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Proceedings of the Royal Society of Edinburgh. [Sess. 
from the actual sines in such a way as to lessen by one-half the amount 
of the labour. Nepair’s arrangement was therefore followed, and the work 
was begun from the sine of 100 c down to 50 c . The calculations were made 
by help of the fifteen-place table of logarithms from 100,000 to 370,000. 
If this table had been continued up to the whole million, the labour would 
have been greatly diminished, but we had to bring the numbers to within 
the actual range of our table by halving or doubling as the case might be. 
The results were then tested by first, second, and third differences, and in 
not a few cases the computation had to be redone, for the sake of some 
minute difference among the last figures. The log sines for the other half 
of the quadrant, that is from 50 c to 0 C , were deduced from the preceding 
by the use of first differences alone. The log tangents from 50 c down to 
0° were also deduced directly by help of the first differences alone. In this 
way the series of fundamental tables needed for the new system has been 
completed, so far as the limit of minutes goes. 
“ While that work was in progress, a circumstance occurred which 
temporarily changed the order of procedure. Kepler’s celebrated problem 
has ever since his time exercised mathematicians, and, sharing the ambition 
of many others, I also sought often, and in vain, for an easy solution of it. 
Accident brought it again before me, and this time, considering not the 
relations of the lines connected with it, but the relations of the areas 
concerned, an exceedingly simple solution was found. In order to give 
effect to this method it was necessary to compute a table of the areas of 
circular segments in terms of the whole area of the circle. That again 
rendered it necessary to calculate the sines measured in parts of the 
quadrant as a unit, instead of in parts of the radius, as usual. This 
computation was effected by using the multiples of twice the versed sine 
formerly employed. From this again the canon of circular segments for 
each minute of the whole circumference was readily deduced. The mean 
anomaly of a planet may be deduced from its angle of position, or as it is 
generally called, its excentric anomaly, by simple additions and sub- 
tractions of these circular segments. The converse problem is very 
easily resolved, particularly when the first estimate is a tolerably close 
one. In order to be able promptly to make this first estimate sufficiently 
near in every possible case, a table of mean anomalies from degree to 
degree of the angular position, and also from degree to degree of the angle 
of excentricity of the orbit, has been computed according to the decimal 
system. 
“The change to this system is inevitable. Each new discovery, 
each improvement in the art of observing, intensifies the need for the 
