» CARNOT’S GEOMETRIE DE POSITION. 
and the question of achromatic glasses, in 
which so much credit. is due to Mr. Dol- 
lond; and on which! so much labour was 
bestowed by the foreign mathematicians, 
is stated and examined in the writer’s 
best manner- Marat, the detested author 
of so much cruelty in the French revo- 
lution, finds a place in this part of the 
work, and his pretended discoveries are 
treated with the contempt they deserve. 
In astronomy we expected much from 
the reputation of the writer, for the 
greater part of this subject fell to the lot 
of the astronomer of France; but whe- 
ther it is that the work not having been 
begun by himself would not add to his 
82i 
reputation, or that he had not time to 
give it sufficient attention, we felt in this 
part of the work considerable disappoint- 
ment. Inthe account of mechanics of 
modern date there are still greater marks 
of carelessness, and at times of partiality. 
But these remarks must be taken with 
great allowance to the merits and gene- 
ral assiduity of the writers. ‘The work 
is a very important one, will add greatly 
to the improvement of mathematics and 
philosophy, and affords a fund of amuse- 
ment and instruction not to be procure 
elsewhere, without great labour and ex- 
pence, by the mathematician. © 
fav Il. Geometrie de Position. Par L. N.M.Carnor, del Institut National de France, 
de P Academie des Sciences, Aris, et Belles Lettres de Dijon, Sc. Ato. 
NEARLY half a century has elapsed 
since Baron Maseres published his use of 
the negative sign, or that mark in algebra 
which denotes the operation of subtrac- 
tion. The work was acknowledged to 
have great merit, and the arguments in 
it were never answered ; but still, such is 
the force of custom, very eminent mathe- 
maticians could not break themselves of 
their old habits, and they continued to 
use'the sign of subtraction without a pre- 
ceding number, from which that to which 
the sign was annexed was to be subtract- 
ed. » Hence a number was said to be less 
than nothing, other numbers were called 
impossible, and these ideal beings became 
the objects of demonstration. ‘lhe diffi- 
culties thus introduced into science gave 
such a mysterious appearance to algebra, 
that few would study it; and it is con. 
ceived at the present moment to require 
transcendant abilities and application to 
make any progress init. The fact, how- 
ever, is quite otherwise. It is the easiest 
and clearest of the sciences, and the whole 
difficulty in the use of the signs may be 
overcome with much less pains than ‘are 
required to learn the common multipli- 
eation table. 
The French and English nations have 
contended with each other on the honour 
of first introducing obscurity into the 
science; but it seems probable that the 
contention will soon be at an end, for in 
both nations men are springing up who 
seem determined to restore algebra to 
that purity with which it was taught ori- 
ginally by Vieta. 
years ago, Mr. Frend published his“ Prin- 
ciples of Algebra,’ in which he excludes 
entirely the nction of either a number 
In England, a few - 
less than nothing, or an imaginary, or an 
impossible number ; and he asserts, that 
whenever such an appearance takes place, 
“the error is either in the person who 
proposed, or in him who attempted to 
solve the proposed equation.’’ This work 
was followed by an appendix by Baron 
Maseres, who overthrew entirely a sup- 
posed demonstration given by Clairaut 
of the existence of negative numbers, and 
the possibility of their producing by mul- 
tiplication 2 positive number, and shew- 
ed that it was in the very outset founded 
on error, 
Carnot, the author of the work before 
us, does not profess to be acquainted with 
the works of either of the writers we have 
mentioned; but he sees in the same point 
of view the confusion that has arisen in 
science by the introduction of a fiction. 
Instead of. rejecting it however entirely, 
as they have done, he wishes to make a 
kindof compromise, and declaring it tobe 
absurd, he would leave it in possession oF 
the rights and privileges with whichfrom 
the time of Descartes it has been indulg- 
ed. Perhaps he thinks this an easier way 
of overthrowing the system in his coun- 
try, for he accumulates instances of the 
false consequences that arise in reasoning 
upon the present plan, and concludes that 
they must have their effect at last in 
opening people’s eyes, and restoring scir 
ence to its ancient footing. 
Among the instances which he adduces 
of the absurdity of considering negative 
numbers as capable of any of the pro- 
cesses of arithmetic, he introduces the 
following: First,from the laws laid down 
by Euclid in the fifth book of Eyelid, since 
it is supposed that " : 
3G3 
