External Ballistics is the study of how an object moves through a medium such as a shell traveling through the air. The most important elements to consider when trying to predict where a flying projectile will land are those which exert forces upon it. The forces produce accelerations according to classical Newtonian rules, and these accelerations alter its velocity over time, and the velocity acts to alter its position over time. That should be obvious to any student of physics.
That gravity acts upon shells is obvious, and rather simple. It was not really even considered explicitly in the design of gunsights, however, as it always acted in concert with other forces that were significant and more subtle in their behaviour.
Drag is the force acting against the shell as it pushes at high speed through the air. It is trickier than gravity, as it depends upon the shape of the shell and the speed of sound at the shell's present location.
Although the scientists of the day were fully capable of computing the effect of such forces at any given moment -- a feat repeatedly demonstrated in manuals in a morass of the most appalling mishmash of systems of measure where angles are measured in degrees, minutes and seconds, velocities variously in feet per second and knots, etc -- they lacked any means of simulating the accumulated effect of these instantaneous influences over the course of a shell's trajectory.
The approach they took was to learn all they could about an idealized shell type through experimentation, and compiled these findings into O. B. Ballistic Tables (where O.B. means "Ordnance Board"; previously, a different set called "Ingall's Tables" were employed). When proving a new weapon system, they fired enough test shots at various angles of elevation that they could see how the gun differed from this idealized weapon system and then provide a few simple fudge factors ("coefficient of reduction") that allowed the idealized data to be distorted into a description of the new weapon despite having conducted very few test firings. Their ability to do this with high precision is a testament to the strong background they had in mathematics, though even a neophyte such as I gets the idea that some seat-of-the-pants guesstimation was woven into the process.
The 1918 Range Tables for His Majesty's Fleet (ADM 186/236) contain preface notes (p. 7) indicating the extent of the Admiralty's confidence in these approximations, specifying that the tables were created based on O. B. Ballistic Tables compiled in June 1903 at Shoeburyness, indicating that discrepancies attributable to the underlying use of Sciacci's formulae can be expected to start at elevations of 10 degrees and to become somewhat onerous past 15 degrees elevation.
Shells fired from rifles would wander to the side as they spiralled down range. The magnitude of this error was proportional to a constant for the weapon system and the time of flight. It was not uncommon for gun sights to automatically correct for this lateral force, although the corrections were often approximations.
Air movement relative to the shell (other than that attributable to its own motion along its length) is the source of an additional force accelerating the shell. Wind came in two flavours, actual wind as might be measurable by a wind vane and anemometer standing at a fixed position in the world, and also the "induced wind" caused when a shell is fired from a moving platform.
Often called "Coriolis Force" despite the fact is merely a geometrical effect caused by firing shells from one latitude to a different latitude on a spinning earth. The result is an apparent deflection error caused by the fact that the shooter and target are moving around the globe's axis of revolution at different radii.