538 
Proceedings of the Royal Society 
seen written 112''*37, rather than r'*52'''-37 ; it would be difficult 
to cite a more forcible example. 
That triumph of skill, patience, and exactitude, the table of 
Lunar Distances, is a protest even stronger; therein the moon’s 
distances from a star are given at intervals of three hours. In 
order to compute the Greenwich time of his observation, the 
mariner compares his observed distance (corrected for refraction and 
j^arallax) with those found in the almanac ; he has therefore to 
make a proportion in sexagesimals. Seamen are understood to be 
so wedded to the present system, that they of all others would 
dislike a change ; yet such are the torments of sexagesimals that, 
for the shunning of them, a column of proportional logarithms is 
contrived, and a special logarithmic system is arranged. Instead of 
having to work out a simple proportion, the seaman is drilled to 
use the proportional logarithm, whose nature, in ninety-nine cases 
out of a hundred, he does not comprehend. 
In the higher branches of astronomical calculations, and in the 
application of trigonometry to mechanical and physical problems, 
the arcs and their various functions have to be compared, the mode 
of comparison being suited to the particular cases. When the arcs 
are homologues of angles measured by help of graduated instruments, 
their natural unit is the entire circumference ; but their sines and 
tangents, having reference to rectilineal measure, are most con- 
veniently compared with the radius. Hence it is that, in ordinary 
trigonometry, two units are employed ; and hence also the con- 
venient though somewhat illogical expression, ‘‘ the sine of an 
angle,” instead of the sine of the arc homologous to an angle.” 
But in many cases, notably in analytical investigations, the radius 
of the circle is made the basis of comparison both for arcs and for 
sines. Also, in computing the anomalies of the planets, the areas 
passed over by the radius-vector have to be considered, and it is 
much preferable to measure the sines in parts of the circumference, 
the areas in parts of the surface of the circle. 
Thus we have often to pass from one unit of measure to another ; 
with no system of subdivision can the transitions be made more 
easily (if at all) than by that of uniform decimal subdivision. 
From whichever point of view the matter may be studied, the 
desirability of the change is clear ; but there are difficulties in the 
