Flamsteed
In his first edition of Principia Mathematica, Newton failed to fully explain the motion of the moon. He wished to correct the omission in subsequent editions but required the observations of the astronomer John Flamsteed to do so. Flamsteed agreed to grant Newton access to his data but requested that they not be made public. Newton agreed. Realizing that he would be unable to complete his desired edits to Principia Mathematic without Flamsteed's notes, Newton conspired to have himself appointed to the governing body of the Royal Observatory. Using his new found authority, Newton seized Flamsteed's catalogs and had them published. Flamsteed learned of the publication and succeeded in obtaining a court order to prevent its distribution. Newton was so enranged that he deleted any mention of Flamsteed in subsequent editions of Principia Mathematica.
Leibniz
Working independently, Newton and the German mathematician Gottfried Leibniz developed calculus. While we have since learned that Newton beat Leibniz to the initial discovery by a few years, Leibniz was the first to publish his findings. The two quarrelled over who should be credited, with each receiving letters of support throughout the field. (It should be noted that many of the letters backing Newton were penned by Newton's friends and even Newton himself!) In what might be considered the ultimate bonehead move, Leibniz appealed to the Royal Society--the very same society of which Newton was president--to settle the dispute. Newton proceeded to stack the supposedly impartial committee with his friends and supporters. To no surprise, the Society found in favor of Newton and formally accused Leibniz of plagiarism. After Leibniz's death, it was said that Newton found great pleasure in "breaking Leibniz's heart."
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