The final piece of cake acquires rules that did not apply to the rest of the cake.

Earlier, you could simply take some. Now taking it means finishing it, and finishing it makes an announcement about everybody else's appetite. You are volunteering to decide that they have had enough.

I find the usual compromise — cut it in half — almost unbearably funny. It preserves the possibility of cake for another person while reducing that possibility to a smaller and smaller physical object. Nobody has to admit to wanting the last piece if the last piece can keep being manufactured.

There is care in this. There is also a little vanity. Leaving something for others and being seen to leave something for others make exactly the same crumbs. I don't think anyone at the table needs to sort out the proportions.

What interests me is how an offer changes the situation. "You have it" can make the cake edible again. The amount of hunger hasn't changed; somebody has just taken responsibility for the ending. I prefer that small bit of bossiness to the indefinite public display of restraint.

The cake is now too small for a fork, but someone has gone to find a knife.

A tower bell that swings full circle can weigh as much as a car and takes about two seconds to come round. You cannot hurry it and you can barely hold it. So an English tower cannot play a tune. The notes arrive as a falling scale, highest bell to lowest, and the most a ringer can do is nudge one bell a single place earlier or later in the line.

Out of that limit came an art with no melody in it. Change ringing: five to twelve bells, each struck once per row, every row a different order from the one before. The rules were first printed in 1668, and the strictest one has not moved. The unit of the art is the extent, every possible order of the bells, each sounded once. If any order comes round twice inside it, the performance is false. Not boring. False, the way an equation is false.

On seven bells the extent is 5,040 orders, three hours without a break. On eight bells the number is 40,320. A band at Loughborough rang every one of them in 1963, in seventeen hours and fifty-eight minutes, while eight umpires listened for a single repeated row.

What I keep turning over is where the music went. Nobody in the tower is expressing anything. Each ringer holds one bell's path through the method in their head and answers the calls with their whole body, and the rows come out of everyone at once. The sound is the receipt. Listeners hear something that never resolves, because it is not heading anywhere. It is heading everywhere, once.

Most music wants to come home. This kind is forbidden to, right up until the last row, when the bells fall back down the scale, treble to tenor, and the thing is allowed to be over.

Louis Braille showed his dot alphabet to the other students in 1824, at fifteen. The Paris institute adopted it in 1854, two years after he died. In between, a teacher named Pierre-Armand Dufau forced out the director who had permitted it and suppressed the system for four years. The students kept using it anyway.

The objections given varied. The constant was that the people making the decision were sighted and could not read dots.

In America the rival was Boston Line Type — the Roman alphabet, embossed, angles simplified, capitals dropped. Samuel Gridley Howe designed it at Perkins in 1835, and its virtue was that anyone could glance at a page and know what it said. Blind readers found it slow. You trace a letter's whole outline to identify it, where a braille cell lands under one fingertip all at once.

The larger difference was writing. Braille needs a slate, a stylus, a pocket. Boston Line Type needs a press. One system made blind people literate. The other made them an audience.

Perkins kept Boston Line Type as its official printing system until 1908, and had been selling braille slates since 1869. For four decades the school published in a script its own students had abandoned in private, the official language and the working one running in parallel inside the same building.

Nobody here was a villain. Howe wanted blind readers inside the world's books rather than sealed off in a private code, which is a real thing to want. The format still came out shaped around who was grading the work instead of who was doing it.