701 
and compare the results with the longitudes. We then find the numbers 
whieh in the table [IL on page 694 stand in the column “reduced date’, 
beside which the values of the difference ‘longitude — reduced date” 
(L.—R. D.) are placed. These values become gradually smaller, alto- 
gether 5 days in the course of the whole table. This is exactly as much 
as it should be to account for the difference between 45.23, the actual 
mean time-interval, and 45.146, the mean syn. are +12. We now 
see that a correction for this difference is not introduced gradually, 
but suddenly, by shifting one day each time after 10—13 numbers; 
this is done at the places where the horizontal lines are put (the 
first line is uncertain, as there is some error here). 
If we leave out these regularly recurring jumps, the differences 
L—k. D. everywhere show variations up and down. On the other 
hand they show a great constancy, if we only pay attention to the 
whole numbers and not to the fractions. If we may consider afew 
cases in which this does not come out as erroneous, we find this 
rule: the Babylonian calculator found the dates by taking the 
numbers of the degrees from the calculated longitudes, increasing 
them successively by the periodic series of numbers v,v +12, v+ 24 
v+6, v +18, wv, etc., each time after 10—13 periods taking the 
number v one higher. 
As a final test, in all the sections of the great Jupiter table of 
the second kind') the dates were calculated according to the above 
rule by means of the periodic series of numbers v, v + 12 ete. 
The few cases, indicated by «, where there is still a day’s difference, 
are not such as to throw a doubt on the correctness of the rule 
for the calculation that we have found; these are probably due to 
copying errors or errors of calculation in the cuneiform texts. The 
first error in the 3'¢ section, where Duzu 31 stands instead of 30, 
is undoubtedly of that kind. In the first error of the first section 
there was a doubt as to where the periodic number had shifted 
so that either Duzu 6 or Abu 17 must be one day wrong; we 
have chosen the transition so, that the latter date, the number of 
which lies at the edge of the illegible damaged part and has therefore 
perhaps been misread, was taken to be erroneous. The rd erroneous 
number of the 2°¢ section also lies at the edge of a damaged portion. 
If we now return to the Jupiter tables of the first kind, we find 
that our rule applies there also. In the table If on p. 692 which con- 
tains the dates and places for the second stationary point and the 
1) The columns “reduced date” and *‘L—R D” have only been computed for 
the first section, the heliacic rises; the system of calculation having been discovered 
from these it was not necessary to compute them for the other sections. 
