during the first second. The velocity acquired _at the end_ of the 1st
second is not 16-1/12, but 32-1/6 feet per second, and at the end of the 2d
second a velocity of 32-1/6 feet has to be added; so that the total
velocity at the end of the 2d second becomes 64-2/6 feet; at the end of the
3d, the velocity becomes 96-3/6 feet, at the end of the 4th, 128-4/6 feet,
and so on. These numbers proceed in the progression 1, 2, 3, 4, &c., so
that it appears that the velocities acquired by a falling body at different
points, are simply as the times of falling. But if the velocities be as the
times, and the total space passed through be as the squares of the times,
then the total space passed through must be as the squares of the velocity;
and as the _vis viva_ or mechanical power inherent in a falling body, of
any given weight, is measurable by the height through which it descends, it
follows that the _vis viva_ is proportionate to the square of the velocity.
Of two balls therefore, of equal weight, but one moving twice as fast as
the other, the faster ball has four times the energy or mechanical force
accumulated in it that the slower ball has. If the speed of a fly-wheel be
doubled, it has four times the _vis viva_ it possessed before--_vis viva_
being measurable by a reference to the height through which a body must
have fallen, to acquire the velocity given.

21. _Q._--By what considerations is the _vis viva_ or mechanical energy
proper for the fly-wheel of an engine determined?

_A._--By a reference to the power produced every half-stroke of the engine,
joined to the consideration of what relation the energy of the fly-wheel
rim must have thereto, to keep the irregularities of motion within the
limits which are admissible. It is found in practice, that when the power
resident in the fly-wheel rim, when the engine moves at its average speed,
is from two and a half to four times greater than the power generated by
the engine in one half-stroke--the variation, depending on the energy