  Connected with his new conception of the continuum is Peirce’s increasingly frequent and sometimes pugnacious defenses of the doctrine of the reality of infinitesimal quantities. The doctrine was not newly taken up by Peirce late in the nineteenth century; indeed, he had held the doctrine for some time, and it had been the doctrine of his father Benjamin. He considered it superior to the newer doctrine of limits for providing a foundation for the differential and integral calculus. What was new was that Peirce began to see the doctrine of infinitesimals as the key to his updated doctrine of the continuum. Thus, adding to his long-standing defense of infinitely large magnitudes (Peirce often used the word “multitudes.”), Peirce began vigorously to defend infinitely small magnitudes, infinitesimal magnitudes.  Many examples of such defenses can be found. Carolyn Eisele collected a number of such examples in her edited work The New Elements of Mathematics by Charles S. Peirce. See, for example, Volume 2, pages 169–170, where Peirce says “My personal opinion is that there is positive evidence of the real existence of infinitesimals; and that the admission of them would considerably simplify the introduction to the calculus.” See also Volume 3, Part 1, pages 121–124, 125–127, 128–131, and 742–750. By the end of the nineteenth century Peirce’s view about infinitesimals was so rare and remarkable that Josiah Royce remarked, in a footnote of his “Supplementary Essay” for The World and the Individual, First Series, that outside of Italy Peirce was virtually the only mathematical philosopher who believed in infinitesimals. (See footnote 2, page 562 of this work by Royce.)