To every one it will be obvious that the two progressions differ in pace; and that the difference between their corresponding terms becomes increasingly larger and larger the farther we go; for instance, the sum of the first six terms of the geometrical progression is 126, whereas the sum of the first six terms of the arithmetical progression is only 42, the difference between the two sums being 84; the sum of 8 terms is 510 for the (_GP_) and 72 for the (_AP_), the difference between these sums (of only 8 terms each) being 438, already much larger than before; if now we take the sums of the first 10 terms, they will be 2046 and 110 having a difference of 1936; etc., etc.
Consider now any two matters of great importance for human weal—jurisprudence for example, and natural science—or any other two major concerns of humanity. It is as plain as the noon-day sun that, if progress in one of the matters advances according to the law of a geometric progression and the other in accordance with a law of an arithmetical progression, progress in the former matter will very quickly and ever more and more rapidly outstrip progress in the latter, so that, if the two interests involved be interdependent (as they always are), a strain is gradually produced in human affairs, social equilibrium is at length destroyed; there follows a period of readjustment by means of violence and force. It must not be fancied that the case supposed is merely hypothetical. The whole history of mankind and especially the present condition of the world unite in showing that far from being merely hypothetical, the case supposed has always been actual and is actual to-day on a vaster scale than ever before. My contention is that while progress in some of the great matters of human concern has been long proceeding in accordance with the law of a rapidly increasing geometric progression, progress in the other matters of no less importance has advanced only at the rate of an arithmetical progression or at best at the rate of some geometric progression of relatively slow growth. To see it and to understand it we have to pay the small price of a little observation and a little meditation.
Some technological invention is made, like that of a steam engine or a printing press, for example; or some discovery of scientific method, like that of analytical geometry or the infinitesimal calculus; or some discovery of natural law, like that of falling bodies or the Newtonian law of gravitation. What happens? What is the effect upon the progress of knowledge and invention? The effect is stimulation. Each invention leads to new inventions and each discovery to new discoveries; invention breeds invention, science begets science, the children of knowledge produce their kind in larger and larger families; the process goes on from decade to decade, from generation to generation, and the spectacle we behold is that of advancement in scientific knowledge and technological power according to the law and rate of a rapidly increasing geometric progression or logarithmic function.