These are not to be applied too strictly, but it is safe to make the general statement about them that they correspond to the three stages of culture,—savagery, half-culture, and full-culture. The simplest rate is by adding one invention or art to another, as does the savage in his lowest stage to-day and as did primitive man for myriads of years. Each such addition is so much gained, but reflects little improvement on the general life. Thus the Australian began with a stone fastened to a wooden handle, and with which he could strike a blow, scratch the earth, or tear flesh. To this he added in time a spear or javelin, a club, and finally that curious weapon, the boomerang. Each of these inventions helped him just to the extent he used it and not more. His general condition was not bettered beyond that amount. It was as if he had added a hundred dollars to his capital and enjoyed the interest of the investment. His was arithmetical progression. This merely arithmetical progression by simple addition, 2 + 2 + 2 + 2=8, explains why the introduction or invention of very important technical procedures have frequently been of no influence on the general culture of a people. Thus, the smelting and forging of iron has been known from time immemorial among the African blacks, and many of them are skilful blacksmiths; but beyond its immediate convenience for weapons, the art did them no benefit. The Chinese knew the compass and gunpowder many centuries before the Europeans, but their methods of war and navigation received no impulse from these potent allies. French physiologists have defined the human brain as “an organ of repetition and multiplication.” So long as its activities are confined to mere imitation, following a set example, it employs the former function only, and the progress of the group must be very slow. This was not Mr. Lewis H. Morgan’s opinion. That thoughtful ethnologist maintained that “from first to last human progress has been in a ratio not rigorously but essentially geometrical.” But the arguments on which he chiefly based this maxim, so far as it applies to primitive conditions were the development of articulate speech and the social, “gentile” organisation; and neither of these resulted from a conscious effort of mind. Progress does proceed in a geometrical ratio—that is, by multiplication, when an invention reacts on the sum of the ethnic possessions to increase their general value—when, as we say, it has an indefinite number of “applications.” This is seen in the recognition of the mechanical powers,—the lever, the pulley, the screw, the weighing-beam, and so on. In ship-building, the oar, the rudder, and the sail improved the whole system of water transportation.