T A N 
TAN 
77 Q T A N 
TANGENT, In geometry, is defined, in ge- 
neral, to be a right line which touches any ar.ck 
©f a curve, in such a manner that no right line 
can be drawn betwixt the right line and the 
arch. See Plate Miscel. fig. 226 '. 
1 he tangent of an arch is a right line drawn 
perpendicularly from the end of a diameter, 
passing to one extremity of the arch, and termi- 
nated by a right line drawn from the centre 
through the other end of the arch, and called 
the secant. 
The tangent of a curve is a right line which 
only touches the curve in one point, but does 
not cut it. 
In order to illustrate the method of drawing 
tangents to curves, let ACG (fig. 227 )be a curve 
of any kind, and C the given point from 
whence the tangent is to he drawn. Then con- 
ceive a right line, mg, to be carried along uni- 
formly, parallel to itself, from A towards Q ; 
and Jet, at the same time, a point / .so move in 
that line, as to describe the given curve ACG: 
also let mm, or C n, express the fluxion of A m, or 
the velocity wherewith the line mg is carried ; 
and let «>S express the corresponding fluxion of 
ti;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, AQ, in F. 
Now it is evident, if the motion of p, along 
the line mg, was to become equable at C, the 
point / would be at S, when the line itself had 
got into the position mSg; because, by the hy- 
pothesis, C n and nS express the distances that 
might be described by the two uniform motions 
in the same time. And if ivsg is assumed to 
represent any other position of that line, and j 
the contemporary position of the point /, still 
supposing an equable velocity of / ; then the 
distances Cn, and vs, gone over in the same 
time by the two motions, will always he to each 
other as the velocities, or as C« to aS. There- 
iOre, since C -j * vs * * C« * »S (which is a known 
property of similar triangles), the point j will al- 
ways fall in the right line FCS fig. 228 ; whence it 
appears, that if the motion of the point / along 
the line mg was to become uniform at C, that 
point would then move in the right line CS, 
instead of the curve-line CG. Now, seeing the 
motion of /, in the description of curves, must 
either he an accelerated or retarded one ; let it 
be first considered as an accelerated one, in 
which ease the arch CG will fall wholly above 
the right line CD, because the distance of the 
point / from the axis AQ, at the end of any 
given time, is greater than it would be if the 
aecek-rarion was to cease at C ; and if the acce- 
leration had ceased at C, the point / would 
have been always found in the said right line 
IkS. But if the motion of the point / is a re- 
tarded one, it will appear, by arguing in the 
same manner, that the arch CG will fall wholly 
below the right line CD, as in fig. 223 . 
This being the case, let the line mg, and the 
point /, along that line, be now supposed to 
move back again, towarcis A and m, m the same 
manner they proceeded from thence: then, 
since the velocity of / did before increase, it 
must now, on the contrary, decrease ; and there- 
fore. as /, at the end of a given time, after re- 
passing the point C, is not so near to AQ, as it 
would have been had the velocitv continued the 
same as at C, the arch C h (as well as CG) must 
fall wholly above the right line FCD : and by 
the same method of arguing, the arch CL, in the 
second case, will fall wholly below FCD. There- 
fore FCD* in both cases, is a tangent to the 
curve at the point C: whence the triangles 
F/«C and CVS being similar, it appears that the 
»ub-tangent wF is always a fourth proportional 
to «S, the fluxion of the ordinate C«, the fluxion 
©f the absciss, and Cm the ordinate ; that is, 
S* «C * * mC * mF. Henco, if the absciss Am 
I = v, and the ordinate mp — y , we shall have 
~ - ; by means of which general expres- 
sion, and the equation expressing the relation 
betvveen A' and y, the ratio of the fluxions x and 
V will be found, and from thence the length 
of the sub-tangent mF, as in the following ex- 
amples. 
Example I. To draw a right line CT (fig. 229 ) 
a tangent, to a given circle BCA, in a given 
point C. Let CS be perpendicular to the dia- 
meter AB, and put AB = <z, BS = .v, and SC 
— J. Then, by the property of the circle, y 2 
| (= CS 2 ) = BS X AS (— .v X~,) — 
— X 2 ; whereof the fluxion being taken, in or- 
der to determine the ratio of x and y, we get 
= ax — 2xx; consequently -7- — — 
i — 1 — 7 ; which, multiplied by y, gives — - — 
* y y 
= 1 = the snb-tangent ST. Whence, O 
X 
being supposed the centre, we have OS (= 
x) * CS (r= y) [ * CS (n= y ) ^ ST ; which 
is also found to be the case from other prin- 
ciples. 
Example II. To draw a tangent to anv given 
point C (fig. 230 ) of the conical parabola ACG 
if the latus rectum of the curve is denoted by 
a, the ordinate MC by y, and its corresponding 
absciss AM by x\ then the known equation ex- 
pressing the relation of a- and y, being ax — y 2 , 
we have, in this ease, the fluxion <?.{- — 2j ; ; 
Whence ~ — dA, and consequently 
y a ' y a 
2 aSc 
— — j — 2.v — MF. Therefore the sub-tan- 
a 
gent is just the double of its corresponding ab- 
sciss AM. 
TANNING is the art of converting the 
raw skins of animals into leather. See Tan, 
and Cutis. 
In a preceding article (Tan), it was stated 
that gelatine with tannin, or the tanning prin- 
ciple of vegetables, formed a combination 
which is insoluble in water. Upon this de- 
pends the art of making leather ; the gela- 
tinous part of the skin combining with the 
tannin of the bark usually employed. 
The process which has long been used in 
this country is as follows : The leather tanned 
in England consists chiefly of three sort's, 
known by the name of butts or backs, hides, 
and skins. Butts are generally made from 
the stoutest and heaviest ox-hides, and are 
managed as follows : after the horns are 
taken off, the hides art laid smooth in heaps 
for one or two days in the summer, and for 
five or six in the winter ; tftey are then hung 
on poles in a close room, called a smoke- 
house, in which is kept a smouldering fire of 
\\ et tan ; this occasions a small degree of pu- 
trefaction, by which means the hair is easily 
got off, by spreading the hide on a sort of 
wooden horse or beam, and scraping it with 
a crooked knife. The hair being taken off, 
the hide is thrown into a pit or pool of water, 
to cleanse it from the dirt, &c. which being 
done, the hide is again spread on the wooden 
beam, and the grease, loose flesh, extraneous 
filth, &c. carefully scrubbed out or taken off; 
the hides are then put into a pit of strong 
liquor, called ooze, prepared in pits kept for 
the purpose, by infusing ground bark in wa- 
ter ; this is termed colouring ; after which 
A 
they are removed into another pit, called 3 
scowering, which consists of water strongly | 
impregnated with vitriolic acid, or with a ve- 1 
getable acid prepared from rye or barley. I 
I his operation (which is cahcd raising), by 1 
distending the pores of the hides, occasions 1 
them more readily to imbibe the ooze, the 1 
effect of which is to combine with the gela- ! 
tinous part of the skin, and form with it lea- 
ther. 1 lie hides are then taken out of the | 
scowering, and spread smooth in a pit com- 
monly fiiied with water, called a binder, with 
a quantity of ground bark strewed between 
each. After lying a month or six weeks, 
they are taken up ; and the decayed bark 
and liquor being drawn out of the pit, it V 
fiiled again with strong ooze, when they are 
put in as before, with bark between each 
hide. 1 hey now lie two or three months, 
at the expiration oi which the same operation 
is repeated ; th y then remain four or five 
months, when they again undergo the same 
process, and alter being three months in the 
last pit, are completely tanned; unless the 
hides, are so remarkably stout as to want an 
additional pit or layer. The whole process 
requires from eleven to eighteen months, and * 
sometimes two years, according to the sub- j 
stance of the hide, and discretion of the tan- 
ner. When taken out of the pit to be dried, 
they are hung on poles ; and after being com- ’ 
presse ! by a steel pin, and beaten out smooth 
by wooden hammers, called battes, the ope- 
ration is complete ; and when thoroughly dry, 
they are fit for sale. Butts are chiefly used 
for the soles of stout shoes. 
1 lie leather which goes under the denomi- 
nation ol hides, is generally made of cow- 
hides, or the lighter ox-hides, which are thus 
managed: After the horns are taken off, and 
the hides washed, they are put into a pit of 
water, saturated with lime ; where they re- 
main a few days, when they are taken out,, 
and the hair scraped off on a wooden beam, 
as before described ; they are then washed 
in a pit or pool of water, and the loose flesh, 
&c. being taken off, they are removed into a 
pit of weak ooze, where they are taken up 
and ^ put down (which is technically termed 
handling) two or three times a day, for the 
first week ; every second or third day they 
are shitted into a pit of fresh ooze, somewhat 
stronger than the former, till at the end of a 
month or six weeks they are put into a strong 
ooze, in which they are handled once or twice 
a week with fresh bark for two or three 
months. r l hey are then removed into ano- 
ther pit, called a layer, in which they are 
laid smooth, with bark ground very tine, 
strewed between each hide. After remain- 
ing here two or three months, they are gene- 
rally taken up, when the ooze is drawn out, 
and the hides put in again with fresh ooze 
and fresh bark, where, alter lying two or 
three months more, they are completely tan- 
ned ; except a very few stout hides, which 
may require an extra layer: they are then 
taken out, and hung 011 poles, and being 
hammered and smoothed by a steel pin, are, 
when dry, fit for sale. These hides are called, 
crop hides; they are from ten to eighteen 
months in tanning, and are used for the soles 
of shoes. 
Skins is the general term for the skins of 
calves, seals, hogs, dogs, &c. These, after- 
being washed in water, are put into lime-pits, 
as before mentioned, where they are taken. 
