*This is a guest post from our very own Mike Schilling!*

Euclid’s Elements begins its study of geometry by listing five self-evident postulates, from which the whole of the science will be proven:

- A straight line can be drawn between any two points.
- A straight line can be extended indefinitely far in either direction.
- A circle can be drawn with any center and radius.
- All right angles equal one another.
- If a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if extended indefinitely, meet on that side on which are the angles less than the two right angles.

You might feel that the fifth is of a different character than the first four. What it says, more or less, is that if two lines appear to be approaching each other to one side, they will eventually meet on that side but not on the other. Or, to say it another way, the plane is infinite (or it might stop before they met) and infinitely flat (or the lines might eventually curve back away on that side, or even curve back together on both sides.) But it’s a very ugly, convoluted way of saying it (though quite useful for the proofs that followed), and geometers spent literally millennia trying to find ways to prove it rather than assume it. Not until two thousand years after Euclid did anyone ask seriously “But what if it’s not true?” and develop the mathematics that describes that situation. About a hundred years after that, Einstein’s Theory of General Relativity concluded that, in the real world, space isn’t perfectly flat:: it’s slightly curved, making Euclid’s geometry the same sort of very good and useful approximation as Newton’s physics. There are some very deep questions brought up by the existence of simple, almost always good enough approximations to the physical world’s actual complexity, e.g. how science might have developed in their absence. But those are for another day. Right now, we’re going to talk about baseball.

I brought up Euclid to illustrate the attraction of building an entire discipline from a few simple principles and their logical conclusions. Baseball comes very close to that with its rules for how runners can be put out. The primary concept is that of a force:

If a runner is forced to go to base, he can be put out by a defender holding the ball and touching that base.

Whether a runner is forced or not, he can be tagged out unless he occupies a base he is allowed to be on.

There are a few situations that create a force:

- A batter who hits a fair ball is forced to go to first base.
- When a batter hits a fair ball, any runner who does not have an unoccupied base behind him is forced to go to the next base until one of the runners behind him is put out.
- When a ball is caught on the fly by a defender:

The batter is out

Any runner who has left the base he begin on is forced to return there.

And that’s it. All the plays you’re used to seeing came from these simple rules:

When there’s a man on first base, and the ball is hit to the first baseman, he throws to second to start a double play. If he stepped on first, he’d remove the force from the runner, making it harder to put him out.

If, in the same situation, the first baseman catches a line drive, the runner is no longer forced to go to second (since the batter is out.: Rule 2) He can make a double play by beating the runner back to first base (Rule 3)

In the unassisted triple play (which is the rarest play in baseball), an infielder catchers the ball (Rule 3), steps on a base than a runner has left (Rule 3) and tags the runner approaching that base (Rule 2). Obviously, the ball was a line drive that the runners expected to go through, or they wouldn’t be moving that way.

There is one case in which these rules lead to an unfortunate conclusion. Let’s consider what happens if the ball is popped up, that is hit almost vertically, to an infielder. He has the choice of catching it or letting it drop and then picking it up, making any runners uncertain about which sort of force exists. Let’s look at the possible situations:

First base is open

This one is easy. A force exists only of the ball is caught, so the infielder should catch it and any runners should stick to their bases.

First base is occupied, but not second.

If the runner tries to advance, the infielder will catch the ball and throw it to first for a double play.

If the runner stays on first, then the infielder can catch it to put the batter out, or drop it and throw to second to put the runner out. Either way, only one out.

So the runner on first should stay there. Likewise, if there’s a runner on third, he should stay there, since he’s not forced regardless.

First and second are both occupied:

For the runner on first, this is the same as the situation above: he should stay out.

If the runner on second advances, the infielder can catch the ball and throw to second for a double play.

If the runner stays on second, the infielder can drop the ball and throw it to third base for one out. From there it can be thrown to second for the second out (since the other runner stayed at first.)

So it’s a double play regardless.

Bases loaded

This is not much different from the situation above, since the same double play is available. Also:

The man on third can’t advance until he knows whether the ball is caught.

If there is one out, the double play ends the inning.

If there are none out and the man on third tries to advance after the ball is dropped, the ball can be thrown home (where a force applies) and the double play completed at third base (where a force still applies).

That is, the logical result of the rules is that in certain circumstances, a pop-up can become a double play, with no poor base-running by the offense or especially skilled play by the defense. This seems unfair. To correct, it the infamous Infield Fly Run was created. It says, simply enough, that when there are fewer than two men out, both first ans second base are occupied, and in the opinion of the umpire an infielder will be able to pull this trick, the batter is declared out. This removes the force and eliminates the double play.

That’s all there is to it.

One story: Frank Robinson is a big, tough, frequently angry man who as a manager was a frequent terror to umpires. One day, his Expos were playing the Giants, who had the bases loaded with one out. Barry Bonds hit a weak pop-up straight up, and was called out by the umpire. The catcher both lost the ball and missed the call. When he finally located the ball, he stepped on home plate, thinking he had forced out the runner at third, who cleverly walked towards home acting as if he’d been put out and was going back to his dugout. But he stayed in the base-path and stepped on home for a run. The catcher who had no clue what had happened, started to argue with the umpire. At this point, out raced the fearsome Frank Robinson, to yell at his own players. “Shut the bleep up! You’re embarrassing me!” And then, with a friendly shrug to the umpire, Frank went back to his bench.

It says, simply enough, that when there are fewer than two men outSo if there are two men out, exactly, the best thing to do is just for the runners to start booking?

Absolutely, especially on a fly ball where the inning is over anyway if it’s caught.

If it wasn’t for that darned no politics rule, I’d have to ask about the Cards-Braves wild-card game.

One caveat, with this part:

First base is occupied, but not second. . . .So the runner on first should stay there. Likewise, if there’s a runner on third, he should stay there, since he’s not forced regardless.

It really depends on where the fly is at in the diamond.

If the fly is away from first base, the runner should run.

One out, regardless of whether at first or second.

If the fielder throws to first, the runner has removed the force behind him.

The throw to first would be longer, and the throw to second would be behind the fielder.

If the fly is toward first base, the runner can tag the base and be on to second, removing the force behind him.