Whilst I do not intend to trouble the reader with any highly technical mathematical arguments, there are a few simple mathematical considerations which anyone of fair education can understand, which are of exceedingly great importance for our purpose, and to which, therefore, I ask the reader’s best attention. One of the ideas is that of an _arithmetical progression_; another one is that of a _geometrical progression_. Neither of them involves anything more difficult than the most ordinary arithmetic of the secondary school or the counting house, but it will be seen that they throw a flood of light upon many of the most important human concerns. Because we are human beings we are all of us interested in what we call progress—progress in law, in government, in jurisprudence, in ethics, in philosophy, in the natural sciences, in economics, in the fine arts, in the practical arts, in the production and distribution of wealth, in all the affairs affecting the welfare of mankind. It is a fact that all these great matters are interdependent and interlocking; it is therefore a fact of the utmost importance that progress in each of the cardinal matters must keep abreast of progress in the other cardinal matters in order to keep a just equilibrium, a proper balance, and so to maintain the integrity and continued prosperity of the whole complex body of our social life; it is a fact, a fact of observation, that in some of the great matters progress proceeds in accordance with one law and one rate of advancement and in others in accordance with a very different law and rate; it is a fact, a fact of observation and sad experience, a fact attested by all history and made evident by reason, that owing to the widely differing laws and rates of progress in the great essential concerns of humanity, the balance and equilibrium among the parts is disturbed, the strain gradually increases until a violent break ensues in the form of social conflicts, insurrections, revolutions and war; it is a fact that the readjustment that follows, as after an earthquake, does indeed establish a kind of new equilibrium, but it is an equilibrium born of violence, and it is destined to be again disturbed periodically without end, unless by some science and art of Human Engineering progress in all the great matters essential to human weal can be made to proceed in accordance with one and the same law having its validity in the nature of man. Taken in combination, the facts just stated are so extremely important that they deserve to be stated with the utmost emphasis and clarity. To this end I beg the reader to consider very carefully and side by side the two following series of numbers. The first one is a simple geometrical progression—denoted by (_GP_); the second one is a simple arithmetical progression—denoted by (_AP_): _GP_: 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, etc.; _AP_: 2, 4, 6, 8, 10, 12, 14, 16, 18, 20, etc.