Barrow and Leibniz on the fundamental theorem of the calculus

At the height of his priority dispute with Newton concerning the invention of the calculus, Leibniz wrote an account, Historia et Origo Calculi Differentialis, describing the contributions by seventeenth century mathematicians that led him to his own development of the calculus (Child 1920, 22). In this account, Isaac Barrow is not mentioned at all, and in several occasions Leibniz denied any indebtedness to his work, particularly during his notorious priority dispute with Isaac Newton 1 . But in Barrow's Lectiones Geometricae (hereafter cited as Geometrical Lectures), which Leibniz had obtained during a visit to London in 1673, the concepts of the differential and integral calculus are discussed in geometrical form, and a rigorous mathematical proof is given of the fundamental theorem of the calculus 2 3 . According to J. M. Child, "a Calculus may be of two kinds: i) An analytic calculus, properly so called, that is, a set of algebraical working rules (with their proofs), with which differentiations of known functions of a dependent variable, of products, of quotients, etc., can be carried out; together with the full recognition that differentiation and integration are inverse operations, to enable integration from first principles to be avoided . . . ii) A geometrical calculus equivalent embodying the same principles and methods; this would be the more perfect if the construction for tangents and areas could be immediately translated into algebraic form, if it where so desired.

Between these two there is, in my opinion, not a pin to choose theoretically; it is a mere matter of practical utility that set the first type in front of the second; whereas the balance of rigour, without modern considerations, is all on the side of the second ( Child 1930, 296). More recently, a distinguished mathematician, Otto Toeplitz, wrote that " Barrow was in possesion of most of the rules of differentiation, that he could treat many inverse tangent problem (indefinite integrals), and that in 1667 he discovered and gave an admirable proof of the fundamental theorem -that is the relation [ of the inverse tangent] to the definite integral (Toeplitz 1963, 128).

Leibniz’s original work concerned the analytic calculus, and he claimed to have read the relevant sections of Barrow’s lectures on the geometrical calculus only several years later, after he had independently made his own discoveries. In a letter to Johann Bernoulli written in 1703 , he attributed his initial inspiration to a “characteristic triangle” he had found in Pascal’s Traité des sinus du quart the cercle. 4 After examining Leibniz’s original manuscript, and his copy of Barrow’s Geometrical Lectures, which were not available to Child, D. Mahnke (Manhke 1926) concluded that Leibniz had read only the beginning of this book, and that his calculus discoveries were made independently of Barrow’s work. Later, J. E. Hofmann (Hofmann 1974), likewise concluded that this independence is confirmed from Leibniz’s early manuscripts and correspondence at the time (Hofmann 1974).

In his early mathematical studies, Leibniz considered number sequences and realized that the operations associated with certain sums and differences of these sequences had a reciprocal relation. Then, by approximating a curve by polygons, these sums correspond to the area bounded by the curve, while the differences correspond to its tangent, (Bos 1973;Bos 1986, 103), which Leibniz indicated 5 , lead to his insight of the reciprocal relation between areas and tangents . For example, Leibniz’s June 11, 1677 letter addressed to Oldenburg for Newton, in reply to Newton’s October 24, 1676 letter to Leibniz (Epistola Posterior), clearly shows that by this time he understood the fundamental theorem of the 4 Leibniz said that on the reading of this example in Pascal a light suddenly burst upon him, and that he then realized what Pascal had not -that the determination of a tangent to a curve depend on the ratio of the differences in the ordinates and abscissas, as these became infinitesimally small, and that the quadrature depended upon the sum of ordinates or inifinitely thin rectangles for infinitesimal intervals on the axis. Moreover, the operations of summing and of finding differences where mutually inverse (Boyer 1949, 203). 5 See Historia et origo calculi differentialis (Child 1920, 31-34) calculus, and its usefulness to evaluate algebraic integrals (Newton 1960, 221; Guicciardini 209, 360). But Child has given considerable circumstantial evidence, based on the reproductions of some of Leibniz’s manuscripts, that Barrow’s Geometrical Lectures also influenced some of Leibniz’s work6 (Child 1920), and he did not concur with Mahnke’s conclusion 7 . To give one example, in the first publication of his integral calculus (Leibniz 1686), Leibniz gave an analytic derivation of Barrow’s geometrical proof in Prop. 1, Lecture 11, (Child 1916, 125), that the area bounded by a curve with ordinates equal to the su

This content is AI-processed based on open access ArXiv data.