The languages of most physical theories and laws use higher mathematics such as analysis, but primarily the evaluation of differential equations. Most breakthroughs in physics have necessitated the use of already existing mathematical techniques. Examples include matrix mechanics for quantum interpretation, and the calculus for early mechanics.
The four principal breakthroughs in the history of physics have been:
- Sir Isaac Newton's laws of motion and his contemporaries' work on such things as planetary motion.
- James Clerk Maxwell's formulation of the laws of electromagnetism, crucial to the next level of understanding of matter after gravity.
- Albert Einstein's seminal work on relativity, and his famous mass-energy equivalence.
- Quantum physics and its counter-intuitive explorations of subatomic physics.
 Early physics
Aristotle believed that everything was made up of one of the five states of matter: earth, fire, air, water, or magic: the "quintessence", or "fifth essence". He was working in the dark.
 Eighteenth century physics
Newton did a pretty good job of mathing up some stuff. That is to say that Newton began attaching a sort of mathematical formalism to the science that had not really existed up until this point.
Some other people pitched in with more details.
 Nineteenth century physics
Maxwell did for electromagnetism what Newton had done for mass and gravity by building it into a coherent theory. One little problem was that the equations said that the velocity of light (etc) didn't depend on the velocity of the source or the observer. Cue much scrabbling around to work out why this wasn't bonkers.
 Twentieth century physics
Einstein had a go at what would happen if Maxwell's constancy of light-velocity wasn't bonkers. His "Special Relativity" refined Newton's laws of motion for object at high relative velocities; gives lots of fun to web-loony watchers (Whaddya mean, time and distances are relative,? You an atheist or summin'?); and spawned a bazillion misstatements about what the 'paradox' in the Twins paradox is supposed to be.
He also had a bit of a think about why an object's inertial mass (resistance to acceleration) is always the same as its gravitational mass (strength of attraction to other masses). Hey, what if acceleration and gravity are just two names for the same thing? Bingo! General Relativity. Mass bends space. Stars bend light. More refining Newton. More loony websites.
Quantum mechanics then made some very strange predictions about how the universe works at a very small scale. Due to the inherent "spookiness" of quantum mechanics, it has been a favorite target of pseudoscientists (to be fair, even Einstein wasn't too chuffed about its use of probabilities). The disagreements over the 'real' interpretation of quantum mechanics, and the lack of an underlying theory linking quantum mechanics and general relativity is a fertile area for people who like to point out that because science cannot answer everything, all science is wrong and therefore [insert flying teapot equivalent here] must be true.
 Twenty-first century physics
One of the more popular of today's physics topics is the attempt to unify the theory of general relativity (GR) with quantum mechanics. When one tries to combine GR and quantum field theory, one encounters a problem of infinities. No one has figured out how to get around this. One of the many proposals is string theory. It essentially "tames" the infinities by spreading out interactions in spacetime. String theory also accounts for the other three interactions of nature. It is thus called a "theory of everything" because it says that all four of the known fundamental forces (gravity, weak-nuclear, strong-nuclear and electromagnetic) are a manifestation of one underlying mechanism.