314 GEOMETRICAL CONSIDERATIONS ON A SOLAR ECLIPSE. [688 
is henceforth selected as being, and is called, the horizon-envelope, and the points on 
this curve are taken to be the points K), viz. P is then a point K on the 
penumbral curve, I call it K x . The successive ovals will in the meantime have 
begun to intersect each other so as to give rise to a northern limit; this will touch 
the critic line (locus of P, P'), and we have a portion of the critic line extending 
from the point of first contact, in one direction to the point of contact with the 
northern limit, and in the other direction to the point K 1 on the horizon-envelope; 
this is the line “ eclipse begins at sunrise.” As the horizon continues to sweep on, 
the other point P', which has not yet reached the horizon-envelope, will gradually 
approach and ultimately arrive at the horizon-envelope, say at the point K 2 ; we 
have thus a second portion of the critic line extending from the contact with the 
northern envelope to the point K 2 ; this is the line “ eclipse ends at sunrise.” The 
horizon continuing to sweep on, the point P beginning with the position Kwhich is 
now on the other side of the point of contact of the horizon with the horizon-envelope, 
will trace out a portion of the critic curve extending from K 1 to a second point of 
contact with the northern limit; this will be the line of “ eclipse begins at sunset.” 
And, finally, the point P from the last-mentioned point of contact, and the point P' 
from its position K 2 , which is now on the other side of the point of contact of 
the horizon with the horizon-envelope, (that is, P, P' have now each passed through 
the point of contact of the horizon with the horizon-envelope, and are both of them 
on the same side thereof, viz. the side opposite to their original side), will come to 
unite at the point of the last contact; we have thus a fourth portion of the critic 
curve extending from K 2 to the second point of contact with the northern limit, viz. 
this is the line “ eclipse ends at sunset.” The description will be more intelligible 
by means of the figure, in which 1, V, 2, 2',..., 8, 8' represent successive corresponding 
positions of the points P, P', the successive positions of the horizon being given by 
the right lines 11', 22', &c., all of them tangents to the dotted circle or horizon-envelope. 
The entire critic line is thus a figure of eight, twice touching the horizon-envelope