All most all modern firearms utilize a center fire, primer in the middle of the casing, cartridge, but for some time in the 19th century rim fire, primer around the circumference of the casing, cartridges were popular. What was the first weapon to make the switch and how long did it take for it to become prevalent?
Or am I wrong in my assumption that cartridges went from rim-fired to center-fires, and they were always used concurrently?
Generally-speaking, most of the first widely-used metal cartridges were rimfire. Small rimfire cartridges for pistols were being produced and sold in the 1850s, at a point where centerfire cartidges were still being refined and tested.
The rimfire .44 Henry (1860) is used in the Henry Rifle (1860) and the Winchester Model 1866 (1866). The .56-56 Spencer (1860) used in the Spencer rifles and carbines (1860) was rimfire.
There was some overlap in service life (for example, the Turks very successfully used Model 1866s in .44 Henry during the Siege of Plevna in 1877), but centerfire quickly became the preferred military cartridge type.
For example, the French 11mm Gras (1874), the British .577/450 Martini–Henry (1870), the American .50-70 Government (1866) and .45-70 Government (1873), and the civilian .44-40 Winchester (1873) were all centerfire. Winchster's decision to take it's premier lever-action rifle from rimfire to centerfire shows how the winds were blowing at the time.
Today, centerfire and rimfire are really the only game in town. But in the mid- to late-1800s, there are plenty of other systems out there.
The cartridge of the Prussian Dreyse rifle had a primer cap inside its paper cartridge, underneath the bullet (it's the small black rectangle in this diagram). When the rifled was fired, the long, needle-like firing pin poked through the bottom of the cartridge and set off the primer. Hence why the Dreyse is often called the "Needle Gun."
The rival French Chassepot rifle also used a paper cartridge, but it had a primer cap near the base of the cartridge (marked in purple in this diagram).
There was the Lefaucheux pinfire system, which essentially had an integral firing pin in each cartridge. The hammer of the gun hit the "pin" that stuck out of the cartridge and drove the pin into the primer cap. You can see this cross-section to get an idea of how it worked.
The cartridge of the 1855 Burnside Carbine didn't even have a primer. It had to be activated by an external percussion cap that was hit by the hammer. The spark of the cap entered a hole in the base of the cartidge and set off the powder.
