 The Roman Empire ended in 476. This started the Dark Ages, a period generally lacking in culture or scientific or technological advances, at least in Europe. Well, Western Europe, if you didn't look too closely. Actually, real historians call this period the Middle Ages. In Latin, this is Medi-Eli, so it's also known as the Medieval Period. An important change that occurred during the Middle Ages was the establishment of a system of primary education. Around 780, the French Emperor Charlemagne, who was actually German, recruited Alcuin of York, an English clergyman, to help create a network of cathedral schools. Alcuin wrote Proposetiones at Acuentes y Ovanes, Problems to Sharpen the Minds of the Young. The book consists of about 50 questions designed to promote critical thinking. Some of the problems are logic puzzles. How many footprints are left in the last furrow plowed by an ox? A man needs to take a wolf, a goat, and a cabbage across a river, but a boat can only carry the man and one of the objects. If left alone, the goat will eat the cabbage and the wolf will eat the goat. How can the man get all three across the river? Then there's this one. A father and son marry a widow and her daughter. The father marrying the daughter and the son marrying the widow. If each couple have a son, what is the relationship between the sons? However, many of the problems in Alcuin are mathematical in nature. Unfortunately, while Alcuin gave solutions, he rarely presented the methods. There are a few exceptions, notably in the geometry questions. For example, a foresighted field is 30 perches along one side, 32 on the opposite, 34 across the top, and 32 across the bottom. How many acres is the field? So one of the features about mathematics is that it has an internal logic. So we can infer that a purge is some unit of length, but what's an acre? Alcuin's computation will tell us. Alcuin's solution is the following. The sum of the two sides is 62, half of which is 31. The sum of the top and bottom is 66, half of which is 33. Multiply 31 by 33 to get 1023, divide by 12, divide by 12 again, so the field has an area of 7 acres. If the sides of the field are A, B, C, and D, Alcuin is using the formula. This is known as the Agromancer's Formula and was used by the Romans, Egyptians, and others to compute areas. This doesn't calculate the area as we know it, but that might not be the point. Instead, the area could be a legal term. In particular, the formula could be computing a cadastral area, an area used to compute the amount of taxes a landowner must pay. Alcuin's mathematical procedure also gives us some insight into medieval culture. The computation should give an area, geometric or cadastral, in square perches. Since Alcuin divides by 12 twice, we can infer that 144 square perches are equal to one acre. So, even if the problem uses archaic or obscure units that no one uses, like perches, acres, feet, or pounds, the mathematics tells us something about the units. Alcuin also computes the area of a circular field 400 perches in circumference. To do that, he divides by 4 to get 100 perches, squares to get 10,000, and again divides by 12 twice to give the number of acres. And we note that in contrast to the Mesopotamian method, it's not clear if the first division is meant to give the diameter. Alcuin also posed to solve the following problem. A staircase has a hundred steps. On the first step is one pigeon, on the second two, on the third, three, and so on. How many pigeons are there all together? To solve this, Alcuin noted that the first and 99th stairs had 100 pigeons. Likewise, the second and 98th, the third and 97th, and so on until the 49th and 51st stairs. These gave 4900 pigeons, and the only pigeons left uncounted are those on the 50th and 100th stairs. Adding those in gave the total 5,050 pigeons.