Pair the Ratio-Speaker

Pair — whose given name was Couplet, the same as the village she was born in, though everyone called her Pair from the time she was nine — grieved with her family for a long winter. She grieved for a year. She grieved, in a smaller quieter way, for the rest of her life. Her parents were patient with her grief. Her parents had grief of their own.

But — and this is the part of the story that matters for ratios — Echo did not simply leave Pair. Echo taught Pair about pairs.

In the years after Echo's death, Pair noticed something. She had always had a partner. In every memory of her early life, there were two of them. Two cradles. Two cots. Two dresses (always identical). Two sets of small footprints in the mud. Two voices when the family sang. Two hands held by their mother on the way to the market. Two.

After Echo died, there was one. Pair held one hand. Pair sang in one voice. Pair walked alone.

And Pair began, very gradually, to notice the world around her in twos.

Shoes came in twos. Eyes came in twos. Hands came in twos. Lungs came in twos. Wheels on a cart came in twos. Horses, when paired for a yoke, came in twos. The villagers' winter-mittens came in twos. The village's church-bells (two of them, north and south) came in twos. The world was, Pair realized when she was nine, organized into pairs. This was, she thought at the time, somehow because of Echo. The world was teaching her about pairs because she had lost one.

What Pair eventually understood — over the slow years of adolescence, and especially during a long summer when she was thirteen and finally read her first book of arithmetic — was that the pair was the irreducible unit of ratio thinking. When the village blacksmith made horseshoes, he made them in pairs because each horse has two front feet and two back feet — a 2-and-2 ratio of front-to-back, and a 4-to-1 ratio of shoes-to-horse. When the village potter made cups and saucers, he made them in pairs because each cup goes with one saucer — a 1-to-1 ratio. When the village weaver made warp and weft, every warp-thread had a corresponding weft-thread — a 1-to-1 ratio of crossings. The world was full of fixed pairings. The fixed pairings were ratios.

Pair, at thirteen, understood this in a way most children would not. She had spent six years thinking about pairs, because Echo had taught her to.

She extended the principle further. She noticed non-1-to-1 ratios. Two wheels per cart — that was a 2:1 ratio of wheels to carts. Four legs per chair — that was a 4:1 ratio of legs to chairs. Eight notes per octave — that was an 8:1 ratio in a sense she did not yet fully understand. The world was full of fixed proportions. Every fixed proportion was a recurring ratio.

When Pair was nineteen, she walked into the RatioRealm academy carrying a small wooden carving of two clasped hands. (Her mother had carved it for her after Echo's death; Pair had carried it in her pocket for twelve years.) She placed the carving on the academy master's desk. She said: "I would like to teach ratios."

The academy master, who knew nothing about Pair's childhood, asked her why.

Pair said: "Because I have been thinking about pairs for twelve years. I think I understand them well enough now to teach them."

The academy master invited her to demonstrate. Pair held up the carving. She said: "Each hand has five fingers. Two hands have ten fingers. The ratio of hands to fingers is 1:5. The ratio of pairs-of-hands to fingers is 1:10. Both ratios describe the same world. Both ratios are true. They are different expressions of the same pairing."

The academy master was, by his own later admission, moved. He had heard a great many lectures on ratios. He had not heard one start from a wooden carving of clasped hands. He invited Pair to join the faculty. She accepted.

That was eleven years ago.

In her classroom, she begins every first-day lesson the same way. She places the carving on the desk. She points at it. She says: "For every one hand, there are five fingers. That is a ratio. It is the same as saying 'one to five' or '1:5'. The colon is just a way of writing 'for every'. For every X, there are Y. That is the foundation of ratio."