T R I 
100 T R I 
bc — tan a , ce — oe x tan a . Hence, cos b — 
o e X tan c . tan c _ a cos B> 
oe X tan a 
As explained in last Article, had we begun the construc¬ 
tion in the figure from b instead of c, we should have ob¬ 
tained an exactly similar formula to the above, the angle c 
taking the place of b, and the side b that of c ; that is to say, 
without further demonstration, tan b — tan a x cos c. q.e.d. 
(11.) In the last figure, retaining the same notation, we 
shall have 
cos a — cos b X cos c. 
Dem. Cos a — cos (of arch) bc — cos boc — — . But 
OE 
OD 
— cos doe — cos (of arch) ba ~ cos c , .\ oe zz od 
X cos c; also, — cos doc — cos (of arch) ac — cos b , 
OD rr OD x cos c , 
.-. oc — , . Hence, cos a zz-—-zz cos ox 
cos b ' 
cos C. Q. E. D. 
OD 
cos b 
The application of trigonometry to practical purposes, has 
already been sufficiently shewn in the articles Astronomy, 
Navigation, Optics, &c„ to which we refer our readers. 
TRIGONUM, an ancient musical instrument. One of 
this kind was taken from an ancient painting in the museum 
of the king of Naples, in which it is placed on the shoulder 
of a little dancing Cupid, who supports the instrument with 
his left hand, and plays upon it with his right. The trigo- 
num is mentioned hy Athenasus, lib. iv. and by Julius Pollux, 
lib. iv. cap. 9. 
TRIGUERA [so named by Cavanilles, from a Spanish 
botanist], in Botany, a genus of the class pentandria, order 
monogynia, natural order of luridse, soloneae (Juss.J —Ge¬ 
neric Character. Calyx: perianth one-leafed, half-five-cleft, 
permanent; segments unequal, acute. Corolla one-petal- 
led, bell-shaped. Tube very short. Border ventricose, plaited, 
five-cleft, longer than the calyx 3 the two upper segments 
reflexed. Nectary membranaceous, short, five-toothed, sur¬ 
rounding the germ. Stamina: filaments five, very short, 
inserted outwardly into the teeth of the nectary. Anthers 
sagittate, converging. Pistil: germ roundish, two-grooved. 
Style filiform, straight, a little longer than the stamens. 
Stigma obtuse. Pericarp: a dry berry, subglobular, grooved, 
four-celled. Seeds two in each cell, roundish, compressed, 
rugged; one above the other.— Essential Character. Co¬ 
rolla bell-shaped, 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. 
1. Triguera ambrosiaca.—Stem grooved and winged; up¬ 
per leaves obovate, toothed, pubescent. Stem angular; some¬ 
what winged. Radical leaves obovate, quite entire; stem- 
leaves toothed, hairy. Peduncles axillary, in pairs. Corolla 
resembling that of hyoscyamus or henbane, of a violet 
colour. 
2. Triguera inodora.—Leaves ovate-lanceolate, quite en¬ 
tire, smooth. This is scarcely winged, the leaves little or not 
at all running down the stalk ; they are also smooth, and 
not toothed.—Both are annual plants, natives of Andalusia, 
in Spain. 
TRILATERAL, adj. [tres and latus, Lat.] Having 
three sides. 
TRILATERAL, adj. \_tres and litera, Lat.] Consisting 
of three letters.—This name is triliteral: —it consists of three 
letters, thau, beth, and he; all which are here symbolical. 
Biblioth. Bibl. 
TRILIX [a tissue of three threads of different colours], in 
Botany, a genus of the class polyandria, order monogynia. 
-—Generic Character. Calyx: perianth three-leaved ; leaf¬ 
lets ovate, acute, spreading, flat, permanent. Corolla: pe¬ 
tals three, lanceolate, acute, less than the calyx. Stamina: 
filaments numerous, capillary, length of the corolla. An¬ 
thers roundish, twin, minute. Pistil: germ five-cornered. 
Style cylindrical. Stigma simple. Pericarp: berry subpen¬ 
tagonal, five-celled, covered with the calyx. Seeds numer¬ 
ous, roundish, minute.— Essential Character. Calyx three¬ 
leaved. Corolla three-petalled. Berry five-celled, many- 
seeded. 
Trilix lutea.—This is a shrub ten or twelve feet high, and 
very much branched. Branches round and somewhat rugged. 
Leaves alternate, petioled, subpeltate, cordate-ovate, serrate, 
acuminate, veined, pubescent. Petioles round, smooth. 
Flowers yellow (not from the corolla but the anthers). Pe¬ 
duncles mostly terminating. Pedicels alternate, one-flow¬ 
ered, round, pubescent.—Native of Carthagena, in America. 
TRILL, s. [ tri/lo , Ital.; drilla, Su. Goth., to utter 
quavering ; tralla, to singh] Quaver; tremulousness of 
music.—I have often pitied in a winter night a vocal musi¬ 
cian, and have attributed many of his trills and quavers to 
the coldness of the weather. Tatler. 
To TRILL, v. a. [ drilla , tralla, Su. Goth. See the 
noun.] To utter quavering. 
Through the soft silence of the listening night 
The sober-suited songstress trills her lay. Thomson. 
To shake. Obsolete. 
What hast thou to do, and I lose my cote ? 
I will trill the bones while I have one grote. Old Morality. 
To TRILL, v. n. [ tri/la , Swed. See To Trickle.] 
To trickle; to fall in drops or slender streams. 
Aye, she took ’em; read em’ in my presence; 
And now and then an ample tear trill'd down 
Her delicate cheek. Shakspeare. 
To play in tremulous vibrations of sound. 
Am I call’d upon the grave debate. 
To judge of trilling notes and tripping feet. Dryden. 
TRI'LLION, s. [A word invented by Locke.] A million 
of millions of millions; a million twice multiplied by a 
million. 
TRILLIUM, in Botany, a genus of the class hexandria, 
order trigynia, natural order of sarmentacese, asparagi 
(Juss.) —Generic Character. Calyx: perianth three-leaved, 
spreading; leaflets ovate, permanent. Corolla: petals three, 
subovate, a little bigger than the calyx. Stamina : filaments 
six, awl-shaped, shorter than the calyx, erect. Anthers ter¬ 
minating, oblong, length of the filaments. Pistil: germ 
roundish. Styles filiform, recurved. Stigmas simple. Peri¬ 
carp: berry roundish, three-celled. Seeds many, roundish. 
— Essential Character. Calyx three-leaved. Corolla three- 
petalled. Berry three-celled. 
1. Trillium cemuum, or drooping trillium.—-Flowers pe- 
duncled, drooping. Root perennial, tuberous. Stem erect, 
a foot high, smooth. Leaves three together, terminating. 
Flower solitary, among the leaves, without bractes.—It grows 
naturally in the woods, in many parts of North America. 
2. Trillium erectum, or upright trillium.—Flower pe- 
duncled, erect. This has a taller stalk. The three leaves are 
placed at a distance from the flower, which stands upon a 
long foot-stalk, and is erect; the petals are purple, larger, 
and end with sharper points.—Native of Virginia, Canada, 
and other parts of North America. 
3. Trillium sessile, or sessile-flowered trillium.—Flower 
sessile, erect. Stalk purple. The three leaves grow at the 
top like the first; but they are much longer, and end in 
acute points. The leaves are mottled.—It grows in shady 
thickets in Carolina aud Virginia. 
Propagation and Culture. —These plants are propagated 
by seeds, which should be sown upon a shady border soon 
after they are ripe, afid then the young plants will come up 
the next spring; but if the seeds are sown in the spring, 
they will remain in the ground a year. When the plants 
come up, keep them clean, and in autumn, after their leaves 
decay, transplant the roots to a moist shady place, where 
they are to remain. 
TRILLO, a small town of the interior of Spain, in the pro¬ 
vince 
