Whilst the greatest effort has been made to ensure the quality of this text, due to the historical nature of this content, in some rare cases there may be minor issues with legibility. The application of mathematics to Belief, the calculus of Probabilities, has been treated by many distinguished writers; the calculus of Feeling, of Pleasure and Pain, is the less familiar, but not in reality 1 more paradoxical subject of this essay. The subject divides itself into two parts concerned respectively with principle and practice, root and fruit, the applicability and the application of Mathematics to Sociology.