate, or subnormal, &c. 
2. Since- - ' - - - -, or V (putting x — 1) de~ 
— xy — y 
notes the radius of curvature of any curve z-, 
whose absciss is x, and ordinate y , if the fluxion. 
of this is divided by is, and is and z are exter- 
minated, the general value of the variation will 
— 3-,-y 3 -{-.y (1 . . . 
come out — — - T; - — then, substi- 
tuting the values of y, y, y (found from the 
equation of the curve) into this quantity, it will 
give the variation sought. 
Ex . Let the curve be the parabola, whose 
equation is ax — y 2 . Here then. 2 yy — aic — a, 
and y = ; hence y — — — — - — — , and 
J 2y J % 4y 3 
y Therefore 
3 . 2 / 8 / 
— 3vy 2 -f- y (1 -f- v 2 ) 
3y -{- y X 
1 +/ 
Gy 
— 3 a 3j 3 aa 16y 6 
=-%r+8,- x (I + i 4 x v>- . 
the variation sought. 
VARIOLfE, the small-pox, in medicine. 
See Medicine, and Vaccination. 
VARNISH, a thick, viscid, shining liquor, 
used by painters, gilders, and various- other 
artificers, to give a gloss and lustre to their 
works ; as also to defend them from the 
weather, dust, &c. See Resins. 
A coat of varnish ought to possess the fol- 
lowing properties: 1. ft must exclude the- 
action of the air ; because wood and metals, 
are varnished to defend them from decay 
and rust. 2. It must resist water ; for other- 
wise the effect of the varnish could not be 
permanent. 3. It ought not to alter suck- 
colours as are intended to be preserved by 
this means. It is necessary, therefore, that 
a varnish should be easily extended or spread 
over the surface, without leaving 1 pores or 
cavities, that it should not crack or scale, and. 
that it should resist water. 
Resins are the only bodies that possess 
these properties, consequently they must 
form the basis of every varnish. For this 
purpose, they must be dissolved, as minutely 
divided as possible, and combined in such a 
manner, that the imperfections of those that 
might be disposed to scale, may be corrected 
by others. 
Resins may be dissolved by three agents ; 
1. bv fixed, or fat oil ; 2. by volatile, or 
essential oil ; 3. by spirit of wine. Accord- 
ingly we have three kinds oi varnish ; fat or 
oilv Varnish, essential oil varnish, and. spirit 
varnish. 
These agents are of such a nature as eitherc 
to dry up and become hard, or to evap rater, 
and lly off, leaving the resin fixed behind. 
