310 
[688 
688. 
GEOMETRICAL CONSIDERATIONS ON A SOLAR ECLIPSE. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xv. (1878), 
pp. 340—347.] 
I consider, from a geometrical point of view, the phenomena of a solar eclipse 
over the earth generally; attending at present only to the penumbral cone, the 
vertex of which I denote by V. It is convenient to regard the earth as fixed, and 
the sun and moon as moving each of them with its proper motion, and also with 
the diurnal motion. The penumbral cone meets the earth’s surface in a curve which 
may be called the penumbral curve; viz. when the cone is not completely traversed 
by the earth’s surface, (that is, when only some of the generating lines of the cone 
meet the earth’s surface), the penumbral curve is a single (convex or hour-glass 
shaped) oval; separated, as afterwards mentioned, into two parts, one of them lying 
away from the sun, and having no astronomical significance; but when the cone is 
completely traversed by the earth’s surface, then the penumbral curve consists of two 
separate (convex) ovals; one of them lying away from the sun and having no 
astronomical significance, the other lying towards the sun. The intermediate case is 
when the cone just traverses the earth’s surface, or is touched internally by the 
earth’s surface; the penumbral curve is then a figure of eight, one portion of which 
lies away from the sun, and has no astronomical significance: there is another limiting 
case when the cone is touched externally by the earth’s surface, the penumbral curve 
being then a mere point. 
It is necessary to consider on the earth’s surface a curve which may for shortness 
be termed the horizon; viz. this is the curve of contact of the cone, vertex V, 
circumscribed about the earth; it is a small circle nearly coincident with the great 
circle, which is the intersection by a plane through the centre of the earth at right 
angles to the line from this point to the centre of the sun. 
Regarding V as a point in the heavens, capable of being viewed notwithstanding 
the interposition of the moon; the horizon, as above defined, is the curve separating