7%e Method of FLUXIONS,
302
midable enough, by reducing
all
difficulties
its
to
the
two former
Cafes.
The
Author's way of demonstrating the Inverfe Method of Fluxions Short, but fatisfactory enough. have argued elfewhere, that from the Fluents given to find the Fluxions, is a direct and fynthetical Operation ; and on the contrary, from the Fluxions And in the given to find the Fluents, is indirect and analytical. order of nature Synthefis mould always precede Analyfis, or Commould before Refolution. But the Terms Synthefis and go pofidon often ufed are a in Analyfis vague fenfe, and taken only relatively, as in this For the direct Method of Fluxions place. being already demonftrated fynthetically, the Author declines (for the reafons he gives) to demonstrate the Inverfe Method alfo, that is, 56, 57.
We
is
fynthetically He contents primarily, and independently of the direct Method. himfelf to prove it analytically, that is, the direct Meby fuppofing
thod, as fufficiently demonstrated already, and Shewing the neceSTary connexion between this and the inverSe Method. And this will always be a full proof of the truth of the conclufions, as Multiplication is a good proof of Division. Thus in the firlt Example we if that the is x1 found, given Equation y -f- xy I,
we
x y=x the truth of which
have the Root
Shall
To
y=^x
1
-j-
fx
-*
3
-f-
6
^.x*
-^r*
,
conclufion, we may hence find, 1 direct i the 2x .i* 3 -f-f#* Method, -{-x _y by T T X', &c. and then fubStitute theSe two Series in the given Equation, as follows; cc.
prove
=
_
_{_
jf 4- ]_ X y -------f_ X Xy ---------1_ X - __ #3 r -f. 2 X A-* -{y 1
.
--
x
_
.
X< 3.4
, _J_ _>_ X
__
^ ^+ _j_
^f -rX'
__ _^ X 6 6 _{_ _?_ X 5
^X
6
1
Now by collecting thefe Series, we mall find the refult to produce the given Equation, and therefore the preceding Operation will be fufticiently proved. 58. In this and the fubfequent paragraphs, our Author comes to open and explain fome of the chief My Steries of Fluxions and Fluents, and to give us a Key for the clearer apprehenfion of their nature and properties. Therefore for the Learners better instruction, I Shall not think much to inquire fomething more into this circumstantially
order to which let us conceive any number of right ae &c. Lines, AE, t as, indefinitely extended both ways, along which a Body, or a defcribing Point, may be fuppofed to move in each matter.
In
Line,