In a work, when you introduce someone by their full name and later refer to them in a context which is not appropriate for a pronoun, do you use their first or last name?
Example: "Eli Whitney is credited with inventing the concept of interchangeable parts. [...] Out of this contract, (Eli or Whitney) developed a sample of guns which he then presented to the United States Congress"
Generally, with names in the Western world that consist of a given name ("first name") and a surname, the surname is used for formal occasions, and the given name is used only in cases of familiarity. Thus in your sentence you'd say "Out of this contract, Whitney developed…".
You would use "Eli" only if you wanted it to appear informal and suggest that you were on a "first-name basis" with Mr. Whitney — knew him intimately — and possibly so was your audience. (E.g. you'd use it if you were toasting your friend "Eli" among an audience of his friends.)
Incidentally, when talking to people, there's a greater assumption of familiarity—you can use the given name in more occasions—in America than in Europe (and in younger people than in older), where using the given name indiscriminately can cause offence or irritate. In general, it's always safe to use the surname, until you're asked to use the given name.
[Caveat: These naming conventions, however, are far from universal. In China, for instance, it's customary to put the surname/family name first, and the given name later. It's the same way in some European countries, I think. Also, many Indian (especially South Indian) names do not have a surname, and consist of just a (given) name followed or preceded by an initial letter (or two) that stands for the given name of one's father (and possibly a town). Some people, forced by the demands of Western convention to have a surname, expand that letter and put their father's name as their last name, in which case if you used "Mr. [last name]", you'd be addressing their father.]
@ShreevatsaR's comment about the level of formality/familiarity is pretty much right for Western names:
"Mr. [Jonathan] Purdy" > "Mr. Purdy" > "Purdy" > "Jonathan" > "Jon".
In an essay or book which recounts historical facts, as in the question, I'd suggest going with the last name. It's not too formal that it sounds stiff, not too personal that it sounds disrespectful. However, there are some cases where you should use the first name; such as a biography of the Wright brothers, since they share a last name it would be confusing to read "Wright handed the letter to Wright...".
In a novel or other similar work it's often the case that characters are referred to by their first names. However sometimes the author mixes and matches; this is one of the things that annoys me about the Malazan series. Most characters with first and last names are referred to by their last name, except a few who are referred to by their first names. Since all these names are unusual made-up names it's hard to remember sometimes if this is a first name or last name. So try to be consistent.
If you're unsure of what's appropriate, using the full name when you need to refer to the name and he/she when you can should probably be polite. (Unless there are cultures where using the whole name is considered impolite).
If it's someone famous, you'd probably be safe using the name, title, or nickname/professional name/pseudonym they're usually known by. The musician Sting would be referred to as "Sting" since that's what he use, a monarch often is referred to by their title name and number and just the title when introduced in the text. If a person is known by a quite short name, such as Bill Gates it's probably best to refer to them as using the full name, if they're know'n by a longer name, especially if the "last name" (family name/place name/patronym etc.) is rather recognizable in it self, I'd use just the "last name".
If the work in question were a news article or an essay, I would use both the first name and last name of a person for every other reference after the first reference to the person. If I were to use a direct quotation in which a person referenced earlier is called by only his or her first name or last name, I would add the first name or last name to the direct quotation between brackets.
In a work- as referring to a formal document like a book / report?This agreement entered between Eli Whitney hereinafter referred to as the Guarantor and ....and then for the rest of the document, they will just refer to Eli Whitney asThe Guarantor. Thats getting rid of the problem at the root itself :-)