Alfred Pringsheim et Jules Molk: Nombres irrationels et notion de
limite. Encyclop.de Sci.Math.Pures et Appl., I , 3 , Paris 1908 .
The following is a translation from my German "Naive Mengen und
Abstrakte Zahlen", vol. II , Mannheim 1968 , pp.201-202 :
Bettazzi's book from 1890, written already in 1888, remained widely
unknown. For non-archimedian domains of magnitudes we find there their
subdivision into classes of archimedicity as well as classifications
based on that. Its principal content, however, is Bettazzi's proof
of the structure theorems for archimedian ordered semigroups and groups
- though still under the assumption of commutativity. If G is an
archimedian totally ordered semigroup without zero then it is either
densely ordered or is isomorphic to the semigroup of positive integers
(para 51); every G can be completed by Dedekind cuts (para 90); if G is
complete and dense then G contains the rational multiples of each of its
elements a , and every element of G is the limit of a sequence of
suitable rational multiples of a (para 89); if G and G' both are
complete and densely ordered and if a is in G and a' is in G' then there
is precisely one isomorphim - corrispondenza metrica - from G onto G'
which sends a to a' : map the rational multiples of a to those of a' and
extend this with help of the approximating sequences to the remaining
elements of G (para 92-93). The positive real numbers are an example of
such complete and densely ordered semigroup; hence any other such is
isomorphic to them (para 102). If G is complete and densely ordered then
the isomorphism theorem permits to define a multiplication: choose u in
G arbitrarily as the unit element and define the product a.m of a and m
as the image of m under the automorphism of G which sends u to a (para
106); this multiplication is distributive (para 107), commutative (para
108), associative (para 109) and remains preserved under order-
isomorphims which map the unit elements into each other (para 110).
Finally, division is possible because automorphisms are invertible (para
111-112). As an application, the theory of proportions is developed and
the existence of the fourth proportional is shown (para 114).
The structure theorems became generally known through Otto Ho"lder's
"Die Axiome der Quantitaet und die Lehre vom Masz". Ber.Verh.Kgl.
Sa"chs.Ges.d.Wiss.Math.Phys.Kl. 53 (1901) 1-64 ; its progress consisted
in making superfluous the hypothesis of commutativity - by showing that
it follows from archimedicity. Ho"lder does quote Bettazzi's book,
though its details do appear not to have been known to him. Ho"lders
construction of the isomorphism of a complete, dense and totally ordered
semigroup onto the positive real numbers avoids the use of sequences:
once u in G has been chosen as unit, an a in G is assigned that real
number which is determined by the cut (A,B) of rational numbers for
A consists of the m/n with n.a > m.u and B consists of the m/n with
m.u >= n.a . Ho"lder expresses this in the terminology of proportions;
with their help - and already making use of the multiplication of real
numbers ! - he then also shows that on G a multiplication can be defined
which has a given element as its unit. The elegant use of the
isomorphism theorem in the manner of Bettazzi escaped Ho"lder's notice
and was rediscovered only decades later, e.g. in Reidemeister's
"Vorlesungen u"ber die Grundlagen der Geometrie", Berlin 1930 .
W.F.