My main thesis is that men have too long allowed themselves to be bemused by the magic of numbers, though the kinds of spell that numbers have cast over us have changed with the ages. In seeking to understand why it is difficult for us to break the spell, I have been led to suspect that schoolmasters (and perhaps especially schoolmistresses) are largely to blame because of the way in which they use school marks. And a special responsibility in this matter rests upon the mathematical staffs of schools, not because they are more “mark-minded” than their colleagues, but because marks appear to be numbers or measures of some kind and the non-mathematicians must look mainly to the mathematician for guidance on the manipulation and interpretation of numbers and measures. Any lead that may be needed to reform our school marking systems must be given, if it is to be given at all, by the mathematicians. That is my justification for addressing my comments on School Marks to the Mathematical Association.

The writer wishes first to acknowledge his indebtedness to Prof. Brodetsky’s book [1] on Nomography; the following observations were, in fact, suggested as a result of reading that work.

Nomography, as most readers will know, is essentially a method of solving problems graphically, but whereas with ordinary graphical solutions a new curve must be drawn each time the data are changed, a nomogram once drawn can be used to solve all problems of a given type (e.g. to solve all equations of the form a cos x +b sin x = l, or to solve all possible pairs of simultaneous linear equations in two unknowns).

The rhombic dodecahedron has been described by Hope-Jones, Math. Gazette, Vol. XX, p. 254 (1936); it can be obtained simply by taking a cube, joining each vertex to the centre, thus obtaining six square pyramids, and placing these pyramids on the faces of a second cube. The solid thus constructed is shown in fig. 1.

A helicoid is the surface generated by a curve which is simultane rotated about a fixed axis and translated in the direction of the axis w velocity proportional to the angular velocity of rotation.[1] An Archime Screw [2] is strictly a finite helicoid in which the generating curve is c (and normally circular) ; but screws, helical in shape, for which the gene curve is a simple finite arc can be—and are—used in practice as Archime Screws, and are considerably easier to manufacture than tubular ones.

The polar equation of the parabola y2=8ax, referred to the origin of coordinates as pole is

r sin2θ = 8a cos θ

or

Inverting with the pole as centre of inversion and 4a as the radius of inversion we find that the equation to the inverse curve is

or r cos θ = 2a sin2θ

or r = 2a sin θ tan θ.

The following error appears in a number of text-books, and tender minds are thereby led astray.

A body (such as a bicycle or railway carriage is travelling in a horizontal circle with uniform speed; G is the centre of gravity of the body, O is the centre of the circle by G, GX is a horizontal line perpendicular to OG and GY is a vertical line.)

Problem. A net is made of string round a unit sphere, so that the sphere cannot slip out of it. How long need the string be?

This problem was offered by me as a “Contest Problem” to undergraduates of Cambridge last Lent Term and a correct solution was given by Mr. R. Schwarzenberger of Trinity College. I am publishing my own solution, as his solution is considerably more complicated.