Did Galileo discover four natural satellites, didn't he? -- No. You can't add a tag question to what is already a question.
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B.J
Galileo discovered four natural satellites, didn't he?
Did Galileo discover four natural satellites?
Didn't Galileo discover four natural satellites?
Are these all the same meaning?
All other things being equal, the first and the third suggest that the speaker expects the statement to be true and is asking for