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Mathematical Treasure: Berkeley's Critique of Calculus

Frank J. Swetz (Pennsylvania State University)

“The ghosts of departed quantities” is a phrase that both haunts and amuses students of the history of calculus. It was conceived by Bishop George Berkeley of the Church of England (1685-1753) in his attack on the logical foundations of Newton’s calculus in a pamphlet entitled The Analyst (1734). Berkeley was a glib and witty orator and writer – he also referred to Newton’s fluxions as “the velocities of evanescent increments” – whose opinions at the time raised serious concerns about the efficacy of the calculus.

The images here of Berkeley’s writings are taken from Alexander Fraser’s The Works of George Berkeley (1871), supplied through the courtesy of the Pennsylvania State University Library. Fraser was a bibliographer and great admirer of George Berkeley. The title page of The Analyst is shown above.

Pages 254-256 give the “Table of Contents” for The Analyst. The listing provides an outline of Berkeley’s arguments against the calculus.

The text of the pamphlet begins on page 257. Fraser’s annotations, given at the bottom of the page, provide valuable information on the contents. The remaining pages supply an insight into Berkeley’s approach.

The pages below present Berkeley’s arguments 21-23 in full, allowing the reader to understand some of the author’s mathematical reasoning.

To read more of The Analyst, see the transcription by David R. Wilkins, Trinity College, Dublin.

To learn more about the history of calculus, watch The Ghosts of Departed Quantities: Calculus and Its Limits, a Gresham College Lecture by Raymond Flood of the Open University, England.

Finally, for more of Berkeley's writings from Fraser's 1871 collection, see the Convergence article Mathematical Treasure: Berkeley on Various Topics.


Alexander Campbell Fraser, The Works of George Berkeley (4 vols.), Oxford Clarendon Press, 1871

Frank J. Swetz (Pennsylvania State University), "Mathematical Treasure: Berkeley's Critique of Calculus," Convergence (August 2015)