Why the numbers on a dartboard are arranged the way they are is a question that comes up frequently. Sven Silow (firstname.lastname@example.org) explained it best:
"The number system on the board is often blamed on a Brian Gamlin, a carpenter from Bury, Lancashire, who is said to have invented it at the age of 44 in 1896. This story is much doubted though, and it is even questioned whether he lived at all.
According to another source, Thomas William Buckle invented the dartboard in 1913. The source in question is his son, Thomas Edward Buckle who 1992 made this statement in Darts World (nr 234).
By the way, the dartboard that is used over most of the world today is not the only one that exists or have existed. Boards with other number systems/sequences exists as well. The Manchester board is still used and there has been many others (ex. the Grimsby board, which had double 14 on top). Actually TE Buckle says that Gamlin could possibly have invented the Manchester board (a bit more geographically probable?).
Without having done any research on the subject, I can't help but believe that darts were played a bit at random in the beginning and that a lot of numbering systems were used, and that they probably changed from time to time and place to place (hence the many different boards that no longer are in use). A few systems survived - and especially so the "London board" which we use today with few exceptions.
I don't think that there has been any greater thought behind the numbering either. Low numbers beside high numbers, it can be done in numerous ways (and probably were). The number-sequence we have today has probably just chance (and that it perhaps was better than most of the other used systems) to thank for it's survival. It's not extremely special though, even if we find it intricate, and could have been done in other ways, not less good."
The text of this article was posted some time ago on alt.sport.darts.
Unfortunately the author's identity has been lost.
First of all, the dart board, the source of all our frustrations, does have a name. It's called the London Board - named for it's initially strong popularity in that city. There are at least five other basically similar boards in use in certain areas of England today. In 1896, Brian Gamlin, a British carpenter, arranged the numbers on the dartboard as we see them today.
The Yorkshire or Doubles board is similar to the London board but has no outer bull, no triples ring and a narrow 1/4" doubles ring. (The London board has 3/8" rings.) Sometimes, it is made with an all-black face.
The Tonsbridge board is similar to the Yorkshire board but with one important difference. The usual doubles become triples and doubles are scored on triangular beds on the inside of them.
The Staffordshire or Burton board is also similar to the Yorkshire board but has two diamond-shaped scoring areas outside the doubles ring. One is placed between the wire numbers 14 and 9 on one side while the other is located between the 4 and 13 on the other. These are worth 25 points each and can be used as an out shot on that number. There is also an outer bull.
The East End board has only twelve segments or pies instead of the 20 found on the London board. They are numbered (clockwise from the top) 10, 20, 5, 15, 10, 20, 5, 15, 10, 20, 5, 15 and have 1/4" (or narrower) doubles and triples rings.
The last, the Manchester or Log-End board is probably the most distinctive one of all. It is numbered (again, clockwise from the top) 4, 20, 1, 16, 6, 17, 8, 12, 9, 14, 5, 19, 2, 15, 3, 18, 7, 11, 10, 13 and is much smaller than other boards with a 10" playing area and 1/4" doubles.
These boards are still in use to some degree in certain areas, but the London board has become the standard in pubs and dart halls around the world. This is far from the complete story. There are several other board patterns known to have existed in the past but which have fallen by the wayside from lack of popularity or use.