Progressive chess

Progressive chess is a chess variant in which players, rather than just making one move per turn, play progressively longer series of moves. The game starts with White making one move, then Black makes two consecutive moves, White replies with three, Black makes four and so on. Progressive chess can be combined with other variants; for example, when Circe chess is played as a game, it is usually progressively. Progressive chess is considered particularly apt for playing correspondence chess using mail or some other slow medium, because of the relatively small number of moves in a typical game.
Rules.
There are two main varieties of progressive chess: Italian progressive chess and Scottish progressive chess (otherwise known as "Scotch chess"). The two have the following rules in common:
Italian and Scottish progressive chess are distinguished by rules on when a player is allowed to give check:
Progressive chess, like orthodox chess, is notated with algebraic notation. However, the numbering of moves is handled slightly differently. Rather than one White and one Black move being given under each move number (leading to notation in orthodox chess like 1.e4 e5 2.Nf3 Nc6 3.Bb5 a6), each turn by each player is given its own move number (leading to notation in progressive chess like 1.e4 2.e5 Nf6 3.Bc4 Qh5 Qxf7#). In this way, the move number is equal to the number of moves in a series available to a player on that turn.
Other variations.
There is another form of progressive chess, English progressive chess, which makes quite a significant change to the rules: within each turn, no piece may be moved twice until every other piece which has a legal move has moved once; no piece may move three times until every other piece which can has moved twice; and so on. These restrictions do not carry over from one turn to the next—so the opening 1.e4 2.e6 f6 3.e5 Nf3 Bc4 is legal (White's e-pawn may move again because its moves are on different turns), but the sequence 1.e4 2.e6 f6 3.e5 Ba6 Bxb7 is not (the bishop has made two moves, but there are many other white pieces which have not moved on that turn). There is no "en passant" capture under English rules, and rules on checks follow the Scottish rules.
Progressive Take-All uses the same rules as Progressive chess, but involves capturing all pieces of your opponent's instead of checkmate. Pawns can also be promoted to Kings.
In Logical progressive chess (by Paul Byway, "Variant Chess" 18, automne 1995) there's no castling or pawn two advance (hence no en-passant capture) since these rules were added to speed up the game, which is not relevant in progressive form.