De plana spera

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De plana spera is an astronomical / mathematical script in which some aspects of the representation of the sphere on the astrolabe are mathematically / geometrically defined and proven. It was written in Latin by Jordanus Nemorarius in the first half of the 13th century .

Purpose and meaning

The units considered in positional astronomy , such as stars , ecliptic , azimuth , etc., have been represented on a single spherical surface since ancient times . In order to make this sphere more accessible for observation and analysis, it was transferred to a plane through a stereographic projection , whereby the astrolabe was developed over the centuries for technical / mechanical realization. This illustration was worked on by several scientists in antiquity. The work Planisphaeium by Claudius Ptolemy is important . This work was available in Europe at the beginning of the 13th century, as were several translations from the Islamic world on the subject into Latin. Jordanus probably used these texts in his treatment of the same subjects. But while Ptolemy always developed his calculations and constructions using acute examples (such as the position of the zodiac or the circulus equinoctalis ), Jordanus dealt with the mathematical / geometric problems (2 circles that intersect at an oblique angle) without the astrolabe or the one on it to mention the objects shown.

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Stereographic projection - great circle inclined to the plane

The work assumes that many terms and definitions are known. The underlying construction is the stereographic projection: a sphere touches a plane at one point, straight lines are led through the diametrical point to this plane; the points of intersection with the plane form a conformal image of the points of intersection with the spherical surface. For this construction, which can only be found in the introduction, which probably does not belong to the first version, 5 proportiones ( sentences ) are set up about the properties of this figure:

  • A circle on the spherical surface is mapped into a circle or a straight line on the plane
  • Circles on the spherical surface above the circular rectus ( great circle parallel to the plane) can be clearly mapped on the plane
  • The position of points on the spherical surface above the circular rectus can be determined.
  • Division of the spherical surface into segments of equal size
  • Points whose position is known in relation to a circular declivus (a circle inclined at an acute angle to the circular rectus , behind which is the idea of ​​the ecliptic ) can be localized on the plane.

For each propositio a proof is offered in a formalized language, whereby the straight lines, angles, circles are described by letter combinations:

  • Cum sint igitur anguli AKB, FYA recti et angulus FAB communis, erit angulus AFY equalis angulo KBA
  • Since AKB and FYA are right angles and angles FAB (both triangles) are common, angle AFY will be the same as angle KBA

However, the evidence is not always complete in a mathematical sense.

Living on and tradition

The text was widely received in the Middle Ages. This can be seen from the fact that there are 3 different versions, because the probably original version was heavily edited by 2 recipients. This is also supported by the number of surviving manuscripts, some from the 13th century. Petrus Peregrinus de Maricourt , who wrote a little later, quotes him in his treatise on the astrolabe, and several other works show parallels. In 1536 the work by Johan Walder (Basel) was included in an astronomical collection edition and in 1558 by Federico Commandino (Venice) in his commentary on the Planisphaerium of Ptolemy. Ron B. Thomson produced an edition in 1978 with translation into English and detailed commentary.

Text editions and literature

  • Burkhard Stautz: The Astrolabe Collections of the Deutsches Museum and the Bavarian National Museum , Munich 1999.
  • Ron B. Thomson: Jordanus de Nemore and the Mathematics of Astrolabes: De plana spera , Toronto 1978.

Individual evidence

  1. ↑ Burkhard Stautz: The Astrolabiensammlungen the German Museum and the Bavarian National Museum , pp 115-117
  2. ^ Ed. By Johan Ludvig Heiberg : Claudii Ptolemaei Opera quae exstant omnia Vol.2, Opera astronomica minora , Leipzig 1907
  3. ^ Ron B. Thomson: Jordanus de Nemore and the Mathematics of Astrolabes: De plana spera , p. 53ff
  4. ^ Ron B. Thomson: Jordanus de Nemore and the Mathematics of Astrolabes: De plana spera , p. 48f
  5. ^ Wolf-Dieter Klix / Heinz Nickel: Darstellende Geometrie , Leipzig 1990, 1.3.1
  6. ^ Ron B. Thomson: Jordanus de Nemore and the Mathematics of Astrolabes: De plana spera , pp. 152f
  7. ↑ Jordanus Nemorarius: De plana spera , propositio 1d
  8. ^ Ron B. Thomson: Jordanus de Nemore and the Mathematics of Astrolabes: De plana spera , on Proposition 1 pp. 136-138
  9. ^ Ron B. Thomson: Jordanus de Nemore and the Mathematics of Astrolabes: De plana spera , pp. 58ff
  10. ^ Ron B. Thomson: Jordanus de Nemore and the Mathematics of Astrolabes: De plana spera , p. 79