> Howard Stein suggested to me the following treatment: The
> 'multiplication' of ratios of a given species of magnitudes requires that
> for every magnitude A of that species, every ratio of magnitudes of the
> species can be put in the form A:X (the existence of so-called 'fourth
> proportionals'). Then the compound ratio (multiplication)A:B x C:D of the
> ratios A:B and C:D is defined as A:E, where C:D=B:E. (Euclid proves that
> this is independent of the particular choices of the representatives of the
> ratios.
A closer analysis shows that one does not need the existence of the fourth
proportional in full. What one really needs is: Given four magnitudes a, b, c,
d, then there are three magnitudes u, v, w such that a : b = u : v and
c : d = v : w. The set of natural numbers has this property. Set u := ac,
v = bc, w := bd. Hence one can also compound ratios of natural numbers and the
composition is what one expects (a : b)(c : d) = ac : bd. I realised this
because I wanted to introduce also the positive rationals by using the ideas
presented in the Elements.
Whenever one has a domain of magnitudes with this property, then one can
construct the positive cone of the quotient field. For the positive rationals,
it is easy to see that multiplication (= composition of ratios) is associative
and that both distributive laws are valid. This is also true in the general
case, but in order to prove it, I had to construct the reals first. Then I
could imbed the wild creatures into the reals taming them in the end.
If one starts with a domain of magnitudes having the property that any three
elements have a fourth proportional, then it is not difficult to show
distributivity and associativity of multiplication.
Heinz Lueneburg