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In Early Modern English, how did 'see' semantically shift to mean 'note/record'?

John McWhorter PhD Linguistics (Stanford). Words on the Move (2016). p. 86. Emboldening mine.

  Commonly we are told that Shakespeare's language is "high," such that the challenge can be met by making a certain effort. Related to this is the idea that Shakespeare's language is poetic, requiring more effort to process than the phraseology of Neil Simon. Then someone will say that the language comes across best with careful acting technique, ideally wielded by British people.
  All claims except the one about Brits are true. However, many will be nagged by a feeling that there is more to the story, and there is. When, in Hamlet, Polonius opens his farewell speech to Laertes ("Neither a borrower nor a lender be") with "And these few precepts in thy memory / See thou character," rising to a challenge can take us only so far. We can indeed process precepts, thy, and thou with the aforesaid rising. But what does Polonius mean by character? Neither intonation, facial expression, being British, nor rising will get across that in Shakespeare's time that word meant "write," as in the characters that one writes. Polonius is telling Laertes, in short, "Note these things well."

Please compare the two emboldened imperatives. I can understand how ‘character’ signified writing. But how did ‘see’ semantically shift to signify ‘note/record’? I.e., even after this explanation, this meaning of 'see' still feels bizarre. Can someone please make it more natural?

I tried http://nfs.sparknotes.com/hamlet/page_44.html.