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. Bishop George Berkeley George Berkeley (British English://; Irish English: //) (March 12, 1685 – January 14, 1753), also known as Bishop Berkeley, was an influential Irish philosopher whose primary philosophical achievement is the advancement of what has come to be called subjective idealism, summed up in his dictum, Esse est percipi (To... Events January 8 - Premiere of George Frideric Handels opera Ariodante at the Royal Opera House, Covent Garden. ... Edmond Halley. ... Sir Isaac Newton, PRS (4 January [O.S. 25 December 1642] 1643 – 31 March [O.S. 20 March] 1727) was an English physicist, mathematician, astronomer, alchemist, and philosopher who is one of the most influential scientists in history. ... Events June 11 - George, Prince of Wales becomes King George II of Great Britain. ...

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. Calculus is a central branch of mathematics, developed from algebra and geometry. ... Fluxion was Isaac Newtons term for the derivative of a fluent, or continuous function (see: Calculus). ... Gottfried Wilhelm von Leibniz (also Leibnitz) (July 1 (June 21 Old Style) 1646, Leipzig – November 14, 1716, Hanover) was a German polymath, deemed a genius in his lifetime and since. ... In mathematics, an infinitesimal, or infinitely small number, is a number that is greater in absolute value than zero yet smaller than any positive real number. ... Natural philosophy is a term applied to the objective study of nature and the physical universe before the development of modern science. ... Since antiquity, people have tried to understand the behavior of matter: why unsupported objects drop to the ground, why different materials have different properties, and so forth. ...

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 rigourously 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. Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... Bernhard Riemann. ... Karl Weierstraß Karl Theodor Wilhelm Weierstrass (Weierstraß) (October 31, 1815 – February 19, 1897) was a German mathematician who is often cited as the father of modern analysis. // The Theory of GOLDFiSH AND CHEESE!!!111oneoneone11!!!! Goldfish are like zomg soo kewl! Yea i knowww how kewl are goldfish?!! Soundness of Goldfish... In mathematics, the derivative is one of the two central concepts of calculus. ... In calculus, the integral of a function is a generalization of area, mass, volume, sum, and total. ... A limit can be: Limit (mathematics), including: Limit of a function Limit of a sequence Limit point Net (topology) Limit (category theory) A constraint (mathematical, physical, economical, legal, etc. ... 1966 (MCMLXVI) was a common year starting on Saturday (link goes to calendar) // Events January January 1 - In a coup, Colonel Jean-Bédel Bokassa ousts president David Dacko and takes over the Central African Republic. ... Abraham Robinson (October 6, 1918 - April 11, 1974) was a mathematician who is most widely known for development of nonstandard analysis, a mathematically rigorous system whereby infinitesimal and infinite numbers were incorporated into mathematics. ... Non-standard analysis is that branch of mathematics that formulates analysis using a rigorous notion of infinitesimal, where an element of an ordered field F is infinitesimal if and only if its absolute value is smaller than any element of F of the form 1/n, for n a natural... Gottfried Leibniz Gottfried Wilhelm von Leibniz (July 1, 1646 in Leipzig - November 14, 1716 in Hannover) was a German philosopher, scientist, mathematician, diplomat, librarian, and lawyer of Sorb descent. ...

The Analyst is available online at David R. Wilkin's 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.