moral
superiority of his opponent and of his own impotence was gained by the
Russians at Borodino. The French invaders, like an infuriated animal
that has in its onslaught received a mortal wound, felt that they were
perishing, but could not stop, any more than the Russian army, weaker
by one half, could help swerving. By impetus gained, the French army was
still able to roll forward to Moscow, but there, without further effort
on the part of the Russians, it had to perish, bleeding from the mortal
wound it had received at Borodino. The direct consequence of the battle
of Borodino was Napoleon's senseless flight from Moscow, his retreat
along the old Smolensk road, the destruction of the invading army of
five hundred thousand men, and the downfall of Napoleonic France, on
which at Borodino for the first time the hand of an opponent of stronger
spirit had been laid.

BOOK ELEVEN: 1812

CHAPTER I

Absolute continuity of motion is not comprehensible to the human mind.
Laws of motion of any kind become comprehensible to man only when he
examines arbitrarily selected elements of that motion; but at the
same time, a large proportion of human error comes from the arbitrary
division of continuous motion into discontinuous elements. There is a
well known, so-called sophism of the ancients consisting in this, that
Achilles could never catch up with a tortoise he was following, in spite
of the fact that he traveled ten times as fast as the tortoise. By
the time Achilles has covered the distance that separated him from the
tortoise, the tortoise has covered one tenth of that distance ahead
of him: when Achilles has covered that tenth, the tortoise has covered
another one hundredth, and so on forever. This problem seemed to
the ancients insoluble. The absurd answer (that Achilles could never
overtake the tortoise) resulted from this: that motion was arbitrarily
divided into discontinuous elements, whereas the motion both of Achilles
and of the tortoise was continuous.

By adopting smaller and smaller elements of motion we only approach a
solution of the problem, but never reach it. Only when we have admitted
the conception of the infinitely small, and the resulting geometrical
progression with a common ratio of one tenth, and have found the sum of
this progression to infinity, do we reach a solution of the problem.

A modern branch of mathematics having achieved the art of dealing with
the infinitely small c