TAN
[ i75 ]
TAN
that is, A C mufl be taken towards E. 5. But when* ever the Affirmative and Negative Quantities are equal in the Numerator, let them be how they will in the Denominator, a will vanish into nothing : And, confequently, the Tangent is either A D itfelf, or EA, or parallel thereto; as will eafily be found by the Data. This he gives plain Examples o£ in reference to the Circle; thus : Let there bea Semicircle, whofe Diameter is E B ; in which there is given any Point ; as D, (Fig. 15.) from which the Perpendicular DA is let fall to the Diameter. Let DA=<y, BA==y, BE=/^ : Then the Equation wi\lbeby—yy=vv, and drawing the 'Tangent
DC, AC, or a = — Now, if b be greater than 2jj',
b — zy the Tangent muft be drawn towards B ; if left, towards E ; if it be equal to it, it will be parallel to EB, as was faid in the firft, fecond and fourth Rules.
Let there be another Semi-circle inverted ; as NDD, C^ig. l6'0 the Points of whofe Periphery, are refcrr'd to the Right Line BB, parallel and — to the Diameter. Let NB becalled^; and all Things elfe as befo.e; then the Equation will be by — y y — dd -\~ v v — 2 d v ; which being
managed according to his Rules, you have a— —
5 a ' ' b -iy
Now fincc v here is fuppofed to be always lefs than d ; if b be greater than 1 y t then the Tangent mult be drawn towards E; if equal, it will be parallel to BB; if lefs, changing all the Sines, the Tnvgtnt mull: be drawn towards B, as by Rules Fourth, Fifth and Third. But there could be no Tangent drawn, or at leaf}, E B would be it, if NB had Jbeen taken equal to the Diameter. Let there be another Semi- circle, whofe Diameter NB, (Fig. 1 7.) is perpendicular to E B, and to which its Points are fuppofed to be referr'd. Let N B be called B, and all Things elfe as above ; the Equation will
be y y =bv — vv, and #— ° 2 . If, now, b be
2.r
greater than 2 v, the Tangent muft be drawn towards B, if lefler, towards E, if equal, DA will be the Tangent, as by Rules 1, 4 and c, appears.
Inverfe Method ef Tangents, is a Method of finding the Equation, or the Conftruftion, of any Curve; from the Tangent or any other Line whofe Determination depends on the Tangent, given.
This Method is one of the great Refults of the New Calculus Integralis, See Calculus.
Its Application we fliall give in what follows.
The differential ExpreiTions of the Tangent^ Sub-tangent^ &c. being delivered under the lair. Article : If you make the given Value equal to the differential Expreffion, and either fum up the differential Equation or, if that can't be, coniiruct it, the Curve required, is had. For Example :
i° Tefindthe Curve Line, whofe Sub-tangent = iyy : a
Since the Sub-tangent of an algebraic Line is = yd x, wc have
ydx :
dy = iyi •■
a
ay dx
= iy 1 dy
a d
x = iydy
The Curve fought, therefore, is a Parabola ; whofe Con- struction is Shewn under the Article Parabola.
2° To find the Curve, whofe Sub-tangent is a Third Tro- fortional tor — x andy.
r — x -.y =.y.y dx
Since
We have
2TK — x x = y The Curve fought therefore, is a Circle.
3 To find the Line -wherein the Sub-tangent is equal to t Semiordinate.
»■•
— X -.y =
-.dy
\dx
rd
»: X d X
—3
dy
rx
- J' «■
= 2'
f
Since,
ydx :dy=y ydx —ydy
■=S
Hence it appears, that the Line fought, is a right Line, that relerrs to the Cathctus of an Equicrural Triangle, as to an Axis or the Hypothenufe of an Equicrural, reflangled Tri- angle. It .•.had been taken for the Arch of a Circle, the fought Line had been a Cycloid. Sec Cycloid.
TANGIBLE. See Tactile
TANlSTRi, or TAN1STR1, in our ancient Cultoros, an Heir preemptive, or fecond Poffeffor. See Heir S?c.
Hence the Law, or Tenure, Taaifiry, or TamftriH, an an- cient municipal Law, or Tenure of England, which allotted the Inheritance of Lands, CaiHes, Fjc. held by this Tenure, to the oldell and molt worthy and capable Perfon of the Deceafed's Name and Blood; without any Regard to Proximity.
This, in reality, was giving it to the Strongeft; which na- turally occafion'd bloody Wars in Families : for which Rea- fbn it was abolifh'd under K. James I. Sir John Davis de- scribes it thus ; gamt afitm 'Perfia moruft feifie des Afcims Cajiks, Manors, Torres eu Tenements del Nature ($ Tenure de Tanillry ; que donques mcfme les Cafiles, &c. dcent def- cender, & de temps dont Memory ne Court oat ttfe de defender Seniori & digniffimo Viro Sanguinis & Cognominis de tiel 'Perfon, &c. The Word is fuppos'd form'd from Thanis.
TANNING, the preparing of Skins or Hides in a Pit, with Tan and Water, after having firft taken off the Hair, by Steeping them in Lime Water. See Tan and Leather.
Methodof Tannino Cows, Calves, andHorfes Hides.
The Skin being Head off the Carcafs if 'tis intended to be kept, 'tis faired with Sea Salt and Alum, or with a kind of Saltpetre, call'd Natron ; if 'tis not for keeping, the Salting is fav'd, as being of no ufe, but to prevent the Hide from corrupting e'er it can be conveniently carry'd to the Tan-lmife.
Whether the Hide have been faired or not, the Tanner begins with taking off the Hotns, the Ears, and the Tail ; after which 'tis thrown into a running Water for about 30 Hours, to wafh off the Blood, and other Impurities adher- ing to the Snfide.
This done, 'tis laid over Night in a Lime Pit, already ufed ; whence it is taken, and left to drain three or four Days on the Edge of the Pit.
This firft and flighted Preparation over, it is teturn'd into a flrong Lime-Pit for two Days ; then taken out for four more ; and thus for fix Weeks alternately, taken out and put in, twice a Week.
At the fix Weeks End, it is put into a fre/h Pit, where it continues eight Days, and is taken out for fo manv ; and this alternately for a Year or eighteen Months, according to the Strength of the Leather, or the Weather : for in great Heats, they put in freSh Lime twice a Week ; and in Fro ft they Sometimes don'r touch them for three Months. Every frefh Lime-Pit they throw them into, is Stronger and Stronger.
At four, five, or fix Weeks end, the Tanner Scrapes "off the Hair, on a Wooden Leg, or Horfe, with a kind of Knife for that Purpofe. And after a Tear or eighteen Months, when the Hair is perfectly gone, he catties it to a River to wafh, pares off the Flefh on the Leg, with a kind of cut- ting Knife, and rubs it briskly wirh a kind of Wher-ftone, to take off any Remains of Flefh or Filth on the Side of the Hair.
The Skin is now put into Tan ; that is, cover'd over with Tan, as 'tis ftretch'd in the Pit, and Water let in upon it: If the Skin be Strong, five Coverings of Tan will be re- quir'd ; for weaker, three or four may fuffice. When the Skin has not been kept long enough in Lime, or the Tatl- Pit; upon cleaving it, in the Middle is Seen a whitifh Streak, call'd the Horn, or Crudity of the Skin ; and 'tis this is the Reafbn why the Soals of Shoes, Boots, fjc. ifretch fb eafily, and take Water.
When the Hides are fufRciently tann'd, they are taken out of the Pit, to be dry'd, by hanging in the Air : Then the Tan is clean'd off them, and they are pur in a Place neither too dry, nor too moifl ; they are well ftretch'd over one another, with Weights a-top, to keep them tight and Straight ; and in this Condition are fold, under the Denomination of Send Leather. This is the Method of tanning Bullocks or Oxes Hides.
Cows, Calves, and Hotfes Skins are tann'd much after the fame Manner as thofe of Oxen, except that the former are only kept four Months, in the Lime-Pit ; and that befote they be put in the Tan, there is a Preparation requir'd, thus : Cold Water is pour'd into a Wooden Fat or Tub, wherein the Skins ate put, which are kept flirring while fome other Water is wanning in a Kettle ; and as foon as that Water is a little more than lukewarm, 'tis pour'd gently into the Fat, and upon this is calf a Basket of Tan ; during which Time, the Skins are ftill kept turning, that the Water and Tan mayn't burn them.
After an Hour, they are taken out, and cad for a Day in cold Water, then return'd into the former Fat, and the
fame
