Descripción
First edition, very rare, of this continuation of Mengoli's most important work, 'Geometriae Speciosae Elementa' (1659), in which Mengoli "set up the basic rules of the calculus thirty years before Newton and Leibniz" (MacTutor). In the present work, Mengoli uses the techniques he developed in the 'Geometriae' to solve the problem of the quadrature of the circle. Mengoli explains that he had found the quadrature of the circle in 1660, but had not published it because he only wanted to publish the mathematics he needed "to explain natural events." "It has been known since the 1920s that Leibniz in April 1676 finally had the opportunity to study Mengoli's book 'Circolo' (1672) and that he made extensive excerpts from this work . . . In 'Arithmetica infinitorum et interpolationum figuris applicata', Leibniz essentially discusses the triangulat tables of Mengoli (3-10), and tries to find a method for the computation of the partial sums of the harmonic series . . . he was able to recognize that Mengoli had already been in possession of the harmonic triangle, used by Leibniz . . . since the end of 1672" (Goethe et al, G.W. Leibniz, Interrelations between Mathematics and Philosophy (2015), 127-130). The most original student of Bonaventura Cavalieri, Mengoli (1625-86) was familiar with his master's theory of indivisibles but wished to develop a method of solving problems of quadratures in an algebraic way. "In his work 'Geometriae,' algebra and geometry are used in complementary ways in the investigation of quadrature problems. At the beginning of this work he claimed that his geometry was a combination of those of Cavalieri and Archimedes obtained using the tools that Viete's 'specious algebra' offered him . . . His principal aim was to square the circle, a goal he achieved by means of his new method in a later work, 'Circolo'" (Massa Esteve, 'Algebra and geometry in Pietro Mengoli (1625-1686)', Hist. Math. 33 (2006), 84). Mengoli's principal tool was his new theory of 'quasi proportions', introduced in the 'Geometriae,' which can be viewed as the first true theory of limits. "Mengoli's aim was to find an algorithm that would enable him to calculate many quadratures at the same time, including some of them, such as the quadrature of the circle, that had never been calculated before. He first attempted this in the 'Geometriae' with the method of indivisibles, but he was only able to obtain the quadratures of geometric figures determined by algebraic expressions of the form y = x^n(1 - x)^(m-n) with exponents m and n that are natural numbers [i.e. positive whole numbers]. He displayed these geometric quadratures in a triangular table. He later identified these geometric figures explicitly with the values of their areas, which were also displayed in another triangular table (now called the harmonic triangle) by means of a proof that employed the theory of quasi-proportions and, to some extent, the method of exhaustion. In the 'Circolo,' for half-integer vales of the exponents, he interpolated both tables, and was thus able to calculate the area of the figure describing a semi-circle [the case m = 1, n = ½] as well as many others not computed before" (Esteve & Delshams, 'Euler's beta integral in Pietro Mengoli's works,' Arch. Hist. Exact Sci. 63 (2009), p. 353). The interpolated tables allowed him to evaluate these quadratures up to the determination of an unknown number 'a' which is closely related to the quadrature of a circle (a = 4/pi). Mengoli was able to obtain a series of successive approximations to 'a' which enabled him to express pi as an infinite product and to calculate it to eleven decimal places - 3.14159265359, stated at the end of the last sentence in the book. No copy listed on ABPC/RBH; OCLC lists four copies in US. Small 4to, pp. [vi], 60 (lacking half-title), with printer's device on title, engraved initials and headpieces, and numerous tables in text, some full-page. 18th century (?) decorated wrappers, rubbed. N° de ref. del artículo ABE-1586263159463
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