812 
T R. A 
TEA 
T It A 
arabic, but in a greater degree. It greatly 
inspissates and ubtunds the acrimony of the 
humours, and is therefore found of service in 
inveterate coughs, and other disorders of the 
breast, arising from an acrid phlegm, and in 
stranguries, heat of urine, and all other com- 
plaints of that kind. It is usually given in 
the compound powder, called the species clia- 
tragacanthi frigidx, rarely alone. It is aho, 
by some, esteemed a very great external re- 
medy for wounds, and in this sense made an 
ingredient in some sympathetic powders, with 
vitriol and other things. It is by some re- 
commended alone, in form of a powder or 
strong mucilage, for cracks and chaps in the 
nipples of women: but it is found, by expe- 
rience, to be a very troublesome application 
in those cases, and to do more harm than good, 
as it dries by the heat of the part, and draws 
the lips of the wound farther asunder than be- 
fore. See Gums. 
TRAGEDY. See Poetry. 
TRAGI A, a genus of the monoecia trian- 
dria class of plants, without any dower-petals ; 
its fruit is a very large tricoccous capsule of a 
roundish figure, containing single and round- 
ish seeds. There are 8 species. 
TRAGOPOGON,goaiVZ>cartZ, a'genus of 
plants of the class syngenesia, and the or- 
der polvgamia acqualis ; and in the natural 
system ranging under the 49th order, com- 
posite. The receptacle is naked, the calyx 
simple, and the pappus plumose. There are 
14 species; of which two are British, the 
pratensis and porifolius. 1. The pratensis, 
or yellow goats’-beard, has its calyxes equal 
witli the florets, and its leaves entire, long, 
narrow, sessile, and grassy. In fair weather 
this plant opens at sun- rising, and shuts be- 
tween nine and ten in the morning. The roots 
are conical and esculent, and are sometimes 
boiled and served up at table like asparagus. 
It grows on meadows. 2. The porifolius, 
or purple goat’s-beard, has the calyx longer 
than the radius of the floret ; the flowers are 
large, purple, single, and terminal ; and the 
leaves long, pointed, and bluish. The root is 
long, thick, and esculent. It grows in mea- 
dows, and is cultivated in gardens under the 
name of salsafy. 
TRAJECTORY, a term often used, ge- 
nerally for the path of any body moving 
either in a void, or in a medium that resists 
its motion; or even for any curve passing 
through a given number of points. Thus 
Newton, IYuicip. lib. 1. prob. 22, purposes to 
describe a trajectory that shall passthrough 
five given points. 
Trajectory of a comet, is its path or or- 
bit, or the line it describes in its motion. This 
path, Hevelius, in his Cometographia, will 
have to be very nearly a right line; but Dr. 
Halley concludes it to be, as it really is, a 
veyy eccentric ellipsis ; though its place may 
often be well computed on the supposi- 
tion of its being a parabola. Newton, in 
prop. 4l of his 3d book, shews how to deter- 
mine the trajectory of a comet from three ob- 
servations ; and in his last prop, how to cor- 
rect a trajectory graphically described. 
TRAMMELS, in mechanics, an instru- 
ment used by artilicers for drawing ovals upon 
boards, Ac. One part of it consists of a cross 
with (wo grooves at right angles ; the other 
is a beam carrying two pins which slide in those 
grooves, and also the describing pencil. All 
fiie engines for turning ovals are constructed 
on the same principles with the trammels: 
the only difference is, that in the trammels 
the board is at rest, and the pencil moves 
upon it ; in the turning engine, the tool, which 
supplies the place ol the pencil, is at rest, and 
the board moves against it. See a demon- 
stration of the chief properties of these instru- 
llam, in the Philos. Trans. 
ments by Mr. Ludlam 
vol. 70, p. 378, Ac. 
Trammel-net, is a long net, where- 
with to take fowl by night in chant pain coun- 
tries, much like the net used for the low 
bell, both in shape, bigness, and mashes. To 
use it, they spread it on the'groimd, so that 
the nether or further end, lilted with small 
plummets, may lie loose thereon ; then the 
other part being borne up by men placed at 
the fore ends, it is thus trailed along the ground. 
At each side are carried great blazing lights, 
by which the birds are raised, and as they rise 
under the net they are taken. 
TRANSCENDENTAL, or Transcen- 
dant, something elevated or raised above 
other things, which passes and transcends the 
nature ot other inferior things. 
Transcendental quantities, among geome- 
tricians, are indeterminate ones, or such as 
cannot be fixed, or expressed by any constant 
equation; such are all transcendental curves 
which cannot be defined by -any algebraic 
equation, or which when expressed by an 
equation, one of the terms thereof is a varia- 
ble quantity. Now whereas algebraists use to 
assume some general letters or numbers, for 
the quantity sought in these transcendental 
problems, Mr. Leibnitz assumes general or 
indefinite equations for the lines sought; 
e.gr. putting „t and ?/ for the absciss and or- 
dinate, the equation he uses fora line sought 
is a±bx+c;/+erif\-fxx-\-g>/y, Ac.=0, by 
the help of which indefinite equation, he seeks 
the tangent : and by comparing the result 
with the given property of tangents, he finds 
the value of the assumed letters, a, b, c, d, 
Ac. and thus defines the equation of the line 
sought. 
If the comparison above-mentioned does not 
proceed, he pronounces the line sought not 
to be an algebraical, but a transcendental one. 
This supposed, he goes on to find the species 
of transcendency; for some transcendentals 
depend on the general division or section of a 
ratio, or upon the logarithms; others, upon 
the arcs of a circle; and others, on more in- 
definite and compound enquiries. He there- 
fore, besides the symbols x and ?/, assumes a 
third, as v, which denotes the transcendental 
quantity ; and of these three forms, a general 
equation for the line sought, from which he 
finds the tangent, according to the dif- 
ferential method, which succeeds even in 
transcendental quantities. The result he 
compares with the given properties of the tan- 
gent, and so discovers, not only the value of 
a, b, c, d, Ac. but also the particular nature 
of the transcendental quantity. And though 
it may sometimes happen, that the several 
transcendentals are so to be made use of, and 
those of different natures too, one from one 
another; also jthough there are transcendents 
of transcendentals, and a progression of these 
in infinitum ; yet we may be satisfied with the 
most easy and useful one ; and for the most 
part, may have recourse to some peculiar ar- 
tifices for shortening the calculus, and re- 
ducing the problem to as simple terms as may 
be. 
This method being applied to (be business of 
quadratures, or to the invention of quadratics, 
in which the property of the tangent is always 
given, it is manifest, not only how it may be 
discovered, whether the indefinite quadra- 
ture may be algebraically impossible; but 
also, how, when this impossibility is discovered, 
a transcendental quadratrix may be found, 
which is a thing not before shewn. So that 
it seems that geometry, by this method, is 
carried infinitely beyond the bounds to which 
Vieta and Des Cartes brought it ; since, by 
this means, a certain and general analysis is 
established, which extends to all problems of 
no certain degree, and consequently not 
comprehended within algebraical equations. 
Again, in order to manage transcendental 
problems, wherever the business of tangents 
or quadratures occurs, by a calculus, there is 
hardly any that can be imagined shorter, 
more advantageous, or more universal, than 
the differential calculus, or analysis of indi- 
visibles and infinites. 
By this method, we may explain the nature of 
transcendental lines, by an equation; e. gr. Let 
a be the arch of a circle, and x the versed sine ; 
then will a ~ . — S — — - 5 and, if the ordinate 
Vox - 
of the cycloid is y , 
s dx 
XX 
* A X 
then will 
\/2a 
y — \/ 2x — 
which equation perfectly 
expresses the relation between the ordinate y 
and the absciss x, and from it all the proper- 
ties of the cycloid may he demonstrated. 
Thus is the analytical calculus extended to 
those lines, which have hitherto been excluded ; 
for no other reason, but that they were thought 
incapable of it. 
TRANSFORMATION of Equations, in alge- 
bra, is the changing equations into others of a 
different form, but of equal value. This opera- 
tion is often necessary, to prepare equations for 
a more easy solution, some of the principal cases 
of which are as follow. 1. The signs of the roots 
of an equation are changed, viz. the positive 
roots into negative, and the negative roots into 
positive ones, by only changing the signs of the 
2d, 4th, and all the other even terms of the 
equation. Thus the roots of the equation 
x* — x 3 — 19.v 2 -{- 49,v — 30 = 0, are j 
4-1, +2, 4-3, -5; 
whereas the roots of the same equation having 
only the signs of the 2d and 4th terms changed, 
viz. of x'' -|- -v 5 — 19* 2 — 49 v — 30 = O, arei 
— 1, — 2, — 3, 4~ A 
2. To transform an equation into another that 
shall have its roots greater or less than the roots 
of the proposed equation by some given differ- 
ence, proceed as follows : Let the proposed 
equation be the cubic x 3 — ax 2 -)- bx — c = 0s 
and let it be required to transform it into ano- : 
ther, whose roots shall be less than the roots 
this equation by some given difference d ; if the 
root y of the new equation must be the less, take 
it y == x — d, and hence x — y d\ then, in- 
stead of v and its powers, substitute y -j- d and- 
its powers, and there will arise tljijrnevv equation. 
(A) / -j- 3<ly 2 4- 3 dy 4- d ' 
— , ay 2 — 2ady — ad 2 j 
4- by 4~ t/d | 
0 ; 
whose roots are less than the roots of the former 
equation by the difference d. If the roots of the 
new equation had been required to be greater 
than those of the old one, we must then, have 
substituted y — x -j- d, or x ' —y — d, &c. 
3. To take away the 2d or any other particu- 
lar term out of an equation 3 or to transform an 
