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GEOMETRICAL CONSIDERATIONS ON A SOLAR ECLIPSE. 
[688 
V (considered as a point in the heavens) is rising, for a point on the other of them 
it is setting; and for the points K and K 1 respectively it is moving horizontally; 
that is, first rising and then setting, or vice versa. 
A solar eclipse is of one of two classes; viz. either the penumbral cone completely 
traverses the earth, so that towards the middle of the eclipse the penumbral curve 
consists of two separate ovals: or the penumbral cone does not completely traverse 
the earth, so that throughout the eclipse the penumbral curve consists of a single 
oval only. In the former case, we have to consider the commencement, during which 
the penumbral curve passes from a mere point to a figure of eight: the middle, 
during which it passes from a figure of eight through two ovals to a figure of eight: 
and the termination, during which it passes from a figure of eight to a mere point. 
In the latter case, we consider the whole eclipse during which the penumbral curve 
passes from a mere point through a single oval to a mere point. 
In an eclipse of the first class: for the commencement, the penumbral curve is 
at first a mere point (point of first contact); it then becomes a convex oval, each 
oval in the first instance inclosing the preceding ones, so that there is not any 
intersection of two consecutive ovals. We come at last to an oval which is touched 
north by its consecutive oval, and to an oval which is touched south by its consecutive 
oval (I presume that the contacts north and south do not take place on the same 
oval, but I am not sure); and after this, the ovals assume the hour-glass form, each 
oval intersecting the consecutive oval in two points north and two points south; the 
ovals thus beginning to form an envelope or limit. There are on each of the ovals 
two critic points, and we have thus a critic curve commencing at the mere point 
(point of first contact) and extending in each direction from this point. The point, 
where an oval is touched by the consecutive oval, is not so far as appears a critic 
point; that is, the critic curve does not at this point unite itself with the envelope 
or limit. But the critic curve comes subsequently to unite itself each way with the 
limit; and, since clearly it cannot intersect the limit, it will at each of these points 
touch the limit; that is, we have a critic curve extending each way from the point 
of first contact until it touches the northern limit and until it touches the southern 
limit. Observe that the penumbral curve, as being at first a mere point or an 
indefinitely small oval, does not at first contain within itself the point K or K x : 
it can only come to do this by passing through a position where the curve passes 
through K or Z x ; viz. K or K x would then be a critic point; and I assume for 
the present that this does not take place. The critic curve at the point of first 
contact is a curve “ eclipse begins at sunrise,” and as not coming to pass through 
a point or K 1} it cannot alter its character; that is, the critic curve, as extending 
each way from the point of first contact until it comes to touch the northern and 
southern limits respectively, is a curve “ eclipse begins at sunrise ”; at the terminal 
points in question, there is a mere contact of the sun and moon, so that they are 
points, where the eclipse begins and simultaneously ends at sunrise. Continuing the 
series of ovals until we arrive at the figure of eight, there are on each of them 
two critic points, which ultimately unite in the node of the figure of eight; these 
constitute a critic curve, extending each way from the node of the figure of eight to