Roots. They run fast and deep through the history of thoroughbred breeding; to 1700 on a formal, structured basis, then on back to the Crusades and the Ottoman Empire for sire blood and to the Roman invasion of Britain for mare lines.

The pedigrees often appear to be hopelessly intertwined, and there are a few blank spots along the way, but it amazing how bloodlines can be traced so readily, to their origin, with the help of computer terminals and the prime reference material found at The Jockey Club Statistical Bureau here.

Take Seattle Slew, for example. The undefeated 3-year-old colt will evoke on Saturday the greatest rooting interest in the history of the Kentucky Derby, in terms of money bet on the favorite. So the time had come to shake down his family tree, John Sparkman, manager of the bureau's pedigree services, needed less than two hours to fill out all the branches.

"Seattle Slew goes back 25 generations on his sire's side, 28 generations in tail-female." Sparkman informed, quickly producing all 53 names, in order, to prove the point. "He traces to The Darley Arabian on the top and to Royal Mare, the foundation mare for Family 13, on the bottom."

Pedigree people pontificate like this. They employ a language of their own when they talk of their favorite subject, and they enjoy every minute of it.

There would be 67,108,864 names, or 2 to the 25th power, in Seattle Slew's complete bloodline diagram. Enough," Sparkman declared, "to require a piece of paper the size of a football field to accommodate a 25 cross on a printout. We did it once, with Secretariat."

The starting point for a bloodlines bloodhounds is Volume One of the General Stud Book, dated 1791 and published in 1793 in England, where thoroughbred racing began in the 17th century.

"The first racing calendar was published in 1723 or 1724," Sparkman noted. Unlike us, the English keep everything. They don't throw anything away, thank heavens. They gave us the basis. The Romans brought the first horses to England during Julius Caesar's time but for more than 1,000 years those horses had no real effect on what later evolved.

"Then came the Crusades, in the 12th, 13th and 14th centuries. The knights took their chargers to help liberate the Holy Lands from the Turks and the Arabs. The Arab horses outran their horses, badly. And it wasn't because the knights' horses were loaded down with equipment. The result was that the knights returned home with better horses than they went down there with, horses captured as chattels of war.

"These Arabian stallions were highly inbred. The Arab sheiks didn't have much of a mixture among their mares. When the Arab stallions were bred to the highly outbred English mares, they produced something even better. Genetically, of course, that's always a good situation.

"There was a gradual buildup of this Arab blood on this English base, a real hybridization for about 400 years, from 1200 to 1600, the pure blood of the Arabian stallions on the cold-blooded English mares."

And so it began, quite loosely at first, but more and more structured with time, the origins of the American racehorse.

Three Arabian stallions. The Byerley Turk. The Darley Arabian and The Godolphin Arabian, are pictured as the three foundation sires.

"Actually, there were more. Many more," Sparkman noted. "There are over 100 foundation sires in Volume One of the GSB (General Stud Book). There were, in fact, others from 1620 to 1797, but three lines in tail-male did better than all the others."

We know them today as Eclipse (Darley Arabian), Herod (Byerley Turk) and Matchem (Godolphin Arabian). The Darley Arabian, Seattle Slew's 25th-generation sire, is described in the stud book as "probably a Turk or Syrian horse, brought over from Smyrna by a brother of Mr. Darley of Yorkshire, who being an agnet in merchandize abroad, became a member of the hunting club by which means he acquired interest to procure this horse."

Matchem's line gave up Man o' War and other great horses of the past, but it is now almost dormant. Herod continues powerful in France but Eclipse - through Pharlaris, Pharos, Nearco, Nasrullah and Bold Ruler - now dominates American and world pedigrees with such comparatively recent sensations as Secretariat and Seattle Slew among others.

"It's always interesting to research some of those horses," Sparkman said. "In Seattle Slew, for instance, the fifth sire from the beginning is Potoooooooo's, a chestnut son of Eclipse, born in 1173. Historians tell us the stable hands called this horse Potatoes. He was to be named Potatoes but one day, before he ran in competition, the trainer told the groom to write the horse's name on the stall door in chalk. The groom, not being too good in spelling, wrote, 'Pot' and eight o's."

At this point in the conversation, Sparkman referred to "The Family Tables of Race Horses," compiled by two Polish Gentlemen. Bobinski and Zamoyski, as the means of tracing Seatle Slew to his 28th dam, Royal Mare, the foundation for Family No. 13.

"This is a good family, as good as any," Sparkman insisted. "No. 1 is considered the best, but I think only because there were more of them. The No. 13 family is excellent, with Frizette and Myrtlewood in there."

There are those who question the staying power or stamina transmitted by Seattle Slew's sire. Bold Reasoning, a grandson of Bold Ruler, Sparkman has no such apprehension.

"I thought Bold Reasoning was the best horse of his generation, except for Hcist The Flag and, maybe, Canonero," he said. "Bold Reasoning, like his sire, Boldnesian, was unsound, but he won eight of 11 races. He was a huge animal, over 17 hands, with brilliant speed, and some of the smartest breeders in this area bred top mares to him as soon as he came to stud at Claiborne Farm."

Ben Castleman, who owns a small farm near here, bred My Charmer to Bold Reasoning to produce Seattle Slew. The colt was sold at auction as a yearling to Mickey and Karen Taylor of White Swan, Wash., for $17,500. The rest, as they say, is history.

Except for the Kentucky Derby. Sparkman likes Seattle Slew's breeding for the mile and a quarter but he remains true to the blue blood in his breeding books when he is asked to pick a winner, on pedigree.

"It's got to be Flag Officer, off the quality of his sire and dam," Sparkman said. "He is by Hoist The Flag out of Batteur, a Bold Ruler mare who was the first of Bold Ruler's fillies or colts to win over a mile and a quarter. You can't beat Flag Officer, on paper."

Which is, of course, another breeding story.