which it must be remembered is a practical science, will not seriously be impaired.13 It is of peculiar interest to notice that Berkeley was influenced to neglect small errors, and to justify his procedure, by the example of the differential calculus. For in the Analyst, written nearly thirty years later, he vigorously attacked the method of ignoring small errors in the calculus. What a triumph for his opponents in the Analyst controversy if they could have seen the Commonplace Book! But though Berkeley made use of the illegitimate method suggested by the calculus, his attitude to the calculus itself was from the first exceedingly critical. And his motive for criticism is not far to seek. If the calculus were sound, then his conception of geometry could not be maintained. For the calculus, whether in the form of Newton's theory of fluxions or Leibniz's method of differentials, rested, Berkeley believed, on the assumption of the existence of infinitely small quantities. Now if these infinitesimals were admitted to exist the significance of his minima sensibilia would disappear, and indeed the foundations of his philosophy as a whole would be seriously shaken. For if quantities could be proved to exist which were neither sensible nor imaginable he would need to revise his theory of knowledge altogether. He therefore had every reason to look with critical eyes on the conception of infinitely small quantities. In the Commonplace Book he says nothing of importance with regard to the use to which infinitesimals are put in the calculus. Yet even then he was certainly acquainted with a good deal of the work that had been done on fluxions and differentials. His notes contain references, on matters connected with infinitesimals, not only to Newton and Leibniz but also to Barrow, in whose Lectiones 13 Ibid., I, 78. opticae et geometricae (1669) was given the chief impulse to Newton's theory of fluxions; to Wallis (1616-1703), whose Arithmetica infinitorum (1656) paved the way for the invention of the calculus; to Keill (1671-1721), who, in addition to his Introductio ad veram physicam (1702), had written of fluxions in the Philosophical Transactions of the Royal Society, and took a prominent part in the famous "Priority controversy" in which he accused Leibniz of having derived the fundamental ideas of the calculus from Newton; to Halley (1656-1742) who in addition to his works on astronomy and magnetism wrote on fluxions in the Philosophical Transactions; to Cheyne (1671-1743), whose Fluxionum methodus inversa (1703) and Philosophical Principles of Natural Religion (1705) gained him admission to the Royal Society; to Joseph Raphson, whose De Spatio reali seu ente infinito (1697) contained a definition of the infinitely small, and who was afterwards to write a History of Fluxions; and also to two more elementary writers, Hayes (1678-1760) who published in 1704 his Treatise of Fluxions, and John Harris whose New Short Treatise of Algebra....Together with a Specimen of the Nature and Algorithm of Fluxions (1702) was the first elementary book on fluxions to be published in England. And that he had not confined his reading to English works is proved by his reference to the Analyse des Infiniments Petits, and to the controversy between Leibniz and Bernhard Nieuwentijt, a Dutch physician and physicist, which took place in 1694-5 in the pages of the Leipsic Acta Eruditorum.14 It is clear, then, that even when the Commonplace Book was written Berkeley was acquainted with much of the work that had been done in the calculus. But at that time he was not in possession of the arguments which he 14 The last-mentioned references occur, not in the Commonplace Book, but in the essay "Of Infinites" (Works, III, 411). afterwards advanced against it in the Analyst.15 In the Commonplace Book he does not venture any criticism in detail of the use of infinitesimals in the calculus.16 What he is concerned to do there is to prove that infinitesimals have no real existence at all. His line of argument is indicated twice over, and is based on his own metaphysic. For the purpose of his proof he posits two axioms: (I) "No word to be used without an idea," and (II) "No reasoning about things whereof we have no idea.". Now we have no idea, Berkeley says, of an infinitesimal. By this he means, if his terminology be translated, that infinitesimals cannot be either objects of sense-perception or objects of representation in imagination. Hence, as we have no idea of an infinitesimal, it is simply a word. Further, according to axiom I, it is a word which means nothing; and, according to axiom II, we have no right to use it in our calculations. We have now considered in outline Berkeley's attitude, as revealed in the Commonplace Book, to contemporary mathematical problems. His willingness to throw overboard the solid achievements of the established geometry simply because they did not accord with an aperçu of his own does not encourage us to rate his mathematical ability very highly. Or perhaps it would be truer to say that when he wrote the Commonplace Book he had not had time to steady his outlook upon science and the world; and allowance may fairly be made for his youthful dreams of a new idea which was destined to revolutionize the sciences, when we remember that it was only about seventy-three years since Galileo expounded the Copernican theory and thus changed entirely the orientation of astronomy and indeed of science as a whole. Another Copernican change, 15 Some of his remarks show that he was at that time, far from understanding its principles and methods (Cf. Commonplace Book, I, 84f). 16 But there is some criticism of the calculus itself in the essay "Of Infinites" (Works, III, 411). And cf. Commonplace Book, I, 83-86. he believed, was not impossible; and in any case he was inclined to think that the wonderful mathematical renaissance of the previous few decades had, among all its triumphs, grown not a few excrescences which it would do no harm but much good to pare off. What he really wished to do was to examine the logical basis of mathematics. He did not advance very far in the Commonplace Book, but it was part of what he attempted, and with greater success, in the Analyst. To the argument of the Analyst we now turn. The Analyst was published in 1734. It is a curious work, and though its purpose is ultimately theological rather than mathematical, it gave rise to a mathematical controversy which lasted for several years and produced more than thirty controversial pamphlets and articles. We have no concern with the theological argument of the Analyst, but before passing to consider its mathematical importance, it may be well to mention that the essay is primarily intended as a defense of Christianity, and that Berkeley, acting on the principle that the best defense is in attack, criticizes the foundations of mathematics on the same lines as those on which Christianity had been opposed by "mathematical infidels." In reply to the criticism that the dogmas of Christianity are mysterious and incomprehensible, Berkeley maintains that mathematics, universally admitted to be the most demonstrable department of human knowledge, is, in that regard, in precisely the same position as Christianity. For it also makes use of mysterious and incomprehensible conceptions, e. g., fluxions and infinitesimals. If mathematicians accept mystery and incomprehensibility in mathematics they have no right to object to it in Christianity. This is the kernel of Berkeley's argument. Berkeley is often regarded, but quite unjustly, as an enemy of the infinitesimal calculus. In reality he had no objection in the world to the calculus as such. What he did was to submit its logical basis to a searching examination. He criticized the conception of infinitely small quantities, which were at that time vaguely conceived as neither zero nor finite, but somehow in an intermediate state. They were said to be "nascent" and "evanescent" quantities, not quite nothing and not quite anything. It was against this "vague, mysterious and incomprehensible notion" that all Berkeley's attacks were directed; and as soon as it was clearly pointed out by one of the parties to the controversy, Benjamin Robins," that the calculus did not necessarily involve this conception of infinitely small quantities, but might be demonstrated by the methods of limits, the controversy was abandoned by Berkeley. He had replied to his other critics, such as Jurin of Cambridge ("Philalethes Cantabrigiensis") and Walton of Dublin, because these mathematicians persisted in trying to defend the conception of infinitely small quantities. But as soon as it became clear, and Robins was the first to make it so, that that conception was not essential to the calculus, the controversy lost interest for Berkeley. For the method of limits, as he seems to have realized, is not incomprehensible; and therefore an attack on it would not have enabled him to use his tu quoque argument, and would thus no longer serve his purpose, which, it must be remembered, was primarily theological.18 17 Robins's contributions to the controversy were contained in his Discourse concerning the Nature and Certainty of Sir Isaac Newton's Methods of Fluxions, and of Prime and Ultimate Ratios (1735), and in a series of articles in the Republic of Letters in 1736 and in the Works of the Learned in 1737. 18 The course of the Analyst controversy, so far as Berkeley was concerned, was as follows. In 1734 the Analyst appeared. It was almost immediately attacked by Jurin in an anonymous tract entitled Geometry no Friend to Infidelity; or a Defence of Sir Isaac Newton and the British Mathematicians. To this Berkeley replied in A Defence of Free-Thinking in Mathematics, published in March, 1735. To this reply Jurin wrote a rejoinder which was published in July of the same year. Berkeley took no notice of it. Berkeley had another critic. This was Walton of Dublin, who produced in 1735 a Vindication of Sir Isaac Newton's Fluxions. It was replied to in an appendix to the second edition of A Defence of Free-Thinking in Mathematics. Walton replied, and Berkeley then published his Reasons for not replying to |