Let's turn to the Pythagorean theorem as to an object-idea. Its seeming simplicity and elementary quality should not hide from us the fundamental significance which contains in itself this so habitual for us a discovery of ancient authors. With the help of algebraic symbolism it is written down in the following form
|a2 + b2 = c2,||(1)|
where: a and b are the real numbers as lengths of legs of the rectangular triangle given by three points in Euclid's space; c – the length of a hypotenuse of this triangle. To prove it geometrically the other form of record is used
|Ssqa + Ssqb = Ssqc, Ssqk := k2,||(2)|
where Ssqk is the area of the square, constructed on an appropriate side of a rectangular triangle. Let us use the well-known algebraic identification property of two quantities and multiply the left and right parts (2) by the common multiplier 4π, and as a result of it, we will have the new equality-theorem
|Sspa + Sspb = Sspc, Sspk := 4πk2,||(3)|
where Sspk is interpreted as the area of a spherical surface of radius k in 3-dimensional Euclid's space E3. The equality (3) can be interpreted as the theorem of a ratio of the areas of the spherical surfaces, constructed on the sides of a rectangular triangle as on the radiuses of respective spheres.
The equality-theorems (2) and (3), which have identical algebraic structure, essentially differ by their geometrical interpretation. The action of the theorem (3) which we will still name the Pythagorean theorem and will designate with Psp, develops in Euclid's 3-space E3. For the standard Pythagorean theorem Psq in the form (2), and the concomitant to it P2, which was displayed in the formula (1), Euclid's plane E2 given by three points-tops of a rectangular triangle is enough.
I believe, there is no necessity to stop on the proof or an explanation of that indisputable fact, that without an absolute correctness of Pythagorean theorem Euclid's space is by no means possible, whether it is the plane E2 or our usual space E3. Let us assume that Euclidness, 3-dimentionality and Pythagorasness of our space E3 are closely interconnected and base on fundamental, defining properties and the role of measure-area Ssp in this space.
Apparently , it was Kant who first noticed, that the laws of return squares for gravitational and electric forces are connected to our three-dimensionality of our space. He wrote: «Three-dimensionality occurs, apparently, because the substances in the existing world affect each other in such a manner that the force of action is inversely proportional to the distance squared».  This innovative idea, probably, for the first time associated properties of space with the specific law of physics. It is possible, that the reader himself while studying the physics, paid attention to seemingly strange coincidences in a number of physical laws. 
Laws of force interaction of two «dot charges» can be written down in the unified form
|Fd12·Ssp12 = qd1·qd2,||(4)|
where: Fd12 – the value of Newton force of d-interaction of two charges qd1 and qd2; Ssp12 - the area of a spherical surface with the centre at the point of location of one of the d-charges, having force influence on the other d-charge laying on this spherical surface with the area Ssp12. For the law of gravitation of Newton the d-charge is body mass; for Coulomb's law in an electrostatics it is a usual electric charge; in magnetostatics it is a magnetic charge. The respective world constants of d-interactions in an explicit form in the law (4) are absent, as also a numerical universal multiplier, because they can be brought directly in the definition of a unit of measurements of the d-charge – qd.
All this is the consequence of 3-dimensionality of our space. This result is easily obtained from 3-dimensional equations of Laplace for potentials of respective fields. 
For example, the Pythagorean theorem and the Newton's law of universal gravity are interconnected, because both of them submit to the same fundamental physical concept – to potential. But for everyone who is familiar with the Einstein's theory of gravity, it is absolutely clear, that both these laws, so different in outward appearance and considered to be so distant earlier, one of which became known in an antiquity and was one of the first theorems studied at school, and another one describes the interaction of masses, are not only the same by their nature, but also are only a part of the same general law.
It is hardly possible to give more amazing an example of fundamental repeatability of geometrical and physical factors. 
|The translation from Russian was made by Masha and Natasha Zazerska|
Last modifications: January 26 2003
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