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On the Notation of the Differential Calculus, adopted in some works lately published at Cambridge.

So long as the notation which forms the subject of this article remained confined to a few elementary works, there was little need for any one who valued the high and increasing scientific character of the University, to do more than regret the unnecessary trouble which would be imposed upon those learners, who should, after being trained in the new system, attempt to read the works of Lagrange, Laplace, or Poisson. But the occurrence of the new language in the public examination papers for the present year, as given in the Cambridge Calendar, has clothed the innovation with another character; and, though we confess we have not much fear, has led us to think there may possibly be danger of the harmony being interrupted which at present exists between the English and foreign scientific language. We cannot abstain from offering some observations upon the inconvenience which would result from such a change; but, to avoid any particular remarks, we shall not mention even the names of the authors to whom we refer. They are very worthy sons of their Alma mater, evidently masters of their subject. but led aside on this particular point, as appears to us, not by any false mathematical views, but by a wrong estimate of the balance of convenience and inconvenience.

The simultaneous invention of the theory of fluxions by Newton, and of the differential calculus by Leibnitz, was rich in useful consequences, by the various lights in which it caused every question to be viewed. But the difference of notation was an evil which, to this country, more than counterbalanced all the advantages. The whole continent adopted the symbols of Leibnitz; the English retained those of Newton, and gradually lost their mathematical character. The reason is obvious enough: our neighbours, with a more general and powerful notation, could easily translate all that was done in England; while we, on the contrary, could not, without great difficulty, make the language of fluxions tell us all that was discovered abroad. They had also the advantage of numbers and international communication; we could hardly read their writings, and could not, or at least did not, introduce their new and powerful methods of investigation. And to increase the difficulty, any attempt

at innovation was considered as a sin against the memory of Newton.

The University of Cambridge first broke through the mist which hid the whole continent from our view. In 1803, Mr. Woodhouse published his Principles of Analytical Calculation, in which the notation of Leibnitz was explained and dwelt upon. The impulse was thus given, though not with very great force. In 1813, it appeared from the Memoirs of the Analytical Society, a body of juniors, among whom were Messrs. Herschel, Peacock and Babbage, that the change had several zealous advocates. In 1816, these gentlemen published a translation of Lacroix's Differential Calculus, with a volume of examples, and in 1817, the second named introduced the Differential Calculus formally into the public examinations. Since that time the new system must be considered as established.

That the differential notation possessed many advantages over the fluxional was soon almost universally admitted. But this, in our opinion, would alone have been but a poor defence of the change. Had the question been upon a matter of reasoning, we need hardly say, that the united opinion of European mathematicians should not have influenced an examiner, further than to induce him to look very narrowly, and be more than commonly sure of his ground, for his own sake, before he ventured to differ from so large a majority of thinking men. But on a question of language, that of the majority is in most cases the best, because it is that of the majority. It is essential that the scientific communication of all countries should be as open as possible; and we feel very sure, that had the notation of fluxions prevailed over the continent at the period of which we have been speaking, no one of those who were instrumental in promoting the Cambridge reformation would have called it by that name, or put his hand to the work.

And it must be observed, that the change which has produced such beneficial consequences appears, at the first glance, a very trifling one. We write the notation of fluxions and differentials in a few cases:

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It should seem then that the two notations may be reduced to perfect coincidence, by so slight a convention as agreeing

that a dot above a letter shall mean that d is to be read before it; which does not apparently require more effort than to recollect, that in old printing of Latin works, a circumflex above a vowel means that m is to be read after it. But this little difference widened the straits of Dover many thousand times, so far as the mathematical sciences were concerned; which is a very strong argument against any variation, however small, from universal practice. At the same time it serves to show, that those who are determined to innovate, may as well at once carry their system to the very furthest point their fancy or judgment will let them go.

Suppose that the inhabitants of continental Europe had, by some circumstance or other, come to speak a common tonguethat the English, sensible of the facilities which their accession to the same would procure for themselves, had, with great pains, educated an entire generation in the new language, so that the old had been entirely thrown aside: what should we say to an attempt to invent and introduce a third, which had never, till then, existed in any country, simply because the framers thought their own invention more expressive or more beautiful? We should laugh, and say they never could succeed; but unfortunately, the case before us, to which the preceding bears great analogy, is one in which they may succeed, if not checked in time. And unless it is to be granted that Cambridge can preserve its scientific reputation and utility by itself, without any communication with foreign, or even with the rest of British science, it behoves those whose influence may be of use, to express their opinion on the subject. We have elsewhere (vol. iii. p. 276) advocated that part of the University system, which consists in non-interference with college tuition, and we should deprecate any public attempt, on the part of the whole corporation, either to aid or obstruct anything of the kind. Discussion and example must be the influences employed, and we hope that there are many in the University, whose authority will have weight, and who will do one of two things either openly declare themselves in favour of the new system, to the end that men of science throughout Europe may know by how great an authority the change is recommended, and may be induced to consider the propriety of imitation; or at once to oppose a measure which cannot be of little consequence either way, but must be either very good or very bad. Neutrality on their parts will lead to an inference-either that they are indifferent to the University system, which is not true-or that they despise the attempt and those who make it, which ought not to be true, for the change certainly comes upon us recommended

by men of talent, and busily employed in writing elementary works for the use of the students in the University.

The new notation is one to which no objection could have been advanced a hundred years ago, though we cannot but declare that we think we should, even upon an unprejudiced examination, have preferred that of Leibnitz. It will be best understood by making a comparison of the two.

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d.y day

dz dzy

New Cambridge system dy We now come to the arguments adduced in favour of the change, which we take from a work entitled On the Notation of the Differential Calculus: Deighton, Cambridge, 1832. On looking through the tract, we see with some satisfaction, that its intelligent author does not seem to feel quite sure the University will see exactly where the beauties of his system lie, unless he points out where and what they are to admire. If there be anything that carries conviction in one moment to the mind, it is a real improvement of mathematical notation. The successive authors who brought a into its present form through the following stages; a cubo-quadratum aaaaaa; a; never needed even a note of exclamation to call the attention of their readers to the convenience of each step. But in the tract before us, we are shown where we are to find 'no ambiguity or clumsiness,'—where there is 'perspicuity and neatness,'-where the latter becomes still more simple and elegant,'—where' it is impossible to conceive a more perfect notation.' And as to the system of Leibnitz we are told, for fear we should not find out -where it is inconsistent with itself,'-where ridiculous subterfuges' have been used-and where it is a matter of wonder that that notation has not been long ago banished.' All this we see with satisfaction, for we always suspect the weak point where we see most parade of attack and defence. We shall now, without a word of either admiration or censure, proceed to state why we think the new system not preferable to that of Leibnitz on any point, and inferior to it in several.

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Firstly, the weak points of the notation of Leibnitz seem to us to be faithfully preserved in the new system. Let us take an example from our tract:

z being a function of x and y, two independent quantities, the foldz dz lowing equation is said to express the connexion between and

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dr dy

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dz and ? dx dy equation (1)

does represent such a connexion, we are told that on the left



hand side of the equation does not mean the same thing as on the right hand side," an explanation not likely to be very satisfactory to a learner.'


This instance is taken from a work' the title of which,' says the author, it is not necessary to mention.' Here we differ: we think it was very necessary to mention the author of the work, that the opponents of the new system might know whether he was one whom they would accept as a fair stater of their case. If any author of reputation has made such an explanation, which may be the case, surely the author of our tract is aware that ninety-nine out of a hundred would not follow him. Has the author of the tract never seen

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purposely employed to distinguish the on the first side from


that on the second? Let him consult the Cambridge transladu du dy

tion of Lacroix, page 158, where he will see +

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dx dy dz


But this is not our strongest objection to the paragraph cited. We see in it that the author has not chosen to bring the new system into juxtaposition even with his own unfair specimen of the old. On looking through the tract we cannot find a single reservation which should hinder us from supposing that his own method of writing the equation (1) would be

d ̧z = d ̧2 + d,z dy

which is liable to his own objection. If he prefer

dx (z) = d2z + dy z dzy

he does no more than, as he ought to have stated, is almost (but for his own quotation, as far as we know, quite) universally done in the old system.

Secondly, our author remarks:

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