There are many questions in mathematics to which an answer can be found by a purely mechanical procedure. For instance we can determine the valid statements of the class of the statements

n is prime

by the sieve of Eratosthenes; the numbers 2, 3, 4 and so on up to any assigned n are written in a row, and we begin by striking out every second number, starting with 2. Of those which are left, the first is 3 and 3 is prime. Starting with 3, we then strike out every third number, and the first of those which remain is 5, which is prime. Starting with 5 we strike out every fifth number, determining the prime 7, and so on.

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.