The Fairfax County library system, which says it can't keep track of all the five million books residents borrow annually, has taken its first significant step toward long-sought automation.

The county board of supervisors agreed this week to earmark $525,000 for an automated circulation system if the county library board gets a $175,000 grant from the National Endowment for the Humanities. Under the terms of the grant, every federal dollar must be matched by three from the local jurisdiction.

This week's decision was a turnaround for the supervisors. In the past, they have repeatedly turned down library budget requests for automation money.

The $525,000 that would be provided by the county ordinarily would go toward the increased cost of new books and other materials. But according to William L. Whitesides, director of libraries, an automated circulatiooin system would save $475,000 worth of books in three years.

According to a county study of library book circulation, the present manual checkout system - built around a card inserted in the jacket of each book - is so overwhelmed that "there is no need to steal a book - just check it out. Unfortunately, many borrowers are aware that the library books are incoocmplete, inaccurate, and sometimes missing . . . you can easily check a book out, 'forget' it, and the library cannot prove who has it."

The study said the present checkout system, although it handles five million transactions annually, was meant to handle only "a few hundred thousand" annually.

Whitesides said the library syysstem would save money not only through fewer books lost but through more fines assessed and collected. The automated system would centralize records on delinquent borrowers.

According to the county study, an automated system would not only reduce delinquency but would provide a faster turnover of borrowed books and permit personnel to be shifted from clerical tasks associated with manual checkout to other duties, such as servicing the public's information requests:

The library system has been trying to automate book checkout since 1972. In 1973, one commercial firm's system got a favorable evaluation by the county, but the firm then went out of business. Subsequent action by the county was delayed by more study and evaluation.

Supervisor Alan H. Magazine (D-Mason) took the lead in getting the supervisors, for the first time, to commit money to automation.

When it appeared at this week's meeting that some of the supervisors might want to delay a decision, as they have in the past, Magazine said, "I think we all know where we stand on this, so why don't we have a vote."

Supervisor James M. Scott privided some additional prodding by saying the library staff was overworked partly because of "the crazy telephone calls for information that it gets from the supervisors' office."