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read in connection with any of them, I have generally assumed as known only those methods which are to be found in all Elementary Treatises. To this, however, there is one exception: it will be seen that I have made constant use of the method known by the name of the Separation of the Symbols of Operation, although the Theory of the process is not usually given in works which are likely to be in the hands of students. I have done so because I think it a matter of some importance that the use of this method should be extended as much as possible, since it shortens and simplifies many of the processes of the Calculus, while at the same time it offers to the student one of the most instructive examples of Analytical Generalization. There seems to have been among writers on the Calculus an unwillingness to consider this method in any other light than as founded on an accidental analogy, and therefore to reject it as not based on a strict logical deduction. This idea I think is formed on a limited view of the nature of Analysis, and I shall be glad if the use which I have made of the Separation of the Symbols may induce others to examine the question closely, and so satisfy themselves of the logical validity of the process. The principles of the method are so simple that I think the short sketch which I have given of them in Chap. xv. will be sufficient to make its application readily understood.

I have adhered throughout to the notation of Leibnitz in preference to that which has been of late revived and partially adopted in this University. Of the Differential notation I need say nothing here, as


it appears to be abandoned as an exclusive system by those who introduced it: but as the use of the suffix notation for integrals has been sanctioned by those whose names are of high authority, I may state briefly some of my reasons for differing from them. In the first place, on considering the subject, I could find no arguments against the use of the notation for Differentials, which did not apply with even greater force against that for integrals: indeed, although there may be some cases in which the use of the former is advantageous, I know of none in which the latter does not appear to me to be inconvenient. In the next place, I fully agree with Professor De Morgan in an unwillingness to lose sight of the analogy to summation which is implied in the old notation; and if it were at any time necessary to consider integration merely as the inverse of differentiation, I should prefer to employ such a symbol as d.-' which expresses the required idea better than lg. But what I look on as a fatal objection to the suffix integral notation is that, like the corresponding one for differentials, it is not applicable to all cases. Of this any one may satisfy himself by attempting to use it in transforming a multiple Integral from one system of independent variables to another, a problem which is of frequent occurrence, but which I have not seen solved analytically in any work in which the suffix notation is employed. So long, therefore, as the old notation adapts itself to all cases in which it is required, while that which is proposed is not so accommodating, there appears to me no doubt which is to be preferred.

The sources from which the Examples have been taken are indicated by the references which will be found in the body of the work. For although I have not thought it necessary to cite an authority for every example, I have done so in all cases in which the student would be likely to wish for more information by consulting the original authors. It has always appeared to me that we sacrifice many of the advantages and more of the pleasures of studying any science by omitting all reference to the history of its progress : I have therefore occasionally introduced historical notices of those problems which are interesting either from the nature of the questions involved, or from their bearing on the history of the Calculus. From a fear of increasing the size of the volume too much, I have not done this to as great an extent as I wished, but these digressions short as they are may serve to relieve the dryness of a mere collection of Examples.


October, 1841.


The first edition of this work, which appeared at the close of the year 1841, having been exhausted, a new edition is, under the sanction of the Proprietors, now presented to the public. The Editor has not attempted to make any alterations in the general arrangement of the treatise, but has confined himself to effecting such occasional changes in the details, as would probably have been thought desirable by the author had he lived to prepare another edition for the press. The last chapter, however, on the Comparison of Transcendents, offers an exception to these remarks, having been in a great measure rewritten: for the alterations in this department of the work, the Editor is indebted to the kindness of Mr Ellis, Fellow of Trinity College


June, 1846.

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