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ART. IV. The Principles of Fluxions, designed for the Use of Students in the Universities. By WILLIAM DEALTRY, B. D. F. R. S. late Fellow of Trinity College, Cambridge. Second Edition. Printed at Cambridge by J. Smith, printer to the University. 1816.
W HEN knowledge is communicated synthetically, the business of an elementary writer is only to establish the principles, or to demonstrate the truths, which are the basis of the science. This is precisely what has been done by Euclid, and the other elementary writers on geometry; a science, of which the elements have always been synthetically delivered. When, on the other hand, a science is to be analytically treated; when an author is to teach the art of investigation; he must follow a plan considerably more enlarged, and, beside the object just mentioned, must direct his attention equally to other two. After explaining and establishing the principles, he must proceed to deduce from them the general rules which are to serve the purposes of investigation; and, lastly, must illustrate those rules by their application to particular examples. As the doctrine of Fluxions is, of all parts of knowledge, the most analytical, this method of treating it is essential to the composition of an elementary work on that branch of science; and the merit of such a work must therefore be judged of from the manner in which all these three things are performed. In reviewing the present treatise, therefore, we shall first consider how it explains the principles, and defines the ideas which are the foundations of the Calculus; next, how it explains the methods or rules derived from those principles; and, lastly, how it applies those methods to particular examples, either in the pure or the mixed mathematics.
In the exposition of general principles, the old method of demonstrating is followed, by the introduction of the notions of VeJocity and Time. This method has, no doubt, the recommendation of being old; but it has not, as Mr Dealtry seems to insinuate in his preface, the merit of being the method of the first inventor. Newton did, it is true, introduce the idea of Motion; but not those of Time and Velocity. The idea of Motion is admitted occasionally even in the Elements of Geometry; and in the higher branches of that science is almost necessarily assumed. But to introduce the other two ideas in a matter purely mathematical, is not strictly scientific; as the notion of Time can never be considered as necessarily involved either in those of Extension or of Number. Indeed, the assistance which the in
troduction of it gives in explaining the principles of Fluxions, is quite imaginary; for, after all, in order to express, or to measure the velocity of a motion which is not uniform, we must have recourse to evanescent or to nascent increments, that is, in other words, to quantities which are infinitely small. The authority of such men as M'Laurin and Simpson, both of whom made use of this method of demonstration, may indeed be pleaded in its defence. But a writer on the subject ought not now to be ignorant, that since the time of the two illustrious authors just named, a new light has been thrown on the principles of the Fluxionary or Differential Calculus, by the discoveries of LAGRANGE. In consequence of these, it has appeared that the method of fluxions, in as far as it is purely algebraical, has no dependence at all on infinitely small quantities. If the meaning of the term Function be rightly defined, it will be found that the second term of the increment of a function is in reality its fluxion, whether the increment, of the root itself be great or small. When the increment of the function is generally and fully expressed, the multiplier of the fluxion of the root in the first term of the increment, is the thing which it is important to ascertain in all the problems to which this Calculus can be applied; and it is not a little singular, that it was not till a hundred years after the invention of the Calculus, that this most important, and, as one would think, most elementary observation, was made concerning it. The second fluxion is the increment of the first fluxion; and in all this we have nothing to do with the magnitude of the fluxion, or the increment of the root; and the only occasion when small or evanescent quantities come necessarily to be considered, is when the general theorems investigated in this method are to be applied to geometrical or physical questions. It is then necessary to consider the increments of two different quantities as evanescent; but as it is the ultimate or limiting ratio of those increments which is then treated of, the reasoning is strictly geometrical, and the problem is resolved by an investigation as rigorous as that which is used in the demonstrations of Euclid and Archimedes.
This is the real state into which the Integral Calculus is now brought, by the researches and discoveries of LAGRANGE. The different views which were taken of the subject, from Newton downwards, were all derived from the last mentioned principle more or less perfectly seen; and it is very remarkale, that the idea of Newton himself approaches, in point of precision and accuracy, much nearer to that of LAGRANGE than any of the intermediate writers, if Carnot alone be excepted. The Calculus itself we conceive to be still susceptible of infinite improvement, both in its ge
neral methods, and in its particular applications; but we are persuaded, that as to what regards the explanation of its principles, and the demonstration of their truth, hardly any thing can ever be added to what has been done by the great geometer just named, in his incomparable work, La Theorie des Fonc tions Analytiques. All this has now been for several years before the scientific world, and is well known to the mathemati cians all over Europe. It is not a little singular, therefore, that a treatise on Fluxions, issuing from the very point which is accounted in this country the centre of mathematical learning, should make no more mention of this most important improvement, than if it were there entirely unknown.
The imperfect and careless manner (for as such we must necessarily consider it) in which the author has laid down the notion of a fluxion, and explained the principles of his method, could not fail to extend its influence to every part of his investigations: But it is particularly felt when he comes to treat of maxima and minima. The reasoning by which he shows that the fluxion of a quantity is equal to nothing when that quantity is a maximum or a minimum, is altogether unsatisfactory; and very far from the accuracy which would easily have been given to it, if, on the one hand, the idea of a fluxion had been accurately stated, and, on the other, the notion of a maximum or a minimum clearly defined. The true definition of these terms is, that a function of a variable quantity is a maximum, when, on increasing or diminishing the variable quantity or the root of the function, by any part however small, the value of the function itself is diminished; and again, that a function of a variable quantity is a minimum, when, on increasing or diminishing that quantity by any part, however small, the value of the function is increased. From this it readily follows, that in both cases the second term of the increment of the function, or that which involves the simple powers of or y, must be entirely wanting. It is not sufficient to say, as our author has done, that the fluxion of the quantity must be equal to nothing, because, as, in every case, the fluxion of a variable quantity, or of the root of a function, may be supposed less than any thing that can be assigned, that quantity may be said, in every case whatever, to be equal to nothing. That which really characterizes the state of a maximum or a minimum, is, that the function into which the simple power of x or y stands multiplied, is then equal to nothing. Our author, however, is not the only writer chargeable with this inaccuracy of thought and of language; many others have fallen into the same error; and it is not a little curious, that they who have thought and spoken, in this instance,
with so little precision, have yet calculated with so good effect. On this subject we must farther remark, that Taylor's theorem might have been applied to demonstrate the general principle of the method of maxima and minima in a manner both simple and unexceptionable. In fact, it has been so applied by Euler, and several of the best writers on the subject of the Differential Calculus and yet this theorem, so important in itself, and so easily investigated, is never once mentioned in the treatise now before us.
We come next to the general rules of the Calculus, or those by which fluxions and fuents are assigned. When a function of a variable quantity, or even of any number of variable quantities is given, it is seldom a matter of any difficulty to determine its fluxion. In general, when there are several variable quantities in the function, the rule is, to regard each as variable in its turn, and all the rest as constant; and the sum of the fluxions thus found is the fluxion when they all vary together. This very simple rule, which always reduces the complicated to the simpler cases, and which paves the way for some very useful generalizations in the inverse method, is not, as far as we can perceive, any where laid down in the work before us. It is however in this inverse method, or in the integration of fluxionary expressions, that the principal difficulty of the Calculus consists: It is here, accordingly, that the greatest improvements have been made: But it is here, also, that the present treatise is the most defective, and the information it conveys the most limited and imperfect. The detail into which we are to enter will prove the truth of this assertion; and we think it material that these defects should be pointed out, because the effect of such imperfect instruction is, to turn aside the attention of the young student from the true sources of sound and extensive information, and to render him contented with what is narrow and partial. A work in science can have but one fault greater than that of concealing the truth; that of substituting error in the room of it. With this last we do not tax the work before us; but of the former, we must accuse it loudly.
In the rules for finding fluents, we hardly meet with any one of which the extent and the limitations are accurately pointed After the integration of Simple Powers, which is attended with little difficulty, the integration of what are called Rational Fractions naturally follows, that is to say, of fractions where the denominator is not a simple power, but a more complicated function, though a rational one, of the variable quanzity. Now, in what is here stated on this subject, it is not ta
ken notice of, that all such fractions, when the denominators can be resolved either into quadratic or simple divisors, can be integrated: And as all quantities can be so resolved, at least by approximation, this method of integration is quite general, and without any exception whatsoever. This very important truth, which has been well known from a very early period in the history of the Calculus, is not once hinted at in the present treatise. Under the same branch of the subject, the use of impossible quantities comes first into view, and forms a very extensive and important field of discussion, about which, however, the reader must not here look for any information.
This branch of the Calculus leads to a great number of curious investigations, and of particular cases where the general rule admits of extraordinary simplifications. To take notice of all these in an elementary treatise, would indeed have been impossible; but to have omitted them altogether is entirely without apology. We would take the liberty of recommending to any one who would obtain a perfect knowledge of this very important though very elementary branch of the subject, the perusal of the second chapter of Euler's incomparable work on the subject of the Integral Calculus.
When our author proceeds to assign the fluents of those fluxionary quantities where the denominator is irrational, or affected by the radical sign, which is, of course, the next object, the defect of his method, and the limitation of his views, compared with those that are now very generally known, become still more apparent. Several examples are given, but no general rule is anywhere laid down. Now, it is well known that, whenever the radical sign on the denominator is that of the square root, and where the variable quantity under that sign does not exceed the second dimension, the fraction may be rendered rational, and reduced, of consequence, to the solution of the preceding problem. No attention whatever is paid to this general truth; though it is certain that upon it depend a vast number of the most elegant solutions that have been given of many of the most important problems, both of the pure and mixed mathematics. It is here, also, that the use of logarithms and of circular arches, for expressing fluents which cannot be otherwise assigned, should be introduced, as constituting the source of much of what is reckoned the most simple and beautiful parts of the higher geometry. Mr Dealtry has, indeed, assigned a considerable number of fluents, both by logarithms and circular arches; yet it is always by particular methods, and never in such a manner as fairly to develop the principle on which the solutions proceed. To find Fluents by Logarithms, is the title of one of his subdivisions. But when