Prog 
gave new solutiéns, and extremely cu- 
rious theorem, for reducing to rectilinear 
triangles, the triangles very little curved 
which are formed on the surface of the 
earth. He has since demonstrated that 
the same theorem is applicable to sphe- 
roidic triangles, His new canons, and 
those of Mr. Delambre, for ail the same 
problems, form the basis of the in- 
struction published by the general Depot 
of the War Department; they have been 
adopted by the astrenomer Swanherg, 
who, in 1802, again measured the degree 
of Sweden. They have chanyed the 
face of this most important part of our 
knowledge, 
_ These ; great operations have diffused 
through Europe a taste for geodesy. 
Fiance is indebted to them for the map 
of its new departments; England for 
thase of its southern provinces; Ger- 
many for several countries partly sur- 
veyed by French engineers ; Switzerland 
for a description of several of its cantons. 
The use of the repeating circle has been 
extended to all the contuent; and there 
1809.] 
is reason to hope, that shertly, all the 
surface of Europe will be covered with 
triangles, and that sovereigns will be 
better acquainted with their dominions, 
than private individuals with their estates, 
The decimal division of the circle, so 
convenient for observers and calculators, 
required new trigonometrical tables. M. 
Provy caused them to be constructed with 
incredible celerity, by means entirely new, 
which enabled him to employ arithme- 
ticians the least informed. 
analysts, with M. Levendre at their 
head, prepared the work, ° and the 
other sections had only the additions to 
perform. Thus were obtained two 
copies of the tables totally independent 
of each other. ‘This vast work, the 
greatest that has ever been executed 
or even conceived, has no fault but its 
immensity, which has hitherto delayed 
its publication, Borda, who was aware 
of the necessity of more portable tables, 
caused them to be calculated under his 
Own inspection; but be could not finish 
the work.” Delambre completed it, and 
gave in his preface methods different 
from those of MM, Prony and Legen- 
dre, which would have led with equal 
rapidity tothe saine end, and they have 
ees d.very curious CeneL at Ona. 
MM. Hobert and Ideler also calcu- 
ey by other means, very accurate 
tables, and still move portable (20). 
If from yeometry we Brasers to com 
pon algebra, we shall fiid advances less 
Blontary Mac. No. 189, 
gress of the Scrvences. 
A section of 
197 
perceptible, but infinitely more difficult. 
The memoirs of M. Lagrange, on the 
complete resolution of literal equations, 
by reducing the problem to its lowest 
terms, show bow dificult it still is. 
M,. Roffini undertook to prove that it is 
inpossible. Lagrange endeavoured 
least to facilitate the solution of nume- 
tical equations, His ingenious analysis 
has reduced the question to finding a 
quantity smaller than the smallest dif 
ference of the roots. He expressed a 
wish, that methods might be found 
within the reach of arithmeticians. Mr, 
Badan, doctor of medicine, hasgiven one 
which employs additionally, a degree of 
simplicity which could not be expected, 
and will not be easily exceeded. 
The lectures at the Normal school 
afforded our great geometricians an oOp- 
portunity of illustrating the most obscure 
theories. M. Lagrange developed the 
analysis of the irreducible case; and 
M. Laplace the demonstration of the 
theorem of d’Alembert on imaginary 
roots. M. Gauss has since decomposed 
into factors of the second degree equa- 
tions, the reduction of which appeared 
impossible; he yave the means of in- 
scribing -a circle without employing the 
rule and the polygonic conipass. The 
number of the sides of which is ex- 
pressed by a primary number (of the 
torm 2n +1.) M. Legendre demon- 
strated thet particular case 2 of the polygon 
of seventeen sides, 
The analysis applied to geometry by 
M. Monge, presents the equations 
of lines of planes, of curves of the 
second degree, the theory of tangent 
planes; in short, the principal cir- 
cumstances of the generation of curve 
surfaces expressed by partial differential 
equations, of which the author makes 
use, to integrate, inanelegant manner,a 
great number of equations, by following 
step by step the details of the geome- 
trical description. As early as the year 
1772 be shewed the connexion existing 
between the curves with a double cur- 
vature and the squarable suriaces. MM. 
Laneret has shewn the ratio between 
the two curvatures, and transferred into 
space the imperfect squares of Reaumur, 
MM. Hachette and Poisson have 
added clegant Eeouems and valuable 
illustrations to the work of M. Monge. 
M. Carnot included in symmetric and 
curious canons all the questions relative 
ta any five points taken in space, 
Fermat had suppressed the demonstra- 
tions of several remarkable theorems of 
2C indeterminate 
