# Infinities

Joseph-Louis Lagrange (in his Théorie des Fonctions Analytiques of 1813), describes retrospectively how the Leibnizian calculus emerged:

The first geometers who employed the differential calculus, Leibniz, the Bernoullis, l’Hôpital, etc. founded it on the consideration of infinitely small quantities of different orders, and on the supposition that one may regard and treat as equal quantities that do not differ amongst themselves except by quantities infinitely small with respect to them. Content to arrive by the procedures of this calculus in a prompt and sure way at correct results, they were not at all worried about demonstrating the principles.

trans. Jacqueline Stedall, Mathematics Emerging (2008, p. 287).

Lagrange (1736-1813) continues to outline the Newtonian approach and to propose a way of avoiding infinitely small quantities (or infinitesimals). The problems of rigour (that is of establishing sound principles) in calculus were to be debated for another sixty yours or so … but, wait, we are now well out of the Early Modern era!

Guillaume François Antoine, Marquis de L’Hôpital, Analysedes infiniment petits: pour l’intelligence des lignes courbes (Paris, 1696), plate 5, figure 67.

Here we see how infinitesimals are at the core of ‘early modern’ calculus. L’Hôpital (1661-1704), writing in the tradition of the Bernoullis and Leibniz (1646-1716), illustrates the analysis of a normal (or perpendicular) to a curve, AMD, at a point, M. If x = AP, y = PM and z = EM, then corresponding infinitesimals are ‘dx’ = Ee and ‘dy’ = ‘dz’ = Rm. (A less complicated example is explained on the L’Hôpital page.) On the one hand, Wallis (1616-1703) already had (in 1656) a ‘modern’ idea of limit, in terms of quantities ‘less than any assignable’, but on the other, George Berkeley (1685-1753), in 1734 (the year after the death of Edward Worth) launched his scathing attack on infinitesimals:

It must, indeed, be acknowledged, that he [Newton] used Fluxions, like the Scaffold of a building, as things to be laid aside or got rid of, as soon as finite Lines were found proportional to them. But then these finite Exponents are found by the help of Fluxions. Whatever therefore is got by such Exponents and Proportions is to be ascribed to Fluxions: which must therefore be previously understood. And what are these Fluxions? The Velocities of evanescent Increments? And what are these same evanescent Increments? They are neither finite Quantities nor Quantities infinitely small, nor yet nothing. May we not call them the Ghosts of departed Quantities?

George Berkeley, The Analyst (1734, paragraph XXXV)

Berkeley was no kinder to Wallis or Leibniz, but tarred all those who traded in the infinitely small with the same brush. In the Edward Worth Library, we can see the works of Wallis, Brouncker, Newton and L’Hôpital as they use their imagination to break new ground in understanding infinitesimals.

Sources

Berkeley, George, The Analyst (1734), edited by David Wilkins (Trinity College, Dublin, 2002), p. 18.

Bos, H. J. M., ‘Newton, Leibniz and the Leibnizian Tradition’, in I. Grattan-Guinness (ed.), From the Calculus to Set Theory 1630-1910: An Introductory History (Princeton University Press: Princeton, 2000), pp 70-73.

Stedhall, Jacqueline, Mathematics Emerging. A Sourcebook 1540-1900 (Oxford University Press, 2008).
Text: Maurice OReilly.