TAN 



gent of any other line, whofc determination depends on the 

 tancent pivcn. 



This mt-thod is alfo one of the great refults of the new 



caliulus intfgralis. .... , r n tu a 



Its application we fliall give m what follows. 1 he Hux- 

 ional exprcffions of the tangent, fub-tangent, &c. being de- 

 livered under the lall article, if you make the given value 

 equal to the fluxional txpreflion, and either fum up the 

 fluxional equation, or, if that cannot be, conftrua it, the 

 curve required is had. For example : 



^ y' 



I. To fnd the curve-line, nuhofe fub-tangent = —^^ 



Since the fub-tangent of an algebraic line is = 



haveJ^ = 



2S- 



and 



ay X =^ ty yt 



and 



a X =■ zy y 



therefore (taking the fluents by the inverfe method of flux- 

 ions) ax = y"". 



The curve fought, therefore, is a parabola ; whole con- 

 ftruftion is (hewn under Parabola. 



2. To find the curve, nuhofe fub-tangent is a third propor- 



yx 

 tiond /o I a - x and y. Since ia - x : y :: y : —.we 



have \a — x:y {-.-.yy -.yx) ::} : i; confequently lai 

 — XX = yj, and, taking the fluents, \ax — i x' = i ^% 

 i.e. ax — x'' =■ y''- The curve fought is, therefore, a 

 circle. 



3. To find a line, wherein the fub-tangent is equal to the 



femiordinate. Since ^ = y ; y x = yy , zni. x = y ; there- 

 fore X ^= y. 



Hence it appears, that the line fought is a right hne, 

 which refpefts the cathetus of an equicrural triangle, as an 

 axis, or the hypothenufe of an equicrural reftangled triangle. 

 If X had been taken for the arc of a circle, the line fought 

 had been a cycloid. 



TANGER, in Geography, a river of Weftphalia, which 

 runs into the Elbe at I'angermunde. 

 TANGERE, ATo/ime. See Noli. 

 TANGERMUNDE, in GfOfra/ij-.atown of Weftphalia, 

 in the Old Mark of Brandenburg, fituated on the Elbe, where 

 feflels pay a toll : the chief trade of the town is brewing ; 

 24 nules N. of Magdeburg. N. lat. 52° 32'. E. long. 

 12° 2'. 



TANGHOO, or Tenhoa, a capital of a province of 

 Tonquin, fituated on a fmall river near the W. coaft of 

 the gulf of Cochinchina. Rice and cattle conftitute the 

 chief riches of the province. The town is called " Cua- 

 bang." N. lat. 19° 40'. 



TANGIA, a town of Arabia, in the province of Hedf- 

 jas ; 50 miles W.N.W. of El Catif. 

 TANGIBLE. See Tactile. 



TANGIER Islands, in Geography, feveral iflafids of 

 the Chefapeak, near the coaft of Maryland, oppofite to the 

 mouth of the Potomack. N. lat. 38° 12'. W. long. 

 76° 12'. 



TANGIERS, anciently called Tinjis and Tingia, and 

 now by the Arabs Tinjiah, a town of Africa, in Fez, fituated 

 at the weftern mouth of the ftraits of Gibraltar, about a day's 

 iourney from Tetuan. This town was firft poflefled by the 

 Romans, who took it under Strtorius ; next by the Goths ; 

 and it was furrendered by count Julian to the Saracens. It 

 was taken in 147 1 by Alonfo, king of Portugal ; and given 



TAN 



to Charles II., king of England, in 1662, as a. marrjagir 

 portion with the princefs Catherine of Portugal. The 

 Englifli abandoned it in 1684, after deftroying the mole and 

 fortifications. Although now almoft. in ruins, it ftilF re- 

 tains fomc batteries, in tolerable condition, facing the bay ; 

 at the bottom of which are a river, and the remains of the 

 bridge of Old Tangiers ; but on account of the accumu- 

 lated fand, the bridge, if it had continued, as well as the rivt-r, 

 would be ufck'fs. The bay of Tangiers, independently of 

 Ceuta, is fo fituated, being the narroweft part of the ftraits, 

 that it muft be favourable to Moorifli piracy ; but Tangiers 

 can never be a commercial town, as it has few produftions iij 

 its vicinity ; the Spaniards, however, formerly (hipped in this 

 place, eggs, vegetables, and fome fruits ; and the Englifli at 

 prefent obtain fupplies for their garrifon at Gibraltar. 

 The bay of Tangiers is not very fafe when the wind is in the 

 weft, having been encumbered by the ruins of the mole and 

 fortification, as the cables are liable to be rent, and the fliips 

 to be driven on (hore. The beft anchorage for frigates and 

 the larger velTels, is at the eaftern point, whence they may 

 eafily fail whatever way the wind fets : however, the bay is 

 only dangerous in winter ; 108 miles N.N.W. of Fez, and 

 38 W.S.W. of Gibraltar. N. lat. 35° 42'. W.long. 5°5o^ 

 Clienier's Morocco. 



TANGLAKE, in Ichthyology, the viviparous blenny of 

 Pennant ; the muftela vivipara of Willughby, Ray, &c. ; 

 and the blennius viviparus of the Linnean fyftem. 



TANGME W, in Geography, a town of the Birman empire, 

 on the right bank of the Ava ; 10 miles N.W. of Prome. 



TANGO, a town of Japan, in the illand of Niphon ; 

 65 miles S.W. of Meaco. 



TANGOLOTANGO, a feaport town of Mexico, in the 

 province of Guaxaca, near the gulf of Mexico ; 100 miles 

 S.S.E. of Guaxaca. N. lat. le" 8'. W. long. 97° 36'. 



TANGONE, a town of New York ; 9 miles "W. of 

 Kingfton. 



TANGOUZI, a town on the eaft coaft of Madagafcai. 

 S. lat. 19° 5'. E. long. 49-^12'. 



TANGOUZLIO, a town of Afiatic Turkey, in Na- 

 tolia ; 70 miles E.S.E. of Smyrna. 



TANGU, a city of Pegu, and capital of a province 

 which was formerly a kingdom ; fituated a conliderable dif- 

 tance to the north of Pegu. 



TANGUEY, or Tonguey, a town of Chili, on the coaft. 

 S. lat. 30° 30'. 



TANGUI A, a river of Chinefe Tartary, which rifes near 

 mount Ilha, and running nearly fouth, falls into the river 

 Ya-lou-kiang. 



TANGULAW, a fmall ifland in the Spanifli Main, near 

 the Mofquito fliore. N. lat. 13° 35'. W. long. 83° 55'. 



TANGUT. See Thibet. 



TANG-YANG, a lake of China, about thirty miles in 

 circumference ; 32 miles N. of Hoai-ngan. 



TANIALA, a town of Hindooftan, in Palnaud ; 25 

 miles E.N.E. of Timerycotta. 



TANIBOUCA, in Botany, a Caribsean name, to be tole- 

 rated only till the genus is properly underftood. — Aubl. 



Guian. 448. Ju(r. 76 Clafs and order, Decandria Mono- 



gynia. Nat. Ord. Elaagni? JulT. 



Gen. Ch. Cal. Perianth fuperior, of one leaf, bell-(haped, 

 internally downy ; its limb in five deep, roundi(h, acute feg- 

 ments. Cor. none. Stam. Filaments ten, thread-fhaped, 

 inferted into the tube of the calyx, as long as its limb ; 

 anthers oval, of two lobes. Pfl. Germen inferior, round- 

 ifli ; ftyle folitary, thread-fliaped, curved ; iiligma fimple. 



Peric 



Eff. 



