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Ancient and Medieval Chronology
Magnitude of a star
The scale of magnitudes in Ptolemy's Almagest is made up of integers from unity (corresponding to the brightest stars) to six (corresponding to the faintest ones). He stated that he had observed all stars up to the sixth magnitude, which seems improbable from today's point of view, since many stars of magnitude 3 to 5 cannot be identified with them, whereas some of magnitude 6 can be identified at once. If we compare the magnitudes of Ptolemy's stars with the modern ones, we shall see that the magnitudes 1 to 2 were always determined correctly, whereas he often made mistakes in the interval of the magnitudes 3 to 6.
We now go on to the problem of dating the observations of the stars which formed the basis for the Almagest star catalogue. Its traditional dating is founded on Ptolemy's statement that the catalogue had been compiled in the second year of the rule of the emperor Pius. According to the ScaligerPetavius chronology, this is the year A.D. 138, though the name "Pius" was common to some other Roman emperors, for example, the famous Maximilian I Pius (14931519), who reigned just at the time of the first Almagest editions (Fig. 9(2)). However, even Delambre noticed that the average error in the star longitudes was about 35 minutes; therefore, the catalogue supplies the coordinates not as they were in A.D. 138 but in ca. in A .D. 60. J. Bode speaks of the year 63, whereas F. Peters and E. Knobel of the year 58. In fact, the error variance for the longitudes of approximately 1,000 stars is about 402 minutes. Hence, the average error in longitude is a random variable with variance o2 = 402/1000 = 1.6'.
Since they must be distributed according to N(0,1), the admissible average error in the longitude should not exceed 5 minutes, which permits us to determine the epoch to which the catalogue dates to the accuracy of 56 years. Thus, in the form it was published in Basel in 1537, the Almagest could not have been made from the observational data of A.D. 138. This last important remark by Delambre led to the hypothesis that Ptolemy had not observed the stars by himself, but rather updated the Hipparchus catalogue to A.D. 138, the latter being dated to 130 B.C. Meanwhile, making use of the incorrect precession value, he actually reduced the catalogue to A.D. 60 instead of A.D. 138. The authors of [105], [106], and [108] were of the same opinion. The important work [102] is devoted just to the proof of this hypothesis. Note that almost all of its conclusions are based on the assumption that the Almagest was written by an ancient astronomer. However, R. Newton's main argument based on the comparison of degree fractional part distribution for latitude and longitude of a star is not related to chronology [102]. In this connection, the investigation of the distributions of degree fractional parts in Ptolemy's catalogue, carried out by Newton, only permits us to state the fact that the catalogue was obtained from the original and is based on direct observations by adding an integral (possibly, negative!) number of one degree and forty minutes.
If, following [108], we assume that the Almagest is related to A.D. 58, and if we take precession into account, then the longitudinal shift through an integral multiple of one degree and forty minutes takes us to the following values for the epochs of observation, namely, ..., 207, 134, 62, 10, 92, 154, 227, 299, 371, 443, 515, 586, 660, 732, 805, 877, 950, 1022, 1094, 1166, 1239, 1311, 1383, 1456, 1528, 1600, ... (years B.C. are indicated by a minus sign).
The conclusion of Newton is that the Ptolemy star catalogue had been "corrected" according to the precession used by Hipparchus. On the basis of statistical investigations, he asserts that Ptolemy was a falsifier. The same thought was even expressed in the book's title, namely, The Crime of Claudius Ptolemy, and in the titles of many of the book's sections. For example, he speaks of equinoxes and solstices allegedly observed by Ptolemy, the fabricated solstice of 431 B.C., observations allegedly made by Ptolemy to determine the ecliptic's slope and Alexandria's latitude, four fabricated lunar eclipse "triads", the proof of a forgery, a swindler, falsifications of calculations and falsifications with oversights, falsification of data, falsification of the Venus data and exterior planet data, and so forth.
Newton writes that all the observations made by Ptolemy himself, so far as they can be checked, turned out to be a forgery, and that many observations ascribed to other astronomers were also part of the falsification. It then becomes clear that none of Ptolemy's statements can be accepted unless they have been confirmed by authors absolutely independent of his data. Be it history or astronomy, all research based on the Almagest should be performed again. Thus, Newton completes his thought, Ptolemy is not the outstanding astronomer of antiquity but a more unusual figure: a most successful swindler in the history of science (see [102]).
However, this estimation of Ptolemy's work by a wellknown astronomer of modern times is only based on the hypothesis of the Almagest's ancient origin. The attitude toward Ptolemy may become quite different if it turns out that we are dealing with a text written in the 1016th century. We have already noted earlier that almost all of Newton's conclusions depend on the a pnon dating of the catalogue to about the year A.D. 1, or the first two centuries before and after. Our goal is the Almagest's analysis based solely on the star catalogue itself and the contemporary idea of the starry sky. Some steps in this direction are made in the sequel. We once again stress the importance of Newton's research and would only like to complete it by suggesting another interpretation.

