252 Proceedings of Royal Society of Edinburgh. [sess. 
some position of the planet, then the surface comprised between 
AS, SQ, and the arc AQ, is proportional to the time elapsed from 
the planet, being in the position A, till its reaching the position 
corresponding to Q, so that this surface is the planet’s mean anomaly. 
Draw now ESF perpendicular to AQ, then the arc AE, which 
has the excentricity OS for its cosine, may be called the arc of 
excentricity ; we shall denote it by e ; while AQ, the arc defining 
the planet’s position, may be denoted by p. Having joined EQ, FQ, 
it is seen, by mere inspection, that the mean anomaly AEQS is half 
the sum of the two circular segments cut off by the chords EQ, EQ, 
or that 
mean anomaly = |{segm(£> + e) + segm(jo - e)} , 
and so a table of circular segments would enable us to determine 
the mean anomaly when the position is given, and conversely the 
position when the mean anomaly is given. 
In order to avoid the frequent multiplication and division by the 
