Cosmographicus liber Petri Apiani mathematici, iam denuo integritati restitutus per Gemmam Phrysium. Item eiusdem Gemmae Phrysij libellus de locorum describendorum ratione, & de eorum distantijs inueniendis, nunquam ante hac visus.

Antwerp: Johannes Graphaeus for Arnold Birckman, 1533.

First edition of Apianus’ cosmography to include Gemma’s highly important treatise on topographical triangulation, a landmark in the history of cartography – copies such as ours which are complete with all the volvelles are very rare. “The work that firmly established Apianus’ academic credentials was Cosmographicus Liber (1524) … it was a lavishly-illustrated treatise on astronomy, navigation, geography, cartography and weather; it contained digressions on various map projections, the shape of the Earth, and descriptions of the use of mathematical instruments … Cosmographicus Liber … was an immediate success enjoying at least 45 editions in four languages by at least 18 different publishers and remained in print for over a half-century after Apianus’ death. Gemma Frisius carried out a careful correction and annotation of the 1524 version; the result was published in 1529 … But it was the 1533 edition of Frisius’ annotated version, including his short works De locorum describendorum ratione (Concerning the method of describing places) and De eorum distantijs inueniendis (On the determination of distances), that earned the book its greatest popularity and secured its place in history” (Barentine, p. 151). On the De locorum describendorum ratione, Haasbroek (p. 11) notes: “its importance can hardly be gauged: for it revealed the final definite way of representing any country with its towns … by means of a series of triangles with one common basis which could be measured with preciseness so that it led to accurate distances and became the beginning of actual geography; subsequent times have only been able to add to it more facilities in the checking and the registering of the various elements”. Van Ortroy (pp. 58-9) notes that the principles of triangulation in it “sont absolument conformes à ceux de la planimétrie ou de la topographie moderne”.“During the first half of the sixteenth century Germany was the principal center of both mathematical and descriptive geography … [The] German school of geographers had its greatest exponents in Peter Apian (1501-52) and Sebastian Münster (1489-1552). Apian was an astronomer and mathematician; in his Cosmographicus Liber … subsequently edited by the great Flemish mathematician Gemma Phrysius [Frisius] (1508-55), he based the whole science on mathematics and measurements, following Ptolemy in making a distinction between geography (the study of the earth as a whole) and chorography (the study of specific areas). His work may best be described as a theoretical textbook; for a hundred years it was a standard source” (Penrose, pp. 308-9). “ … it is the book’s volvelles that represent its main selling point and principal innovation. Whereas earlier books of similar content were largely constructed around sets of tabular information, Apianus’ volvelles turned the pages of Cosmographicus Liber into functional computers, enabling skilled users to make calculations involving navigation, distances and time” (Barentine, p. 152). Apianus used volvelles so effectively in his work that they are sometimes known as Apian wheels. The paper volvelles also served a commercial purpose – they provided his readers with a model of the real instruments he could produce. ABPC/RBH list only four copies complete with all the volvelles and pasted-on woodcuts in the last 50 years, all in 19th or 20th century bindings.

“Petrus Apianus, also known as Peter Apian, Peter Bennewitz, and Peter Bienewitz, was one of the foremost mathematical publishers, instrument makers and cartographers of the sixteenth century. Born on 16 April 1495 in Leisnig, Saxony, he was one of four sons of Martin Bienewitz, a shoemaker of comfortable middle-class extraction. He was educated first at the Latin school in Rochlitz, and then from 1516 to 1519 at the University of Leipzig where he studied astronomy, mathematics, and cosmography. While at Leipzig, he Latinized his surname to ‘Apianus’, deriving from apis (‘bee’) and equivalent to Biene in German. Apianus relocated to Vienna in 1519 to complete his degree at the University of Vienna, taking a B.A. two years later during an outbreak of plague. Fleeing the city, he landed first in Regensburg before settling in Landshut. He married Katharina Mosner, the daughter of a local councilman, in 1526 and by her had fourteen children. Among his sons was Philip Apianus, born 1531, who would later follow his father into the study of mathematics.

Apianus was fascinated first and foremost by cosmography, a broad science of the Renaissance which set out to explain everything in the universe within a mathematical framework. He excelled in its study and later became one of its most famous practitioners; by modern standards, he can be thought of as one of the best applied mathematicians of his day. His interest in cartography was stimulated during one of the most momentous periods in European history: the Age of Exploration, witnessing the trailblazing voyages of the likes of da Gama, Columbus, and Magellan.

“The Cosmography instructs its readers on how to determine latitude, longitude and the time with the aid of instruments. For the measurement of longitude it proposes a recent technique involving an instrument for measuring angles called the Jacob’s staff. After having expanded the construction of a personal Jacob’s staff, Apian explains how longitude can be derived with the aid of the staff, astronomical tables and a little calculation. In itself his explanation is quite difficult to interpret, and fortunately Apian inserts an illustration of the construction and use of this instrument [f. XVIv]. However, on closer inspection this picture is more than a mere illustration. It provides a valuable compliment to the text by incorporating a visual explanation of the geometrical basis of this technique of finding longitude which the text does not even hint at. Apian makes two lines of vision intersect in the moon and shows by graduated marks on either side of the intersection that the difference between the angles of vision of two observers is equal to their difference in longitude. The geometry is extremely simple (the picture clearly shows the quality of opposing angles without verbalising it) but driving it from Apian’s text would be very difficult.

“The Cosmography also incorporates five volvelles, i.e. paper instruments with moving parts. The first of these is intended to show that the terrestrial latitude of a place is equal to the observed height of the world [f. IXv]. The accompanying text explicitly refers to Sacrobosco’s Sphere (book II), which contains a proof that ‘the elevation of the pole of the world above the horizon is as great as the distance of the zenith from the equator’ … The relevance of this proposition to the investigation of latitude was recognised by only some of Sacrobosco’s commentators, which indicates the disciplinary separation between late medieval astronomy and cosmography. Apian therefore translates Sacrobosco’s proposition to one concerning latitude by clarifying that the latitude of an observer is the arc between the pole of his horizon (zenith) and the terrestrial equator. More important for the matter at hand is the fact that Apian does not incorporate Sacrobosco’s verbal proof but instead uses the volvelle to make a visual point. The somewhat laborious reading of a simple but non-illustrated mathematical proof is replaced with a device which immediately shows the relation between terrestrial latitude and the height of the pole in a small model.

“Two other volvelles offered by Apian are two-dimensional models of the quantified path of the Sun across the sky over the Earth’s surface. Determining the quantity of solar movement in Apian’s time could proceed in two general ways: calculating the motion in astronomical tables or visualising it on an instrument. A distinguishing characteristic of mathematical instruments in Apian’s time is the visualisation of the relations between quantified data.

“These volvelles are conceptually based on familiar astronomical calculating instruments, but are modified to suit a didactic and cosmicgraphical purpose. The first of both volvelles, the so-called organum Ptolomei, may be regarded as a universal sundial designed for instruction [f. XIIv]. It has three movable parts: a rectangle representing the horizon, a triangle measuring the Sun’s altitude, and a rotatable disk indicating the Sun’s declination, its hour-lines and the altitude of the celestial pole. The double didactic efficacy of the instrument is remarkable. First of all it provides a two-dimensional model of the Sun’s annual and daily path across the heavens (the rotatable disk), related to the perspective of a terrestrial observer (the horizon). Compared to a regular sundial, this rare instrument employs a much more transparent projection which distinctly models and visualises its astronomical basis. As such it provides a powerful introduction to visual thinking about astronomical phenomena and relations. Learning this skill most probably was indispensable to a proper understanding of the principles of mathematical instruments in this period. On this basis the organun Ptolomei secondly instructs on the specific techniques of time measurement by means of the Sun’s movement. If properly positioned for a specific latitude, the intersection of the triangle with a grid of parallel- and hour-lines on the rotating disc works like an equatorial sundial, while the intersection of the horizon-line with this grid can easily be related to the set-up of a horizontal sundial. Apianus however did not provide a practical instrument: inserting it in a book and producing it in paper made it quite impossible to keep his instrument in the horizontal plane as was required.

“The second volvelle of this type is a paper astrolabe equipped with a geographical or map-plate [f. XXXIIr]. This exclusively cosmographical variant of the age-old astrolabe indicates the latitude and longitude of a region, the movement of the Sun as seen from the Earth, and the relative time in different parts of the Earth. Notwithstanding its typical plate, this model referred the reader of the Cosmography to a type of projection different from that of the organum Ptolomei but very common for designing astrolabes. It thus literally offered a different perspective to the forementioned visual thinking about astronomical concepts as well as a modest reference to more common astronomical astrolabes. In both cases the latter function can be interpreted as a reference to commercially available mathematical instruments. A remarkable example of this is a brass astrolabe made in 1560 by a genius Egidius Coignet from Antwerp, which has Apian’s geographical place on the front and the organum Ptolomei on the back. The link between this extraordinary instrument and Apian’s Cosmography is evident. The precise nature of the relationship however is not. Coignet seems to have made this instrument for a rich commissioner with a non-professional interest in instruments and exquisite design, who was acquainted with Apian’s Cosmography. The symbiosis between cosmography and instrument-design not only made cosmographical treatises depict actual instruments, but also led to occasional brass implementation of paper instruments contained in these treatises” (Vanden Broecke, pp. 139-141).

The two unused volvelles pasted onto f. 55r relate to the Instrumentum Syderale illustrated on the preceding page. This was an early form of ‘nocturnal’, an instrument used to determine the local time by measuring the relative positions of two or more stars in the night sky, in this case the last two stars of the Plough in relation to the Pole star. Apianus not only depicted the image of the Instrumentum, but also reproduced the individual components of a “nocturnal” to be cut out and assembled as illustrated. These two components were originally loosely inserted in the book, and so are often missing, but were often pasted in to avoid their being lost. Nocturnal’s had been used since the 12th century at least, but Apianus’s is perhaps the earliest printed illustration of the instrument in use.

Gemma Frisius was born in Dokkum in Friesland (now in the Netherlands) in 1508. He studied medicine in the Collegium Trilingue at Louvain (modern Belgium) from 1525, achieving an M.D. in 1536. Although Gemma is most remembered for his contributions to instrument making and cosmography, his formal training and occupation in medicine shows that most mathematical cosmography was then practiced outside universities. At 21 and while still a student, Gemma published his corrected edition of Apian’s Cosmographia (to be followed by several subsequent enlarged editions), and in the following year Deprincipiis astronomiae et cosmographiae. During these years and after, Gemma produced a great number of mathematical instruments, maps and globes. His new designs included the astronomical rings described in his editions of the Cosmographia and the new kind of cross-staff described in De radio astronomico (1545). During and after his schooling, Gemma ran an instrument-making workshop that achieved great renown. As late as the end of the century, Tycho commented on the quality and accuracy of Gemma’s instruments. Among his students were Gerard Mercator, John Dee, his nephew Arsenius and possibly the early English instrument maker Gemini. Like Apian, Gemma was patronized by the Emperor Charles V. Gemma Frisius died in 1555 at the age of 47 in Louvain.

The Libellus de locorum describendorum ratione has seven chapters. “In the first and most important chapter Gemma gives first a definition of what we call nowadays a magnetic bearing. Then he treats the principles of triangulation. They are illustrated with some drawings but there is not a single formula in the text … Two more accurate methods are described in the third and the fourth chapter” (Haasbroek, pp. 11 & 13).

“Chapter I. Topographical survey – with unknown longitudes, latitudes, and distances of the stations – was a novelty … The horizontal use of an astrolabe for the determination of azimuths of celestial bodies was centuries old; Gemma was the first to realize that by determining the bearings of terrestrial landmarks, and by repeating the observations at several stations, a network could be drawn on paper which would give, by the intersections of corresponding pointings, a map of the country surveyed. The instrument devised by Gemma for this graphic triangulation is a link between the astrolabe and our plane table … The choice of Antwerp as the first station was highly appropriate; so was the choice of Louvain, in the illustration of the ‘angulis positionis’ counted from the (magnetic) meridian of Antwerp. The value of the clear and methodical exposition is enhanced by the inclusion of stations (Middelberg and Bergen op Zoom) invisible from Brussels, the consideration of the case of the alignment of three stations, and the determination of the scale without measuring the baseline Antwerp-Brussels. Gemma points out that the illustration on fol. 59v is not a map …

“Chapter II. The elementary problem of plotting the positions of stations whose mutual distances – not directions – are known, is discussed here as a counterpart to the previous problem of mapping stations determined by their bearings only. Gemma is aware of the fact that the solution of the problem will meet with surveying difficulties if the mutual distances of the stations are to be measured ‘in oceano & inter montes.’ The author considered, apparently, chapter II as a ‘second method’ of making maps, and the two or three following chapters merely a special cases of surveying practice …

“Chapter III. Gemma gives here a solution of the surveying problem of determining – by means of similar triangles – the distance of an inaccessible object … Gemma informs the reader who should desire a mathematical proof that he has it in readiness for him …

“Chapter IV. The use, for surveying purposes, of the altimetric or shadow scales, in the lower quadrants on the reverse side of an astrolabe suspended by its ring, was not new. Gemma introduces here the new idea of using the altimetric scales in a horizontal plane for determining the distance of a remote object” (Pogo, pp. 475-7). For this method, “a ‘mathematical instrument’ must be used, the so called scala altimetra or scala geometrica … It consists of a cross-sight vane with which right angles can be set out. It is fastened on a staff ‘with a length of five or six feet’” (Haasbroek, p. 13).

“Chapter V. The relative positions and distances of the inaccessible points a, b, and c are determined by means of their bearings from the stations d and e at the end of the measured baseline de” (Pogo, p. 478) “After the angles have been drawn on the map the distances between arbitrary points can be scaled-off on an arbitrary scale. If one distance is known in the terrain all other distances can be computed by a proportion … he states that the distances can also be computed ‘with the tables of sine, but I omitted this intentionally as it is too difficult for the common man’” (Haasbroek, pp. 13-14).

“Chapter VI. The method of triangulation by distances and bearings … is an applications and combination of the precepts given in the previous chapters …

“Chapter VII. Approximate solution of the following problem: knowing the difference of latitude of two places, and their distance, to find their difference in longitude … No great precision could be expected from the solution offered: the two places had to be sufficiently close to justify the fiction of a plane rectangular triangle, and this implied a rather unreliable value of the difference in longitude” (Pogo, pp. 478-9). Gemma recognizes that “without deformation, the spherical earth cannot be represented on a flat map, not even ‘if Ptolemy would come back’” (Haasbroek, p. 14).

It appears that, possibly for reasons of ill health, Gemma himself never applied his triangulation techniques in practice. The first important practical application was made by Tycho Brahe (1546-1601), who carried out a triangulation of The Sound in Denmark, in which his island of Hven was situated. Tycho wanted to determine the longitude difference between his observatory Uraniborg on Hven and Copenhagen. “It is important to state that Tycho Brahe had connections with members of Gemma’s family. For, in his ‘Description of his instruments and scientific work’ he mentions that the radius astronomicus (cross-staff) and annulus astronomicus (astronomical ring) which he uses ‘are not constructed by myself but by Walter Arscenius, a grandson of the eminent mathematician Gemma Frisius who at one time lived in Louvain in Belgium’ … From the quotation ‘eminent mathematician Gemma Frisius’ it is clear that Tycho must have known Gemma’s work. Moreover, it was written in the language (Latin) which was accessible to him. If he had not known it from his own investigation – which is improbable – Arscenius would have drawn his attention to it. Therefore Tycho’s triangulation over The Sound in Denmark could probably be carried out because he knew the principles of triangulation which were published 45 years earlier in Gemma’s remarkable Libellus” (Haasbroek, p. 15). “Tycho’s triangulation from 1578 and 1579 included targets in the towns Helsingør, Køge, Copenhagen, Helsingborg, Landskrona, and Malmö. To determine the scale of the network, a baseline was measured on Hven. It was established between the center of Uraniborg observatory and the old Saint Ib church (middle of eastern tower). The length was 1,287.90 metres … Tycho published a map of Hven [which] is the earliest map of any part of the Scandinavian countries based on a geodetic survey” (Borre, pp. 75-6).

Harrisse, Bibliotheca Americana Vetustissima, 179; Houzeau & Lancastee 2392; JCB I, page 106; Sabin I, 1742; Van Ortroy, Apian 27 & Gemma 8. Barentine, Uncharted Constellations: Asterisms, Single-Source and Rebrands, 2016. Borre, ‘Fundamental triangulation networks in Denmark,’ Journal of Geodetic Science 4 (2014), pp. 74-86. Haasbroeke, Gemma Frisius, Tycho Brahe and Snellius and their Triangulations, 1968. Penrose, Travel and Discovery in the Renaissance 1420-1620, 1952. Pogo, ‘Gemma Frisius, his Method of Determining Differences of Longitude by Transporting Timepieces (1530), and his Treatise on Triangulation (1533),’ Isis 22 (1935), pp. 469-506. Vanden Broecke, ‘The use of visual media in Renaissance cosmography: the Cosmography of Peter Apian and Gemma Frisius,’ Paedagogica Historica 36 (2000), pp. 131-150.



Small 4to (207 x 155mm), ff. LXVI, title-page with large woodcut illustration of a globe, numerous astronomical and geographical woodcuts in text, those on ff. C1v, C4v, H4r, and O1r with intact and working volvelles, two unused woodcuts for the ‘Instrumentum Syderale’ pasted to f. LVr (light browning). Eighteenth-century stiff buntpapier wrappers (a little rubbed).

Item #5361

Price: $17,500.00