TAN 
TAN 
tlie fluxion of Am, or the velocity where- 
with the line, mg, is carried ; and let n S 
express the corresponding fluxion of m p, in 
the position mCg, or the velocity of the 
point, p, in the line, mg: moreover, through 
the point, C, let the right line, SF, be 
drawn, meeting the axis of the curve, A Q, 
in F. 
Now, it is evident, that if the motion ofp, 
along the line, mg, was to become equable at 
C, the point, p, would be at S, when the line 
itself had got into the position, mSg ; be- 
cause, by the hypothesis. Cm and «S ex- 
presses the distances that might be described 
by the two uniform motions in the same time. 
And if wsg be assumed to represent any 
other position of that line, and s the con- 
temporary position of the point, p, still sup- 
posing an equable velocity of p ; then the 
distances, Cv, and vs, gone over in the 
same time by the two motions, will always 
be to each other as the velocities, or as Cm 
to M S. Therefore, since Cv:vs::Cn:nS, 
(which is a known property of similar tri- 
angles), the point, s, will always fall in the 
right line, FCS, (fig. 11): whence it ap- 
pears, that if the motion of the point, p, 
along the line, mg, was to become uniform 
at C, that point would then move in the 
right line, C S, instead of the curve line, 
CG. Now, seeing the motion ofp, in the 
description of curves, must either be an ac- 
celerated or retarded one ; let it be first 
considered as an accelerated one, in which 
case, the arch, C G, will fall wholly above 
the right line, C D, as in fig. 10 ; because 
the distance of the point, p, from the axis, 
A Q, at the end of any given time, is greater 
than it would be if the acceleration was to 
cease at C ; and if the acceleration bad 
ceased at C, the point, p, would have been 
always found in the said right line, FS. 
But if the motion of the point, p, be a re- 
tarded one, it will appear, by arguing in 
the same manner, that the arch, C G, will 
fall wholly below the right line, CD, as in 
fig. 11. 
This being the case, let the line, m g, and 
the point, if, along that line, be now sup- 
■ posed to move back again, towards A and 
m, in the same manner they proceeded 
from thence ; then, since the velocity of p 
did before increase, it must' now, on the 
contrary, decrease; and therefore, as p, at 
the end of a given time, after repassing the 
point, C, is not so near (o AQ, as it would 
have been had the velocity continued the 
same as at C, the arch, C h (as well as C G) 
must fall wholly above the right line, F C D; 
and by the same method of arguing, the 
arch, C/r, in the second case, will fall 
wholly below FCD. Therefore FCD, in 
both cases, is a tangent to the curve at the 
point, C ; whence the triangles, F m C, and 
CmS, being similar, it appears that the sub- 
tangent, »iF, is always a fourth propor- 
tional to mS, the fluxion of the ordinate, 
Cm, the fluxion of the absciss, and Cm, the 
ordinate ; that is, S m : mG jm C : mi F. 
Hence, if the absciss, Am = x, and the or- 
dinate mi p == y, we shall have MtF = ^; 
y 
by means of which general expression, and 
the equation expressing the relation be- 
Iween x and y, tire ratio of the fluxions, x 
and y will be found, and from thence the 
length of the sub-tangent, m F, as in the fol- 
lowing examples. 
1. To draw a right line, C T, a tangent 
to a given circle, (fig. n) B C A, in a given 
point, C. Let CS be perpendicular to the 
diameter, A B, and put AB =a, B S = x, 
and S C = y. Then, by the property of 
the circle , y^ (= C S^) = B S x A S (= x 
X a — x) = a X — x* ; whereof the fluxion 
being taken, in order to determine the ra- 
tio of X and j, we get 2 y^ z=a x — 2xx; 
consequently ~ = 
3 y 
a — 2x 
?/ 
— X 
; which 
multiplied by y, gives ^ f = , ^ the 
y la — X 
sub-tangent, ST. Whence, O being sup- 
posed the centre, we have O § (= i a — x) 
• C S (= 2t) " C S (= y) : S T ; which is also 
found to be the case from other princi- 
ples. 
2. To draw a tangent to any given point, 
C, (fig. 13) of the conical parabola, A C G. 
If the latus rectum of the curve be denoted 
by a, the ordinate, M C, by y, and its cor- 
responding absciss, A M, by x ; then the 
known equation, expressing the relation of 
X and y, being a x = y^, we have, in this 
case, the fluxion a x = 2 y y ; whence 
= and consequently, 
= 2 X r= M F. Therefor/ the sub-tangent 
is just the double of its corresponding ab- 
'sciss, A M. And so for finding the tangent* 
of other species'of curves. 
TANNING, the art of manufacturing 
leather from raw hides and skins. Befora 
we detail the process, it may be proper to 
observe, that raw hides and skins being 
composed of minute fibres intersecting each 
other in every direction, the general opera- 
tion of tanning consists chiefly in expanding 
