## Mental Exercise The Distribution of Measurements

You measure stellar latitudes with a device marked in fifths of a degree, and a colleague measures latitudes of other stars with a device marked in quarters of a degree. Next, you two combine your data.

Assuming a random distribution of the stars, at what fractional degree marking would you expect the peak number of measures to fall?

At what fractional degree marking would you expect the minimum number of measures to fall?

If latitudes for 2,000 stars were recorded, what quantitative distribution of fractional degrees would you expect to see?

What distribution might you expect if the observing devices were marked in fourths and thirds of degrees?

The distribution evidence is suggestive, but not conclusive. Suppose that Ptolemy actually made his reported measurements against a reference star, later found that his longitude for the reference star was short by a fractional amount of 2/3 degree (or over by 1/3 degree), and then added this 2/3 degree to (or subtracted 1/3 degree from) all his genuine measurements? Or perhaps he only determined the absolute longitude of the reference star after making the relative measurements, and it had a fractional part of 2/3 degree, which he then added to each star's relative longitude? Indeed, 3 of the 4 bright stars that Ptolemy might conveniently have used as reference stars have fractional 2/3 degree longitudes in the Almagest. The addition of a reference star's fractional longitude can account for the distribution of fractions in longitude, without denying the genuineness of Ptolemy's reported observations.

Interpretations of the distribution of fractional values in Ptolemy's stellar longitudes are numerous and ingenious but so far none has proven decisive. For every conclusion so far proposed, subsequent examination by another historian of astronomy has uncovered one or more possible and plausible alternative explanations.

In too many instances, agreement between Ptolemy's theory and observation is too good to be true. But if there is a hint of fraud in the Almagest, there is also scientific greatness. Agreement between Ptolemy's numerical parameters and modern values is too close to be fortuitous. Probably, Ptolemy had a large number of observations, and errors largely canceled each other out in calculations of a general theory. Next, Ptolemy might have selected from among his observations a few in best agreement with the theory, and then presented these examples to illustrate the theory.

Ptolemy lacked our modern understanding of error ranges, standard deviations, and the use of mean values from repeated observationsâ€”concepts that would have enabled him to present a theory not necessarily in total and absolute agreement with every data point obtained, but in a less strict agreement with all the data points to within a statistically defined interval around a mean value. Instead, absent any tolerable fluctuation in the agreement between theory and observation, every measurement would be understood by Ptolemy and his contemporaries as an exact result. Consequently, judicious selection from among many measurements was required.

We should also recognize that the Almagest is not a modern research paper. Rather, it was a textbook. Ptolemy was attempting to demonstrate a new type of science, in which specific observational data were converted into the numerical parameters of a geometrical model. The lesson taught in the Almagest would enable astronomers in the future to add their own observations over a longer temporal baseline and obtain an even more accurate theory or model of planetary motions. Any fudging or fabrication by Ptolemy might well have been understood by him as little fibs allowable in the neatening up of his pedagogy, not as lies intended to mislead his readers about crucial matters. The Greek astronomical tradition was far more concerned with general geometrical procedures than with specific numerical results. Modern norms of science did not yet exist.

Gingerich challenges any attribution of criminal and nefarious motives to Ptolemy but at the same time acknowledges that Newton deserves credit for bringing to our attention inconsistencies and anomalies in Ptolemy's work. History is an activity and an argument, not merely a chronological collection of facts, and in pursuing questions of possible fraud, scholars, including Newton and Gingerich, are making history.

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