Bologna: Giovanni Battista Ferroni, 1659.

First edition, and a fine copy from the library of Pietro Riccardi, of this very rare and highly original work in which 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). Mengoli develops here a new method of quadratures, i.e., finding the areas of geometrical figures, based on François Viète’s symbolic, or ‘specious’ algebra, and on his own new theory of ‘quasi proportions,’ the first true theory of limits. Like the Geometria indivisibilibus (1635) of Mengoli’s master Bonaventura Cavalieri, the Geometriae is an important precursor of the integral calculus. Cavalieri’s method of indivisibles, which was entirely geometrical, had been much criticized by his contemporaries for having uncertain foundations, and Mengoli wrote the Geometriae in part as a defense of Cavalieri. Mengoli’s ‘specious geometry’ is, however, an entirely new method with a largely algebraic basis. This is an extremely rare book: OCLC records just one copy in the US (New York Public Library), and we have located only one copy in auction records.

In the first decades of the seventeenth century mathematicians began to assimilate the symbolic algebra introduced by Viète in his great work In Artem Analyticem Isagoge (1591). The first important consequence was Descartes’ La Géométrie (1637), which used Viète’s specious algebra to solve geometrical problems (essentially those that reduce to the determination of the points of intersection of two curves). Mengoli’s work applies specious algebra to the summation of power series and a different class of geometrical problems, the quadrature of curves. “In his work Geometriae Speciosae Elementa, 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, p. 84). “Mengoli believed that the basis of Cavalieri’s method of indivisibles was not sufficiently sound. He wanted to provide a solid foundation for the application of this method to square the given figures, new figures, and, especially, the circle … First, using his own system of coordinates, he expressed geometric figures by algebraic expressions. Second, to classify these algebraic expressions he placed them in a triangular table. Third, he used these algebraic expressions as part of a method for the geometrical construction of coordinates of these figures, and finally, he used triangular tables and quasi proportions to find new quadratures and to produce general demonstrations of quadrature results” (ibid., p. 94).

The Geometriae opens with a long introduction, the Lectio elementario, which provides an overview of the six chapters, or Elementa. The first two chapters contain the calculations of numerous summations of powers and products of powers, and the proofs of several identities. These calculations are based on three triangular tables. “Mengoli’s originality stems not from the definitions of these tables but from his treatment of them. On the one hand he uses these tables and Viète’s algebra to create other tables with letters expressing additions of powers and products of powers; on the other he employs the relations between these summations and the combinatory numbers of the arithmetic triangle to prove one of the most important results of his book, namely the sum of the pth powers of the first t – 1 integers” (Massa, pp. 261-2). This formula had been stated earlier by Fermat and Pascal, but without proof.

The third Elementum introduces Mengoli’s theory of ‘quasi proportions.’ In this chapter, “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 … 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 &infinity;; 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’.” (DSB).

In the fourth and fifth Elementa, Mengoli developed “a rigorous basis for the logarithm on the model of the Eudoxian definitions of equality and inequality for reals” (Whiteside, pp. 192-3). He “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 ‘Logarithmotechnia’)” (DSB). The series expansion for the logarithm was apparently included at the last minute, as it appears in the introduction (appendix: pp. 73-75).

“Finally, the sixth Elementum, De innumerabilibus quadraturis, involves calculating the quadratures of curves that correspond to functions represented by y = x^s(1 – ^x)^p-s [where p and s are non-negative integers]. Mengoli does this by using the theory of quasi proportions explained in the Elementum tertium” (Massa, p. 260). “The triangular table of quadratures that Mengoli constructed could be extended indefinitely. He knew the value of these quadratures and looked for a rule that allowed him to associate any geometric figure to an algebraic expression. Putting these expressions in the table with the appropriate coefficients, the quadratures of the new curves were given. He classified the curves that determine them into three types and studied the properties of each group, again using the theory of quasi proportions” (Massa Esteve, p. 110).

“In spite of these innovations, Mengoli was scarcely understood. In a letter to Collins, Isaac Barrow said that Mengoli’s style was harder than Arabic, and that if Mengoli had found something new, he did not have the time to investigate it. Even though his reputation was strong during his lifetime, it seems that Mengoli died isolated and ignored. The reasons for this are not clear. It is possible that his complex and confusing writing style and the complicated nature of his notation made his works too hard to read; perhaps, for this reason, he had no followers. It is equally possible that his introduction of algebra into geometry failed to accord with the prevailing mathematical practice of the 17th century” (Massa, p. 278).

Riccardi I (2), 150 (this copy). R. Massa, ‘Mengoli on “quasi proportions”,’ Historia Mathematica 24 (1997), pp. 257-80; R. Massa Esteve, ‘Algebra and geometry in Pietro Mengoli (1625-1686),” Historia Mathematica 33 (2006), pp. 81-112; D. T. Whiteside, ‘Patterns of mathematical thought in the late seventeenth century,’ Archive for History of Exact Sciences 1 (1961), pp. 179-388.

4to (198 x 147 mm), cContemporary Italian vellum. Provenance: Ex libris of the Biblioteca Riccardi to front pastedown, pp. 80; 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).

Item #6697

Price: $22,500.00