254 
Proceedings of Royal Society of Edinburgh. [sess. 
orbit of some particular planet, we are computing the positions at 
equal intervals of time, attention to the differences reduces the 
labour to little more than that of writing out the results. It is 
only when a sporadic case is presented that the approximation is 
attended with any difficulty. 
Mean Anomalies. 
In order to obviate even this difficulty, a table has been con- 
structed of the mean anomalies for orbits of each degree of excen- 
tricity, and for every degree of the angle of position, up to 200° in 
each of these orbits. This table enables us, in every possible case, 
to get at once a first assumption so near as to make the subsequent 
approximation quite easily. 
This table is presented in two forms. In the volume marked 
mean anomalies A, the values are given to four decimal places of a 
second. In the corresponding volume marked B, they are written 
only to the nearest second ; but the differences and the variation 
from one orbit to another are inserted. Hence, by the ordinary 
method of interpolation for two variables, we can solve both the 
direct and the inverse problem with precision sufficient for all the 
purposes of practical astronomy. 
My intention was to have computed also the radii vectors and 
the true anomalies. For this, however, the only available trigo- 
nometrical tables were those to seven places printed in a most 
inconvenient form, by Callet, in his Tables Portatives. The work 
was scarcely begun when it became apparent that the precision 
attainable was not commensurate with the labour. Therefore, 
putting that work aside, I preferred to undertake the hopeless- 
looking task of computing the logarithmic sines and tangents to a 
greater number of places. This work is fortunately accomplished, 
although there still remain the transcription in the order usually 
adopted for convenient reference. 
The application of these tables to the computation of the true 
anomalies, is a task far too great to be undertaken at the close of a 
long life, and, not without reluctance, it is left to the zeal of other 
computers. Enough, that I have been enabled to place within the 
reach of mathematicians some contributions to the progress of exact 
science. 
