In 1841, the U.S. Army fielded the first homegrown artillery. The 1841 model 6 pound cannon & 12 pound howitzer were significantly lighter than the cast iron guns that were the previous standard.
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The standard battery consisted of a section (4 guns) of flat trajectory cannon & a section of howitzers that would lob exploding shells. By the start of the Civil War, rifled cannon were introduced. At Stones River, Parson's Battery of four rifles & four howitzers anchored the Union line. Parson's fired almost 2,000 rounds that day.
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In this image, the battery commander in the upper left, ordered,"fire by piece from the left, fire!" From that position, he could observe the fall of shot & make adjustments accordingly. You can see the cannoneers reloading their pieces. Firing in sequence like this meant that a continuous fire could be maintained. The battery never fired in a volley (all at once) because that would left the battery vulnerable & made it all but impossible to observe the fall of shot.
(In this photo, the left gun of the section nearest the camera had experienced a misfire. The number one of that gun (me) has not moved to sponge the piece. A misfire was a common occurrence during the war & still happens these days. The gunner calls, "Do not advance, the primer has failed!.... Re-prime the piece!" During the war, they re-primed almost immediately. We are a little more circumspect & wait a bit.)
At Stones River, Rosecrans' 14th Army Corps, soon to be the Army of the Cumberland, had an artillery menagerie. Six pound smoothbores, 12 pound howitzers, Wiard rifles, 3" rifles, 2.9" Parrott rifles, James rifles & Rifled James & 12 pound cannon/howitzers all used unique ammunition. Needless to say, attempting to resupply the right battery with the right ammunition during combat was a recipe for chaos.
Lessons were being learned. Eight guns when deployed in battery was about the size of a football playing field. Rosecrans reduced the battery to two sections of three & homogenized them. It would take a while, but the goal was to have batteries consist of 12 pound Napoleon cannon/howitzers or 3" rifles. In time, the rough mountainous country in which the AoC operated & experience resulted in a half battery of 4 guns to become the best solution. An 8 gun battery in column was a mile long, halving that was obviously going to be more nimble.
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In the latter stages of the war, a section of two guns became common. Here is where terminology gets a bit difficult. A battery still consisted of eight guns. The 4 gun & 2 gun units were still part of the battery. As a generic term, battery also refers to any line of guns of 2 or more, which can be a bit confusing. When it became necessary, the various elements of the battery could coalesce & line up with other batteries to form a large battery of as many guns as were needed.
Needless to say, there were any number of variables. The Chicago Board of Trade Battery salvaged an abandoned cannon & overnight refurbished it during the Battle of Stones River. So, it became a nine gun battery. Eli Lilly's Battery that was assigned to Wilder's Lightening Brigade had four 3" rifles & 4 12" mountain howitzers known as the Jackass Battery.
As with all things Army of the Potomac, things were a bit more ponderous back east. Their small area of operation & other operational factors resulted in a different deployment of artillery than the Army of the Cumberland & Sherman's army group. I will leave that topic to someone who studies that part of the war.