Astronomical Images : Problem 10: Measuring a comet's parallax

Johannes Regiomontanus

Astronomical Images

<p style='text-align: justify;'>Johannes Regiomontanus died in 1475, leaving behind a printing press, instruments and a library containing printed books and manuscripts. Most of the library was bought by Bernhard Walther (1430-1504), the merchant-astronomer at Nuremberg and then ended up in the possession of Willibald Pirckheimer (1470-1530), the patrician friend of Albrecht Duerer. Pirckheimer sold on several of Regiomontanus's works to Johannes Schoener (1477-1547), who taught mathematics at the gymnasium in Nuremberg. Regiomontanus's work on comets, which was listed in his own printing advertisement, was first edited and published by Schoener in 1531 as <i>Sixteen Problems on the Magnitude, Longitude and True Position of Comets</i>. It was printed again, with several other works of Regiomontanus in 1544. The tenth problem is to measure the distance of the comet from the centre of the World and the centre of vision of the observer. The diagram is an extension of the one used in the first problem and is set up as follows: ABCD: great circle H: the centre of vision of the observer HL: the surface of the Earth E: the centre of the Earth EH: the radius of the Earth EH is extended on both sides and intersects with ABCD at point A at the top (zenith) and at point D at the bottom. G: the centre of the comet Two lines are extended through G from points E and H, to intersect with the great circle at B and C. B: the true place of the comet C: the apparent place of the comet K: a point on the great circle that intersects with the line drawn from E parallel to HC. It can be taken for C, as the size of the Earth is negligible in relation to the great circle. N: the point at which the extension of CH intersects perpendicularly with a line drawn from E; thus angle ENH is a right angle. BK/angle BEK: parallax value derived using preceding methods (a) Angle AHG: known from observation (b) The task is to find EG (the distance of the comet from the centre of the Earth) and HG (the distance of the comet from the centre of vision) For right-angled triangle EHN: Angle NHE = angle AHG (known from b) EH is the radius of the Earth Thus: The ratio of EN to HE and NH to HE are known (in modern parlance, NE = EH x sin EHN (1); NH = EH x cos EHN (4)) For right-angled triangle GNE: Angle NGE = angle BEK (alternate angle, known from a) Thus the proportion of GE to lines EN and NG are known (in modern parlance: EN = GE x sin EGN (2); NG = GE x cos EGN (3)) From (1) and (2) the ratio of EG and EH is known, thus EG can be expressed in terms of the Earth's radius (in modern parlance: EG = EH x sin EHN / sin EGN, where EH is the radius of the Earth). Furthermore NG can also be expressed in terms of the Earth's radius (using 3; thus in modern parlance: NG = EH x sin EHN x cos EGN / sin EGN). Subtracting NH (from 4) from NG yields HG.</p>