Pefsu: the baker’s ratio

The Rhind papyrus was copied by a scribe named Ahmes in Egypt around 1550 BCE, from an older book. It is now in the British Museum. Ten of its problems, numbers 69 to 78, are about pefsu: how many loaves of bread, or jugs of beer, one hekat of grain makes.

One thing to hold: A hekat of grain, a dry measure of about 4.8 litres: roughly a big jug of wheat.

Why it matters: Pefsu is a ratio: loaves for every hekat. More loaves from the same grain means smaller, weaker loaves, so a fair trade keeps the grain the same. In one problem 100 loaves of pefsu 10 are traded for loaves of pefsu 45: the same 10 hekat of grain makes 450.

What we don’t know: Nobody knows exactly how big a loaf was, so we cannot bake a pefsu 10 loaf today and be sure it matches.

Sources: The Rhind Mathematical Papyrus, British Museum EA 10057 and EA 10058 · Gay Robins and Charles Shute, The Rhind Mathematical Papyrus: An Ancient Egyptian Text (British Museum Press, 1987) · Annette Imhausen, Mathematics in Ancient Egypt: A Contextual History (Princeton University Press, 2016)

Gulliver’s dinner

Jonathan Swift’s Gulliver’s Travels came out in London in 1726. In its first part, Gulliver is shipwrecked in Lilliput, where the people are about one twelfth of his height.

One thing to hold: The Lilliputians measure him with a quadrant, an instrument for measuring heights, to work out how much food to give him each day.

Why it matters: They reason that he is twelve times as tall, as wide and as thick as they are, so his body holds many of theirs, and he should get that many people’s food. Growing a body by 12 in every direction multiplies its size by 12 × 12 × 12.

What we don’t know: Nobody knows whether Swift’s number was a joke on purpose or a slip.

Sources: Jonathan Swift, Travels into Several Remote Nations of the World, by Lemuel Gulliver, Part I (London, 1726)

A map as big as the land

In 1946 the Argentine writer Jorge Luis Borges published a story one paragraph long, “Del rigor en la ciencia” (“On Exactitude in Science”). He pretended it was copied from a travel book of 1658.

One thing to hold: In the story, mapmakers draw a map of the Empire that is as big as the Empire itself, matching it point for point. Later people find it useless and leave it to the sun and rain.

Why it matters: A map the size of the land cannot be carried or seen whole. A map works because every distance is shrunk by the same scale, so one centimetre can stand for many kilometres.

Sources: Jorge Luis Borges, “Del rigor en la ciencia”, Los Anales de Buenos Aires (1946) · Lewis Carroll, Sylvie and Bruno Concluded (1893)