There are two things that men should never weary of, goodness and humility; we get none too much of them in this rough world among cold, proud people.
Is this the same as "It should be sunny tomorrow?"
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Should is a modal auxiliary verb, which is a very special category of auxiliary verb in English.
can, could, shall, should, may, might, will, would, must, and sometimes need and dare
a deductive or inductive logical possibility or necessity, called the Epistemic meaning.
a moral permission or necessity, originating from natural or social forces, called Deontic.
The epistemic sense of should is something like "likely by induction", not all that different from the "likely by deduction" of epistemic must. Both are used to announce hypotheses derived from theory or experience.
The deontic sense of should is something like "mild obligation", contrasting with the "strong obligation" of deontic must. It's usually a matter of reminder and nagging, rather than order and obligation.
Reminder: All modals have two senses, not just should. They're easy to tell apart once you learn.
The first statement is more like a moral standpoint, in the sense what,in an ideal state, should or should not happen.Something like a commandment.
One should not steal.
The second statement is more like an educated guess:
It should be sunny tomorrow.
would read something like:
*Given the weather conditions today, I assume it would be sunny tomorrow.
(Given my performance), I should clear the interview.*
Both should never weary and should be sunny can be read as predictions of future events, based on past observation of men and weather.
There is a subtle difference why each should play out as predicted.
I read the author as arguing there is value in men not wearying, perhaps for their own good. The author is hopeful or even commanding for a specific future event. This is distinct from men being likely to not worry, as the weather is likely to be sunny in the future, based on evidence.