Harvard and Yale played a rugby challenge match in 1875. The second meeting came in 1876 and was played a few days before the formation of the Intercollegiate Football Association, which established the first standard rules for American football.
Freshman Walter Camp played for Yale in the 1876 game and had not yet fathered football. Still, there are some interesting elements in the game reports.
The passage below from the New York Daily Herald tells us Yale won the game by kicking one goal, while Harvard scored two touchdowns. At that point in the game’s development, the team scoring the most goals won the game. The primary value of a touchdown was that it provided the opportunity for a free kick, but you still had to make them, and Harvard did not make the kick after either touchdown.
Things got a bit chippy near the end of the first half, but the boys settled their differences like gentlemen.
Thompson of Yale kicked the game-winning goal in the second half (referred to as an inning at the time). The celebration following the goal lasted twenty minutes.
Details are few as to just how the lady friends in the carriage congratulated Thompson. If anyone else has additional information, please let us know.
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Based on the photo and the description of Yale placing the ball on the ground before the goal-from-field during open play, I assume they were playing with a round soccer like ball.
I also read years ago that pre-1876, a goal from the field (field goal) wasn't even permitted, that the try-at-goal after a touchdown was the only way to score, though I haven't found much else about this since the rules during that era are murky.