The best proof of 'not a ball of the stated size' are visual. (or light doesn't travel in straight lines)
So, you take the govern-cement's website data on how far away certain light houses can be seen. Do the math, and find that they should be well over the horizon.
And, on a clear, still night, you can see these light houses from further.
Then their are the photographs you see, taken on a rare chance of clear, still days. Of cities that are so far, they should be over the horizon.
Today, there are many people taking such photos with IR cameras on the regular.
Other things, that are harder to prove, but are way more relevant is that there is too much land area below the equator. You measure points across the land mass... and they are longer than they should be measured on a globe.
One popular scientist who was confronted with this fact said "The earth is pear shaped"
It can be bent.
When I have seen people "show their math," they use the wrong formulae, usually a parabolic formula to disprove a spheroid. It doesn't add up because it literally can't.
Was it a mirage? These occasions are notable for their rarity and the wavering vertical distortion they exhibit based on what I have seen when such evidence is presented. Sometimes objects may even seems to float or look upside-down, because mirages are weird and atmospheric refraction is a thing.
[citation needed]
Slightly pear-shaped. Very slightly. A minutely lopsided oblate spheroid. Not a literal pear. I'm no fan of NdGT, but quoting him out of context is dishonest.
I have been to the Pacific Ocean, Lake Superior, and the large lakes in North Idaho. My own admittedly limited observations reflect the predictions of the sphere model. One of my vague plans involves visiting the northeast shore of Lake Pend Oreille near Owens Bay and looking SSW from the shore. What would you predict I should see?
In case you're curious about where some of these claims originally came from, it will interest you that Giovanni Cassini thought that the Earth was egg-shaped; he came into conflict with Isaac Newton, who maintained that the Earth bulged out slightly at the equator because of angular inertia. Because Cassini was a cartographer, his claim still holds some sway even now, despite observations overwhelmingly confirming Newton's hypothesis.