In his paper The search for unity: Notes for a history of quantum field theory [Daedalus 4, 106 (1977)], Steven Weinberg writes:
The first successful classical field theory was based on Newton's theory of gravitation. Newton himself did not speak of fields - for him, gravitation was a force which acts between every pair of material particles in the universe, "according to the quantity of solid matter which they contain and propagates on all sides to immense distances, decreasing always as the inverse square of the distances." It was the mathematical physicists of the eighteenth century who found it convenient to replace this mutual action at a distance with a gravitational field, a numerical quantity (strictly speaking, a vector) which is defined at every point in space, which determines the gravitational force acting on any particle at that point, and which receives contributions from all the material particles at every other point.
Which mathematical physicists is he referring to?