206 
appears that the climate of France will 
not allow the fugar-cane to ripen. 
CELLs terminated his labours by indi- 
cating the means by which the maturity © 
of different parts of vegetables may be af- 
certairfed, and by laying down fome gene- 
ral ideas on the abortion of grains, or 
other parts of the fruétification of divers 
vegetables. 
PREADAU-CHEMILLY has difcufied, 
~in an extenfive memoir, the advantages 
al 
which cultivators may derive trom the in- 
clofure of their lands. TeEssrER has 
commenced a great labour on agriculture, 
and has already executed that part of 
his plan in which he propofed the extent 
of land, the variety of crops, and the 
number of ufeful animals which a well- 
conitituted. farm requires, confidered 
with relation to the induftry of the culti- 
vator, the laying out of the fields, the 
nature of the foil, and the temperature 
of the climate. 
Some of the members of the clafs, de- 
voting themfelves to chemical labours, 
have been feeking, in a number of mine- 
ral fubftances, the {pecies of earth newly 
recognized, which bears. the name of 
fivonthian, and of which the celebrated 
KLAPROTH, of Berlin, has difcovered a 
combination, almoft pure, with the ful- 
furic acid, in a! mineral brought from 
Pennfylvania.. Guyton has alfo found 
this combination of /froxthian. He has 
extracted it from a native fulfate of 
baryte fent him from Saxony ; and PreL- 
LETTER has colletied a fufficiently large 
quantity of muriate of ftronthian, from 
the white opaque baryte of Hartz, and 
rom the heavy ftone known by the name 
of Bologna-ftone. After having corrobo- 
rated, in another paper, the experiments 
of the fame chemilt, KL APROTH, on the 
facility with which pewter may, after the 
manner of acids, unite with fixed alka- 
lics, GuyTON proves that it is not the 
duperoxigenation of this metal in certain 
mines which oppofes its diffolution in 
acids, but oaly the ftate of aggregation 
of the oxide. a force which refifts very 
powerful affinitics, and which it is fo 
much the more important to recognife, 
as it often injures the fuccefs of the 
analyfis of bodies. GuyTON points out 
alfo, in a third memoir, the manner of 
employing the combined calculation 
(le crkeul rapproché) of very fenfible che- 
mical effets, to verify and bring to per- 
tection, the ufeful tables which are 
known by the names of ‘ables of cowpofi- 
tion of Wilts. 
[Thé Proceedings of the To other Claffes 
will be infcried in the neat Number. | 
National Inftitute... Mathematical Correfpondence. 
[ Sept. 
MATHEMATICAL CORRESPONDENCE: 
To the Editor of the Monthly Magazine. 
STIR; ; 
[ AM happy to fee, by your laft Magazine, 
that the queftions on infinite and imaginary 
quantities. now the fubject of much difcuffion 
amongft mathematicians, are likely to be brought 
fairly berore the public. You will permit me 
to make fome remarks on the obfervations of 
your correfpondents. 
The one is an advocate for imaginary quan 
tities in algebra, and founds his veafoning upon 
the following definition: * Def. The fquare 
“ root of =a is expreffed thus, 4/—a; of 
*© courfe, the fquare of this quantity 4/—<¢ 
‘¢ muft be —-a, being juft the reverfe.” 
This fentence contains two things, a defini- 
tion, and a corollary from that definition. I 
feak generally upon this head, not being will- 
ing to cavil upon the term definition. The 
firft part, then, or the definition, I do not ex- 
cept againft. The fquare root of —a is ex- 
“ preffed thus: 4/—a,”° that is, in plain lan- 
guage, a term which has no meaning may be 
exprefied by another term which has no mean- 
ing. The corollary fhall be examined on the 
principles allowed on all fides by the defenders, 
as well as the oppugners of the admiffion of 
imaginary or impoffible quantities. ‘ The 
‘ fquare of this quantity ,/—a muft be —a, 
“ being juft the reverfe.”? - 
Againft this corollary I thus argue: The 
fecond power of a term @ is a, becaufe 
aXaza?, The fecond power of a—d is 
at— 2ab -- b3, becaufe a—b x a—b 0? —~ 
246-152, When, in two fums to be multiplied 
together, there are terms having the fign of 
fubtra€tion before them, the produét of two 
fuch terms has always before it the fign of ad- 
dition. Thus the term 47, in the infiance 
above, has the fign of addition before it, 
though both the terms which produce it had 
befure them the fign of fubtraftion. The rule 
is eafily proved; and, as it is generally allowed, 
if it were poffible to confider —a and —é as 
feparate independant terms, and to multiply 
them together, the product muft be +25, and 
in the fame manner, / —aFxX /—?= 
af —a? x —b* = 4/ a2 4, Upon the fame 
principle, 4/ —a X 4/ —a=4/—a X —a= 
4/a% Thus the fecond power of 4/—a is not 
—a, but a. This appears to me evident from 
the principles laid down by the vindicators of 
the negative quantities, as they are called ; but 
as the corollary of the definition is proved to be 
falfe, let us now fee whether the definition does 
not offend againft the rules of right reafoning, 
and contain within itfelf terms incompatible. 
“«* The fquare root of —a is thus exprefied, 

«<< ,/—a;"" but —a is’ an imaginary number, 
therefore it cannot have any fquare root; fdr 
the fquare root of a term means a number 
whic 
