ye 
823 
v 
Y:—l:i:—1:1 
and — 1 being less than nothing, Lis evi- 
dently greater than —- 1; therefore in 
the proportion the first term is greater 
than the second, but the third is not 
greater than the fourth.’ ‘The propor- 
tion then is evidently false, — 1x —1 
cannot be equal to + 1, fegaine — BAs 
less than 2; therefore — 3*is less than 2°, 
But — 3", according to the prevailing 
doctrine, is equal to.9, and 9.is, greater 
than 4. Therefore the square, of — 3 is 
greater than the square of. 2, which jis 
absurd ; and, as the author says, “Cette 
theorie est done complettement .fausse.”’ 
This theory is absolutely false. In the 
application to geometry the same is evi- 
dent ; for taking a point in the diameter 
of a circle produced, and drawing a line 
from that point through the, circle, we 
have two values according to the sup- 
posed. doctrine of negative, quantities ; 
for the distances between the point of the 
convex and concave parts of the circle 
derived fromthe équationyx* +x = ab, 
and these.two values are; the one x => 
VRC Habs he; the other x = — 
Vee ab — 4c. The first is evident 
hy true, but the latter Heing’ a negative 
value caimot be taken in the same direc- 
tion, and-consequeéntly carinét here have 
an existence. In the same manner it 
will follow, from an aleébraical. expres- 
sion, for the radius of the circle, and al- 
lewing the usual ddctriné of negative 
quantities, that the radius of a circle may 
be both negative and posrive: 
From the above and similar instances 
the author concludes, that 
«© Every negative quantity standing by itself 
is A mere creature of the mitd, and that those 
which are met with in. calenlations are only 
mere algebraical forms, incapable of represent- 
ing any thing real and effective. 2."That each 
of these algebraical forms being taken, with a 
proper consideration of its sign, is nothing else 
but the difference of two other absolute quan- 
tities, of which the greatest in the case 9a 
which the reasoning depends, is the least in 
the case in which the result of the calculat'91 
18 to be applied. 
«To say of a quantity that it becomes ne- 
gative, is to employ an improper expression,, 
and one leading, as has been seen above, inio 
error ; and the true meaning of the expression 
is, that this absolute quantity does not belong 
to the system on which the reasonings have 
been established, but io another with which 
itis related ; so that to apply the forms ay pli- 
cable to it for this first syetem, the sign + 
must be changed into the sign —.in the j fe- 
¢eding form. ' 
MATHEMATICS AND NATURAL PHILOSOPHY. 
The idea of a negative quantity being 
thus overthrown, what is to be done in 
future ? 
«« Nothing is to be changed,” says the au- 
thor, in the usual processes, ‘* we have only 
to substitute a clear and true idea in the room 
of one faulty and useless ; and this is the ob~ 
ject of this work. I think that I have accom- 
plished it by substituting for the notion of 
positive and negative quantities, which I have 
opposed, that of quantities which I cail direct 
and inverse; and the geometry of position is 
that where the notion of positive and negative 
quantities standing by themselves is supplied 
by that of quantities direct and inverse.” 
The mathematical reader will at once 
perceive the embarrassment in which the 
author was placed. He had reprobated 
the doctrine of negative quantities ; but 
he could not fail of perceiving how often 
he was liable to run into them in the ap- 
plication of algebra to geometry. Here 
he conceived that the usual processes 
might be retained ; whereas if the notion 
itself, as he has evidently proved, of ne- 
gative quantities is “ faulty and useless” 
in the science of algebra itself, it ought 
by.no means to be permitted to stand one 
moment in its application to another 
science. The fact is, algebra knows nos 
hing of position; thatis peculiar to geo- 
metry; and when from the consideration 
of lines an equation ‘is formed in algebra, 
the rules of algebra alone can be used in 
the solution of it; and the geometrician 
must previously tell what quantities he 
chooses to be greater or less than others, 
before the algebraist can give an answer 
to the question. 
An instance occurs which the writer 
might have made clear to his purpose; 
but by not having: rejected entirely in 
practice, though he has in his mind, the 
old theory, he runs into the same absur4 
dities with common writers, and is then 
obliged to enter into an explanation, 
Suppose a problem to have brought us 
to this equation.x* —2ax +a*—b=o 
Then, says the author, I deduce that «— 
a=+ W/ b;and of course he is obliged 
to shew us how the “ unintelligible’ 
phrase — 4/6 is to be applied to any 
purpose. Whereas the algebraist who 
has never been shackled by negative 
numbers, would, on the equation being 
proposed, deduce all the conclustons in 
the simplest and easiest manner. He 
would first-observe that a* must be either 
greater than, equal to, or less than 6. Tn 
the first case the equation becomes’ i 
2ax— wma —fh, 
