Magic squares might reasonably be supposed to be amongst the most useless of mathematical diversions. Yet, like other apparently fruitless amusements such as Eulerian squares and Kirkman’s school-girls problem, they have found applications in the field of the design of experiments. In an investigation of the effects of two variables upon a phenomenon, where the observations are obtained one after the other, magic squares make it possible to take account of linear trend in the results. For example (the writer is only competent to offer illustrations in the field of psychology : others in bis own field will doubtless occur to the reader) Miller and Selfridge investigated how much subjects could remember of series of words which varied both in their length and the degree to which they resembled everyday English. Again Fitts and Annett, Golby and Kay investigated how fast a subject could alternately tap with a metal stylus two separate metal plates, and how his speed was affected by the width of the plates and the distance between them. In both cases the subjects performed the task a number of times under different combinations of levels of the variables (length and resemblance to English, and width and distance respectively), so that any analysis of the data is liable to be vitiated by the fact that each subject will probably improve with practice throughout the experiment, unless special account is taken of this trend.