1816.] Royal Institute of France. 147 
all those of M. Gauss, otherwise demonstrated and brought to a 
notation, which it has been attempted to render more clear. All 
these formulas are illustrated by numerical examples. The same 
attention has been paid to the parallaxes, the eclipses, and to every 
thing that concerns the planets and the comets. In the article 
Tables of the Sun will be seen the mearis employed to construct the 
last tables, the reductions applied to observations made with the 
repeating circle, the method of observing an equinox and a sclstice, 
with tables to facilitate all these operations. The Theory of Eclipses 
is explained in different manners, partly new, which have the ad- 
vantage of being easier and more general. The application of 
them has been made to the eclipses of 1764, of which it has served 
to determine the curves of entrance and exit. 
In the chapter on The Planets is shown how by means the most 
simple we obtain the first approximation of an unknown orbit, and 
how this first sketch may be afterwards perfected. ‘The Transits of 
Mercury and Venus are treated in a way altogether new, which 
leads by a more direct and certain way to a knowledge of the paral- 
jaxes. In the article Rotation, seven different solutions of that 
problem are collected, and, for an example, are taken eleven obser- 
vations of aspot of the sun. The Theory of Saturn’s Ring, 
which terminates this chapter, offers the most exact and simple 
means, either to predict the phenomena, or to determine the ele- 
ments from observations. The chapter on The Comets is long. The 
known formulas are presented so as to serve equally for the parabola 
and the ellipse. The methods are explained properly to facilitate the 
construction and ensure the accuracy of the general tables. Ex- 
amples are given, calculated according to the most accredited 
methods; and one is explained, which recommends itself by the 
following advantages: it employs only the calculations most fami- 
liar to the astronomer; the longitudes and latitudes observed are, as 
jnall the other methods, the primary data; but they enter only into 
the first calculations ; the succeeding ones employ only heliocentric 
places; and the elements found may be perfected by the totality of 
the observations, by equations nearly of the same conditions as 
those of the planets: the calculations are made’ as the observations 
succeed, and the astronomer who has discovered a comet, may on 
the day itself of the third observation, ascertain its elements. 
This chapter is terminated by a catalogue of the orbits of all the 
comets hitherto observed, and by the general tables of the para- 
bolic movements of different form, according as we take for data 
the anomaly or the days since the perihelion. In the chapter on 
‘The Measure of the Earth will be found new calculations on the 
irregularities of the are of the meridian, measured in France and 
in Spain. li the chapter on Nautical Astronomy there are nume- 
yous solutions of the most important problems; and the work ter- 
minates by a chapter on The Calendar, in which we must point out 
a fault in the drawing up, which may in some rare circumstances 
occasion an error of seven days in the result of the calculus. We, 
