in interior ballistics: The equation of state of the propellant The equation of energy The equation of motion The burning rate equation The equation of the form function History Prior to the mid-1800s, before the development of electronics and the necessary mathematics, (see Euler), and material science to fully understand pressure vessel design, internal ballistics did not have a lot of detailed objective information. Barrels and actions would simply be built strong enough to survive a known overload (Proof test), and muzzle velocity change could be surmised from the distance the projectile traveled. In the 1800s test barrels began to be instrumented. Holes were drilled in the barrel, fitted with standardized steel pistons, acting to deform a standardized small copper cylinder pellet which is crushed when the firearm discharged. The reduction in the copper cylinder length is used as an indication of peak pressure. Industry standards defined "Copper Units of Pressure", or "CUP" for high pressure firearms. Similar standards were applied to firearms with lower peak pressures, typically common handguns, with test cylinder pellets made of more easily deformed lead cylinders, hence "Lead Units of Pressure", or "LUP". The measurement only indicated the maximum pressure that was reached at that point in the barrel. By the 1960s, piezoelectric strain gauges were also commonly used. They allow instantaneous pressures to be measured and did not need a pressure port drilled into the barrel. More recently, using advanced telemetry and acceleration-hardened sensors, instrumented projectiles were developed by the Army Research Laboratory that could measure the pressure at the base of the projectile and its acceleration. Priming methods Through the years, several methods of igniting the propellant have been developed. Originally, a small hole (a touch hole) was drilled into the breech so that a fine propellant (black powder, the same propellant used in the gun) could be poured in, and an external flame or spark was applied (see matchlock and flintlock). Later, percussion caps and self-contained cartridges had primers that detonated after mechanical deformation, igniting the propellant. Another method is the use of an electric current to ignite the propellant. Propellants Black powder Gunpowder (Black powder) is a finely ground, pressed and granulated mechanical pyrotechnic mixture of sulfur, charcoal, and potassium nitrate or sodium nitrate. It can be produced in a range of grain sizes. The size and shape of the grains can increase or decrease the relative surface area, and change