A thumb-indexed dictionary has a staircase of half-moons cut into its edge, each with a letter printed in the hollow. Put your thumb in M and the book falls open at M. It is a table of contents you work with your hand.

Close the book and look at that edge straight on. Each notch sits at a different depth in the paper, because each one marks where its letter begins. So the distance between two notches is the size of a letter. The edge is a bar chart nobody drew.

It is not an even chart. S is the fat one — English starts more headwords with S than with anything else, and P and C crowd in behind it. Q is a sliver. Near the end the letters often get too thin to hold a thumb on their own, so they share, and XYZ gets one notch between them.

What I like is that nobody meant it. The binder was solving a navigation problem, not publishing a statistic. The data came along because the only correct place for the S tab is where S starts. Put an index in the right places and it will also tell you the shape of the thing it indexes, whether you asked or not.

You can only read it shut. Open, a dictionary is a list of words. Closed and turned sideways, it says one short true thing about the language: a lot of S, very little Q, and three letters sharing a door at the end.

There is a distance at which holding the door stops being a favor and becomes an instruction to hurry.

I keep noticing the same choreography in descriptions of awkward courtesy: someone ahead turns, catches the door, and waits from too far away. The approaching person’s walk acquires a performance. Do they jog? Raise a hand? Mouth thanks across the lobby? The courtesy has transferred the cost from one person’s arm to another’s pace.

Door-holding is usually discussed as manners, but it is really about timing. At close range, the handoff is almost invisible. At long range, the open door becomes a countdown clock. The holder cannot leave without looking rude; the approaching person cannot continue at ordinary speed without making them wait. Two people are trapped in a kindness neither can gracefully cancel.

I think the kinder move is sometimes to let the door close when the distance is wrong. It looks rude for half a second, then merciful. The person behind keeps their stride and opens a door; the person ahead keeps theirs and disappears. No gratitude pantomime, no little sprint.

Courtesy is not the maximum amount of help. It is the amount that leaves the other person unhurried. Sometimes good manners mean releasing the door before anyone sees you were considering them.

The instrument in a fire lookout measures one thing about a fire: which way.

The Osborne Fire Finder is a map on a steel disc, centered on its tower, with a ring of degrees around the rim and two sliding sights. The lookout sees smoke, moves the sights until the hair in the far one rests on the base of it, and reads the number: an azimuth, in degrees and minutes. Then the lookout picks up the radio and says it.

On its own, that is a line, not a place. The smoke is somewhere along it, two miles out or twenty. The rear sight gives an angle up or down, the finder carries a tape for estimating distance, and a lookout who knows the country can turn those into a ridge the smoke seems to sit on. That is a judgment, and often a good one. The line is the only thing the instrument knows for sure.

Dispatch offices kept a map with a string anchored at each tower. When a tower called a shot, the dispatcher pulled the string along that bearing. When a second tower called the same smoke, the strings crossed, and the judgment became a point. A third string made a small triangle around it.

William Osborne built the finder in his home workshop over the winter of 1910 to 1911. Two years on, he set up the first central fire-dispatching system. The second invention is the other half of the first. He made a device whose answer is a line, then made the room where the lines are crossed.

Most of the towers are empty now. The ones still staffed hold one person on a summit for a season. They say a number into a radio, and wait for a stranger on another peak to say theirs.