A good example is Archimedes' Method of Mechanical Theorems, where the principle of the balance is used to find results in pure geometry.
[2] The science of kinematics created a need for mathematical representation of motion and has found expression with complex numbers, quaternions, and linear algebra.
[3] "... [N]ew books which appeared in the mid-eighteenth century offered a systematic introduction to the fundamental operations of the fluxional calculus and showed how it could be applied to a wide range of mathematical and physical problems.
The strongly problem-oriented presentation in the treatises ... made it much easier for university students to master the fluxional calculus and its applications [and] helped define a new field of mixed mathematical studies..." An adventurous expression of physical mathematics is found in Maxwell's A Treatise on Electricity and Magnetism, which used partial differential equations.
The text aspired to describe phenomena in four dimensions, but the foundation for this physical world, Minkowski space, trailed by forty years.