With a mushrooming circulation, several new branches and a growing list of holdings, the Fairfax County library system has been counting its blessings.

Just one problem: Until recently, the library system had very little idea of how many blessings it had. Using a system of microfilmed book checkouts, the library found it was having a hard time keeping track of what it had on its shelves, and was falling victim to book thieves and habitual overdue offenders.

Now the library is fighting back. It's installing a fully automated computer system that will keep track of what books are where, how long they have been there and what books aren't there at all.

The new system is known as ALIS, (pronounced ALICE), for Automated Library Information System, and will cost almost $500,000 to install. Library officials characterize the change as a giant leap into the technological world for the library system.

"We're going from a one-horse shay to a spacecraft," said Nancy Woodall, acting coordinator for public relations.

Four of the 18 branch libraries -- Lorton, Woodrow Wilson, Tysons-Pimmit and Fair Oaks -- already are on line and three more -- Burke Center, Great Falls and Richard Byrd -- are to be hooked up by September. By 1983, library officials hope, all branches will be part of the system, which is similar to one now in use in the District and one Prince William County is installing.

Until then, the majority of the branches will have to make do with the old system. Under that system, a microphotograph is taken of each book checked out along with the patron's library card. The film is then processed and sent to the county's main computer.

Under that system, library workers had a difficult time keeping track of overdue books. In fact, the process is so time-consuming and considered so unwieldy that overdue notices stopped going out altogether last summer. Instead, patrons are on an honor system to pay their fines.

"We have so little control of our books now," Woodall acknowledged."It's a problem for us."

Just how inadequate the current system is was underscored recently when the manager of an Arlandria apartment building found a closeful of library books in a vacant apartment. The books -- 308 in all and most from the Fairfax library system -- were valued at more than $1,800.

"What don't we know?" asked Woodall. "How many people are throwing out their books that we never find out about? That's what scares me. We don't even know how many books these (Arlandria) people took with them when they left."

The new computer system, while it won't stop book thieves, will help the library keep track of what it has and what books are overdue. For example, under the new system, overdue notices will be produced automatically. Seven days after the due date patrons will be sent a letter notifying them that their books are overdue. Two weeks later, another notice will be sent, noting the amount owned on the overdue books. Overdue rates will remain the same: 2 cents a day or 12 cents a week, up to a maximum charge of 50 cents a book.

Library patrons will not need new cards for the computerized system. Rather, a computer-readable label will be pasted on the backs of the current cards, which carry codes for the user's name and address.

Under the conversion process, library workers will label every item in the system's collection with a similiar label stamped with a special code number and then enter the number into the computer system. The current holdings of the library are estimated at 1.5 million books, with an average annual circulation of 5.5 million, up from 5.08 million a year ago.

"I'm really looking forward to the day when everything is all hooked up and running," Woodall said. "Our lives will be much, much easier."