I must not be understood as advancing anything new to mathematicians, though I know of no English text-book which consistently expounds and maintains the theories of number and algebra here presented. The work is ad dressed, not to mathematicians, but to inquiring students and teachers. A sound doctrine of number and its algebra seems to be left by our text-books to chance inference, or deferred to stages seldom reached in undergraduate courses of study. A straightforward development, comprehensible by beginners, of the number concept would be of immense service in mathematical instruction.