Logos

Although there are moments here and there where Lincoln waxes a little poetic in the "House Divided" speech, really he's doing his bestest to prove his point through cold, hard evidence, so his method of rhetoric really is logos.

He really doesn't want emotion muddying the crystal clear waters of logic, guys.

A ton of the speech is occupied discussing why there's a risk of the North having to accept slavery. He says,

Let anyone who doubts carefully contemplate that now almost complete legal combination—piece of machinery, so to speak—compounded of the Nebraska doctrine and the Dred Scott decision. (12)

Then he spends a loooong time reviewing both of those things, in case anyone had forgotten any of the details.

The whole time, he carefully points out the ways these details are steps (or "points") towards the horrible outcome of slavery being legalized everywhere. For example, when introducing the Kansas-Nebraska Act:

This opened all the national territory to slavery and was the first point gained. (16)

It's all very cause-and-effect. This pattern continues. A great example of the logical progression of evidence to claim is this:

In what cases the power of the states is so restrained by the U.S. Constitution, is left an open question, precisely as the same question, as to the restraint on the power of the territories was left open in the Nebraska act. Put that and that together, and we have another nice little niche, which we may, ere long, see filled with another Supreme Court decision, declaring that the Constitution of the United States does not permit a state to exclude slavery from its limits. (84-85)

This plus this equals that. (Maybe they should have called him "Logical Abe"?)

Even when he gets flowery, there's still reason behind it:

We shall lie down pleasantly dreaming that the people of Missouri are on the verge of making their State free; and we shall awake to the reality, instead, that the Supreme Court has made Illinois a slave State. (89)

It's quite an image he paints—who doesn't like to think about pleasantly dreaming?—but it's straightforward and serves a clear purpose in supporting his idea.

He must have done a pretty good job convincing people. Yeah, he didn't win this election, but he did get the next one in 1860…which included a pretty big promotion.