  Not only did Peirce defend infinitesimals. He furthermore claimed that he had proved the consistency of introducing infinitesimals into the system of real numbers in such a way as to form a new system in which there were infinitely many entities that were not equal to zero and yet were all smaller than any real number r that is not equal to zero, no matter how small r might be. To use modern terminology, Peirce was claiming to have shown the existence of ordered fields that were non-Archimedean. It was these non-Archimedean fields that Peirce now wanted to call genuine continua.  Additionally, Peirce wanted to use his notion infinitesimal quantities and his revised concept of the continuum in order to justify the traditional pre-Gaussian definitions and underpinnings of the differential calculus.