The following is extremly boring unless you have any interest in the process through which Issac Newton's Principia was born.
Newton published his Principia in 1687. The last problem he encountered before publishing was with gravitation. The current year is 1679. Based upon Kepler's Third Law of Gravtitaion Newton, as had many other physicists and astronomers of no little regard, assumed that the force of gravity held to an inverse square law. Newton then hypothesized that the local value of g (the acceleration due to gravity) for all falling objects and the force of gravity itself should vary accordingly. To confirm this theory, Newton thought to calculate the centripital acceleration of the moon, and compare that to the falling motion of an apple. The theory was this: Since the moon orbits at a distance of approximately 60 Earth radii, the force of gravity which acts on the moon should be weaker than the force of gravity on the apple by a factor of 3600, or the distance covered by a falling apple in one second equals the distance the moon travels in one minute.
Newton began his calculations assuming that an angle of 1 degree subtended an arc length of 60 miles. This was, unfortunately, the only information available to him, taken from a sailor's manual. In actuality, the distance is 60 nautical miles, equivalent to 69 English miles. Based upon these data he computed that the moon travelled .0036 feet per second, or 13 feet per minute. Earlier Galileo's experiments had proved the falling time is independant of mass of 15 feet in one second. The results, 13 ft/min compared to 15 ft/s were remarkably close, but far enough apart that Newton abandonded his ideas and began searching for new theories.
Move forward to Autumn of 1683. A physicist named Robert Hooke began an investigation into gravitation, writing coorespondences with Newton concerning the trajectories of objects being affected solely by the force of gravity. Hooke became certain that he could crack the problem.
Forward to January of 1684. Sir Edmond Halley, Robert Hooke, and Christopher Wren were eating lunch. Halley began pointing out many logical, qualitative reasons for the inverse square law of gravity, however was dismayed that none could prove it mathematically. He said that part of the problem was that no one but Newton could even prove that the gravitational forces of spherical bodies could be considered to eminate and terminate as if the spheres were points, and Newton had not been heard from on the subject publically in these past five years. Hooke brashly declared that he could certainly prove the orbits of planets were elliptical, but refrained from publishing so that others might struggle through the problems and appreciate the magnitude of the difficulty. Wren at once offered 40 shillings for Hooke to draw it up on the table. When Hooke said it would take some time, Wren made the offer public, 40 shillings for proof of a planet's elliptical orbit, good for two months. Not suprisingly, two months later the 40 shillings went uncollected.
Forward to August 1684. While visiting colleagues in Cambridge, Halley stopped by to visit with Newton in person. He asked Newton dierctly what the orbit of the planets would be if they were indeed subject to the sun's gravity should that be of an inverse square nature. Newton casually replied it would be an ellipse. Halley was taken aback that Newton should know this offhand, and Newton admitted doing the calculations five years prior. The two searched through thousands of Newton's papers but could not find the calculations. Newton promised to redo them. Due to his past coorespondences with Hooke, Newton sent Hooke several sets of finalized calculations. Hooke found several errors, and an angry Newton threw his entire effort into the problem, attacking it with renewed vigor. Before the year was through, Newton had correctly derived all three of Kepler's Laws using an inverse square model of gravitation. Thus having overcome his final worrisome obstacle, Newton proceeded to publish his Principia