He was a mathematician, thus walking in the footsteps of Roger Bacon. This science, since the days of Euclid, had been pursued with untiring ardor, and many who neglected to study, or who, by their own imagination, distorted the actual phenomena of nature, addicted themselves to the investigation of the abstract properties of magnitude and number. Copernicus, in his knowledge of mathematical principles, and in his skilful application of them to astronomical inquiries, probably surpassed all his contemporaries. And, at the same time, he had that inventive genius which is fruitful in suggestions, such as become pioneers in the path of scientific demonstration. His independence of mind, his real originality, and his boldness in the pursuit of truth are quite as remarkable as the qualities just noticed; indeed, they are involved in or they led to the latter of these. "I beg you," says one of his admiring disciples, "to have this opinion concerning that learned man, my preceptor, that he was an ardent admirer and follower of Ptolemy; but when he was compelled by phenomena and demonstration, he thought he did well to aim at the same mark at which Ptolemy had aimed, though with a bow and shaft very different from his." We must recollect that Ptolemy says 'He who is to follow philosophy must be a freeman in mind.' Copernicus knew very well that there were many prepared to challenge his conclusions, and perhaps to bring theological objections to the principles of science which he had been constrained to adopt. "If, perchance," it is said in the preface to his book on astronomy, "there be vain babblers who, knowing nothing of mathematics, yet assume the right of judging, on account of some place of Scripture, perversely wrested to their purpose, and who blame and attack my undertaking, I heed them not, and look upon their judgments as rash and contemptible."