I am very grateful to have found this geometry learning video on UA-cam. I saw the discussion of rotation and shift in the comments section, and I tried to offer my understanding of this. Euclid did not mention rotation and translation in the fundamental theorem, and I think it is the use of congruent triangles that achieves rotation and translation. In the case of moving and rotating triangles, moving a particular Angle is difficult, but we can rely on the third video to shift a line segment of a particular length to a line in either direction. This completes the translation of DE. For the translation of AC, we can only rely on α and then transfer the length of DF to the circle with point A as the center of the circle. And then the proof is the same as in the video.
Up to this point, Euclid has established making a given line equal to another. As well, we have the notion of equality from the common notions. What we don’t have is the concept of coincidence or rotation to establish either congruence or equality. (Please don’t mistake my critique of this proposition as a critique of this channel’s presentation. I’m only commenting on the concepts, not the presentation. The presentation, unlike this proof , is cogent and compelling!)
I always love how Euclid usually insists on perfectly rigid proofs, but then for this one he's just like "I dunno man, just eyeball it." The whole proof is based on the notion that the two triangles appear to coincide, which just strikes me as a bit cheap.
Certainly the mathematicians who study Euclid have determined that his Axioms were not always sufficient. I just want to explain his proofs as best as I can in modern terminology. sometimes it is me who makes mistakes my translations (using the word 'move' for example). This was one of my least favourite proofs, because even if you read Euclid directly, it still seems a bit wishy-washy
There's a better explanation if you "read Euclid between the lines": Recall what he did in two previous propositions - they were all about translating line segments to other places with Euclidean constructions only. So if we want to "move" one triangle to some other place in order to superimpose it on top of another triangle, we can do that simply by "moving" each of its line segments there, and we can do that with constructions from Props. 2 and 3. For example, if we want to "move" AB to superimpose it on top of DE, we have the original line segment AB, and a point D where we want the new copy of AB to start from, so we can use Prop. 2 to verify that this line segment can in fact be moved there. And we can do the same with other segments.
I think there comes the problem of how do your properly translate an angle? I could conceive using the parallel postulate to do this but Euclid didn't like using it very much.
Use of the first three propositions can be used to prove equals for sides but not for angles. Hence the difficulty of providing a definitive proof other than by transposition.
Alternatively, you can think of it as the "physics part" of geometry, that is, it is empirically verifiable that you can move a triangle so that it lies on top of another.
Mm, I think we can use equatorial sundials and gnomums' analemma shadows to confirm 360 degrees in a circle, then show full rotations, half rotations, quarter rotations equal to arbitrary degrees, hours, minutes, and arc seconds, convert to rads or grads, then prove equal angles from rotations against equal sides. Simply transposing removes the requirement of constraints to first boundaries, those, in this proposition, being the triangles' perimeters, themselves. Even accommodating post-diluvian spin rates and solar orbit processions not exactly equal to 360 days or 24 hours, and even acknowledging, yet foregoing, atomic valence iscillations for temporal candidates, we can still use 360 degrees as a NATURAL ROUND number, verified IN NATURE HERSELF, to approximate an axiom of circular measurements. This transposition is wrong. But don't get ME wrong: this presentation is sorely needed and this instructor is EXCELLENT. THANK YOU for putting this together and for FREELY offering this on UA-cam.
I know this demonstration is in the spirit of Euclid and it's OK. However, to be perfectly honest, in Euclid there is no notion of "moving a figure." This proof is a pseudo-proof. This is not an accusation, of course, it's just stating the fact that one can't prove this theorem rigorously using the machinery of "moving an object" since Euclid DOES NOT DEFINE this notion. It just does not exist in his axioms; it might be obvious to us but it's not rigorously defined. I know it took David Hilbert (in the 20th century) to give a full and complete set of axioms for Geometry that enables to either prove this theorem or it might be (can't remember right now) that this theorem is one of the axioms. However, Hilbert's development of Geometry is far, far, FAR more advanced and sophisticated than what Euclid left us and such development is not for the faint of heart. This is the reason Euclid is still popular.
The axiom of the free-mobility of figures, is, indeed, an implicit axiom which is necessary to prove the possibility of congruence relations. The problematic nature of this axiom was first discovered by Arthur Schopenhauer. Then it is carefully scrutinized by Herman von Helmholtz and later Henri Poincaré.
@@cancoteli9669 Sounds good, but I think they're missing the sundial and assignment of 360 degrees around a full rotation. Understanding that MOVEMENT around a point is different than TRANSPOSITION across a plane. You don't get the plane without the point.
@@akbaer60 Could you elaborate on this? It seems a sound and interesting argument but it's also such a broad statement that it's almost impossible to say you are wrong
@@karabomothupi9759 Common Notion 4 basically gives you the end result after some sort of movement occurs but never actually states how the movement is to be done. So yeah It's all kinds of confusing.
I am very grateful to have found this geometry learning video on UA-cam. I saw the discussion of rotation and shift in the comments section, and I tried to offer my understanding of this.
Euclid did not mention rotation and translation in the fundamental theorem, and I think it is the use of congruent triangles that achieves rotation and translation.
In the case of moving and rotating triangles, moving a particular Angle is difficult, but we can rely on the third video to shift a line segment of a particular length to a line in either direction. This completes the translation of DE. For the translation of AC, we can only rely on α and then transfer the length of DF to the circle with point A as the center of the circle. And then the proof is the same as in the video.
for 2 points just 1 straight line pass through and just 1.
pretty clear!! tanks very much!!
Great videos. What software or program do you make these constructions on?
Up to this point, Euclid has established making a given line equal to another. As well, we have the notion of equality from the common notions. What we don’t have is the concept of coincidence or rotation to establish either congruence or equality.
(Please don’t mistake my critique of this proposition as a critique of this channel’s presentation. I’m only commenting on the concepts, not the presentation. The presentation, unlike this proof , is cogent and compelling!)
Bertrand Russell once said "The fourth proposition is a tissue of nonsense".
I always love how Euclid usually insists on perfectly rigid proofs, but then for this one he's just like "I dunno man, just eyeball it." The whole proof is based on the notion that the two triangles appear to coincide, which just strikes me as a bit cheap.
@@leftylizard9085 🤣🤣🤣
Move the triangle? Can you really do that? It's not any of the propositions nor it's defined, so it's not even clear what "move" means.
Certainly the mathematicians who study Euclid have determined that his Axioms were not always sufficient. I just want to explain his proofs as best as I can in modern terminology. sometimes it is me who makes mistakes my translations (using the word 'move' for example).
This was one of my least favourite proofs, because even if you read Euclid directly, it still seems a bit wishy-washy
Ok, thanks.
There's a better explanation if you "read Euclid between the lines":
Recall what he did in two previous propositions - they were all about translating line segments to other places with Euclidean constructions only. So if we want to "move" one triangle to some other place in order to superimpose it on top of another triangle, we can do that simply by "moving" each of its line segments there, and we can do that with constructions from Props. 2 and 3.
For example, if we want to "move" AB to superimpose it on top of DE, we have the original line segment AB, and a point D where we want the new copy of AB to start from, so we can use Prop. 2 to verify that this line segment can in fact be moved there. And we can do the same with other segments.
Genius
I think there comes the problem of how do your properly translate an angle? I could conceive using the parallel postulate to do this but Euclid didn't like using it very much.
you are a genius!!!!!!!!!!!
magnific videos!!!!!!!!!!!
Use of the first three propositions can be used to prove equals for sides but not for angles. Hence the difficulty of providing a definitive proof other than by transposition.
You could use propostion 23, but that hasn't been proved yet at this point.
Alternatively, you can think of it as the "physics part" of geometry, that is, it is empirically verifiable that you can move a triangle so that it lies on top of another.
Mm, I think we can use equatorial sundials and gnomums' analemma shadows to confirm 360 degrees in a circle, then show full rotations, half rotations, quarter rotations equal to arbitrary degrees, hours, minutes, and arc seconds, convert to rads or grads, then prove equal angles from rotations against equal sides.
Simply transposing removes the requirement of constraints to first boundaries, those, in this proposition, being the triangles' perimeters, themselves.
Even accommodating post-diluvian spin rates and solar orbit processions not exactly equal to 360 days or 24 hours, and even acknowledging, yet foregoing, atomic valence iscillations for temporal candidates, we can still use 360 degrees as a NATURAL ROUND number, verified IN NATURE HERSELF, to approximate an axiom of circular measurements.
This transposition is wrong.
But don't get ME wrong: this presentation is sorely needed and this instructor is EXCELLENT.
THANK YOU for putting this together and for FREELY offering this on UA-cam.
These are very helpful. Thank you!
I know this demonstration is in the spirit of Euclid and it's OK. However, to be perfectly honest, in Euclid there is no notion of "moving a figure." This proof is a pseudo-proof. This is not an accusation, of course, it's just stating the fact that one can't prove this theorem rigorously using the machinery of "moving an object" since Euclid DOES NOT DEFINE this notion. It just does not exist in his axioms; it might be obvious to us but it's not rigorously defined. I know it took David Hilbert (in the 20th century) to give a full and complete set of axioms for Geometry that enables to either prove this theorem or it might be (can't remember right now) that this theorem is one of the axioms. However, Hilbert's development of Geometry is far, far, FAR more advanced and sophisticated than what Euclid left us and such development is not for the faint of heart. This is the reason Euclid is still popular.
The axiom of the free-mobility of figures, is, indeed, an implicit axiom which is necessary to prove the possibility of congruence relations. The problematic nature of this axiom was first discovered by Arthur Schopenhauer. Then it is carefully scrutinized by Herman von Helmholtz and later Henri Poincaré.
@@cancoteli9669 Sounds good, but I think they're missing the sundial and assignment of 360 degrees around a full rotation.
Understanding that MOVEMENT around a point is different than TRANSPOSITION across a plane.
You don't get the plane without the point.
POV: you're in Ms. Neacsu's class :)
Didn't know the rules Euclid laid out let him translate and rotate at will
I am also confused.
Translating and rotating is basically the proposition 2
@@akbaer60 Could you elaborate on this? It seems a sound and interesting argument but it's also such a broad statement that it's almost impossible to say you are wrong
@@michelef406 here you go bud
ua-cam.com/video/3ixdsyh_twc/v-deo.html
@@karabomothupi9759 Common Notion 4 basically gives you the end result after some sort of movement occurs but never actually states how the movement is to be done. So yeah It's all kinds of confusing.
No definition of movement