246 NOTICES OF THE MEETINGS {Feb. 4, 
revolving so as to complete a revolution in about 193 years; and that 
an eclipse of the sun can happen only when the line of nodes is 
turned nearly towards the sun (as, in other cases, the shadow falls 
above or below the earth). If fora given day of the year, (when 
the sun is in one certain position), the moon is in that part of its 
orbit most nearly in the direction of the sun, the shadow of the moon 
will fall upon a certain point of the earth; but now if the place of 
the node be changed, the effect will be that of driving a wedge under 
the moon, and she will be thrown further north or south, and the 
shadow upon the earth will be thrown further north or south. Thus 
the place of the node will define the part of the earth on which the 
shadow will be thrown; and conversely, a knowledge of the part of 
the earth on which the shadow is thrown will give information on 
the place of the node. Thus the alteration of the lunar elements, 
which is necessary to throw the shadow further north in the eclipse 
of Agathocles, consists in an alteration of the place of the node 
(other elements being supposed moderately correct); and this re- 
quires an alteration in the annual motion of the node, reckoning 
backwards from the present time when the position of the node is 
well known; and applying the same annual correction by the rule 
of three backwards to the place of the node at an assumed time of 
the eclipse of Thales, the corrected place of the node at that time 
is found, and then the corrected track of the moon’s shadow can be 
found. 
Subsequently to the time of the calculations of Baily and Oltmanns, 
the improvements in astronomy have been very great. Many 
advances have been made in theory, and one of the secular changes 
(that of motion of perigee) has been greatly modified. The Green- 
wich Lunar Observations from 1750 to 1830 (which are the founda- 
tion of Lunar Astronomy) have been completely reduced, on one 
uniform plan. Improvements have been made in the details of con- 
struction, but still greater improvements in the principles, of astro- 
nomical instruments, Our knowledge, also, of the geography of the 
countries to which the eclipses before us have relation, is much more 
accurate and extensive than it was. 
Still there remain causes of uncertainty in the results of any calcu- 
lations made for such distant periods. 
First, in the theory. No person who has not fairly entered into 
the details of the Lunar Theory can conceive the complexity of the 
algebraical expressions and the operations which occur init. Besides 
the usual chances of error from mistake of figures and mistake of 
signs, there is the risk of mistake in the selection of some terms to 
the exclusion of others, and the possibility of positive error in the 
metaphysical reasoning which guides some of the operations. And 
we are driven at last to admit that what is sometimes called mathe- 
matical evidence is after all but moral evidence. And thus it has 
happened that the conclusions of different theorists on some very im- 
portant points are by no means accordant. 
