By the 1860's why did Americans no longer seek to resolve their political dispute through compromises?
By the 1860's, North and South were so polarized that there wasno middle ground. It is important to note that both sides were firmly convinced of the justice of their cause and the fallacy and unreasonableness of the opposing side. Ministers on each side of the slavery issue often cited lengthy passages of scripture to support their respective positions. The Baptist and Methodist Churches split North and South over the slavery issue. A second factor to be considered is Harriet Beecher Stowe's novel, Uncle Tom's Cabin, written to protest the Fugitive Slave Act of 1950. The novel is quite melodramatic, and presented an unrealistic description of slavery. It further polarized opinions on both sides, with each seeing in the novel that which it wished to see: In the North, the horrors of slavery; in the South, an unrealistic portrayal of Southern life. Additionally, the failure of Popular Sovereignty to solve the issue in Kansas and the violence that accompanied it; plus the tendency of the North to portray John Brown as a latter day prophet whereas the South considered him a terrorist and vigilante meant there was no possibility of a peaceful solution.
There are two reasons why Americans stopped trying to reach compromises to resolve their political disputes. (I assume you are talking about the issue of slavery.)
First, they had tried various compromises and none of the compromises managed to satisfy both sides. Both the Missouri Compromise and the Compromise of 1850 helped for a while, but both eventually broke down.
Second, and perhaps more importantly, the two sides no longer trusted one another and were not really all that interested in trying to get along. The events of the last two decades or so had hardened the two sections' attitudes towards one another. Compromise relies on the two sides having at least somewhat positive attitudes towards one another. The North and South no longer had this sort of attitude.