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From his analysis of that data Kepler determined that the path of Mars is an ellipse, with the sun located at a focal point. Motion of the planet Mars from the Danish astronomer Tycho Brahe. In 1601, Kepler obtained a detailed set of observations of the The direct problem that challenged mathematicians in the seventeenth century was determining the nature of the gravitational force acting on planets. Conversely, solutions to problems that proceed in the inverse direction using descriptions of the force function to find the path about the force center were called the "inverse" problems. The solutions to such problems proceed directly from the observed path to the unseen force, and therefore the problems were called "direct" by the early investigators. The fundamental type of problem addressed in the tract On Motion and in the first three sections of Book One of the Principia may be stated as follows: Given a path of a particle and a center of force relative to that path, find the dependence of the force upon the distance between the particle and the center of force.
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In December of 1684, Newton sent Edmund Halley a tract containing a set of theorems and problem solutions this tract was the basis for the opening sections of the 1687 edition of the Principia.