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Astronomical Images : Regiomontanus's addition to Al-Battani on trigonometry


Astronomical Images

<p style='text-align: justify;'>The Nuremberg astronomer, Johannes Schoener (1477-1547), is well known for bringing certain of Regiomontanus's uncompleted publications into print. In this important edition, he published a manuscript of Al-Battani's <i>De motu stellarum </i>previously owned by Regiomontanus, and incorporated Regiomontanus's marginal annotations. Albategnius (Al-BattÄ?nÄ«) was one of the greatest Islamic astronomers, active from the second half of the ninth century to the beginning of the tenth. Schoener's edition is the translation of the twelfth-century scholar, Plato of Tivoli. It included six of Regiomontanus's own comments, the one in the present illustration being an important addition on trigonometry. In his addition to Al-Battani's explanation of a method to find the azimuth (direction in the horizontal plane) of a celestial object, Regiomontanus explains how the azimuth is connected to, and can be derived from, the altitude of the Sun (measured in degrees above the horizon). In the diagram, ABGD represents a vertical meridian circle, so that A is a point directly above the observer's head (at E) and G is directly below his feet. The horizon, which is of course circular, appears as a straight line BD on this diagram because it is perpendicular to the page. FR is the celestial equator (a projection of the Earth's equator on the sky) and LP the path of the Sun on a given day, which is parallel to the celestial equator. YZ is the Earth's axis, so that Z is the northern celestial pole. At sunrise the Sun is on the horizon at point K. Regiomontanus first uses similar triangles to find the angle on the horizon between north and the line EK, which runs from the observer to the point where the Sun rises (since angle NEK is equal to the observer's latitude, and ENK is 90 degrees). He then posits an almicantarath circle MOSQ, which is a horizontal circle with its centre at some point directly above the observer's head. O, where LP crosses this circle, is the location of the Sun at some undetermined time of day; L is the location of the Sun at noon, when it crosses the meridian. Taking the semidiameter (i.e. radius) of the celestial sphere to be 1, the distance from the observer to O is 1, so the straight line length of OH is the sine of the Sun's altitude (measured in degrees from the horizon). Again using the unit semidiameter, he points out that XD is the sine of the observer's latitude NEK, and EX is the sine of the complement of the observer's latitude, NKE. He then notes the similarity of triangles OHK and XED: they both have right-angles, and angle OKH is equal to XDE. The ratio of lines EX:XD is thus the same as the ratio OH:HK; since we know EX, XD and OH, HK may be easily worked out. If we subtract EK (which is the sine of the angle on the horizon we found earlier) from HK, we are left with EH, which is equal to OS. It can be seen from Regiomontanus's diagram that, if we again take the radius of meridian circle ABGD to be 1, the radius of MOSQ (i.e. MS) is equal to the sine of angle SEM, which is the Sun's zenith distance (the complement of its altitude) when it is at O. If we take O as a point between M and S on the line MS (which is a radius of the almicantarath circle), the length of OS can be related to what Regiomontanus calls 'the arc in the almicantarath circle', i.e. the angle at S between east and the point on the circle that lies due east of O. The sine of this angle equals OS divided by the radius of the almicantarath circle. Since we know both the radius (MS) and OS, we can calculate the angle at S, which will tell us the azimuth of the Sun when it passes O.</p>

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