Ptolomaei Planisphaerium. Jordani Planisphaerium. Federici Commandini Urbinatis in Ptolemaei Planisphaerium Commentarius. In quo universa Scenographices ratio quam brevissime traditur, ac demonstrationibus confirmatur.

Venice: Aldus [Paolo Manuzio], 1558.

First edition of Commandino’s important treatise on perspective, together with the first printing of the treatises on stereographic projection of Ptolemy (fl. 2nd century AD) and Jordanus (fl. 13th century) with Commandino’s commentary. “[Commandino’s] section on perspective [in the second part of the present work, with separate title-page and foliation] … is remarkable in that it is the first example of an entirely geometrical and rigorous approach to perspective. Thus, he involved not a single argument from optics in his proofs and he did not assume the convergence rule – nor did he apply it” (Andersen, p. 141). Stereographic projection is a method of representing the celestial sphere on a plane. As the Earth rotates on its axis, the stars trace out circular paths on the celestial sphere. Stereographic projection represents these paths as circles (or occasionally straight lines) in the plane. (Stereographic projection also preserves angles, although this was probably not known at the time.) This representation was at the heart of the method construction of the astrolabe, a physical representation of part of the plane on which the celestial sphere is projected. Ptolemy’s Planisphaerium survives only in Arabic translation; a revision of this translation was made by the Spanish Islamic astronomer Maslama al-Majriti (d. 1007/1008) and was in turn translated into Latin by Hermann of Carinthia in 1143. It treats the problem of mapping circles on the celestial sphere onto a plane. Ptolemy projects them from the south celestial pole onto the plane of the equator. This projection is the mathematical basis of the plane astrolabe, the most popular of medieval astronomical instruments. Since the work explains how to use the mapping to calculate rising times, one of the main uses of the astrolabe, it is highly likely that the instrument itself goes back to Ptolemy (independent evidence suggests that it goes back to Hipparchus)” (DSB). The treatise of Jordanus was based on that of Ptolemy. Both of these treatises were first published in Latin in 1536, but this is the first separate and best edition, edited by Commandino and with his valuable commentary. “In the sixteenth century, Western mathematics emerged swiftly from a millennial decline. This rapid ascent was assisted by Apollonius, Archimedes, Aristarchus, Euclid, Eutocius, Hero, Pappus, Ptolemy, and Serenus – as published by Commandino” (DSB).

“Commandino (1509-75) is best known for his scholarly editions of classical Greek works, among them are Ptolemy’s Planisphaerium (1558) and Apollonius’s Conics (1566). Actually Commandino’s contribution to perspective is part of a commentary he wrote on the former Greek work applying a result from the latter. The Planisphaerium has as its theme the stereographic projection, which is a central projection with its centre at the celestial south pole. Ptolemy used it for mapping points on the celestial sphere upon the plane of the equator, applying that all the circles of the sphere – apart from great circles passing through the poles – are projected upon circles. He did not prove this result, probably assuming that his readers would know how to derive it from Apollonius’s work. By the 1550s such an assumption was no longer reasonable, so Commandino provided a proof (f. 20r); this was presumably very close to the one Ptolemy had in mind …

“While working with the stereographic projection and conic sections, Commandino remarked that the sections can also be considered sections in visual cones or pyramids, in other words, as perspective images. This inspired him to take up a study of perspective and to devote the first nineteen folios of his comments on Ptolemy’s Planisphaerium to the subject …

“Commandino kept to the tradition of relating a perspective projection to figures rather than to points. The figures he treated were quadrangles, circles, and segments of circles. For each type of figure he went through many special cases, ending with the situation in which a figure is located arbitrarily in relation to the picture plane – and he even included the possibility that the figure is located in front of the picture plane …

“Having been motivated to study perspective by considering projections of circles, Commandino naturally paid much attention to the perspective images of circles, listing the conditions under which these would be either circles or ellipses (fol. 6r-6v, 12v-13r, 15v-16r, 17v). He also looked at segments of circles and included cases in which they would be mapped as parabolas or hyperbolas (fol. 6v-8v, 13v-14r, 16r). Although Commandino’s main concern was to throw plane figures into perspective, he also touched upon methods for constructing the perspective image of a polyhedron such as a pyramid or a cube …

“Since Commandino’s contribution to perspective was part of an academic work on an astronomical projection, written in Latin in a mathematically demanding style, its natural fate would have been to exist unobserved by practitioners of perspective, unknown to all but a handful of mathematicians. However, this does not seem to have happened. I have not noticed other mathematicians than Danti mentioning the work, whereas several mathematical practitioners did mention it – a phenomenon that can probably be attributed to Barabaro. In his Practica della perspectiva (1568), he referred scholars to Commandino’s work and warned non-scholars that it was difficult reading. The architects and artists presumably took Barbaro’s warning seriously and did no read Commandino, but they still mentioned him. They may have found it reassuring that a mathematician had proved at least some perspective procedures to be correct” (Andersen, pp. 138-145).

“The dedication to Ranuccio Farnese explains that fellow-mathematicians had told Commandino of the extreme difficulty encountered in reading Ptolemy’s Planisphaerium. Such difficulty arose in part out of the loss of the original Greek text; the extant treatise was merely a Latin translation (done by Hermann of Carinthia in 1143 on the basis of the ninth century Arabic version of Maslam of Cordoba, or Messal as Commandino calls him). However, one Balthasar Turrius Metinensis, skilled in medicine, philosophy and mathematics, had persuaded Commandino to read through the book and try to render it intelligible. Commandino had then found that the book required some elucidation of its scenographic (perspectival) ideas. Yet no classical treatment of scenography was extant, apart from the present treatise and this omitted or neglected much of the topic. Commanidno’s commentary seeks to remedy this lack. (These studies were later furthered by Commandino’s pupil Guidobaldo, whoc published his own treatments of the planisphere and scenography in 1579 and 1600.)” (Rose, p. 197).

The importance of spherical astronomy is reflected in the manifold of mathematical tools developed in order to solve its main problems. Eventually spherical trigonometry superseded all other methods which were, in all probability, invented before spherical geometry was sufficiently far advanced to reach numerical results. At that early stage the methods of descriptive geometry, reflected in the Analemma, might seem the most natural way to transform arcs on the sphere into arcs of one plane. Another work of Ptolemy, the Planisphaerium, shows, however, that also stereographic projection was known, in particular its important quality of mapping circles into circles, straight lines included … The projection chosen maps the whole sphere onto the plane of the equator with the south pole as center of projection. In order to determine the center and the radius of the image of a circle, descriptive methods are again employed …

‘The main goal of the whole procedure is the determination of the rising times for the zodiacal signs. In order to find the representation of these signs one has only to construct the parallel circles of given declination … Finally it is easy to construct the circle which represents the horizon for a given latitude. The variable positions of the ecliptic with respect to the horizon at different times of the year are in our projection represented by different positions of the horizon circle with respect to the fixed image of the equator-ecliptic system. In order to find the rising times of a given arc of the ecliptic we have only to construct the two positions of the horizon passing through its end points. These two horizon circles intersect the equator in two points whose angular distance is the rising time in question. Because angles on the equator are represented without distortion, our problem is solved by this construction.

“The above description shows that the ‘planisphaerium’ could be used for a purely graphical or mechanical solution of problems of spherical astronomy. This was indeed the use made of this method especially by the Arabs whose ‘astrolabes’ are based on the projections described here. In Ptolemy’s treatise, however, a different attitude is taken. The geometric constructions are only used for transforming spherical problems into problems of plane geometry which then are solved numerically by means of plane trigonometry. Obviously we have here before us the method used before spherical trigonometry was invented, that is, before the Menelaus theorem was known.

‘Much speaks in favor of the assumption that the planisphaerium was the tool of Hipparchus. All computations are based on the latitude of Rhodes, where Hipparchus made his observations. Synesius of Cyrene ascribes the invention of the ‘astrolabe’ to Hipparchus; this statement is certainly to be taken seriously in view of the fact that Synesius was a pupil of Hypathia, who collaborated with her father Theon on the commentaries to the Almagest. Finally, Hipparchus determines the positions of stars by a combination of ecliptic and equator coordinates; he takes the longitude of the point where the circle of declination through the star meets the ecliptic and then uses the remaining declination as second coordinate. This system finds its direct explanation in the planisphaerium: the first coordinate is given on the image of the ecliptic whereas the circles of declination are mapped into radii” (Neugebauer, pp. 1034-37).

“Although Jordanus has been justly proclaimed the most important mechanician of the Middle Ages and one of the most significant mathematicians of that period, virtually nothing is known of his life. That he lived and wrote during the first half of the thirteenth century, is suggested by the inclusion of his works in the Biblionomia, a catalogue of Richard de Fournival’s library compiled sometime between 1246 and 1260. In all, twelve treatises are ascribed to Jordanus de Nemore … The appellation ‘Jordanus de Nemore’ is also found in a number of thirteenth-century manuscripts of works attributed to Jordanus. The meaning and origin of ‘de Nemore’ are unknown. It could signify ‘from’ or ‘of Nemus,’ a place as yet unidentified (the oft-used alternative ‘Nemorarius,’ frequently associated with Jordanus, is apparently a later derivation from ‘Nemore’), or it may have derived from a corruption of ‘de numeris’ or ‘de numero’ from Jordanus’ arithmetic manuscripts” (DSB).

“The text [of Jordanus’s Planisphaerium] may be resolved into five propositions:

  1. A demonstration that circles on the sphere become, when projected, circles on the plane;
  2. and 3. On constructing parallels of given declination;
  3. On the equal division of an oblique circle (although it is not so specified, we may consider this circle the ecliptic or the horizon);
  4. On finding the position of a point whose coordinates with respect to a given oblique circle are known.

“Much of the material is clearly based on corresponding passages in Ptolemy's Planisphaerium, which was translated into Latin in 1143 by Hermann de Carinthia from the Arabic. For the proof that circles become circles, which unfortunately appears to be not quite sound, no sources have yet been found – al-Farghânî, for instance, supplied a different proof, based on a proposition in Apollonius' Conics. For Jordanus' fourth proposition, on the equal division of an oblique circle, three methods are given: by means of ascensions (if we may use the ordinary astronomical term), by declination circles, and by a special method involving the plane through the equinoxes which bisects the angle between the equator plane and the ecliptic plane … The first two methods may be taken from Ptolemy’s Planisphaerium. But all three methods are given in an extra chapter written by Maslama al-Majriti. This chapter, which is extant in Arabic, was translated into Latin in the twelfth century and is almost certainly the ultimate source, if not the immediate source, of Jordanus' three methods” (Folkerts & Lorch, p. 11).

Born in Urbino, Commandino studied Latin and Greek at Fano, then returned to Urbino where he studied mathematics. Later he studied medicine at Padua, and after returning home again he became personal physician to the Duke of Urbino. There he met Cardinal Ranuccio Farnese, the brother of the Duke's wife, who was to become his most important patron. In the early 1550s the Cardinal persuaded Commandino to move to Rome as his personal physician; while there he became friendly with Cardinal Cervini, who was elected Pope Marcello II in 1555. But following Cervini’s death shortly after his election, both Commandino and Farnese returned to Urbino, where Commandino continued in the service of the Duke and Cardinal. But Commandino’s true love was mathematics, and in 1558 he published his edition of Archimedes’ Opera, which he dedicated to Farnese (this did not contain any of Archimedes’ works on mechanics). Also in 1558 Commandino published Commentarius in Planisphaerium Ptolemaei, a work he had begun in Rome, and in 1562 he published his edition of Ptolemy’s work on the calibration of sundials, De Analemmate.

Adams P2242; Censimento 16; CNCE 28281; Renouard 1558/4; Ahmanson-Murphy/UCLA 449; Houzeau & Lancaster 769; Honeyman 2557; Riccardi I, 360, 1; Sarton I, 277 and II, 616; Andersen, The Geometry of an Art, 2006; Folkerts & Lorch, ‘The Arabic sources of Jordanus de Nemore,’ Foundation for Science, Technology, and Civilization (2007), pp. 1-15; Neugebauer, ‘Mathematical methods in ancient astronomy,’ Bulletin of the American Mathematical Society 54 (1948), pp. 1013-41; Rose, The Italian Renaissance of Mathematics: Studies on Humanists and Mathematicians from Petrarch to Galileo, 1975

Small 4to (211 x 156 mm), ff. [4, the last blank], 37 [Ptolemy 1-25, Jordanus 26-37], [1, printers device, recto blank]; [Commandino’s Commentary, with separate title-page and imprint:] 28. Contemporary limp vellum.

Item #2236

Price: $7,500.00