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BERKELEY'S LOGIC OF MATHEMATICS.*

THAT was Commonplace Book a great

HAT Berkeley was keenly interested in mathematics

deal of attention is paid to mathematical questions; it is noticeable, indeed, that in its pages Berkeley refers to mathematicians far more frequently than to philosophers. The extent of his interest in mathematics is indicated also by a group of early writings, Arithmetica absque Algebra aut Euclide demonstrata, and Miscellanea Mathematica which includes papers "de Radicibus Surdis," "de Cono Aequilatero et Cylindro eidem Sphaerae circumscriptis," "de Ludo Algebraico," and "Paraenetica quaedam ad studium matheseos praesertim Algebrae." Both of these tracts were written in 1705 and first published in 1707. Belonging to the same period is the essay "Of Infinites," which is in part concerned with the infinitesimal calculus. Berkeley deals with mathematical questions also in The Principles (1710) and in De Motu (1721), and his criti

*The following article contains, in its treatment of Berkeley's early work which was not published for generations after it was written, a new and important contribution to the history of mathematics. It will also be of interest to our readers to know that editions of the books by Barrow and Wallis mentioned in this article are in preparation. They are edited by Mr. J. M. Child and will appear in the "Open Court Classics of Science and Philosophy." Further, in the same series a small volume by Prof. Florian Cajori on the history of fluxional concepts from the time of Newton is also in preparation. It will contain a detailed account of the Analyst controversy. Finally it is to be noticed that Berkeley's doctrine of "compensation of errors" in the calculus was later advocated by the eminent mathematicians Lagrange and Lazare Carnot.-Proofs of this article did not reach the author who was absent on military service.-ED.

cisms of the logical basis of the infinitesimal calculus in The Analyst (1734) and A Defence of Free-Thinking in Mathematics (1735) are of considerable importance in the history of mathematics.

In this paper I propose to consider the mathematical views stated in Berkeley's Commonplace Book and Analyst. In both cases he is concerned mainly with the logical basis of mathematics.

Berkeley very clearly perceived that his "new principle" involved difficulties with regard to the nature of mathematics. The "new principle" implied that lines consist of a finite number of points, that surfaces consist of a finite number of lines, and that solids consist of a finite number of surfaces. Thus ultimately all geometrical figures are composed of complexes of points, which are regarded by Berkeley as ultimate individualities. These indivisibles are minima sensibilia, the minutest possible objects of sense. It is impossible that the minimum sensibile should be divisible, because in that case we should have something of which our senses could not make us aware, and that, Berkeley believes, is simply a contradiction.1

Sensation, then, is the test of all geometrical relations. Thus equality depends simply on our inability to distinguish in sense-perception. "I can mean nothing by equal lines but lines which it is indifferent whether of them I take, lines in which I observe by my senses no difference." He explicitly considers the claims of imagination and pure intellect to judge of geometrical relations, and summarily rejects their pretensions. Imagination, he points out, is based on sensation, and has no other authority than that of the senses. It has no means of judging but what it derives from the senses, and, as it is removed by one stage from immediate sense-perception, and has its knowledge

1 Berkeley's Works, Oxford, 1901, Vol. I, p. 2 Ibid., I, 22.

86.

only at second-hand, it is in fact not so well fitted as sensation to judge and discriminate. Pure intellect, Berkeley continues, has no jurisdiction in mathematics, for it is concerned only with the operations of the mind, and "lines and triangles are not operations of the mind."3

Now this view of the nature of geometry is the direct consequence of Berkeley's early metaphysical doctrine, but it is interesting to note that it also connected itself in his mind with the method of indivisibles maintained by the Italian mathematician Cavalieri. "All might be demonstrated," he says, "by a new method of indivisibles, easier perhaps and juster than that of Cavalerius."4 What precisely Cavalieri meant by his conception of indivisibles is open to doubt, but it is certain that Berkeley's sympathy would be elicited by his demonstration that quantities are composed of indivisible units, a line being made up of points, a surface of lines, and a volume of surfaces. It is possible, though he is very obscure, that he regarded areas as composed of exceedingly small indivisible atoms of area. Berkeley's conception is very similar to this; but whereas Cavalieri maintained that the number of points in a line is infinite, Berkeley was convinced that no line or surface can contain more than a finite number of points, points for him being minima sensibilia. This, then, is Berkeley's "new method of indivisibles."

It will follow that geometry must be conceived to be an applied science. The only pure science will be algebra, for it alone deals with signs in abstraction from concrete things. Geometry may be regarded as an application of arithmetic and algebra to points, i. e., the minima sensibilia which constitute the whole of concrete reality, Berkeley admits that it is difficult for us "to imagine a minimum," but the reason is simply that we have not been accustomed to take note of it separately. In reading we 3 Ibid., I, 22; cf. I, 14.

4 Ibid., I, 87.

5 Ibid., I, 85.

do not usually notice explicitly each particular letter; but the words and pages can be analyzed down to these minimal letters. Similarly, though we are not explicitly aware of the minima sensibilia, they do exist separately, and may be analyzed as indivisibles in the complex sense-datum presented to us in perception. Geometry, then, is an applied science dealing with finite magnitudes composed of indivisible minima sensibilia.

If this conception of geometry be adopted, it immediately follows, as Berkeley very clearly perceived, that most of the traditional Euclidean geometry must be rejected. (1) In the first place, on the new theory, not all lines are capable of bisection. Only those lines which consist of an even number of points can be bisected. If the number of points composing the line be odd, then (supposing bisection to be possible) the line of bisection would have to pass through the central point. But the point is ex hypothesi indivisible; hence the line does not admit of bisection. (2) Again, the mathematical doctrine of the incommensurability of the side and diagonal of the square must be rejected.3 For since both the side and the diagonal of the square consist of a finite number of points, the relation between these lines will always be capable of exact numerical expression. Berkeley even makes the general statement, "I say there are no incommensurables, no surds." (3) It follows that one square can never be double another, for that is possible only on the assumption of incommensurables. And it also follows that the Pythagorean theorem (Euclid, I, 47) is false." (4) Further, it is no longer possible to maintain that a mean proportional may be found between any two given lines. A mean pro

6 Ibid., I, 79, 80.

7 Ibid., I, 60, 78, 79.

8 Ibid., I, 14.

• Ibid., I, 19.

portional will be possible, on Berkeley's theory, only in the special case where the numbers of the points contained in the two lines will, if multiplied together, produce a square number.10 (5) Finally, the important work that had recently been done on the problem of squaring the circle is, in Berkeley's view, quite useless. Any visible or tangible circle, i. e., any actually constructed circle, may be squared approximately; and it is therefore time thrown away to invent general methods for the quadrature of all circles.

That his new doctrine necessitated such a clean sweep of important mathematical propositions, most of which had been accepted for hundreds of years, might well have given pause to an even more confident man than Berkeley; for (to take only one instance) apart from its startling theoretical aspects, serious practical difficulties would arise if some lines should prove incapable of bisection. Berkeley therefore suggests that for practical purposes small errors may be neglected. Though we cannot bisect a line consisting of 5 points, we can divide it into two parts, one containing 3 points, the other 2; and, as the minimum sensibile is so minute, it makes no practical difference that the lines are only approximately equal. Berkeley was influenced to make this suggestion by the method of neglecting differences practised in the calculus." If differentials, which are admitted to be something, are overlooked under certain circumstances in the calculus, are we not justified in the new geometry, Berkeley asks, in neglecting everything less than the minimum sensibile?12 The resulting errors will be so slight that the usefulness of geometry,

10 Ibid., I, 14.

11 Ibid., I, 85.

12 It might seem that in our approximate bisection of the line we have neglected a whole minimum sensibile. But from the point of view of the error involved in each of the resulting parts we are not guilty of that. Each of the parts ought to contain 22 points. Now each of the lines obtained by the approximate method differs from this by only 1⁄2 a point. Hence the error to be neglected in each case is less than a minimum sensibile. And this is the condition laid down by Berkeley.

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