[SeekHer]

On the Prairie you would use the materials handy to make your snares, usually woven grasses and such...
stakes would have been used, to prevent the animal running off, as well as tying to the hardier bushes and small trees...which would be used to make spring traps that flicked the animal into the air, hopefully breaking their neck but high enough that their body weight would strangle themselves.

I remember from my youth, (long long ago in a galaxy etc) that you would place the snares on the animal's runs and the more out there the better your chances...but what I recall was that you would place crossed sticks in the path to make the animal go over or under to ensnare themselves...but this meant that the animal had to keep running to choke themselves...

Pick up any of the books from A.R.Harding which would describe the process far better than from someone who hasn't done it in 35+ years...I checked and the volume you'd want is titled Deadfall & Snares by A.R.Harding published 1935 by Harding who published the magazine Fur-Fish-Game

Other titles, by the same author, of interest would be
Camp & Trail Methods
Trails To Successful Trapping
The Trapper's Companion
50 Years A Hunter & Trapper

other authors published by them
The Science Of Trapping by E. Kerr
Canadian Wilds by Martin Hunter
Home Tanning And Leather Making Guide by A.B. Farnham
Home Manufacture of Furs and Skins by A.B. Farnham

All of these were published from the 1900s to the 1930s so the techniques described are from the mid 1800s which is the time frame you desire...they should be available at the library--as they're mostly still in print--otherwise a used book store or you could try one of the trapping supply companies like Legget's etc...

You might want to pick up a copy of Fur-Fish-Game, still an excellent magazine, and see if they have them...I don't know where mine have gotten to and besides they'd be 15+ years old.