Can you compare Othello's empricism in asking Iago for "ocular proof" of Desdemona's infidelity in Othello to the empirical and scientific language Galileo uses in his scientific paper The Starry Messenger?
Villain, be sure thou prove my love a whore,
Be sure of it; give me the ocular proof:
Or by the worth of man's eternal soul,
Thou hadst been better have been born a dog
Than answer my waked wrath!
Make me to see't; or, at the least, so prove it,
That the probation bear no hinge nor loop
To hang a doubt on; or woe upon thy life! (III.iii)
In order to know what we are comparing, "empiricism" is a philosophy that asserts that concepts of knowledge are attained through experience alone and is opposite of "rationalism" that asserts some concepts of knowledge are gained through reasoning faculties separately from experience. Thus your question is asking about Othello's philosophical position on gaining concepts of knowledge. Othello's philosophical language, at some points, discloses his position as an empiricist, not as a rationalist (i.e., one who expects to reason their way to knowledge) (Stanford Encyclopedia of Philosophy).
Similarly, yet nonetheless differently, "empirical" is a system of scientific investigation that discovers factual reality through experience and observation alone (Collins Dictionary), though most definitions add through experiment as well. Thus your question is asking about Galileo's scientific methodology in observing the solar system through the spyglass he developed (1609) after learning about Hans Lipperhey 1608 spyglass invention (patented 1609), observations which Galileo made public in his 1610 paper we call The Starry Messenger.
It is clear your question is asking for a comparison between two related though different fields of learning, being philosophy and science. Bear in mind that Othello preceded The Starry Messenger by at least six years having been written between 1601 and 1604 though first published twelve years after in 1622. So though there can be a comparison of the language usages, it cannot be supposed that Shakespeare was in any way alluding to or reflecting Galileo's spyglass or observations in any way, as Galileo had neither invented nor written these things yet.
Othello says only a few things that reveal empiricism. One is the phrase "ocular proof"; another is "prove/proof"; another is "living reason" [i.e., experience]; the other is two iterations of "see":
Make me to see't;...
Now do I see 'tis true. ...
This is ironically tragic because, with the first utterance of "see't" following after "ocular proof," we believe he means physically to see it, yet, with the second utterance of "see 'tis" following after Iago's fabricated word-images, we realize he is yielding to a metaphorical sense of "see" and wish he were a strong empiricist who would not so readily be led into rationalism in which reasoning alone can prove a concept (since Iago has just reasoned Othello into a change of perspective). Thus Othello's language of empiricism is limited and it is weak since it turns to the language of opposing rationalism too quickly as represented in "denoted" "conclusion" "doubt" and "dream":
But this denoted a foregone conclusion:
'Tis a shrewd doubt, though it be but a dream.
Galileo, on the other hand, uses empirical scientific language consistently all throughout from beginning to end:
treatise I propose for observation ... because of the instrument by means of which they have been revealed to our senses (1)
our own eyes show us four stars which wander around Jupiter
as does the moon around the earth (15)
Moreover, Galileo's empirical scientific language is particular unlike Othello's philosophical empiricist language, which is general and limited in breadth. Galileo's explanation of "a method of measuring distances apart" is very particular and detailed and illustrates the contrast with Othello's generalized language:
the rays would reach the object FG along the straight lines ECF and EDG. But when the lenses have been inserted, the rays go along the refracted lines ECH and EDI;