Ethelfleda the mound on which the present ruins of the castle stand, but the ruins themselves are of later date. An old ditch, yet visible, called 'the king's dyke,' which surrounds the town on three sides, is supposed by Shaw to be of yet greater antiquity than the time of Edward. In the Saxon Chronicle' the town is called Tamaworthige, Tameweorthige, Tamanweorthe, or Tamweorthe: in other antient writings the orthography is still further varied. The place is not described in Domesday ;' but the 'burgenses' (burgesses) of Tamworth, are mentioned in that record, in the notice of other places. After the Conquest, the castle and adjacent territory were granted to Robert Marmion, hereditary champion to the dukes of Normandy; and afterwards, on the extinction of the male line of his family in the time of Edward I., passed to the family of Frevile. The castle now belongs to Marquis Townshend. Sir Walter Scott has enumerated Tamworth tower and town' among the possessions of his fictitious Marmion: but the family had become extinct long before, as observed by Sir Walter in the Appendix to poem. his The town stands on the north bank of the rivers Tame and Anker, just at their junction, and consists of several streets not very regularly laid out. The streets are paved, but had not been lighted when the Municipal Boundary Commissioners' Report was drawn up (Parl. Papers for 1837); the inhabitants were however about to assess themselves for the purpose. The church is a large and handsome edifice, with a fine tower, and a crypt under part of the church. Some portions are of decorated date, and some perpendicular, and both good: some of the windows have had very fine tracery. In the tower is a curious double staircase, one from the inside and one from without, each communicating with a different set of floors in the tower.' (Rickman's Gothic Architecture.) The remains of the castle are on a mound close to the Tame: they are of various periods, and some modern buildings have been added to adapt the whole to the purposes of a modern residence: the castle commands a fine prospect. There are some Dissenting places of worship; an almshouse, founded by Guy, the founder of Guy's Hospital in Southwark; a town-hall, with a small and inconvenient gaol beneath; and two bridges, one over the Tame, the other over the Anker. The population of the municipal borough in 1831 was 3537, that of the whole parish (containing several hamlets and townships) 7182. Some manufactures are carried on; but the whole number of men employed in them in the parish was, in 1831, only 38. Some coals and brick-earth are dug in the neighbourhood, and bricks and tiles are made. The market is on Saturday: there are three chartered fairs for cattle and merchandise, and several new fairs for cattle only; some of them held at Fazeley in the parish. The Coventry Canal passes near the town. Tamworth was a borough by prescription; but the town having declined and ceased to be regarded as a corporation, was incorporated anew by letters patent of Queen Elizabeth: the governing charter is one of Charles II. By the Municipal Reform Act the borough has four aldermen and twelve councillors, but is not to have a commission of the peace except on petition and grant. The criminal jurisdiction of the corporation had fallen into disuse before the passing of that act, as well as the court of record: quarter-sessions were held, but for civil purposes only. Tamworth first sent members to parliament in the reign of Elizabeth: it still returns two members. The number of voters on the register in 1835-6 was 531: in 1839-40, 501. The living of Tamworth is a perpetual curacy, of the clear yearly value of 1701., with a glebe-house. There are in the parish the perpetual curacies of Fazeley, Wigginton, and Wilnecote, of the clear yearly value of 2351. (with a glebe-house), 921. and 907. respectively: the curate of Tamworth presents to Wigginton and Wilnecote. There are also in the parish two chapelries, Amington and Hopwas. There were in the borough, in 1833, three endowed and three unendowed day-schools, with 183 children, namely 142 boys, 21 girls, and 20 children of sex not stated; and three Sunday-schools, with 203 children, viz. 97 boys and 106 girls. In the rest of the parish were one infant-school, partly supported by subscription, with 88 children, namely 41 boys and 47 girls; ten day-schools of all kinds, with 96 boys, 80 girls, and 80 children of sex not stated, making 256 children in all; and three Sunday-schools, with 288 children, namely 150 boys and 138 girls. (Shaw's Stuffordshire; Parliamentary Papers.) TANA-ELF. [TRONDHEIM.] TANACETUM, a genus of plants belonging to the natural order Compositæ, and the suborder Corymbiferæ or Asteraceæ. The involucre is imbricated and hemispherical. The receptacle is naked; the flowers of the ray are 3-toothed, those of the disk 5-toothed, tubular, and hermaphrodite. The fruit, an achenium, is crowned with a membranous margin, or pappus. The flowers are yellow. The most common species is the Tanacetum vulgare, common Tansy. It has bipinnatifid leaves, with serrated sections or lacinia. This plant is abundant in Great Britain and throughout Europe, on the borders of fields and road-sides. It possesses in a high degree the bitterness of the whole order Compositæ, which, in the section Corymbiferæ, is combined with a resinous principle. It is recommended and has been extensively used in medicine as an emmenagogue and anthelmintic. Although the flavour and smell of this plant are both at first disagreeable, a taste for it may be acquired, and it has been used in cookery for the purpose of flavouring puddings and sauces. The young shoots yield a green colouring-matter, and are used by the Finlanders for the purpose of dyeing their cloths of that colour. It is said that if meat be rubbed with the fresh leaves, it will not be attacked by the flesh-fly. TA'NAGERS. The genus Tanagra of Linnæus stands, in the 12th edition of the Systema Naturæ, between Emberiza and Fringilla, in the order Passeres. Cuvier characterises the genus as having a conical bill, triangular at its base, slightly arched at its arête, and notched towards the end: wings and flight short. He observes that they resemble our sparrows in their habits, and seek for seeds as well as berries and insects. The greater part, he remarks, force themselves upon the attention of the spectator of an ornithological collection by their vivid colours. He places the genus between the Drongos (Edolius, Cuv.) and the Thrushes (Turdus, Linn.), thus subdividing it: 1. The Euphonous or Bullfinch Tanagers (Euphones, ou Tangaras Bouvreuils). These have a short bill, presenting, when it is seen vertically, an enlargement on each side of its base: tail short in proportion. Examples, Tanagra violacea, Cayennensis, &c. 2. The Grosbeak Tanagers. Bill conic, stout, convex, as wide as it is high; the back of the upper mandible rounded. Examples, Tanagræ magna, atra, &c. 3. Tanagers, properly so called. Bill conic, shorter than the head, as wide as it is high, the upper mandible arched and rather pointed. Examples, Tanagra Talao, tricolor, &c. 4. Oriole Tanagers (Tangaras Loriots). Bill conic, arched, pointed, notched at the end. Examples, Tanagræ gularis, pileata, &c. 5. Cardinal Tanagers. Bill conic, a little convex, with an obtuse projecting tooth on the side. Examples, Tanagra cristata, brunnea, &c. 6. Ramphocele Tanagers. Bill conic, with the branches of the lower mandible convex, backwards. Examples, Tanagra Jacapa, Brasilia, &c. The views of Mr. Vigors on the subject of this group will be found in the article FRINGILLIDE. Mr. Swainson remarks that the Tanagrinæ, or Tanagers, form that group which is probably the most numerous, as it certainly is the most diversified of all those in the comprehensive family of the Fringillida. As the dentirostral division of that family, it is, he observes, typically distinguished from all the others by the bill having a distinct and well-defined notch at the end of the upper mandible, the ridge or culmen of which is much more curved than the gonys; or, in other words, the culmen is more curved downwards than the gonys is upwards: this inequality, he further states, as in the genus Ploceus, very much takes off from that regular conic form of bill so highly characteristic E 2 of the greater number of the finches; so that the combina- | second, Pipillo would stand intermediate between Aglara tion of these two characters is, he thinks, perhaps the best and Tanagra, and thus constitute the rasorial genus of the distinction of the whole group. Another peculiarity, he whole subfamily; and this latter arrangement appears to adds, of these birds consists in their geographic range; for him to be the natural one. He considers that the two the whole, as far as has hitherto been ascertained, are na- typical groups or genera are Tanagra and Phænisoma; tives of the warmer parts of America, being most abundant while those which he thinks aberrant are Nemosia, Aglaia, in those regions nearest to the equinoctial line. They and Pipillo. It was only between the two last of these are,' says Mr. Swainson in continuation, in general small that he had not as yet discovered any affinity sufficiently birds, the largest being intermediate between a sparrow strong to justify the belief that these five genera form a and a thrush, while the majority do not exceed the size of circle more or less complete; the difficulty being how to a linnet; some few are even smaller. It is quite evident, connect Aglaia with Pipillo. He then takes a review of from the great strength of bill possessed by some, and the the genera, for which we must refer our readers to the work notch which is conspicuous in all, that these birds feed itself; and, in the Synopsis at the end of the volume, upon seeds and creeping insects picked from the branches makes the Tanagrinæ, which he places between the Coccoof trees, for very few of them are ever seen upon the thraustine and the Fringillinæ, consist of the following ground. Their colours in general are bright; and, in a genera and subgenera, all of which he characterizes :— large number, particularly rich and beautiful. The little Tanagrinæ. birds forming the genus Aglaia, in fact, are ornamented with the most vivid hues or glossed with rich reflections of gold, rendering them inferior only to the Humming Birds. Some possess considerable vocal powers; and the notes of the subgenus Euphonia, as its name implies, are said to be particularly musical. The impossibility however of pro-nagra, and Ramphopis). Phænisoma (with the subgenera viding the Tanagers with their native sweet food has prevented them from ever being brought alive to the European menageries, to which their beauty would render them the greatest ornaments.' Subfamily Character.-Bill equally conic; the upper mandible more or less arched, and very distinctly notched. Feet formed for perching. Claws broad and fully curved. Genera. Tardivola, Tanagra (with the subgenera Pitylus, TaPhænisoma, Tachyphonus, and Leucopygia). Nemosia. Aglaia (with the subgenera Euphonia and Tamagrella). And Pipillo (with the subgenus Årremon). (Classification of Birds.) The Prince of Canino (Bids of Europe and North America) places the Tanagrina between the Fringillinæ and the Emberizinæ. Pyranga is the only genus recorded as belonging to the Tanagrinæ. Mr. G. R. Gray makes the Tanagrine the third subfamily of the Fringillida, arranging it between the Coccothraustina and Fringilline. The following genera are enumerated by Mr. Gray as belonging to the third subfamily: Emberizoïdes, Temm.; Pipilo, Vieill.; Embernagra, Less.; Arremon, Vieill.; Cissopis, Vieill.; Pitylus, Cuv.; Tanagra, Linn.; Saltator, Vieill.; Spindalis, Jard. and Selby; Ramphopsis, Vieill.; Lamprotes, Sw.; Pyranga, Vieill.; Lanio, Vieill.; Tachyphonus, Vieill.; Nemosia, Vieill. Tanagrella, Sw.; Euphonia, Desm.; Calaspiza, G. R. Gray; Stephanophorus, Strickl.; Cypsnagra, Less. Mr. Gray, with his usual industry, gives the numerous synonyms of each genus. (List of the Genera of Birds, 2nd edition, 1841.) We select Nuttall's description of the Scarlet Tanager, or Black-winged Summer Red-Bird, Tanagra rubra, Linn. (subgenus Pyranga). Mr. Swainson then dwells on the obscurity which attends the examination of this group, which he states to be one of the most difficult to be understood in the whole circle of ornithology. He points out, for instance, that the comparative strength of the bill is so variable in the same subgenus, that such variation, indicative of genera in other families, is in this no more than a discrimination of sections or species. Nothing, according to him, can illustrate this fact more than the affinity between Pitylus and Tardivola. Looking to the types of each, he observes, we should say that they did not belong even to the same subfamily; for the bill of the first is nearly as large as in the hawfinches (Coccothraustes, HAWFINCH), while that of Tardivola is so comparatively slender that it seems more akin to the LARKS than to the Tanagers; and yet, he remarks in continuation, between these two extremes or types, he had, when he wrote, before him such a perfect series of graduated forms, wherein not only the bill, but all the other subordinate characters of the two groups, progress in such a perpetual and almost imperceptible manner, that he was actually at a loss to know where Tardivola ends and Pitylus begins. The foregoing affinity being admitted, and it should be remembered that some of the best ornithological The male is scarlet-red, with the wings and notched tail writers have placed it as a genus in a totally different black: the base of the plumage is ash, then white. The family, Mr. Swainson next proceeds to inquire into the female, young, and male in autumn, are dull green, incause of so remarkable a variation in the bill of such closely-clining to yellow in the latter; yellow beneath; wings and united species. He first states that nearly the whole of the tail dusky. Length about six inches and a half; alar exseed-eating birds of Tropical America are composed of the tent ten inches and a half. Tanagers, which, in those regions, supply the place of the This splendid and transient resident,' says Nuttall, other finches so abundant in all parts of Europe. The in-accompanying fine weather in all his wanderings, arrives numerable small and hard fruits produced in the American from his winter station in tropical America from the beforests are, he observes, the appointed food of the Tanagers, ginning to the middle of May, and extends his migrations the parrots living principally upon the larger nuts, and the probably to Nova Scotia as well as Canada. With the shy, bill of the former birds is constructed accordingly. After unsocial, and suspicious habits of his gaudy fraternity, he noticing the disparity of the bills in the finches, taking the takes up his abode in the deepest recesses of the forest, common linnet and the hawfinch for example, he remarks where, timidly flitting from observation, he darts from tree how little reliance can be placed on such diversity in de- to tree like a flashing meteor. A gaudy sylph, conscious termining genera: but this, he observes, will not explain of his brilliance, and the exposure to which it subjects the great difference which often exists in the size and him, he seems to avoid remark, and is only solicitous to be plumage of species which all writers agree in arranging known to his humble mate, and hid from all beside. He within the limits of the same subgenus; and he takes the therefore rarely approaches the habitations of men, unless restricted genus Pitylus, Cuv., as an example. Some of perhaps the skirts of the orchard, where he sometimes the species of that genus are green, some black, others however builds his nest, and takes a taste of the early and grey; and in size they vary from the dimensions of a spar-inviting, though forbidden cherries.' row to those of a small thrush. Among the thick foliage of the tree in which he seeks The doubts which, in Mr. Swainson's opinion, hang over support and shelter, from the lofty branches, at times, we the correctness of the views which he entertained with re-hear his almost monotonous tship-witee, tship-idee, or spect to the natural affinities of these birds, may, he says, be said to hinge almost entirely upon his not having been able to examine specimens of Fringilla Zena, which has certain peculiarities which lead him to expect that it forms the type of one of the principal divisions among the Tanagers, or that it connects his genus Aglaia with Pipillo. On the first supposition, F. Zena would, according to Mr. Swainson, constitute the passage from the true sparrows (Pyrgita) to the subgenus Tanagra proper; while by the tshukadee, tshikadee, repeated at short intervals, and in a pensive under-tone, heightened by the solitude in which he delights to dwell. The same note is also uttered by the female when the retreat of herself and young is approached; and the male occasionally utters, in recognition to his mate, as they perambulate the branches, a low whispering 'tait, in a tone of caution and tenderness. But besides these calls on the female, he has also, during the period of his incubation, and for a considerable time after, a more musical strain, resembling somewhat, in the mellowness of its tones, the song of the fifing Baltimore. The syllables to which I have hearkened appear like 'tshoove 'wait 'wait, 'vehowit wait, and 'wait, 'vehowit vea wait, with other additions of harmony, for which no words are adequate. This pleasing and highly musical meandering ditty is delivered for hours, in a contemplative mood, in the same tree with his busy consort. If surprised, they flit together, but soon return to their favourite station in the spreading boughs of the shady oak or hickory. This song has some resemblance to that of the Red-eyed Vireo in its compass and strain, though much superior, the 'wait 'wait being whistled very sweetly in several tones, and with emphasis; so that, upon the whole, our Pyranga may be considered as duly entitled to various excellencies, being harmless to the farmer, brilliant in plumage, and harmonious in voice.' Nest, Food, &c.-The same author describes the nest (which is built about the middle of May, on the horizontal branch of some shady forest-tree, commonly an oak, but sometimes in an orchard tree) as but slightly put together, and usually framed of broken rigid stalks of dry weeds or slender fir-twigs, loosely interlaced together, and partly tied with narrow strips of Indian hemp (Apocynum), some slender grass-leaves, and pea-vine runners (Amphicarpa), or other frail materials; the interior being sometimes lined with the slender, wiry, brown stalks of the Canadian cistus (Helianthemum), or with slender pine-leaves; the whole so thinly platted as to admit the light through the interstices. The three or four eggs are dull blue, spotted with two or three shades of brown or purple, most numerous towards the larger end. As soon as their single brood, which is fledged early in July, is reared, they leave for the south, generally about the middle or end of August. The female,' says this interesting author in continuation, 'shows great solicitude for the safety of her only brood; and, on an approach to the nest, appears to be in great distress and apprehension. When they are released from her more immediate protection, the male, at first cautious and distant, now attends and feeds them with activity, being altogether indifferent to that concealment which his gaudy dress seems to require from his natural enemies. So attached to his now interesting brood is the Scarlet Tanager, that he has been known, at all hazards, to follow for half a mile one of his young, submitting to feed it attentively through the bars of a cage, and, with a devotion which despair could not damp, roost by it in the branches of the same tree with its prison.' The food of this species consists mostly of winged insects, such as wasps, hornets, and wild bees, the smaller kind of beetles, and other Coleoptera. Seeds are supposed to be sometimes resorted to, and they are very fond of whortle and other berries. It is in August that the moult of the male, when he of the female,' commences. (Manual of the Ornithology exchanges his nuptial scarlet for the greenish-yellow livery of the United States and of Canada.) TANAGRI'NÆ. [TANAGERS.] TANCRED, of Hauteville in Normandy, was a feudal paron who lived in the latter part of the tenth and beginning of the eleventh century. After doing military service for some years under Richard the Good, duke of Normandy, he retired to his hereditary mansion, where he lived poor, and reared up a numerous family of twelve sons and three daughters. All his sons were remarkable for their comeliness, their great strength, and their courage. The eldest, Serlon, followed William the Bastard in his conquest of England, and the others went successively to seek their fortune in Apulia, where Rainulf, another Norman adventurer, had already obtained the countship of Aversa from Sergius, duke of Naples. William, one of Tancred's sons, called 'Fier à bras,' or strong of arm, became count of Apulia, and after his death, his brother Robert, called Wiskard, or the wise,' became duke of Apulia and Calabria, and the founder of the Norman dynasty of Sicily. [SICILIES, TWO, History of.] Their father Tancred died at a very great age at Hauteville. Traces of the château of Tancred, according to old popular tradition, were still seen a few years since in a pretty valley near Hauteville, four miles north of the town of Marigny, in the arrondissement of Coutances department of La Manche. (Gaultier d'Arc, Histoire des Conquêtes des Normands en Italie, en Sicile, et en Grèce.) TANCRED, son of Eudes, a Norman baron, and of Emma, sister of Robert Wiskard, duke of Apulia, according to some (Gaultier d'Arc, Histoire des Conquêtes des Normands en Italie, en Sicile, &c.), and nephew of Bohemund, son of Wiskard, and prince of Tarentum according to others (Giannone and the authorities he quotes), was serving with Bohemund under Roger, duke of Apulia, son and successor of Wiskard, at the siege of Amalfi, A.D. 1096, when the report of the great crusade which was preparing for the East determined Bohemund, who was not on good terms with Duke Roger, to join the Crusaders. Tancred followed him with a vast number of men from Apulia and Calabria. The exploits, true or fabulous, of Tancred, in Syria and Palestine, have been immortalized by Tasso in his poem of La Gerusalemme.' TANCRED, king of Sicily. [SICILIES, TWO, History of.) TANGENT. In the article CONTACT We have given the first notion on this subject, which we now resume in a somewhat more general manner, annexing the usual details of formulæ, but without proof. It is usual to apply the word tangent to the tangent straight line only, on which see DIRECTION: generalizing the definition, it will be as follows:-Of all curves of a given species, or contained under one equation, that one (B) is the tangent to a given curve (A) at a given point, which passes through that given point, and is nearest to the curve (A): meaning that no curve of the given species can pass through the given point, so as to pass between (B) and (A), immediately after leaving the point at which the two latter intersect. To ascertain the degree of contact of two curves which meet in a point, proceed as follows. Let y = pr and y=x be the equations of the curves, and a the abscissa at the point of contact; so that paya. At the point whose abscissa is a+h, the difference of the ordinates of the curves is, by Taylor's theorem, h2 m+1 ha (p'a-y'a) h + (p"a-4′′a)+("a-4""a)2.3+ as to which, generally speaking, it will be found that h can be taken so small that the series shall be convergent: if this be not so, the method of arresting the series given in TAYLOR'S THEOREM must be employed. Now of two which m is the greater will diminish without limit as comseries of the form Ah"+Bh" +.... the value of that in pared with the other, when h diminishes without limit. Consequently, every curve y=x, which has 'a d'a, will approach, before the point of contact is attained, nearer to y=pr than any other in which 'a is not p'a. Again, when d'a 'a, those cases of y=x in which "a="a, will approach nearer to y=4x than any in which p'a is values to the constants in y=x as will satisfy as many as not "a; and so on. Hence, to make y=4r have the closest possible contact with y=pr when xa;-give such possible of the equations paya, p'a=y'a, p′′a=4′′a, &c. consecutively from the beginning. This is a brief sketch, which can be filled up from any elementary work; and the following are the principal results: 1. When the string of equations is satisfied up to (n) (n) a=4a, the contact is said to be of the nth order. 2. In contact of the nth order, the deflection (a+h)— ‚n+1 and vanishes in a finite ratio a+h) diminishes with h to it. 3. In contact of an even order, the curves intersect at the point of contact; in contact of an odd order, they do not intersect at that point. 4. When curves have a contact of the nth order, no curve, having with either a contact of an order inferior to the nth at the same point, can pass between the two. 5. A straight line, generally speaking, can have only a contact of the first order with a curve; and the equation to the tangent straight line of the curve y=pr, when xa, is y-pa-p'a(x-a). But if it should happen that (n) "a=0, ""a=0, &c., up to 4a=0, then for that point the tangent has a contact of the nth order. Thus, at a point of contrary flexure the tangent has a contact of the second order, at least, with the curve. 6. A circle, generally speaking, can be made to have a contact of the second order with a curve, and the equation p"a -2 This circle cuts the curve, generally speaking: if not, as for example, at the vertices of an ellipse, it is evidence that the circle has a contact of some higher and odd order. The centre of the circle of curvature is a point on the normal, being that at which the normal touches the evolute. [INVOLUTE AND EVOLUTE.] Not only is the term tangent most generally applied to the closest straight line only, but frequently only to that portion of the straight line which falls between the point of contact and the axis of x. Again, the normal is a straight line perpendicular to the tangent, drawn through the point of contact: but this term also is frequently applied only to that portion which falls between the point of contact and the axis of x. It is with reference to this limitation that the terms subtangent and subnormal are to be understood: the first meaning the distance from the foot of the tangent to the foot of the ordinate; the second that from the foot of the ordinate to that of the normal. The formula for the subtangent is pa÷p'a; that for the subnormal pax p'a. Let ẞ be the angle made by the tangent with the axis of; usually the angle made by that part of the tangent which has positive ordinates with the positive side of the axis of x. Then ẞ, at the point whose abscissa is x, is determined by the equation Unless the mode of attributing signs be carefully attended to, these last equations, though always considered as universally true, are not so in reality. We now come to the consideration of a surface. The mode of defining contact of a given order resembles that adopted with reference to a curve. Thus if = (x, y) and = (x, y) be the equations of two surfaces coinciding when xa, yb, so that op (a, b) = ↓ (a, b), then if the point be taken at which =a+h, y=b+k, the contact of the two surfaces is of the nth order, when the deflection (a+h, b+k)-(a+h, b+k) being developed in powers of h and k by Taylor's Theorem, shows no terms lower than those of the form Ah”+Bh”¬1k+...+Mk". This is tantamount to the following: two surfaces have a contact of the nth order when any plane whatever drawn through the point of contact cuts the surfaces in two curves which have a contact of the nth or a higher order. Every surface has at every point a plane which has a complete contact of the first order. If z= (x, y), and x, y, z be the co-ordinates of the point of contact, and ,, those of any point in the tangent plane, then the equation of the tangent plane is dz dz dx dy -2= (E-x)+ (n-y). - x n-y Z 2 аф аф dx dy dye - (addy)" , say U, at the point of contact. Imagine a plane to pass through the normal, cutting the surface in the curve (C) and the tangent plane in the straight line (L). Then, while the plane revolves about the normal, (L) is always tangent to (C). 1. Let U be positive. Then (L) has never more than a contact of the first order with (C), the surface nowhere passes through the tangent plane, and we have only such contact as is seen at any point of a sphere or ellipsoid. 2. Let U=0. Then (L) has never more than a contact of the first order with (C), except when the plane is in one position, in which there is a contact of a higher order. If U=0 at the point of contact only, and begin to take value at all adjacent points, nothing more would appear than in the last case, except that in one particular direction from the point of contact, and in its opposite, the surface would seem to grow nearer to the tangent plane than in any others. But if U=0 at all points of this surface, this approach to the tangent plane in one particular direction becomes more marked: for the surface lies on that plane in a straight line, that is to say, every tangent plane meets the surface in a straight line infinitely ex tended both ways; and the plane is tangent to the surface at every point of that straight line. Such surfaces, namely those in which U is always =0, are developable, or can be unrolled without any overlapping, rumpling, or tearing. Cones and cylinders are instances. Again, if U=0, not throughout the whole surface, but throughout one particular line upon it, that line will be a plane curve, and its plane will be tangent to the surface at every point in which it meets the surface. 3. Let U be negative. Then (L) has never more than a contact of the first order with (C), except in two different positions, in both of which there is contact of a higher order. Draw lines marking out these two positions of (L), and consequently dividing the tangent plane into four parts, with four angles round the point of contact. In one pair of the opposite angles, the surface lies on one side of the tangent plane, and in the other on the other. Again, as the plane which revolves round the normal takes its different positions, the curvature of the section (C) changes. The two positions of the revolving plane in which the curvatures are greatest and least (algebraically) are at right angles to one another. We shall not enter into the mathematical formulæ connected with this subject, but shall only endeavour to give a popular illustration of this remarkable point. Suppose an eggshell, unbroken, to be placed with either vertex uppermost. The descent will be equally rapid in all directions, or the curvature at the highest point of all the vertical sections will be the same. But suppose the shell to be so placed that some point intermediate between the two vertices is uppermost. The descent will not then be equally rapid in all directions, or the curvatures of the vertical sections will not be the same. The direction of most rapid descent But if the equation be given in the form (x, y, z) = 0, will be at right angles to that of least rapid descent. The it is dp аф dp (−x) + dx dy dz In the first case, the equations of the normal, a line drawn through the point of contact perpendicular to the tangent, are tangent plane has here a contact of the first of the three kinds above mentioned. If there be a contact of the second kind, all the circumstances are the same, except that the direction of least rapid descent gives, comparatively speaking, no descent at all at the first instant. If we take a cylinder, or other developable surface, and make a tangent plane horizontal, there is absolutely no descent in one direction, or, by going along the tangent plane, we can remain entirely on the surface, in one certain direction, as before observed. And the direction of most rapid descent is at right angles to this direction of no descent. To put a case of the third kind, suppose a saddle placed on a horse, and we take the lowest point of the seat. The tangent plane then cuts through the saddle horizontally. In some directions there is descent, in others ascent, with two directions in which there is, comparatively speaking, neither ascent nor descent. The direction of most rapid ascent, which is from the lowest point of the seat directly towards the head or tail of the animal, is at right angles to the direction of most rapid descent. Mathematically speaking, the curvatures of the vertical sections are sometimes positive, and sometimes negative, and the direction of the greatest negative (or algebraically least) curvature is at right angles to the direction of the greatest positive (or algebraically greatest) curvature. 107° E. long. To the south of it is Tibet; to the west Chinese Turkistan, or the government of Thian-Shan Nanlu; and to the north Mongolia, of which also a portion is in cluded within the lately erected province of Kansi. As the boundary-lines of the country are not politically deter mined, it is not possible to give an estimate of the area. The southern portion of Tangut, or that which lies south of 38° N. lat., is one of the most mountainous tracts on the globe, and extends over the upper course of the river Hoang-ho and the basin of the lake of Khookhoo-nor. Along its southern border there is a very elevated range, which divides the upper courses of the rivers Hoang-ho and Yan-tse-kiang, and is called the Bayan Khara range. [BAYAN KHARA MOUNTAINS.] Another elevated range traverses the country in the same direction from east to west near 38° N. lat. This range rises at a short distance from the banks of the Hoang-ho north of the town of Lantcheou, and in its eastern part is called Kilian Shan; but farther west it takes the name of Nan Shan (or Southern Chain). It rises to a great elevation, especially towards the As to points connected with the apparent physical cha-west, where many of their summits are covered with snow and racter of the tangent, which have been in various places united by extensive glaciers. This mountain-chain is supreferred to this article, it will be more convenient to con-posed to be connected with the Kuenluen range near 92° E. sider them under the word VELOCITY. long. These two ranges above mentioned occupy a great portion of the country between 33° and 38° N. lat., and nearly the whole of the remainder of the country is filled up by a third range, which connects these two ranges, and extends from south-east to north-west, being on the north united to the Nan Shan, and on the south to the Bayan Khara Mountians. This chain bears the name of Siue Shan, or Snowy range, on account of the numerous summits which rise above the snow-line. The river Hoang-ho breaks through this range, but the huge rocky masses compel the river to make a great bend towards the west between 34° and 36° N. lat., and the circuit which the river makes shows the immense extent of these masses of rock. In this part of its course the river is said to be hemmed in by lofty mountains, so that no communication can be established along the banks. Its course above this bend is very imperfectly known, and the fabulous accounts of its sources show that they have never been visited even by Chinese geographers. The river enters a wide valley by a narrow gorge formed by two very elevated mountains a little above the town of Ho-cheou (36° N. lat. and 102° E. long.). At the opening of this gorge is a fortress, called Tsy-shy-kuan. TANGHI'NIA, the name of a genus of plants belonging to the natural order Apocynaceæ. This name was given by Aubert du Petit Thouars to the plant which produces the celebrated Tanghin poison of Madagascar. The genus possesses an infundibuliform corolla, with a clavate tube, and 5-toothed throat: the anthers are subsessile; the fruit is a drupe, with a fibrous ligneous putamen or stone, which contains one or two seeds. The specific name T. venenifera was given to the plant which yields the poison. It has dense leaves, with erect branches, and paniculated terminal flowers. At the time Du Petit Thouars described this plant, he stated that it was closely allied to the Cerbera Manghas; and since its cultivation by Mr. Telfair in the Mauritius, there can be no doubt of its belonging to the genus Cerbera, and the plant is now called C. Tanghin. In its native island this plant attains the size of a tree, and has a hard wood which may be used for many kinds of carpentry. But the part which yields the poison is the kernel of the fruit. Although this kernel is small, not much larger than an almond, Mr. Telfair says that it contains enough poison to kill twenty persons. Its great use in Madagascar was as a means of trial, the innocent being supposed able to resist its action, whilst the guilty suffered under its influence. Radama, the late king of Madagascar, was desirous of abolishing its use, but found great difficulty in doing so on account of the prejudices of the natives. Mr. Telfair witnessed a sad instance of its use. The king Radama was taken ill, and got well by the use of mercury; but this medicine affected his mouth, so that the impression produced upon his skid,' or physician, was that the king had been poisoned. He therefore insisted that the Tanghin should be administered to himself and all the servants of the household, in order to ascertain the guilty party. The king protested against the procedure, but in vain. The whole household were shut up during the night without food, and in the morning were brought out for trial. The presiding 'skid,' or physician, then pounded the Tanghin bean to a pulp between two stones, and applied a small quantity to the back of the tongue of each individual. The effects varied in different individuals. In some it produced vomiting, and the poison being ejected from the stomach, they recovered. In others convulsions were brought on with violent efforts at vomiting, which soon destroyed life. (Botanical Magazine, fol. 2968.) TANGIER. [MAROCCO.] TANGLE. [SEA-WEEDS.] TANGUT is the historical name of a country in Asia, which occupies the centre of the eastern, more extensive, and more elevated table-land of that continent [ASIA, vol. i., p. 464], where a nation, which originally inhabited Tibet, and was called Tang, founded an empire in the seventh century, which was very powerful for a long time, and was overthrown by Genghis Khan in 1227. The country still goes by the name of Tangut, though at present a part of it is incorporated in the Chinese province of Kansi, whilst another is mostly in possession of two Mongol nations, the Olöth Tshoros and the Torbod Mongols. Tangut borders on China Proper on the north-west, extending between 33° and 42° N. lat., and between 94° and Tangut is separated from China Proper by a fourth range, the mountains of Sifan, which run south and north, being connected at their southern extremity with the Bayan Khara Mountains and the Siue Shan by an extensive mountain-knot, which is in the country formerly called Sifan, whence the chain has obtained its name. Though this range is less elevated than the Siue Shan, it rises in several places above the snow-line, and occupies a considerable width. It is supposed to terminate near the banks of the Hoang-ho, a few miles south of 38° N. lat. Opposite to it and on the northern banks of the river rises another chain, which may be considered as the continuation of the mountains of Šifan; but this range, which continues along the western bank of the river as far north as 42° N. lat., rises only to a moderate elevation, and is stated to occupy in many places only three or four miles in width: it is called Holang Shan, and slopes on the west down into the steppe of the Olöth Tshoros. This range is distinguished from all the other ranges of Tangut by being thickly wooded on its eastern declivity. Only a small portion of the countries enclosed by these mountain masses is fit for cultivation. It does not appear that there is any cultivation in the upper valley of the Hoang-ho above the fortress of Tsy-shy-kuan. Below that place and as far as Lan-tcheou, the valley is wider, and narrow tracts along the banks of the river are cultivated and fertile. This part of the valley is compared with that of the Adige in Tyrol. Farther down, and as far as the neighbourhood of Ning-hia, a town built on the western banks of the Hoang-ho, at the eastern declivity of the Holang Shan (38° 32′ N. lat.), the valley has not been visited by Europeans. At this place the river runs in a wide valley which has been rendered fertile by numerous canals, which are fed by the waters of the river, and in which rice is extensively cultivated. There are also numerous plantations of fruit-trees. The soil contains much saltpetre. The town of Ning-hia, the antient capital of Tangut, is of considerable extent, being fifteen li (equal to five miles) in cir |