The investigations of Professor Karl Pearson and his coadjutors on the applications of the theory of errors to statistics have greatly increased the importance and extent of the subject.

It is hoped that the following version of one of the methods of arriving at the so-called normal law of error may be of interest from an elementary point of view: affording, as it does, a simple application of the binomial theorem and some interesting graphical work.

The set of solutions to the equation xy = yx has been studied extensively over the past three centuries, including work by well known mathematicians such as Daniel Bernoulli (1700–1782), Leonhard Euler (1707–1783), and Christian Goldbach (1690–1764). Various mathematicians have focused on the integer, rational, real, and complex solutions. For example, it has been shown (see [1]) that the equality 24 = 42 gives the only distinct integer solutions. Our exposition below presents some of the key ideas behind the positive real solutions to this equation and illustrates how rational solutions can be found. To learn more about the various solutions, the reader can consult the articles listed at the end of this paper, as well as the extensive references given in these articles. It is also possible to find some of this material on the Web.

The circles C1, & , Cn form a chain of length n if Ci touches Ci + 1, for i = 1, & , n − 1, and the chain is closed if also Cn touches C1. A cyclic chain is a chain for which all the circles touch another circle S, the base circle of the chain. If Ci touches S at Pi, then P1, & , Pn are the base points of the chain. Sometimes there may be coincidences among the base points; in particular, if Pi = Pj, then the line PiPj should be interpreted as the tangent S to at Pi.

The seven circles theorem first appeared in [1, §3.1], and some historical details of its genesis can be found in John Tyrrell's obituary [2]. The theorem concerns a closed cyclic chain of length 6, and says that, if a certain extra condition is satisfied, then the lines P1P4, P2P5, P3P6 joining opposite base points are concurrent. Here and throughout, ‘concurrent’ should be read as ‘concurrent or all parallel’, that is, the point of concurrency might be at infinity.

The term measuring number is acquiring currency as a popular synonym for non-negative real number. This is good. Yet the belief persists that the real numbers, by whatever name, must remain fundamentally inaccessible to school children, obscured behind a veil of Dedekind cuts or Cauchy sequences. (Thus Griffiths and Howson, in what is generally a humane book [1], devote several pages to an ingenious, but ultimately condescending, attempt to foreshadow the mysteries of real numbers in terms of points of a line.)