TRA 
discovers, not only the values of a, b, c, d, 
&c. 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 
another ; also, though there be transcen- 
dents of transcendentals, and a progression 
of these in wfinitum ; yet vre may be satis- 
fied with the most easy and useful one ; and 
for tlie most part, may have recourse to 
some peculiar artifices for shortening the 
calculus, and reducing the problem to as 
simple terms as may be. 
This method being applied to the busi- 
ness of quadratures, or to the invention of 
quadratices, in which the property of the 
tangent is always given, it is manifest, not 
only how it may be discovered, whether 
the indefinite quadrature may be algebrai- 
cally impossible ; but also^ how, when tliis 
impossibility is discovered, a transcenden- 
tal quadralrix may be found, which is a thing 
not before shown. So that it seems, that 
geometry, by this method, is carried infi- 
nitely beyond the bounds to which Vieta 
and Des Cartes brought it ; since, by this 
means, a certain and general analysis is es- 
tablished, which extends to all problems of 
no certain degree, and consequently not 
comprehended within algebraical equa- 
tions. 
Again, in order to manage transcenden- 
tal problems, wherever the business of tan- 
gents or quadratures occurs, by a calculus, 
there is hardly any that can be imagined 
shorter, more advantageous, or more uni- 
versal, than the differential calculus, or ana- 
lysis' of indivisibles and infinites. By this 
method we may explain the nature of trans- 
cendental lines, by an equation ; e. gr. let a 
be the arch of a circle, and x the versed 
Sdx 
sine; thenwdla=^|== ; and if 
the ordinate of the cycloid be y, then will 
y = ^ 2x - XX -f ; which 
equation perfectly expresses the relation 
between the ordinate, y, and the absciss, x, 
and from it all the properties of the cycloid 
may be demonstrated. 
Thus is the analytical calculus extended 
to those lines which have hitherto been ex- 
cluded ; for no other reason, but that they 
were thought incapable of it. 
TRANSCRIPT, a copy of any original 
writing, particularly that of an act or in- 
strument, inserted in the body of another. 
TRA 
TRANSFER, in commerce, &c. an act 
whereby a person surrenders his right, iii- 
terest, or property, in any thing moveable 
or immoveable, to another. The term, 
transfer, is chiefly used for the assigning 
and making over shares in the stocks, or 
public funds, to such as purchase them of 
the proprietors. 
TRANSFORMATION, in general, de- 
notes a change of form, or the assuming a 
new form different from a former one. Tlie 
chemists were a long time seeking the trans- 
formation of metals ; that is, their transmu- 
tation, or the manner of changing them into 
gold. See Transmutation. 
Transformation of equations. The 
doctrine of the transformation of equations, 
and of exterminating their intermediate 
terms, is thus taught by Mr. Mac Laurin. 
The aflirmative roots of an equation are 
changed into negative roots of tlie same va- 
lue, and the negative roots into affirmative, 
by only changing the signs of the terms al- 
ternately, beginning with the second. Thus, 
tlie roots of the equation, x'' — x’ — 19 x^ 
+ 49x — 30=0, are-j-l,-j- 2 , ^ 
whereas the roots of the same equation, 
having only the signs of the second and 
fourth terms changed, viz. x'' + X* — I9x^ 
— 49 X — 30 = 0, are — 1, — 2, _ 3, _j_ 5 . 
To understand the reason of this rule, let 
us assume an equation, as x~a X x b 
Xa: c X ^ d X oc — e, &c. = 0, whose 
roots are + a, -f 6, + c, + d, -j- e, &c. ; 
and another, having its roots of the same 
value, but affected with contrary signs, as 
X -I- a Xx + C X x-H-f X x-J-d X x^^ 
&c. = 0. It is plain, that the terms taken 
alternately, beginning from the fiist, are 
the same in both equations, and have the 
same sign, being products of an even num- 
ber of the roots ; the product of any two 
roots having the same sign as their product 
when both their signs are changed ; as -4-a 
X — bz=. — a x -1-6. 
But the second terms, and all taken al- 
ternately from them, because their coeffi- 
cients involve always the products of an 
odd number of the roots, will have contrary 
signs in the two equations. For example : 
the product of four, m. aft cd, having the 
same sign in both, and one equation in the 
fifth term having abedx+e, and the 
other abedx — c, it follows, that their 
product, aft ede, must have contrary signs 
in tlie two equations ; these two equations, 
therefore, that have , the same roots, but 
with contrary signs, have nothing different 
