I am very interested in your question. I do not yet have the precise
references you would like; but have a few comments.
1. First a remark about the reference to Newton. What he writes is
"Mathematical quantities I here consider not as consisting of least
possible parts, but as described by a continuous motion. Lines are
described and by describing generated not through the apposition of parts
but through the continuous motion of points ...".
I have understood this, not as a repudiation of the idea of a line as
consisting of "least possible parts", but as merely stating that, in that
paper, he is going to consider only *parameterized* lines (surfaces, etc.),
i.e. lines armed with a *particular* parameterization by time. This reading
of him fits the text and explains why he made this remark: for his
subsequent discussion always assumes a particular parameterization, i.e.
motion.
2. It is useful to ask what it means for a line to be a set of points.
Certainly it would have been agreed very early on that two curves with the
same sets of points on them are identical. So it has seemed to me that the
issue always had to be whether the relations of <= of curves qua magnitudes
could be defined in terms of a relationship between their sets of points.
(Of course, this is possible only if we regard the sets, not as abstract
sets, but as point-sets.) Anyway, let me assume that this is the issue.
3. This suggests that adopting L=set of points on L amounts to adopting the
measure-theoretic conception of the integral. So maybe Jordan's work on
integration is the first instance---if I am right that he was the first to
treat the integral measure-theoretically.
4. But I think that there is a lower bound, too. There is an old argument,
due to Zeno, via Simplicius, that the line cannot be the set of its points
(in the sense I have indicated). The argument begins with the fact that the
line (segment) has an infinite number of points. The argument is that, if
each point has (or better, is) no magnitude, then neither has (is) the
line; and if each point has (is) a magnitude, then the line is
infinite---the implicit assumption of the first horn of the dilemma being
that,even in the infinite case, magnitudes should be additive, and the
implicit assumption of the second horn being Archimedes' axiom (maybe its
first appearance).
So now it is obvious what my lower bound is: it is 1874, the date of
Cantor's proof that the set of real numbers (and so the set of points on
the line) is uncountable. The property that a countable set of points has
measure 0 if each (unit set of a) point has measure 0 certainly had to be
preserved. So the viability of line L = set of points on L depends on being
able to distinguish countable from uncountable sets.
Sorry: this is the kind of a priori history we philosophers do. But I hope
it will help some.
Bill Tait