Rectification problem in the development of analysis. (Q2856942)

The author traces in this substantial review the controversies around rectification of curves from the ancient Greeks until the mid-nineteenth century. The dichotomy between curved and rectilinear lines was for a long time considered essential and any attempt to express the former in the terms of the latter as illegitimate. This was Aristotle's as well as Descartes' position, in contrast to both ancient sophists and early modern mechanical philosophers. Many mathematicians however attempted to square the circle and rectify other special curves. The author brings together careful, concise reconstructions of these attempts by Archimedes (circle, spiral, helix, 3 c. BCE), Pappus (quadratrix, 3 c. CE), Thomas Hobbes and Gilles Personne de Roberval, recorded by Blaise Pascal (equivalence between spiral and parabola, ca. 1641--1643), William Neil recorded by John Wallis (semicubic parabola, 1657), Pierre Fermat (general rectification method demonstrated on semicubic parabola, ca. 1655), Thomas Harriott and Evangelista Torricelli (rhumb line or loxodroma, 1590's and 1630's, respectively), Christopher Wren (cycloid, 1659), and Christian Huygens (evolute of a cycloid, 1659). The widely shared interest in rectification led in the mid-seventeenth century to almost simultaneous publication of related results and to subsequent priority disputes. The author moves on to show how the changing methods of rectification reflected the evolving philosophy of mathematics, concluding with a more cursory treatment of rectification by founders of rigorous analysis Joseph-Louis Lagrange, Augustin-Louis Cauchy, Bernard Bolzano and, finally, Jean-Marie Constant Duhamel, who established the current approach to rectification.NEWLINENEWLINEThe author mostly goes directly to the sources of the seventeenth century rectifications, drawing also on previous scholarship by \textit{D. M. Jesseph} [Squaring the circle. The war between Hobbes and Wallis. Chicago, IL: The University of Chicago Press (1999; Zbl 0938.01001)], \textit{K. Moller Pedersen} [Centaurus 15, 26--43 (1970; Zbl 0251.01007)], \textit{J. G. Yoder} [Unrolling time. Christian Huygens and the mathematization of nature. Cambridge: Cambridge University Press (1989; Zbl 0687.01005)] and others.NEWLINENEWLINEFor the entire collection see [Zbl 1330.01007].