The k2p blog

I like 60. Equilaterals. Hexagons. Easy to divide by almost anything. Simple integers for halves, quarters, thirds, fifths, sixths, tenths, 12ths, 15ths and 30ths. 3600. 60Hz. Proportions pleasing to the eye. Recurring patterns. Harmonic. Harmony.

The origins of the use of base 60 are lost in the ancient past. By the time the Sumerians used it about 2,500 years ago it was already well established and continued through the Babylonians. But the origin lies much earlier.

I speculate that counting – in any form more complex than “one, two, many….” – probably goes back around 50,000 years. I have little doubt that the fingers of one hand were the first counting aids that were ever used, and that the base 10 given by two hands came to dominate. Why then would the base 60 even come into being?

The answer, I think, still lies in one hand. Hunter-gatherers when required to count would prefer to use only one hand and they must – quite early on and quite often – have had the need for counting to numbers greater than five. And of course using the thumb as pointer one gets to 12 by reckoning up the 3 bones on each of the other 4 fingers.

a hand of 12 – image sweetscience

My great-grandmother used to count this way when checking the numbers of vegetables (onions, bananas, aubergines) bought by her maid at market. Counting up to 12 usually sufficed for this. When I was a little older, I remember my grandmother using both hands to check off bags of rice brought in from the fields – and of course with two hands she could get to 144. The counting of 12s most likely developed in parallel with counting in base 10 (5,10, 50, 100). The advantageous properties of 12 as a number were fortuitous rather than by intention. But certainly the advantages helped in the persistence of using 12 as a base. And so we still have a dozen (12) and a gross (12×12) and even a great gross (12x12x12) being used today. Possibly different groups of ancient man used one or other of the systems predominantly. But as groups met and mixed and warred or traded with each other the systems coalesced.

And then 60 becomes inevitable. Your hand of 5, with my hand of 12, gives the 60 which also persists into the present.  (There is one theory that 60 developed as 3 x 20, but I think finger counting and the 5 x 12 it leads to is far more compelling). But it is also fairly obvious that the use of 12 must be prevalent first before the 60 can appear. Though the use of 60 seconds and 60 minutes are all pervasive, it is worth noting that they can only come after each day and each night is divided into 12 hours.

While the use of base 10 and 12 probably came first with the need for counting generally and then for trade purposes (animals, skins, weapons, tools…..), the 12 and the 60 came together to dominate the measuring and reckoning of time. Twelve months to a year with 30 days to a month. Twelve hours to a day or a night and 60 parts to the hour and 60 parts to those minutes. There must have been a connection – in time as well as in the concepts of cycles – between the “invention” of the calendar and the geometrical properties of the circle. The number 12 has great significance in Hinduism, in Judaism, in Christianity and in Islam. The 12 Adityas, the 12 tribes of Israel, the 12 days of Christmas, the 12 Imams are just examples. My theory is that simple sun and moon-based religions gave way to more complex religions only after symbols and writing appeared and gave rise to symbolism.

Trying to construct a time-line is just speculation. But one nice thing about speculation is that the constraints of known facts are very loose and permit any story which fits. So I put the advent of numbers and counting at around 50,000 years ago first with base 10 and later with base 12. The combination of base 10 with base 12, I put at around 20,000 years ago when agricultural settlements were just beginning. The use of 60 must then coincide with the first structured, astronomical observations after the advent of writing and after the establishment of permanent, settlements. It is permanent settlements. I think, which allowed regular observations of cycles, which allowed specialisations and the development of symbols and religion and the privileged priesthood. That probably puts us at about 8 -10,000 years ago, as agriculture was also taking off, probably somewhere in the fertile crescent.

Wikipedia: The Egyptians since 2000 BC subdivided daytime and nighttime into twelve hours each, hence the seasonal variation of the length of their hours.

The Hellenistic astronomers Hipparchus (c. 150 BC) and Ptolemy (c. AD 150) subdivided the day into sixty parts (the sexagesimal system). They also used a mean hour(¹⁄₂₄ day); simple fractions of an hour (¹⁄₄, ²⁄₃, etc.); and time-degrees (¹⁄₃₆₀ day, equivalent to four modern minutes).

The Babylonians after 300 BC also subdivided the day using the sexagesimal system, and divided each subsequent subdivision by sixty: that is, by ¹⁄₆₀, by ¹⁄₆₀ of that, by ¹⁄₆₀of that, etc., to at least six places after the sexagesimal point – a precision equivalent to better than 2 microseconds. The Babylonians did not use the hour, but did use a double-hour lasting 120 modern minutes, a time-degree lasting four modern minutes, and a barleycorn lasting 3¹⁄₃ modern seconds (the helek of the modern Hebrew calendar), but did not sexagesimally subdivide these smaller units of time. No sexagesimal unit of the day was ever used as an independent unit of time.

Today the use of 60 still predominates for time, for navigation and geometry. But generally only for units already defined in antiquity. A base of 10 is used for units found to be necessary in more recent times. Subdivision of a second of time or a second of arc is always using the decimal system rather than by the duodecimal or the sexagesimal system.

If we had six fingers on each hand the decimal system would never have seen the light of day. A millisecond would then be 1/ 1728th of a second. It is a good thing we don’t have 7 fingers on each hand, or – even worse – one hand with 6 fingers and one with 7. Arithmetic with a tridecimal system of base 13 does not entice me. But if I was saddled with 13 digits on my hands I would probably think differently.

Tags: counting, number systems

This entry was posted on September 14, 2015 at 10:08 am and is filed under Development, Mathematics. You can follow any responses to this entry through the RSS 2.0 feed. Both comments and pings are currently closed.