66

AN UNPUBLISHED ESSAY BY BERKELEY.

NOTWITHSTANDING the very careful search in

the Library of Trinity College, Dublin, made by Prof. A. C. Fraser when collecting materials for his account of Berkeley, two early essays of the great Idealist escaped his notice. One of these essays is a description of the cave of Dunmore, near Kilkenny; the other bears the title, "Of Infinites." Both are in Berkeley's own handwriting, and are contained in that miscellaneons collection of MSS. of varying dates ranging from the middle of the seventeenth century to the middle of the eighteenth, known as the Molyneux Papers." The collection contains many of the contributions to the proceedings of the Dublin Philosophical Society, the predecessor of the present Royal Dublin Society in one sense, and of the Royal Irish Academy in another. To William Molyneux, the patriot and friend of Locke, is due the credit of founding, in 1683, what was then called simply the Dublin Society. Molyneux was its first secretary, and managed its affairs successfully until the political disturbances of 1687-1690, by banishing its members, put a stop to its meetings. In 1692, Molyneux brought about a reconstitution of the society, but it had not sufficient energy to survive his death in 1698. Towards the close of 1707 the Society was revived, and the post of secretary was passed on to the son, Samuel Molyneux, then an undergraduate in Trinity College. The "Molyneux Papers" contain essays read at each of these three periods of the society's existence; and it is exceedingly probable that it is in this way, as contributions to the proceedings of the Society of 1707, the two essays by

Berkeley have been included in the collection. The endorsement on each essay is in the handwriting of Samuel Molyneux, at least as far as it is safe to conclude from the few words used in each case.

The description of the Dunmore cave, however, bears date, January 10, 1705/6: while the revived Dublin Society held its first meetings at the end of 1707. Now Prof. Fraser found, in Berkeley's "Commonplace Book," codes of rules of two societies, the earlier code being headed thus:-"The following Statutes were agreed to and signed by a Society consisting of eight persons, January 10, A.D. 1705." One of these rules limits the membership to eight persons. The second of the two Societies, an enlargement of the first, appears to date from the end of 1706. So the history of these gatherings runs somewhat as follows:In January, 1705/6, a small coterie of College men arranged meetings for discussing subjects of common interest. A successful session caused them to widen their lines for the following year, and finally, at the end of 1707, to attempt the much more ambitious task of reviving the Dublin Society. Berkeley's "Description of the Dunmore Cave" may have been the inaugural essay of the first stage, while certain corrections and additions that are in the copy seem to be a retouching for a subsequent reading at the more public meetings of the 1707 revival. Of the contents of this essay it is not necessary to treat, for it is practically identical with the description of the cave found by Prof. Fraser in Berkeley's "Commonplace Book," and printed by him in the biography.

There is no date attached to the essay "Of Infinites." However that it is among the Molyneux Papers and endorsed by Samuel Molyneux points unmistakably towards the conclusion that this essay belongs to the same period as its companion. The only internal chronological mark of any importance is the reference to Cheyne's

Philosophical Principles of Natural Religion, which was published in London in 1705. There is a passage in the "Analyst" which may possibly refer to this essay :Section 50. "Of a long time I have suspected that these modern analytics were not scientifical, and gave some hints thereof to the public about twenty-five years ago." Professor Fraser, however, regards this as an allusion to certain sections in the "Principles." The essay "Of Infinites" has much in common with the "Analyst," though it does not carry the argument into the religious province by the ad hominem method so prominent in the later attack on the mathematicians of the calculus.

As the reference to Cheyne fixes 1705 as the earliest limit of the time of composition, so possibly 1709 may be assigned as a late limit, for in that year Samuel Molyneux left Dublin. The somewhat disconnected character of the essay, and a certain impression of crudeness in the manner of thought, suggest the earlier portion of this period as the more probable.

SWIFT P. JOHNSTON.

OF INFINITES.

Tho' some mathematicians of this last age have made prodigious advances and open'd divers admirable methods of investigation unknown to the ancients, yet something there is in their principles which occasions much controversy and dispute to the great scandal of the so much celebrated evidence of Geometry. These disputes and scruples arising from the use that is made of quantitys infinitely small in the above mentioned methods, I am bold to think they might easily be brought to an end by the sole consideration of one passage in the incomparable Mr. Locke's "Treatise of Humane Understanding," b. 2. ch. 17. sec. 7, where that authour, handling the subject of infinity with that judgment and

clearness which is so peculiar to him, has these remarkable words :-"I guess we cause great confusion in our thoughts when we joyn infinity to any suppos'd idea of quantity the mind can be thought to have, and so discourse or reason about an infinite quantity, viz., an infinite space or an infinite duration. For our idea of infinity being, as I think, an endless growing idea, but the idea of any quantity the mind has being at that time terminated in that idea, to join infinity to it is to adjust a standing measure to a growing bulk, and, therefore, I think 'tis not an insignificant subtilty if I say we are carefully to distinguish between the idea of infinity of space and the idea of space infinite."

Now if what Mr. Locke says were, mutatis mutandis, apply'd to quantity infinitely small, it would, I doubt not, deliver us from that obscurity and confusion which perplexes otherwise very great improvements of the Modern Analysis. For he that, with Mr. Locke, shall duly weigh the distinction there is betwixt the infinity of space and space infinitely great or small, and consider that we have an idea of the former but none at all of the later, will hardly go beyond his notions to talk of parts infinitely small or partes infinitesimae of finite quantitys and much less of infinitesimae infinilesimarum and so on. This, nevertheless, is very common with writers of fluxions or the differential calculus, &c. They represent, upon paper, infinitesimals of several orders, as if they had ideas in their minds corresponding to those words or signs, or as if it did not include a contradiction that there should be a line infinitely small and yet another infinitely less than it. 'Tis plain to me we ought to use no sign without an idea answering it and 'tis as plain that we have no idea of a line infinitely small, nay, 'tis evidently impossible there should be any such thing, for every line how minute soever, is still divisible into parts less than itself, therefore there can be no such thing as a line quavis data minor or infinitely small.

Further it plainly follows that an infinitesimal even of the first degree is merely nothing from what Dr. Wallis, an approv'd mathematician, writes at the 95th proposition of his "Arithmetic of Infinites," where he makes the asymptotic space included between the two asymptotes and the curve of an hyperbola to be in his stile

a series reciproca primanorum, so that the first term of the series, viz., the asymptote, arises from the division of 1 by o. Since, therefore, unity, i.e. any finite line divided by o gives the asymptote of an hyperbola ie. a line infinitely long, it necessarily follows that a finite line divided by an infinite gives o in the quotient i.e. that the pars infinitesima of a finite line is just nothing. For by the nature of division the dividend divided by the quotient gives the divisor. Now a man speaking of lines infinitely small will hardly be suppos'd to mean nothing by them, and if he understands real finite quantitys he runs into inextricable difficultys.

Let us look a little into the controversy between Mr. Nieuentiit and Mr. Leibnitz. Mr. Nieuentiit allows infinitesimals of the first order to he real quantitys but the differentia differentiarum or infinitesimals of the following orders he takes away making them just so many noughts. This is the same thing as to say the square, cube, or other power of a real positive quantity is equal to nothing, which is manifestly absurd.

Again Mr. Nieuentiit lays down this as a self evident axiom, viz., that betwixt two equal quantitys there can be no difference at all, or, which is the same thing, that their difference is equal to nothing. This truth, how plain soever, Mr. Leibnitz sticks not to deny, asserting that not onely those quantitys are equal which have no difference at all, but also those whose difference is incomparably small. Quemadmodum (says he) si lineae punctum alterius linea addas quantitatem non auges. But if lines are infinitely divisible, I ask how there can be any such thing as a point? Or granting there are points, how can it be thought the same thing to add an indivisible point as to add, for instance, the differentia of an ordinate in a parabola, which is so far from being a point that it is itself divisible into an infinite number of real quantitys whereof each can be subdivided in infinitum, and so on, according to Mr. Leibnitz. These are difficultys those great men have run into by applying the idea of infinity to particles of extension exceeding small but real and still divisible.

More of this dispute may be seen in the Acta Eruditorum for the month of July, A.D. 1695, where, if we may believe the French author of Analyse des infiniments petits, Mr. Leibnitz has sufficiently