1875.) Selenography : its Past, Present, and Future. 209 
had 105 points of the first order, only two of which rested on 
under five measures, as well as a considerable number of 
auxiliary points of the second order. 
Madler’s measures were made with a parallel wire position 
micrometer, with a power of 140 and a field of 143’, or about 
20 revolutions in diameter, the thickness of the wires being 
1‘91”, but as the edge of the field was imperfect, distances 
exceeding I4 to 15 revolutions could not be measured direct ; 
consequently, objects more than ro’ or 11’ from the limb 
were measured from an intermediate spot, generally Posi- 
donius, Piccolomini, Messier, Gassendi, or Seleucus, whose 
distance from the limb was also measured. To eliminate 
as much as possible incidental errors, and to simplify the 
computations, measures were only made when the moon 
was near the meridian and possessed a greater altitude 
than 18°. The distances were found by bisecting the spot 
with one wire and bringing the other tangentially to the limb, 
a somewhat difficult operation to perform in a right ascen- 
sion direction, owing to the rapid motion of the moon, and 
the difficulty of putting the two distant objects in position 
at the same time. Each result consisted of two measures 
in a right ascension dire¢tion, and one at right angles. 
Beer and Madler considered that from the difficulty of 
exactly adjusting the wires of the micrometer so as to keep 
both in position, and from the irregularity of the moon’s 
limb and variability in the amount of irradiation at the 
edge, the results of a single measure could not be regarded 
as giving the position of the point within 30’ of longitude 
or latitude near the centre, and proportionately greater 
towards the limb, the uncertainty of a single result being 
about 8” to g”, a result borne out by their measures, which, 
rejecting the unfavourable points, show the uncertainty of 
a single measure to be 7:2”. From this the average pro- 
bable error of one of their points of the first order, resting 
as already mentioned on from 8 to 12 measures, usually 
might be expected to be from 2°2” to 1°8”, or 8’ to 7 of sele- 
nographical longitude or latitude near the centre, a result 
sufficiently approximate to the actual probable errors for 
the more favourable points. Inthe above no account has 
been taken of the considerable personal equation now known 
to exist in the observations of the lunar limb; and this 
is a weak place in Encke’s method of determining positions 
of the first order, as employed by Madler. Of necessity, 
the position of all the points were measured by Madler, 
mainly from one pair of limbs, either S. and W. or S. and E., 
or else N. and W. or N. and E., and therefore the resulting 
