Draper keeps in touch almost daily with Michael Fox, a guru whom he credits with transforming his negative attitude and turning him into a competitor considered by seasoned Australian observers to have the potential to become a second Rod Laver.
That is not the sort of judgement to be made lightly - "flattering, but a bit ridiculous and embarrassing," in Draper's view - Laver being the last man to win all four Grand Slam singles titles in 1969. Sampras, who idolises Laver, had hoped to complete his collection before running into Schaller.
Until this year, Draper had only two things in common with the great one: both are Queenslanders and both are left-handed. After spending 12 months under the guidance of Fox, an American with an Australian wife, Draper has become the latest player to demonstrate how big a part the mind plays in this game.
He simply refused to be beaten by Schaller, saving a match point in the third set, coming back from 0-3 in both the fourth and fifth sets and saving two more match points before advancing to the third round with a 4-6, 4-6, 7-6, 7-5, 8-6 victory.
It was his second consecutive five-set win - he defeated Sweden's Jonas Bjorkman the previous day - a performance which added substance to the words of praise being heaped upon Draper, who is ranked No 135 in the world and will be 21 on Monday.
Even Andre Agassi's coach, Brad Gilbert, declared the Australian to be one of the most promising players in the game, after he had lost to the world No 1 in the quarter-finals in Tokyo in April.
Although on the small side for the modern game at 5ft 10in, Draper has an impressive all-court game and served well enough yesterday to hit 21 aces past his Austrian opponent. He also had the nerve and imagination to caress a winning drop shot from the baseline at 4-5, 30-30 in the fifth set.
"I can't really believe I've won my first five-set matches," Draper said. "When you're down you've got nothing to lose, and you've just got to go out and take it. I can only be aggressive. I've never been the type to rally and work my points for 50 hits."
As for the help he has received from Fox, he said: "I've been through every stage, experiencing ups and downs, and I think I've learned that a lot of things that go on on the court are just unnecessary. I've matured, and I try to think clearly and be as positive as I can."
Draper now meets the American Richey Reneberg, ranked No 46, and both players are aware of the opportunities opened up in their quarter of the draw. Marc Rosset, the 16th seed, joined Sampras yesterday, losing to the Brazilian Fernando Meligeni.
Stefan Edberg's latest quest for the French title - and one of his last - ended in the second round with a 7-6, 6-3, 6-2 defeat by Michael Stich, a match of serve and volley that seemed almost incongruous among the baseline duels.
Stich was only in any danger after winning the final point. He celebrated by hitting a ball high into the stand, and was met by a hail of balls thrown by a group of youths, who thought they would add a little fun to the proceedings. On this occasion, Stich smiled.
Sergi Bruguera, the champion for the past two years, also enjoyed a wry touch, hitting a winning volley when his opponent, Emilio Alvarez, served underarm on match point.
Mary Pierce, according to her German opponent, Christina Singer, was so nervous on the Centre Court that her breathing could be heard from the other end of the court when she served. Pierce explained that she had a cold and would not have been surprised if spectators at the top of the stand had heard her breathing.
Perhaps Pierce, in turn, was conscious of the nervous crowd holding its breath whenever she went for a big shot, which was almost every time a ball came her way. The problem is that they are just as likely to boom out as in.
Most of them missed the lines by yards in the first set, which Singer ought to have won, but the adoptive Frenchwoman managed to find her range in time to advance to the third round, 7-5, 6-0.
The 14-year-old Martina Hingis will play Lindsay Davenport, the seventh seed, after defeating Mariaan de Swardt 6-1, 6-7, 6-2.
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