I have long been convinced that the teaching of mathematics suffers from the premature introduction of abstraction without adequate attention to the concrete facts from which the abstractions are derived. Perhaps I cannot better begin my argument than by taking as text a passage from the end of the first chapter of Todhunter’s Differential Calculus :

There is probably no other item of the arithmetic syllabus which is confronted by such a diversity of earnestly supported method as is the case with the multiplication of decimals.

The problems which form the basis of this investigation are :

1. (a) What are the special advantages and disadvantages which bring about the adoption or rejection of these methods?

2. (b) What is the extent of these advantages and disadvantages?

3. (c) And finally a consideration of the methods of overcoming the disadvantages.

A closed conic C is completely contained inside a closed conic S. From any point 1 on S we draw a tangent 1, 2 to C meeting S again at 2. From 2 we draw the tangent 2, 3 meeting S again at 3, and so on ; for any n, the tangent to C from the point n on S meets S again at n + 1.