Mr. W.G. Horner on Congeneric Surd Equations. 45 
Lindley Murray, we admit the “the” when the same quantity 
appears a second time under the same radical; and that 
without a greater latitude still, we should never be able to 
prove that + YW w—ax—W x+a made —¥V x*—a’*, and so 
on. So that the professed ambiguity is subject, after all, to a 
conventional permanence. 
The source of the whole mystery, in my judgement, is to 
be found in the almost unavoidable imperfection of the man- 
ner in which we are taught to transform equations when we 
are at school. The operations consequent on /ransposition 
are correct as far as their principles are resolvable into Eu- 
clidian axioms. Beyond that they are liable to fallacy; and, 
generally speaking, we are infallible in our judgement only 
as long as every term is on one side. We may then deter- 
mine satisfactorily i in what cases zero is admissible as the ag- 
gregate value. An instance of the hazard attending the neg- 
lect of this principle is given in my paper in the ‘Lond. and 
Edinb. Phil. Mag. for September 1834, (vol.v.) p. 189. In the 
management of surds, instances might easily be accumulated. 
And whence this hazard? and why consequent upon trans- 
position? Because, from the nature of analysis, we are con- 
tinually arguing from the direct to the converse. An equa- 
tion is formed hypothetically. We trace out certain direct con- 
sequences, in the form of equations also, and so on; until an 
equation is obtained, such that if the first be true, the last is 
therefore true. But the converse is that which we wished to 
ascertain. Is the hypothetical equation true, because the re- 
sulting equation is so? ‘To determine this, a similar query 
must be instituted from link to link throughout the chain of 
reasoning. Is each equation in succession true, because the 
next in succession is so? If each of these subordinate in- 
quiries admits of a decided affirmative, the reply to the ge- 
neral query is satisfactory; otherwise, it isnot. Now in the 
management of equations, we have been taught, either vir- 
tually or in direct terms, to rely upon certain “axioms, which 
for the present purpose will be most effectually stated in 
pairs, viz. 
If equals be added to equals, the wholes are equals) 
and, 
If equals be subtracted from equals, the remainders 7 I. 
are equal. 
If equals be multiplied by equals, the products are) 
equal; and iy 
If equals be divided by equals,the quotients vi ’ 
equal. 
