Now Mr. Plumer's improvements proceed from two series of data. _First_, certain theoretical inferences from the facts above named. Finding the arches liable to break down, he supports the transverse arch by making the inner surface of the sole corresponding to it _convex_ instead of concave transversely; he makes the middle portion of the sole convex again in both directions to support the longitudinal arch, and for the same reason extends the heel of the boot or shoe forward, so as to support the anterior portion of the heel of the foot. _Secondly_, Mr. Plumer takes an old shoe that has done good service, and studies the reliefs and hollows-which the foot has shaped on the inner surface of its sole. Comparing the empirical results of this examination with those based on the anatomical data above given, and finding a general coincidence in them, he constructs his last in accordance with their joint teachings. Theoretically, Mr. Plumer is on somewhat dangerous ground. If the arches of the foot are made to yield like elliptical springs, why support them? But we subject them to such unnatural conditions by pressure from above over the instep, by adding high heels to our boots and shoes, by taking away all yielding qualities from the soil on which we tread, that very probably they may want artificial support as much as the soles of the feet want artificial protection. If, now, we find that an old, easy shoe has worked the inside surface of its sole into convexities which support the arches, we are safe in imitating that at any rate. We shall have a new shoe with some, at least, of the virtues of the old one.