(FORTUNE Magazine) – Armed with his billion-dollar checkbook and a keen vision of the oncoming digital age, Microsoft Chairman Bill Gates began courting the world's major museums in 1989. But when he asked for the digital reproduction rights to their masterpieces, he was spurned. After four years of rejection, Gates wisely recruited J. Carter Brown, director emeritus of Washington's National Gallery of Art, to bang the drum for digital. Since then, several august institutions, including the Philadelphia Museum of Art, have signed over their rights to Gates' privately held Continuum Productions. Among these art works: Picasso's Three Musicians and Cezanne's Large Bathers. ! Still, wooing the major museums to the digital altar remains a formidable task for Gates and his major software rival, DCI of Berkeley, California, which recently published a CD-ROM featuring Rembrandt's Self-Portrait and other old masters from the Frick collection in New York City. Says Jessica Schwartz, information director at New York's Museum of Modern Art: ''Many museums fear their customers will become mouse potatoes: They will learn about art on their home computers and never go to museums anymore.'' Digital proponents counter that the emerging technology will have the opposite effect and will hardly mean the end of those visiting the Louvre to see the Mona Lisa. In addition, software companies like DCI are doling out about three times the royalties of book publishers. Since many benevolent corporate sponsors like IBM have drastically cut back contributions to the arts in recent years, museums could surely use the extra income.