If I am grasping your question in the right direction, this will definitely
be a hard one, because the process was fairly gradual. But in order to have
a better understanding of the scope of your research, which is your assessment
of the following Cartesian thoughts? ...
"I shall not stop to consider in detail the curves corresponding
to the other cases, for I have not undertaken to give a complete
discussion of the subject; and having explained the method of
determining an infinite number of points lying on any curve, I
think I have furnished a way to describe them.
"But the fact that this method of tracing a curve by determining
a number of its points taken at random applies only to curves that
can be generated by a regular and continuous motion does not
justify its exclusion from geometry.
...
"When the relation between all points of a curve and all points
of a straight line is known, in the way I have already explained,
it is easy to find the relation between the points of the curve
and all other given points and lines ..."
On top of that, and as far as your quest is concerned, which would be the
role of the earliest 'definitions' of the GRAPH of a 'function'? ...
With best regards,
Julio
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On July 4th, 1999, Gregory H. Moore wrote:
"Does anyone on the list have any information about when and how lines
(also curves, planes, etc.) started to be treated as sets of points?
(I distinguish this from the idea that a line is generated by the
motion of a point, as in Newton"s Tractatus de quadratura curvarum.)"