THE ANALYST

'''The Analyst''', subtitled ''A DISCOURSE Addressed to an Infidel Mathematician'', is a book published by George Berkeley in 1734. The "infidel mathematician" is believed to have been Edmond Halley or Sir Isaac Newton. In the latter case, no reply would have been possible, as Newton died in 1727.
''The Analyst'' was a direct attack on the foundations and principles of the calculus, specifically on Newton's notion of fluxion and on Leibniz's notion of infinitesimal change. Berkeley sought to defend religion by showing that the calculus, which grounded religion's new rival, natural philosophy (the predecessor of today's physics), led to paradox and absurdity.
Most frequently quoted passage:

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?

A more modern paraphrase:

What are these "instantaneous" rates of change? The ratios of vanishing increments? And what are these "vanishing" Increments? They are neither finite quantities nor "infinitesimal" quantities, nor yet nothing. May we not call them the ''ghosts of departed quantities''?

Nothing much came of Berkeley's criticisms in the 18th century, if only because Berkeley was neither mathematician nor natural philosopher. But beginning around 1830, first in the hands of Augustin Cauchy, later in those of Bernhard Riemann, and Karl Weierstrass, the derivative and integral were redefined using a rigorously defined new concept, that of limit. But only in 1966, with the publication of Abraham Robinson's book ''Non-standard Analysis'', was the object of Berkeley's strongest ridicule, Leibniz's intuitive notion of the infinitesimal, made fully rigorous, thus showing another way of overcoming the difficulties which Berkeley pointed out in Newton's approach.
''The Analyst'' is available online at David R. Wilkins' website, which includes links to some responses by Berkeley's contemporaries. ''The Analyst'' is also reproduced, with commentary, in:
Ewald, William, ed., 1996. ''From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Vol. 1''. Oxford Uni. Press.
Ewald believes that Berkeley's objections to the calculus of his day were, by and large, well taken.