AT the outset I desire to express my deep appreciation of the honour of being invited to lay before your Association some of the investigations into educational practice which I have undertaken during the past few years. I ask only that they be regarded aa any scientific experiment, as a search for some true principles, and that at any stage of that quest failure was to be welcomed as warmly as success, as affording suggestions for further enquiries. That the experiment has been attended with a measure of success does not make me ready to proclaim it as the ne plus ultra : I submit the facts to your judgment in the hope that they may afford material for a profitable discussion and stimulate some abler investigators to pursue the matter further.

Mr. Fletcher begins his article in the January number of the Gazette by giving two reasons why a logarithm ought not to be defined as the index of the power of the base which is equal to the number. But are these reasons well-founded? One is that it does not afford a basis for the more advanced parts of the theory of logarithms. That statement would require a proof that might be very difficult or perhaps impossible to produce.

In putting forward a plea for the inclusion of “Arithmetic of Citizenship” in the regular arithmetic course in secondary schools, there are two points which I want to make clear : (i) the purpose of such a course, and (ii) suggestions as to how it may be carried out in practice.

The title of my lecture is rather misleading. When I was asked to deliver it I thought I could profitably indicate how very definitely mathematics is used in a particular activity, such as gunnery, the application being neither complicated nor varied. If one was asked to write a paper on the utility of mathematics in engineering or in wireless electricity, the problem would be much more diffuse. The man who makes or fires a gun is only, after all, making one particular machine or doing one particular job. Hence it is possible briefly to state what mathematical difficulties he encounters.