OF AGATHOCLES, THALES, AND XERXES. 
191 
the Southern position ; 4. that the Southern boundary will touch the Northern posi- 
tion. The shaded irregular ovals are intended merely to show the breadth of the dark 
shadow in the direction transverse to the path of shadow, without any relation to the 
extent of shadow at any one instant in the direction of the path. 
31. From an inspection of this map, with some geographical measures, the follow- 
ing conclusions will be easily deduced : — 
{a.) With Unvaried Elements, the dark shadow passes over the Southern position 
of Agathocles, but not over the Northern position. 
{h.) In order that the eclipse may be total at the Southern position, the change of 
Mean Argument of Latitude must be included between those corresponding to Con- 
dition 1 and Condition 3, or between 
— 0'55X Variation and -f lT4x Variation. 
(c.) In order that the eclipse may be total at the Northern position, the change 
must be included between those corresponding to Condition 2 and Condition 4, or 
between 
+0*68 X Variation and -l-2'38x Variation. 
The latter is the system of limitations which, for reasons already explained, I am 
disposed to adopt. 
32. Assuming the uncertainty in the circumstances of an eclipse to be due entirely 
to uncertainty in the place of the moon’s node, these statements supply us with an 
easy method of correcting (within certain limits) the elements for any other eclipse, 
according as we adopt one or the other position of Agathocles. It is only to be 
remarked that if, in making the application to another eclipse, we adopt in the cal- 
culations for that eclipse (as will be convenient) a Variation of 0^*20, then the factors 
corresponding to these Conditions must be increased in the same proportion as the 
interval of time backwards from the present age (when the place of the node is well 
know n) to the age of the eclipse in question. Thus in applying them to the eclipse 
of Thales we must increase the factors by about yth part. 
Section IV. Eclipse o/* Thales, as recorded hy Herodotus. 
33. The account of this eclipse given in the first book of Herodotus is in sub- 
stance as follow^s. “ Upon the refusal of Alyattes (king of the Lydians) to give up 
some Scythian fugitives to Cyaxares (king of the Medes), the Lydians and the Medes 
were at war for five years ; during which the Medes often defeated the Lydians and 
the Lydians often defeated the Medes; they had also in this war a sort of night- 
battle ; and while they were still carrying on the war with equal success, and met 
for battle in the sixth year, it happened that on the battle being joined the day 
suddenly became night. Thales the Milesian predicted to the lonians that this 
change would happen, fixing beforehand this very year, in which the change did 
oceur. The Lydians and the Medes, when they saw that it was night instead of day, 
ceased from fighting, and on both sides endeavoured more anxiously to obtain peace. 
