[Moderator comment: on the off-chance that an artilleryman reads this and acts on it, I've corrected the mils/degrees confusion; original text in [], corrected text in italics]
Lets make one thing clear we are talking about the British Mils system which is now the NATO Mil, different countries use different Mil systems, Russia being notable.
Basically there are 18 mils to 1° more accurately there is 17.777 mils to 1°.
A 1 degree deviation from a bearing over 1 Km is 18 meters or more accurately 17.777 meters this is well within the effective kill zone of an Artillery shell which differs between weapon type but for the purpose of this conversation is a radius of 75 m. And it is very rare for 1 gun to fire one round as a fire mission, normally the very least is 3 rounds from 6 guns and like with darts it is hard to have the rounds fly without some deviation in flight so they don't hit the ground in the exact place, meaning they are an area weapon.
Depending on the charge used we could shoot up to 30 Km so a deviation of 1 degree would mean a loss of accuracy of around 540 meters add to that deviations in the fight and other tolerances the accuracy could drop to around 1 km.
This is unacceptable by any ones standards so you can start to see why the Artillery are masters of bearings and navigation. In the OP's the Artillery observers when a gun fires you expected to see the splash of the round with in the field of view of the binoculars, if you did not see the splash you would ask your team if they saw the splash without using bins if they did not something very wrong has happened. If you have not heard the round splashing then your ass was seriously on the line because the round had dropped somewhere unknown and you are never sure it has splashed safely.
[Moderator comment:
The length of a circular arc of radius r, and subtended angle, theta radians, is simply r*theta. Now, since the mil is essentially a milliradian*, the length of a circular arc of radius r and subtended angle 1mil is r.1/1000. So, for a radius of 1000m, the arc length is 1000*1/1000 = 1m. This is the reason why mils are useful for artillery purposes, because the arithmetic for correcting shot fall angle is simple.
*There is a small error arising from the approximation of 6400 mil in a circle, compared with 6283 milliradians; about 1.25% error.]
Cheers CP for spotting that the second correction should be degrees because i am trying to explain the size of the error using degrees. If that makes sense. (I think, it was ages ago since i posted that)