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La Geometria del Compasso.
Heirs of Pietro Galeazzi, Pavia 1797 - First edition of Mascheroni’s most important book, in which he proves that any geometrical construction that can be performed using ruler and compasses can be carried out by means of compasses alone. The history of the problem goes back at least as far as Euclid, the restriction to ruler and compass occurring in Book I of the Elements. The problem of carrying out constructions under additional restrictions was considered by da Vinci, Dürer, Cardano and Tartaglia, among others. "In the preface Mascheroni recounts the genesis of his work. He was moved initially by a desire to make a contribution to elementary geometry. It occurred to him that ruler and compass could perhaps be separated, as water can into two gases; but he was assailed by doubts and fears often attendant upon research. He then chanced to reread an article on the way Graham and Bird [who supplied instruments to Maupertuis] had divided their great astronomical quadrant, and he realised that the division had been made by compass alone, although, to be sure, by trial and error. This encouraged him and he continued his work with two purposes in mind: to give a theoretical solution to the problem of constructions with compasses alone and to offer practical constructions that might be of help in making precision instruments. The second concern is shown in the brief solutions of many specific problems and in a chapter on approximate solutions. "The theoretical solution (see especially §191) depends on the solution of the following problems: (1) to bisect a given circular arc of given center; (2) to add and subtract given segments; (3) to find the fourth proportional to three given segments; (4) to find the intersection of two given lines; (5) to find the intersection of a given line and a given circle" (DSB).There is some debate as to whether Mascheroni’s result was anticipated by the Danish mathematician George Mohr in his Euclides Danicus (1672), although Mascheroni explicitly denied that anyone had previously treated the matter. "When the works of Mascheroni and Mohr are compared, it is apparent that the main ideas of their solutions of individual problems are in most cases quite different. In particular, this can be said for the bisection of a given segment. Moreover, the problem of bisection plays no role in Mascheroni’s general solution, whereas it is central in Mohr’s constructions. Still more significantly, the general problem is not formulated in Mohr’s book. Thus, any suggestion of Mascheroni’s direct reliance on Mohr would be quite inappropriate Mohr’s book is quite remarkable and contains the basis for a simple proof of Mascheroni’s result, but there is no evidence within the book itself that Mohr formulated the problem of constructions with compass alone in complete generality" (ibid.).Bound after the title is the dedication in verse to Napoleon. The connection with Napoleon is of special interest as 1797 was the year Bonaparte was elected a member of the Académie des Sciences. His interest was attracted by Mascheroni’s work and, to amuse himself, he selected geometrical problems from it and presented them to his confreres at the Académie to solve. They were, however, unable to do so. The most difficult of these problems was to determine the centre of a drawn circle, by means of just a compass, which is still known as the Problème de Napoleon. "Mascheroni (1750-1800) was the son of Paolo Mascheroni dell’Olmo, a prosperous landowner, and Maria Ciribelli. He was ordained a priest at seventeen and at twenty was teaching rhetoric and then, from 1778, physics and mathematics at the seminary of Bergamo. His Nuove ricerche su l’equilibrio delle volte (1785) led to his appointment as professor of algebra and geometry at the University of Pavia in 1786. In 1789 and 1793 he was rector of the university and, from 1788 to 1791, was head of the Accademia degli Affidati. Mascheroni was a member of the Academy of Padua, of the Royal Academy of Mantua, and of the Società [Attributes: First Edition]
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