Ned,

Thank you for your inquiry about the dictionary definition
"zwei Teile = 2/3, drei Teile = 3/4" etc. I agree that the
expressions are not colloquial, yet find them not difficult 
to explain.

"Teilen" spelled, until the language reform of 1901, "Theilen"
related to English, "deal", means to separate or to part.
Although it is easy to design and construct a grid with which
one may simultaneously split an object into many parts, the
primary (existential) experience of separating is by division
as with a knife, into two parts which are not necessarily equal.
The description of their relative sizes is perhaps more a matter
of linguistic convention than we realize. When we say "a third",
we mean one of three parts of the whole; but it seems to me not
unreasonable to say "three parts" (drei Teile) with the implication
that the other part (Teil) of the division (Teilung) is but one.
Hence drei Teile would be equal to 3/4. I hear a cook saying:
"put three parts of flour into a bowl, add sugar and stir" with
the understanding that the sugar constituted the remaining fourth
of four equal parts. It's easy for me to imagine a community of
chemists who adopted such nomenclature, especially since separation
in chemistry, if I understand correctly, is more commonly of liquids
than of solids, and while a solid may be broken into many fragments
with a single blow, pouring out a portion of a liquid is inherently
productive of only two parts, the aliquot and the remainder.
Am I writing nonsense?

Jochen