On the northern hemisphere, we can at any location find the direction to Polaris.
From this we can get our latitude by getting the vertical angle from level to Polaris, which should match your latitude location on a map.
On the southern hemisphere, we can do the same as on the northern hemisphere, just using the Sigma Octantis to get our latitude.
At any location on the earth, we can at equinox (when the sun travels straight above the equator), measure the angle from level to the sun at high noon (when the sun is highest in the sky, and directly north or south from your location)
From this we can also get our latitude by negating our result from 90 degrees, which should also match your latitude on a map.
At equinox, you can always find your latitude using the sun at high noon (you need an offset for the tilt at any other time, but this is also easily doable).
At any time, you can always find your latitude using Polaris or Sigma Octantis.
At equinox, we can conclude and observe that the angle between the Polaris and the sun is 90 degrees, anywhere they can be observed on earth.
At equinox, we can conclude and observe that the angle between the Sigma Octantis and the sun is 90 degrees, anywhere they can be observed on earth.
Even when not at the equinox, we can observe that the angles between the sun and the Polaris and Sigma Octantis combine to 180 degrees.
This is the case and can be observed at any location on earth, and will only be possible on a spherical earth.If you can show these observations to be false, you would disprove this argument which clearly indicates a spherical earth.
You are also welcome to try and explain these observations on a flat earth, but I highly doubt you can do this with an explanation that fits all locations on earth.
TLDR;At equinox, on the northern hemisphere, you can measure the elevation-angle to the sun when it is directly south. You can at night also measure the elevation-angle to Polaris.
These angles will be 90 degrees apart from each other at equinox due to being measured north and south, no matter where you are on the northern hemisphere.
In the southern hemisphere, you can do the exact same measurements, but instead you would measure north to the sun and south to Sigma Octantis.
There is no way this phenomenon can occur on a flat earth, but it is easily explained on a spherical earth.
To give you a visual of the difference between flat and globe models:
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