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Published online by Cambridge University Press: 17 January 2013
Article 1. The introduction of the notion of a function of a variable quantity into the mathematics, without any regard to its particular form, has given vast extension to the science, and been the germ of some of its most important theories. The doctrine of curve lines, no doubt, produced that of functions, for the former may be made the visible expression of the latter: thus, either of the coordinates of a curve being taken as the representation of the variable, the other co-ordinate is a function of the variable; so also are the arc of the curve, and its area. Indeed, in contemplating functions, and discussing their properties, it is convenient to substitute in our reasonings the geometrical representation for the abstract notion of the function.
page 635 note * Leslie's Geometrical Analysis. The Catenary.
page 636 note * Philosophical Transactions, No. 231, (vol. i. p. 39 of Lowthorp's Abridgment), Gregory's Memoir, which was in Latin, was translated and published in Miscellanea Curiosa, edited (I believe) by Dr Derham.
page 636 note † His articles in that edition of the Encyclopædia and its Supplement, were, in 1822, collected and published in 4 vols. 8vo. The article on Arches is republished in the seventh edition of the Encyclopædia, to which I added a short supplement on Equilibrated Curves.
page 640 note * See my paper in this volume, page 436.
page 641 note * For the mode of deduction, see the paper just quoted.
page 650 note * Euler, Calculus Differentialis, Pars ii. cap. viii.; also Legendre, Exercices de Calcul Integral, tome ii. p. 144.
page 652 note * Mendoza Rios' Collection of Tables for Navigation; or any treatise on navigation.
page 654 note * Philosophical Transactions, as quoted at art. 21.
page 655 note * See my Treatise on Conic Sections, Part i. proposition 14.
page 657 note * Philosophical Transactions for 1826, Part iii. I have been told that the very ingenious author of this memoir did not himself compute the numbers, which are almost all incorrect.
page 664 note * These are nearly the dimensions of the middle arch of Blackfriars' Bridge, London.