Professor Stewart's Hoard of Mathematical Treasures Read Online Free

wild speculation, but I quite like it.
    Much earlier, around 3100 BC, the Babylonians had been even more ambitious, using base 60. Babylon is almost a fabled land, with biblical stories of the Tower of Babel and Shadrach in Nebuchadnezzar’s furnace, and romantic legends of the Hanging Gardens. But Babylon was a real place, and many of its archaeological remains still survive in Iraq. The word ‘Babylonian’ is often used interchangeably for several different social groupings that came and went in the area between the Tigris and Euphrates rivers, who shared many aspects of their cultures.
    We know a lot about the Babylonians because they wrote on clay tablets, and more than a million of these have survived, often because they were in a building that caught fire and baked the clay rock-hard. The Babylonian scribes used short sticks with shaped ends to make triangular marks, known as cuneiform, in the clay. The surviving clay tablets include everything from household accounts to astronomical tables, and some date back to 3000 BC or earlier.
    The Babylonian symbols for numerals were introduced around 3000 BC, and employ two distinct signs for 1 and 10, which were combined in groups to obtain all integers up to 59.

    Babylonian numerals from 1 to 59.
    The 59 groups act as individual digits in base-60 notation, otherwise known as the sexagesimal system. To save my printer having kittens, I’ll do what archaeologists do and write Babylonian numerals like this:
    5,38,4 = 5×60×60 + 38×60 + 4 = 20,284 in decimal
    notation
    The Babylonians didn’t (until the late period) have a symbol to play the role of our zero, so there was a degree of ambiguity in their system, usually sorted out by the context in which the number showed up. For high precision, they also had a symbol equivalent to our decimal point, a ‘sexagesimal point’, indicating that the numbers to its right are multiples of,×=, and so on. Archaeologists represent this symbol by a semicolon (;). For example,
in decimal (to a close approximation).
    About 2,000 astronomical tablets have been found, mainly routine tables, eclipse predictions, and so on. Of these, 300 are more ambitious - observations of the motion of Mercury, Mars, Jupiter, and Saturn, for instance. The Babylonians were excellent
observers, and their figure for the orbital period of Mars was 12,59;57,17 days - roughly 779.955 days, as we’ve just seen. The modern figure is 779.936 days.
    Traces of sexagesimal arithmetic still linger in our culture. We divide an hour into 60 minutes and a minute into 60 seconds. In angular measure, we divide a degree into 60 minutes and a minute into 60 seconds, too - same words, different context. We use 360 degrees for a full circle, and 360 = 6×60. In astronomical work, the Babylonians often interpreted the numeral that would usually be multiplied by 60×60 as being multiplied by 6×60 instead. The number 360 may have been a convenient approximation to the number of days in a year, but the Babylonians knew that 365 and a bit was much closer, and they knew how big that bit was.
    Nobody really knows why the Babylonians used base 60. The standard explanation is that 60 is the smallest number divisible by 1, 2, 3, 4, 5 and 6. There is no shortage of alternative theories, but little hard evidence. We do know that base-60 originated with the Sumerians, who lived in the same region and sometimes controlled it, but that doesn’t help a lot. To find out more, good sites to start from are:
    en.wikipedia.org/wiki/Babylonian_numerals
www.gap-system.org/~history/HistTopics/Babylonian_numerals.html

Magic Hexagons
    You’ve probably heard of magic squares - grids of numbers that add up to the same total when read horizontally, vertically or diagonally. Magic hexagons are similar, but now the grid is a honeycomb, and the three natural directions to read the numbers are at 120° to each other. In Cabinet (page 270) I told you that there are only two possible magic hexagons,