Evolution of mathematics

This 3,700-year-old Babylonian clay tablet (Si.427) uses applied mathematics to map land and measures its boundaries with extreme precision.

Babylonian clay tablet (YBC 7289). The diagonal displays an approximation of the square root of 2 in four sexagesimal figures, 1 24 51 10, which translates in decimal format to (1 + 24/60 + 51/60² + 10/60³)= 1,41421296.

Mesopotamian clay tablets. The left clay tablet shows a mathematical problem, as well as the intermediate and final (wrong) solution. The other tablets depict tables which are to be consulted during the exercise.

This is the Plimpton 322 tablet.

Column 2 in row 10

- reading in Base 60: 1 22 41

- in Base 10: (1 x 60²) + (22 x 60¹) + (41 x 60⁰) = 3600 + 1320 + 41 = 4961 (=A)

Column 3 in row 10

- reading in Base 60: 2 16 1

- to Base 10: (2 x 60²) + (16 x 60¹) + (1 x 60⁰) = 7200 + 960 + 1 = 8161 (=C)

Calculation

With: C² = A² + B²

We get: B² = C² - A²

We get: B = √(8161² = 4961²) = 6580

As B is an integer number, we known that column 2 and 3 express (large) Pythagorean numbers.

How did an Egyptian mathematics course look like? This partial replica of the Rhind papyrus (c. 1550 BC) shows what math students in ancient Egypt were reading circa 3500 years ago.

Dated to circa 100 AD, this is one of the oldest fragments of the Euclid's Elements (from the Oxyrhynchus papyrus).

The diagram belongs to Book II, Proposition 5.

Games of chance in ancient times: different types of dice.

Aeneas Tacticus describes the following technique for encrypting a message: each hole in these disks represents a letter of the alphabet. The disk on the top right shows which letters correspond to which hole. Only the sender and receiver have such a disk and therefore know which letter belongs to which hole.

On another disk there are no markings, only the holes are present (the disk on the bottom). A wire is passed through the holes in the order of the letters of the message to be encrypted. Once the entire message has been encrypted, the disk without markings can be sent to the receiver.

However ... the ancients made an important cryptographic error. Can you find it?

A short hint: statistics are involved.