The professional training of the Mathematical Teacher after his degree course is finished is a principle which has come to stay, and it therefore behoves all who are concerned in the practical administration of this principle to do their utmost to make the course of professional training as valuable as possible for the future teacher. So far as I know nothing has as yet been done by the Mathematical Association as a body in helping forward this movement, but surely the Mathematical Association will fail in its fundamental aims if it does not take a very active part in helping to decide what are the best courses of instruction to give to the student-teachers while in training. The object of the present paper is not to lay down dogmatic assertions as to what ought to be done, but rather to put forward suggestions, and thus to act as a nucleus round which the discussion can revolve. No attempt is made to deal with the training of anyone except the specialist type of mathematical teacher,—the person who in time to come is destined to teach in the Advanced Departments of the Secondary Schools in addition to taking his share in the general mather matical work of the school, and who, if weighed in the balance and not found wanting, will be called upon to act as Head of a Mathematical Department, and thus to guide and mould the mathematical destinies of the institution to which he is attached. In almost every University nowadays there exists a Department of Education, and within that Department of Education a School of Mathematical Pedagogy. It is for us to discuss now what ought to be the type and scope of instruction given in this School of Mathematical Pedagogy.

As a student in the Department of Textile Industries of the University of Leeds—then the Yorkshire College—I was taught to emulate the successes of my forefathers in the spinning of yarns and in the weaving of fabrics,—and also to perpetuate some of their errors. Empirical work, without any discrimination or “imaginative insight,” was the fundamental basis of practically all instruction.

Geometrically considered the sateen arrangement is a certain regular method of arranging points on squared paper, each point being placed at the centre of a square. In textile work the sateen arrangement is used, (1) as a basis for actual weaves, (2) as a mode of distributing over the surface of a cloth the repeats of an ornamental design.

Before commencing his purely Textile Course, the student has had, at least, a two years’ course consisting of Practical Mathematics and Drawing, English, and Laboratory Work in Mechanics and Physics. The syllabus for Mathematics includes: Mensuration of the rectangle, parallelogram, triangle and oircle; surface areas and volumes of the cylinder, cone and sphere; measurements are made from everyday objects and models, and symbolical representation of the results leads to formulae and substitution of numbers in these formulae. From this we pass to simple equations. Squared paper is used to plot results of measurements made (either in the Mathematics lessons or the Laboratory), such as comparing:—circumferences and diameters of circles; areas of triangles of equal altitudes and different bases; extensions and loads for a spring; corresponding readings of temperatures in C.° and F.°, and from these we pass to proportionality, or variation. Powers and roots are dealt with, and the use of Logarithms taught. The Drawing, which is taught alongside the Mathematics, consists of: Hand-sketches, plans, elevations and sections of models and parts of machinery used by the students.

I am not going to inflict a sermon on you; but if I had to give out a text for my discourse I should, I think, choose the words, “The letter killeth, the spirit maketh alive.” This is a very general truth, but it seems to me to have special force in connection with dynamical science. I shall leave the application to be traced by yourselves as I proceed.

My first duty in approaching this subject is to make it clear that I am not a mathematician seeking to demonstrate the application of mathematical methods in the field of economics. My aim is to indicate the phases of the general economic problem which demand the co-operation of mathematician and economist as the only means of arriving at sound judgments and giving the scientific study of industry and commerce a value which will command increasing respect and attention.

James Stirling, who takes rank with Maclaurin and Napier among the great Scottish mathematicians, belonged to the Garden branch of the family of Stirling. The Stirling family is one of the oldest of the landed families of Scotland. They appear as proprietors of land in the twelfth century.