2. Egypt

I know I don’t have to tell you about Egyptian civilization, but did you know the Egyptians had maths?

Problem number 56 from the Rhind Mathematics Papyrus (dated to around 1650 BC):

Seked of the Great Pyramid

If you construct a pyramid with base side 12 [cubits] and with a seked of 5 palms 1 finger; what is its altitude?^[1]

Most Egyptian geometry questions appear to deal with more mundane matters, like the dimensions of rectangular fields and round granaries, rather than pyramids. (The Egyptians had not yet worked out an exact formula for the area of a circle, but used octagons to approximate it.)

A “pefsu” problem involves a measure of the strength of the beer made from a heqat of grain, called a pefsu.

pefsu = (the number of  loaves of bread [or jugs of beer]) / (number of heqats of grain used to make them.)

For example, problem number 8 from the Moscow Mathematical Papyrus (most likely written between 1803 BC and 1649 BC, but based on an earlier manuscript thought to have been written around 1850 BC):

Example of calculating 100 loaves of bread of pefsu 20:

If someone says to you: “You have 100 loaves of bread of pefsu 20 to be exchanged for beer of pefsu 4, like 1/2 1/4 malt-date beer,”

First calculate the grain required for the 100 loaves of the bread of pefsu 20. The result is 5 heqat. Then reckon what you need for a des-jug of beer like the beer called 1/2 1/4 malt-date beer. The result is 1/2 of the heqat measure needed for des-jug of beer made from Upper-Egyptian grain.

Calculate 1/2 of 5 heqat, the result will be 2¹⁄_2. Take this 2¹⁄₂ four times.

The result is 10. Then you say to him:

Behold! The beer quantity is found to be correct.^[1]

“Behold! The beer quantity is found to be correct,” is one of the most amusing answers to a math problem I’ve seen.

The Egyptians also used fractions and solved algebraic equations that we would write as linear equations, eg, 3/2 * x + 4 = 10.

But their multiplication and division was really weird, probably as a side effect of not yet having invented a place value system.

A. Let’s suppose you wished to multiply 9 * 19.

B. First we want to turn 9 into powers of 2.

C. The powers of 2 = 1, 2, 4, 8, 16, 32, 64, etc.

D. The closest of these to 9 is 8, and 9-8=1, so we turn 9 into 8 and 1.

E. Now we’re going to make a table using 1, 8, and 19 (from line A), like so:

1        19
2        ?
4         ?
8         ?

F. We fill in our table by doubling 19 each time:

1        19
2        38
4         76
8         152

E. Since we turned 9 into 1 and 8 (step D), we add together the numbers in our table that correspond to 1 and 8: 19 + 152 = 171.

Or to put it more simply, using more familiar methods:

9 * 19 = (1 +8) * 19 = (19 * 1) +(19 * 8) = (19 * 1) + (19 * 2 * 2 * 2) = 171

Slab stela of Old Kingdom princess Neferetiabet (dated 2590–2565 BC), with number hieroglyphs

Now let’s do 247 * 250:

The closest power of 2 (without going over) is 128. 247 -128 = 119. 119 – 64 = 55. 55 – 32 = 23. 23 – 16 = 7. 7 – 4 = 3. 3 – 2 = 1. Whew! So we’re going to need 128, 64, 32, 16, 4, 2, and 1, and 250.

Let’s arrange our table, with the important numbers in bold (in this case, it’s :

1       250
2        ?
4         ?
8         ?
16       ?
32       ?
64       ?
128      ?

So, doubling 250 each time, we get:

1       250
2       500
4       1000
8        2000
16     4000
32     8000
64     16,000
128    32,000

Adding together the bold numbers in the second column gets us 61,750–and I probably don’t need to tell you that plugging 247 * 250 into your calculator (or doing it longhand) also gives you 61,750.

The advantage of this system is that the Egyptians only had to memorize their 2s table. The disadvantages are pretty obvious.

Berlin Papyrus

See also the Lahun Mathematical Papyri, the Egyptian Mathematical Leather Roll, the Akhmim wooden tablets, the Reisner Papyrus, and finally the Papyrus Anastasi I, which is believed to be a fictional, satirical tale for teaching scribes–basically, a funny textbook, and the Berlin Papyrus 6619:

The Berlin Papyrus contains two problems, the first stated as “the area of a square of 100 is equal to that of two smaller squares. The side of one is ½ + ¼ the side of the other.”^[6] The interest in the question may suggest some knowledge of the Pythagorean theorem, though the papyrus only shows a straightforward solution to a single second degree equation in one unknown. In modern terms, the simultaneous equations x² + y² = 100 and x = (3/4)y reduce to the single equation in y: ((3/4)y)² + y² = 100, giving the solution y = 8 and x = 6.