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Geometriae Speciose Elementa.The Most Original Pupil of Bonaventura Cavalieri. Bologna: Giovanni Battista Ferroni, 1659.
Very rare first edition, and a fine copy from the library of Pietro Riccardi, of this important work on limits of geometrical figures. In this work Mengoli "set up the basic rules of the calculus thirty years before Newton and Leibniz. Both of these were influenced by his contribution, in the case of Leibniz the influence was direct as he read Mengoli's work while in the case of Newton he knew of it indirectly through studying Wallis." (MacTutor History of Mathematics).

"In the 'Geometriae speciosae elementa' (1659), Mengoli set out a logical arrangement of the concepts of limit and definite integral that anticipated the work of nineteenth-century mathematicians. In establishing a rigorous theory of limits, he considered a variable quantity as a ratio of magnitudes and hence needed to consider only positive limits. He then made the following definitions: a variable quantity that can be greater than any assignable number is called 'quasi-infinite'; a variable quantity that can be smaller than any positive number is 'quasi-nil'; and a variable quantity that can be both smaller than any number larger than a given positive number a and greater than any number smaller than a is 'quasi-a.'

Using these precise concepts of the infinite, the infinitesimal, and the limit, and working from simple inequalities valid between numerical ratios, he demonstrated (as Agostini recognized by translating his obscure exposition into modern symbols and terminology) the properties of the limit of the sum and the product, and showed that the properties of proportions are conserved also at the limit. The proofs obtain when such limits are neither 0 nor for this case Mengoli set out the properties of the infinitesimal calculus and the calculus of infinites some thirty years before Newton published them in his 'Principia.'

Mengoli's predecessors (among them Archimedes, Kepler, Valerio, and Cavalieri) had assumed as intuitively evident that a plane figure has an area. By contrast, he proved the existence of the area by dividing an interval of the continuous figure f(x) into n parts and considering, alongside the figure to be squared (which he called the 'form'), the figures formed by parallelograms constructed on each segment of the interval and having the areas (in modern notation):

sn= ∑li(xi+1-xi), i=1, 2, ... n (inscribed figure)

Sn= ∑Li(xi+1-xi) (circumscribed figure)

σn= ∑f(xi)(xi+1-xi) or

σ'n= ∑f(xi+1)(xi+1-xi) (adscribed figure)

where li and Li denote, respectively, the minimum and maximum of f(x) on the interval (xi, xi+1). Drawing upon the theory of limits that had worked so well in the study of series, Mengoli demonstrated that the sequences of the sn and Sn tend to the same limit to which the sequences of the σn and σ'n compressed between them, also tend. Hence, since the figure to be squared is always compressed between the sn and the Sn, it follows that this common limit is the area of the figure itself.

Mengoli also used this method to integrate the binomial differentials Zs(a-x)rdx with whole and positive exponents. (He had, preceding Wallis, already integrated these sometime before by the method of indivisibles.) Before publishing his results, however, he wished to give a rigorous basis to the method of indivisibles or to develop in its stead another method that would be immune to criticism. He therefore set out a purely arithmetic theory of logarithms; having given a definition of the logarithmic ratio similar to Euclid's definition of ratio between magnitudes, he then extended Euclid's book V to encompass his own logarithmic ratio. Mengoli also did significant work in logarithmic series (thirteen years before N. Mercator published his 'Logarithmotecnia'). (DSB: IX, p.303-304).. 4to: 198 x 147 mm. Contemporary Italian vellum. Provenance: Ex libris of the Biblioteca Riccardi to front pastedown. Fully complete: 80 (introduction); 392 pp. Internally clean and crisp, a fine and unsophisticated copy. Riccardi I (2), 150. Very rare: OCLC records just one copy in the US (New York Public Library)

      [Bookseller: Sophia Rare Books]
Last Found On: 2012-12-27           Check availability:      Antikvariat    

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