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.

The question of sign constitutes a real difficulty to the intelligent boy at the outset of his study of Coordinate Geometry. At the beginning of his Trigonometry he is told OP must be considered always positive, but later on he will find some authorities giving a point in the third quadrant as ( - r, θ), while others prefer (r, θ + π). The perpendicular distance of (h, k) from ax + by + c = 0 is given by ± (ah + bk +c)/(a2 + b2)½, and sign seems to matter, but usually the pupil is told that he only wants to know how far off (h, k) is, and he is advised to stick to the absolute value. But a little later on he wants the equations of the bisectors of the angles between two given lines, and then he is blamed for not remembering that signs matter a good deal.

It is hard to understand why the problem of tracing a conic from its equation is usually subordinated to if not identified with the problem, intrinsically more difficult, of calculating axes, or why students are not encouraged to bring their knowledge of geometrical properties of the curve to the drawingboard. The essence of tracing a curve is to render the sketching of it as accurate as may be desired, and I venture to assert that for this purpose the discovery of axes and vertices is not worth a tithe of the labour which it demands. Prom the advice in some of our text-books one would imagine that the ellipse could be drawn accurately from its four vertices alone, the hyperbola from its vertices and its asymptotes, and the parabola from its axis and vertex and one or two other points. The truth is, that in any but expert hands these details are wofully insufficient; what is wanted is a multitude of points on the curve, and it matters little whether or not the vertices are included. The student finds satisfaction in drawing the curve from its equation and observing the symmetry that appears when the axes are inserted subsequently.

In the Report of the Committee of this Association published a year ago, it was asserted that a boy’s education “should enable him not only to apply Mathematics to practical affairs, but also to have some appreciation of the greater problems of the world, the solution of which depends on Mathematics and Science….” Again, “the material side of modern civilisation stands on a foundation of Applied Mathematics. This statement will not be controverted by anyone familiar with the work of modern scientists and engineers and with the part played by Mathematics in their achievements…. The modern man, therefore, should have at least some conception of the means by which these results, so vital to him, are obtained.” We have to give to our boys a training in logical accuracy, a demonstration of the beauty and interest of mathematical ideas and methods in themselves, and in the words which I have quoted, “some appreciation of those problems of the world, the solution of which depends on Mathematics and Science.” “The majority of boys will not make any extensive or profound use of Mathematics in after life … but they can be taught to realise the tremendous potentialities of the study whose elements they are commencing…. A public must be created able to realise what Science and Mathematics are doing for the world and to form some general conception of the means employed.”

The section on “Ratio, Proportion and Variation” in the text-books is by far the least satisfactory part of the standard course of Algebra. And it is probably regarded by many readers of these books as of no particular importance. But, when seen in the right perspective and given proper treatment, this part of the subject may fairly be regarded as the most far-reaching in importance—a contention for which the reasons will appear in the following pages.

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.