of adfeSled Equations . 457 
through in bringing the folution of a problem to an equation, 
inftead of performing the operations themfelves. 
Befides the advantages which this method of procedure 
affords to the mode of refolution now more particularly under 
confideration, it has fo many others over that which is com- 
monly made ufe of, that I am much furprized the latter fhould 
ever have been adopted. By preferving thus the original fub- 
ftitutions diftindt, all .the way through an operation, every 
expreffion, even to the final equation, will exhibit the whole 
procefs up to that ftep ; and it will appear as clearly, how every 
expreffion has been derived, as it does in that mode of analyfis 
which was ufed by the ancient geometricians : whereas, whe$ 
the feveral original expreffions are melted down into one mafs by 
the multitude of a&ual additions, fubtradtions, multiplications, 
and divifions, which they generally undergo, in a long alge- 
braical procefs, conducted in the ufual manner, it is impoffible 
to trace the fmalleft veftige of the original quantities in the 
final equation, except fuch as are reprefented by a fingle letter. 
Of courfe,- however obvious the feveral fteps might be at the 
time when they were taken, every idea of them muft be totally 
loft in the refult ; and it will be utterly impoffible to trace 
them back again, in the manner they are done in the compofi- 
tion of a problem, the folution of which has been inveftigated 
by the geometrical analyfis*. Let me add,, that it is to this 
caufe 
* This fubjeft, if ever I am bleffed with more leifure than is at prefent my 
lot, Ihall bepurfued farther in another paper: in which I fhall endeavour toihew, 
that, notwithftanding the great difference which there appears to be between 
algebra and geometry, they are really but one fcience, differently treated ; and 
that the operations of the former may be rendered as clear and perfpicuous as 
thofe of the latter are allowed to be. A difquifition of this nature will at leail 
have the merit of refcuing a very ufeful and expeditious mode of inveffigation 
from 
