12.8 Dr . Robertson's expeditious methods of calculating 
and some others which had occurred to me in the course of 
my reading. The result of this search is a belief that no one 
before has aimed at a direct solution through the same small 
angle, and, by means of an equation in which this angle and its 
powers are the only unknown quantities, obtained a quickly 
converging series for its value in known terms. The small 
angle being found, with due precision, the eccentric anomaly 
readily becomes known. 
M. Delambre, in his Astronomy, published at Paris in 
1814, in three quarto volumes, calculates the excentric ano- 
maly by a method founded on those of Cassini,* * * § La Caille^ 
Simpson^ and Cagnoli.§ This eminent astronomer says of 
it, “ Ce procede, le plus directe que je connaises, est aussi le 
plus precis ; il n'est qu'approximatif, mais il est toujours 
exact au-de 3 a des dixiemes de seconde pour toutes les planefes 
de notre systeme.” That the reader may readily judge how 
far the third method, which I now propose, deserves atten- 
tion, I have annexed to my investigation the calculation by it 
of M. Delambre's two examples, and also one relating to the 
comet of 1682 and 1759. * 
Each of the following methods of solution is to be considered 
as direct, although it proceeds through the medium of what 
is commonly called Cassini's approximation. This approxi- 
mation, as here used, can only be considered as the first cer- 
tain step in the computation. No hypothesis is introduced 
* Memoiresde l’Academie de Pannee 1719. 
f Le£ons Elementaires d ’Astronomic, Paris, 1761. 
% Essays on several curious and useful subjects, London, 1740, 
§ Trigonometric, Paris, 1786. 
