Articles before "covenant"?

The article seems to be left off for several different reasons.

1. Why do we leave off the in "by covenant"? For the same reason that we leave off the in "traveled by train." We're not talking about a specific train; we're talking about a generic means of travel. We can use this construction even if there is a specific train or covenant involved — I can say I traveled by train to New York and be grammatical, even though I clearly took one specific train. Similarly, even though Hobbes is talking about a specific covenant, he drops the article in this sentence because he intends to reference a generic means of becoming king.
2. These are definitions in 17th century English. Hobbes generally uses either uppercase or all capitals for words when he defines them. He seems to be leaving off the article in these definitions, even though we'd put definitely put it in today. You really can't judge 17th century grammar by today's standards. Hobbes seems to be inconsistent about whether he puts in or leaves off the article. Lots of nouns, like duty, can be both uncountable and countable, and it's possible that Hobbes is treating covenant as one of these.
3. Again, as in (1), Hobbes is using covenant generically. Unlike (1), this does seem strange if you evaluate it in terms of today's grammar. Many nouns, like duty, are uncountable when they are used generically, and countable when they are used for specific instance of a duty. I suspect Hobbes is treating covenant as one of these. I don't know whether this is specific to Hobbes, or occurred more broadly in 17th century Englich.
4. Covenant is a countable noun, and you should generally put an article before it. There is nothing at all unusual about this usage, the way there is about (2) and (3).