IF YOU SHOULD meet a mermaid, so any Greek sailor will tell you, she will ask you a question and you had better give her the right answer. She is the sister of Alexander the Great and like Robin Lane Fox, the Greek Ministry of Culture and Sciences, the National Gallery of Art, Time Incorporated and Mobil, she is engaged in the search for her brother. "Where is Alexander the Great?" she will say. If you tell her the truth -- that he died at Babylon in 323 B.C., that his corpse was hijacked by his chief of staff, Ptolemy, and burned at Alexandria in Egypt and that no trace of it or his tomb has ever been found -- she will be furious; what is more, she will raise a storm and sink your boat. You must tell her what she wants to hear: "Alexander the Great is alive and reigns as king." He is certainly alive and well in Washington these days, and will be until the splendid exhibition at the National Gallery leaves for Chicago in April.

But the search, as a final wall poster informs visitors on their way out of the exhibition, continues. It is a search for the truth about Alexander, for definitive answers to the riddles posed by his personality, his actions and intentions -- riddles which have drawn from modern historians a wide range of answers, mutually contradictory and fiercely polemical. There is disagreement, for example, about his involvement (if any) in the murder of his father Philip -- which certainly came at a most convenient moment for him; about the strategic objectives (if any) of his intention (frustrated by a mutinous army) to push on into India as far as the Ganges; abut his plans (if any) for the organization and administration of the vast area he had conquered. This is hardly surprising in view of the fact that the sources are themselves contradictory; what is more, they are depressingly late. The earliest continuous narrative of Alexander's career, that of Diodorus, was written some three centuries after the king's death; Plutarch and Arrian, our other main sources, wrote their books some 200 years later still. All of these writers used (and sometimes cited by name) sources contemporary with Alexander, but most of the time we can only guess which contemporary source (if any) stands before us in their pages. The search for the real Alexander has to be made through a distorting screen.

About the importance of his achievement there is no doubt at all: he changed the world. Though he died at the age of 32, and, though his huge empire broke up at once into warring kingdoms ruled by his generals, the cities he had founded all over the area we know as the Middle East imposed on the inhabitants a Greek culture which remained dominant until the end of antiquity. It is because of Alexander's conquests, for example, that the sayings of Jesus of Nazareth, who addressed his hearers in Aramaic, were circulated as written documents in Greek, the language of culture, commerce, and administration, the lingua franca of the cities located in the lands which are now called Greece, Turkey, Syria, Iraq, Lebanon, Jordan, Israel, and Egypt.

But the real Alexander remains an elusive figure. He has been seen as an idealist who wished to create in his empire a brotherhood of peoples and so proclaim the "brotherhood of men"; as a ruthless conqueror whose wars were waged to satisfy an insatiable appetite for personal power; as a pragmatist whose determination to be master in his own unruly kingdom led him step by step to mastery of half the world; and as a romantic, a self-appointed reincarnation of Homer's Achilles, with his insatiable appetite for martial glory. He has even been billed, in a recent study, as an alcoholic; that may explain the burning of Persepolis and the murder of his friend Cleitus (both events the climax of a wild party) but if liquor was the secret of his brilliant victories, one is inclined to echo President Lincoln and wish that our own generals could be given some of the stuff he was drinking.

Robin Lane Fox's Alexander is a complex combination: a pragmatist skilled in the art of survival (indispensable for any member of the Macedonian royal family, where prompt elimination of potential competitors was standard operating procedure) but also a romatic rival of Achilles, one who might well have answered the question "Why do you want to conquer the world?" with the modern cliche: "Because it's there." His book introduces the reader to the world of the fourth century B.C. which Alexander was to transform by his Asian conquests, locates the young prince in the context of the northern kingdom of Macedon, which his father Philip II had made master of the Greek world and equipped with a matchless army, and traces the young conqueror's astonishing march, which in 10 years of constant fighting took him more than 11,000 miles, over some of the most formidable territory on the face of the earth -- the barrier of the Hindu Kush, the awesome wastes of the Makran desert -- and left him, on his deathbed in Babylon, the ruler of a kingdom one million square miles in extent.

Robin Lane Fox has told this story before. His Alexander the Great (1977) is a book remarkable for its dismissal of scholarly impedimenta -- "I do not like the proper names of nonentities, numbered dates of unknown years or refutation of other men's views" -- and also for its narrative skill and elegant prose. It was a difficult act to follow and this shorter version, evidently written in haste, suffers by comparison; compression has sometimes produced obscurity (the account of the sources in the first chapter, for example) and the reader is often momentarily checked by awkward phrasing and loose transitions. In compensation, the 222 illustrations (135 of them in full color) provide a magnificent visual accompaniment to the text; noteworthy among them are photographs (known only to specialists until now) of the Greek town excavated by French archaeologists at Ai Khanum on the northeast border of Afghanistan.

Even more exciting are the photographs contained in the catalogue of the exhibition, also entitled The Search for Alexander . It presents, with scholarly comment and preceded by three illuminating essays, photographs of 173 object, some 60 of them from United States museums and over 80 from Greece. About 50 out of the total are representations of Alexander and his contemporaries but they all, like the written sources, date from much later periods. The rest of the exhibition, by far the most interesting part of it, displays a selection from the archaeological finds which in the last few decades have revealed an unexpectedly high level of culture and technique in the Macedonian kingdom. These superb pieces, bronze, silver and gold, are chance survivors, treasures the tomb robbers of 20 centuries failed to locate; in their time they were obviously choice items, but far from unique. The magnificent mixing-bowl from Derveni, for example, three feet high and covered with gilded figures in high relief and in the round -- one's first impression is of a gigantic work by Benvenuto Cellini -- was placed in the grave of an unknown young man from Larissa called Astion. It was found in 1962 and at that time we could only imagine what the objects in a royal grave would be like; now we know. In 1977, at Vergina (ancient Aigai, the burial place of the Macedonian kings) Manolis Andronikos opened an unpillaged tomb which he identifies, in his brief but illuminating account of the find, as that of Alexander's father Philip II. The treasures it contained are breathtaking and they are splendidly illustrated and described in the catalogue -- the royal diadem, the gold chest which contained the ashes, above all, the astonishing gold crown of oak leaves and acorns.

This catalogue, however, is more than a masterly pictorial and scholarly record of an exhibition; it is a souvenir of a rare event. When Greece finally won its independence in 1829, most of the available antiquities had already been sold by Turkish pashas and carted off by European connoisseurs to adorn the museums of London, Paris and Munich. Since then the museums of Greece have been filled with fresh treasures from its past but the government, understandably, has been reluctant to let them out of the country. It was only in 1976, in fact, that legislation was passed to make temporary exportation for exhibition possible; in 1979 the Metropolitan in New York put on display just over 100 items from Greece in a show entitled "Greek Art of the Aegean Islands." But these objects were already well known to visitors and by photographs; the treasures from Vergina are fresh out of the ground and, in fact, have not yet been exhibited in Athens. The American public which will be privileged to see this extraordinary show in Washington, Chicago, Boston, San Francisco and New York over the next two years owes a vote of thanks to the Greek Ministry of Culture and the Greek people.