Galilei’s Famous Experiment of Gravitation Assesses the Universal Law with the Pythagorean Theorem

A Fictitious but Scientifically Very Truthful Report Beyond Time and Space

Georgi Stankov, March 4, 2017


My idea to write this playful essay was to show that knowledge is eternal and exists beyond time and space. It is an excerpt from Volume II on Physics and Mathematics (page 381 – 386). Galileo Galilei’s experiment I am referring to in this essay has really happened and marked the beginning of modern experimental physics. I saw the presentation of this experiment in 1997 in a special exhibition in the world famous “Deutsches Museum”  in Munich that is dedicated to science, engineering and technology throughout the ages.

I am talking about the famous Galileo’s inclined plane experiment of gravitation of which there are numerous variations. The one I saw used a geometric presentation of a series of rectangular (right) triangles with the same perpendicular hypotenuse and varying sides (cathetus) placed in a circle so that the hypotenuse was the diameter of the circle.

I searched on the Internet for a visual presentation of this specific experiment I saw in the museum but could not find one. There are many other versions of this experiment which are rather confusing. Therefore I made a drawing of this experiment as I remember it and have added it to the text below.   

When I wrote this essay I was fully channelled by the Source and I could hear the giggle of the angels that were thrilled by the simplicity and incredible clarity of my humorous scientific argumentation that spanned a bridge from the major scientific ideas in Antiquity to Modern Times when science first emerged as applied physics in this famous experiment of Galileo Galilei on gravitation, who since then is considered to be the father of modern physics. 


“ All truths are easy to understand once they are discovered; the point is to discover them.“   Galileo Galilei

Before Galilei starts with his experiment, he argues as follows: “The theorem of Pythagoras which I have used for the construction of this experiment says that: c² = a² + b². According to this equation, it does not make any difference if the ball is falling to the earth in a free fall along the perpendicular hypotenuse c or along the inclined path consisting of the sides (a+b). If I define the work which my assistant does to carry the ball to the top of the triangle as “energy“ with respect to my favourite philosopher, Heraclitus, this would say that the energy of the falling ball will be the same, no matter which way it falls down to the same point on the earth. From the geometry of the triangle, I can assert that the energy (work) remains unchanged, independently of how the ball moves from one point to another.


To prove this hypothesis, I must measure the falling times in a, b and c and compare them. To ensure that I do not commit any mistake, I shall change each time the length of the inclined tubes as the sides of the right triangle and measure the falling times of the ball for various side lengths of a and b of any right triangle in the circle.

After the experiment, Galilei analyses the results ad alta voce: “My experiment on gravitation shows that the falling time, tempo t, of the ball, which I have chosen as a representative object of matter, materia m, is independent of the slope of the inclined tubes: the falling time for the perpendicular hypotenuse c is equal to the falling times for any length of the inclined tubes a and b as the sides of the right triangle. Therefore I can write this practical result as follows:

 tc = ta = tb = t = constant

In this case, I can use the famous Pythagorean theorem, which I have already employed for the construction of my experiment, to present the results in a simple mathematical equation. This method has recently become quite popular, after that French youngster Descartes and his followers, the Cartesians, are keen in explaining the world from the mind by employing the geometric method – they call it boisterously the “Cartesian method“. Why not! This may be a good idea.

As far as I remember, it was Descartes who wrote about the conservation of movement in the universe? This is exactly what I have observed in my experiment on gravitation. Indeed, it would be “una buona idea“ to test if the theorem of the old grand master also holds for earth’s gravitation. If I am lucky to prove it, I will at the same time present evidence that the Aristotelian system of forms, which is based on the Pythagorean school, also holds in gravitation. This will be an excellent confirmation of the validity of ancient Greek science in the spirit of Italian Rinascimento (Renaissance).

On the one hand, the system of Aristotle has not been challenged since antiquity; it is generally accepted among scholars and does not need any additional confirmation. On the other hand, I have read that most Greeks were contemptuous to experiments and did not bother  much about scientific experience – for them Geometry was the ultimate Truth. If I could now prove that Geometry holds for earth’s gravitation – this divine force of matter – I will be the first scholar to show convincingly that Nature operates according to Geometry.

Pythagoras teaches us that “everything is number“. Could it be that his theorem is also valid for the new system of Copernicus, as my intuition whispers me when I reflect on my recent astronomic observations of the planets’ movement? In this case I have to refute the Ptolemaic system, to which this god-damned church sticks without any grounds. Take care, old chap! The spies of the inquisition have flooded even the free town of Florence. You better solve this problem for yourself and keep it secret during your lifetime. Let future scientists re-discover the mechanism of gravitation and the motion of planets when life will be less dangerous than in our turbulent times.

Let us now order the results of the experiment in a logical manner. If the time t of the falling ball m is constant in any of the tubes a, b and c, I can introduce the falling time t and the ball m as mathematical symbols in the Pythagorean theorem. For this purpose, I have to multiply the hypotenuse c and the sides of the right triangle a and b with the term m/t² :

c² = a² + b² ∖ × m/t² .

This artificial mathematical operation will not alter the initial validity of the famous theorem. On the contrary, it will bring a real physical meaning to this abstract theorem of Geometry – from now on, it will also hold in gravitation:

m(c²/t²) = m(a²/t²)  + m(b²/t²)  (259)

This is a pretty good result, but my intuition tells me that I have to present this mathematical equation in a more adequate form. Let us try it now! The hypotenuse and the sides of the right triangle are straight lines. According to Euclid, they have only one dimension, which I can present as “1d “. I can express these straight paths with the symbol [1d-spazio] for one-dimensional space. The time t measures how “quick“ the movement of the falling ball is. As the ball needs the same time to fall in c as in each of the sides, a and b, of the right triangle, the movement of the ball is the “quickest“ during the free fall in the hypotenuse because c is longer than any of the sides, a or b.

If I now build a quotient of space (spazio) and time (tempo) I will have an adequate measure to compare how “quick“ the movement of the ball is. This is, indeed, a brilliant idea! As far as I know, nobody has come to this idea before. I will call this new mathematical quantity “velocita“ (velocity) and express it mathematically with the first letter of the word “v“. I can now write the following equation:

v = velocita = [1d-spazio] / [tempo] =  [1d-space] / t

(Nota bene: Before Galilei the concept of velocity (speed) did not exist and humans were unable to measure how quick a movement was but only used verbal descriptions such as “quick” and “slow”. This physical quantity v = s/t was first introduced by Galilei in this experiment and since then it is the backbone of classical mechanics and physics as a whole. I have proved that velocity is a universal geometric presentation of one-dimensional space-time as energy which all physicists use in an unconscious manner without understanding the epistemology of this quantity as they have not grasped the essence of energy as consisting of only two dimensions/constituents – space and time – as proven beyond any doubt in the new Theory of the Universal Law.).

Not bad, but I am not satisfied with this presentation. Building quotients like this one takes a lot of space and paper is expensive nowadays. I can solve this practical problem by defining the reciprocal time 1/t as tempo fisico (physical time) and use the first letter of the word “fisico“ as a mathematical symbol for this quotient:

f =  1 /[tempo]  = 1/t.

Thus, physical time f can be easily distinguished from (t)empo ordinario t (conventional time). Now, I can write for the velocity: v = [1d-spazio] f , or simply:

v = [1d-spazio-tempo] = [1d-space-time].

I think this is a simple expression, which any educated man with a modest knowledge of mathematics will immediately understand. I shall now express the Pythagorean theorem with the new symbols, so that everybody can learn this equation of gravitation by heart without realizing that I have borrowed it from Pythagoras. This is a good method to hide my initial source of inspiration:

m(c ²/t ²) = m (a ²/t ²) + m(b ²/t ²) = mvc²=  mva² +  mvb² =

m[2d-spazio-tempo]c = m[2d-spazio-tempo]a + m[2d-spazio-tempo]b = cons.   (260)

Galilei contemplates for a long time before he speaks again: “If I am honest, it is unfair to hide the name of the greatest scholar of antiquity, to whom I owe my entire scientific knowledge. I must find an elegant solution of paying reverence to Pythagoras without going into troubles with the inquisition, which looks with a bad eye upon his Geometry.“ He thinks intensively: „Now, I got it! I will substitute the symbol for the ball m with a new symbol of abbreviation: “SP(A)“ for “il Supremo Pythagoras di (A)ntiquita“. I like this very much! (In the new Theory of the Universal Law I use this symbol for the “statistical probability of the event A – SP(A)” in order to show that statistics is another adequate mathematical method of assessing the physical events of space-time = energy in addition to Geometry. Note, George)

Similarly, I will express the constant (e)nergy of the ball in a free fall mc² /t² with the first letter “E“ of the name of its first discoverer – “il grande filosofo di Efeso – Eracliteo.“ In this way, I will pay tribute to the two greatest philosophers of ancient Greece in my General equation of gravitation:

E = SP(A)[2d-spazio-tempo] = SP(A)[2d-space-time] = cons. (261)

Strange! I have an awkward feeling that I have met this equation before. I am sure that it can’t stem from another contemporaneous physicist. As there are only few physicists like me in Italy and North Europe, I am well acquainted with their works. Could it be that I have met this equation in the works of that wizard – an excellent mathematician and astrologist with an incredible virtue of prophecy – who had died in Salon-de-Provence only two years after I was born. What was his name?

Ah, yes, I got it, they called him Nostradamus! I must have hidden his apocryphal books somewhere in my private library. I remember that I bought them from a beggar who knocked on my door some years ago. He was selling beautiful books written partly in Latin and partly in French. I had never seen such books before. I must find them and check their content again.“

He is searching in his library: “Ah, here they are! Let me see (he reads). What an ambiguous and secret language! Poor guy! His life must have been as insecure as mine. Yes, I have found what I am looking for. Nostradamus foretells the arrival of an unknown scholar of Byzantine origin who will come to the West and will (re)discover the Universal Law of nature at the end of the second Millennium“

(Nota bene: Bulgaria was the first Slavonic and Christian state on the Old Continent since the 7th century and was a cultural mirror image of Byzantine with which it fought numerous wars, in many of which the Byzantine army was crucially defeated. My birthplace Plovdiv was the capital of the rich Roman province Thracia for many centuries and then an important city in the Byzantine empire after the Reptilian Emperor and founder of the state church of Christianity as Caesaropapism (for more information read my recent comments here) Constantine “the Little” moved the capital of Rome to Constantinople on the Bosphorus. Plovdiv is the oldest city in the world with an uninterrupted history that goes back to the 5th Millennium B.C. based on excavations and material facts.).

Galileo reads from Nostradamus’ book:

“After much “trial and error“ in science, lasting for more than four centuries from now on, this man will unify science and will trigger a new renaissance of Greek Logic, similar to that we observe in arts and literature in Western Europe after the fall of Constantinople.“

Galilei murmurs: “What a coincidence! This man uses the same equation for Heraclitus’ primordial energy (flux) as myself. Excellent! It was a very good idea to think of Nostradamus. One never knows where one’s inspiration will come from.“ Galilei is excited. He turns the pages of Nostradamus’ book forth and back: “Ah, what do I see? This Byzantine scholar must have had some predecessors during Novecento (20th century). Their names are Lorentz, Einstein and some more, especially Einstein is often mentioned by Nostradamus. But this is incredible! How is it possible that so many physicists are working on the same problem? This will never happen in Italy today. All these scholars are using geometric formulae to solve physical problems. Here, Nostradamus gives us an example.“

Galilei reads further with an expression of incredulity on his face: “Mamma mia! They also use the Pythagorean theorem, but what a complicated mathematical expression have they chosen! Vergogna! Now wait! How do they call this equation? – the right triangle theorem of the total relativistic energy in relation to momentum and rest energy:

= (pc)² + (moc²)²    (262)

Dio mio, this is my geometric theorem of gravitation – only written with other symbols! I must scrutinize it.“ He reads further: “Now, I see. These scholars depart from the equation of the relativistic energy (231) and the equation of the relativistic momentum p, which is obviously a mathematical iteration of the above equation.

What does the future Byzantine scholar say about this result? Yes, he is in accordance with me. He proves that the equation of the relativistic energy is an application of the universal equation of Heraclitus’s primordial fire as obtained by myself for gravitation. The same is also true for the relativistic momentum, which is a mathematical quantity of the primordial energy and has no real existence. That’s good! It seems that I am on the right track.

This scholar shows that the above equations are mathematical abstractions that merely assess the “continuum of numbers or probabilities“. This expression is new to me. I only know of the continuum of geometry – Plato and Aristotle tell us about the ideal forms of the geometric continuum that assimilate real forms, but why not use the continuum of numbers for the same purpose. Most probably, both terms are identical. Anyway, it is a well-known fact that we can express any geometric solution in numbers and vice versa.

Take for example the irrational number √2 , which follows from the Pythagorean theorem. Plato says that this number symbolizes the incommensurability of the geometric continuum. Therefore the continuum of numbers expresses the continuum of Geometry with different symbols – we can replace any geometric symbol with a mathematical one and vice versa. This is exactly what I have done in my equation on gravitation.“

Galilei turns the pages hastily and reads at random. He is bewildered: “This is, indeed, a pure nonsense! Lorentz and Einstein, or whatever their names will be, assert that the aforementioned relativistic equations of the Pythagorean theorem prove that the velocities of the particles cannot be greater than the speed of light because otherwise their solutions “will give imaginary numbers“. What a stupid argument! Aren’t they aware of the fact that all numbers are imaginary signs? They are symbols of the mind – the Platonic shadows of the real world. Why don’t these guys study Greek philosophy! This will help them avoid such stupid conclusions.

As I see, the Byzantine scholar also disproves their conclusion. Good! He proves that the aggregated velocity of the particles is greater than the speed of light (equation (189c)). If velocity is a mathematical quantity of energy, as I have defined it for gravitation, it follows that the particles of matter must have a greater energy than light. This physical fact was predicted by the famous Thracian atomist – Democritus. He teaches us that atoms have emerged from light – they are condensed light and must have more energy than light. In this case, their velocity is greater than that of light. Democritus is, indeed, a good student of great Heraclitus who says: „Da tutte le cose ne sorge una sola, e da una sola possono sorge tutte (217)“

This is an exciting idea. I will have to work it out, after I have finished with this experiment and, if I may hope, the inquisition will no longer bother me. Heraclitus idea that all objects emerge from light (flux) and disappear into light seems to be a key idea of this Byzantine scholar who also comes from Thracia. Indeed, to believe that the speed of light is the maximal possible speed, only because a mathematical solution of an artificial equation will render imaginary numbers is not at all convincing to me. I wonder how many physicists will earnestly believe this nonsense in the future. I suppose that such erroneous conclusions stem from a misapprehension of the fact that physics is applied mathematics.

Only when this fact is well understood, can we perceive why most non-mathematical interpretations of physical results are not true. I recommend all future scholars to consider my advice seriously, not only because I am the founder of modern physics, but because I am in the first place an excellent mathematician.“

Galilei scrutinizes Nostradamus’ books silently for a while, then exclaims: „There it is! Lorentz, Einstein & Co. seem to realize this truth too. They argue that if E is much greater than the mass at rest mo in equation (262), that is, if moc² → 0, then E = pc; this would say that if the side of the right triangle b approaches zero b →0, then a will approach c: a → c. Evidenza! In this case, the energy in a is equal to the energy in c. Questo lo chiamo “instinto di conservazione“ (218). Ecco la! Energy cannot be destroyed. How right was Heraclitus to say:

“Il mondo che abbiamo intorno, e che è lo stesso per tutti, non lo creò nessuno degli Dei o degli uomini, ma fu, è, e sempre sarà, Fuoco vivente. Un bel Fuoco che divampa e si spegne secondo misura (219).“


217. One thing emerges from all things, and all things can emerge from one thing.

218. “I call it the “conservation of momentum“. This is it!“

219. “The world which surrounds us is the same for everybody, no God or humans have created it, but it was, is, and will always be a living fire. A wonderful fire that extinguishes and ignites to a precise measure.“

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