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.]