TAN

TAN

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TANDAYE, one of the most easterly of the Philippine islands,separated from Manilla by a- narrow strait. Lou. 124. 56. E.■Lat. 12. 36. N.

TANDLA, a town of Hindooslan, in Malwa ; 72 miles W. ofOugein, and 130 N. E. of Surat . Lon. 74. 30. E. Lat. 23. 5. N.

TANGA, a silver-coin, used by the Usbeck Tartars, worth-about is. 3d.

TANGENT , in geometry, is defined, in general, to be a rightline ET, (Plate CLV1IL fig. 10,) which touches any arch of acurve HE in E, in such a manner that no right line can be drawnthrough E betwixt the right line ET and the arch EH, or withinthe angle IIET that is formed by them. The tangent of an archis a right line drawn perpendicularly from the end of a diameter,passing to one extremity of the arch, and terminated by a rightline drawn from the centre through the other end of the arch, andcalled the secant. And the co-tangent of an arch is the tangent ofthe complement of that arch. See Geometry , and Trjgono-METRy. The tangent of a curve is a right line which only touchesthe curve in one point, but does not cut it. In order to illustratethe method of drawing tangents to curves, let ACG, (fig. 11,) bea curve of any kind, and C the given point from whence the tan-gent is to be drawn. Then conceive a light line, mg, to be car-ried along uniformly, parallel to itself, from A towards Q; andlet, at the same time, a point/) so move in that line, as to describethe given curve ACG: also let mm, or Cn, express the fluxion ofAm, or the velocity wherewith the line mg is carried ; and let nSexpress the corresponding fluxion of mp, in the position mCg, orthe velocity of the point p, in the line mg: moreover, through thepoint C let the right line SF be drawn, meeting the axis of thecurve, AQ, in F. Now it is evident, if the motion of p, alongthe line mg, was to become equable at C, the point/) would be atS, when the line itself had got into the position mSg ; because, bythe hypothesis, Cn and nS expresses the distances that might bedescribed by the two uniform motions in the same time. And ifit sg be assumed to represent any other position of that line, and sthe contemporary position of the point p, still supposing an equa-ble velocity of p ; then the distances Co, and vs, gone over in thesame time by the two motions, will always be to each other as thevelocities, or as Cn to nS. Therefore, since Cr>: vs: : Cn : nS,(which is a known property of similar triangles) the point* willalways fall in the right line FCS, fig. 12: whence it appears,that if themotion of the point/) along the line mg was to become uniform atC, that point would then move in the right line CS, instead of thecurve-line CG. Now, seeing the motion of p, in the descriptionof curves, must either be an accelerated or retarded one; let it befirst considered as an accelerated one, in which case the arch CGwill fall wholly above the right line CD, as in fig. 11, because thedistance of the point p from the axis AQ, at the end of any giventime, is greater than it would be if the acceleration was to cease atC; and if the acceleration had ceased at C, the point p wouldhave been always found in the said right line FS. But if the mo-tion of the point p be a retarded one, it will appear, by arguing inthe same manner, that the arch CG will fall wholly below the rightline CD, as in fig. 12. This being the case, let the line mg, andthe point p, along that line, be now supposed to move back again,towards A and m, in the same manner they proceeded fromthence: then, since the velocity of p, (fig. 11,) did before increase,it must now, on the contrary, decrease; and therefore as/), at theend of a given time, after repassing the point C, is not so near toAQ, as it would have been had the velocity continued the same asat C, the arcli Ch (as well as CGy must fall wholly above the rightline FCD : and by the same method of arguing, the arch Ch, inthe second case, will fall wholly below FCD. Therefore FCD,in boll) cases, is a tangent to the curve at the point C : whencethe triangles F;aC and C'wS being similar, it appears that the sub-tangent mF is always a fourth proportional to 7iS the fluxion ofthe ordinate C«, the fluxion of the absciss, and Cm the ordinate;that is, Sn: nC :: mC : mF. Hence, if the absciss Am = x, and

the ordinate mp — y, we shall have m F =—; by means of which

general expression, and the equation expressing the relation be-tween x and y, the ratio of the fluxions x and_y will be found, andfrom thence ttie length of the subtangent mF, as in the followingexamples:

Eiamp. I. To draw a right line CT, (fig. 13,) a tangent to agiven circle BCA, in a given point C. Let CS be perpendicularto the diameter AB, and put AB = a, BS = x, and SC = y.Then by the property of the circle, y 2 , (= CS 2 ) = BS X AS(= x x a—x) = ax— x 2 ; whereof the fluxion being taken, in or-der to determine the ratio of x and y, we get 2 yy = ax — 2xx ;

consequently —; which, multiplied by y,

■ W.v v *

gives'-j = ———= the subtangent ST.

Whence, O being sup-

posed the centre, we have OS (=.\a — x) : CS (=y); : CS(= y ): ST ; which is also found to be the case from other princi-ples. See Circle .

Examp. II. To draw a tangent to any given point, (fig. 14.) ofthe conical parabola ACG. If the latus rectum of the curve bedenoted by a, the ordinate MC by y, and its corresponding abscissAM by or; then the known equation, expressing the relation of xand y, being ax = y 2 , we have, in this case, the fluxion ux = 2 yy ;

whence4-=—, and consequently — = -SL. —

Sax

a '

= 2x =

MF. Therefore the subtangent is just the double of its corre-sponding absciss AM. And so for finding the tangents of otherspecies of curves. See Fluxions.

TANGEUMUNDE, a town of Brandenburg, ql miles W. ofBerlin. Lon. 13. 30. E. Lat. 52. 46. N.

TANGIER, a port-town of Morocco , in the kingdom of Fez,at the W.entrance of theStraits of Gibraltar ; 108 miles N. N. W.of Fez, and 38 W. S. W. of Gibraltar . Lon. 5. 50. E. Lat. 38.49. N.

Tangier, Bay of, a bay of the coast of Tangier , peculiarlyfavourable to Moorish pirates, who from this,' the narrowest part-of the strait, can easily surprise merchant-ships while passing.

TANGROLIPIX, sultan of the Turks, in the eleventh cen-tury, a barbarous but not ungenerous conqueror, and the first ofthe 1 urks who made inroads upon the eastern empire. He andhis nephew', Cutlu-Moses, ravaged Iberia. Tangrolipix also con-quered Persia, in 1030, and lounded a new dynasty of Turkish sultans, who reigned there fur a century.

TANJORE , orTANJOUR, a province of Hindoostan, on thecoast of Coromandel, included in the Carnatic, but governed byits own rajah.

Tanjore , or Tanjovr, the capital of the above province, 205miles S. W. of Madras. Lon. 79. 12. E. Lat. 10. 46. N.

TANK, in the language of Hmdoslan, is a place inclosed forreceiving and retaining the rain. During the periodical rains thetanks are filled, and thus in the dry season furnish water for therice-field* and cattle.

TANKERDSONG, a town of Thibet , 548 miles E. of Lassa,Lon. 100. 7. E. Lat. 29. 56. N.

'TANK I A, or TIN KI A-LI NG, a town and fort of Thibet , 275miles S. W. of Lassa. Loo. 87. 22. E. Lat. 29. 5. N.

TANNA, an island' in the South Sea, near Anattom. Lon.1611° 44' 35" E. Lat. 19° 32" 251" S.

TANNIN, or TAN, a vegetable principle employed in thetanning of leather. I is to Mr. Proust that we are indebted forthe investigation of the nature and properties of tannin, and of themethods ot obtaining it in a separate state. Much curious andimportant information has likewise been obtained by the experi-ments of Mr. Davy on the const'lluent parts of astringent vegeta-bles, and on their operation in tanning. 'Tannin exists in a greatnumber of vegetable substances; but it may be procured mostreadily, and in the greatest pnriiy, from nptgalis and catechu,Nulgalls are excrescences formed on the leaves of the oak by thepuncture of an insect which deposits its eggs on them. The bestare known by the name of Aleppo galls, imported in large quanti-ties in this country for Ihe use ot the dyers, calico-printers, &c.'They are in a great measure soluble in water; what remains be-hind is tasteless, and possesses the properties of the fibre of wood.A very great proportion ot water is necessary to carry off everytiling soluble. Deyeux found that a French pound of nutgalls re-quired 96 French pints of water, applied in 20 different portion*

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