What's in a number?
most of their history, roads had local names, mostly referring to the towns they were headed for - the Daventry Road, the Banbury Road, the Burford Road - all the way to the Glastonbury Road. Of course, in the other direction, the names are different. You'll still find these names in use, especially where 20th century development has created ribbons of housing along them. Names are useful for postal addresses.
But as the internal combustion engine replaced the horse and cart in the period after the Great War, and it became possible to travel further than the next town in a day, it became clear that the bureaucrats would have to step in.
Plans would have to be made, rules would have to be invented, signs would have to be erected. So the Ministry of Transport decided to do away with all those inefficient, confusing names (which everyone understood and cherished) and replace them with nice, modern numbers.
They began in London - because that's where the Ministry of Transport is - and numbered the great roads radiating out of the capital. Clockwise, of course.
In the sectors defined by these spokes, the major roads were given two-digit numbers. So, between the A3 and A4, there were the A30, A31, A32.... up to the A39. Some of these roads also radiated from London, and were numbered clockwise, but others were just fitted in to the system as convenient. For instance, the A36 runs from Southampton to Bath.
But that still left a lot of roads un-numbered - none of them particularly important, but perfectly serviceable roads, nonetheless. So these roads were allocated 3- and 4-digit numbers, designed to reflect the major roads they connected. The road which connects the A40 to the A4 became the A404, the road which connects the A26 to the A2 is the A262.
The A361 connects the A36 to the A1 (near as dammit). The system is far from consistent. It is often incomprehensible. One suspects the men from the Ministry rather lost interest after a while.
That is how the road from Watford Gap to Camelot became the A361 - in the fevered brain of a digitally-obsessed bureaucrat.