Short answers, pulled from the story.
Newton's Philosophiæ Naturalis Principia Mathematica was published on the 5th of July 1687. The book did not contain the modern equation F equals ma or use vector notation that defines physics today.
Isaac Newton constructed his arguments using the rigorous geometry of Euclid to ensure his proofs were unassailable. He deliberately avoided using the mathematical tools he had invented, which he called the science of fluxions.
Newton defined momentum as the product of a body's mass and its velocity. He stated that the change in this quantity is proportional to the impressed force.
Émilie du Châtelet played a key role in clarifying the distinction between potential and kinetic energy. She was a French physicist and philosopher who clarified terms that were previously confused with what Newton called dead and living force.
Josiah Willard Gibbs and Oliver Heaviside pioneered the modern vector algebra used to describe forces and motions in the late nineteenth century. They replaced the earlier system of quaternions to transform Newton's laws into a powerful mathematical framework.