Did Lambert Prove That Saccerei's Acute Angles Contradict Euclidean Geometry?

[Thoth]

You guys know of any good site that explains fully about how Lambert proved that Saccerei’s acute angles does not result in a contradiction to the fifth postulate of Euclidean geometry? Thank you for any help

 

[HallsofIvy]

I did a "google" on "Lambert Saccheri" and at the top was:
http://www.math.uncc.edu/~droyster/math3181/notes/hyprgeom/node41.html
That looks like it will give what you want.

 

[HallsofIvy]

I don't know that it is correct to say that Lambert "gave a proof" that Saccheri was wrong. He just noted that ``the hypothesis of the acute angle is absolutely false; because it is repugnant to the nature of straight lines.'', Saccheri's conclusion, does NOT prove anything. "Straight lines" are geometric figures that satisfy certain properties. In order to show that something is "repugnant to the nature of straight lines", you would have to show which of those properties are violated.

 

[Thoth]

Thank you HallsofIvy for your help, but I have seen that site before. From here the Internet is heavily censured and I was hoping that you might have better access to a larger library on the net, but perhaps I was wrong.

As you probably are aware of, Saccheri wanted to proof Euclidean fifth postulate by showing that the only accurate answer to three cases of Saccheri’s quadrilateral is when the summit angles are equal to 180 degree. So he was trying to find a contradiction in acute case to settle his objective.

According to this site:
http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Non-Euclidean_geometry.html

Lambert investigated the hypothesis of the acute angle without obtaining a contradiction. I just want to know how Lambert was able to do that, but unfortunately I have remained unsuccessful in my search.

 

[Hurkyl]

Lambert proved no such thing. He merely explored the fields of neutral and hyperbolic geometry. He was unable to derive a contradiction in hyperbolic geometry, but he certainly did not prove hyperbolic geometry was contradiction free.

(note: I don't mean to imply his work was of little importance)

His basic program (that he was unable to complete) is:
Assume Euclid V is false.
Derive contradiction.
Conclude Euclid V is true.

(he was unable to do the second step)

It wasn't until... Klein I think... that it was proven that hyperbolic geometry is consistent (relative to Euclidean geometry).

But in no way is it true that "The hypothesis that Saccheri quadrilaterals have an acute angle is consistent with the parallel postulate."


Glossary:
neutral geometry - Euclidean geometry, minus the parallel postulate.
hyperbolic geometry - neutral geometry, plus the axiom that the parallel postulate is false.

 

[HallsofIvy]

Lambert proved no such thing. He merely explored the fields of neutral and hyperbolic geometry. He was unable to derive a contradiction in hyperbolic geometry, but he certainly did not prove hyperbolic geometry was contradiction free.

Hurkyl, no one said he did. The original poster asked about his "proof" that Saccheri's work was invalid.