Nova Stereometria doliorum vinariorum, in primis Austriaci, figurae omnium aptissimae; et usus in eo virgae cubicae compendiosissimus & plane singularis. Accessit Stereometriae Archemedeae supplementum...

Linz: Johann Planck, 1615.

First edition, the fine Huet-Honeyman-Tomash copy, of Kepler’s contribution to the mathematics of integration techniques, an important precursor to the calculus. This is one of very few copies with the errata leaf. This copy is also printed on superior paper stock, quite white and crisp, whereas most copies are on a poor quality paper and often browned. The Nova Stereometria is “generally regarded as one of the significant works in the prehistory of the calculus” (DSB). Kepler “made wide application of an old but neglected idea, that of infinitely great and infinitely small quantities. Greek mathematicians usually shunned this notion, but with it modern mathematicians completely revolutionized the science” (Cajori). Kepler “employs primitive integration techniques in attempting to find volumes of bodies with curved surfaces, his researches in this area having been spurred by comparison of the current methods used to find the volume of wine casks with the work of Archimedes on volume measurement. Kepler views solids as composed of infinitesimal pieces and proceeds to determine volumes of various solids of revolution, some not considered by Archimedes” (Parkinson, Breakthroughs). “Desiring to outfit his new household with the produce of a particularly good wine harvest, Kepler installed some casks in his house. When he discovered that the wine merchant measured only the diagonal length of the barrels, ignoring their shape, Kepler set about computing their actual volumes. Abandoning the classical Archimedean procedures, he adopted a less rigorous but productive scheme in which he considered that the figures were composed of an infinite number of thin circular laminae or other cross sections. Captivated by the task, he extended it to other shapes, including the torus” (DSB). “Kepler opens his work on curvilinear mensuration with the simple problem of determining the area of a circle. In this he abandoned the classical Archimedean procedures. He did not substitute for these the limiting consideration proposed by Stevin and Valerio, but had recourse instead to the less rigorous but more suggestive approach of Nicholas of Cusa. Like Stifel and Viète, he regarded the circle as a regular polygon with an infinite number of sides, and its area he therefore looked upon as made up of infinitesimal triangles of which the sides of the polygon were the bases and the center of the circle the vertex. The totality of these was then given by half the product of the perimeter and the apothem (or radius). Kepler did not limit himself to the simple proposition above, but with skill and imagination applied this same method to a wide variety of problems” (Boyer, The History of the Calculus and its Conceptual Development). Boyer also examines the influence of this work on Fermat, Cavalieri, Guldin, and finally Leibniz and Newton: “It is Kepler’s mode of expression which appeared in the work of Fermat. Although the Scholastic views on variation played a significant role in the anticipations of the calculus, the static approach of Kepler predominated. Increments and decrements, rather than rates of change, were the fundamental elements in the work leading to that of Leibniz, and played a larger part in the calculus of Newton than is generally recognized”.

Provenance: Pierre Daniel Huet (1630-1721), bishop of Avranches, with printed shelfmark label ‘XLVII.C’ on front pastedown; living gift of his library to the Jesuit order of Paris in 1692, with manuscript inscription of gift on title and the Jesuits printed label ‘Ne extra hanc Bibliothecam efferatur. Ex obedienta’ on foot of title, with manuscript shelfmark on front flyleaf; Honeyman sale, Sotheby’s 12 May 1980, lot 1791 (with Honeyman ex libris on front pastedown); Bernard Quaritch collation note dated May 1981 on rear pastedown. Huet, ‘un des hommes les plus savants de France’ (NBG), was an accomplished mathematician and celebrated scholar and author. He studied mathematics and wrote a critique of Cartesian philosophy. ‘His great library and manuscripts, after being bequeathed to the Jesuits, were bought by the King for the Royal Library’ (Encyclopaedia Britannica 11th edn.); Erwin Tomash (bookplate on front pastedown); his sale, Sotheby’s 18 September 2018, lot 311.

Kepler (1571-1630) became interested in stereometry as a result of a serendipitous event that took place in November 1613 in Linz, where Kepler was then living. Kepler purchased some barrels to lay in a supply of wine for his family and had them delivered to his house. When the wine dealer came to the house to measure the volume of wine the barrels contained, he used the standard gauger’s technique which in effect meant approximating the barrel by a cylinder of the same height as the barrel but with cross-sectional area equal to the average of the area of the ends of the barrel and that of its middle bulge. Thus, the approximate formula for the volume of the barrel was

V = ½ x height x (end-diameter2 + bulge-diameter2) V0,

where V0 was the known volume of a cylinder of unit height and diameter. To simplify the calculation a gauging rod was used. This was a rod marked with a quadratic scale (i.e., 1 at the first mark, 4 at the second, 9 at the third, etc.); by laying it across the end of the barrel, then inserting it through the bung hole in the middle of the bulge, and reading off the numbers on the scale, the gauger could then calculate an approximation to the volume of wine in the barrel by using the above formula. Kepler was more than sceptical about the accuracy of this method of volume determination, especially how it could work for barrels of any shape and size, and he immediately decided to try to find a better mathematical method, and one that would also deal with the case of partly empty barrels, for which the gauger had no solution.

By December 17, 1613 Kepler believed that he had reached his goal. He had composed a short six-page manuscript with about ten theorems, and with a dedication to Prince Maximilian of Liechtenstein and Baron Helmhard Jörger as a New Year gift. Around February 1, 1614 Kepler sent the manuscript to Markus Welser in Augsburg to have it printed, there being at that time no printer in Linz. On February 11, Welser replied that he had received the manuscript and had discussed publication with the Augsburg bookseller Hans Krüger. Although he accepted that Kepler’s name was highly respected in academic circles, Krüger felt that the work would sell poorly, especially in Latin. Krüger accepted Welser’s suggestion that the book be printed at Kepler’s expense, but before a printer could be found Welser died on June 23. The manuscript remained with Krüger and may have been forgotten by Kepler had a new printer not arrived in Linz.

In the spring of 1615, the printer Hans Blanck or (Planck) arrived in Linz from Erfurt and offered his services. Kepler remembered his manuscript, which was still in Augsburg, and demanded it back. He received it with much effort at the end of May (1615), but immediately realised that it was unsatisfactory. As well as being too brief, there was also a serious mistake in it, and he was forced to rewrite it. The new version of Stereometria grew considerably compared to the original, the Supplementum ad Archimedem was added, as well as the entire second part. But by July 15, after only six weeks, the work was done. The Stereometria doliorum, the first book printed in Linz, was available at the autumn 1615 book fair at Frankfurt.

“In this work Kepler uses a wide range of methods including visual imagery, geometric transformation, analogy and tabulation but most of all the cutting of small sections varying in size and shape, parallel to no given direction and chosen at will in the most convenient form to meet the needs of a particular problem. Probably no one since Kepler has used infinitesimals quite so freely” (Baron, p. 110). Kepler’s stereometrical work “exerted such a strong influence in the infinitesimal considerations which followed its appearance, and which culminated a half century later in the work of Newton, that it has been called [by Moritz Cantor] the source of inspiration for all later cubatures" (Boyer, p. 110). It also contains the germ of the differential calculus: Kepler “showed, among other things, that of all right parallelepipeds inscribed in a sphere and having square bases, the cube is the largest, and that of all right circular cylinders having the same diagonal, that one is greatest which has the diameter and altitude in the ratio of √2 : 1. These results were obtained by making up tables in which were listed the volumes for given sets of values of the dimension … He remarked that as the maximum volume was approached, the change in volume for a given change in the dimensions became smaller” (ibid.). Kepler had noted, in modern terms, that when a maximum occurs the rate of change becomes zero, a basic principle of the differential calculus that is usually credited to Fermat later in the century.

In the first and longest part of the Nova stereometria Kepler introduces an ingenious ‘unfolding’ technique for calculating areas and volumes, with which he obtains certain results which Archimedes had obtained by a reductio ad absurdam argument (Kepler refers explicitly to Archimedes’ Dimension of the Circle, Quadrature of the Parabola, etc.). The simplest case is that of a circle, which can be divided into an infinitely large number of triangles, each with one vertex at the centre of the circle, the other two vertices being infinitesimally nearby points on the circumference. Unfolding the circumference of the circle onto a straight line gives a series of triangles all with the same height equal to the radius of the circle. Since the area of a trangle equals ½ x base x height, the total area of the unfolded triangles is ½ x circumference x radius, which is the correct formula for the area of a circle. “The properties of circular cylinders, cones and prisms can all be derived by similar considerations. The sphere can be regarded as made up of infinitely many small cones, each with its base on the surface of the sphere and its vertex at the centre …

“[In the Supplementum ad Archimedem,] Kepler considers the rotation of certain closed figures about axes in the plane. Whereas Archimedes had confined himself to rotations about the principal axes, Kepler generates a variety of different solids by rotating about lines in the plane other than the principal axes. A solid formed by rotating a plane figure about a line in the plane which does not intersect the curve is called a ring. If the line is parallel to an axis of symmetry the volume of the ring is equal to that of a cylinder with height equal to the circumference of the circle described by the centre of the figure and with base equal to the section itself” (Baron, pp. 110-111). Kepler proved this by cutting the ring into infinitely many thin slices by planes passing through the axis. Similar but more difficult techniques can be used when the axis of rotation does intersect the figure being rotated. As examples, Kepler determined the volume of solids obtained by the rotation of circular segments around chords of the circle, which produced a variety of solids which he named after fruits: rotating a circle about a chord of the circle gave an apple; rotating the minor segment cut off by the chord about the chord gave a lemon; there was also a plum, a spindle, etc. “In all Kepler considers the volumes of 96 different solids obtained by rotating segments of conic sections (ellipses, parabolas, hyperbolas) about various axes” (Baron, p. 115).

In Part 2 Kepler explored the solid geometry of the Austrian barrel, approximating the barrel as a cylinder or as two truncated cones joined at their larger base. He studied the dependence of the volume of a barrel with fixed diagonal on the ratio of the stave length to the diameter of the bottom, and of the largest diameter of the belly to that of the bottom. For Austrian barrels, it was the rule among coopers that the radius of the barrelhead should be one-third of the length of the staves, whereas for Rhineland casks the radius of the barrelhead was equal to one-half of the length of the staves. Kepler showed that the volume of the ‘truncated lemon’ (or ‘hyperbolic spindle’), which he was able to calculate, gave the best approximation for the Austrian barrels.

In part 3, Kepler showed that the Austrian barrel was the most reliable shape because slight errors in construction would have minimal effect on its volume as measured by the gauging rod. Other shapes were less reliable, a conclusion that must have delighted Kepler’s Austrian patrons. In discussing how to find the volume of a barrel of arbitrary shape Kepler described a tool for recording the cross-sectional shape of a surface, very like the modern ‘contour gauge.’

“Kepler denounces the imprecise use in these [wine] casks of the ‘cubic ruler’ designed to measure their content. He saved the explanation of this ruler – a wooden stick, which, he says, is supplied with each cask without variation or calculation, and regardless of the cask’s shape – for the book he went on to publish a year later, Ausszug auss der uralten Messekunst Archimedis” (Radelet-de Grave, p. 60).

The Nova stereometria did not sell well. “Kepler had presented a copy of the Stereometria to the Dantzig mathematician Crüger [i.e., Peter Crüger (1580-1639), who had been a student of Kepler] with the request to advertise his work. Crüger replied (letter dated July 31, 1616) that apart from him only his Königsberg colleague, the Königsberg library and a Prussian noble named Niewieschinsky were buyers. Nevertheless, Kepler apparently got his money’s worth. On June 17, 1616, he wrote back to Crüger: ‘You are giving me to understand that there is no prospect of selling copies. But that does no harm, I had already forgotten about this matter. I wrote for the Austrians; they have already paid and will pay a little more’” (Hammer, our translation).

Kepler’s patrons were also unimpressed with the work, and they advised their mathematician to concentrate on the matter specified in his contract: the completion of the Rudolphine Tables. Kepler, however, immediately began to rework the book for those not proficient in Latin or advanced mathematics, recognising that it was results rather than derivations that were important for gaugers. This appeared in the following year as Ausszug auss der uralten Messekunst Archimedis vnd deroselben newlich in Latein aussgangener Ergentzung.

The difficulties Kepler faced in promoting the sale of the Stereometria accounts in part for its rarity. As Casper notes in his bibliography, Kepler found only four purchasers for the work in the whole of Prussia. A few copies, such as this one, contain a quarter-sheet errata leaf.

BL 17th C. German K116; Caspar 48 (43 copies located, the great majority in German and other European libraries); Cinti 55; Parkinson p. 68; Smith Rara p. 416. Baron, The Origins of Infinitesimal Calculus, 1969. Boyer, The History of the Calculus and its Conceptual Development, 1949. Hammer, ‘Nachbericht,’ in: JohannesKepler Gesammelte Werke IX, 1960. Radelet-de Grave, ‘Kepler, Cavalieri, Guldin. Polemics with the departed,’ pp. 57-86 in: Seventeenth-Century Indivisibles Revisited (Jullien, ed.), 2015.

Folio (306 x 198mm), ff. [56], with additional errata leaf at end (printed on shorter paper), with numerous woodcut illustrations in the text (the large woodcut on H4 verso slightly cropped at foot as usual, small piece missing from upper margin of title). Contemporary vellum, spine titled in manuscript with shelf-mark ‘8’ at foot (spine darkened, small piece of vellum missing from foot of front board, front inner joint cracked but holding firm). Preserved in a characteristic red morocco-backed slipcase (made for Honeyman). An excellent copy with no restoration and fine provenance.

Item #5949

Price: $150,000.00