Nova stereometria doliorum vinariorum, in primis Austriaci, figurae omnium aptissimae; et usus in eo virgae cubicae compendiosissimus & plane singularis. Accessit stereometriae Archimedeae supplementum.
Linz: Johannes Plancus, for the author, 1615. First edition, the Huet–Honeyman–Tomash copy on superior paper and with the rare quarter-sheet errata leaf, of Kepler’s great treatise on the volumes of solids of revolution — “one of the significant works in the prehistory of the calculus” (DSB). The first part is on Archimedean stereometry together with a supplement of some ninety-six solids not treated by Archimedes; the second on the gauging of Austrian wine casks; the third on the principles by which the Austrian shape minimises the error of the gauger’s rod. The treatise was the first book ever printed at Linz, where Kepler had served since 1612 as Mathematician of the Upper Austrian Estates. The genesis is anecdotal and characteristic. In November 1613 Kepler, freshly remarried and laying in a supply of wine for his household, watched the Linz wine merchant measure each cask with the standard gauger’s rod — a stick slipped diagonally through the bung-hole and read off against a quadratic scale, by which the volume of any barrel was approximated as half the height multiplied by the squared sum of the end-diameter and the bulge-diameter. The merchant’s formula treated barrels of every shape alike, and made no provision for the partly empty cask. Kepler thought he could do better. By 17 December he had drafted a six-page manuscript of about ten theorems, with a New Year dedication to Prince Maximilian of Liechtenstein and Baron Helmhard Jörger, and the following February he sent it to Markus Welser at Augsburg in the hope of having it printed there. The Augsburg bookseller Hans Krüger thought the work would sell poorly, especially in Latin, but accepted Welser’s suggestion that it be printed at Kepler’s expense. Welser died on 23 June 1614 before a printer was found, and the manuscript languished with Krüger until the spring of 1615, when the printer Hans Blanck (or Planck) arrived in Linz from Erfurt and offered his services. Kepler demanded the manuscript back, recovered it at the end of May, and on rereading it judged it both too brief and seriously mistaken. He rewrote it on the spot. The new version expanded considerably, the Supplementum ad Archimedem and an entire second part were added, and within six weeks the work was complete. The Stereometria doliorum was on the autumn 1615 book fair tables at Frankfurt and was the first book ever printed in Linz. The treatise opens with the simplest case of curvilinear measurement, the area of a circle. Kepler abandons the classical Archimedean reductio ad absurdum and follows instead the more suggestive approach of Nicholas of Cusa, regarding the circle as a regular polygon with infinitely many sides. The area is then made up of infinitesimal triangles whose bases are the sides of the polygon and whose common vertex is the centre, so that, by the triangle formula, the total is half the product of the perimeter and the radius. Unfolding the circumference onto a straight line gives a row of triangles all of the same height equal to the radius, and the formula is read off directly. Cylinders, cones, prisms, and the sphere all yield to similar reasoning: the sphere can be regarded as composed of infinitely many small cones, each with its base on the surface and its vertex at the centre, and its volume is one-third the surface times the radius. The same divide-and-sum technique extends to the segments and zones of the sphere, the cone, and the conoids, and Kepler is careful in each instance to compare his results against those of Archimedes (whom he cites by chapter and verse) and Stevin and Valerio. He does not, like the limit-takers Stevin and Valerio, attempt to make the proofs rigorous in the modern sense; the strength of the method is not its rigour but its imagination, and like Stifel and Viète before him Kepler is content to read off the answer once the figure has been imagined apart into its infinitesimal pieces. In the Supplementum Kepler turns to solids generated by rotation. Whereas Archimedes had confined himself to rotations about a figure’s principal axes, Kepler generates a wider variety of solids by rotating about lines in the plane other than the principal axes. A solid formed by rotating a plane figure about an axis that does not intersect the curve he calls a ring; if the axis is parallel to an axis of symmetry, the volume of the ring equals that of a cylinder whose height is the circumference described by the centre of the figure and whose base is the section itself — an anticipation of what would later be called Pappus’s or Guldin’s theorem. Kepler proved this by cutting the ring into infinitely many thin slices by planes through the axis. The harder case in which the axis of rotation intersects the figure receives similar but more elaborate treatment, with the slicing adapted to the geometry at hand. The most charming feature of the Supplementum is Kepler’s nomenclature for the solids so generated. Rotating a circle about a chord gives an apple; rotating the minor segment cut off by the chord, about that chord, gives a lemon; further variations yield a plum, a spindle, and others. In all, Kepler computes the volumes of ninety-six different solids obtained by rotating segments of conics — circles, ellipses, parabolas, hyperbolas — about a variety of axes. Baron observes that probably no one since has used infinitesimals quite so freely. Kepler’s catalogue of fruit-shaped solids has remained part of the technical vocabulary of solid geometry: the apple and lemon surfaces still bear his names in modern reference works. The work also contains the germ of the differential calculus. By tabulating the volumes of right parallelepipeds inscribed in a sphere with square bases, Kepler showed that the maximum volume occurs when the inscribed solid is a cube; for right circular cylinders inscribed with a fixed diagonal, the maximum is reached when the diameter and altitude are in the ratio √2 : 1. He observed that as the dimensions approached the optimum the corresponding change in volume became smaller and smaller. In modern terms he had noticed that at a maximum the rate of change vanishes, the basic principle of differential calculus that is usually credited to Fermat later in the century. The observation is unobtrusive and is buried in the table; but it is there. The second part of the Nova stereometria applies these methods to the Austrian barrel itself. Kepler models the cask as a cylinder, or as two truncated cones joined at their larger bases, or as a hyperbolic spindle — the ‘truncated lemon’ whose volume he is in a position to calculate exactly. He examines the dependence of the volume of a barrel of fixed diagonal on the ratio of stave-length to the diameter of the bottom, and on the ratio of the diameter at the bulge to the diameter at the head. The Austrian rule was that the radius of the head should be one-third of the length of the staves; for Rhineland casks, the proportion was one-half. Of all the cask shapes he examined, Kepler showed that the truncated-lemon was the best approximation to the Austrian. The third part demonstrates a more delicate result: the Austrian barrel was the most reliable shape because slight errors in its construction would have minimal effect on its volume as measured by the gauger’s rod. Other shapes were less robust, and small departures from the cooper’s prescription could produce gross discrepancies in the volume read out by the rod. The conclusion was a neatly turned compliment to Kepler’s Austrian patrons. In passing, in his discussion of how to measure casks of arbitrary shape, Kepler describes a tool for recording the cross-sectional outline of a curved surface that is essentially the modern contour gauge. Through it all Kepler denounces, as Radelet-de Grave observes, the imprecise use of the gauger’s ‘cubic ruler’, supplied with each cask without variation or calculation regardless of the cask’s shape; he saved the explanation of this rod for the German vernacular abridgement he published the following year. Cantor, quoted by Boyer, called the Nova stereometria the source of inspiration for all later cubatures, and Boyer himself shows that Kepler’s mode of expression reappears in the work of Fermat and that the static, infinitesimal-increment approach of Kepler — rather than the kinematic Scholastic doctrine of variation — predominated in the work leading to Leibniz, and played a larger part in the calculus of Newton than is generally recognised. The Supplementum was the immediate point of departure for Cavalieri’s Geometria indivisibilibus of 1635, which generalised Kepler’s technique into the method of indivisibles; Cavalieri’s Jesuit critic Paul Guldin, while attacking the rigour of indivisibles in his Centrobaryca (1635–1641), attacked Kepler with equal vigour, and the polemic between them helped to define the terms in which the seventeenth-century calculus would be debated. The line of descent through the seventeenth century runs from Kepler to Cavalieri, from Cavalieri to Torricelli and Roberval, and through them to Wallis’s Arithmetica infinitorum of 1656 and Barrow’s Lectiones geometricae of 1670 — the works which Newton read as a Cambridge undergraduate and which placed before him the integral calculus in a form ready to be united with the differential calculus he was about to invent. Leibniz, working independently on the Continent, drew on the same Cavalierian inheritance through the works of Pascal and the Huýgenian school. The Nova stereometria stands at the head of this tradition; the result, as Boyer notes, is that the static approach — the resolution of a figure into infinitesimal pieces and the summing of those pieces — rather than the kinematic approach of fluxion and rate of change, became the fundamental conceptual idiom of the integral calculus until Newton and Leibniz reunited the two strands at the close of the century. The book also bears on the prehistory of the theorem usually attributed to Pappus and rediscovered by Guldin: that the volume of a solid of revolution is equal to the area of the generating figure multiplied by the circumference traced out by its centre of gravity. Kepler’s slicing of a ring into thin parallel sections, each treated as a thin cylinder of small height, is the geometrical kernel of that theorem in a special case, and it was Guldin’s rediscovery and proof of the more general statement — published in the Centrobaryca shortly after Kepler’s death — that gave the theorem its modern name. Whether Guldin had taken the idea from Kepler, as some of his contemporaries supposed, or had reached it independently from Pappus, was already a matter of polemic in the 1630s. The Nova stereometria did not sell. Kepler had presented a copy to Peter Crüger of Danzig — a former pupil — with a request to advertise the book; Crüger replied on 31 July 1616 that beyond himself only his Königsberg colleague, the Königsberg library, and a Prussian noble named Niewieschinsky had bought the work. Kepler’s reply of 17 June 1616 is unbothered: he had forgotten the matter, having written for the Austrians, who had paid and would pay. Kepler’s patrons were unimpressed, and pressed him to attend instead to the matter specified in his contract, the Rudolphine Tables. Kepler began at once to recast the work for those without Latin or advanced mathematics, and the result appeared the following year as the Ausszug auss der uralten Messekunst Archimedis — results without derivations, written for gaugers and coopers, and supplying the explanation of the gauger’s rod that Kepler had withheld in the Latin text. The German Ausszug was no more commercially successful than the Latin original, and Kepler returned to the labour of the Tables with the consolation that he had given the Austrian cooper a more accurate technique than he had asked for, and the European mathematician something rather more important than that. The poor sale accounts in part for the rarity. Caspar’s bibliography locates forty-three copies, the great majority in German and other European libraries; only a handful, of which the present is one, contain the quarter-sheet errata leaf. Most surviving copies are on a poor-quality paper which has darkened severely; the present copy is on superior paper stock, crisp and quite white throughout. The large woodcut on H4 verso is, as usual, slightly cropped at the foot. Cinti, Smith, and the British Library’s catalogue of seventeenth-century German books all record the work; the present copy belongs to a small group with the errata that includes the Honeyman copy now identified as the same. Provenance: Pierre-Daniel Huet (1630–1721), bishop of Avranches, with the printed shelfmark label ‘XLVII.C’ on the front pastedown; donated in his lifetime, in 1692, to the Jesuit professed house in Paris, with the manuscript inscription of gift on the title and the printed Jesuit caution ‘Ne extra hanc Bibliothecam efferatur. Ex obedientia’ at the foot of the title, with the Jesuit manuscript shelfmark on the front flyleaf; passed to the King after the dissolution and the dispersal of the Jesuit library; Honeyman, his sale, Sotheby’s 12 May 1980, lot 1791 (with Honeyman ex libris on the front pastedown); Bernard Quaritch, with their collation note dated May 1981 on the rear pastedown; Erwin Tomash, with bookplate on the front pastedown; his sale, Sotheby’s London, 18 September 2018, lot 311. Huet himself, ‘un des hommes les plus savants de France’ in the words of the Nouvelle biographie générale, was an accomplished mathematician, the editor of the Delphin Classics, and a celebrated anti-Cartesian; he studied mathematics seriously and wrote a critique of Cartesian philosophy. He bequeathed his great library and manuscripts to the Jesuits, and they were bought by the King for the Royal Library on the dissolution of the order. The Honeyman, Tomash, and present provenance places this copy among the small handful of large-scale scientific libraries through which the book has descended over the four centuries since Linz; the combination of the Huet shelfmark, the Jesuit caution, the Honeyman ex libris, the Quaritch collation note, and the Tomash bookplate make for one of the most fully documented provenances of any copy of the work. Kepler’s wider career — the Mysterium cosmographicum of 1596, the Astronomia nova of 1609 with its first two laws of planetary motion, the Harmonices mundi of 1619 with the third, and the great Rudolphine Tables finally completed at the end of 1623 and printed at Ulm in 1627 — is here represented by the unassuming-looking folio in plain contemporary vellum that, as much as any of the more famous works, opened the road to the calculus. References: BL 17th C. German K116 — Caspar 48 (locating 43 copies, the great majority in German and other European libraries) — Cinti 55 — Parkinson, Breakthroughs, p. 68 — Smith, Rara Arithmetica, p. 416 — Baron, The Origins of the Infinitesimal Calculus, 1969 — Boyer, The History of the Calculus and its Conceptual Development, 1949 — Hammer, ‘Nachbericht’, in Johannes Kepler Gesammelte Werke IX, 1960 — Knobloch (ed.), Nova stereometria doliorum vinariorum / New Solid Geometry of Wine Barrels, Paris, 2018 — Radelet-de Grave, ‘Kepler, Cavalieri, Guldin: Polemics with the departed’, in Seventeenth-Century Indivisibles Revisited, ed. Jullien, 2015, pp. 57–86 — DSB VII, pp. 289–312 (Gingerich) — MacTutor, ‘Johannes Kepler’. Folio (306 × 198 mm), ff. [56], with the 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.
Item #5949
Price: $125,000.00
