Over four years, Gostkowski made 70 of 92 field-goal attempts and 159 of 165 extra points. He showed his versatility by recovering his own onside kick and his foresight by using an N.F.L.-size tee on kickoffs. He left Memphis as the 13th most productive scorer in N.C.A.A. Division I-A history.

Schoenrock thought football was Gostkowski’s best bet for the future. Nonetheless, baseball scouts had inquired about Gostkowski, impressed by his ability to throw hard sliders in the 90-mile-per-hour range and apparently undeterred by his career record of 7-22 with a 6.04 E.R.A.

“It was harder for me with baseball,” Gostkowski said. “I didn’t do it year round.”

Because of football, he missed the fall workouts with the baseball team. In the spring, during the baseball season, Schoenrock had his pitching coach bring a bag of footballs to practice.

The Patriots were among a handful of teams that worked him out at the Liberty Bowl. They were about to part ways with Adam Vinatieri, their kicker since 1996.

“I had no idea what was happening,” Gostkowski said. “They don’t give you any inclination. I thought I did pretty well. I didn’t do the combine. I went to the Senior Bowl. I pitched my last game a week before the draft. I was in my own world.”

He is still very much in his world, one in which he has surpassed Vinatieri, a likely Hall of Famer, as the Patriots’ scoring leader. His accuracy is astounding. He has converted an N.F.L. record 432 consecutive extra points since his only miss, during his rookie season. He has made more than 87 percent of his field-goal attempts, including all 12 this season. Among them is a career-best 57-yarder.

Not only is he the leading scorer in franchise history, he is No. 2 in continuous service behind Tom Brady and, at 31, second in age to Brady. A case could be made that not only are they the two oldest players on the Patriots, they are also the two most secure. Belichick has not even bothered to bring in a kicker during training camp to push Gostkowski.