37o Mr. Woo'd 071 the 
of considerable importance, and its truth ought to be established 
on surer grounds. The various transformations of equations, 
the dimensions to which they rise in their reduction, and the 
circumstances which attend their actual solution, are most 
easily explained, and most clearly understood, by the help of 
this principle. Mr. Euler appears to have been the first wri- 
ter who undertook to give a general proof of the proposition ; 
but, whatever may be thought of his reasoning in other re- 
spects, as he carries it no further than to an equation of four 
dimensions, and it does not appear capable of being easily ap- 
plied in other cases, it gives us no insight into the subject. Dr. 
Waring’s observations upon the proposition are extremely 
concise ; * and, to common readers, it will still be a matter of 
doubt, whether a quantity of any description whatever will, when 
substituted for x in the expression x 8 — fx 1 + qx 6 — . . . . + w * 
cause the whole to vanish. 
In the investigation of the proof here offered, it became ne- 
cessary to attend to the method of finding the common measure 
of two algebraical expressions ; and to observe particularly, in 
what manner new values of the indeterminate quantities are 
introduced; and how they may again be rejected. It ap- 
pears, that these values are necessary in the division; and, 
when they have been thus introduced, they enter every term 
of the second remainder, from which they may be discarded. 
This circumstance enables us, not only to determine the na- 
ture of the roots of every equation, but also affords us a direct 
and easy method of reducing any number of equations to one, 
and obtaining the final equation in its lowest terms. 
• Meditationes Alg> p. 272. 
