of the class syngenesia, order polygamia æqualis. Receptacle naked; seeds crowned with a slight margin; calyx imbricate, hemispherical; florets of the ray obsolete, or three-cleft; sometimes wanting. Eighteen species; chiefly natives of the Cape, the rest of Europe or the Levant: one common to our own wastes. The following are cultivated.

1. T. vulgare. Common tansy. Leaves doubly pinnatifid, deeply serrate; root fibrous, creeping, spreading to a great distance; flowers in terminal corymbs, of a gold colour, and fattish. The leaves and flowers have a strong, not very disagreeable smell, and a bitter, somewhat aromatic taste. The virtues of tansy are tonic, stomachic, anthelmintic, emmenagogue, and resolvent. It has been much used as a vermifuge; and testimonies of its efficacy are given by many respectable physicians.

2. T. annuum. Annual tansy.

3. T. balsamita. Costuiary: formerly employed as a carminative in medicine under the name of BALSAMITA MAS, which see.

4. T. Sibericum. Siberian tausy. 5. T. suffruticosum. Shrubby tansy. 6. T. flabelliforme. Fan-leaved tansy. The herbaceous kinds are increased by seeds and by parting the roots; the shrubby sorts by cuttings of the branches in spring and summer. TANÆCIUM, in botany, a genus of the class didynamia, order angiospermia. Calyx cylindrical, truncate; corol tubular, nearly equal, five-cleft; rudiment of a fifth filament; berry very large, covered with a bark. Three species; two, natives of Jamaica, climbing plants; one of Mozambique, a vast tree, with fruit the size of a man's head.

TANAGER. See TANAGRA. TANAGRA. Tanager. In zoology, a genus of the class aves, order passeres. Bill conic, pointed, notched, almost triangular at the base, a little inclining at the tip. Fortyeight species; all of foreign extraction, and nearly all belonging to the West Indies and America. Buffon, indeed, confines the whole tribe to South America; nor will such a multiplicity of distinct species of the same genus in one country appear surprising when we recollect that, in these warm climates, where the food is abundant, the forests larger, and the lands more thinly peopled, birds are ten times more numerous than with us, where one severe winter almost exterminates whole tribes, both of land and water fowls. The tanagers, in South America, may be regarded as the representatives of the sparrow of Europe; a genus which they resemble in almost every particular, excepting the colour, and the small groves hollowed out of the sides of the upper mandible towards the point. Like the sparrows, they make short and heavy flights; like them, too, they are granivorous: both cultivate the same familiarity with man; both are dependent on his labours, and are destructive of his property. They are sociable with regard to each other, as well as to man; for they assemble in flocks upon the dry and open fields, around the villages. The tanagers, however, lay only two

eggs at a brood, while our sparrows lay five; a circumstance pretty frequent among birds of warm countries, whose apparent infecundity is so amply compensated by the frequency of their breeding. In the mild and equal temperature of these climates, every season is a season of love, and every brood is quickly produced, and as quickly succeeded by another. The following will serve sufficiently as specimens: 1. T. jacapa. Red-breasted tanager. Black; front, throat, and breast scarlet. Female: purplish-brown, beneath reddish; wings and tail brown. Inhabits South America; six and a half inches long; frequents inhabited places, and builds a pendulous, cylindrical, and somewhat curved nest; feeds on fruit; eggs white, with sinall reddish spots.

2. T. violacea. Golden tanager. Violet; beneath, and hind-head fine yellow; middlequill and lateral tail-feathers within white. Another variety, black instead of violet. Fem.: above olive; young bird blue and olive. Inhabits Brasil and Cayenne; three and a half inches long; is very destructive to rice-plantations. Variable in its colours.

3. T. Mexicana. Black and blue tanager. Black, beneath yellowish; breast and rump blue. Another variety, with tail-coverts green, the body beneath white. Inhabits South America: five inches long; sings very finely.

TANDA, or TANRAH, a town of Hindustan Proper, in Bengal, of which soubah it was the capital in the 17th century. There is little remaining of it but the rampart; and the period when it was deserted is not certainly known. It is seated on the Ganges, 120 miles N.W. of Dacca. Lon. 87. 56 E. Lat. 23. 25 N.

TANG. s. (tanghe, Dutch, acrid.) 1. A strong taste; a taste left in the mouth (Locke). 2. Relish; taste (Atterbury). 3. Something that leaves a sting or pain behind it (Shaksp.). 4 Sound; tone (Holder).

To TANG. v. n. Tosing with (Shakspeare). TANGENCIES (Problem of). This ge geral problem in geometry furnishes the subject of one of the twelve treatises described by Pappus in the preface to the 7th book of his Mathematical Collections. In Dr. Halley's translation of Pappus the problem is thus enumerated: "E punctis rectis et circulis, quibuscunque tribus positione datis, circulum ducere per singula date puncta, qui, si fieri possit, contingat etium datas lineas." This is naturally subdivided into ten distinct propositions, which, if a point be represented by (.), a line by (1), and a circle by (0), may be stated very briefly according to the several data, in the following order :-(.. 1), ( . 11), (110), (.10), (100), (....0), (.00), (000), (...), (111).

The treatise on tangencies was restored by Vieta, under the title of Apollonius Gallus, and many of his deficiencies were supplied by Mari

nus Ghetaldus. These have been translated with the addition of a supplement by Mr. Lawson, and a farther addition of Fermat's Treatise on Spheri

cal Tangencies. Mr. Leslie has given in his Geometry, solutions to all except the 5th, 7th, and 8th, of the preceding enumeration; but they are inelegant and defective.

TANGENT, in geometry, a right line

which touches a circle; that is, meets it in such manner, as that though infinitely produced, it would never cut the same; that is, never come within the circumference.

It is demonstrated in geometry; 1. That if a tangent and a secant be both drawn from the same point, the square of the tangent will be equal to the rectangle, under the whole secant, and that portion thereof which falls without the circle.

2. That if two tangents be drawn to the same circle from the same point, they will be equal to each other.

As a right line is the tangent of a circle, when it touches the circle so closely that no right line can be drawn through the point of contact between it and the arc, or within the angle of contact that is formed by them; so, in general, when any right line touches any are of a curve, in such a manner that no right line can be drawn through the point of contact, betwixt the right line and the arc, or within the angle of contact that is formed by them, then is that line the tangent of the curve at the said point.

The tangent of an arc is the right line that limits the position of all the secants that can pass through the point of contact, though strictly speaking, it is no secant.

In the ellipse and hyperbola, the semiaxis is a mean proportional between the distance from the foot of the ordinate let fall from the point of contaet, and the distance from the centre to the point where the tangent cut the axis produced. In the parabola, the distance from the vertex to the foot of the ordinate is equal to the distance from the vertex to the point where the tangent cuts the axis produced.

TANGENT, in trigonometry.-A tangent of an arc, or of the angle measured by that are, is a right line, raised perpendicularly on the extreme of the diameter, and continued to a point, where it is cut by a secant, that is, by a line, drawn from the centre through the extremity of the arc whereof it is a tangent.

Co-tangent, or tangent of the complement, is the tangent of an arc which is the complement of another arc to a quadrant.

The tangent of an arc is a 4th proportional to the cosine, sine, and radius.

If A denote any arc, then we have A = tan. A

-tan. A tan. A Whence since tan. 45°

tan. 7A+ &c.

1, we have arc 45° = 1 − } + } − ¦ + ¦ + &c. And, conversely,

tan. A = A +

+

+

62A9

+

2A5 17A7 3Ra 15R4 315R6 2835R8

+

See the treatises on trigonometry by Emerson, Caguoli, Mauduilt, Bonnycastle, Woodhouse, &c. See also our article TRIGONOMETRY.

Artificial tangents, or logarithmic tangents, are the logarithms of the tangents of arcs; so called, in contradistinction from the natural tangents, or the tangents expressed by the natural numbers.

Line of tangents, is a line usually placed on the sector, and Gunter's scale; the description and uses of which see under the article SECTOR.

Sub-tangent, a line lying beneath the tangent, being the part of the axis intercepted by the tangent and the ordinate to the point of contact.

Method of tangents, is a method of determining the quantity of the tangent and subtaugent of any algebraic curve; the equation of the curve being given.

This method is one of the great results of the doctrine of fluxions. It is of great use in geome try; because that in determining the tangents of curves, we determine at the same time the quadrature of the curvilinear spaces: on which ac count it deserves to be here particularly treated

on.

To draw the tangent, or to find the sub-tangent, of a curve. If AE be any curve, and E any point in it, to which it is required to draw a tangent TE. Draw the ordinate DE, (fig. 4. pl. 167.) then if we can determine the subtangent TD, by joining the points T and E, the line TE will be the tangent sought.

Let dae be another ordinate indefinitely near to DE, meeting the curve, or tangent produced, in e; and let Ea be parallel to the axis AD. Then is the elementary triangle Eae similar to the triangle TDE;

and therefore. ea: aE:: ED: DT;
but.......... ea: E:: flux. ED: flux. AD;
therefore..... flux. ED: flux. AD:: DE: DT;
ух

that is........ ÿ : i : : y :

=

ᎠᎢ,

y

1382A11

i

லுல்

sin. A

and when re

y

=

cot. A

cos. A sin. A √(1-sin. 2A)

2 tan. A

1 – tau. A

=

= cot. A 2 cot. 2A =

sin. 2A

1 + cos. 2A

tan. (45 + 1A)—tan. (45–JA)

duced to its simplest terms, it will be the value of the subtangent sought. This we may illustrate in the following examples.

Er. 1. The equation defining a circle is 201-11 =y, where a is the radius; and the fluxion of this

is Lei — 2ri – Zyÿ; hence

multiplied by y, gives

க்

=

y ; this

the subtangent TD, or CD: DE:: DE : TD, which is a property of the circle we also know from common geometry,

The

29

=

2yy; hence

=

a

2ax

=

TD; =

a

that

conseq. is, the subtangent TD is double the absciss AD, or TA is = AD, which is a well-known property of the parabola.

Er. 3. The equation defining an ellipsis is

from CD or a + 1, gives CT = CA: CA: CT.

or CD:

The preceding examples relate to curves whose ordinates are parallel to each other. We shall now briefly illustrate the method of drawing tangents to curves of the spiral kiud, all whose ordina'es issue from a point: such as the spiral BAG (fig. 5.) whose ordinates CB, CA, CG, are referred to the point C, called the centre of the spiral. Let SAN be a tangent to the spiral at any point A, and let CT be perpendicular to it, and let the are CBA (considered as variable by the motion of A towards G) be denoted by z, and the ordinate CA by y. Then : ý :: AC (y): AT Hence, if upon CA, as a diameter, a semicircle be described, and in it, from A, a right line equal to be inscribed, that right line will be a tan

gent to the spiral at the point A.

с

in the direction Ae, as to unity, or as y to b. Consequently CT and AT are in the same ratio, and AC: CT::√ yy + bb : y; and AC AT :: yy+bb: b; whence CT and AT, are given, 42 equal to respectiveyy + bb ly; from either of which expressions the tangent AT may be drawn: and, in the same manner, may the position of the tangent of any other spiral be determined.

and

by yy + bb

In the business of finding tangents by fluxions there is an interesting case arises when the exIt is not pression for the subtangent becomes. to be concluded from this circumstance that the sub-tangent itself vanishes. The ratio repre0 sented by ought to arise in those 0 y points wherein two or more branches of a curve intersect each other. The difficulty is removed by the method of multiplying by arithmetical progressions, and by other processes; for a descripton of which, see Agnesi's Analytical Institutions, vol. ii. p. 50-56; and Bossut Traite de Calcul Differentiel, &c. toin, i. p. 167–174. The latter author applies this case of tangents very ingeniously to the theory and processes respecting vanishing fractions.

The inverse method of tangents-This is the reverse of the foregoing, and consists in finding the nature of the curve that has a given subtangent. The method of solution is to put the given subtangent equal to the general expression which serves for all sorts of curves; then the equation reduced, and the fluents taken, will give the fluential equation of the curve sought.

Er. 1. To find the curve line whose subtangent 2y? 242 yx

is =

a

Here

a

y

; hence 2yy = ax, and

the fluents of this give y =ar, the equation to a parabola, which therefore is the curve sought. Ex. 2. To find the curve whose subtangent is

TANGIBILITY. s. (from tangible.) The quality of being perceived by the touch. TAʼNGIBLE. a. (from tango, Lat.) Perceptible by the touch (Locke).

TANGERMUNDE, a town of Germany, in the old marche of Brandenburg, with a castle, seated on the Tanger, where it falls into the Elbe, 24 miles N.W. of Brandenburg, and 28 N.E. of Magdeburg. Lon. 13. 30 E. Lat. 52. 46 N.

TANGIER, a seaport of the kingdom of Fez. It was taken by the Portuguese, in 1471, and given as a dower to the princess Catharine, on her marriage with Charles II. of England; but he did not think it worth the expence of keeping, and therefore, in 1683, caused the works to be blown up, and withdrew the garrison. It is 130 miles N. of Fez. Lon. 5. 50 W. Lat. 35. 49 N.

To TANGLE. v. a. (See ENTANGLE.) 1. To implicate; to knit together. 2. To ensnare; to entrap (Milton). 3. To embroil; to embarrass (Crashaw).

To TANGLE. . . To be entangled. TANGLE. s. (from the verb.) A knot of things interwoven in one another (Milton).

TANJORE, a province of Hindustan, on the coast of Coromandel. It is an appendage of the Carnatic, but subject to its own rajah, who pays an annual subsidy to the English East India Company.

TANJORE, a city of Hindustan, in the Carnatic, capital of a province of the same name. It is seated on the Cauvery, 156 miles S. by W. of Madras, and 166 S.E. of Seringapatam. Lon. 79. 12 E. Lat. 10. 46 N. TA'NISTRY. 8. The Irish hold their lands by tanistry, which is no more than a personal estate for his lifetime that is tanist, by reason he is admitted thereunto by election (Spenser). TANK. s. (tanque, Fr.) A large cistern or basin (Dryden).

TANKARD. s. (tankaerd, Dutch.) A large vessel with a cover, for strong drink (Arbuthnot).

TANNA, a fertile island in the Pacific Ocean, one of the New Hebrides, on which is a volcano. The inhabitants are brave and hospitable; and their arms are bows and arrows, slings, spears, and clubs. Lon. 169. 46 E. Lat. 19. 30 S.

TAʼNNER. s. (from tan.) One whose trade is to tan leather (Moxon).

TANNER (Thomas), a learned English prelate, was born at Market Lavington, in Wiltshire, in 1674. He received his academical education in Queen's college, Oxford, where he took his degree of B. A. In 1695 he was elected fellow of All-saints, and after passing through various church preferments, was consecrated bishop of St. Asaph, in 1732. He died in 1735, and was buried in the cathedral of Christ-church, Oxford. Bishop Tanner is known by an excellent work, entitled Noticia Monastica; or, an Account of all the Religious Houses in England and Wales, folio, 1744. In 1741 appeared his Bibliotheca Britannico Hibernica.

TANNIN. See TAN.

TANNING, the process of preparing hides and skins by means of tan, so as to give them the full benefit of its effects; becoming harder, thicker, heavier, more impervious to water, and capable of resisting putrefaction. Hides and skins thus prepared are called LEATHERS and to this article we refer our readers for the process of tanning.

TANSY, in botany. See TANACETUM. TANSY (Wild). See POTENTILLA. TANTALIUM, in mineralogy, a genus of the class metals. Blackish-grey, softish, of a granular fracture, not magnetic; specific gravity 6.500: not soluble in any acid, nor altering its colour when heated to redness; melting with phosphat of soda and borax iuto a colourless glass. Two species.

1. T. manganesiatum. Tantalite. Found at Kimito in Finland, in irregular crystals; colour between blueish-grey and blackish-grey; surface smooth with metallic lustre; very hard; fracture com act.

2. T. ytriatum. Yttrotantalite. Consisting of oxyd of tantalium, combined with oxyds of iron and yttria. Found, also, at Kimito in Finland, in small kidney-form masses of inconsiderable hardness; fracture granular, irongrey, of metallic lustre; may be scratched with a knife, and gives a grey powder.

To TA'NTALIZE. v. a. (from Tantalus, whose punishment was to starve among fruits and water which he could not touch.) To torment by the show of pleasures which cannot be reached (Addison),

TANTALUS, a king of Lydia, son of Jupiter, by a nymph called Pluto. He was father of Niobe, Pelops, &c. by Dione, one of the Atlantides. He is represented by the poets as punished in hell, with an insatiable thirst, and placed up to the chin in the midst of a pool of water, which flows away as soon as he attempts to taste it. There hangs also above his head a bough, richly loaded with delicious fruit, which, as soon as he attempts to seize, is carried away from his reach by a sudden blast of wind. He is thus punished either for theft, cruelty, and impiety, or lasci viousness, for the causes are variously ex plained.

TANTALUS. Ibis. In zoology, a genus of the class aves, order gralla. Bill long, subulate, rounded, subarched; face naked; tongue short, broad; jugular pouch naked; nostrils oval; feet four-toed, palmate at the base. Twenty-three species, scattered over the warmer climates of the globe. The fol lowing are chiefly worthy of notice.

bill

1. T. ibis. Egyptian ibis. Face red; pale yellow; quill-feathers black; body whitish-rufous. From thirty to forty inches long; inhabits in vast numbers the lower parts of Egypt.

This bird, so faithful in the service of its native country, was made the emblem of it. Its figure, which we find wrought on all the ancient Egyptian monuments, represents Egypt; where divine honours were paid to it by the

superstitious inhabitants. The intention of the legislator in consecrating this bird was, no doubt, to preserve and to multiply animals that destroyed the noxious reptiles with which Egypt abounds. Even after it is satiated, the ibis is continually occupied, on the banks of the Nile, in destroying locusts, caterpillars, and serpents. But however useful the ibis may be, some have doubted whether this bird be the same with that which had divine honours paid to it by the ancients. Storks, kites, and vultures, are all hostile to serpents; and the figure, marked in their hieroglyphics, is not sufficiently distinct, to determine exactly for what species of the serpentivorous birds it was intended.

The consecration of brute animals, which at first sight appears so monstrous and degrading to human reason, was, however, in the early periods of the Egyptians, attended with circumstances that favoured its establishment, and in some measure palliated its absurdity. The early stages of society in that country were periods of misery. The noisome and destructive tribes, with which those wretched savages were surrounded, were too numerous, and too powerful, to be repelled by men, in a rude, solitary, and defenceless state, who had neither arms, nor any of those arts that are necessary to render human strength of avail against the animal creation. There cannot be a stronger proof of the wretched imbecility of man in his first stage of social union than his being obliged to humble himself, even to adoration, before the brutes. Those numerous tribes, that we afterwards behold his slaves, were originally his masters, or, at least, his formidable rivals for dominion. Fear and hope are the great pillars upon which superstition builds those monuments of weakness and ignorance, which are deemed so mortifying to pride, and disgraceful to reason. By uniting these together, she has made deities of almost every animal, either useful or destructive.

In Egypt, accordingly, the worship of animals was early established, and rigidly adhered to, for a long succession of ages. This the ancient monuments fully evince; and seem to be a proof of the painful struggles between man and the noxious animals around him, before his dominion over them was established. That country, indeed, seems peculiarly favourable to the growth of serpents, crocodiles, locusts, and every species of impure animals. Among the vast masses of fat mud with which that flat and fertile district is constantly covered, by the overflowings of the Nile, such animals teem with fecundity, and almost cover the surface of the earth. The numberless myriads of noxious animals which are thus generated under the invigorating influence of a vertical sun acquire also a size, that renders them truly formidable to man; as well as to the nobler animals, to which, however, they are at last obliged to yield. In this manner was the worship of animals established in Egypt, and, probably, in every country in which it has prevailed; and, in the

general veneration that was paid to certain animals by the Egyptians, the ibis came in for a share, proportioned to the services it rendered them in destroying these armies of reptiles.

In the progress of knowledge and the arts, reason might have checked this extravagance, had not the priests and legislators encouraged it, to give greater sanction to those laws, that were made for the preservation of this useful bird. Such is the weakness of the human mind, that, in all ages, wise legislators have found it necessary to call in the aids of superstition to strengthen their laws. In Egypt they alleged, that the ibis was a favourite of heaven; and that, if ever the gods should deign to manifest themselves to man, it would certainly be in the shape of that animal: and that in the great metamorphosis of the gods, Mercury, the inventor of laws and arts, and the tutelar deity of Egypt, had already undergone this transformation.

Herodotus informs us, that he went to be an eye witness of the labours of the stork; and that, near Butus, on the confines of Arabia, where the mountains open to the vast plains of Egypt, he saw the fields covered with an incredible quantity of the bones and fragments of serpents, that had been devoured by this bird. Cicero and Pliny confirm this account.

In return for these services, the Egyptians ordained, that the killing of the ibis should be held a capital crime; and, being skilled in embalming, that gloomy art of perpetuating the images of death, they rewarded, as they imagined, the good offices of the ibis, by preserving its carcase after its decease. In the plain of Saccara, there are several pits, in which the dried skeletons of birds are found preserved, as mummies; and, among others, have most generally been recognised those of the ibises. Buffon had several of the vessels, in which these skeletons were contained, sent to him for his inspection.

2. T. loculator. Wood ibis. Face blueish; bill reddish; legs, quill, and tail-feathers black; body white. Two other varieties; in one the head and neck diversified with yellow; in the other, the wing-coverts white. Inhabits New Holland and the warmner parts of America; three-feet long; is very slow in flight and stupid; sits on trees, and feeds on herbs, seeds, fruits, fishes, and reptiles: the flesh good. Bill nine inches long; irids reddish.

3. T. leucocephalus. White-headed ibis. Head, neck, and body white; bill and face yellow; legs pale flesh colour; rump with long rosy feathers. Inhabits India; the largest of its tribe; every year before the rainy season sheds its rosy feathers.

4. T. ruber. Scarlet ibis. Face, bill, and legs red; body scarlet; wings tipt with black. Inhabits South America; twenty-one inches long; sits on trees, but lays its greenish eggs on the ground. The young are at first black, then grey, just before they fly whitish, and afterwards grow gradually red. 5. T. igneus. Glossy ibis. Head and neck black; legs green; body varied with glossy