204." -, > French Nasional Inflitute. 
tricians are not fatisfied unlefs their refults are more general, — 
They. defire, for each degreé, a.final equation, which may 
reprefent i in what manner each of the given) numbers) con- 
curs to the different ‘values of the unknown quantity.;\ fo 
that with thefe expreflions, or formulz., nothing remains to! 
be gone through but fimple arithmetical operations to de- 
termine the unknown part of an equation, whatever:may be 
the abfolute values of the known quantities which enter into 
its compofition,. Hitherto we had none of thefe formule 
beyond the fourth degree. It was neceflary then ‘to: recur 
to means for refolving individually numerical equations, 
and to bring out, one by one, the values of the unknown 
quantity, Newton gave a method for this purpofe ; but it 1s 
only approximative, even when there exift exact values’ of 
what is fought for, and in certain cafés it gives nothing. 
In,1767 Cit. Lagrange, whofe name is’ here naturally 
coupled with that of the Englifh geometer, gave a new 
method, exempt from the inconveniences of that of Newton, 
Analyfis was then in a more advanced ftate, and the author 
took advantage of thofe degrees of perfection which he had 
himfelf given to the analytic art. The queftion then is to 
keep the number fought between two determined fractions, 
which go on decreafing, one of which is greater and the’ 
other lefs than that number, however fmall the fra@tiongs ° 
may be; and thus, by exhaufting a frational difference, 
we arrive at exact values of the unknown quantity, ‘if there 
are any poflible, or at leaft we approach fufficiently near it. 
Thefe firft refearches of Cit, Lagrange are but a fmall part of’ 
the volume juft publifhed. Thofe which form this  laft 
work will add a new degree to the efteem and gratitude to 
which he has been long entitled by his numerous and 
learned labours. 
Cit. Duc Lachapel, an affociated member, read a memoir 
on an obfervation which he made at Montauban on the 
appulfe of the Moon and Mars. ‘Aftronomers have given 
this name to phenomena where the moon paffes fo near to” 
, a ftar, 
