The modern world uses a number system built around 10, the *decimal system*. Things are counted and measured in tens and powers of ten. Thus we have 10 years in a decade, 100 years in a century and 1000 years in a millennium. On the other hand, the number 60 pops up in some interesting places; most notably, there are 60 minutes in an hour and 60 seconds in a minute. The only challenge to this setup came during the decimal-crazed French Revolution which introduced a system with 10 hours in the day, 100 minutes in the hour and 100 seconds in the minute. Their decimal metric system (meters and kilos) prevailed but the decimal time system, despite its elegance, was soon abandoned. But why have there been 60 minutes in an hour and 60 seconds in a minute in a numerical world dominated by the number 10? And that for thousands of years. Mystère?

The short answer is Babylon. In the ancient world, Babylon was renowned as a center of learning, culture, religion, commerce and riches. It gave us the code of Hammurabi and the phrase “an eye for an eye”; it was conquered by Cyrus the Great and Alexander the Great. The Israelites endured captivity there and two books of the Hebrew Bible (Ezekiel and Daniel) were written there. The Christian Bible warns us of the temptations of the Whore of Babylon. It was the Babylonian system of telling time that spread west and became the standard in the Mediterranean world: 24 hours in the day, 60 minutes in the hour, 60 seconds in the minute. So far, so good; but still why did the Babylonians have 60 minutes in an hour? Mystère within a mystère.

Here the answer is “fractions,” a very difficult topic that throws a lot of kids in school but one that will not throw the intrepid sleuths getting to the solution of this mystery. The good news is that one-half of ten is 5 and that one-fifth of ten is 2; the bad news is that one-fourth of ten, one-third of ten and one-sixth of ten are all improper fractions: 5/2, 10/3, 5/3; same for three-fourths, two-thirds and five-sixths. A number system based on 60 is called a “sexagesimal” system. If you have a sexagesimal number system, you need different notations for the numbers from 1 through 59 rather than just 1 to 9, but it makes fractions much easier to work with – the numbers 1,2,3,4,5,6 are all divisors of 60 and so one-half, one-quarter, one-third, one-fifth, and one-sixth of 60 are whole quantities, nothing improper about them. This also applies to two-thirds, three-quarters and other common fractions.

This practice of using a base different from 10 for a number system is alive-and-well in the computer era. For example, the base 16 hexadecimal system is used for the addresses of memory locations rather than the verbose binary number system that computers use for numerical computation. The hexadecimal system which uses 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F as its sixteen “digits,” is also used to describe colors on web-pages and you might come across something like <body bgcolor = “#FF0000”> that is instructing the page to make the background color red.

Having a good system for fractions is especially important if you are measuring quantities of products or land area: thus a quarter of a pound of ham, two-thirds of an acre, … . The Babylonians of the period when the Book of Daniel was written did not invent the sexagesimal system out of whole cloth; rather they inherited it from the great Sumerian civilizations that preceded them. At the birth of modern civilization in Mesopotamia, Sumerian scribes introduced cuneiform writing (wedges on clay tablets) and then sexagesimal numbers for keeping track of accounts, fractions being important in transactions between merchants and buyers. At first, the notations for these systems would differ somewhat from city to city and also would differ depending on the thing being quantified. In the city of Uruk, 6000 years ago, there were at least twelve different sexagesimal number systems in use with differing names for the numbers from 1 to 60, each for working with different items: barley, malt, land, wheat, slaves and animals, beer, … . It is as though they were using French for one item and German for another; thus “cinq goats” and “fűnf tomatoes.” What this illustrates is that it requires an insight to realize that the *five* in “cinq goats” is the same as the *five* in “fűnf tomatoes.” Eventually, these systems became standardized and by Babylonian times only one system was in use.

The Sumerians gave us the saga, *The* *Epic of* *Gilgamesh*, whose account of the great flood is a brilliant forerunner of the version in Genesis. In fact, so renowned were the Sumerian cities that the Hebrew Bible tells us the patriarch Abraham (Richard Harris in the TV movie *Abraham*) came from the city of Ur, a city also known as Ur of the Chaldees; at God’s bidding, he left Ur and its pagan gods and moved west with his family and retinue to the Land of Canaan. The link to Ur serves as a *hommage* on the part of the Israelite scribes to the Sumerians, one designed to ascribe illustrious origins to the founder of the three Abrahamic religions.

In addition to being pioneers in literature, agriculture, time-telling, accounting, etc. the Sumerians pushed beyond the boundaries of arithmetic into more abstract mathematics. They developed methods we still use today for solving quadratic equations (*x*^{2} + *b* *x* = *c*) and their methods and techniques were imported by mathematicians of the Greco-Roman world. Moreover, very early on, the Sumerian scribes were familiar with the mighty Pythagorean Theorem: an ancient clay tablet, known as “ybc 7289,” shows that these scribes knew this famous and fundamental theorem at least two thousand years before Pythagoras. For a picture of the tablet, click HERE . The image shows a square with each side of length 1 and with a diagonal of length 2 written in sexagesimal with cuneiform wedges (so this satisfies the Pythagorean formula *a*^{2} + *b*^{2} = *c*^{2} in its most simple form 1 + 1 = 2). In his *Elements*, Euclid gives a proof of the Pythagorean Theorem based on axioms and postulates. We do not know whether the Sumerians thought of this remarkable discovery as a kind of theorem or as an empirical observation or as something intuitively clear or just a clever *aperçu* or something else entirely. This tablet was on exhibit in NYC at the Institute for the Study of the Ancient World in late 2010; for the mathematically sensitive the event was an epiphany.

The Sumerians were also great astronomers – they invented the constellations we use today and the images associated with them – and their observations and techniques were used by geometers and astronomers from the Greco-Roman world such as Eratosthenes and Ptolemy. Indeed, the Babylonian practice of using sexagesimal numbers has persisted in geography and astronomy; so to this day, latitude and longitude are measured in degrees, minutes and seconds: thus a degree of north latitude is divided into 60 minutes and each minute is divided into 60 seconds. James Polk was elected to be the 11^{th} president of the United States on the platform “Fifty-Four Forty or Fight” meaning that he would take on the British and push the Oregon territory border with Canada up to 54°40′ . (Polk wisely settled for 49°00′).

The Sumerians were also great astrologers and soothsayers and it was they who invented the Zodiac that we still use today. If we think of the earth as the center of the universe, then of course it takes one year for the sun to revolve around the earth; as it revolves it follows a path across the constellations, each of which sits in an area called its “house.” As it leaves the constellation Leo and rises in front of the constellation Virgo, the sun is entering the house of Virgo. According to the horoscopes in this morning’s newspaper, the sun is in the house of Virgo from Aug 23 until Sept 22.

Recently, though, NASA scientists have noted that the Sumerian Zodiac we employ is based on observations of the relative movements of the sun, earth, planets and stars made a few thousand years ago. Things have changed since – the tilt of the earth’s axis is not the same, measurements have improved, calendars have been updated, etc.; the *ad hoc* Sumerian solution to keeping the number of signs the same as the number of months in the year no longer quite works. So the constellations are just not in the locations in the heavens prescribed by the ancients on the same days as in the current calendar; the sun might just not be in the house you think it’s in – if you were a Capricorn born in early January, you are now a Sagittarius. So far, the psychic world has pretty much ignored the implications of all this – people are just too attached to their signs.

It must be admitted that the NASA people did get carried away with the numbers and the science. They stipulated that there should actually be 13 signs, the new one being Ophiuchus, the Serpent Bearer (*cf*. Asclepius the Greek god of medicine and his snake entwined caduceus); this is a constellation that was known to the Sumerians and Babylonians but one which they finessed out of the Zodiac to keep the number of signs at 12. Click HERE for a picture. However, it is a sign of the Zodiac of Vedic Astrology and, according to astrologer Tali Edut, “It’s a pretty sexy sign to be!”

But why insist on 12 months and 12 signs, you may ask. Again mystère. This time the solution lies in ancient Egypt. The Egyptians started with a calendar based on 12 lunar months of 28 days each, then moved to 12 months of 30 days each with 5 extra days inserted at the end of the year. This solved the problem encountered world-wide of synchronizing the solar and lunar years (the leap year was later added). And this 12 month year took root. We also owe the 24 hour day to the Egyptians, who divided the day into 12 day hours and 12 night hours; at the outset, the length of an hour would vary with the season and the time of the day so as to insure that there were 12 hours of daylight and 12 hours of nighttime. The need for simplicity eventually prevailed and each hour became one-twenty-fourth of a day.

The number 12 is also handy when it comes to fractions and to this day it is the basis for many measuring systems: 12 donuts in a dozen, 12 dozen in a gross, 12 inches in a foot, 12 troy ounces in a troy pound. One that recently bit the dust, though, is 12 pence in a shilling; maybe Brexit will bring it back and make England Great Again.

Once again we are reminded that we learn from the past, and continue to need to learn of the past. I appreciate understanding now how we wound up with a different system for time management. I will use that the next time I need to explain fractions. Thanks!

How many ounces in a pint of guinness?