Indirect fire
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Indirect fire is the use of artillery to fire at targets out of the crew's line of sight, by firing in a high arc out to long distances and/or over blocking terrain. Usually an observer in closer proximity to the target reports the target's estimated position back to the crew or their headquarters, and the weapon is then fired at that point on the basis of mathmatical calculations for the necessary horizontal and vertical angles and shell velocity.1 The observer then takes note of any error in the resulting fire, and if necessary requests adjustments relative to the impact point of the initial rounds, e.g. "right 400 add 2000", in units of distance understood by the army's conventions.
Direct fire, in contrast, is the use of artillery to fire at targets that can be observed, aimed at, and corrected for by the crew itself.
Note 1: Throughout most of the history of indirect-fire artillery the aiming was done on the basis of pre-calculated firing tables, which took account of wind velocity in addition to the relative position of the enemy and allowed crews to aim their guns by means look-up tables rather than their own mathematical calculations. The creation of the tables was a labor-intensive task, usually performed by teams of non-combatant women known as "computers". When digital computers were first invented their primary use was to automate the calculations for firing tables, and thus in the anglophone world the machines came to be known as "computers" as well.