## Nova Stereometria doliorum vinariorum

SCARCE FIRST EDITION of one of the most significant works in the prehistory of calculus. With the rare errata leaf present in two variant states.

"The task of writing a complete treatise on volumetric determination seems to have been suggested to Kepler by the prosaic problem of determining the best proportions for a wine cask. The result was the Nova stereometria, which appeared in 1615. This contains three parts, of which the first is on Archimedean stereometry, together with a supplement containing some ninety-two solids not treated by Archimedes. The second part is on the measurement of Austrian wine barrels, and the third on applications of the whole" (Boyer, The History of the Calculus).

Kepler's basic method was to regard the circle as a polygon with an infinite number of sides and its area as being composed of an infinite number of infinitesimal triangles with vertex at the centre of the circle and base one of the sides of the polygon. Similarly, the volume of a sphere was made up of an infinite number of pyramids, the cone and cylinder of infinitely thin circular discs or of infinitesimal wedge-shaped segments radiating from the axis. "Kepler then extended his work to solids not considered by the ancients. The areas of the segments cut from a circle by a chord he rotated about this chord, obtaining solids which he designated characteristically as apple or citron-shaped, according as the generating segment was greater or less than a semi-circle... Kepler's Doliometha... 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).

Kepler's book on integration methods also contains the germ of the differential calculus. "The subject of the measurement of wine casks had led Kepler to the problem of determining the best proportions for these. This brought him to the consideration of a number of problems on maxima and minima ... he 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 [square root 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" (Boyer). 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.

Nova Stereometria doliorum vinariorum, in primis Austriaci, figurae omnium aptissimae; et usus in eo virgae cubicae compendiosossimus & plane singularis. Accessit Stereometriae Archimedae Supplememtum. Linz: J. Plancus for the author, 1615. Folio, contemporary calf sympathetically rebacked. With two errata leaves, woodcut on H3v shaved at foot as usual, occasional foxing, small closed tears to final leaf; a very good crisp copy. RARE.

Price: \$48,000 .

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