The History of Non-Euclidean Geometry – Sacred Geometry

From the Nile and therefore the Euphrates
Flowed knowledge of an art on which numerous other arts are based. From the Fertile Crescent up to Greece
mathematics began to flow
Sometime within the 6th century BCE, the good Geometer, Pythagoras of Samos visited Egypt
He returned even more fascinated with geometrical ideas than he had been when he had left
He knew there was wisdom and possibilities he had to share
He saw geometry as a part of a bigger whole. a part of a philosophy about the perfection of the Universe
He needed to share this too and he knew the way to roll in the hay .
He envisioned the study of geometry together of the disciplines that might lead a person’s being to be more in-tuned with truth
perfection of the universe
So he visited Magna Graecia. The Greek colonies in what we might now call
Italy and found out a mystery cult to review philosophy and practice the sacred art of
Geometry and his cult did great
But the thing about mystery cults is well, they like their mysteries
So they’re not always great at you recognize
Writing a bunch of stuff down, thus while the pythagoreans
Taught and shared their knowledge and weren’t nearly as secretive as most of those groups
They were more curious about the philosophy of Pythagoras and therefore the ways
Mathematics pointed to a gorgeous perfection underlying the universe than they were in providing a unified mathematical system
So enter Euclid, a figure we all know surprisingly little about, but whose work had an almost
indescribable impact on human history
Euclid wrote a book or rather within the parlance of the time thirteen books called the weather for over two
thousand years this work would stand because the height of logical rigor
This book right here is that the root of just about all mathematics Modern Geometry, Algebra
Calculus, all of them founded during this work to the present day
It is the second most republished add history after the Bible. during this book
Euclid brought together all of the geometric knowledge of the traditional World
transcribing the discoveries of the Pythagoreans et al. and increasing them adding his own proofs and
discoveries to the present great catalogue of the known
But what makes this work truly one among the pinnacles of human achievement is how he put it together how it had been organised
because the book begins with alittle number of definitions,
postulates, and customary notions and says that with those everything else, every single thing in Geometry
Follows logically. He then organises his proofs, the varied geometric problems
he presents in order that all of them build off of 1 another
No proof within the entire book would require knowledge beyond those initial
definitions and therefore the proofs that came before it.
Showing just how far we will accompany a couple of simple ideas
the elements is that the
Foundation of mathematical thinking and during a lot of the way the inspiration for a way we expect of logic today. it had been an enormous achievement
But there was one small issue
That bothered a number of those studying this text. a problem that appears to possess bothered even Euclid himself
And that was the 5th Postulate. Most of the postulates within the book are fairly simple and easy
They say things such as you can draw a line between any two points or alright angles are equal
But the fifth postulate isn’t simple within the slightest
It’s more complex and it just feels different than any of the remainder . How complex is it?
The 5th Postulate states, quote “If a line falling across two straight lines makes internal angles on an equivalent side
less than two right angles
The two straight lines if produced indefinitely meet thereon side on which are the angles but two right angles.”
*UGH! That felt gross to mention . Feels tons messier than alright angles are adequate to each other right?
So let’s just break it down real quick. A line falling across two straight lines
That’s just a lines crossed by two other line somewhere. “Makes internal angles on an equivalent side
less than two right angles.” and this is often basically saying if the interior or
Interior angles, these angles which face one another right here made by the 2 lines crossing that third line add up to but two
Right angles or 180 degrees, “Then the 2 straight lines if produced indefinitely meet on the side where the angles are but two
right angles, so, okay.
If that thing I said about the inside angles before is true, then if you extend those two lines forever
They are getting to intersect at some points on the side where the inside angles are but right angles
putting all that together, if you draw a line and you’ve got two other lines cross it if their
Interior angles add up to but 180 degrees
Those lines are eventually getting to intersect if you draw them out far enough or put even more simply lines
angled towards one another are getting to
Intersect if you draw them out far enough and once you put it that way it actually seems quite obvious, right?
In fact, we are so wont to that idea that it barely even seems worth annunciated
But Euclid was nothing if not thorough and hidden during this concept is another
All-important one because let’s check out those two lines crossing the third line again. What are the chances here?
Well, if their interior angles on a side are but 180 degrees

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