The idea of the sum to infinity of a geometrical progression admits of a simple but very effective illustration already very much in vogue among teachers.

We take a line OA two units long, and from it cut off in succession segments

The following notes deal mainly with the Arithmetic of the primary school, but the subject under consideration is only a single aspect of a very wide question.

It is a common remark that Oral (or Mental) and Written Arithmetic should be as nearly as possible identical: that pupils ought to be able to do mentally with small numbers whatever they can do in writing with larger numbers, and vice versa. Unfortunately, experience shows that, as a rule, they are not able to do so; that the child fails to identify written calculation completely with mental work. It is here proposed to try to discover some of the reasons why this is so.

The inclusion of Dynamics in a school curriculum is a topic that affords always much scope for discussion among interested teachers, and widely opposed opinions are held. I venture to state my own opinion, and to suggest a somewhat fuller treatment in dealing with one or two sections of the subject than is usual in the text-book.

The following is a scheme of classroom work designed to introduce a pupil to quadratic equations, graphically and algebraically.

Newton's Chord Rule is applied in (3) to calculate by Arithmetic close approximations to the roots of an equation. One graph suffices for any number of equations, as in (2), (5) and (7). Both the method of completing the square and the method of factors are required as algebraic equivalents, as in (4), (6), (7). The general rules for solving quadratics sum up the lessons.

The parts of the figures are numbered to indicate a proof of the theorem by dissection. The congruence of parts on which the same number is marked can be demonstrated by geometry. The steps of the proof are in order (i), (ii), (iii). Sufficient importance is, perhaps, not attached to (i) and (ii) at this stage; for they give (1) the construction of a square equal to a given rectangle (2) the graphical construction of √2, √2 + 1.etc.