Planisphaeriorum universalium theorica.

Pesaro: Girolamo Concordia, 1579.

First edition, very rare, the Jones-Macclesfield copy, signed by William Jones, of the first modern theoretical treatise on the planisphere, an instrument for representing the celestial sphere on a plane surface. Guidobaldo was one of the most prominent figures in the renaissance of the mathematical sciences, and is famous for his Mechanicorum liber (1577), generally regarded as the most important treatise on mechanics since Archimedes and a critical influence on his friend and disciple Galileo. “Two years after the publication of the Mechanicorum Liber, Guidobaldo released the Planisphaeriorum Universalium Theorica (1579), re-edited after another two years later at Cologne. The treatise attended to the mathematical branch in which Guidobaldo seems to have been most interested besides mechanics, namely to perspective … Guidobaldo’s treatise of 1579 is subdivided in two books and is dedicated to the explication of various types of planispheres. Planispheres had the function to represent in the plane the celestial sphere with all his significant circles – a procedure that obviously posed problems relative to stereographic or orthogonal projection, according to the type of planisphere. Fundamental for their construction were the empirical guidelines given in Ptolemy’s Planisphaerium. A different way of stereographic projection had been found by Gemma Frisius (1508-55), exposed in De astrolabio catholico (1556), by assuming the centre of projection on the equinoctial circle, while in Ptolemy it was fixed in one of the two poles. The advantage of this method consisted in the possibility to adapt the planisphere to an arbitrary latitude (for that reason it was called ‘universal’), while Ptolemy’s was valid only for a specific horizon.In the first book, Guidobaldo attends to comment on Gemma Frisius’s planisphere furnishing geometrical demonstrations of what had remained unproven by the Dutch mathematician. Furthermore, he gave the necessary indications to construct the described device. In the second book, Guidobaldo approached the analysis of the planisphere of Juan De Rojas, considered the inventor of the universal astrolabe. As the projection exposed by the Spanish mathematician refers to a point of observation at infinite distance, his planisphere poses problems relative to orthographic projection. Guidobaldo addressed himself to the topic with the usual mathematical rigour, inter alia proving that the section of a cylinder and a plane (not parallel to the axis of the cylinder) is generally an ellipse – a fact unknown both to De Rojas and to Frisius. Here, again, he exposed a scientific instrument appropriate to draw ellipses, with a clear and detailed theoretical justification. This fact confirms Guidobaldo’s interest in the practical aspects connected with mathematics, besides his unquestioned skill to present theorisations of mathematical fields” (Frank, pp. 40-41). ABPC/RBH lists only one other complete copy in a contemporary binding in the last 40 years.

Provenance: William Jones (1675-1749) (signature on title), probably acquired by him from John Collins (1626-83); the Earls of Macclesfield (South Library bookplate on front paste-down, embossed stamp on first two leaves); sold Sotheby’s, 13 April 2005, lot 1434, £3,360 ($6,372). As the Macclesfield catalogue notes (A-C, p. 12), “It is not unreasonable to suppose that anything published before 1683 [in the Macclesfield library] belonged to him [i.e., Collins]”. It would seem very likely that Collins would have owned a copy of this book, given his own interest in mathematical instruments – he wrote The sector on a quadrant (1658); Geometrical dyalling (1659); and The mariner’s plain scale (1659). Jones acquired Collins’s books and papers some time before he edited Newton’s Analysis per quantitatum series (1711).

Painters were well acquainted with the representation of a three-dimensional object by three two-dimensional drawings, in each of which the object is viewed along parallel lines that are perpendicular to the plane of the drawing (plan, front and side elevations). For example, a projection by means of parallel lines was used by Albrecht Dürer in his Von Menschlicher Proportion in 1528. Dürer used an orthographic projection to foreshorten the heads of humans, showing a top, front and side view, a method already used by Piero della Francesca in his unpublished manuscript De Prospectiva Pingendi. Thus, as a drawing method, a projection along parallel lines must have been common practice in the painter’s workshop. However, it was left to the designers of mathematical instruments around the middle of the sixteenth century to understand the orthographic projection as a projection along parallel lines perpendicular to the plane of projection with the center of projection at infinity.

“Astrolabes had a plate showing a map of the celestial sphere. The equator, the tropics, the ecliptic, the lines of equal azimuth and the almucantars were projected onto the astrolabe plate. Most common, also in the sixteenth century, was the use of a stereographic projection, with the south celestial pole as center of projection and the plane of the equator (or a plane parallel to it) as the plane of projection. This type of stereographic projection was used by Stöffler in his Elucidatio fabricae ususque astrolabii (1513).

“However, a major drawback was that the astrolabe plate made with this type of stereographic projection could only be used at the particular latitude it was designed for – each latitude asked for a different plate. The sixteenth century saw the publication of two different types of projection that were considered to be a solution to this problem. These projections allowed a ‘universal’ application of the astrolabe, that is, only one plate was needed, whatever the latitude of the observer or user of the astrolabe.

“In 1556, the ‘astrolabum catholicum’ of Gemma Frisius was posthumously published. Gemma revived a universal astrolabe type known as the ‘sapheae arzachelis’, originally invented in the eleventh century by Ibn az-Zarqellu, an astronomer of Toledo. Gemma Frisius used a stereographic projection, but shifted the center of projection to one of the equinoxes. The colure of the solstices functioned as plane of projection. In this projection, the parallels and meridians became arcs of circles. Gemma’s ‘astrolabum catholicum’ was presumably meant as an alternative to another type of universal astrolabe, published by one of his students at Louvain, Juan De Rojas. With the help of Hugo Helt, De Rojas designed a universal astrolabe that used the orthographic projection to map the celestial sphere. The plane of projection was the solstitial colure, while the center of projection was considered to be at infinity and perpendicular to the plane of the solstitial colure. In this projection, the parallels became straight lines and the meridians were elliptical arcs, although the latter were only recognized as such by Guidobaldo del Monte in his Planisphaeriorum Universalium Theorica, published in 1579.

“The De Rojas astrolabe became popular in Florence around 1570, when Egnazio Danti had several of them made in the workshop of Giusti and devoted a chapter to it in his Trattato dell’ uso e fabbrica dell’ astrolabio (1567). Typically, the map of the celestial sphere on the orthographic projection appeared on the back of an ‘ordinary’ astrolabe with a map of the celestial sphere on the stereographic projection designed for a specific latitude on the front of the instrument” (Dupré, pp. 193-196).

Scientific instrumentation is an aspect of Guidobaldo del Monte’s work that is of considerable importance. His interest in this subject was undoubtedly influenced by the development in Urbino, in the mid-sixteenth century, of a workshop specializing in the construction of scientific instruments which soon gained wide fame. Simone Barocci (1525-1608) was the founder of the workshop and forefather of an important group of craftsmen-mechanics from Urbino. It is well-known that an important part of Guidobaldo’s work was spent, on a theoretical level, in an attempt to systematize the theory of simple machines. But in this axiomatic-demonstrative programme he never lost sight of the need to highlight the relation between theory and practice, thereby underlining the practical utility of the ‘mathematical disciplines.’ The need for scientific instruments was stimulated primarily by three factors: the existence of a high-level local scientific environment; the simultaneous formation of an advanced technical environment which produced a substantial increase in the demand for such instruments from topographers, agronomists, civil and above all military architects; and finally, wealthy clients, notably the Dukes of Urbino. Francesco Maria II himself owned clocks and mathematical instruments built by Barocci, and many of them were requested by nobles and prelates, or sent as gifts to the courts of Italy and Europe. It is in this favorable political framework, and of technical-scientific skills, that Guidobaldo’s interest in scientific instruments must be seen.

“In 1579 Guidobaldo published at Pesaro a work of considerable scientific depth, the Planisphaeriorum universalium theorica, divided into two books, where he tackles the much debated problem of representing the celestial sphere on the plane.The question had assumed practical importance with the spread of the astrolabe in the West by the Arabs.The use of the polar stereographic projection had made it necessary to equip astrolabes with an increasing number of discs, each of which provided the representation of the celestial sphere for a given latitude.This projection, used at the beginning of the sixteenth century by Johannes Stöffler in the construction of the astrolabe, had its historical roots in the Ptolemaic works De Analemmate (1562) and Planisphaerium, the latter later revived with a ‘commentarius’ by Commandino in 1558. It was therefore a subject that Guidobaldo knew well, having been extensively treated by his teacher [Commandino] with the geometric rigour that characterized Commandino’s work and that Guidobaldo had well assimilated. Attempts, around the middle of the sixteenth century, to introduce a graphical representation of the sphere on a plane that was independent of latitude prompted Guidobaldo to investigate the subject. The theories of the universal planisphere of Juan de Rojas Sarmiento and Gemma Frisius had been proposed by their authors without geometric proofs and for purely practical purposes. The Planisphaeriorum programme was created precisely to overcome these theoretical deficiencies. Guidobaldo tackles Gemma Frisius’s theory of the planisphere in the first book and the one proposed by Juan de Rojas in the second. The two constructions differ in the choice of the projective pole, which in the first case is identified with the equinoctial point, and in the second case with the same equinoctial point brought to infinity. Leafing through the work, one is struck by the almost maniacal precision with which he demonstrates the mathematical principles underlying the two theories, perhaps meticulously treated for educational needs and to facilitate greater dissemination. In this theoretical reworking, Guidobaldo does not neglect the aspect of the practical operation of the new planispheres by proposing the creation of two new drawing tools to facilitate the tracing of curves such as large radius arcs of a circle and arcs of an ellipse.

“In the first book of the Planisphaeriorum, Guidobaldo demonstrates for the first time that the projective elements of Gemma Frisius’s planisphere are straight lines and circles. In particular he provides the geometric demonstration that some meridians and parallels, those close to the diametrical axes of the planisphere, are arcs of circles with a large radius. Tracing these curves with precision, especially in planispheres of medium or considerable size (by medium size Guidobaldo means the diameter of a foot – the Pesaro foot measured 34cm), involved serious technical difficulties, because of the rather uncertain operation of identifying the center when there are large radial distances, and because of the lack of suitable instruments. To overcome these difficulties Guidobaldo invents an ad hoc instrument capable of drawing arcs of a circle passing through three given points when they are almost aligned. Its mechanical apparatus consists of a pair of wedge-shaped prisms and a triangular articulated system [figure, p. 54] formed by two rulers hinged at a pin B where a pointed stylus is also positioned. (Guidobaldo cites iron, bronze, or hard wood as the most suitable materials for the construction of the instrument.) To trace the arc of the circle passing through points V, I, X, first position the tip of the stylus at point I with the edges Y and Z of the two prisms respectively in contact with points V and X. By rotating the system, taking care to maintain constant contact between the edges and sides of the rulers, the stylus will trace the desired curve. Guidobaldo describes and draws the instrument very accurately, to make its construction as easy as possible.

“In the second book of the Planisphaeriorum Guidobaldo demonstrates for the first time that in the equinoctial orthographic projection of the de Rojas planisphere the meridian circles are arcs of ellipses. (Gemma Frisius, while acknowledging that these are not circles, speaks of ‘anomalous curves’;De Rojas, on the other hand, leaves them without a name but at least has the merit of drawing them in the correct form.) He then describes two graphic methods for drawing an ellipse, the first of which, classical, uses one stylus and a line of constant length. In providing the second method, Guidobaldo describes the ‘ellipsograph’, an instrument he designed to draw arcs of an ellipse. Hitherto these curves had been drawn point by point, or with inaccurate methods. Among his contemporaries who used the point by point method, Guidobaldo cites Dürer and Commandino.Albrecht Dürer had described in his Institutionum Geometricarum (1532) a method for drawing conics.Commandino had done the same in De Horologiorum Descriptione (1562), in the description of horizontal sundials, where he had provided a theorem to draw the ellipse point by point.Commandino had limited himself to providing a geometric theorem; Dürer had built a real apparatus which, according to him, drew ellipses but which in reality produced other curves that twisted on themselves. Guidobaldo, aware of this, proposed a new apparatus capable of accurately drawing quarters of an ellipse in a continuous way [figure, p. 126]. Simon Stevin mentions in his Mémoires mathématiques (1605-1608), vol. II, book I, the method used by Guidobaldo, showing that Guidobaldo’s work was known and appreciated.

“The instrument consists of a square and a ruler in which a groove is made in which two cursors can slide and be locked in any position.A stylus can be fixed at will in one of the two holes at the ends of the ruler depending on whether you want to draw the upper or lower quarter of an ellipse (the distance of each hole from the farthest cursor is the length of the semi-major axis of the ellipse).Initially, the distance between the sliders is adjusted so that they are spaced according to the difference between the two axle shafts.Once the square is positioned along the half-axes of the ellipse, one simply slides the cursors along the sides of the square to trace the quarter of the ellipse.

“One thing that is especially notable is the way in which Guidobaldo presents the ellipsograph, providing images of the complete instrument, of some of the parts that compose it, even using exploded views (figures, pp. 105-106); this is almostunique in a text of this period. In short, Guidobaldo provides detailed information on the construction of the instrument in the workshop. The clear intention is to increase its diffusion.Perhaps this is why the Planisphaeriorum was reprinted in Cologne two years later.

“The second point to note is the ‘derivation’ of Guidobaldo’s ellipsograph from the instrument devised by Nicomedes (d. 74 BCE) to trace the conchoid. An image of this instrument can be found in the General Trattato (1560) by the Brescia mathematician Nicolò Tartaglia (1499-1557). This instrument had inspired the construction of a special compass by a pupil of Regiomontano, namely Johann Werner (1468–1528), described in a libellus of 1522. Probably this last apparatus had also provided the inspiration for Dürer’s. It is evident that analytically the conchoid has very little to do with the ellipse, but in mechanical configuration the two instruments are very similar.In fact, it is enough to make the fixed point mobile for Nicomedes’ instrument to trace ellipses rather than conchoidal ones.This is interesting because it shows how multipurpose tools were developed. Finally, it should be emphasized that Guidobaldo’s instrument has the important advantage of drawing the ellipse starting from knowledgepredetermined by the semi-axes rather than by the two foci ‘quia vero in astrolabio ellipsis describendae semper dati sunt axes’ (p. 102)” (Gamba & Mantovani, pp. 211-217, our translation).

Monte (1545-1607) studied mathematics at Padua and later at Urbino became the friend and pupil of Federico Commandino, whose translation of Pappus he edited and published. Galileo wrote to him, and thus began an exchange of correspondence on scientific matters. Monte secured for Galileo an appointment to the chair of mathematics at Pisa and later at Padua. “Guidobaldo was Galileo’s patron and friend for twenty years and possibly the greatest single influence on the Mechanics of Galileo. In addition to giving Galileo advice on statics, Guidobaldo discussed projectile motion with him, and both scientists reportedly conducted experiments together on the trajectories of cannonballs” (DSB).

Censimento 16 CNCE 16712; Cinti 4; Riccardi II, 179; not in Vagnetti or Mortimer, Italian. Dupré, Galileo, the Telescope, and the Science of Optics in the Sixteenth Century, 2002. Frank, Guidobaldo dal Monte’s Mechanics in Context, 2011. Gamba & Mantovani, ‘Gli strumenti scientifici di Guidobaldo del Monte,’ pp. 209-128 in: Guidobaldo del Monte (1545-1607). Theory and Practice of the Mathematical Disciplines from Urbino to Europe (Becchi, Meli & Gamba, eds.), 2013.

4to (264 x 183mm), pp. [vii], 128, [1, errata & colophon], [3, blank], with woodcut of a planisphere on title and numerous fine diagrams and illustrations in the text (title page a little soiled and with small old repair at foot). Sixteenth-century English blind-ruled calf. A fresh and exceptionally large copy, with some deckle edges.

Item #5426

Price: $12,500.00