The problem of the greatest term of a binomial expansion is a favourite one in elementary text-books, and its solution is often difficult to a beginner. The difficulty, at least in the case where the index is negative or fractional, seems to be caused by the fact that a “formula” is provided which gives a value for r, such that the (r + 1)th term is the greatest. Moreover, this formula is not always the same. Sometimes it is sometimes ; and unless the student has a very good memory he is sure sometimes to make mistakes. Elementary mathematics ought not to be a memory exercise. It is a platitude to say that the educational value of the teaching of mathematics lies in its training of the powers of reasoning. This element is eliminated when processes of reasoning are reduced to a rule of thumb. As well might one use “Molesworth” as a text-book of the principles of mechanics.