CARNOT’s GEOMETRIE DE POSITION. 
and subtracting both sides from mo 
a— Pant x= b. ; 
Then taking the roots of both sides, (since 
the root.on one side may be either 
a@—nx,ore —4;) 
a=—x= Vb 
or rc —az= J b5 ae 
That is, x =a— /%, orxn=at Vb 
If a* is equal to 4, then 
ee s='9 a'x® 
and consequently x = 2a. 
If a is less than J, 
then x? —2ax=b—a’; 
therefore r =a + /b—<a’. 
Thus the various values of x are ascer- 
tained without any interference of nega- 
tive quantities: the reasoning is clear 
and just, and the geometrician takes the 
solution which suits the conditions of his 
of 
Oar author not having seen these evi- 
dent truths, is obliged, after a long de- 
duction, to lay down these tedious con- 
clusions. 
‘In order that the solution. ofa problem given 
by apariicular root of an equation should be 
effective, the conditions proposed, the supposi- 
tions on which the reasoning is established, 
and the constructions or operations indicated 
by the root of this equation, should all be 
consistent with each other, 
‘© That if there exists any incompatibility 
between the one and the other, this root can 
be considered only as the simple indication 
of another question analogous to the former ; 
and to have its true meaning, we must renew 
the calculation upon other conditions or other 
ssuppositions, tll consistency is established 
between them, or with the operation which 
the root modified on these changes would in~ 
dicate. 
© That consequently negative and imagi- 
nary roots are never true solutions of the pro- 
posed question, but simple indications. of 
questions differing more or less fro 2 the for- 
mer; that often they are only algebraical 
forms of no signification, which algebraical 
transformations have amalgamated with the 
real roots. 
« That real and positive roots do not any 
more than negative or imaginary roots express 
sactual solutions, and are only, like them, 
simple indications of analogous questions, 
when the constructions or operations to 
which they conduct us are not entirely con- 
sistent with the conditions proposed, and the 
suppositions on which the reasoning was 
established, . 
“That any equation, orroot of an equation, 
eannot give an actual sclution whilst it con- 
tains absurd quantities or operations incapable 
ef execution, unless they destroy each, other 
respectively, as in the gase of real roots of the. 
third degree, 
828 
«« That however these unintelligible forms 
ought not to be neglected, and that they may 
be employed like real forms, because they 
may be made to disappear by simple-alges 
braical transformations; and there will then 
remain only explicit forms immediately ap- 
plicable to the object proposed, provided, as 
it has been before observed, that these forms, 
conditions proposed, and suppositions on 
which the reasoning is grounded, are all cons 
sistéat with each other.” 
We have now then to apply our au« 
thor’s notions to the geometrical posi 
tion. An equation conducted in the 
common mode leads him to certain false 
conclusions, from which he discovers 
some new relations of the proposed un- 
known quantity to some new conditions. 
This may be obviated by his direct and 
inverse quantities, by which from quana 
tities given in a certain positiow he will 
discover the relation of each to the other 
in another position, Thus to take an 
easy instance, for the want of figures and 
of room will not permit us to introduce 
the more complicated, we will suppose 
the value of the square of the side of a 
triangle, required in terms of the squares 
of the other two. sides. In this case, sup~ 
posing the square of the side required to 
be opposite to the obtuse angle, and its 
value discovered upon that supposition 5 
then to make it-answer for the case of an 
acute angle, to which it is opposite, we 
must invert certain quantities. Instances 
are given, and very ingenious ones, of 
this inversion in a variety of cases. The 
study of them will.be found very useful 
to the higher mathematician, and parti- 
cularly to those entangled with tae doc- 
trine of negative quantities; but we must 
retain our doubts whether every question 
would not be more. easily solved by the 
algebraical formula once. established, 
being accommodated to every supposi- 
tion, asin the case we have adduced, and 
then leaving the geometrician to adapt 
his cases to the solutions afforded him. 
The work cannot fail of creating con+ 
siderable interest in France, and among 
mathematicians in-general .throughout 
Europe. Ic will tend assuredly to the 
overthrow of the modern use of negative 
quantities ; and it is very probable that 
after a little time foreigners will acqui- 
esce in the opinion which has now made 
some progress in this country, that nega~ 
tive quantities are injurious to science, 
and that every useful deduction mty be 
produced without their assistance. 
$G4 * 7 
