Of all the problems which have perplexed teachers of mathematics in this generation, probably none has been more irritating and insistent than the choice of assumptions which must be made in each branch of the science. In geometry, in analysis, in mechanics, one and the same difficulty arises. Are we to prove that any two sides of a triangle are greater than the third? That the limit of the sum of a finite number of functions is equal to the sum of their limits? That the total momentum of two bodies is uninfluenced by their mutual action? And in every such case, on what is the proof to depend? A clear understanding of the answers to such questions, or better still, a clear understanding of principles by which answers may be found, would go far to co-ordinate and simplify elementary teaching; the object of this paper is to state such principles, and indicate their application.