"Leonardo da Vinci: Drawings of Horses from the Royal Library at Windsor Castle" is a spiraling exhibit. It starts here on our own plane, and then it spins beyond.

The horses we encounter in these 50 precious drawings begin as evocations of the smoky battleground, the stable yard, the meadow, the statue-decorated square. But then, to our amazement, they transcend such earthly themes. Leonardo's horses are not animals exactly. They are beings made round by light. They are vessels that contain -- in addition to the veins and sinews that he measured so exactly -- the harmonies that rule the universe itself.

They go on view today in the West Building of the National Gallery of Art, and there are many ways to read them. Leonardo's horses certainly are weapons. They scream and bare their teeth and trample men beneath their hooves. ("To represent a battle," the master told his pupils, "paint first the smoke . . . mixed in the air with the dust raised by the horses . . . Paint them with little clouds of dust as far apart, one from the other, as are the intervals between the strides . . . Paint a horse dragging his dead master, and leaving behind in the dust and mud, the trace of the dragged body . . . " He knew the arts of war.

And those of antiquity. Leonardo's horses often arch their necks and lift their hooves as do those of Greek statues. Imitating "antique work is . . . praiseworthy," he wrote.

He was a scientist as well. In a day when speed and power, and victory in war and the grandeur of the prince, were dependent on horses, Leonardo gave them his meticulous attention. He kept them, and he raised them and, according to Giorgio Vasari, "trained them with great kindness." He also skinned them and dissected them and measured them with care. Many of the master's ruled and measured drawings are included in this show.

In the last years of his life, horses sometimes rode riot through his dreams. Some appear to fly. Some are giant beasts with trunks, half-elephant, half-charger. Yet even these monsters are informed by a lifetime of observation. We know, from his notebooks, that he often drew -- from life -- horses he admired. They rear, they graze, they amble, they toss their heads about. The raised forefoot of one stallion, called the Siciliano, is depicted in this show. So is a Spanish jennet bred in the same stable and described by one contemporary as "verye seemely to the eye."

When Leonardo studied horses chomping grass before him, he did not see what we might see. Or did not see that only. His eye searched for perfection and drilled its way beyond the surfaces of things to detect the ideal reined in by the real. Beauties and strengths, the powers and the graces, unite in these drawings.

Leonardo's horses, like his angels and madonnas, are true to life -- and yet too good to be true. We know that they've been measured. They also have been blessed.

Can perfection be enumerated? Can it be defined by geometry? The notion that the beauty of the human body resides in the harmony of its proportions is one that Leonardo borrowed from antiquity and all his life pursued. The Mona Lisa's smile has a circle hidden in it. The curling locks that grace the necks of Leonardo's angels also seem to yearn to partake of the circular. The naked -- and ideal -- man who spreads his arms and legs in the famous drawing in the Academy in Venice has been inscribed in both a circle and a square.

A similar sort of beauty -- mystically arrived at -- is felt throughout this show. And it is that eerie rightness -- that perfection derived half from careful observation and half from speculation -- that gives this show its rising spin.

That spiraling intensity is apparent in one sketch here from about 1503 (reproduced, in part, here on the front page of Show at upper left). Its horses were not drawn from life, but from some dream of frenzy. Surely, they are steeds of war. Their eyes bulge and their nostrils flare in sketch after sketch on the same sheet. One almost hears their screams. But then those screams transmute. As the viewer's eyes proceed, following the artist's hand, those enraged screams become -- not a horse's whinny, but a lion's roar. And the transmutation has not ceased. A man's face, just as fierce, also seen in profile, with eyes dark and mouth open suddenly appears.

The drawing is two-sided. There is one horse on the back. The notes and arcs and circles scattered all around it do not treat of war, nor of those emotions shared by horses, men and lions. They deal with eclipses, with the squaring of the circle and with an explanation of the blueness of the sky.

The mind that just a thought ago was considering a screaming horse has been sent into the cosmos.

The viewer cannot help but feel that sense of whirling progress in many of these drawings. The animal they summon up is more than just a horse.

Leonardo is, of course, not the first grand artist to drench likenesses of horses with otherworldly meanings.

One small horse of large power was carved, of mammoth ivory, in Vogelherd, in Germany, 30,000 years ago. Another, riding on the wind, the "Flying Horse of Kansu," was made in China in the 2nd century A.D. The Greeks and Romans filled their art with horses, as did Rubens, Delacroix, George Stubbs, Degas, Eakins and countless more. There is a screaming horse in the "Guernica" of Picasso.

Certain telling images are sufficiently elastic to carry hugely complex meanings. They are essential to the magic that resides in works of art.

The wind-waved cloth is one of these. The judge's robe, the hero's cape, the pennant and the flag partake of the same image. It is rich enough to represent the turning cycle of one's life from swaddling cloth, to matrimonial veil, to shawl, to burial shroud. The red thorny rose is another poignant image, the sailing moon a third.

As countless little girls learn each year anew, no image in the realm of art contains more complex thought, or power, than that of the horse.

The unicorn of spring, and the proudly prancing steed of summer, and the horrid, skinny nags of the dead time of year share a pedigree that's magical. Horseshoes bring us luck, nightmares bring us terror. The Four Horsemen destroy us, the Man on Horseback saves us.

Leonardo knew all that. Many of these drawings, and some of the most beautiful, were made in preparation for the casting of two grand equestrian bronzes. One -- 24 feet high and weighing 158,000 pounds -- was to "endue with immortal glory" Francesco Sforza of Milan. The first studies show a rearing horse, the next a horse parading. A full-scale model was constructed, but the bronze was never cast. Neither was the life-size horse-and-rider designed by the master as a monument to Sforza's enemy, Giovanni Giacomo Trivulzio.

The famed fresco of the "Battle of Anghiari," which Leonardo designed for the Palazzo Vecchio in Florence, was similarly ill-fated. Michelangelo was to have produced a wall painting for the same hall, but he never started. Leonardo's was begun in 1505, but never finished. Rubens sketched it, so we know its composition. Many other artists admired it immensely. Scholars have been searching for its trace now for years, but none has yet been found. It now seems likely that Leonardo's fresco, or, rather, the portion he completed, was destroyed at the Grand Duke's instigation in 1563.

Leonardo as a young man occasionally drew unicorns, those images of chastity and spring. One (not in the show) depicts a unicorn bowing gracefully to purify with its horn the waters of a spring. Leonardo as an old man conjured many monsters. A number of his dragons, and one beast with many wens and a loosely hanging lower lip, are included in this show.

These horses, too, are symbols. But most of them are symbols of something much less obvious. Look, for instance, at the splendid drawing on blue prepared paper -- "Right profile of a horse and details of its fore-legs and chest, in frontal and three-quarters views" (circa 1490), the one that's reproduced at center left on the front page of Show. In the exceptional catalog, which he wrote with Windsor's Jane Roberts, Carlo Pedretti notes that the large horse, in profile, fits into a square almost exactly. But that is not the source of this image's great power. Its power comes from light.

It is likely that Leonardo prepared the paper -- that is, painted it blue before he drew on it. Look closely and you see that his brushwork is not entirely even. The whiteness of the paper shows through here and there. And it brings the horse alive, giving to its coat an amazing glossy shimmer. Leonardo must have seen that horse, and the highlights on its flanks, before he began to draw.

Another red-chalk drawing here, an image of a rearing horse related to the "Battle of Anghiari" fresco (seen on first page of Show at lower right) combines that sense of holy and otherworldly light with a whirling action, a violent, twisting spin.

The Queen owns perhaps 600 Leonardo drawings. Most have been in England for 400 years. Ninety-two of these show horses and 50 of the best have been borrowed for this show.

Paul Mellon, the Gallery's chairman and its founder's son -- and a man whose loves include horses, art and England -- paid for its exhibition here. Another show of horses, also Mellon-funded, is currently on view at the Yale Center for British Art, the New Haven museum he gave to his alma mater. The Yale exhibition is devoted to the motionless, serene horse paintings of England's George Stubbs (1724-1806). Many of these canvases, and others brought from Britain, will be displayed here in the Gallery's West Building from May 3 to June 9.

The horses of Picasso, most of them at least, evoke the victims of the bullfight. Those of Edgar Degas, drawn to suggest speed, look like bullets or torpedoes. Stubbs' are like statues, and still in their perfection. They feel like hollow Chinese jars.

Leonardo's horses move. Leonardo's horses glow. They certainly suggest grace, correct proportion, speed and immense power. But they also conjure more. In their whirlings and their shinings, their harmonies and roundnesses, one can hear them sing the music of the spheres.

Only a few of them have been displayed before in America, although this exhibit was seen last summer in Florence. It will travel to Houston and to San Francisco after closing here June 9