Magnum in finis

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“When one rows, it is not the rowing which moves the ship: rowing is only a magical ceremony by means of which one compels a demon to move the ship.”

 Friedrich Nietzsche, Human All Too Human

Marbles and Ramp

Section I


"Parvus error magnum in finis", small error big in the end, for a circular principle that things come back around, even when wrong, and it fits heliocentrism and Big G and little g(9.8).

Galileo first discovered and charted his way into abstract little "g"(9.8) by rolling metal balls down ramps. He conducted hours and hours of painstaking research, rolling balls down inclined planes to study acceleration and establish that, according to him, objects with different weights accelerate equally under the influence of gravity. Even when the fact is they do not, he would maintain that they do, because the differences in acceleration discovered at times, according to his terms in the lab, were conveniently small and only so slight for comparison in view of the greater differences in weight.
In defense of the principle of uniform acceleration, due to "the force of gravity", exercised universally over all objects, he said, “So far as I know, no one has yet pointed out that the distance travelled in equal intervals of time, by a body falling from rest, stand to one another in the same ratio as the odd number beginning with 1."(1) One of these to one of those, and what a measure of proportionality that would be for counting all things. The golden mean or whatever fraction for a ratio divided by one is always the golden mean or whatever fraction, as the golden mean or whatever fraction multiplied by one is what it is. An ingenious perhaps if stupid way of saying that in natural units the "force of gravity", as a physical construct in nature, may also be set equal to one, for constancy in details, when inserted parenthetically into an equation to represent rate by distance of fall.
The wonderful cape of gravitation, the primary basis of heliocentric fiction, is not actually any active force at all, other than what would be present by some form of impetus and the development of momentum. And division or multiplication by one, leaving everything as it is, is not saying very much for the theory of gravity or Galileo, since the heavier weights still fall and roll faster, even if the differences in some cases can be marked down as comparitively slight for Galileo's argument. Yet the harder and heavier they come, the harder and heavier they fall. Tickled with a feather or hit with a hammer, let the toes and nose of more honest science be the guide.
To express his principle of perpetual horizontal motion, he also said, “I mentally conceive of some movable projected on a horizontal plane, all impediments being put aside" ...[and] "that the equable motion on this plane would be perpetual if the plane were of infinite extent"(2). The image was a key component of the way he developed a foundation for gravity, little g(9.8), and the world of heliocentrism. It may seem odd that this has anything to do with whether the Sun orbits the Earth, or the Earth orbits the Sun, but it became an important talking point to Galileo, and would be a way to fit the infinite into a place that it might othewise not go. 
The crafty discussion he made of his experiments with metal balls and ramps was to construct in the mind an image of equivalence so convincing that people could be led to believe that rest is the same quality as motion and motion is the same quality as rest. By hypnotic effects and induction, they could at least confuse the two; and these stories of Galileo would later form the seed of Newton's own counter-intuitive and erroneous first and third "laws" of motion.
Imagine that a metal ball or marble is let go to roll down one ramp, across a table, and then up a second ramp. To begin the picture of equivalence, imagine that, if the two ramps are identical, the ball will reach virtually the same height in the second ramp as that from which it started in the first. In the same way, if a marble is let go to roll down the inside of a large bowl, it will roll through the bottom and then up the other side virtually level with its starting height. To put an image of equivalence of motion in the mind, the metal ball and the marble are coming close to the same height when they finish as when they started. However, the truth is that the marble and metal ball do not really finish at the same height as that from which they started.
Never mind the little part, Galileo had a more crucial observation. Use another receiving ramp that is less steep and longer than the first second one. Once again, the image of equivalence is that the ball let go ends up at virtually the equivalent height from where it started, though this time it had to roll farther to get there. Do the experiment again with another receiving ramp that is less inclined and longer than the second third one, that is tilted up only slightly, and the ball will eventually reach its starting height, said Galileo, but it will have to roll and roll more to get there on the other side. The lower the angle of the receiving ramp the longer the ball will roll to reach the same height from where it started, and for the sake of equivalence it should be imagined that the ball invariably will roll that far.
Now suppose that the receiving ramp was perfectly flat, not tilted at all. Then, in that case, said Galileo, the metal ball and the marble would roll on and on horizontally forever. But the flatest ramp that is the last image in the series is really a form of mildly hypnotic suggestion. It is a thought experiment, not a real one, to introduce a new law of nature: that any object moving horizontally will continue moving horizontally forever, at the same speed, unless something happens to interfere, like "friction". Newton's first law of motion is a generaliza
tion of the principle. However, it is not scientific or logical. It only is a thought experiment that presents a subtle and mildly hypnotic form of suggestion.

Ignore experience and common sense then, and what can actually be seen of the inclined planes and marbles. Rather listen to Galileo intone about infinity, and more important than the limiting details there, in the ramps, that actually can be verified, there is a greater and idealized "truth", an abstract and mathematical world of the mind, where marbles and metal balls c
an roll on and on in horizontal planes forever. To the mind under the control of Galileo, in that mind's eye, things may be reversed and there could be a plane beyond measure, in "absolute space", where, contrary to Aristotle, purely natural and horizontal motion is perpetual.

Constant horizontal motion outside the curve of a circle was important to Galileo and the development of heliocentrism for a few reasons:
a) it provided a theoretical basis for gravity, even as a lateral force in perpetuity; and gravity needed to be a constant and powerful lateral force to imagine that the Earth could be orbiting the Sun.
b) it provided a theoretical basis for relativity and heliocentric "invariance"(undetectable "unaccelerated" motion) and equivalence in various relative inertial frames of reference, et cetera.
c) it contradicted Aristotle's common sense logic and scientific observation that things loosed across the Earth tend also to come to rest on the Earth, as the Earth is always at rest before them: and that like goes to like, simile gaudet simili, as likeness will increase, and "that like is known as well as perceived by like"(3).
Aristotle would have noticed that the rolling metal balls and marbles did not really finish, even on the first roll, exactly as high as where they had started, and that they all also exhausted themselves over little time and came to a state of typical and completely authentic rest.
Galileo would howl at Aristotle's naive honesty, insult him for stupidity, invoke Lenin and the central bank of the communist party, then say he was missing the point of his revolutionary new science, that by gravity "distance is proportional to times squared."

In some cases maybe, but everything like that is proportional to multiplying or dividing by 1, as well as it goes with one of these to one of those, on and on. Aristotle would have to respond that distance is equivalent to rate by time, of course; however, the rate of acceleration is not affected by "gravitation", as much as bricks are not feathers, and gravity and acceleration are not equivalent; but always concede to divide or multiply by one, to figure the constant force of Galileo's "gravity" in natural units, in lab experiments of limited contrivance, to be safe.
Galileo probably would not appreciate Aristotle for common sense or paleolithic wisecracks in the avant garde laboratory. Ptolemaic "Syntaxis Mathematica" conservatives would have to go. There was trouble and time to develop a proper liberal background for little g(9.8), and one day perhaps dogs would be able to land on the Moon too, like Kepler or Lady Gaga, if spaghetti could fly ... like bat wings ... that far ... like 252,000 or 253,000 miles away. If one can land on the Moon at apogee, he should be able to land on it at perigee, of course, and vice versa.
But before flying to the Moon in dreams, or like Kepler with demons, which was an old-fashioned way, he would need to consolidate new breakthrough insights about the cosmos, including a feeling that time, not space or energy, is the essential variable that governs the world. Putting time first would open a better abstract plane for interpretations of gravity, and putting time first would be the best way to sustain his experiments with motion. So when Galileo was working in the lab with inclined planes and rolling metal balls, it was not an ordinary level of geometry or terrestial affairs. "As above, so below" astrologers, mystics, and occultists have said. When he was taking notes, Galileo would be fiddling with the cosmos as much as the infinite.
In the beginning, the set-up was bare-bones: a wooden ramp with a thin groove down the middle, a bag of metal balls and marbles, and a series of movable catgut strings. Like frets on the neck of a guitar, the strings lay within the groove and pulled tight across the surface of the ramp at a right angle to the downward roll of the balls. When a ball crossed a string, it would make an audible click.
After kicking old Aristotle and Ptolemy out of the lab, Galileo could hear the ball cross each string in turn ... because of "gravity"! It was little "g" at work, like one of the little people nobody can see. Painstakingly, as a fool would be wise, he would roll the balls again and again down the ramp, trying to position the strings so that the travel time between each pair of strings was equivalent. To support his theories, the strings needed to be arranged so the time interval it took a ball to descend rolling from section to section was equal ... equivalent for the clicks.
He had five strings, A,B,C,D, and E  positioned at different lengths of separation along the surface of the ramp, and he finally fixed it so well that a ball let go from the top would move to string A in the same time that it would then move to string B ... and so on to C, D, and E at the bottom. Out of this comparison he was able to build a math table and equation that would form the basis of little g(9.8), and his smashing theory of uniform gravity in free fall acceleration, to explain why the Earth orbits the Sun.
Measuring the distance between strings in little increments yielded results in two columns and five rows where:
                 Time(in seconds)      Distance(in inches)
Start to A          1                   1
A to B               1                   3
B to C               1                   5
C to D               1                   7               
D to E               1                   9

In Galileo's opinion, a game of marbles of limited contrivance in a lab could be expressed as a function of time and the force of gravity alone. Since there were five divisions in the ramp for a total of 25 inches, (1 + 3 + 5 + 7 + 9), and the pivotal roll was five seconds, he could say that a ball rolling down one of these ramps at gravity's command traveled precisely t^2 inches.


"In 1 second, a ball rolled 1^2 inches, in 2 seconds 2^2 inches, in 5 seconds 5^2 inches and so on."(4) What was equivocally equivalently important was that the math table, the law, and its equation did not say anything about radius or weight, or anything further of the what and how in the elemental conditions employed.


                  Time(in seconds)       Distance(in inches)


Start to A           1                    1
Start to B           2                    4    
Start to C           3                    9
Start to D           4                    16
Start to E           5                    25

Therefore, Galileo thought that he could extrapolate from this business that all objects around the Earth, regardless of their weight and actual heavy density, fall at exactly the same rate. So according to Galileo, roll a cannonball and a BB down a ramp side-by-side, and they would fall rolling alongside each other at uniform rates of acceleration ... "ad infinitum" due to gravity. Yet force and rate of acceleration are not separable terms, since they always go together, since more force will mean more acceleration. All other things being equal, down the same or similar ramps, a bowling ball with greater radius and weight will pass a BB, if there is enough time and space given for the difference in force to develop.
But not to Galileo: for any given ramp the same law of "equivalence" between all objects would always hold, and the distance the ball traveled would be proportional only to the time squared and the "force of gravity". All that counted in the design was the height above the ground from where the ball would be released. With the right ball and the right ramp, Galileo could say that due to gravity alone a weight always falls as it rolls, a distance proprotional to seconds of time squared and as a function of universal gravitation.
Of course, in time, space, and energy there is always some proportion, but natural proportionality of things and Galileo's best contrived ramp do not prove there is an interstitial "force of gravity" innate to matter that causes motion. The manipulated interpretation was outcome engineered. Put a gnarly rock on the ramp, for example, and it will not follow the rule. It would not fit the structure of gravitation hidden in the trick.

If not as much as setting the ramp on fire, and with charcoal lighter or lighter fluid, a rock stuck on the ramp would act like another anti-gravity device, because it lacks a smooth and uniform surface for rolling like a metal ball. As much as it does not roll away, it would tend to nullify the set-up force-field equation for gravity in the game.  

Aside from dropping rocks and cannon balls, melons and garbage, from the Leaning Tower of Pisa, this was how the distance formula for free fall acceleration and little g(9.8) was developed. Roll marbles but drop rocks, of course, and from this it is said that all objects with greater density and weight than air supposedly fall according to a precise rule of uniform acceleration, written in English units as d = 16t^2, where d stands for distance and t stands for time (and d = 1/2gt^2, where g is 9.8 meters equivalent).
Due to the putative interstitial force of gravity, it is said that acceleration of an object in free fall is independent of weight, and that in one second, any rock falls a distance of 16 x 1 feet. In two seconds, a distanceof 16 x 4 feet, or 64 feet, in three seconds, 16 x 9 feet, or 144 feet, et cetera by the seconds squared.

Similar to ramp and marble, the universal table for any abstraction of rock or cannon ball goes:
                  Time(seconds)            Distance(feet)             
Start to A           1                        16
Start to B           2                        64
Start to C           3                       144
Start to D           4                       256
Start to E           5                       400

Start to A           1                         16
A to B                1                         48
B to C                1                         80
C to D                1                        112
D to E                1                        144

However, even without interference from feathers or atmospheric friction, according to Galileo's formula, the rate of acceleration is not really uniform. In his own math, falling is not constant but accelerating exponentially. The rate of increase in the real world is not independent of density and weight either; and it does not go on and on ad infinitum. Every free falling weight eventually reaches a maximum acceleration in an environment, and none accelerate infinitely, even in a so called vacuum.
Even with Newton's calculus, a finite weight cannot accelerate  infinitely, and absolute infinity can never be fathomed by parts, not by the segments of any curve or pattern of exponents. And the putative intersitial "force of gravity" is not accelerating any weights anyway, since gravity is not a lateral or vertical force. Impetus and momentum cause motion and acceleration.
Anvils of styrofoam, foot-long matches, and ping pong balls and ramps all covered in velcro, and other anti-gravity devices, demonstrate that what is really at work in these disingenuous experiments of Galileo are motions and forces that are all predicated in quale quid ... not in undetectable occult action-at-a-distance like "gravitation".
What makes a bb and cannon ball roll side by side down a ramp, for a little while, is in the limited details of coincidence. Funny coincidence too, yet motion and force overall become a matter of the factors involved, like radius and weight, as well as density and constitutive material, and the quality of smoothness of the ball and the surface used for free rolling movement, etc. Something marginal at play, not gravitational, affects the results, and specific weight clearly does, in fact, contribute to the development of force and acceleration in free falling objects.
For instance, not because of gravity, people do not play volley ball with volley balls made of lead or plutonium, but because they are too heavy for sporting material and do not bounce.