The problem of finding rational integral numbers which could be made the sides of right triangles, is said by Proclus to have been solved by Pythagoras, by a method which is equivalent to the use of the formula,

where m is any odd integer.

Heath (1) quotes Proclus’ statement as follows:

“The method starts from odd numbers. For it makes the odd number the smaller of the sides about the right angle; then it takes the square of it, subtracts unity, and makes half the difference the greater of the sides about the right angle; lastly it adds unity to this and so forms the remaining side the hypotenuse.”

Is a quadrilateral in which one pair of opposite sides are equal and one pair of opposite angles are equal necessarily a parallelogram? This question arose in class recently and led me to discover much that I did not know about an elementary topic. Possibly others may be interested in the results of my investigation.

The following ‘proof’, for which I am indebted to my colleague, E. H. Lockwood, answers the question in the affirmative. The subsequent paragraph contradicts this.

I have never seen any explanation of why it is possible for a fast bowler to make a new cricket ball swerve, and why it becomes more difficult or impossible to do this when the shine has worn off the ball. Of course a sliced golf ball swerves because of the spin it has, and this swerve is a well-known illustration of the Magnus effect. Spin cannot be the explanation in the case of fast swerve bowling, because if it were the effect would increase as the ball became older and rougher. In any case a fast bowler imparts little or no spin to the ball.

In cases where successive integration of F(x) is not practical, a solution of the equation d2y/dx2=F(x) is obtained by approximation. The solution attempted is to find the value of y at selected points in the range of integration. A common procedure is to reduce the problem to the solution of a set of linear simultaneous equations, and these equations are then solved by “relaxation”. It does not appear to be generally known that an explicit and simple solution of these simultaneous equations exists. In this note the detail of such a solution is carried out for functions F(x) as occur in the theory of thin beams.

In the Gazette for December 1953, I gave an account of all that I knew about Schur’s inequality, namely

If μ≥0 and x, y, z are all positive, then

with equality occurring when and only whenx = y = z.