Commerce is the great conservator of standards. These may become altered by the ill-advised action of rulers, by municipal or parochial carelessness, even by the desire of profit on short measure, or occasionally, as seen to a slight extent in our old Bushel, by the faulty dimensions of a standard; but wholesale trade, supported, in weights at least, by the goldsmith and the apothecary, preserved the integrity of many standards during the Middle Ages and up to modern times. Commerce conveyed to the West the standards that had developed in the great Oriental Kingdoms, sometimes with the modifications due to Roman influence. Masons and architects also preserved the standards of length and, allowing for variations inevitable under the feudal system, the principal linear measures can generally be traced to their sources as surely as weights. But there are two, yea three, striking exceptions among the linear standards of the West: the English foot, and the Rhineland foot, and also the Pán of Marseilles. These are quite unconnected with any ancient measures, and there is no record of their origin. The only clue to it is found in the simple relation of each to the corresponding weights and measures of capacity, the origin of which can be satisfactorily traced. This leads to the hypothesis that these linear measures were ‘involved,’ that is produced by a method of involution the inverse of that which had evolved the measures of weight and capacity. 1. THE ENGLISH FOOT There seem three hypotheses for the origin of the English foot. 1. That it was the Olympic foot = 12·16 inches, its standard diminished by the accidents of time. But we know that the Romans established their measures in Britain, and our mile of 8 stadia and of 5000 feet (first Roman, then English) up to Tudor times, shows that it was originally 1000 Roman paces of 5 feet; and our early wine-bushel, of which the wine-gallon was 1/8, is referable to the cube of the English foot, not to that of the Olympic foot. There is no trace of the Olympic foot in Northern Europe except the possibility (mentioned under Foreign Linear Measures) of the Amsterdam local foot, = 11·146 inches, being 11 inches of the Olympic foot. 2. It happens that the mean of the Roman foot = 11·67 inches, and of the Rhineland foot = 12·356 inches, gives 12·013 inches. But there is no instance of a new standard being formed from the mean of two older ones; moreover this hypothesis begs the question of the Rhineland foot.