ataclysmic changes--such "jumps," as we may call them--such violent
readjustments in human affairs and human relationships--are recorded
throughout the history of mankind. And I would have him see clearly that,
because the _disparity_ which produces them increases as we pass from
generation to generation--from term to term of our progressions--the "jumps"
in question occur not only with increasing violence but with increasing
frequency. This highly significant fact may be graphically illustrated in
the following figure:

[ A horizontal line, with numbers 2, 4, 8, 16, 32 above, and markings
_a'_, _a_, _b_, _c_, _d_ in the corresponding locations below. ]

Geometric evolution of the natural and technological sciences.--Peaceful
progress.

[ A horizontal line, again marked with numbers 2, 4, 6, 8, 10, but with
gaps between 6 and (unprinted) 7, 8 and (unprinted) 9, 10 and (unprinted)
11. Above are markings: _A'_ above 2 and _A_ above 4; _B_ above 6 and _C_
above 7; _D_ above 8 and _E_ above 9; _F_ above 10 and _G_ at the far
right end. The steady progress areas, 2 to 4, 7 to 8, 9 to 10, are marked
"Peaceful", while the gaps are labeled "Jump, Revolution or War." ]

Arithmetical evolution of the so-called social "sciences," accelerated by
violent "jumps."--Non-peaceful social progress.

_a'2_, _2a_, _ab_, _bc_, _cd_, represent the geometrical law of
progression in the natural and technological sciences (peaceful
evolution).

_A'2_, _2A_, _AB_, _CD_, _EF_, represent the lagging arithmetical law of
progression in the so-called social sciences (peaceful evolution).

Both of these during the same periods of time.

_BC_, _DE_, _FG_, represent revolutions or wars, with the aftermath of
revolution of ideas--the "jump"--violent readjustment of ideas to
facts--forced by events.

_ab_, _bc_, _cd_, and _AB_, _CD_, _EF_, take the same amount of time, but
the second progression being much slower than the first one, the "jumps"
or revolutions occur at shorter intervals as time goes on and thus more
frequently force us to coordinate our ideas to facts. Periods of peace or
seeming peace alternate more and more frequently with periods of violence;
the mentioned _disparity_ of progress in peaceful times is the hatching
seed of future violence.(1)

As a matter of fact these few mathematical considerations can hardly be
called mathematics or mathematical philosophy; nevertheless, without
bringing attention to these very simple m