Only three football rules in use today remain unchanged from those laid down by the Intercollegiate Football Association (IFA) in 1876. Every other rule was created or evolved to address specific problems or otherwise improve the game. In many cases, the concerns that led to creating a rule are no longer relevant, but the rule proved useful for other reasons and remains in effect today. Such is the case with the longstanding penalty for roughing the punter. The rule was not put in place to protect the punter during the act of punting, but to protect the punter after he punted the ball. Why would football need to protect the punter after kicking the ball? To make that connection, we must step back to understand punting and the concept of "onside" during the early days in football's history.
Early American football was rugby. The IFA rules of 1876 copied many of England's rugby rules word-for-word with only a few minor modifications of substance. As in rugby, any offensive player could punt the ball at any point in a play and from anywhere on the field, but that changed with the addition of downs and the line of scrimmage in 1880. Once offenses had three downs to make five yards, punts could occur only behind the line of scrimmage. Teams commonly punted on third down just as they do today on fourth down, but they also punted on first or second down, particularly when inside their 20-yard line. That practice remained common until the 1920s when offenses developed that allowed teams to consistently move the ball. Only then did offensive play overtake the field position and punting-oriented game that prevailed to that point.
Whereas teams have specialist punters and punt formations today, teams originally punted from the T formation. The center snapped the ball to the quarterback, who tossed it to the fullback, who then executed the punt. It was not until the late 1890s that teams snapped the ball directly to the fullback in punting situations.
Unlike today's football rules, rugby and early football rules allowed the punter's teammates to recover the punted ball, provided they were "onside" or behind him when he kicked the ball. Those teammates who were behind the punter could run downfield and recover the ball, much like occurs with onside kickoffs today. (The halfbacks were typically behind the fullback after he took several steps and punted the ball.) The punter could also run downfield and, by advancing beyond his other teammates, place them back onside, making them eligible to recover the punt. Of course, the receiving team did not want the punter running downfield, so they commonly assigned a few players to knock him down, preventing him from heading downfield. Since fullbacks generally punted the ball, the defense's tactic was known as "roughing the fullback."
To take advantage of the offense's ability to recover the punt, Penn's coach, George Woodruff, developed the "guards back" formation in 1892. Woodruff positioned his guards behind the scrimmage line, increasing their ability to shift and block on mass and momentum plays. When defenses moved their safeties and defensive halfbacks close to the line of scrimmage to defend the mass and momentum plays, Woodruff had the quarterback "quick" punt the ball short and toward a sideline, away from the deep safety. Both guards and the three backs were behind the quarterback and eligible to recover what became known as the "quarterback kick."
The threat of the offense recovering a quick punt/quarterback kick forced defenses to keep a safety deep while other defenders had to remain spread out enough to be in a position to recover a wide kick, making defenses more vulnerable to inside runs.
Seeking to make the game safer in 1903, the rules committee required teams to have seven men on the line of scrimmage at the snap and eliminated the punter's ability to make their teammates eligible to recover punts by running past them downfield. The latter rule meant receiving teams could no longer justify knocking the punter to the ground, so the punter received special protection with the "roughing the fullback" penalty.
The rules changes of 1903 did not dramatically change the game, resulting in the need for many rule changes in 1906, including the legalization of the forward pass. Another rule change expected to open up the game was to expand the quarterback kick rule by creating the "onside kick from scrimmage." The onside kick rule made every player on the punting team eligible to recover the ball, not just those positioned behind the punter. In turn, this increased the likelihood of the offense recovering the kicked ball, forcing defenses to spread out and increasing the offense's ability to move the ball.
The onside kick from scrimmage received relatively wide usage in 1906 -some teams used it more often than the forward pass- and the next few seasons. It was made more challenging in 1910 when punts had to travel at least twenty yards downfield to be recovered by the punting team, and the 1912 rule book removed all versions of the onside kick. The old "quarterback kick" rule was reestablished in 1913 and survived until 1923, when it was eliminated from the American game due to its limited use. Though seldom used, the onside kick from scrimmage remains legal in Canadian football.
The shift to a rule protecting the punter as we conceive the rule today came in 1915 when the rule makers distinguished between roughing the kicker and running into the kicker. Since then, roughing the kicker has evolved into a foul that occurs during or immediately after the act of punting, and the penalty has been reduced from disqualification to fifteen-yard and five-yard penalties, respectively.
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You got the general idea right, but nearly all the details (especially the years) wrong. Let's see how many I can straighten out:
Until the changes of 1900, you didn't have to be onside at the time of the kick, only at the time of the recovery. That is, the kicker could put you onside AFTER kicking the ball by going past you upfield. The change you state as having been made in 1880 was from 1900, which was also the date of the roughing-the-fullback rule's introduction, for almost the reason you gave. Also beginning in 1900, the kicker, although onside, was ineligible to recover the ball. The three changes were to go together, removing the incentive and adding the penalty.
7 on the offensive line were not required until 1906. In 1903, a minimum of 6.
The other form of onside kick you describe both came in and went out later than the dates you gave, and required the ball to hit the ground beyond, and later 20 yards beyond, the line. The original form of onside kick didn't require the ball to hit the ground for recovery, and it continued until 1923, while the later version was abolished a few years later; they coexisted.