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GEOMETRICAL CONSIDERATIONS ON A SOLAR ECLIPSE. 
the contacts with the northern and southern limits respectively. There is, as before, 
no passage through a point K or K 1} the curve in question thus retains throughout 
the same character; and by consideration of the two terminal points it at once appears 
that it is a curve “ eclipse ends at sunrise.” The above-mentioned critic curves form 
together an oval touching the northern and southern limits respectively; say this is 
the sunrise oval. 
The termination of the eclipse is similar to this, only the events happen in the 
reverse order; we have a critic line starting from the node of the figure of eight 
and extending each way until it comes to touch the northern and southern limits 
respectively, viz. this is the line “ eclipse begins at sunset ”; and then, extending each 
way from the points of contact to reunite itself at the point of last contact, this 
being the line “ eclipse ends at sunset,” and the two portions together form an oval 
touching the northern and southern limits respectively; say this is the sunset oval. 
It is to be noticed that certain portions of the two limits are generated as the 
envelope of the penumbral curve during the commencement and during the termination 
of the eclipse. 
For the middle of the eclipse; the penumbral curve, in the first instance a 
figure of eight, breaks up into two ovals, but only one of these is attended to; 
and ultimately the oval unites itself with another oval so as to give rise to a new 
figure of eight. There is thus throughout the middle of the eclipse a single oval; 
this has, north and south, an envelope which joins itself on to the portions enveloped 
during the latter part of the commencement and the former part of the termination 
of the eclipse, and constitutes therewith the northern and southern limits respectively, 
viz. each of these is considered as extending from a point of contact with the sunrise 
oval to a point of contact with the sunset oval. 
The line K 1 VK, or say the meridian line through V, travels westwardly, while 
the penumbral curve travels eastwardly; the two come to touch each other, and there 
are then two intersections which ultimately come to the northern and southern limits 
respectively: the locus of these is a line of “ eclipse commences at midday ”; as the 
motion continues, the points of intersection move away from the two limits respectively 
and ultimately unite at the point where the line KVK 1 again touches the penumbral 
curve; the locus is the line of “ eclipse terminates at midday,” the two lines together 
forming an oval which touches the northern and southern limits respectively and which 
may be termed the midday oval. In all that precedes, no distinction has been made 
between the two portions of the horizon-envelope, or the points K and K lf and either 
curve and point indifferently may be alone attended to. 
Considering now an eclipse of the second kind, the penumbral curve is at first 
a mere point (the point of first contact) and it then becomes an oval, the successive 
ovals not at first intersecting each other, but each oval inclosing within itself the 
preceding ones. Any oval is met by the corresponding horizon in two points P and F, 
at first coinciding with each other at the point of first contact, and then separating 
from each other, one of them, say P, moving down towards and ultimately arriving 
at one of the horizon-envelopes, say to fix the ideas the southern one (which curve 
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