356 Proceedings of Royal Society of Edinburgh. [sess. 
from being edge on to the light, its shadow will be an ellipse which 
will correspond with the sun’s path projected on the plane of the 
horizon in lat. 20°. In the same manner, whatever may be the 
obliquity of the disc, its shadow will be an ellipse of the same 
relative dimensions as that representing the sun’s path, in the lati- 
tude corresponding to that obliquity. It will at once be seen that, 
constructed on this principle, the major axes of all the ellipses 
would be equal to twice radius, consequently to each other, and 
would run into each other at the east and west points, and the 
minor axes would be equal to the sine of the angle of obliquity, or 
sine of lat. 
This arrangement would answer very well while the declination 
was 0°, but in allowing for declination the position of observer 
would have to be shifted in the opposite direction by the following 
quantity: — (tan of dec!, x cos of lat.). This would require a sepa- 
rate scale of declination to be laid down for each ellipse of latitude, 
which, to say the least of it, would be extremely inconvenient ; so, 
to make the scale of declination available for all latitudes, I decided 
to vary the size of the ellipses instead of the scale of declination. 
That is to say, instead of multiplying the tan of declination by the 
cos of latitude, I divide both major and minor axes of any particu- 
lar ellipse by the same quantity. The formula would therefore be 
as follows : — - 
Radius S. lat. 
Q og j at r = sec lat. = major axis, ^ = tan lat. = minor axis. 
It will easily be seen that this preserves the relative lengths of the 
major and minor axes of the ellipse for any degree of latitude, as illus- 
trated by the shadow of the disc, because as radius : sec : : sine : tan. 
This arrangement also allows the declination to be measured 
and marked off on the same scale as the latitude ; and further, it 
locates the foci of all the ellipses in the ends of the straight line 
which represents the sun’s path in lat. 0°, a great consideration when 
^he ellipses have to be drawn with pins and a thread as I have 
always done them. 
I have taken it for granted that the sun’s path, projected on the 
plane of the horizon, will be exactly the same at all times of year, 
the whole ellipse representing said path being shifted north or 
