il to observe in this connection the following two facts. One of
them is that the magnitude of the terms of any geometric progression whose
ratio (no matter how small) is 2 or more will overtake and surpass the
magnitude of the corresponding terms of any arithmetical progression, no
matter how large the common difference of the latter may be. The other
fact to be noted is that the greater the ratio of a geometric progression,
the more rapidly do its successive terms increase; so that the terms of
one geometric progression may increase a thousand or a million or a
billion times faster than the corresponding terms of another geometric
progression. As any geometric progression (of ratio equal to 2 or more),
no matter how slow, outruns every arithmetic progression, no matter how
fast, so one geometric progression may be far swifter than another one of
the same type.

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 mere