251 
1888 - 89 .] Dr E. Sang on Fundamental Tables. 
Canon, had experienced its utility, had received the approval of the 
scientific world; and yet, foreseeing the advantage of accommodating 
his plan to the notation in common use, he at once recommended 
the putting of it aside for a far better plan. No stronger evidence 
can he adduced in favour of discarding the time-honoured division 
by 90 and by 60, and substituting the decimal division throughout. 
Kepler’s Problem. 
While the Canon of Sines was still in progress, circumstances led 
to a repetition of the often fruitlessly made attack on Kepler’s 
famous problem, and this time an unexpectedly simple solution 
presented itself. The Royal Academy of Sciences of Turin did me 
the very great honour of giving that solution a place among their 
memoirs. The subject, however, may be treated more generally 
and even more simply, thus : — 
Let us suppose ourselves to be studying the apparent relative 
motion of a binary system of stars; each one seems to describe 
round the other an ellipse, and the areas passed over by the radius 
vector are proportional to the elapsed times. But, since the actual 
orbit may be inclined anyhow to the plane on which it appears to 
be projected, the one star does not appear to be in the focus of the 
orbit of the other ; nor is the diameter drawn through its apparent 
place, necessarily be the major axis. If we divide the periodic 
time into equal portions, the corresponding vectors will similarly 
divide the area of the ellipse, and hence the problem virtually comes 
to be this, — “ to subdivide the area of an ellipse by lines diverging 
from some point within it.” 
The line from the eye to the revolving star defines the surface of 
a cone, in our imaginary case sensibly of a cylinder, and the planes 
passing through the eye, and along the vectors, subdivide this 
cylinder into wedges. If now this system be cut by any plane, the 
section so made will have its area also subdivided ; now we can 
always cut a cylinder so that its section may be a circle, and thus, 
ultimately, the problem becomes this, “ to subdivide the area of a 
circle by lines diverging from a point within it.” 
If S represent the point given within the circle described round 
the centre O, the diameter ASOa will represent the line of apsides, 
A being the perihelion, a the aphelion. Let now Q correspond to 
