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GEOMETRICAL CONSIDERATIONS ON A SOLAR ECLIPSE. 
311 
the portions of the earth’s surface for which V is visible and invisible respectively. 
The horizon does or does not meet the penumbral curve, according as this last 
consists of a single oval or of two distinct ovals; viz. in the latter case the horizon 
lies between the two ovals, in the former case the horizon traverses the area of the 
oval (separating this area into two parts), thus meeting the oval, or penumbral curve, 
in two points, or say these points separate the oval into two parts; from any point 
of the one part V is visible, from any point of the other part V is invisible; and 
from each of the two points themselves V is visible as a point on the horizon in 
the ordinary sense of the word; that is, there is an exterior contact of the sun 
and moon visible on the horizon. It is to be observed that, in the limiting cases 
where the penumbral curve is a mere point and a figure of eight respectively, the 
horizon passes through the mere point and through the node of the figure of eight 
respectively. 
The two points of intersection of the penumbral curve with the horizon may 
for shortness be termed critic points. The lines which present themselves in a diagram 
of a solar eclipse, (see Nautical Almanac) are the “northern and south lines of simple 
contact,” say for shortness the “ limits ”; viz. these are the envelope or, geometrically, 
a portion of the envelope of the penumbral curve; and the lines of “eclipse begins 
or ends at sunrise or sunset,” say for shortness the critic lines; viz. these are the 
locus of the critic points. 
The point V considered as a point in the heavens is a point occupying a position 
intermediate between those of the centres of the sun and moon; hence referring it 
to the surface of the earth by means of a line drawn from the centre, its position 
on the earth’s surface is nearly coincident with that point to which the sun is then 
vertical; and its motion on the earth’s surface is from east to west approximately 
along the parallel of latitude = sun’s declination, and with a velocity of approximately 
15° per hour. For any given position of V on the earth’s surface, describing with 
a given angular radius nearly = 90° a small circle (nearly a great circle), this is the 
horizon; as V moves upon the surface of the earth, the horizon envelopes a curve 
which is very nearly a parallel, angular radius = sun’s declination (there are two such 
curves in the northern and southern hemispheres respectively, but I attend only to one 
of them in the proper hemisphere, as will be explained), say this is the horizon-envelope; 
the horizon in each of its successive positions is thus a curvilinear tangent (nearly 
a great circle) to this horizon-envelope. If for a given position of V, and also for 
the consecutive position we consider the corresponding horizons, these intersect in a 
point K on the horizon-envelope, and the horizon for V is the circle centre V and 
angular radius VK; Z is a point which is very nearly upon, and which may be 
taken to be upon, the meridian through V; the horizon may be regarded as a 
tangent which sweeps round the horizon-envelope; to each position thereof there 
corresponds a position of V, and consequently also a penumbral curve; and (when 
this is a single oval) the horizon meets it in two points, which are the critic points. 
It is to be added that, if for a given position of the horizon we consider as well 
K as the opposite point K 1} (viz., K x lies on the great circle KV), then the points 
K and K 1 divide the horizon into two portions; for any point on one of these portions