Why is Euclid's SAS proof considered not valid?
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pka
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You ask about "Why is Euclid's SAS proof considered not valid"
Please tell more about the question.
What leads you to ask that particular question?
To what proof of Euclid does it refer?
Can you refer us to text material that is the basis for you quarry?
Please tell more about the question.
What leads you to ask that particular question?
To what proof of Euclid does it refer?
Can you refer us to text material that is the basis for you quarry?
In the 19th century, mathematicians began to demand far more rigor in proofs than previously. In part this was because of the development of non-Euclidean geometries early in the century, and in part because of concerns about the logical validity of calculus, a branch of mathematics that was then historically recent and far from rigorous in its development.
When these much higher standards of rigor were applied to Euclid, it was found that a number of his early theorems did not meet the new standards. (This is not really a criticism of Euclid; his standards were considered good enough for thousands of years. And the average high school student is likely to consider the deficiencies mere pettifogging.) So Euclid was done over. The first answer in the link below gives more details with respect to the specific issue of SAS, but I point out that the very first proof in Euclid is also considered to not meet modern standards of rigor though common sense will not even notice the defect.
math.stackexchange.com
When these much higher standards of rigor were applied to Euclid, it was found that a number of his early theorems did not meet the new standards. (This is not really a criticism of Euclid; his standards were considered good enough for thousands of years. And the average high school student is likely to consider the deficiencies mere pettifogging.) So Euclid was done over. The first answer in the link below gives more details with respect to the specific issue of SAS, but I point out that the very first proof in Euclid is also considered to not meet modern standards of rigor though common sense will not even notice the defect.
Is triangle congruence SAS an axiom?
I was wondering if there is a way to prove SAS in triangle congruence with Euclidean axioms. Thank you for your help!
math.stackexchange.com
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In the 19th century, mathematicians began to demand far more rigor in proofs than previously. In part this was because of the development of non-Euclidean geometries early in the century, and in part because of concerns about the logical validity of calculus, a branch of mathematics that was then historically recent and far from rigorous in its development.
When these much higher standards of rigor were applied to Euclid, it was found that a number of his early theorems did not meet the new standards. (This is not really a criticism of Euclid; his standards were considered good enough for thousands of years. And the average high school student is likely to consider the deficiencies mere pettifogging.) So Euclid was done over. The first answer in the link below gives more details with respect to the specific issue of SAS, but I point out that the very first proof in Euclid is also considered to not meet modern standards of rigor though common sense will not even notice the defect.
Is triangle congruence SAS an axiom?
I was wondering if there is a way to prove SAS in triangle congruence with Euclidean axioms. Thank you for your help!
The link didn't help :/
Dr.Peterson
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Here is the proof you are asking about, with comments: https://mathcs.clarku.edu/~djoyce/java/elements/bookI/propI4.html
I also found this paper, whose introduction discusses various flaws in Euclid's proofs, without singling out I.4: https://link.springer.com/content/pdf/10.1007/s10472-018-9606-x.pdf
I also found this paper, whose introduction discusses various flaws in Euclid's proofs, without singling out I.4: https://link.springer.com/content/pdf/10.1007/s10472-018-9606-x.pdf
Wow. The second link is a powerhouse. It would take me a week to grasp it fully. Thank you for the link.
By the way, I have an historical supposition about the logical deficiencies in Euclid. He was idealizing compass and straight-edge geometry and may never have asked himself about things that are obviously true about physical reality. That is, his geometry may have been about idealized points, lines, circles, and polygons, but he missed axioms needed to translate from the physical to the ideal. Platonism makes it easy to make that mistake. If reality is the shadow of the ideal, then the physical fact that two approximate circles scratched in the sand intersect means that the corresponding ideal circles must intersect in the ideal world of the imagination.
pka
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thank you