Mathematics

Horcher, Philipp | Libri tres: In Quibus Primo Constructio Circini Proportionum Edocetur

  • First edition of this rare work on the proportional compass, the first to describe both its construction and application. A very attractive copy in handsome contemporary calf. Rare, only one copy appears in recent auction records, at Sothebys in 2018, and WorldCat locates around fifteen copies.

    The proportional compass, a forerunner of the Galilean compass, was one of the first mathematical calculating instruments. It allowed volumes of solids to be calculated and compared, lines to be divided according to a given proportion, circles and curves to be divided proportionally, surfaces of a given shape to be multiplied or reduced, π to be approximated, shapes to be transformed into other shapes of equal surface area, and spheres and the five regular solids to be transformed. It was invented in antiquity, and later developed by Leonardo da Vinci who referred to it specifically as a proportional compass.

    The instrument described by Horcher had its genesis in the observatory of Wilhelm IV, Landgraave of Hesse, at Kassel. He employed Joost Bürgi as an instrument maker, and it is believed that Bürgi devised the instrument described by Horcher. It was first publicised in a work by Levinus Hulsius entitled Dritter Tractat der mechanischen Instrumente... Beschreibung und Unterricht dess Jobst Burgi Proportional Circkels (Frankfurt 1604). Hulsius had been a student of Galileo’s at Padua. However, Hulsius withheld details of the compass’s construction, as he was offering the instrument for sale. Horcher here gives the first account of its construction, as well as numerous examples of its use in calculating and scaling.

    ”Important for our purposes is how these new instruments effectively mechanized the basic processes of the geometric game: two-dimensional quadrature of the circle, three-dimensional cubature of the sphere, problems of doubling the volume of a cube or transforming one regular solid into another were now operations which could be analyzed quantitatively. They were physical, mechanical problems which could be reduced to numerical ratios and these could happen without the aid of three-dimensional representations. Hence it was paradoxically the very study of the regular and irregular solids as concrete physical models that brought about a new level of abstraction, which resembled the earlier neo-Platonic interpretation but was in fact fundamentally different because it assumed a new mechanistic view of the universe. Indeed, where the geometrical game had been an intellectual play of geometrical forms in mediaeval times, it now involved nature itself” (Kim H. Veltman, Geometric Games: A Brief History of the Not so Regular Solids).

  • Mainz: Balthazar Lipp, 1605.

    Quarto (204 x 150mm). Contemporary mottled calf, rebacked to style, spine gilt in compartments with pomegranate tools, red morocco label, double gilt fillets, new endpapers, edges dyed red. Woodcut illustration of the compass to title, folding woodcut plate, numerous woodcut diagrams and figures in the text. Upper corner of binding bumped, B2 and 3 defective in lower blank margin, not affecting text. A very good copy, the contents fresh.