French National Inflitute. 208 
tions, with itfelf, and with the known quantities: thefe are _ 
called determinate problems. The indeterminate are-thofe, 
the folution of which depends on an equation where there are 
found two or more unknown quantities, fufceptible of all 
the values which the analyft may aflign to them, fo that 
the unknown quantities remain indeterminate, at leaft in 
certain limits. The procefies are very different in thefe 
two forts of problems, and hence arife two kinds of analyfis. 
Among thofe queftions which cannot be folved but by the 
indeterminate analyfis, are fome refpecting the nature of 
numbers; queftions of a delicate nature and difficult to be 
treated, which require intenfe thought and exceedingly 
delicate and varied refources of mind, but which, at the 
fame time, excite the curiofity more, and become more 
engaging. Cit: Legendre publithed, in the Tranfaétions of 
the Academy of Sciences for 1785, a memoir upon this 
fubje&. In the laft quarter he has made public new 
refults refpecting his further refearches in this branch of the 
analytic art, under the title of Effaz fur la Théorie des 
Nombres. This modeft title promifes much lefs than the 
work affords. It is a complete treatifeof every thing known. 
on the theory of numbers, or even of indeterminate analyfis. 
But thefe theories would be much lefs ufeful if they wantéd 
the demonftrations and new theorems difcovered by Cit. 
Legendre, 
A new work on determinate analyfis engages, at this 
moment, alfo the attention of analyfis. Cit. Lagrange has 
publifhed a work on the refolution of numerical equations. 
He gives this name to equations where, with the unknown 
quantity, there are only numbers which retain their nu- 
merical value. In the laft analyfis every determined problem 
may be reduced to expreffions of this kind. It is of great 
importance, therefore, to have methods given for refolving 
them; and we might even be inclined to believe that it 
would be fufficient for mathematicians to confine thenafelves 
to refearches for difcoyering thefe methods. But geome- 
trigians | 
