"If a straight line falling in two straight lines make the interior angles on the same side less than 2right angles, the two straight lines of produced indefinitely meet on that side on which the angles are less than 2right angles"
-Euclid's fifth postulate,Elements
The fifth postulate of Euclid always have created problems for mathematicians. Because there wasn't any proof. The statement itself seems to be true. Mathematicians considered it as a consequence of the first 4 postulates and ofcourse if that of geometry itself.
NB: The postulate says nothing but "if we have two straight lines,and we keep another one over them, we will get two angles on any side of them,whose sum will be less than 180. Then if the two lines are extended they will meet at some point somewhere on that same side"
It feel common to us and sensible. But yet there wasn't any direct cause for it.
The same postulate was later interpreted by mathematicians in many ways to make it perfect.
Ptolemy,Proclos,Borelli,Al tusi,Vitale,Wallis,Saccheri,Lambert,Legendre etc...
See what Proclos interpreted it : "If a straight line intersects one of the two parallels,it will intersect the other also"
Wallis interpreted as : "Given any figure there exists a figure similar to it,of any size"
These may sound simple and nonsense. but not.
However you might be familiar with what legendre interpreted.
Legendre : " There is a triangle in which the sum of the three interior angles is equal to 2right angles"
///////////
See how differently people interpret and create new amazing theories and concepts. This is not because Euclid was a god and he stated some prophecy.
He stated something crude and infant and confusing and ofcourse unclear. So,people may have chance to discover more from such unclear statements. It's a lucky thing after all.
Point to be noted : Euclid and Ptolemy lived and died sometime in BC 400-100 before even Christ and Mohammed came.
/////////////
Amazing fact: Later in 18th-19th century Carl Frederich Gauss re-examined this same fifth postulated and found serious problem with that.
Yes he found Mistakes in the work of Euclid,Ptolemy,Legendre and many.
I can tell problem in a simple way : "what is the validity of a triangle if it has its sum of interior angles less than or greater than 180?"
This was actually produced from Saccheri's 'hypothesis of acute angle' which states "A straight line being given,there can be drawn a perpendicular to it and a line cutting it at an acute angle,which do not intersect each other"
This statement seems nonsense for most of the mathematicians. But later from the works of Bolyai, Gauss confirmed the existence of Triangles with an interior angles sum greater or lesser than 180.
/////////
Now what does this have so importance?
Well it's tremendous.
All the mathematics we have so studied,mostly upto Bsc Physics consists of Euclidean. Every mathematics we studied works in Euclidean space.
Gauss's work proved the existence of Non-Euclidean spaces and of course Non-Euclidean Geometry.
Which is simply the geometry of curved surfaces. While Gauss examined the Curvature of normal surfaces (Triangles are formed in surfaces), later Riemann found geometry of the Curved surfaces in higher dimensions. With the works of David Hilbert in Higher dimensional spaces and with Minkowskis's Four dimensional space time which eventually ended up for in the discovery of Special Relativity and General Relativity by Albert Einstein.
Let me ask now.
Does Euclid made something which ended up in Relativity?
Does the discovery of Relativity now can be attributed to Euclid?
///////////
This is a historic time line to understand that philosophical literary works when mixed with facts could eventually lead to something fruitful,but by editing and modifying at a larger scale including a larger time span.
So,it better if we consider rational thoughts born from anxiety and curiosity rather than following up literary works like Texts followed by people.
-Euclid's fifth postulate,Elements
The fifth postulate of Euclid always have created problems for mathematicians. Because there wasn't any proof. The statement itself seems to be true. Mathematicians considered it as a consequence of the first 4 postulates and ofcourse if that of geometry itself.
NB: The postulate says nothing but "if we have two straight lines,and we keep another one over them, we will get two angles on any side of them,whose sum will be less than 180. Then if the two lines are extended they will meet at some point somewhere on that same side"
It feel common to us and sensible. But yet there wasn't any direct cause for it.
The same postulate was later interpreted by mathematicians in many ways to make it perfect.
Ptolemy,Proclos,Borelli,Al tusi,Vitale,Wallis,Saccheri,Lambert,Legendre etc...
See what Proclos interpreted it : "If a straight line intersects one of the two parallels,it will intersect the other also"
Wallis interpreted as : "Given any figure there exists a figure similar to it,of any size"
These may sound simple and nonsense. but not.
However you might be familiar with what legendre interpreted.
Legendre : " There is a triangle in which the sum of the three interior angles is equal to 2right angles"
///////////
See how differently people interpret and create new amazing theories and concepts. This is not because Euclid was a god and he stated some prophecy.
He stated something crude and infant and confusing and ofcourse unclear. So,people may have chance to discover more from such unclear statements. It's a lucky thing after all.
Point to be noted : Euclid and Ptolemy lived and died sometime in BC 400-100 before even Christ and Mohammed came.
/////////////
Amazing fact: Later in 18th-19th century Carl Frederich Gauss re-examined this same fifth postulated and found serious problem with that.
Yes he found Mistakes in the work of Euclid,Ptolemy,Legendre and many.
I can tell problem in a simple way : "what is the validity of a triangle if it has its sum of interior angles less than or greater than 180?"
This was actually produced from Saccheri's 'hypothesis of acute angle' which states "A straight line being given,there can be drawn a perpendicular to it and a line cutting it at an acute angle,which do not intersect each other"
This statement seems nonsense for most of the mathematicians. But later from the works of Bolyai, Gauss confirmed the existence of Triangles with an interior angles sum greater or lesser than 180.
/////////
Now what does this have so importance?
Well it's tremendous.
All the mathematics we have so studied,mostly upto Bsc Physics consists of Euclidean. Every mathematics we studied works in Euclidean space.
Gauss's work proved the existence of Non-Euclidean spaces and of course Non-Euclidean Geometry.
Which is simply the geometry of curved surfaces. While Gauss examined the Curvature of normal surfaces (Triangles are formed in surfaces), later Riemann found geometry of the Curved surfaces in higher dimensions. With the works of David Hilbert in Higher dimensional spaces and with Minkowskis's Four dimensional space time which eventually ended up for in the discovery of Special Relativity and General Relativity by Albert Einstein.
Let me ask now.
Does Euclid made something which ended up in Relativity?
Does the discovery of Relativity now can be attributed to Euclid?
///////////
This is a historic time line to understand that philosophical literary works when mixed with facts could eventually lead to something fruitful,but by editing and modifying at a larger scale including a larger time span.
So,it better if we consider rational thoughts born from anxiety and curiosity rather than following up literary works like Texts followed by people.
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