TRI 
base, is to the tangent of half their diffe- 
rence, so is the tangent of half the vertical 
angle to the tangent of the angle which the 
perpendicular CD makes with the line CF, 
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 equi- 
lateral, isocelar, or scalene, according as it 
has its three angles all equal, or two ot 
them equal, or all three unequal. 2d. The 
greatest side is always opposite the greatest 
angle, and the smallest side opposite the 
smallest angle. Sd. 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, ac- 
cordingly, all less than 90”, or 90“, or greater 
than 900. 5th. If from the three angles 
A, B, C, of a triangle ABC (fig. 28); as 
polgs, there be described on the surface of 
the sphere, three arches of a great circle 
DE, DF, FE, forming by their intersec- 
tions a new spherical triangle DEF; each 
side of the new triangle will be the supple- 
ment of the angle at its pole; and each 
angle of the same triangle will be the sup- 
plement of the side opposite to it in the 
triangle ABC. 6th. In any triangle ABC 
(fig. 29), or A 6 C, right-angled in A : 1st, 
The angles at the hypothenuse are always 
of the same kind as their opposite sides, 
gdly. The hypothenuse is greater or lesser 
than a quadrant, according as the sides, in- 
cluding the right angle, are of the same, or 
different kinds ; that is to say, aecording 
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, 
gdly. The sides, including the right angles, 
will be of the same, or different kinds, ac- 
cording 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 po- 
pular 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; par- 
ticularly those by Simpson, Bonycastle, 
Payne, &c. 
TRIGUERA, in botany, a genus of the 
Pentandria Monogynia class and order. 
Natural order of Lurid®. Solane®, Jus- 
sieu. Essential character: corolla bell- 
TRl 
shaped, with an unequal border; nectary 
short, five-toothed, surrounding the germ; 
filaments inserted into llie nectary ; bf-rry 
four-celled, with two seeds in each cell. 
There are two species ; viz. T. ambrosiaca, 
and T. inodora: these arc borh annual 
plants, and natives of Andalusia, in Spain. 
TRILIX, in botany, a genus of the Po- 
lyandria Monogynia class and order. ES“ 
sential character: calyx three-leaved; co-. 
rolla three petalled ; berry fi ve-celled, many- 
seeded. There is only one species ; viz. T. 
lutea, a native of Carthagena, in America, 
TRILLION, in arithmetic, a bill’ on of 
billions. See Arithmetic, Numeration. 
TRILLIUM, in botany, a genus of the 
Hexandria Trigynia class and order. Na- 
tural order of Sarmentace®. Asparagi, 
Jussieu. Essential character: calyx three- 
leaved ; corolla three petalled ; berry three- 
celled. There are three species. 
TRIM of a ship, her best posture, pro- 
portion of ballast, and hanging of her masts, 
&e. 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 w’elb 
holes for stairs. 
TRJNGA, the sand-piper, in natural his- , 
tory, a genus of birds of the order Grail®. 
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 thb 
following are the principal. 
T. pugnax, or the ruff, is twelve inche.s 
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 ap- 
pearance somewhat resembling the fashion- 
able neck-ruff of the age of Queen Eliza- 
beth. These feathers are not acquired till 
the second year, and continue only during 
the season of spring ; after which, also, the 
caruncles whicli previously rise on the face 
of the male shrink back and disappear. The 
males of these birds are thought far more 
