- Axel Harnack's Preface.
The present work owes its origin to the special circumstances of the Professorship which I hold at the Polytechnikum. Having to give instruction in the Differential and Integral Calculus to classes of technical students, who require a knowledge of Analysis chiefly as an instrument for the solution of mechanical problems, I have had to adopt a strictly limited conception of a function in order that the proofs of the general theorems might be simplified. But I could not avoid feeling it all the more due to such of my hearers as desire to devote themselves to the study of Mathematics, that I should put within their reach a supplement to my lectures, giving greater prominence to the systematic parts of Analysis.
Such a supplement may not, I think, be altogether superfluous even for a larger circle of readers, since most treatises on Higher Analysis (one exception is the recent work of Dr Lipschitz), are so occupied with its practical applications that they enter but inadequately into any discussion of the principles on which Analysis is founded. The student first realises the necessity of discussing these fundamental problems, when he comes to study treatises introductory to the Theory of Complex Functions, where much that he had probably become accustomed to regard as established by Analysis appears once more called into question. That the scientific discussion of its principles should thus be severed from the practical applications of Analysis has no justification in the nature of the subject, and any such severance is quite unsuitable in teaching it. I cannot indeed claim to have wholly avoided this in the following Essay. Even in the necessary division of its contents into four Books a separation is apparent which is based upon the fact that in the Theory of Real Functions the data are much more detailed than in that of Complex Functions. But besides the purely didactic aim, my guiding wish has been that my work might contribute towards laying the foundations on which the Differential and Integral Calculus may some day come to be treated with perfect unity of system.
In the selection and ultimate limitation of the contents it was not always easy to decide: there may be a diversity in judgments respecting what belongs to the "Elements" of the Calculus. My purpose will be fulfilled, if this Introduction be found useful as a preparation for the study of differential equations, of algebraic curves and of algebraic integrals. The theory of these integrals might have been joined on immediately to the last chapter; but I had to omit it, since a short sketch would have been of little use, and an investigation in detail would have completely displaced the centre of gravity of my work.
I desire here also to express my thanks to my friend and colleague Dr. Voss for all the benefit my work has derived from the interest he has taken in it.
Dresden, February 1881.
- Note by the Translator George L Cathcart.
An acquaintance of some years with Prof. Harnack's work: Die Elemente der Differential- und Integralrechnung, Leipzig, B. G. Teubner, 1881, led me to desire that the subject should be made accessible to English readers in the manner in which he has presented it. I therefore wrote to the Author, on March 12, 1888, and asked his sanction to publish a translation.
In his reply dated Dresden, March 15, 1888, Prof. Harnack gave me his permission, and proceeded to say:-
I must add however, that my book, written rather rapidly and with somewhat youthful lightness of heart eight years ago, seems to me now in many places not sufficiently exact. ... Since its publication I have made a considerable list of corrections, and noted many points on which attention should be bestowed in any future Edition. These portions of the work I shall therefore ask you to allow me to recast for your English Edition. I shall make the alterations as concise as possible, so as to enable you to insert them into your manuscript with no great labour or loss of time. I hope to be able to send you as a sample next week, the alterations I propose to make in the first Book. Those in the second Book, I cannot send for about six weeks, as I shall be in North Italy during the Easter Vacation. The rest will follow soon after, so that the printing of your work need not be delayed. Hoping that you will be able to accept my proposal, as it will conduce to the improvement of the work,I replied, stating that I too was starting for Italy, and suggested that we might possibly meet and confer upon the subject. But in London, I received a post-card from Dr Harnack, as follows:-
I remain yours, etc.
Dresden, 21 March 1888.Soon after my return from Italy, I was shocked by receiving the following letter:-
I can propose no definite engagement to meet you in Italy, as I have been ill for a week, and my journey has therefore become uncertain. Hence also I have been able to work out very little as yet of the supplementary matter; but I hope to be able to send you the notes to the first Book by April 16, and the rest as you require it.
Munich, 23 April, 1888.In answer to my reply to the above, Professor Voss wrote as follows, on May 6:-
You will have heard ere this of the unexpectedly sudden death of Prof A Harnack, on April 3. A few days previously he had, as I learn, been in correspondence with you about the publication of an English translation of his work on the Differential and Integral Calculus, and had specially expressed his wish to be allowed to make some needful improvements and alterations. In the papers he has left, which I received yesterday from his Widow, there are unfortunately only the beginnings of a revision, as far as page 40. ...
Dr A Voss.
Prof. d. Mathematik.
I send you now the few sheets my dear friend put together shortly before his death, (mostly in connection with his edition of Serret's 'Cours de calcul différentiel'). They show that he contemplated considerable changes in the subsequent parts of his work, but nothing further has been found.These notes by Professor Harnack, as well as a few of his own referring to the same part of the work, which Professor Voss sent along with them, I have adopted and incorporated in the translation. He kindly offered also to take part in more extensive alterations, but on full consideration I determined to adhere to the work in its original form, and my decision was confirmed by the receipt of the following letter from Professor Voss:-
I have come to the opinion that the work should be kept exactly in its present form, and in this my friends, Professor Klein in Göttingen and Professor Nöther in Erlangen, also agree. Only I should regard it as due to the author, that it should be stated in the preface to the English Edition, that Professor Harnack contemplated essential changes and improvements, which after his death could not be made save at the risk of losing somewhat of the admitted freshness and originality of the work. I believe that Professor Harnack, in the further course of his own revision, would have given up the idea of recasting it.
In conclusion, I most thankfully acknowledge my obligations to the Rev George Salmon, D.D., Provost of Trinity College, Dublin, for much kindly counsel relative to my work, of which I have availed myself throughout its progress.
25, Trinity College, Dublin.
George L Cathcart.
- First paragraphs of Harnack's Introduction.
Real numbers and functions of real numbers.
The conceptions of space and of number are the subject matter of mathematical investigations. These investigations accordingly diverge into two main branches: Geometry and Analysis. It thus appears that Mathematics are of fundamental importance to all our knowledge of Nature: for our representations of space contain the simplest properties which are common to all things in the surrounding world; and accurate comparison or measurement of quantities leads always to concrete numbers of the units employed: in order to understand the result, we require a knowledge of numbers and of their combinations.
Nature in its phenomena is perpetually exhibiting change; the simplest changes we perceive externally are changes of place, motions. The representation of motion is necessarily combined with that of continuity, i.e. of an uninterrupted connection in space and of an uninterrupted sequence in time. To describe thoroughly the phenomena of motion is to assign every circumstance in numbers of concrete units: so that if the series of numbers is also to enable us to describe motion, it must contain a continuous series of quantities. Thus the first problem of Analysis is: to develop the conception and the properties of the continuous series of numbers.
- First Chapter. Rational numbers.
1. The natural series of numbers, which arises by adding on a thing to others in counting, advances always by unity; each number is defined by the preceding number and by unity. This series of integers starting from unity can be continued on indefinitely. Now as each several number is a sum of repeatedly added units, such a sum of units can be composed of different given numbers. This arithmetical operation, merely a continued reckoning up of groups of units, is called Addition; it embraces all other operations, from it all others arise. The fundamental proposition for addition is: the sum of given numbers ...
- Second Chapter. Complex numbers.
64. The contrasted epithets "real" and "imaginary" favor the erroneous impression, which indeed has impeded the systematic introduction of imaginary numbers into analysis, that numbers of the first kind possess a practical reality which those of the second have not. Considering the arithmetical operations merely, without application to physical quantities, fractions, irrational numbers, imaginaries, all form legitimate extensions of the conception of number, that are connected with the integer by determinate arithmetical operations. In the applications of these operations on the other hand everything depends on the kind of numbers introduced at the outset in framing a problem analytically. If, for example, in the case of discrete quantities, from the way in which the problem is proposed only integers are admissible, the proposed problem is seen to be impossible when the result is a fraction. Likewise in the result of a calculation referring to physical quantities, a negative number will have a meaning applicable to these quantities, only when from the first the quantities were distinguished in the sense of positive and negative. In analogy with this, even when the result of calculation is imaginary, its meaning is no longer unreal, when the actual quantities considered, are characterised not only by real, but also by imaginary numbers. The simplest example of a representation of intuitive quantities by imaginary numbers is the geometric interpretation, which we shall deal with as we go on. "As mathematical science strives towards doing away with exceptions to rules and towards contemplating different propositions from one point of view, it is often compelled to enlarge its conceptions or to establish new ones, and this nearly always denotes a progress in the science. A great example of this is the introduction of imaginary quantities into analysis." (von Staudt, Beitrige zur Geometrie der Lage. Heft I. Vorwort.)