Till 
same parallels, are equal. If a jtefpondi- 
ciilar be let fall upon the base of an ob- 
lique-angled triangle, the difference of the 
squares of the sides is equal to the double- 
rectangle under the base, and the distance 
of the perpendicular froin the middle of 
the base. The side of an equilateral tri- 
angle, inscribed in a circle, is in power 
triple of the radius. The sides of a tri- 
angle are cut proportioiiably, by a line 
■drawn parallel to its base. A whole tri- 
angle is to a triangle cut off by a right line 
drawn parallel to the base, as the rectangle 
tinder the cut sides is to the rectangle of the 
two other sides. In a right angled triangle, 
a line drawn from the right-angle at the 
top, perpendicular to the hypothenuse, di- 
vidfes the triangle into two other rigiit- 
angled triangles, which are similar to the 
first triangle, and to one another, In every 
right-angled triangle, the square of the liy- 
potheuuse is equal to the sum of tlie squares 
of the other two sides ; and, in general, any 
figure described on the liypothenuse, is 
equal to tlie sum of two similar figures 
described upon the two sides. In an isos- 
celes triangle, that is a triangle having two 
of its sides equal, if a line be drawn from 
the vertex to any point in the base ; the 
square of that line together with the rect- 
angle of the segments of the base, is equal 
to the square of the side. If one angle of a 
triangle be equal to 120" ; the square of the 
base will be equal to tiie squares of both 
sides, together with the rectangle of those 
sides ; and if those sides be equal to each 
other, then the square of the base will be 
equal to three times the square of one side, 
e'r equal to twelve times the square of the 
perpendicular from the angle upon the 
base. 
If any angle of a triangle be bisected, 
the bisecting line will divide the opposite 
side in the same proportion as the legs of 
tlie angle are to one another. Eveiy tri- 
angle is one half of a parallelogram of the 
same base and height. The area of any tri- 
angle may be had by adding all the three 
sides together, and taking half the sum, 
and from that half subtracting each side 
severally, and multiplying tliat halt sura 
and the remainder continually into one 
another, and extracting the square root of 
the product. See Trigonometiiy. 
Triangle, in astijouomy, one of the 
forty-eight ancient constellations, situated 
in the northern hemisphere. There is also 
a soutbern triangle, in the other hemis- 
pherPi According to the British cfitalogue. 
Till 
there arc sixteen stars in the northern; and 
in Sharp’s catalogue there are five in the 
southern triangle. 
TRIANGULAR compasses, are such as 
have three legs, or feet, whereby to take off 
any triangle at once ; much used in the 
construction of maps, globes, &c. 
Triangular numbers, are a kind of po- 
lygonal numbers, being the sums of aiitli- 
metical progressions, the difference of whose 
terms is 1. 
Thus from the arithmetical numbers 1, 2, 
3, 4, 5, 6, are formed the triangular num- 
bers 1, 3, 6, 10, 15, 21. The sum of any 
number n of the terms of the said triangular 
, . n n-Ll n -4- 2 . „ . 
numbers is = - x — I — X — s — ; it » be 
12 o 
5, the sum will be 35, which is also equal to 
the sum of the number of shot in a triangu- 
lar pile of balls, the number of row's, or the 
number in each side of the base being n. 
The sum of the reciprocals of tlie tdaii- 
gular series infinitely continued is equal to 
2 = 1+3-1-H-il. &c. 
TrianguliVH canon, the tables of artifi- 
cial sines, tangents, secants, &c. 
Triangular quadrant, is a sector fur- 
nished with a loose piece, whereby to make 
it an equilateral triangle. 
The calendar is graduated thereon, with 
the sun’s place, declination, and other use- 
ful lines ; and' by the help of a string and a 
plummet, and the divisions graduated on 
the loose piece, it may be made to serve 
for a quadrant. 
TRIANTHEMA, in botany, a genus of 
the Decandria Digynia class and order. 
Natural order of Succuleutae. Portiilace®, 
Jussieu. Essential character : calyx mu- 
donate below the tip ; corolla none ; sta- 
mina five or ten ; germ retuse ; capsule 
cut round. Tliere are seven species. 
TRIBOME'rER,a terra applied by Mns- 
chenbroek and others to an instrument in- 
vented for measuring the friction of me- 
tals. It con.sists of an axis formed of 
hard steel, passing through a cylindrical 
piece of w'ood ; the ends of the axis, which 
aie highly polished, are made to rest on the 
polished semicircular cheeks of various me- 
tals, and the degree of friction is estimated 
by means of a weight suspended by a fine 
silken string or ribband over the wooden cy- 
linder. 
TRlBUr.US, in botany, caltrops, a genus 
of tlie Decandria Monogynia class and or- 
der. Natural order of Gruinales. Rutaceie, 
JiLssieu. Essential character; calyx five- 
parted ; petals five, spreading ; style none , 
