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An item that delays the functioning of an explosive device for a given period, or the that actuates the device when a condition is met.

In the case of a fuse for an artillery shell , it could be either a fuse, MT, PD or Proximity.

The MT fuse actuates after a period of time has elapsed, making the shell explode.

The PD fuse actuated when the shell hits something (a condition is met).

The Proximity fuse actuates when the shell reaches a certain set distance from the ground or an aircraft, and causes the shell to explode.

A long time ago

Fuses used to be a simple affair; basically a tube of paper or a straw, filled with gunpowder, the powder burnt at a reasonably regular pace so the fuse was set by cutting a bit off. If your fuse burnt at 1" per second, and you needed 12 seconds to run away, you needed a fuse 12" long.

This type of fuse soldiers on in certain types of demolition, in fireworks and in cowboy films where it is often to be found sticking out the top of a stick of dynamite.

It was refined into a train of gunpowder, held in a series of circular channels within the nose cone of a shell that was set by piercing a membrane with your spike, fuse setting.

The fuse was lit by the wash of hot gas passing round the body of the shell when the gun was fired.

A reasonable time ago

Clockwork MT fuses began to replace gunpowder time fuses as being more reliable. They are set with a fuse setter (really!) before the shell is loaded, then they begin their countdown on firing as the safety mechanism " sets back" due to the launch acceleration of firing the shell.

The clock ticks away, and at the appointed time, detonates the shell.

A PD fuse is armed in a similar way, but the sudden deceleration of the shell hitting something causes it to actuate on impact, and detonates the shell. This detonation can be delayed by a short amount (about 0.2 seconds IIRC) so the shell penetrates the ground before exploding. All the better to cave in your trench walls my dear.

Not so long ago

During WW2, the Proximity fuse was developed to try and answer the knotty problem of shooting shells at aircraft and getting them to explode somewhere in the vicinity. The difficulty is that unless you know the altitude and speed of the aircraft, it's heading and the time of flight of your shell to the point where the trajectory of shell and flight path of plane will meet, you can't sensibly set your fuse to go off near the plane. All this isn't as easy as it sounds . . . . .

If you have a Proximity fuse, all you need to do is get the shell to pass close to the plane, and it will go off as it did so. You still have to get the shell near the plane in the first place, but it at least removes a level of complexity.

Thus the Proximity fuse had a small radar transmitter/receiver combo on board. It flew through the air with the greatest of ease until the point technically known as "blammo!!!" arrived.

It didn't take very long to realise that the ground also has a radar reflection, so you could use Proximity fuses to make shells detonate a few metres above the ground.

Why do you want your shell to do this?

A shell bursting in the air as described above, has a more lethal effect on troops in the open, or hiding in trenches.

Modern Proximity fuses can be set to go off at a certain altitude, as PD or as PD Delay, and are thus very versatile.