have been observed, as the Almagest reports. But this is fallacy, the so 

 often mentioned conclusio in circulo. All Lunar Tables from the Alphon- 

 sian down to Lalande, Burckhardt, and Damoiseau, were constructed for 

 the purpose of corresponding both with future eclipses and with those in 

 the Almagest. Consequently, as often as we recalculate Ptolemy's eclipses 

 by means of Lunar Tables based on the same eclipses, the Babylonian 

 eclipses must of necessity agree with the Tables. This is the fatal self- 

 delusion. Buerg was the only astronomer who, in spite of the Almagest, 

 based his Tables on 3,200 Greenwich observations. According to the 

 same Tables, three of the seven Babylonian eclipses were invisible ones, 

 wherefore Buerg's Tables happened instantly to be banished from all ob- 

 servatories. And yet, on occasion of the total eclipse in 1851, it came to 

 light that Buerg's Tables, not being based on Ptolemy's eclipses, were 

 much more correct than those of Damoiseau, Burckhardt, or Lalande, of 

 which the Almagest was the groundwork. This fact, together with Heiss's 

 computations of the Greek eclipses visible during the Peloponnesian war, 

 puts beyond question that the whole Historical Canon of Ptolemy is erratic, 

 and that his eclipses must be referred to other years. Had Hansen's Ta- 

 bles not agreed with the Almagest they never Avould have been printed, or 

 would have been soon after cast away like Buerg's. Had the latter taken 

 the secular acceleration of the moon a few seconds greater than he did, 

 his Tables would have remained practicable down to this day. 



Besides, it is a gross delusion that Hansen's Tables harmonize with 

 Ptolemv's Almagest. Hartwich's computations have evidenced (see these 

 Transactions iii. 512) that one of these eclipses was invisible; another one, 

 amounting to one-quarter of an inch, could not be seen with the naked 

 eye; others were two, even three times larger or smaller; some of them 

 happened 55m., 54m., 27m., ih. 15m. later than the Almagest says. Sup- 

 pose the Babylonian astronomers had observed these eclipses themselves 

 to the nicety of minutes, then they must have been in possession of astro- 

 tronomical instruments for measuring minutes of time and parts of inches, 

 But in this case, however, they could commit no blunders of ihr. 15m., 

 and never determine the minutes of eclipses being invisible. In short, it 

 is Ptolemy who referred the eclipses in his Almagest to wrong years, and 

 calculated their times and magnitudes. 



To this the astronomers object that it would have been impossible for 

 Ptolemy, by means of his very imperfect theory of the moon's motions, to 

 compute eclipses being 800 years older. This impossibility, I confess, is 

 not very plain to me. Ptolemy, having had the opportunity of astronomi- 

 cally observing four lunar eclipses in 125, 133. 134 and 136 A.c, could, by 

 means of them, easily fix the longitudes both of the moon and her nodes 

 for the epoch 100 a.c. Further, comparing these results with the 300 years 

 older observations of Hipparch, notoriously the best astronomer of the 

 ancients, he determined the longitudes of the moon and her nodes for the 

 epoch — Boo, and hence the secular mean motions of the moon and that of 

 her nodes. The secular accelerations of the latter being unknown to Ptol- 



