“The publication of Berkeley’s Analyst was the most spectacular mathematical event of the eighteenth century in England. Practically all British discussions of fluxional concepts of that time involve issues raise by Berkley.” -Cajori
"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 ghosts of departed quantities?” -George Berkeley, The Analyst, p.59
VERY RARE FIRST EDITION OF ONE OF THE MOST INFLUENTIAL WORKS IN THE HISTORY OF CALCULUS.
In 1731 Berkeley returned to England from the New World, where he had spent years working to open a new college with the support of a royal charter. Despite having funds earmarked for the college, no funds were ever released, and he had grown tired of the whole affair – and of those who held the money he was meant to receive. Moreover, “[r]eports of growing infidelity in English society, to which he was always liable to give credence, were fueled by the continuing bad faith of the government in failing to lodge the funds he considered legally his.” (Stewart).
Perhaps pushed by this diminishing opinion of the English gentry, he revisited his earlier attacks on the secular ‘freethinkers’ and composed The Analyst, “an acute and influential critique of the foundations of Newton's calculus.” (Downing). As Stewart explains, “Berkeley considered [the theory] incoherent and a disservice to mathematics, but one which, if unchecked, might reinforce prevailing views on the divisibility of matter and support infidelity.”
Within the criticism, Berkeley raises careful arguments, which often employ sophisticated philosophical distinctions. For example, as Andersen explains, “Berkeley acknowledged that mathematicians who applied Newton’s method of fluxions or Leibniz’s calculus ended up with valid results. However, … he considered their calculations to be based on incorrect assumptions and to violate the rules of logic.” As such, he wished “to explain why this may come to pass, and [show] how Error may bring forth Truth, though it cannot bring forth Science.” (Berkeley).
However, between these arguments we find marvelously snide, often comedic complaints about the whole approach. For example: “Now to conceive a Quantity infinitely small, that is, infinitely less than any sensible or imaginable Quantity, or than any the least finite Magnitude, is, I confess, above my Capacity. But to conceive a Part of such infinitely small Quantity, that shall be still infinitely less than it, and consequently though multiply'd infinitely shall never equal the minutest finite Quantity, is, I suspect, an infinite Difficulty to any Man whatsoever”. (Berkeley).
The concepts of the infinitesimal quantities of calculus haunted Berkeley, and he returned to it repeatedly in his criticism, most notably in his famous passage near the end of the book where he memorably referred to such infinitesimals as “ghosts of departed quantities”.
Note: This is the true first edition, printed in London in 1734. A Dublin edition was also published in 1734, but appears to be a reprinting (with some changes) of the London first (see Wilkins, 2002). These were the only editions published in Berkeley’s lifetime.
References: Andersen, K. (2011) “One of Berkeley’s arguments on compensating errors in the calculus.” Historica Mathematica, 38. Cajori, F. (1919) A History of Mathematics. (2nd ed. revised and enlarged). Macmillan 1919; Downing, Lisa, (2020) "George Berkeley", The Stanford Encyclopedia of Philosophy (Spring 2020 Edition), Edward N. Zalta (ed.); Stewart, M. (2005). Berkeley, George (1685–1753), Church of Ireland bishop of Cloyne and philosopher. Oxford Dictionary of National Biography.; Wilkins, D. (2002) edited version of The Analyst.
London: J. Tonson, 1734. Octavo, contemporary full calf; custom box. Without errata leaf and final blank, but with fragment of interesting binder’s scrap (showing ghost of part of the title page) and partial blank bound in rear. With two manuscript corrections, as usual, on p. 85. Repairs to joints and spine; some spots of scattered foxing, but text generally very clean. RARE.
Price: $12,500 .