My attention was first directed toward the considerations which form the
subject of this pamphlet in the autumn of 1858. As professor in the Polytechnic
School in Zurich I found myself for the first time obliged to lecture upon the
elements of the differential calculus and felt more keenly than ever before the
lack of a really scientific foundation for arithmetic. In discussing the notion of
the approach of a variable magnitude to a fixed limiting value, and especially
in proving the theorem that every magnitude which grows continually, but not
beyond all limits, must certainly approach a limiting value, I had recourse to
geometric evidences. Even now such resort to geometric intuition in a first pre-
sentation of the differential calculus, I regard as exceedingly useful, from the
didactic standpoint, and indeed indispensable, if one do es not wish to lose to o
much time.
But that this form of introduction into the differential calculus
can make no claim to being scientific, no one will deny. For myself this feel-
ing of dissatisfaction was so overpowering that I made the fixed resolve to keep
meditating on the question till I should find a purely arithmetic and perfectly
rigorous foundation for the principles of infinitesimal analysis. The statement is
so frequently made that the differential calculus deals with continuous magni-
tude, and yet an explanation of this continuity is nowhere given; even the most
rigorous expositions of the differential calculus do not base their pro ofs upon
continuity but, with more or less consciousness of the fact, they either appeal
to geometric notions or those suggested by geometry, or depend upon theorems
which are never established in a purely arithmetic manner. Among these, for ex-
ample, belongs the above-mentioned theorem, and a more careful investigation
convinced me that this theorem, or any one equivalent to it, can be regarded in
some way as a su cient basis for infinitesimal analysis. It then only remained to
discover its true origin in the elements of arithmetic and thus at the same time
to secure a real definition of the essence of continuity. I succeeded Nov. 24, 1858,
and a few days afterward I communicated the results of my meditations to my
dear friend Dur‘ ege with whom I had a long and lively discussion. Later I ex-
plained these views of a scientific basis of arithmetic to a few of my pupils, and
here in Braunschweig read a paper upon the subject before the scientific club
of professors, but I could not make up my mind to its publication, because, in
the first place, the presentation did not seem altogether simple, and further, the
theory itself had little promise. Nevertheless I had already half determined to
select this theme as subject for this o ccasion, when a few days ago, March 14,
by the kindness of the author, the paper Die Elemente der Funktionenlehre by
E. Heine (Crelle’s Journal, Vol. 74) came into my hands and confirmed me in
my decision. In the main I fully agree with the substance of this memoir, and
indeed I could hardly do otherwise, but I will frankly acknowledge that my own
presentation seems to me to be simpler in form and to bring out the vital point
more clearly. While writing this preface (March 20, 1872), I am just in receipt
of the interesting paper Ueber die Ausdehnung eines Satzes aus der Theorie der
trigonometrischen Reihen, by G. Cantor (Math. Annalen, Vol. 5), for which I
owe the ingenious author my hearty thanks. As I find on a hasty perusal, the
axiom given in Section II. of that paper, aside from the form of presentation,
agrees with what I designate in Section III. as the essence of continuity.
But what advantage will be gained by even a purely abstract definition of real num-bers of
a higher type, I am as yet unable to see, conceiving as I do of the domain of real numbers as complete in itself.
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