Cartesian-you are leaving out the last phrase of Tom's question,"at the exact moment in time someone else is doing it on a distant point on earth?" It is a crucial phrase that Alex ignores.
That's the point I was trying to make, but not to Alex, it was for Tom.
This is how I understand the conversation between Tom and Alex. Tom's question was how many people were measuring the angle of the sun. There are two ways of understanding this question.
First. If you consider that question on its own (not in the context of the sun distance) as I thought the question was, then the answer would be a lot (similar to Alex's answer). Sailors (a lot less when GPS became available) measure the angle of the sun using sextant to navigate. They do exactly what Tom asked; measure the angle between the sun and the horizon (to determine their latitude). But then, in the next question, Tom asked Alex to prove that all those people were performing accurate triangulation experiments on the sun with another distant partner simultaneously. This is when the discussion fell apart. Those sailors do (or did) measure the angle of the sun to calculate their position but not to do triangulation experiment with another partner simultaneously.
Second. If you consider that question in the context of the sun distance then the answer would be why would ordinary people measure the sun distance. They leave that job to scientists. Some scientists may do or repeat the experiment to measure the sun distance but not using the method described by Tom (triangulation experiment with another partner simultaneously).
So in both cases, Tom question didn't make sense. At least to me. That's why I asked him whether he manipulated the question probably with an intention to steer this discussion to an area where he is more comfortable with.