What is the difference between "amazed at" and "amazed by"?

Good day to all!

Could you please explain me in which cases it's better to use a preposition at and in which we should use by?

• amazed at / amazed by
• astonished at /astonished by

Thank you in advance

Amazed at would be used to describe amazement of a single instance of something.

Amazed by would be more likely to be used as a more general description of amazement.

Julie was amazed/astonished at Ted's impromptu juggling display.

Julie was amazed/astonished by Ted's juggling skills each time he displayed them.

There is a very subtle difference.

The first one would be used to indicate that Julie's amazement is caused by the juggling, whereas in the latter example Ted is causing Julie's amazement.