Antonio Leon
"New Elements of Euclidean Geometry" Revised 2023
"Apparent Relativity" Revised 2023
Both books available free at Academia.edu

He demonstrates Euclid's 5th postulate, something supposedly ridiculous to claim, since many attempted to do this for over 2,000 years without success.

The non-Euclidean geometries take as a pretext the alleged inability to prove the 5th postulate to embark on their unproven claims.

Jeremiah Joseph Callahan already accomplished the same in his 1931 book, "Euclid or Einstein: A Proof of the Parallel Theory and a Critique of Metageometry"

LaurenceClarkCrossen <clzb93ynxj@att.net> wrote:

> Antonio Leon
> "New Elements of Euclidean Geometry" Revised 2023
> "Apparent Relativity" Revised 2023
> Both books available free at Academia.edu
>
> He demonstrates Euclid's 5th postulate, something supposedly ridiculous to
> claim, since many attempted to do this for over 2,000 years without
> success.
>
> The non-Euclidean geometries take as a pretext the alleged inability to
> prove the 5th postulate to embark on their unproven claims.
>
> Jeremiah Joseph Callahan already accomplished the same in his 1931 book,
> "Euclid or Einstein: A Proof of the Parallel Theory and a Critique of
> Metageometry"

Great! Show us the proof,

Jan

You might first try to understand Callahan's proof if you need help. I'm kind of busy reading Leon's second book, which criticizes relativity.

I think non-Euclidean geometry is recognizable as necessarily involving the reification fallacy, so it is not true. It is necessary to attribute qualities to abstract space to suppose that parallel lines meet. Contrary to Tom Roberts, in physics, one cannot use models that involve reification fallacy because they cannot account for causation.

On 03/16/2024 01:46 PM, LaurenceClarkCrossen wrote:
> I think non-Euclidean geometry is recognizable as necessarily involving
> the reification fallacy, so it is not true. It is necessary to attribute
> qualities to abstract space to suppose that parallel lines meet.
> Contrary to Tom Roberts, in physics, one cannot use models that involve
> reification fallacy because they cannot account for causation.

I think it's "there's a metric everywhere and it's a norm".

So, we have different metrics and norms and coordinate settings,
in our patch-work piece-wise models, all about spherical symmetries,
and the "un-linear", which is simpler in a world where the geodesy
is "world-lines" is "orbit-lines".

The "coordinate-free" and that tensors are objects that
preserve the geometric relations the affine their products,
gets into why that because space-time is continuous,
and Space-Time is pretty much flat, that, the ideas of
there being "non-Euclidean" geometries, are really
"super-Euclidean projection in Euclidean geometry".

I think it's more relevant that geometry gets appended
a "continuity postulate", for example as has been around
a long time and is usual in the modern point-set models.

I.e., some of Euclid's results have "line-drawing" implicitly.

On 03/16/2024 02:02 PM, Ross Finlayson wrote:
> On 03/16/2024 01:46 PM, LaurenceClarkCrossen wrote:
>> I think non-Euclidean geometry is recognizable as necessarily involving
>> the reification fallacy, so it is not true. It is necessary to attribute
>> qualities to abstract space to suppose that parallel lines meet.
>> Contrary to Tom Roberts, in physics, one cannot use models that involve
>> reification fallacy because they cannot account for causation.
>
> I think it's "there's a metric everywhere and it's a norm".
>
> So, we have different metrics and norms and coordinate settings,
> in our patch-work piece-wise models, all about spherical symmetries,
> and the "un-linear", which is simpler in a world where the geodesy
> is "world-lines" is "orbit-lines".
>
> The "coordinate-free" and that tensors are objects that
> preserve the geometric relations the affine their products,
> gets into why that because space-time is continuous,
> and Space-Time is pretty much flat, that, the ideas of
> there being "non-Euclidean" geometries, are really
> "super-Euclidean projection in Euclidean geometry".
>
>
> I think it's more relevant that geometry gets appended
> a "continuity postulate", for example as has been around
> a long time and is usual in the modern point-set models.
>
> I.e., some of Euclid's results have "line-drawing" implicitly.
>
>

The quantum mechanics has a very similar "un-linearity",
that particle/wave duality results from the "everywhere
at once", of the behavior of the "particle", which is
observable in terms of quantities of particles, which
results the stochastic interpretation of QM, "probability
waves", has another interpretation as super-classical,
where there is "real wave collapse", that there is
the flow model and and the flux model, as of the viscous
flow of a fluid model, and the flux of a light model, and
the extensions of those again, superfluid and "superflux",
and the statifications of those, the "subfluid" and subflux".

So, "un-linearity" is an idea that while the classical is
linear, that is yet a singularity in the overall theory,
which results as a sort of sum-of-histories in sum-of-potentials,
then for entropy in complementarity and entropy in unitarity,
it makes for that the linear is the sum of the un-linear,
as as that is what is real.

LaurenceClarkCrossen <clzb93ynxj@att.net> wrote:

> You might first try to understand Callahan's proof if you need help. I'm
> kind of busy reading Leon's second book, which criticizes relativity.

Proof, not words about it,

Jan

LaurenceClarkCrossen <clzb93ynxj@att.net> wrote:

> I think non-Euclidean geometry is recognizable as necessarily involving
> the reification fallacy, so it is not true. It is necessary to attribute
> qualities to abstract space to suppose that parallel lines meet. Contrary
> to Tom Roberts, in physics, one cannot use models that involve reification
> fallacy because they cannot account for causation.

FYI, all this talk of // lines meeting at infinity is obsolete.
In modern presentations Euclidean geometry is defined
as that geometry in which the Pythagorean theorem holds.
The intersection at infinity, or better non-intersetction in the finite
can then be proven as a theorem.

The two definitions can be shown to be equivalent,

Jan

W dniu 17.03.2024 o 12:45, J. J. Lodder pisze:
> LaurenceClarkCrossen <clzb93ynxj@att.net> wrote:
>
>> I think non-Euclidean geometry is recognizable as necessarily involving
>> the reification fallacy, so it is not true. It is necessary to attribute
>> qualities to abstract space to suppose that parallel lines meet. Contrary
>> to Tom Roberts, in physics, one cannot use models that involve reification
>> fallacy because they cannot account for causation.
>
> FYI, all this talk of // lines meeting at infinity is obsolete.
> In modern presentations Euclidean geometry is defined
> as that geometry in which the Pythagorean theorem holds.
> The intersection at infinity, or better non-intersetction in the finite
> can then be proven as a theorem.
>
> The two definitions can be shown to be equivalent,

And communism can be shown to be the best. If an idiot
buys it...

Callahan and Leon prove it if you look through the telescope by reading their books...

Jan just declared that parallel lines can be defined as not meeting at finite distances but meeting at infinity. Jan, we disagree. Callahan and Leon prove they don't meet at infinity. My explanation above is enough because making them meet requires attributing qualities to space, which involves the reification fallacy.

What relativist or non-Euclidean geometer has ever directly replied to Callahan's book/argument?

On 03/17/2024 09:39 AM, LaurenceClarkCrossen wrote:
> Jan just declared that parallel lines can be defined as not meeting at
> finite distances but meeting at infinity. Jan, we disagree. Callahan and
> Leon prove they don't meet at infinity. My explanation above is enough
> because making them meet requires attributing qualities to space, which
> involves the reification fallacy.

No he said the opposite and that non-Euclidean geometries
aren't quite the raging banger they once were.

It's not so much that space time is curved far away -
as that it curves to get there.

The "linear, vanishing, non-linear, highly non-linear,
spaghettified ,asymptotic, ", sort of reflects extremes
in regimes of dynamics, and it's a continuum mechanics,
about that "matter is mostly space" and, you know,
"ponderable matter's inertial systems are in frames",
then, frame-spaces space-frames are sort of ubiquitous,
so that what's "local" is "local' and what remains "spooky"
is also "local".

"Well then, by vacuity I mean intangible and empty space.
If it did not exist, things could not move at all."

- Lucretious, circa 50 BC the 0'th century

"Eppur si muove" - Galileo

So, "non-Euclidean geometries", make for of course
projective and perspective and what result from
the right angle, two ways of looking at things.

When two observers happen to have the same derivation
of what they "see" and moreso the forces they observe,
yet going either way, they just diverge, that "the
classical is only the limit all else held constant",
has that the classical actually just is a singularity,
you know, mathematically.

It's a continuum mechanics.

"Nothing is necessitated whose opposite is possible" - Leibniz

"And just as the same town, when looked at from different sides,
appears quite different and is, as it were, multiplied in perspective,
so also it happens that because of the infinite number of simple
substances, it is as if there were as many different universes,
which are however but different perspectie representations of a
single universe from the different point of view of each monad."

"For all bodies are in perpectual flux like rivers,
and parts are passing in and out of them continually."

"The mind is not only capable of knowing [innate ideas],
but further of finding them in itself; and if it had only
the simple capacity to receive knowledge ... it would not be
the source of necessary truths...". - ibid

Obviously "Multiple-Worlds Interpretation" is un-scientific,
and, "discontinuous continuous" is a contradiction in terms,
and dark matter and energy are placeholders because "data
don't fit". Don't get me wrong, the "new wobbly" nonsense is
just another linearisation of a linearisation of a linearisation.

So anyways, Einstein doesn't have any problems at all,
his unified theory of Relativity, that Mathematical Physics
doesn't have, according to "Theory and Philosophy of" as
what results "Empirical Standing and Dogma of".

LaurenceClarkCrossen <clzb93ynxj@att.net> wrote:

> Callahan and Leon prove it if you look through the telescope by reading
> their books...

Really? Then why not repeat?

Jan

LaurenceClarkCrossen <clzb93ynxj@att.net> wrote:

> What relativist or non-Euclidean geometer has ever directly replied to
> Callahan's book/argument?

<https://en.wikipedia.org/wiki/Jeremiah_J._Callahan>

Duquesne University tries to deny that they ever had such a president,
<https://www.duq.edu/about/history/index.php>

Jan

On 2024-03-17 19:58:02 +0000, J. J. Lodder said:

> LaurenceClarkCrossen <clzb93ynxj@att.net> wrote:
>
>> What relativist or non-Euclidean geometer has ever directly replied to
>> Callahan's book/argument?
>
> <https://en.wikipedia.org/wiki/Jeremiah_J._Callahan>
>
> Duquesne University tries to deny that they ever had such a president,
> <https://www.duq.edu/about/history/index.php>
>
> Jan

Really? "Rev. Jeremiah Joseph Callahan
President, 1931 - 1940" at the link you give.

--
athel -- biochemist, not a physicist, but detector of crackpots

Why not listen (see above)?
Jan claims that it has been proven that parallel lines meet at infinity without giving proof.
No such proofs do not presume what they want to conclude.
Why don't they diverge at infinity?
Because Jan wants space to curve one way and not the other.

Callahan, in Euclid or Einstein, says,
p.222 "he is himself struggling to give, in a hazy way, some kind of reality to his mathematics by clothing his formulae with some interpretation or other....clarity ends, and we step into a region of mistiness and fog. We certainly cannot consider Einstein as one who shines as a scientific discoverer in the domain of physics, but rather as one who in a fuddled sort of way is merely trying to find some meaning for mathematical formulae in which he himself does not believe too strongly..."

BTW Jan, when parallel lines meet at a distance, that's an optical illusion...

Yes, Jan, we know about those. So there are no refutations of Callahan's proof because relativists are not able to.

Einstein and his defenders are crackpots.
You can detect a crackpot by their inability to reason exhibited in their use of logical fallacies.

Callahan says,
Callahan page 285: "only another example of the lack of clearness of conception and love of the incongruous that everywhere characterizes the position of Einstein...how can we explain such a confused mentality in anyone professing himself to be a scientist? ... Einstein is neither a physicist nor a metaphysician capable of dealing clearly with physical and metaphysical entities in a scientific way... he is hampered by a load of contradictory and absurd assumptions... above all he utterly lacks the scientific sagacity or instinct to choose the proper physical, metaphysical and mathematical basis.... as a result his beginnings are as bad as his conclusions... even worse... is the absence of the power of criticism to enable him to see the glaring contradictories which he is embracing, and the lack of logical insight to understand the use and force of language. No wonder then he blunders, and in floundering..."

LCC Skeptic of UFO's Bigfoot & Relativity

On 03/17/2024 02:18 PM, LaurenceClarkCrossen wrote:
> Callahan, in Euclid or Einstein, says,
> p.222 "he is himself struggling to give, in a hazy way, some kind of
> reality to his mathematics by clothing his formulae with some
> interpretation or other....clarity ends, and we step into a region of
> mistiness and fog. We certainly cannot consider Einstein as one who
> shines as a scientific discoverer in the domain of physics, but rather
> as one who in a fuddled sort of way is merely trying to find some
> meaning for mathematical formulae in which he himself does not believe
> too strongly..."

That's "Mathematical Physics".

Consider for example the Cartanian developments
and Geometric Algebra. Didn't exist. Got figured
out, added to physics, discovered a particle, and
a remarkable virtual anti-particle at that. (That
it's so said the contrivance the configuration the
energy the experiment so found not falsified,
or so it is said and relayed according to the
interpretation of the data.)

Science is still figuring out there's an
"upper, middle, and lower" sky, that the
Babylonians named in hazy antiquity, that
"peripheral parallax" is still sort of a
toss-up between Fresnel and Huygens.

Now, Einstein is lionized and that means
beyond even what he deserves, at the same
time he sort of pushes off from his followers
writ large, who if they "know" the theory
probably learned it ten different ways,
most of which giving absurd consequences,
that it's also so, that, Einstein was still
searching for a "total field theory", and
even just a practical "bridge" from the linear
to and from the rotational, "Einstein's bridge",
even out of his later years.

So, these days, the, "nonconservative", the
"pseudomomentum", the "quasi-invariant",
the "pseudodifferential", these aren't just
names for resonance theory and rest-exchange
and resonance theory and rest-exchange,
they represent conceptual concepts in the
mathematics, that most people sort of naively
have, but don't intuitively formalize.

Or, if they don't, it's like "well, there
were only three people who know relativity,
Einstein, Eddington, and nobody knows who
else, since then at least two of those are
wrong, so, what it's figured is that Einstein
knows less now than was then".

The usual idea is "physics has three great
edifices, GR and gravity and QM, somehow
they're all supposed to be one theory".
I imagine you may have heard of the idea
of "crisis" in physics, what it means is
that GR and QM sort of, stop talking to each other,
and gravity isn't in the picture at all.

Then, how to unify those, seems is going to
require just enough super-classical mathematics,
to result that a fall gravity unites with
strong nuclear force, where "QM is a particle/wave
theory usually with a stochastic interpretation
in the fields a field theory", and "GR/SR is space
and kinetic frame with L-principle, Maxwell's fields
and allowed a non-zero while vanishing cosmological
constant, in the fields a field theory", with
conservation about energy, then that it's unified
the particles to make a continuum mechanics, and
diversified the linear and rotational to make
an orbital mechanics, it seems pretty simple
to describe it this way.

Then, that it's, "un-linear", has that lots of
things in "the usual derivation" are linear,
linear, linear, and linear again, with regards
to explaining, pretty much to explaining,
"triangle rule", and, "inverse square",
winding around the clock, the sum-of-histories,
the sum-of-potentials.

Then of course there's interpreting all the
derivations and all the data in that, ...,
though at least it's all already sort of
organized this way.

On 03/17/2024 05:59 PM, Ross Finlayson wrote:
> On 03/17/2024 02:18 PM, LaurenceClarkCrossen wrote:
>> Callahan, in Euclid or Einstein, says,
>> p.222 "he is himself struggling to give, in a hazy way, some kind of
>> reality to his mathematics by clothing his formulae with some
>> interpretation or other....clarity ends, and we step into a region of
>> mistiness and fog. We certainly cannot consider Einstein as one who
>> shines as a scientific discoverer in the domain of physics, but rather
>> as one who in a fuddled sort of way is merely trying to find some
>> meaning for mathematical formulae in which he himself does not believe
>> too strongly..."
>
> That's "Mathematical Physics".
>
> Consider for example the Cartanian developments
> and Geometric Algebra. Didn't exist. Got figured
> out, added to physics, discovered a particle, and
> a remarkable virtual anti-particle at that. (That
> it's so said the contrivance the configuration the
> energy the experiment so found not falsified,
> or so it is said and relayed according to the
> interpretation of the data.)
>
>
> Science is still figuring out there's an
> "upper, middle, and lower" sky, that the
> Babylonians named in hazy antiquity, that
> "peripheral parallax" is still sort of a
> toss-up between Fresnel and Huygens.
>
>
> Now, Einstein is lionized and that means
> beyond even what he deserves, at the same
> time he sort of pushes off from his followers
> writ large, who if they "know" the theory
> probably learned it ten different ways,
> most of which giving absurd consequences,
> that it's also so, that, Einstein was still
> searching for a "total field theory", and
> even just a practical "bridge" from the linear
> to and from the rotational, "Einstein's bridge",
> even out of his later years.
>
> So, these days, the, "nonconservative", the
> "pseudomomentum", the "quasi-invariant",
> the "pseudodifferential", these aren't just
> names for resonance theory and rest-exchange
> and resonance theory and rest-exchange,
> they represent conceptual concepts in the
> mathematics, that most people sort of naively
> have, but don't intuitively formalize.
>
> Or, if they don't, it's like "well, there
> were only three people who know relativity,
> Einstein, Eddington, and nobody knows who
> else, since then at least two of those are
> wrong, so, what it's figured is that Einstein
> knows less now than was then".
>
>
> The usual idea is "physics has three great
> edifices, GR and gravity and QM, somehow
> they're all supposed to be one theory".
> I imagine you may have heard of the idea
> of "crisis" in physics, what it means is
> that GR and QM sort of, stop talking to each other,
> and gravity isn't in the picture at all.
>
> Then, how to unify those, seems is going to
> require just enough super-classical mathematics,
> to result that a fall gravity unites with
> strong nuclear force, where "QM is a particle/wave
> theory usually with a stochastic interpretation
> in the fields a field theory", and "GR/SR is space
> and kinetic frame with L-principle, Maxwell's fields
> and allowed a non-zero while vanishing cosmological
> constant, in the fields a field theory", with
> conservation about energy, then that it's unified
> the particles to make a continuum mechanics, and
> diversified the linear and rotational to make
> an orbital mechanics, it seems pretty simple
> to describe it this way.
>
>
> Then, that it's, "un-linear", has that lots of
> things in "the usual derivation" are linear,
> linear, linear, and linear again, with regards
> to explaining, pretty much to explaining,
> "triangle rule", and, "inverse square",
> winding around the clock, the sum-of-histories,
> the sum-of-potentials.
>
>
> Then of course there's interpreting all the
> derivations and all the data in that, ...,
> though at least it's all already sort of
> organized this way.
>
>
>
>
>
>
>
>
>

https://en.wikipedia.org/wiki/Robert_Hermann_(mathematician)

I have a couple of these books on jet bundles,
they help explain at least there are more modern
formalisms for the same things.

Wikipedia quotes Hermann thusly:

"It is a deeply regrettable fact
that the flow of information back and forth
between "modern" geometric and algebraic mathematics
and classical applied mathematics has been so minimal,
even though there is clearly a solid basis for such
interaction. One of my overall motives in writing my
series of books "Interdisciplinary Mathematics" was
to facilitate this flow...[despite] high structural
and mental barriers to such cross-fertilization."

On 03/17/2024 07:49 PM, Ross Finlayson wrote:
> On 03/17/2024 05:59 PM, Ross Finlayson wrote:
>> On 03/17/2024 02:18 PM, LaurenceClarkCrossen wrote:
>>> Callahan, in Euclid or Einstein, says,
>>> p.222 "he is himself struggling to give, in a hazy way, some kind of
>>> reality to his mathematics by clothing his formulae with some
>>> interpretation or other....clarity ends, and we step into a region of
>>> mistiness and fog. We certainly cannot consider Einstein as one who
>>> shines as a scientific discoverer in the domain of physics, but rather
>>> as one who in a fuddled sort of way is merely trying to find some
>>> meaning for mathematical formulae in which he himself does not believe
>>> too strongly..."
>>
>> That's "Mathematical Physics".
>>
>> Consider for example the Cartanian developments
>> and Geometric Algebra. Didn't exist. Got figured
>> out, added to physics, discovered a particle, and
>> a remarkable virtual anti-particle at that. (That
>> it's so said the contrivance the configuration the
>> energy the experiment so found not falsified,
>> or so it is said and relayed according to the
>> interpretation of the data.)
>>
>>
>> Science is still figuring out there's an
>> "upper, middle, and lower" sky, that the
>> Babylonians named in hazy antiquity, that
>> "peripheral parallax" is still sort of a
>> toss-up between Fresnel and Huygens.
>>
>>
>> Now, Einstein is lionized and that means
>> beyond even what he deserves, at the same
>> time he sort of pushes off from his followers
>> writ large, who if they "know" the theory
>> probably learned it ten different ways,
>> most of which giving absurd consequences,
>> that it's also so, that, Einstein was still
>> searching for a "total field theory", and
>> even just a practical "bridge" from the linear
>> to and from the rotational, "Einstein's bridge",
>> even out of his later years.
>>
>> So, these days, the, "nonconservative", the
>> "pseudomomentum", the "quasi-invariant",
>> the "pseudodifferential", these aren't just
>> names for resonance theory and rest-exchange
>> and resonance theory and rest-exchange,
>> they represent conceptual concepts in the
>> mathematics, that most people sort of naively
>> have, but don't intuitively formalize.
>>
>> Or, if they don't, it's like "well, there
>> were only three people who know relativity,
>> Einstein, Eddington, and nobody knows who
>> else, since then at least two of those are
>> wrong, so, what it's figured is that Einstein
>> knows less now than was then".
>>
>>
>> The usual idea is "physics has three great
>> edifices, GR and gravity and QM, somehow
>> they're all supposed to be one theory".
>> I imagine you may have heard of the idea
>> of "crisis" in physics, what it means is
>> that GR and QM sort of, stop talking to each other,
>> and gravity isn't in the picture at all.
>>
>> Then, how to unify those, seems is going to
>> require just enough super-classical mathematics,
>> to result that a fall gravity unites with
>> strong nuclear force, where "QM is a particle/wave
>> theory usually with a stochastic interpretation
>> in the fields a field theory", and "GR/SR is space
>> and kinetic frame with L-principle, Maxwell's fields
>> and allowed a non-zero while vanishing cosmological
>> constant, in the fields a field theory", with
>> conservation about energy, then that it's unified
>> the particles to make a continuum mechanics, and
>> diversified the linear and rotational to make
>> an orbital mechanics, it seems pretty simple
>> to describe it this way.
>>
>>
>> Then, that it's, "un-linear", has that lots of
>> things in "the usual derivation" are linear,
>> linear, linear, and linear again, with regards
>> to explaining, pretty much to explaining,
>> "triangle rule", and, "inverse square",
>> winding around the clock, the sum-of-histories,
>> the sum-of-potentials.
>>
>>
>> Then of course there's interpreting all the
>> derivations and all the data in that, ...,
>> though at least it's all already sort of
>> organized this way.
>>
>>
>>
>>
>>
>>
>>
>>
>>
>
>
> https://en.wikipedia.org/wiki/Robert_Hermann_(mathematician)
>
>
> I have a couple of these books on jet bundles,
> they help explain at least there are more modern
> formalisms for the same things.
>
> Wikipedia quotes Hermann thusly:
>
> "It is a deeply regrettable fact
> that the flow of information back and forth
> between "modern" geometric and algebraic mathematics
> and classical applied mathematics has been so minimal,
> even though there is clearly a solid basis for such
> interaction. One of my overall motives in writing my
> series of books "Interdisciplinary Mathematics" was
> to facilitate this flow...[despite] high structural
> and mental barriers to such cross-fertilization."
>
>

Basically most all the coolest things that are
ultimate answers in analysis point to Poincare.

https://en.wikipedia.org/wiki/Poincar%C3%A9_half-plane_model

https://en.wikipedia.org/wiki/Angle_of_parallelism
https://en.wikipedia.org/wiki/Ultraparallel_theorem

That's pretty involved, I'm still trying to figure
out singular analysis with "the identity dimension"
in the "descriptive differential dynamics".

Then "Newton's Zero'eth Law(s)" and "Einstein's
attack on same" are rather relevant things when
it results they're all classical and all.

The Poincare "rough" plane basically has the idea
whether the "grain" of the plane is rough or smooth,
that it's Euclidean and it's smooth.

The "Poincare disc" and "Poincare sphere" sort of
sit atop after the Dirichlet problem how that the
harmonic functions, which are usually called in usual
potential theory "the objects of the theory of potentials"
but aren't quite complete, has that the Dirichlet
problem to the Poincare Sphere is one of the usual
highest sort of goals in analysis.

"The Poincaré half-plane model is named after
Henri Poincaré, but it originated with Eugenio
Beltrami who used it, along with the Klein model
and the Poincaré disk model, to show that hyperbolic
geometry was equiconsistent with Euclidean geometry.

This model is conformal...."

https://en.wikipedia.org/wiki/Category:Angle
https://en.wikipedia.org/wiki/Curvature

I ponder the identity dimension what isn't
so much about geometry as original analysis,
where it's sort of so that the identity function
is used as the prime coordinate, or, ordinate,
then though it intends to employ all sorts the
usual analysis of the first quadrant about and
around it. https youtube /@rossfinlayson

Athel Cornish-Bowden <me@yahoo.com> wrote:

> On 2024-03-17 19:58:02 +0000, J. J. Lodder said:
>
> > LaurenceClarkCrossen <clzb93ynxj@att.net> wrote:
> >
> >> What relativist or non-Euclidean geometer has ever directly replied to
> >> Callahan's book/argument?
> >
> > <https://en.wikipedia.org/wiki/Jeremiah_J._Callahan>
> >
> > Duquesne University tries to deny that they ever had such a president,
> > <https://www.duq.edu/about/history/index.php>
> >
> > Jan
>
> Really? "Rev. Jeremiah Joseph Callahan
> President, 1931 - 1940" at the link you give.

Yes, and that is all they want to know about him,

Jan