By Giovanni Ferraro
The concept of sequence within the seventeenth and 18th centuries poses numerous fascinating difficulties to historians. many of the effects derived from this time have been derived utilizing tools which might be discovered unacceptable this day, and consequently, whilst one seems again to the speculation of sequence sooner than Cauchy with no reconstructing inner motivations and the conceptual history, it seems that as a corpus of manipulative innovations missing in rigor whose effects appear to be the difficult fruit of the brain of a magician or diviner instead of the penetrating and intricate paintings of significant mathematicians.
This monograph not just describes the full complicated of seventeenth and 18th century methods and effects touching on sequence, however it additionally reconstructs the implicit and specific ideas upon which they're dependent, attracts cognizance to the underlying philosophy, highlights competing ways, and investigates the mathematical context the place the speculation originated. the purpose here's to enhance the knowledge of the framework of seventeenth and 18th century arithmetic and steer clear of trivializing the complexity of old improvement by means of bringing it into line with smooth recommendations and perspectives and by means of tacitly assuming that sure effects belong, in a few experience, to a unified concept that has come all the way down to us today.
Giovanni Ferraro is Professor of arithmetic and background of arithmetic at collage of Molise.
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Extra resources for The Rise and Development of the Theory of Series up to the Early 1820s
An . I shall return to the notion of ﬁctitious quantities in Chapter 7, where I shall show that it can be used to improve our understanding of 18th-century mathematics. 2 Power series Although Leibniz considered certain numerical series interesting in their own right, he concentrated above all on what is referred to today as power series. However, his investigations originated in the ﬁeld of geometry and this gave a peculiar aspect to series theory. Leibniz’s analysis was not based upon the notion of function (either in the modern or in the 18th-century sense) but upon curves.
In his James Gregory Tercentenary Memorial Volume [GT, 26], Turnbull hypothesized that Gregory might have used a procedure corresponding to the use of the Taylor formula to obtain some of his expansions. He based his argument upon an error of calculation of the series for arctangent (in 3968a9 3233a9 eﬀect the ﬁfth term is 181440r 8 and not 181440r 8 ). In his James Gregorie on Tangents and the “Taylor ” Rule of Series Expansions , Malet states convincingly that Gregory calculated the coeﬃcient of any of the expansions sent to Collins by means of the following recursive procedure: Let x2 x3 + A∗3 2 + etc.
15) log(1 + x) = x − 2 3 which he derived by using the results of Saint-Vincent and de Sarasa (namely, 1 ) and the fact that the logarithm is the area under the hyperbola y = 1+x by applying Wallis’s method of quadrature (see Mercator [1668, 28–33]). 29 To compute the area 1 and above the segment [1, A], one divides this under the hyperbola 1+x segment into an equal number of parts (in aequales partes innumeras). , 1 , 1 + nh where nh = A. By expanding these fractions by long division (I shall clarify the meaning of this term a few lines below), one obtains 1 1+h 1 1 + 2h 1 1 + 3h = 1 − h + h2 − h3 + .