Euclid's Elements are about the 5 Elements
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In Book 1 of “Elements ”, Euclid gives 5 Postulates which people think are about geometry.
This is why they get so confused with Postulate 5 which is about 2 straight lines meeting. This makes people, such as Gauss, think that Euclid was talking about Geodesics and spherical geometry which they call as non-Euclidean.
This non-Euclidean thinking was then used by Einstein to justify Riemann curvature, which is very complicated.
Johannes Kepler denounced such shallowness. In fact, he bases his planetary orbits in Euclidean geometry, specifically in Book 10. There is no need for geodesics is Kepler’s theories.
We side with Kepler and emphasize that Euclid’s Elements are about the dynamics of the 5 Elements: Aether, Air, Fire, Water, and Earth.
These were called Akasha, Vayu, Tejas, Jala, and Pritvi by the Hindus, and by 水,木,火,土,and 金 by the Chinese. Nowadays, these are called Aether, Spacetime, Electromagnetism, Weak Force, and Strong Force by Physics.
- Postulate 1 is about how the aether links two discrete identities. We call this an aethereal band. The details of this is given by Books 8 and 9
- Postulate 2 is about how aethereal links spread out indefinitely, creating spacetime. The details of this is given by Books 5, 7, 10
- Postulate 3 is about how aethereal links spread in all directions from a source, as seen in the light of Electromagnetism. The details of this is given by Books 3 and 4.
- Postulate 4 is about how some aethereal identities have a ‘right angle’ which is now the right handedness of a particle in the Weak Interaction. The details of this is given by Book 2, 6.
- Postulate 5 is about how the meeting of aethereal links create matter (Strong Force) or interaction from non-right-handed particles. The details of this is given by Books 11-13.
Euclid and the Greeks got these principles from the Egyptians who were expert in geometry and math.
A shallow person will say that there is no way for the Egyptians to know about left and right handedness of particles.
We reply that left and right hand chirality manifests in chemical compounds, especially those that are used in Medicine. The Egyptians were also skilled with medicine, and so it is totally possible for them to derive Postulates 4 and 5 by experimenting with how chemicals work.
Proof is that the Chinese also used the principles of the 4th Element to invent gunpowder.
Euclid’s Elements are therefore a guide on how the 5 Elements work, explained in a logical manner. It is not a mere book about how to draw shapes.
Younger generations are not necessarily wiser
The main problem is that the later generations did not know about the 5 Elements and so their brains interpreted Euclid as writing about mere shapes.
Newton added to the problem by using his Calculus (which is ignorant of the 5 Elements) to bulldoze over the underlying propositions of Kepler’s Laws.
Therefore, saying ’non-Euclidean geometry’ is as absurd as saying ’non-universal universe’. Spherical (non-Euclidean, Riemann) Geometry and geodesics are really a mix of Postulates 2 and 3, just as Einstein used c for spacetime.