What is the meaning of "on" in the following sentence,

Question

What is the meaning of "on" in the following sentence,

Einstein arrived one day at Brussels on a visit to the Belgian Queen.

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Does it mean "Einstein arrived one day at Brussels for the purpose of a visit to the Belgian Queen" ?

Does "on" in the sentence mean "for the purpose of" ?

If so, Can we replace "He went to Italy for the purpose of studying music" with "He went to Italy on studying music" ?

Can we replace "I came to Paris with the view of studying painting" with "I came to Paris on studying painting" ?

If so, What is the difference in the nuance of the meaning between "on" and "for the purpose of" ?

Answer 1

According to the Cambridge Dictionary, there are over twenty different meanings of on: "for the purpose of" is not one of them. The most appropriate is

used to show that a condition or process is being experienced

In this case, the process that is being experienced is "a visit to the Belgian queen". Another example might be:

I met Dr Sykes on a walking tour of Norway

This means that I was in the process of a walking tour of Norway, when I met Dr Sykes.

We also use on for holidays:

I'm afraid Dr Sykes is on holiday at the moment
I met Dr Sykes while I was on holiday in Norway

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on can never mean "for the purpose of". The most widely used expression for "for the purpose of" is to. The Cambridge dictionary offers this meaning:

used with an infinitive to express use or purpose

Note that to must be used with an infinitive. Here are the two examples that you suggested, adapted to use to.

He went to Italy to study music
I came to Paris to study painting

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Note that we normally use at for buildings (a supermarket, a football stadium, a house, a gate) but we use in for cities.

Einstein arrived one day in Brussels on a visit to the Belgian Queen.

Answer 2

In the example sentence:

Einstein arrived one day at Brussels on a visit to the Belgian Queen.

"on" could mean "for the purpose of" or "during the course of", more likely the latter. This is a subtle difference, and in many cases it won't matter much. However, this does not mean that "on" can always be substituted for "for the purpose of" -- it depends on the construction.

"on a visit" like "on a trip" is something of a set phrase, generally meaning "during the course of" or "in the process of" a visit or trip. Or it can otherwise refer to a trip/visit as a whole:

They went on a trip to Europe.

In regard to the other suggested sentences:

"On" does not work in the example:

He went to Italy for the purpose of studying music.

This is because " a visit to the Belgian Queen" is a noun phrase which can be the object of the preposition "on" but "studying music" is not. This could be shortened to:

He went to Italy to study music.

The third example

I came to Paris with the view of studying painting.

is at best awkward. "intent" or "plan" works much better than 'view" here. Again, "studying painting" is not a NP, and the "on" form cannot be used without significant recasting.