542 
Proceedings of the Roycd Society 
When the position of a planet in its orbit is given, the mean 
anomaly is obtained directly and almost by inspection ; but when 
the mean anomaly is proposed, the position has to be got by the 
inverse use of the tables, that is by approximation. When the first 
estimate is reasonably near, the work is scarcely more laborious than 
an ordinary interpolation ; and when, as in preparing the equations 
of the centre, the computations are to be made at stated intervals, 
the labour is insignificant. 
For the purpose of guiding the first estimate in sporadic cases, 
the mean anomalies corresponding to each degree of position, and in 
orbits of every degree of eccentricity, are given in the volume A, 
titled Mean Anomalies, the results being given to ten decimal places, 
and in volume B to eight places, with differences and variations. 
In Kepler’s time the details of only six elliptic orbits needed to 
be worked out ; now we have forty times as many. The motions of 
the cloud of specks, so small as to be seen only by help of the 
telescope, afford an opportunity of verifying and correcting our 
estimates of the relative masses of the major planets, so much the 
more valuable that the disturbances exerted by these miniature 
worlds upon their giant neighbours escape our power of detection. 
This new mode of calculation vastly reduces the labour of comparing 
the purely elliptic with the observed motions. 
For all analytical investigations the arc, as well as its sine, cosine, 
and tangent, is reckoned in parts of the radius — an arrangement also 
suited for several other applications of trigonometry. From this 
point of view, the sine and cosine take their place among functions 
with recurring derivatives : they are most easily and rapidly com- 
puted in this connection, without reference to the properties of the 
circle, being regarded as functions equal to their second derivatives 
with the signs changed. The volume titled “ Eecurring Functions ” 
contains their values to twelve decimal places, for each thousandth 
part of the radius, up to two radii. 
For facilitating the change from the one unit to the other in the 
measurement of arcs, a table is here presented of the “ Lengths of 
Circular Arcs ” both for the ancient and for the modern graduation. 
The contrast in the arrangement of the two parts of this table affords 
an excellent exam]3le of the power and conciseness of the decimal 
system. 
