Washington really uses all three forms of rhetoric, including pathos (as most good politicians do), but the one he leans on the most heavily is ethos—with a side of logos.
The fact that Washington dispenses advice while stepping down from the presidency is a pretty clear sign that his role as president is significant. He plays it off modestly, but despite claiming inadequacy, he keeps on dishing out the lessons.
Obviously everyone knew that he was president, but Washington also gently reminds them that he had wanted to retire earlier, but "mature reflection on the then perplexed and critical posture of our affairs with foreign nations, and the unanimous advice of persons entitled to my confidence, impelled me to abandon the idea" (3.3).
In other words, lots of people wanted him to stay on as president, so that means he was doing something right. So, you should listen to him.
When he transitions to his giant advice column, he really lays on the ethos:
But a solicitude for your welfare […] and the apprehension of danger, natural to that solicitude, urge me […] to offer to your solemn contemplation, and to recommend to your frequent review, some sentiments which are the result of much reflection, of no inconsiderable observation, and which appear to me all-important to the permanency of your felicity as a people. These will be offered to you with the more freedom, as you can only see in them the disinterested warnings of a parting friend […]. Nor can I forget, as an encouragement to it, your indulgent reception of my sentiments on a former and not dissimilar occasion. (7.2-4)
This paragraph does several things. It tells his audience that he (while being president) has really given this a lot of thought. It flatters the American people by calling them his friends, which makes him look a bit more egalitarian. Finally, he reminds them that they've listened to his advice in the past (again, while he was president…and also a war hero). They wouldn't have done that if he weren't important.
He also helpfully reminds the audience:
This government, the offspring of our own choice […] has a just claim to your confidence and your support. Respect for its authority, compliance with its laws, acquiescence in its measures, are duties enjoined by the fundamental maxims of true liberty. (16.4-5).
That's a not-so-subtle hint that people should listen to the government that they, ahem, chose.
At the same time, Washington doesn't just offer up words of wisdom without some logic to back it up. For example, when he discusses staying away from alliances with foreign nations, he lays it out like this:
Antipathy in one nation against another disposes each more readily to offer insult and injury […] and to be haughty and intractable, when accidental or trifling occasions of dispute occur. (32.4)
That's a pretty solid reason for being wary of choosing sides.
Similarly, political parties are problematic because the "spirit of encroachment tends to consolidate the powers of all the departments in one, and thus to create, whatever the form of government, a real despotism" (26.2). It's just that he doesn't like factions and party politics; he sees a real danger in that based on human behavior and past experience.
Washington's farewell address definitely wouldn't hold the same weight nor would it be so well remembered if he weren't the legendary George Washington—war hero and father of our country. His importance in early American political history makes his advice worth listening to.
Being the upstanding gentleman that he is, though, he only offers up advice that he can support with some solid logical thinking.
Washington's farewell address is a bit of a strange one—it's written like a speech, but he never presented it publicly. It was always meant to be published in print. So, the structure mimics a speech, but it's presented in essay format.
The text transitions from subject to subject, but not with dividing headers or the same kind of transition sentences your English teachers taught you to use in your essays. The transitions are sometimes a little abrupt as he changes topics, without any written division between sections.
Also, the way the paragraphs are broken up almost feels like that's where he would have paused if he were speaking the words. And yet, he never did. It's also a bit too fluid and consistently eloquent to be a speech since those generally take the live nature of the event into account.
Granted, old-timey speeches were generally pretty sophisticated. Overall, it's really a hybrid, but since it was published in a written format, we'll have to call it an essay.
Washington gets some business out of the way right at the start—telling everyone he's stepping down from the presidency and retiring. He throws in some complimentary stuff about the American people and some self-deprecating moments where he claims he's really not that great.
The first big section of advice Washington dishes out is about the rise of factions and political parties in the United States and how people should resist it. He talks about how it leads to tyranny and despotism, and how liberty and the strength of the country will be protected by unity, not division.
One of the divisions in the country in 1796 was about which foreign country the United States should be allied with. Washington talks at length about how the United States shouldn't form allegiances with one foreign country over another because that leads to foreign entanglements that will threaten the nation.
Washington ends in a similar way to his start, but with even more flattery to the American people and self-deprecation. He just hopes that some of his advice is intelligent enough that some people listen to it.
Washington spends most of his farewell address giving the American people some sage advice, despite his self-professed incompetence. That means he sees something is wrong and thinks he's got some good ideas, but he also isn't brazenly shouting them from the rooftops. He always tempers the advice with solid logic and a bit of flattery toward his audience.
That flattery also serves as a way to make his American audience feel that they are fully capable of rising to the occasion and doing what Washington asks. Otherwise, why bother trying?
Take, for example, the moment where he transitions into talking about parties and factions:
The unity of government which constitutes you one people is also now dear to you. It is justly so, for it is a main pillar in the edifice of your real independence, the support of your tranquility at home, your peace abroad; of your safety; of your prosperity; of that very liberty which you so highly prize. But as it is easy to foresee that, from different causes and from different quarters, much pains will be taken, many artifices employed to weaken in your minds the conviction of this truth. (9.1-3)
Look how great the government (which you voted for) is—but also, it's at risk. So, they should be wary but also know that they have something worth protecting. That theme continues throughout the address, like when he talks about foreign alliances:
Excessive partiality for one foreign nation and excessive dislike of another cause those whom they actuate to see danger only on one side, and serve to veil and even second the arts of influence on the other. Real patriots who may resist the intrigues of the favorite are liable to become suspected and odious, while its tools and dupes usurp the applause and confidence of the people, to surrender their interests. (35.3-4)
He confidently tells his audience the risks but gives options for "real patriots" to stop the worst from happening. When there's a will, there's a way!
George Washington, Alexander Hamilton, and James Madison were not ones for short, direct sentences. They tend to go for eloquence, which can mean some very, very long sentences. We're talking entire paragraphs.
Take this one, for instance:
I beg you, at the same time, to do me the justice to be assured that this resolution has not been taken without a strict regard to all the considerations appertaining to the relation which binds a dutiful citizen to his country; and that in withdrawing the tender of service, which silence in my situation might imply, I am influenced by no diminution of zeal for your future interest, no deficiency of grateful respect for your past kindness, but am supported by a full conviction that the step is compatible with both. (2.1)
Translation: Don't think I made this decision without thinking of your welfare, and I'm confident your future is still secure.
Washington also doesn't skimp on the ideas. He goes on at length about each of his pieces of advice, laying out a number of reasons why he's saying what he's saying. For example, on the issue of factions, he starts off by reminding people:
With slight shades of difference, you have the same religion, manners, habits, and political principles. You have in a common cause fought and triumphed together; the independence and liberty you possess are the work of joint counsels, and joint efforts of common dangers, sufferings, and successes. (10.4-5)
But he doesn't stop there. He outlines how every part of the country benefits from being connected to the others, to show how "every part of our country thus feels an immediate and particular interest in union" (13.1). Plus:
One of the expedients of party to acquire influence within particular districts is to misrepresent the opinions and aims of other districts. (15.2)
He even brings up the recent treaty with Spain as "decisive proof how unfounded were the suspicions propagated among them of a policy in the General Government and in the Atlantic States unfriendly to their interests" (15.4). And for a little scare: "and sooner or later the chief of some prevailing faction, more able or more fortunate than his competitors, turns this disposition to the purposes of his own elevation, on the ruins of public liberty" (22.3).
This anti-faction section of the speech goes on for about 20 paragraphs. He probably could have shortened that a bit, but one has to cover all their bases.
The title here is pretty straightforward. The address was published, but it's written like a speech, and speeches didn't go for punchy or catchy titles in the colonial era.
The main thing about the title is that it says up front what the purpose of the speech is: Washington wants to say goodbye. Yeah, he says a lot of other stuff, but he's saying it because he's stepping down and wants to pass on some advice before he goes. The fact that it's a goodbye speech is pretty crucial to the historical context.
Sometimes you don't need to find a good pun or a famous song lyric to create a clever title. Sometimes you just tell it like it is.
Washington begins his farewell address by getting the primary business out of the way: telling people he's not going to be president again. It's a long sentence, but it gets the job done:
The period for a new election of a citizen to administer the executive government of the United States being not far distant, and the time actually arrived when your thoughts must be employed in designating the person who is to be clothed with that important trust, it appears to me proper, especially as it may conduce to a more distinct expression of the public voice, that I should now apprise you of the resolution I have formed, to decline being considered among the number of those out of whom a choice is to be made. (1.1)
Basically, there's an election coming up, and he's not going to be one of the options on the ballot. To him, this seems like a good time to break the news to America.
Sometimes people bury the lede, hiding their main point or big news farther down in order to build up to it. Other times, you put the main point up front, like in a thesis, and that's what grabs people's attention. Washington clearly opts for the second option, and he uses the rest of the speech to talk about his feelings about retirement and what ideas he wants to pass on to the American people before he's out of office.
At least he doesn't keep us in suspense.
Washington ends his speech on a personal note, looking forward to what lies ahead when he retires:
Relying on its kindness in this as in other things, and actuated by that fervent love towards it, which is so natural to a man who views in it the native soil of himself and his progenitors for several generations, I anticipate with pleasing expectation that retreat in which I promise myself to realize, without alloy, the sweet enjoyment of partaking, in the midst of my fellow-citizens, the benign influence of good laws under a free government, the ever-favorite object of my heart, and the happy reward, as I trust, of our mutual cares, labors, and dangers. (50.1)
George has worked hard, and he's excited to relax on his plantation as a "reward" for all that he's done. Which seems pretty fair for the "Father of Our Country."
Washington probably closes on such a personal note because this is his official goodbye to the American people. The fact that he was president at all, and stayed on for two terms when he didn't want to, shows that he cared about the country and the people in it. That might explain why he wants to share a bit of himself and his feelings at the end.
The effect is to humanize this giant of early-American history. In the end, he's just an older gentleman (and war hero) who really wants to retire.
Washington doesn't try to be cryptic or do anything really fancy with his language. However, politicians generally used pretty advanced language and long sentences, and Washington is no exception here.
Also, to really understand some of his points and why he's making them, you need to have a basic understanding of the political situation of the time since he references events but doesn't explicitly explain them.
George Washington was naturally brown-haired (and a borderline redhead). The iconic white hair in all his portraits was not a wig, like others wore at the time, but was just his normal hair—powdered. (Source)
Washington was given a key to the Bastille after it was destroyed in the early days of the French Revolution. The storming of the Bastille was a major event at the start of the revolution, and Thomas Jefferson witnessed it as ambassador to France. But it was Washington's old friend and ally from the American Revolution, the Marquis de Lafayette, who gave him a key from the French royal prison. (Source)
The last letter Washington ever wrote was to Alexander Hamilton. A couple days before Washington died, he sent Hamilton a letter praising his idea for a national military academy. (Source)
Aaron Burr once helped prevent a duel between Alexander Hamilton and future president James Monroe. Burr, the man who would later kill Hamilton, was also an old friend of Monroe's, and when Monroe and Hamilton were preparing to duel in 1797 over an argument and the publishing of some very private papers, Burr was asked to step in. (Source)
George Washington had to deal with pirates during his presidency. Legitimate, old-school pirates. In 1785, a ship with 23 men was captured by Barbary pirates from Algiers. Between the changing government and problems getting funds, the men weren't ransomed until 1795, when Washington finally secured enough money from Congress to get back the 10 men who were still alive. (Source)
The "whiskey tax" suggested by Hamilton (which caused the Whiskey Rebellion of 1794) affected Washington himself. He ran a distillery on his estate at Mount Vernon, which at one point was actually the largest whiskey distillery in the country. And yes, he paid the taxes. (Source)