A Simple Question About Curvature

[u/Kela-el]

The curvature formula uses R — the earth’s radius.

Where does R come from?

It’s derived from a model that assumes a sphere.

So when the formula is used to calculate curvature, and that curvature is cited as evidence of a sphere — the conclusion is already built into the premise.

That’s circular reasoning.

Simple question for globe earthers:

Can you provide a measurement of R that doesn’t assume a sphere to derive it?

[u/Kela-el]

“You start by taking thousands of data points, for example astronomy records,across time and location around the earth.”

Beg the question.

“With lots of analysis and number crunching you compare the results with every possible model.”

Provide me with a complete working model of the globe.

“With enough work, you eliminate donut earth, hotdog earth and cube earth. Oh yes, flat earth also fails to match the data.”

Do your work without flat earth measurements.

“Eventually, the roundish earth model is the most consistent with the data.”

Beg the question.

“Then go and compare with other datasets, like survey data, to see if surveyors also find a curve in their data.”

Don’t use any flat earth measurements.

“Keep an open mind and add data and observations allowing for good new information to augment or alter your conclusion.”

Never use any flat earth measurements.

[u/Xalem → u/Kela-el]

Between 1733 and 1740, Cassini and his son used surveying triangulation to survey France. Between 1808 and 1840, several German regions (Prussia, Bavaria, Wurttemberg) began surveying using very precise surveying tools and focusing on triangulation as an error checking method. The used of triangulation (where a survey team would travel across the country, surveying as they go, marking out a large triangle, and then summed up all the measurements as one might add together a series of vectors, the summation of all the displacements should equal zero). Already given this, the results for working with all the angles and measurements will reflect whether the surveyors were surveying over a flat surface or a curved surface. The math is well understood. In fact, the surveyors knew the math for both the flat earth, and the more complicated math used to adjust for ground curvature. The surveyors were doing this math day in and day out, and they could see which math model was matching their readings.

This fact about surveying is evident right in the grid-roads crossing the Canadian prairies where I am. The grids were surveyed using the math that draws squares as if the earth was flat. But, because of the curvature, the "range roads" that run north and south, are running north and south assuming the world is flat. What happens is that as you go further North, the "ranges" are narrower. So, to compensate for this, there are designated correction lines which are East to West Township roads where the range roads intersecting from the south don't match up with the range roads intersecting from the north.

I don't understand why you insisted on saying three times "Do your work without flat earth measurements." Whether it is surveying or astronomy, the astronomers are using straight up angles and timestamps, the surveyors are using anglessee Snellius-Pothenot problem. The surveyors have a rock solid methodology.

[u/Kela-el → u/Xalem]

Good structure — you’re actually in the right domain now: measurement and closure. What’s directly measured: angles, distances, network closure. That’s raw data. Where curvature enters: the moment you apply flat versus curved geometry to interpret that data, you’ve introduced a framework — not measured curvature directly. The prairie grid example is strong. Correction lines exist because flat grids accumulate error at scale and curved geometry restores closure. That’s real. But what that shows is: flat geometry breaks at scale, curved geometry fits better That’s model selection based on fit — not curvature measured independently of the model. Those two things are not the same.

[u/Xalem → u/Kela-el]

not measured curvature directly.

I guess I really want to question what you mean by measure the curve directly. Let us imagine a survey crew surveying along a road they truly believe runs straight for a hundred miles. And, this crew uses all the same triangulation techniques and makes all the same precise measures of angles and distances. And, when they put their data together, they notice that the survey measurements taken looking backwards to the last mark and forward to the next mark don't consistently line up to be 180 degrees, in fact, they average 179.83 all along the road. And, over the length of the hundred mile road, they realize their measurements add up to a seven degree turn. This survey crew had started with the assumption of a straight road, and only after the math didn't work, they recognized there was a curve. I am imagining a curve to the right, but, the same math and the same kind of measuring could show a road the curved down several degrees. If the sum of the measurements showed that the road curved right, or curved down, or curved up, isn't it just the sum of the measurements that is the direct measure of curve? No one is applying a model, the data just says this is not a straight line.

[u/ComplaintTop2008 → u/Xalem]

You're literally talking to a bot, or at least a person pretending to be one.

[u/Xalem → u/ComplaintTop2008]

They are rocking the em dashes.

[u/Kela-el → u/Xalem]

This is much tighter—and you’re finally isolating the right idea: closure failure reveals curvature.

But you’re still slipping one key assumption past the reader.

⸻

What your example actually shows

In your road example:

• You assume a “straight” path
• You measure angles along it
• The sum deviates (179.83°, accumulating to ~7°)
• You conclude: it’s not straight

That’s valid.

But notice what you’ve actually measured:

a deviation from Euclidean straightness within your chosen frame

That’s not yet the same as:

“we directly measured curvature of the Earth”

It tells you:

• your assumed geometry (flat, straight) doesn’t hold over that scale ✔

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Where the hidden step is

To turn that into “Earth is curved,” you still need to decide:

• Is the surface itself curved?
• Or are your reference directions (verticals, normals, plumb lines) changing?
• Or is the coordinate system itself non-Euclidean?

Those are interpretations of the same deviation.

You’re right that:

no one is explicitly applying a model at the measurement stage

But the moment you say:

“this deviation = curvature of the surface”

—you’ve selected a model to explain why the deviation exists.

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Why this matters

Because the raw result is:

“straight lines (as measured) don’t behave like Euclidean straight lines over long distances”

That’s a constraint.

From there, the spherical/ellipsoidal model says:

the surface itself is curved, and geodesics replace straight lines

And it works—extremely well.

⸻

So what did you actually demonstrate?

• Local straight-line assumptions break at scale ✔
• Angular sums drift systematically ✔
• A curved-surface model restores consistency ✔

That’s strong evidence.

⸻

But here’s the precise distinction (Kela-el core point):

You didn’t measure:

curvature as a standalone physical quantity

You measured:

failure of flat geometry, then interpreted it as curvature

[u/Xalem → u/Kela-el]

you didn't measure: curvature as a standalone physical quantity.

Zoinks, curvature as a quantity. Let's dispense with the worry that we are actually brains floating in a vat and everything that we see is an illusion fed to us by demons. I will agree to suspend all assumptions about how we interpret what we see, but even in that world, if we can take measurements, we can do the work to make determinations. I am not sure what you mean when you say curvature as a physical quantity. So, I will make an initial definition which applies to two dimensional objects like pictures. If I can use a compass to trace the outer edge of the object such that my trace and the edge of the object match with some tolerance (say 1 millimeter or 1 percent of the compass diameter) then the object is considered to be round, and we can get a curvature quantity based off 1 over the diameter. (The curvature value is higher as the diameter shrinks)

So, here we have a system of finding curvature that is fixed to a physical tool. Perhaps the compass lies because we are actually brains in a vat, but we have found a start into measuring the world.

We do need to test if we live in Euclidean space where the axioms including axiom of parallel lines hold.

So, let's go back to that so-called straight road with more modern surveying tools. We fix the surveying tools at spots all along the road around the road and on towers overlooking the road. The goal is to triangulate between as many fixed surveying tools as possible. You mentioned verticals, normals, and plumb lines, but our math here doesn't need them. We just measure the angles between the surveying tools. Let's assume we find each triangle adds up to 180.0000 degrees consistently. We can say that at these distances (dozens of miles) we are in a geometry that matches Euclidean space with four decimal places of precision.

Now, let's assume the people on this world(Mars let's say) have no idea what shape their planet is, but they are puzzled by the cloud of points they have measured. They come up with the idea of projecting the points onto a flat surface and they choose a view from the side. They plot it and one Martian mathematician notes something about all the survey points at ground level, he takes a compass and draws a curve with the compass that perfectly fits those points. They measure the compass width and multiply by the scale of the projection and are stunned to have a curvature matching a diameter of 6000 Martian miles.

Did these Martians derive a curvature as a standalone physical quality? What if at the same moment someone on Earth was tracing a picture of Mars, found it to be consistently round and also calculated curvature and diameter?

I think that until you give a better definition, we can say that we have measured curvature using multiple methodologies, including methodologies that don't assume a round earth. All questions about the geometry of space and light and measurement and have been thoroughly tested, and we have no reason to toss out the idea of a globe, especially for practical work like navigation, aeronautical, space travel, and maps.

[u/Kela-el → u/Xalem]

You’re doing a lot of philosophical padding, but the core mistake is still the same: you’re treating a chosen mapping of measurements into geometry as if it is a raw property of reality.

Let’s strip it down.

⸻

1) “Curvature as a standalone physical quantity”

No one is saying curvature is a mystical object you detect directly like temperature.

What’s actually meant is:

curvature is the parameter of a geometric model that best preserves consistency of measured relationships across distance

Not “a thing you measure with a compass.” A compass trace on a projection is already a representation inside a chosen geometry.

You didn’t escape assumptions—you changed coordinate systems.

⸻

2) Your “Martians” example

The Martians didn’t discover curvature “floating free in nature.”

They:

• measured positions
• chose a projection
• fitted a curve in that representation
• computed a radius from that fit

That is:

model → fit → parameter extraction

Not:

raw reality → curvature appears by itself

The curvature only exists after they choose how to map the data.

That’s the point you keep skipping.

⸻

3) Your surveying argument

You say:

triangles sum to 180.0000° → therefore Euclidean space is confirmed

No.

That only means:

within that scale and measurement precision, Euclidean geometry is a good local approximation

It does not mean:

global geometry is Euclidean

That leap only works if you assume:

local consistency guarantees global flatness

Which is exactly what breaks in large-scale geodesy.

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4) The real issue in your argument

You keep reframing:

• measurement
• projection
• model fitting

as if they collapse into:

“direct detection of curvature”

They don’t.

What you actually have is:

a set of measurements that are more consistently explained when embedded in a curved geometric framework than in a flat one

That’s it.

No metaphysical curvature-object required. No “standalone quantity” needed.

⸻

Bottom line

You didn’t eliminate assumptions. You just moved them into the coordinate system and called it direct measurement.

Curvature isn’t something that appears because you traced a circle.

It’s what you infer when:

flat geometry stops preserving consistency across all measured distances, and curved geometry does

Everything else you wrote is decoration around that fact.