Tusi couple

Abstract: The geometrical theorem known as the ‘Tusi couple’ was first discovered by Persian astronomer Nasir al-Din al-Tusi (1201–1274). The Tusi couple was believed to be discovered for Europeans by Nicolaus Copernicus (1473–1543) and it played an important role in the development of his planetary system. It has been suggested by Willy Hartner, that Copernicus borrowed it from al-Tusi, however, a particular way of transmission is not known. In this article I show that Spanish-Jewish author Abner of Burgos (1270–1340) was familiar with the Tusi couple and followed Tusi’s notation in his diagrams. This may provide a missing link in the transmission of the Muslim astronomic knowledge to Europe and advance our understanding of the European Renaissance as a multicultural phenomenon.

Abstract: In the first edition De revolutionibus in the Linda Hall Library in Kansas City (Missouri) there is laid a curious paper instrument with a double volvelle. Because any paper devices relating to Copernicus's work are rare, I this modest device is of more than passing interest. The instrument (Figures 1 (a) and 1 (b)) consists of a paper roughly 10 em square, to which a circular volvelle 6·6 cm in diameter is attached with a thread and green wax. Fastened to this circle is a second volvelle of the same size. What we have is clearly a Tusi couple, the device that Copernicus illustrates on ff. 67 and 67v for producing a straight-line movement from the uniform motion of two circles. This mechanism was much admired in the sixteenth century as part of Copernicus's ingenuity in purging astronomy of philosophically objectionable procedures such as the equant and in restoring uniform circular motions. Such a combination had been introduced into astronomy by Nasir ad-Din at-Tusi in the thirteenth century, but it is not known whether Copernicus learned of it through Arabic sources or whether he invented it independently; in any event, the name \"Tusi couple\" is a modern coinage. Copernicus, in his theory of the Earth's rotations, uses two pairs of Tusi couples at right angles to each other. One set controls the inequality of the precession (i.e. the trepidation) with a period of 1717 Egyptian years, the other, the variation in obliquity with a period of 3434 Egyptian years. The device itself is introduced in De revolutionibus 111.4. This is the first heavily technical innovation and departure from Ptolemy in his book, one that calls for high concentration on the part of the reader. The Linda Hall instrument follows exactly the lettering of Copernicus's diagram of the Tusi couple, and the circles can be moved to generate the straight-line motion. It appears that the volvelles were made for the instruction of its maker, rather than as a more finished model for the instruction of others. On each side are a number of small notes. For example, the motion of the lower disk, I, is specified as \"in precedentia\", in contrast to the motion of the upper disk, II, which is labelled \"in consequentia\", thus indicating the opposing motions of the circles in any Tusi couple. One of the notes on the sheet points out that the mean motion of precession is specified at the bottom of f. 65 as lOin seventy-one years; another quotes from De revolutionibus 111.6 that the motion of the obliquity is twice as slow as the precession of the equinoxes. That these are learning notes rather than a polished instrument is even more obvious from the verso, where a circular diagram appears. According to the text of 111.6, the maker has divided a circle with the letters a, b, c, d, and, still following Copernicus's instructions, has written \"Timocarus\" in the quadrant da, \"Ptolemeus\" in ab and \"Machometes\" (al-Battani) in cd, and has identified the final position as \"Nicolaus\" in cd. However, in 111.6 Copernicus immediately refines this initial placement, so that \"Nicolaus\" should fall between \"Timocarus\" and \"Ptolemeus\", with very different spacings from those shown here. Although the maker has added Copernicus's specified