George Berkeley, answered from the texts and cited to the page.
The differentials, fluxions, and infinitesimals of the calculus are not clearly conceived. The mathematicians proceed as follows: when computing a derivative, they assume a small increment; they perform the algebraic manipulation that uses this increment; then, when the manipulation is complete, they let the increment vanish and treat the result as if the increment had never been there.
They cannot have it both ways. If the increment exists, the result depends on its magnitude and is approximate; if the increment is zero, the algebraic manipulation that depended on dividing by it is invalid. The "vanishing differential" is neither something nor nothing; it is the ghost of a departed quantity. I do not deny the practical utility of the calculus; the mathematicians compute, and the computations work.
What I deny is that they have given a rigorous account of why the computations work, and I observe that mathematicians who ridicule religious mysteries on the ground that they cannot be clearly conceived are at the same time entertaining mathematical mysteries fully as obscure as those of the Trinity or the Eucharist.1 Halley, the infidel mathematician to whom the Discourse is addressed, mocks priests for what he himself does in his own discipline.
The diagnosis is not a critique of mathematics as such but of mathematicians who pretend to a clarity their own foundations do not possess.2345
TEXT WI and is acquainted with the humour of the times and the characters of men, is well aware there are too many that deride Mysteries, and yet admire Fluxions; who yield that faith to a mere mortal which they deny to Jesus Christ, whose religion they make it their study and business to discredit.Works Vol 4 - De Motu Analyst and Mathematical Writings, p. 121
All which, you insist, 'appears very strange to you and the rest of that famous University, who plainly see of how great use mathematical learning is to mankind. Hence you take occasion to declaim on the usefulness of mathematics in the scveral branches, and then to redouble your surprize and amazement (p. 19 and 20). To all which declamation I reply that it is quite beside the purpose.Works Vol 4 - De Motu Analyst and Mathematical Writings, p. 120
It appears from hence, how unjustly you blame me (p. 32) for omitting to give any account of that first section of the first book of the Principia, wherein (you say) the foundation of the method of fluxions is geometrically demonstrated and largely explained, and difficulties and objections against it are clearly solved.Works Vol 4 - De Motu Analyst and Mathematical Writings, pp. 134–135
I, on the contrary, affirm, the increments must be understood to be quite gone, and absolutely nothing at all. My reason is, because without that supposition you can never bring the quantity or expression = מ a. παπα ox"-? + &c. down to nx"-!, mn the very thing aimed at by supposing the evanescence. Say whether this be not the truth of the case? Whether the former expression is not to be reduced to the latter?Works Vol 4 - De Motu Analyst and Mathematical Writings, p. 135
You, who are a mathematician, must acknowledge there have been divers such methods admitted in mathematics, which are not demonstrative. Such, for instance, are the inductions of Dr. Wallis, in his Arithmetic of Infinites; and such what Harriot, and, after him, Descartes, have wrote concerning the roots of affected equations.Works Vol 4 - De Motu Analyst and Mathematical Writings, pp. 125–126
The most-vindicated of Berkeley's diagnostic claims at the level of the history of mathematics. The eighteenth-century defenders (Jurin, Walton, Maclaurin in the Treatise of Fluxions 1742) attempted to ground the calculus on rigorous foundations and acknowledged in the attempt that what Berkeley had identified was a real gap. The nineteenth- century rigorisation by Cauchy (Cours d'analyse 1821) and Weierstrass (epsilon-delta limits) replaced the fluxional apparatus with rigorous limit-definitions and was the mathematical community's eventual answer to exactly what Berkeley had attacked. Abraham Robinson's Non-standard Analysis (1966) rehabilitated infinitesimals on rigorous foundations using model-theoretic apparatus, vindicating the intuition behind the original differential calculus while acknowledging that the eighteenth-century formulations were not rigorous. Cajori's History of the Conceptions of Limits and Fluxions (1919) traces Berkeley's role in the foundational history. Contemporary historians of mathematics (Boyer, Grattan-Guinness) credit Berkeley with identifying the foundational gap that took mathematics a century and a half to close.