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Noel M. Swerdlow Professor Emeritus, Department of Astronomy and Astrophysics; and the College Education: Ph.D., Yale University, 1968 Phone: 6263951750 Location: California Institute of Technology, MC 10140, Pasadena, CA 91125 Email: nmsoddjob.uchicago.edu Research Publications: ADS  arXiv Research Fields: I have continued to work on the book on Renaissance astronomy, in particular the chapter on Tycho, the last to be written, and also making any number of revisions and additions to the chapters on Regiomontanus, Copernicus, Kepler, and Galileo. I am hoping to finish the book in the next few months. It will be published by the Dibner Institute for the History of Science and Technology through MIT Press, and as I had noted earlier will contain many plates from books in the Berndy Library of the Dibner Institute, so will be a large and elaborate publication. I finally completed a paper that has been in progress off and on for a number of years, an examination of the astronomical and astrological chapters of Ptolemy’s Harmonics and their relation to a list of the tones of the planets in Ptolemy’s Canobic Inscription. The paper also contains a review of the earlier parts of the Harmonics, on harmonics itself, that is, tunings and scale forms, necessary to understand the astronomical part. My principal goal in this paper was modest enough, to explain the astronomical chapters, which are, to say the least, obscure, and I must say that after it was done I could only reach the conclusion that the chapters contain nothing that can be called profound or on the level of Ptolemy's better work. Still, they are the most extended treatment surviving from antiquity of what is commonly, although not accurately, called the 'music of the spheres,' and so are at least worth understanding, and it appears from the relation to the tones in the Canobic Inscription, which is known to precede the Almagest, that the Harmonics is an early work of Ptolemy's, perhaps his earliest that survives. What is of interest about this is that the Harmonics contains Ptolemy’s most detailed statement, or even only detailed statement, of what he considers proper method in the applied mathematical sciences, rigorous mathematics combined with rigorous empiricism, and it is significant for understanding his work as a whole if this statement is already present in his early or earliest work. Last year I wrote a short paper with Patrick Scott of the Department of Special Collections at the library of the University of South Carolina concerning a second known copy of a small book by Melchior Jöstellius in which he computes a lunar eclipse of 31 January 1599 from Tycho’s early lunar theory, preceding the final theory published in the Progymnasmata (1602). That paper was a comparison of the copy in Columbia with the only other known copy in Hannover, of which a facsimile was published by Victor Thoren in 1972, for the Columbia copy contains revisions, corrections of errors, that must have been made between 28 and 31 January. I have now written a full description of Tycho’s early lunar theory with an analysis of its application by Jöstellius to the computation of the eclipse, of his initial errors and his, not entirely successful, attempt to correct them. Tycho’s early lunar theory is primitive compared to his final theory, being in principle a modification of Copernicus’s lunar theory by the addition of the variation, the annual equation, and the inequalities of the inclination of the lunar orbit and the motion of the lunar nodes. For the annual equation, Tycho has the correct value, 11', and appears to consider it equivalent to omitting the entire equation of time, which can reach about 13' in the motion of the moon. Jöstellius became confused in applying the annual equation and the equation of time, and made errors which were not entirely corrected in the revised printing. When Tycho found that the observed time of the beginning of the eclipse was about twofifths of an hour earlier than the computed time, he decided that the entire lunar theory required revision, which was carried out in 1600 by his assistant Christian Longomontanus. In fact, the main problem with Jöstellius's computation had nothing to do with the annual equation or the equation of time, but resulted from making the apparent diameter of the earth’s shadow too small. 