The image of Isaac Newton as a lonely mathematician is problematized by a palaeographic study of his mathematical manuscript archive. A study of Newton's mathematical archive, now made possible by several online projects, reveals that he corresponded on mathematical matters with practitioners, mathematics tutors and librarians, mostly based in London, and with colleagues in Cambridge, Oxford, and Scotland. Further, Newton was actively engaged in public lecturing and disputations in Cambridge. Our historical understanding of Newton the mathematician should include a due consideration of the above aspects of his work, which are somewhat obscured in the print edition of his mathematical papers.

In 1669 Newton wrote De Analysi to claim the priority of the method of infinite series. Newton did not want to publish his method of fluxions, so he introduced the moment instead of the ratio of fluxions, which is the differential quotient in modern calculus, and expressed the fluent, which is the antiderivative in modern calculus, in terms of the pair of the indefinite region drawn by the motion of the ordinate and its signed area. In the priority dispute, Newton suspected that Leibniz had read De Analysi in 1676, so he used De Analysi as evidence that Newton was the first inventor of the method of fluxions and that Leibniz had plagiarized differences from the moments of De Analysi. However Newton did not use fluxions when he wrote De Analysi and the moment in De Analysi was differential quotient, not differential, in modern calculus. In 1715, in his anonymous An Account of the Book, Newton explained how he had represented the fluxions in De Analysi and rewrote the note he had given in De Analysi regarding the unit of moment so that moment meant differential.

The “New Image of Newton,” proposed in the 1980s, casted light on some unknown aspects of Newton until then, including his musical study. Penelope Gouk related Newton's musical study to symmetry and regarded it as one of the characteristics of his approach to music. This paper reconsiders whether Newton was guided by symmetry through an in-depth examination of his musical manuscript. First, a brief summary of themes concerning Newton's musical thought is provided. Then this paper analyzes the parts in the manuscript that are related to just intonation, equal temperament or the grading of musical intervals and scales from a musical mathematical point of view. This leads to the following three findings. (1) Newton's twelve semitones are not strictly symmetrical. Newton seems to have tried to make symmetrical thirteen semitones only once, but he abandoned the attempt and never returned to it. (2) He preferred just intonation to equal temperament, although equal temperament was perfectly symmetrical. (3) His gradings of intervals and modes are not based on a symmetrical criterion. Thus, it may be suggested that Newton did not pursue symmetry very enthusiastically. His approach to music was more likely mathematical deduction from a posteriori principles and just intervals.