I. _A heavy body, that in the first second of time has fallen the
height of a feet, has acquired such a velocity, that, moving uniformly
therewith, will in the next second of time move the length of 2 a feet._
II. _The spaces run thro’ by falling bodies are proportional to one
another as the squares of their last or acquired velocities._
These two principles are demonstrated by the writers on mechanics.
III. _Water forced out of a larger chanel thro’ one or more smaller
passages, will have the streams thro’ those passages contracted in the
ratio of 25 to 21._
This is shewn in the 36th prop. of the 2d book of Newton’s Principia.
IV. _In any stream of water, the velocity is such, as would be acquired
by the fall of a body from a height above the surface of that stream._
This is evident from the nature of motion.
V. _The velocities of water thro’ different passages of the same
height, are reciprocally proportional to their breadths._
For, at some time, the water must be delivered as fast as it comes;
otherwise the bounds would be overflowed.
At that time, the same quantity, which in any time flows thro’ a
section in the open chanel, is delivered in equal time thro’ the
narrower passages; or the momentum in the narrow passages must be
equal to the momentum in the open chanel; or the rectangle under the
section of the narrow passages, by their mean velocity, must be equal
to the rectangle under the section of the open chanel by its mean
velocity.
Therefore the velocity in the open chanel is to the velocity in the
narrower passages, as the section of those passages is to the section
of the open chanel.
But the heights in both sections being equal, the sections are
directly as the breadths;
Consequently the velocities are reciprocally as the breadths.
VI. _In a running stream, the water above any obstacles put therein
will rise to such a height, that by its fall the stream may be
discharged as fast as it comes._
For the same body of water, which flowed in the open chanel, must
pass thro’ the passages made by the obstacles:
And the narrower the passages, the swifter will be the velocity of
the water:
But the swifter the velocity of the water, the greater is the height,
from whence it has descended:
Consequently the obstacles, which contract the chanel, cause the
water to rise against them.
But the rise will cease, when the water can run off as fast as it
comes:
And this must happen, when, by the fall between the obstacles, the
water will acquire a velocity in a reciprocal proportion to that in
the open chanel as the breadth of the open chanel is to the breadth
of the narrow passages.
VII. _The quantity of the fall caused by an obstacle in a running
stream is measured by the difference between the heights fallen from
to acquire the velocities in the narrow passages and open chanel._
For just above the fall, the velocity of the stream is such, as would
be acquired by a body falling from a height higher than the surface
of the water:
And at the fall, the velocity of the stream is such, as would be
acquired by the fall of a body from a height more elevated than the
top of the falling stream; and consequently the real fall is less
than this height.
Now as the stream comes to the fall with a velocity belonging to a
fall above its surface;
Consequently the height belonging to the velocity at the fall must be
diminished by the height belonging to the velocity, with which the
stream arrives at the fall.