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Here in the U.S., we keep having stupid, ill-informed discussions about our electoral college. In an effort to inject a little information into this, I'd like to take a little time and go through Federalist Paper #68, "The Mode of Electing the President"; this document is attributed to Alexander Hamilton, and presents the Federalists' justification for, well, the mode of electing the president, as originally specified in the then-new Constitution. We'll see that some of the justifications EC supporters like to use are not mentioned, and that most of the benefits claimed do not currently apply to the EC/direct popular election debate.

Before we dig in, a few things worth noting:

• The original system, discussed here, differs in a few regards from the modern system, most notably in that the electors originally each cast one ballot, rather than a ballot for president and a separate ballot for vice-president. As the runner-up became vice-president, this did not work well with the two-party system that emerged (Imagine either Hillary as Trump's VP, or Trump as Hillary's! So each party runs two candidates, but then those two candidates could accidentally tie...), and the Constitution was amended to fix this issue. Also there's a difference in n when the House votes on the top n candidates in case of no EC majority.
  The big difference is not one of law, but of custom -- the Federalists envisioned independent electors being chosen for their wisdom and integrity, and then evaluating the candidates and deciding how to vote. The eventual reality of the party system is that electors are pre-committed (whether by pledge, state law, or merely party loyalty) to vote for their party's candidate, and less than 1% of them break that commitment.
• The Federalist Papers as a whole were an organized campaign to persuade the state of New York to ratify the new Constitution. In some cases, they justify or clarify provisions of the Constitution in general terms (most of #68 is of this nature); in others, they respond to specific criticisms made by others, notably the Anti-Federalist Papers. And the Anti-Federalist Papers likewise contained both general criticisms of the Constitution, and specific rebuttals to some points in the Federalist Papers; this public dialogue went on for some years. So this is really propaganda, and should not be trusted implicitly; however, the EC was not actually controversial.
• While the current discussion is all about EC vs. direct popular vote, it was not so simple then -- alternatives suggested in the Constitutional Convention included election by Congress or by the state governors. Thus some of the points made in favor of the EC, as opposed to these plans, apply just as well (or even better) to direct popular election.

To the People of the State of New York:

THE mode of appointment of the Chief Magistrate of the United States is almost the only part of the system, of any consequence, which has escaped without severe censure, or which has received the slightest mark of approbation from its opponents. The most plausible of these, who has appeared in print, has even deigned to admit that the election of the President is pretty well guarded. [1] I venture somewhat further, and hesitate not to affirm, that if the manner of it be not perfect, it is at least excellent. It unites in an eminent degree all the advantages, the union of which was to be wished for.

1. Vide FEDERAL FARMER.

If you're confused, "the Chief Magistrate of the United States" is a description of the President, and "Federal Farmer" was the pen name of an Anti-Federalist who authored a pair of Anti-Federalist Papers critiquing the Constitution.

It was desirable that the sense of the people should operate in the choice of the person to whom so important a trust was to be confided. This end will be answered by committing the right of making it, not to any preestablished body, but to men chosen by the people for the special purpose, and at the particular conjuncture.

This is one of those arguments that serves direct popular election at least as well.
Inapplicable, or maybe even reversed.

It was equally desirable, that the immediate election should be made by men most capable of analyzing the qualities adapted to the station, and acting under circumstances favorable to deliberation, and to a judicious combination of all the reasons and inducements which were proper to govern their choice. A small number of persons, selected by their fellow-citizens from the general mass, will be most likely to possess the information and discernment requisite to such complicated investigations.

This is rendered irrelevant by the party system -- rather than choosing electors who we trust to assess the candidates and vote on our behalf, we're choosing electors based on their party allegiance, and trusting them to vote for that party's candidate. They do no "analyzing" or "deliberation", and they are not in any meaningful sense "selected by their fellow-citizens from the general mass" -- in fact, their names are not even on the ballot in many states.
Inapplicable because the parties have corrupted the EC's intent.

It was also peculiarly desirable to afford as little opportunity as possible to tumult and disorder. This evil was not least to be dreaded in the election of a magistrate, who was to have so important an agency in the administration of the government as the President of the United States. But the precautions which have been so happily concerted in the system under consideration, promise an effectual security against this mischief. The choice of SEVERAL, to form an intermediate body of electors, will be much less apt to convulse the community with any extraordinary or violent movements, than the choice of ONE who was himself to be the final object of the public wishes. And as the electors, chosen in each State, are to assemble and vote in the State in which they are chosen, this detached and divided situation will expose them much less to heats and ferments, which might be communicated from them to the people, than if they were all to be convened at one time, in one place.

This is actually an argument against direct popular vote. I'm not sure it's true, given our current electoral system, where seeing a few electors flipped one way or the other (Florida 2000, anyone?) is clearly seen as equivalent to flipping the outcome. (Again, it would be different if electors were chosen for their wisdom and integrity, and left to vote their conscience, rather than pre-committed tot their parties' candidate.)

At best, sometimes it will be non-tumultuous because the race wasn't close enough to be flipped -- but then, 50 states are 50 times as likely to produce a near-tie in some state than a single national election. It's not clear that these effects should precisely cancel, but I believe they do in large part.

Applicable, but not (to me) very persuasive.

Nothing was more to be desired than that every practicable obstacle should be opposed to cabal, intrigue, and corruption. These most deadly adversaries of republican government might naturally have been expected to make their approaches from more than one quarter, but chiefly from the desire in foreign powers to gain an improper ascendant in our councils. How could they better gratify this, than by raising a creature of their own to the chief magistracy of the Union? But the convention have guarded against all danger of this sort, with the most provident and judicious attention. They have not made the appointment of the President to depend on any preexisting bodies of men, who might be tampered with beforehand to prostitute their votes; but they have referred it in the first instance to an immediate act of the people of America, to be exerted in the choice of persons for the temporary and sole purpose of making the appointment. And they have excluded from eligibility to this trust, all those who from situation might be suspected of too great devotion to the President in office. No senator, representative, or other person holding a place of trust or profit under the United States, can be of the numbers of the electors. Thus without corrupting the body of the people, the immediate agents in the election will at least enter upon the task free from any sinister bias. Their transient existence, and their detached situation, already taken notice of, afford a satisfactory prospect of their continuing so, to the conclusion of it. The business of corruption, when it is to embrace so considerable a number of men, requires time as well as means. Nor would it be found easy suddenly to embark them, dispersed as they would be over thirteen States, in any combinations founded upon motives, which though they could not properly be denominated corrupt, might yet be of a nature to mislead them from their duty.

This is an argument against election by the legislature or similar body. It also looks like a good argument against the current system. The claimed benefit that electors cannot be pre-corrupted "without corrupting the body of the people" is only valid if electors are chosen from the body of the people -- with the rise of the two-party system, you only need to corrupt twice the number of electors, that is, the R slate of electors, and the D slate of electors, both groups known well in advance. And in many states, you can predict which party wins with good confidence, and only need to corrupt one slate of electors.

As far as the dispersion over "thirteen" (or fifty) states, that certainly seems no obstacle to party loyalty (which, AIUI, is exactly the sort of motive referred to as "not properly ... corrupt, [but] of a nature to mislead them from their duty").

But of course, nobody does buy electors -- it's pointless. Not only would it be transparently obvious (because there are clear expectations who each elector should vote for), but it's also easier to just corrupt the primary process, and see your favored candidate gets the party nomination.
Inapplicable

Another and no less important desideratum was, that the Executive should be independent for his continuance in office on all but the people themselves. He might otherwise be tempted to sacrifice his duty to his complaisance for those whose favor was necessary to the duration of his official consequence. This advantage will also be secured, by making his re-election to depend on a special body of representatives, deputed by the society for the single purpose of making the important choice.

Simple enough. If we were talking about initial election, I'd point out that our political parties represent just such a group "whose favor was necessary", but we're actually talking about reelection, where political parties don't want to give up the incumbent advantage; AFAIK, they make comparatively few demands on a sitting president compared to an up-and-coming politician currently or soon-to-be seeking the nomination. (This might be different without Washington's example of stepping down after two terms, and its eventually codification in term limits.)
Inapplicable

All these advantages will happily combine in the plan devised by the convention; which is, that the people of each State shall choose a number of persons as electors, equal to the number of senators and representatives of such State in the national government, who shall assemble within the State, and vote for some fit person as President. Their votes, thus given, are to be transmitted to the seat of the national government, and the person who may happen to have a majority of the whole number of votes will be the President. But as a majority of the votes might not always happen to centre in one man, and as it might be unsafe to permit less than a majority to be conclusive, it is provided that, in such a contingency, the House of Representatives shall select out of the candidates who shall have the five highest number of votes, the man who in their opinion may be best qualified for the office.

The process of election affords a moral certainty, that the office of President will never fall to the lot of any man who is not in an eminent degree endowed with the requisite qualifications. Talents for low intrigue, and the little arts of popularity, may alone suffice to elevate a man to the first honors in a single State; but it will require other talents, and a different kind of merit, to establish him in the esteem and confidence of the whole Union, or of so considerable a portion of it as would be necessary to make him a successful candidate for the distinguished office of President of the United States. It will not be too strong to say, that there will be a constant probability of seeing the station filled by characters pre-eminent for ability and virtue. And this will be thought no inconsiderable recommendation of the Constitution, by those who are able to estimate the share which the executive in every government must necessarily have in its good or ill administration. Though we cannot acquiesce in the political heresy of the poet who says: "For forms of government let fools contest That which is best administered is best," yet we may safely pronounce, that the true test of a good government is its aptitude and tendency to produce a good administration.

This is a conclusion, rather than an argument, but it's the closest we get to the idea espoused by some EC supporters, that the Founding Fathers intentionally picked a method requiring a President to win a majority (however slim) in many states, rather than a huge majority in a few states. In point of fact, no such concern is established here -- the concern that Hamilton believes has been defeated is about whether the methods that could make a politician well-known and liked in one state would allow such a politician to win nationally, not that there was ever any concern of a candidate winning only one (or a few) states, and yet winning presidency. The phrase "so considerable a portion of [the Union] as would be necessary to make him a successful candidate", in contrast to a single state, could apply just as well to "a majority of voters" as to "a sufficient number of states to have a majority in the EC", and thus this same statement could have been made, if the Constitution had implemented direct popular election.

But speaking of a politician becoming known and well-liked in one state (or a few) by "the little arts of popularity", and somehow taking the presidency as a result -- doesn't that sound like exactly what all the candidates are trying for when they descend on the early-primary states, hoping to get "momentum" from a win there, and ride it to the party's nomination?
Again, the electoral college could be a good thing, stopping such a candidate's momentum cold in favor of "characters pre-eminent for ability and virtue", but the two-party system means electors are first chosen by their party for loyalty to that party, and then, even if suddenly overwhelmed by a sense of personal duty, have little choice but to hold their nose and vote for their own candidate (low character, but presumably on the "right" side of the issues) or to vote for the opposite party (even if better character, still on the "wrong" side) -- right or wrong, you're a fool to bet on the latter.

The Vice-President is to be chosen in the same manner with the President; with this difference, that the Senate is to do, in respect to the former, what is to be done by the House of Representatives, in respect to the latter.

The appointment of an extraordinary person, as Vice-President, has been objected to as superfluous, if not mischievous. It has been alleged, that it would have been preferable to have authorized the Senate to elect out of their own body an officer answering that description. But two considerations seem to justify the ideas of the convention in this respect. One is, that to secure at all times the possibility of a definite resolution of the body, it is necessary that the President should have only a casting vote. And to take the senator of any State from his seat as senator, to place him in that of President of the Senate, would be to exchange, in regard to the State from which he came, a constant for a contingent vote. The other consideration is, that as the Vice-President may occasionally become a substitute for the President, in the supreme executive magistracy, all the reasons which recommend the mode of election prescribed for the one, apply with great if not with equal force to the manner of appointing the other. It is remarkable that in this, as in most other instances, the objection which is made would lie against the constitution of this State. We have a Lieutenant-Governor, chosen by the people at large, who presides in the Senate, and is the constitutional substitute for the Governor, in casualties similar to those which would authorize the Vice-President to exercise the authorities and discharge the duties of the President.

Well, nobody today seems to want to abolish the Vice-Presidency, so I guess we don't need to get into that bit...

PUBLIUS

And done!

Finally, there's an argument that's not (AFAIK) stated by the Federalists with regards to the EC, but is often mentioned by EC, and not totally without basis. Namely, that the EC strikes a balance between giving each state equal say, and giving each person equal say; as each state's number of electors is equal to their representation in Congress, i.e. 2 for their 2 Senators, plus as many as their Representatives (proportional to population). On one hand, this isn't wrong -- while it's not mentioned (again, AFAIK) for the EC, they certainly didn't follow the same compromise used for Congress by accident. Other proposals were contemplated, and they must have chosen this for the same reason.

But on the other hand the reason the Senate has equal representation per state and the House has equal representation per person was not because that's the one true logical way to run an entity that's half a confederacy of coequal states and half a nation of coequal citizens. It was because the low-population states would have rejected a Constitution calling for both houses of Congress to be proportional to population, and the high-population states would have rejected one calling for both houses to be equal per-state.

Likewise, the use of the same numbers for electors was not for ideological reasons, but because that was the compromise that everyone could live with. And I, for one, have no problem revisiting old compromises -- as long as everyone can live with whatever setup we might replace the EC with, it doesn't matter that it's not the same as it was, 2 centuries and 37 states ago. But if we choose to hold the old compromise sacred, it's simple enough to establish a weighted popular vote, where votes in each state are weighted differently. This keeps the extra impact of small states, but still removes the EC, and still lets voters in non-swing states have an effect on the election. So this argument, valid or not, is no argument for the EC, only against uniformly-weighted popular vote.

(BTW, I've gotten into historical stuff I've not studied in any depth here -- if the simplified version I've no doubt been taught is misleading here, I'd welcome correction.)

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Now maybe you find that third argument more persuasive than I do, and support the EC because you fear a popular vote will cause rioting. If so, cool -- at least you have a reason.

I don't find it persuasive, but I'm still weakly opposed to abolishing the EC. Why?

1. I don't see it as a real problem. The parties campaign knowing the rules. Change the rules, and you'll change how they campaign -- so it's invalid to look at an election where both parties were campaigning for the EC, and use the popular vote totals to say the "wrong" candidate won, or that the election would really have turned out different if we had abolished the EC ahead of time. Of course such a change would inevitably flip some races, but I'm not convinced it does so in any systematic or unfairly exploitable way -- If I was, of course I'd support ditching it.
2. Because some people (usually including whichever political party most recently lost the EC while winning the popular vote) really want to get rid of it, and I really want to use a voting system that does not create and enforce a 2-party system^* (unsurprisingly, the 2 parties benefiting from the current system don't want this change), so I'd rather keep the EC until we can use it as leverage to do both at once. Secondarily, because it's a compromise everyone is used to -- once we start trying to change it, we may have a bigger project than we bargained for getting everyone to agree on a replacement. (Though maybe states have diminished enough it won't be a big deal.)

^*I really don't have time to go in-depth here, but briefly: range/score, approval, and probably some Condorcet methods (ones supporting equal-ranking) are all good in this regard, while FPP (our current system), IRV (sometimes misleadingly called "ranked-choice voting", and an inexplicably popular idea to "fix" our voting system), and such Condorcet methods as forbid equal-ranking, all support 2-party lock-in.

• I personally prefer score, but am willing to accept approval -- both simple, elegant, linear systems that voters can have complete confidence in.
• Condorcet methods can be "good enough", but they're terribly complicated, and thus unsuitable -- if regular voters can't understand it, they can't have confidence in it, and that's a disqualifier.
• IRV is the worst of all worlds -- 2-party lock-in, plus horrible non-linear behavior, plus too complicated for voter confidence, plus a nightmarish proclivity for ties due to the recursive way it's counted.
  Oh, and that nonlinearity? it means IRV can't be counted until you have every ballot in the USA tabulated (including absentee ballots that don't arrive till days after election day) -- add 1 more ballot, and you have to do a full recalculation, because eliminating candidates in a different order can totally change the outcome. In contrast, FPP, score, and approval all let you count the votes you have, determine the current margin of victory, know how many ballots would be required to change the outcome, and keep on adding new ballots to the count as they arrive; this also means you count votes in your state/county/precinct, I count votes in mine, and we simply add our results to get the total.