148 Proceedings of Philosophical Societies. [Frs. 
allude to the formulas given by M. Gauss to find Easter for any 
year of the Italian and Gregorian calendar. We satisfy ourselves 
with noticing here that the formulas are not complete, we shall 
republish them elsewhere with more exactness, and we shall join 
to them new formulas still more expeditious, and which, besides, 
give us the dominical letter, the golden number, and the epact of 
the year. See the Connoissance Des Temps of 1807, p. 307. 
Exercises in the Integral Calculus. Fourth Part, by M. le Che- 
valier Legendre. This new part is divided into two sections. In 
the first M. Legendre has completed the theory which he explained 
in the second part of his exercises. He has particularly attached 
himself to explain with all the requisite minuteness the properties 
of a function, which is the mutual connecting link of a multitude 
of transcendental quantities, and the source from which flow all 
the formulas which concern the comparison of these transcend- 
entals, their reduction, and their evaluation. The author has 
already satisfied himself, that he was not mistaken when he hoped 
that this theory, considered under a new point of view, and aug- 
mented by a great number of new formulas, might fix the atten- 
tion of mathematicians, and that they would see a new branch of 
analysis brought almost to perfection, 
To extend further the applications of this theory, he has in- 
serted at the end of the first section a more extended table than 
that which terminated the second part. The new logarithms are 
exact to the 12th decimal, in order that the cipher of the last 
order may never be in error more than one unity, or at most two. 
This has given occasion to rectify, and almost to double in extent, 
a table which Euler had given in his differential calculus, for the 
sums of the reciprocal powers of the natural numbers. 
.In the second section will be found different researches, which 
form a sequel to the third part. A great number of formulas are 
demonstrated, either entirely new or recently discovered. Among 
the last are several definite integrals given by M. Bidonc, in the 
Memoirs of Turin. The author has also presented some new 
views on the summation of different series, and on the formulas 
which serve to give the sum of a series of which the general term 
is given. 
[t is impossible for us to dwell longer on a work of pure analysis, 
and almost entirely composed of formulas. See what we have said 
formerly on the first part of this work. We shall take this oppor- 
tunity to rectify a passage in our notice of 1810. 
In giving an account of the second memoir on Elliptical Trans- 
cendentals, we have denoted by the word Loxodromic, a species of 
spiral which M. Legendre there considers, and one of the pro- 
perties of which is to be the shortest road between two points 
situated under two different meridians or parallels. This accep- 
tation of the word Loxodromic is not that of navigators and mathe- 
maticians ; but it appeared to us more conformable to the ety- 
mology of the word, which is an oblique route. There is no straight 
! 
