The Almagest, Probabilities, and the Method of Least Squares
The moon's acceleration was only one disagreement between Robert Newton and Anatoly Fomenko. They also strongly differed on the Almagest, the most influential astronomy book ever written. Claudius Ptolemy, one of the greatest scientists of antiquity, wrote it in Alexandria during the reign of the Roman emperor Antoninus Pius, traditionally set from AD 138 to 161. Any firm evidence for a different dating of this treatise would affect the chronology of Rome and consequently most ancient history. This opus touches on the main problems of astronomy, from the nature of the universe to lunar and planetary motion, and contains detailed star catalogs and records of eclipses, occultations, and equinoxes, all of which are prone to mathematical dating. The original version of the Almagest has been lost, but in its many translations the work has been in circulation since ancient times.
In The Crime of Claudius Ptolemy [10], a book published in 1977, Robert Newton accused the ancient astronomer of fabricating evidence. Newton argued that many of the coordinates presented in the Almagest as observations are nothing but fraud. Fomenko disagreed, so he took on the task of dating the book.
His first attempt was based on the fact that every star has a proper motion that is unrelated to the apparent one due to precession. The discovery of this phenomenon is attributed to Edmund Halley, who described it at the beginning of the eighteenth century. Ptolemy had also asked if stars moved independently of each other, but he missed the correct answer.
The motion of stars can be detected only by hundreds of years of precise observations of their tangential components. Using the relative positions given in the Almagest and comparing them with the present ones, Fomenko wanted to find out when the book had been written. But that goal was not easy to achieve. One hurdle was the use of Ptolemy's catalog for tracing the motion of some stars. If the catalog's dating was incorrect, the computed speeds of these stars were also wrong. Fomenko had, therefore, to trace the history of those determinations and eliminate from his analysis the stars related to the Almagest. But the most difficult process was to identify the cataloged stars, a problem that had preoccupied many astronomers starting with the sixteenth century.
In ancient and medieval times the shapes of constellations were not standardized, and their description was often vague. Therefore, telling which star from the catalog corresponds to the one we see in the night sky is difficult. Ptolemy provided positions and magnitudes. For bright objects identification is easier because there are few to choose from, but with faint stars, things get complicated: in the Almagest their coordinates, and also their magnitudes, are often incorrect.
Research done on this problem assumed that the observations were made in the second century AD, a fact that influenced the identification of the stars. The outcome changes for different suppositions. This leads to a circular argument. Fortunately, identification is easier for the stars of zodiacal constellations because they have been studied more carefully for astrological purposes and there is more historical information about them. Of the 350 zodiacal stars recorded in the Almagest, Fomenko chose to focus on the very fast ones, with an individual motion of at least one arc second per year, because slower objects could have traveled distances that were less than those resulting from Ptolemy's observational errors.
Fomenko then applied the method of least squares. He took the distance between the position of a star as recorded in the Almagest and its real position in a given year, as determined by computations. He then summed up the distances for all stars and repeated the procedure for all years within some interval long enough to avoid bias, from 500 BC to AD 1800. Finally, he compared the results and chose the year corresponding to the minimum sum. Estimates for each century pointed out that the only interval in which the errors were smaller than Ptolemy's ten-arc-minute precision was from AD 600 to 1300, with the highest probability around AD 800.
This conclusion depends on several assumptions, and Fomenko checked the reliability of his result. His estimate showed a very small, but nonzero, probability that the Almagest had been written outside this interval. With admissible (but unrealistic) changes in the parameters, the interval could have been extended as far back in time as AD 350, a date still two centuries after the traditional dating. The good news was that the outcome didn't change when slightly varying the data. To gain more confidence in this procedure, he also tested star catalogs from the sixteenth and seventeenth centuries, as well as some computer-generated ones. The results proved more than satisfactory: he recovered the known dates within a ten-year margin of error.
The Almagest contains other resources, such as occultations and lunar eclipses, phenomena that are prone to independent dating. Fomenko and his collaborators devised methods to check the dates, which then led them to the time when the Almagest was written. The estimates they obtained were consistent with the previous dating of the Almagest to about AD 800 [3], [4].
So far, historians have ignored these studies, which are published in a mathematics journal that has a reasonably good ranking.
