BD85

CD=50

135

35

52 the half of great side.

3.67440

2.02119 30 the lesser segment.

Having divided the right-angled triangle into two right-angled triangles, the hypothenuses and bases of which are given, to find the angles by Gunter. 1. The extent from 105 to 135 will reach from 35 to 45 on the line of sines. 2. The extent from 85 to 75, on the line of num

bers, will reach from radius to 61° 56', the angle B D A on the line of sines. 3. The extent from 50 to 30, on the line of numbers, will reach from radius to angle A D C 36° 53', on the line of sines.

TRIGONOMETRY, spherical, relates to triangles, or figures which are reducible to

triangles, whose sides are segments of circles. Thus, if we describe a triangle on any spherical body, say a globe, it is evident that all the sides must be composed of curved lines; and it is the same in the case of a series of circles, or of orbits intersecting each other. When two equal circles intersect, they will give a parabolic spindle, more or less acute, according as the centres of the two circles may be more or less distant. When three circles mutually intersect, there will be formed a great variety of spherical triangles, of which the areas and the properties could not be ascertained by plane trigonometry, but come under consideration as parts of spherical surfaces. The following definitions should be clearly understood; they are simple in the extreme, but highly important: 1st. The poles of a sphere are two points in the superficies of the sphere, that are the extreme of the axis. 2d. The pole of a circle in a sphere is a point in the superficies of the sphere, from which all right lines that are drawn to the circumference of the circle are equal to one another. 3d. A great circle in a sphere, is that whose plane passes through the centre of the sphere; and whose centre is the same as that of the sphere. 4th. A spherical triangle is a figure comprehended under the arcs of three great circles in a sphere. 5th. A spherical angle is that, which, in the superficies of the sphere, is contained under two arcs of great circles; and this angle is equal to the inclinations of the planes of the said circles. It is particularly to be held in mind, that although we can, upon any actual sphere, describe triangles at pleasure, which may nearly embrace the whole circumference, yet that such cannot be laid down, so as to be represented on paper; for every side of a spherical triangle is less than a semicircle.

With respect to spherical triangles, the learner may generally entertain a correct opinion of their value, if he considers that every arc or segment of a circle may have a chord drawn from one to the other extremity; and that the triangle which can be contained within such arc or segment, taking the chord for a hypothenuse, will determine how much of that circle has been cut off, and is included between the extremes of the segment. It is utterly impossible to produce any two measurable segments, taken from two different circles, which, having chords of equal length, will contain the same angle. A semicircle, having the diameter for its chord, will give a right angle; for if to

any point within that semicircle two lines be drawn from the ends of the chord respectively, their union at such assumed point will form a right angle. In proportion as the chord is less than a diameter, so must the segment be a less part of the whole circle, and the angle contained therein will be more acute. Spherical triangles may be acute, right-angled, or obtuse, the same as on plane-trigonometry. In all right-angled spherical triangles, the sine of the hypothenuse: radius:: sine of a leg: sine of its opposite angle. And the sine of the leg: radius :: tangent of the other leg: tangent of its opposite angle. In any right-angled spherical triangle, A B C (fig. 25,) it will be as radius is to the co-sine of one leg, so is the cosine of the other leg to the co-sine of the hypothenuse. Hence, if two right-angled spherical triangles, A BC, CBD (fig. 26,) have the same perpendicular, B C, the cosines of their hypothenuses will be to each other directly as the co-sines of their bases. In any spherical triangle it will be, as radius is to the sine of either angle, so is the co-sine of the adjacent leg to the co-sine of the opposite angle. Hence, in right-angled spherical triangles, having the same perpendicular, the co-sines of the angles at the base will be to each other directly as the sines of the vertical angles. In any right-angled spherical triangle, it will be, as radius is to the cosine of the hypothenuse, so is the tangent of either angle to the co-tangent of the other angle. As the sum of the sines of two unequal arches is to their difference, so is the tangent of half the sum of those arches to the tangent of half their difference and as the sum of their co-sines is to their difference, so is the co-tangent of half the sum of the arches to the tangent of half the difference of the same arches. In any spherical triangle, A B C (fig. 27,) it will be, as the co-tangent of half the sum of the angles at the base is to the tangent of half their difference, so is the tangent of half the vertical angle to the tangent of the angle which the perpendicular C D makes with the line Ċ F, bisecting the vertical angle.

The following propositions and remarks concerning spherical triangles will render their calculation perspicuous, and free from ambiguity. 1st. A spherical triangle is equilateral, isoscelar, or scalene, according as it has its three angles all equal, or two of them equal, or all three unequal, 2d. The greatest side is always opposite the greatest angle, and the smallest side opposite the smallest angle. 3d. Any two

sides, taken together, are greater than the third 4th. If the three angles are all acute, or all right, or all obtuse, the three sides will be, accordingly, all less than 90°, or greater than 90°. 5th. If from the three angles, A, B, C, of a triangle ABC (fig. 23,) as poles, there be described on the surface of the sphere, three arches of a great circle, DE, DF, FE, forming by their intersections a new spherical triangle DEF; each side of the new triangle will be the supplement of the angle at its pole; and each angle of the same triangle will be the supplement of the side opposite to it in the triangle ABC. 6th. In any triangle A B C (fig. 29,) or A b C, right-angled in A: 1st. The angles at the hypothenuse are always of the same kind as their opposite sides. 2dly. The hypothenuse is greater or lesser than a quadrant, according as the sides, including the right angle, are of the same or different kinds; that is to say, according as the same sides are either both acute, or both obtuse: or, as one is acute, and the other obtuse. And vice versa: 1st. The sides including the right angles are always of the same kind as their opposite angles 2dly. The sides including the right angles will be of the same or different kinds, according as the hypothenuse is less or more than 90°; but one at least of them will be of 90°, if the hypothenuse is so.

Considering it impossible to give a popular idea of this highly important branch of mathematics in any brief form, we must refer those readers, who wish to become proficients therein, to the various excellent treatises published on that subject; particularly those by Simpson, Bonycastle, Payne, &c.

TRIGUERA, in botany, a genus of the Pentandria Monogynia class and order. Natural order of Lurida. Solaneæ, Jussieu. Essential character: corolla bellshaped, with an unequal border; nectary short, five-toothed, surrounding the germ; filaments inserted into the nectary; berry four celled, with two seeds in each cell. There are two species, viz T. ambrosia ca, and T. inodora : these are both annual plants, and natives of Andalusia in Spain.

TRILIX, in botany, a genus of the Polyandria Monogynia class and order. Essential character: calyx three-leaved; corolla three-petalled; berry five-celled, many-seeded. There is only one species, viz. T. lutea, a native of Carthagena, in America.

TRILLION, in arithmetic, a billion of

TRILLIUM, in botany, a genus of the Hexandria Trigynia class and order. Natural order of Sarmentacea. Asparagi, Jussieu. Essential character: calyx threeleaved; corolla three-petalled; berry three-celled. There are three species.

TRIM of a ship, her best posture, proportion of bailast, and hanging of her masts, &c. for sailing. To find the trim of a ship is to find the best way of making her sail swiftly, or how she will sail best. This is done by easing of her masts and shrouds, some ships sailing much better when they are slack, than when they are taught or fast; but this depends much upon experience and judgment, and the several trials and observations which the commander and other officers may make

aboard.

TRIMMERS, in architecture, pieces of timber framed at right angles to the joints, against the ways for chimneys, and wellholes for stairs.

TRINGA, the sand-piper, in natural history, a genus of birds of the order Gralla. Generic character: bill round, straight, slender, and about the length of the head; nostrils small and linear: tongue slender; toes very slightly, if at all, connected at the base by a membrane; hind-toe weak. There are thirty-seven species, of which the following are the principal.

T. pugnax, or the ruff, is twelve inches long. The male is distinguished by a ruff, differing in colour on almost every bird, composed of long feathers, standing out in a peculiar manner, and constituting an appearance somewhat resembling the fashionable neck ruff of the age of Queen Elizabeth. These feathers are not acquired till the second year, and continue only during the season of spring; after which, also, the caruncles, which previously rise on the face of the male, shrink back and disappear. The males of these birds are thought far more numerous than the females. Frequent conflicts between the formerare occasioned from this circumstance; and in the commencement of spring, a male sand-piper is said to take his station near some water, and run round a particular spot such a number of times, that at length he bares a circular path upon the herbage. On the appearance of a female near this spot, the males engage in the most animated and ferocious contests, and, occupied solely by the idea of triumphing over their rivals, they suffer

themselves to be taken by the net of the fowler, who avails himself of these opportunities for their destruction. In England they are migratory, and are found frequently in Lincolnshire and the Isle of Ely, where, after being taken, they are fed for sale, till they at length become nearly a mass of marrowy substance, and are sent to the markets of the metropolis.

T. vanellus, or the lapwing, is thirteen inches long, and of the weight of eight ounces. It remains in England the whole year; lays its eggs on the ground; and the female bird exercises various arts to attract the attention of mischievous and depredating school-boys from the discovery of her nest, and is said, with this view, even to pretend lameness, to direct their pursuit to herself. In winter these birds appear in flocks of several hundreds, and are caught in great numbers, being highly esteemed for food. They live chiefly upon worms, which appear to constitute their delicious banquet, and are sometimes familiarized, and kept in gardens to clear them of slugs and worms, in search for which, both in the morning and evening, they are extremely assiduous.

T. hypoleucos, or the common sand-piper, breeds in England, but soon withdraws after the summer. It is about eight inches long, and is distinguished by its piping note. It is found in France and Siberia.

The T. canutus is about ten inches in length, and weighs four ounces, and frequents the coasts of Lincolnshire, England, where it is taken in considerable numbers, and fattened for the London market. By some these birds are preferred to the ruff.

TRINITY house, a kind of college belonging to a company or corporation of seamen, who, by the King's charter, have power to take cognizance of those persons who destroy sea-marks, and to get reparation of such damages; and to take care of other things belonging to navigation. At present, many in the first rank of society are members of that community. The master, wardens, and assistants of the Trinity House may set up beacons and marks for the sea, in such places, near the coasts or forelands, as to them shall seem meet. By a statute of Queen Elizabeth, no steeple, trees, or other things standing as sea marks, shall be taken away or cut down, upon pain that every person guilty of such offence shall forfeit 1007.; and if the person of fending be not possessed of the value, he shall be deemed convict of outlawry. VOL. XII.

TRINOMIAL, or TRINOMIAL root; in mathematics, is a root consisting of three parts, connected together by the signs + or, as x + y + z, or a +b- e. See BINOMIAL and ROOT.

TRIO, in music, a part of a concert wherein three persons sing; or more properly a musical composition consisting of three parts. Trios are the finest kinds of composition, and these are what please most in concerts.

TRIOPTERIS, in botany, a genus of the Decandria Trigynia class and order. Natural order of Trihilatæ. Malpighia, Jussieu. Essential character: calyx fiveparted, with two honey pores at the base on the outside; petals roundish, clawed filaments cohering at the base; capsules three, one-seeded, three or four-winged. There are eight species.

TRIOSTEUM, in botany, a genus of the Pentandria Monogynia class and order. Natural order of Aggregatæ. Caprifolia, Jussieu. Essential character: calyx length of the corolla; corolla one-petalled, almost equal; berry three-celled, inferior; seeds solitary. There are three species.

TRIPARTITE, something divided into three parts, or made by three parties, as indenture tripartite.

TRIPLE time, in music, a time consisting of three measures in a bar; the two first of which are beat with the hand or foot down, and the third marked by its elevation.

TRIPLARIS, in botany, a genus of the Triandria Trigynia class and order. Natural order of Polygoneæ, Jussieu. Essential character: calyx very large, three or six-parted; corolla three-petalled, or none: nut three-sided, within the ovate base of the calyx. There are two species, viz. T. Americana, and T. ramiflora.

TRIPLICATE ratio, the ratio which cubes bear to one another. This ratio is to be distinguished from triple ratio, and may be thus conceived: in the geometrical proportions 2, 4, 8, 16, 32, as the ratio of the first term (2) is to the third (8) duplicate of that of the first to the second, or of the second to the third, so the ratio of the first to the fourth is said to be triplicate of the ratio of the first to the second, or of that of the second to the third, or of that of the third to the fourth, as being compounded of three equal ratios. See RATIO.

TRIPOLI, in mineralogy, a species of the Clay genus, is of a greyish colour: it occurs massive, is soft and friable, feels meagre, and does not adhere to the tongue. It occurs in veins and beds in R

floëtz rocks, and perhaps in alluvial land. It is found in beds in the coal works of Thuringia: in Derbyshire, it occurs in veins: in Tripoli, whence its name is derived, it also forms veins. It is also found in Russia, Westphalia, Flanders, Hessia, Bohemia, and Switzerland. When reduced to powder, it is employ. ed for polishing metals, marbles, and other minerals, and likewise for polishing glass. Formerly it was supposed to be a volcanic production, which has been long since disproved, and it appears to be an extremely fine mixture of clay and sand.

TRIPPANE, in mineralogy, is of an apple-green, or greenish white. It occurs in mass, is moderately hard, and easily frangible. Specific gravity is 3.21. Before the blow-pipe it becomes yellow, and splits into thin plates, and then melts into a thin transparent glass. It has hitherto been found in Sweden, in veins of quartz and mica.

TRIPPING, in heraldry, denotes the quick motion of all sorts of deer, and of some other creatures, represented with

one foot as it were on a trot.

TRIPSACUM, in botany, a genus of the Monoecia Triandria class and order. Natural order of Gramina, Gramineæ, or Grasses. Essential character: male, calyx glume four-flowered; corolla glume membranaceous: female, calyx glume with perforated sinuses; corolla glume two-valved; styles two; seed one. There are two species, viz. T. dactyloides, and T. hermaphroditum.

TRISECTION, or TRISSECTION, the dividing a thing into three. The term is chiefly used in geometry, for the division of an angle into three equal parts. The trisection of an angle geometrically, is one of those great problems, whose solution has been so much sought by mathematicians for these two thousand years, being, in this respect, on a footing with the quadrature of the circle, and the duplicature of the cube angle.

TRISPAST, in mechanics, a machine with three pullies, or an assemblage of three pullies for raising of great weights. TRITICUM, in botany, wheat, a genus of the Triandria Digynia class and order. Natural order of Gramina, Gramineæ, or Grasses. Essential character: calyx two-valved, solitary, subtriflorous; corolla blunt, with a point. There are nineteen species. T. æstivum, or spring wheat, has four flowers in a calyx, three of which mostly bear grain. The calyces stand pretty distant from each other,

on both sides a flat smooth receptacle. The leaves of the calyx are keel-shaped, smooth, and they terminate with a short arista The glumes of the flowers are smooth and bellying, and the outer leaf of three of the glumes in every calyx is terminated by a long arista, but the three inner ones are beardless. The grain is rather longer and thinner than the common wheat. It is supposed to be a native of some part of Tartary. The farmers call it spring-wheat, because it will come to the sickle with the common wheat, though it should be sown in February or March. T. bybernum, winter or common wheat, has also four flowers in a calyx, three of which are mostly productive. The calyces stand on each side a smooth flat receptacle, as in the former species, but they are not quite so far asunder. The leaves of the calyx are bellying, and so smooth that they appear as if polished, but they have no arista. The glumes of the flowers too are smooth, and the outer ones, near the top of the spike, are often tipped with short arista. The grain is rather plumper than the former, and is the sort most generally sown in England; whence the name of common wheat. T. turgidum, thickspiked or cone wheat, is easily distinguished from either of the former; for though it has four flowers in a calyx, after the manner of them, yet the whole calyx, and the edges of the glumes, are covered with soft hairs. The calyces, too, stand thicker on the receptacle, and make the spike appear more turgid.Some of the outer glumes, near the top of the spike, are terminated by short arista, like those of the common wheat. The grain is shorter, plumper, and more convex on the back than either of the former species. Its varieties are numerous, and have various appellations in different counties, owing to the great affinity of several of them.

TRITOMA, in natural history, a genus of insects of the order Coleoptera. Antennæ clavate, the club perfoliate; lip emarginate; anterior feelers hatchetshaped; shells as long as the body. There are ten species, found in different parts of the world. T. bipustulata is black; shells with a lateral scarlet spot. It inhabits England, and is found on tree fungi. The glabra is found in Sweden, under the bark of trees.

TRITON, in natural history, a genus of the Vermes Mollusca class and order. Body oblong; mouth with an involute spiral proboscis; tentacula twelve, six