High hopes continued to fall as the day moved on. Carlos Moya, the defending men's singles champion, was unable to contain a resurgent Andre Agassi, who recovered from shaky start on the Centre Court to overcome the Spaniard 4-6, 7-5, 7-5, 6-1, showing glimpses of the form that took him to consecutive finals here in 1990 and 1991.
Agassi, the No 13 seed, has yet to add the French title to the Wimbledon, United States Open and Australian Open championships that marked him out as a winner earlier in the decade.
Young Hingis, too, is keen to complete her collection of Grand Slam singles titles. She will be aware of the left-handed Schwartz's determination and impressive cross-court backhand, but is no doubt relieved not to be facing Williams, the No 5 seed, whose growing confidence on clay accounted for her in the semi-finals of the Italian Open.
Williams, the form player of the clay court season, had never seen Schwartz play until they walked on court.
"I was occupied playing my own matches," the American said. "I couldn't go. I didn't know who exactly I was playing. I didn't watch the draw so much."
Considering that Williams had won 22 of her previous 24 matches and had not failed to advance to the quarter-finals, at least, of the last six Grand Slam championships, her decision not to have Schwartz scouted could be taken either as a sign of self-confidence or of complacency.
"It didn't happen, what else can I say," Williams said. "I mean, what's the difference? You got out there, you compete, you play the ball, you play the point. I have to be prepared with my game."
To be fair, the 18-year-old American did not need a dossier to dominate her opponent in the opening set, and although the second set was closer, Williams created three match points at 6-5. Schwartz showed impressive resolve in salvaging the situation and forcing the tie-break, which she won on her third set point, 9-7. "I wasn't afraid," the 5ft 11in Schwartz said, "because I have my own power."
The forehand was believed to be Schwartz's best weapon until she began to torment Williams with the one-handed backhand in the concluding set. Schwartz broke for 3-1, was broken back for 3-2, but was encouraged when Williams netted a forehand, enabling Schwartz to serve for the match at 5-3. The Austrian suffered a couple of blips from 40-0, before producing a winning backhand down the line to convert her third match point.
Although Jennifer Capriati's hearty comeback from semi-obscurity was interrupted by Lindsay Davenport, the No2 seed, 6-2, 6-3, but the former prodigy has achieved enough during the past few weeks to suggest that her career, which was launched in sensational fashion in 1990, is far from finished.
"It wasn't a case of me running out of steam, or whatever," Capriati said. "She played really well. It's been a while since I've played her, and she's improved since the last time I played her."
The 23-year-old American hopes to continue her progress at Wimbledon. "I'm really looking forward to going there," she said. "Grass is one of my favourite surfaces. I'm just going to take a few days off and get back into working hard - working harder. I'm very optimistic about the future. This is just like a starting point for me."
Davenport next plays Steffi Graf, the No6 seed, who wore down Anna Kournikova 6-3, 7-6. In the semi-finals, the winner will meet either the third-seeded Monica Seles, last year's runner-up, or the unseeded Conchita Martinez. Arantxa Sanchez-Vicario, the defending champion, had a walk-over to the last eight when her fourth-round opponent, the American Mary Joe Fernandez, withdrew because of a thigh strain.
Marcelo Rios recovered from two sets down to defeat Alberto Berasategui, of Spain, who had a similar escape before eliminating Tim Henman in the previous round. Rios, the No 9 seed, overcame Berasategui 3-6, 3-6, 6- 3, 6-4, 6-3. The Chilean former world No1 plays Dominik Hrbaty, of Slovakia, in the quarter-finals.
Pat Rafter's prospects of becoming the world No 1 here disappeared in a flurry of groundstrokes delivered by the Brazilian, Fernando Meligeni, in the third round on Saturday, leaving the Australian No 3 seed to join the likes of Pete Sampras and Tim Henman contemplating the grass courts.
Although Rafter's attacking game would appear to suit the pace of Wimbledon as much as the concrete courts of New York, where he has won the United States Open title for the past two years, he seldom makes an impact at the All England Club. "My movement is probably my biggest downfall on the grass," he said. "I prefer to move on clay than I do on grass. The game is there, I've just got to learn to try to stand up a little bit. I've lost the last three years there in the fourth round. I've been consistently ordinary."
Rafter, who intends to work his way towards Wimbledon via grass-court tournaments in Halle, Germany, and Rosmalen, in the Netherlands, sees Sampras as the man to beat again, in spite of the American's slow start to the year.
"Pete's got an excellent game for grass," Rafter reminded us. "His serve is very difficult to get back. He returns well. He's quick. He's strong. The points are short. You don't have very long matches. That's everything Pete needs, especially at Wimbledon. Although he hasn't had very good results this year, I'd still put him as a heavy favourite. I wouldn't like to play him."