ARTS AND SCIENCES AND SOCIAL MILIEU 7oi
must evacuate the province of Euclidean Geometry as the price of sain
ing an entry into the province of a Cartesian Calculus, or alternative^
must renounce all hope of mastering the Calculus if he is unwillW to
relinquish his hold upon Geometry. *—^ to

The divers provinces of Mathematics, Natural Science, and Techno
logy that have been successively conquered by the Collective Intellect
of Mankind do not stand to one another in the same relation as the
divers systems of government or law that succeed one another in the
history of a human society. A new ministry, new ruler, new dynasty
new regime, or new state cannot come into power without replacing a
predecessor; a new law cannot be enacted without abrogating the law
previously in force on the same subject. In short, successively estab-
lished political and legal institutions cannot coexist side by side. They
are mutually exclusive because they are incompatible, and they are
incompatible because each of them has, as we have seen, to be geared to
the particular circumstances of some local and temporary social situa-
tion. There is no room in the dimension of social life for more than
one such situation at a time, and therefore there is no room there, either
for more than one institution at a time, since every institution's reason
d'gtre is to provide a solution for some social problem here and now
Thus, on the plane of social affairs, all human experience testifies
with one accord that

Time, like an ever rolling stream,

Bears all its sons away;
They fly forgotten, as a dream

Dies at the opening day.1

But a 'law of Sin and Death',2 which reigns inexorably in this social
human realm of government, legislation, and the rest, has no dominion
over the abstract non-human realm of Mathematics. In the struggle for
mastery between Time and Man in this bloodless intellectual arena,
Man has succeeded in making Time Man's servant instead of allowing
Time to make Man Time's victim. The monument of Man's victory
over Time here is a Collective Human Intellect's cumulative achieve-
ment; and this exception to Time's rule, which Watts overlooked and
Spengler ignored, had long since been divined by Fontenelle and been
noted by Gibbon.

'The Mathematics are distinguished by a peculiar privilege that, in the
course of ages, they may always advance and can never recede.*3

If we are right in thinking that we have now disposed of Spengler's
contention that Mathematics are subject to the same law of social
relativity as social human affairs,4 we may now go on to examine
Perrault's contention that the Fine Arts are subject to the same law of
cumulative growth as Mathematics.

1 Watts, Isaac, quoted in I. i. 459.
* Rorn. viii. a.

3 Gibbon, Edward: The History of the DecUne and Fall of the Roman Empire, chap. Hi.
+ This thesis of Spengler's receives short shrift at Collingwood's hands (see Colling,
wood, R. G~ The Idea of History (Oxford 1946, Clarendon Press), pp. 225-6),