ROGERS HORNSBY: A BIOGRAPHY By Charles C. Alexander Henry Holt 366 pp., $27.50 His .358 lifetime average is second only to Ty Cobb's .366, his .424 mark in 1924 is unmatched in this century, and he is generally acknowledged as the greatest right-handed hitter ever to play the game. Baseball aficionados know all this and more about Rogers Hornsby, and so did the general public in his heyday, when he won several National League batting titles and was that circuit's principal answer to Cobb and Babe Ruth. But while they and others have been immortalized via print and film, ''the Rajah'' and his feats have been left pretty much to serious fans and historians. Now in this excellent and informative biography, Charles C. Alexander attempts to rectify the situation. Alexander, a history professor at Ohio University, has written on subjects ranging from the Ku Klux Klan to Project Mercury. Lately, however, his focus has turned to baseball, with biographies of Cobb and John McGraw, a history of the game, and this effort. He does a remarkable job of evoking a bygone era; some of the most fascinating chapters depict Hornsby's boyhood in Texas. The early 20th century was a time when baseball was woven into the fabric of American life, with millions playing the game in school, commercial, industrial, municipal, and semipro competition. Hornsby went all these routes, but despite his determination he ''hadn't awed anybody with his talent.'' A scrawny 5 ft., 11 in. and 135 pounds, he hit only .232 and .277 in the low minors, but a series of unlikely breaks landed him with the St. Louis Cardinals anyway. The last of these came after a brief trial in 1915 when manager Miller Huggins told the rookie he intended to ''farm him out'' (i.e., to the minors). The youngster took him literally and spent the winter on a farm, where the strenuous regimen built him into an imposing physical specimen. His hitting in spring training opened all eyes; he made the team as a regular infielder. While he was widely applauded for his abstention from alcohol and tobacco, the rest of Hornsby's personal life was a bit checkered, including three marriages, a mistress who committed suicide, various legal battles, and a gambling habit that kept him in constant financial straits. He had difficulty getting along with people, a problem perhaps best described by St. Louis sportswriter John B. Sheridan, who noted that Hornsby was, ''as the French say, deficient in the social relation.'' Hornsby prided himself on not being a hypocrite or a ''yes-man,'' but his bluntness and tactlessness made him a lot of enemies. A .358 average can overcome a lot, though, and he was able to spend his life in the game he loved, even though it was sometimes a vagabond existence. His 23-year playing career embraced five teams. He also managed five clubs (most notably as player-manager of the Cardinals in their historic 1926 World Series victory over the New York Yankees), and he spent many other years managing in the minors or coaching in the majors up to his death in 1963. As columnist Red Smith put it, ''he made hitting look easy ... but he made everything else look hard because to him baseball was not a game, it was a crusade.''