A Perfect Watch is always Fast

Let's say that you bought a perfect watch on January 1, 1972. By perfect, we mean that each and every second measured by this hypothetical watch is exactly equal to a SI second - defined in 1967 as "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom". Let's say also that, comforted by this extreme accuracy, you've never adjusted the watch or had to change to battery or wind it up or whatever. After all it is a perfect watch, right?

Sorry to tell you that your watch is now 24 seconds fast. Yep. You see, measuring the passage of time is not quite the same as telling the time. Time itself is but a concept, a human observation of change if you will. On the other hand, "the time" [of day] is our expression of an agreed instant in the passage of time. In fact, the time of day, or the definition thereof, has been messed around with ever since the first non-sundial clock was invented.

You would think that the invention of atomic clocks would clear everything up. After all, atomic clocks are almost as good as the perfect watch that you bought back in 1972. Unfortunately, we still have the same problem in modern times that the ancients had. Days are still varying in length - not just the period of daylight versus night, but also the actual length of each day varies during the year. Not to mention that the wobble of our planet (about once every 26,000 years) causes a change in the length of each year and that the planet itself is slowing down due to tidal friction.

Before the second was re-defined in 1967, it was defined with respect to the mean solar day and thus measured the passage of ephemeris time. However, the first atomics clocks proved that even ephemeris time was not perfectly constant so, up until 1967, they were actually adjusted to keep ephemeris time, known quaintly as "ET".

These days we still match "the time" to our planetary motion but now it is done by adding leap-seconds when required by our slowing planet. It is only by adjusting atomic time with leap-seconds do we arrive at the familiar UTC which is now truly "the time", as far as everyone on the planet is concerned. In 1972, when UTC was invented, it was already slow to the atomic clock by 10 seconds and since then 24 leap-seconds have been added - which is why your perfect watch is now 24 seconds fast!

Best regards, xpatUSA

Sexagesimal Time

Old measure wasn't much of a problem BC (before computers). We mixed together all kinds of units and bases but we were quite used to them. The Brits coped quite admirably with pounds, shillings, pence and even the occasional sovereign up until the early 70's when Heath took them kicking and screaming into the E.U., or EEC as it was known back then. To this day, Americans still struggle on bravely with tons, pounds, ounces and grains in everyday life - not to mention gallons, pints and fl. ounces!

Although most countries are metricated now, the time of day has remained supremely sexagesimal to this very day. The time of day is represented as hh:mm:ss. This format is based on the Babylonian positional numbering system, with colons separating the positions. The Babylonians inherited sexagesimal counting from the Sumerians and improved the system by adding positional significance (by powers of 60) and the concept of the number zero. They still used chicken-scratchin's (cuneiform) to represent the values of each position and each value was represented in base 10 notation using a grouping of the two symbols for one and ten. This symbol grouping was not itself positional and so simple addition gave the value, see table below. For example 23 is represented by 2 eyeballs and 3 wine-glasses, i.e. 2x10 + 3x1 = 23.



Because the number of hours in a day is less than 60 (on this planet, anyway) our time-of-day representation is truly sexagesimal and follows the Babylonian system with the addition of colons as separators and the use of Arabic numerals instead of cuneiform.

There is an interesting property of the hh:mm representation. It is either a fraction or an integer number depending on how you look at it, but the representation itself does not change. For example, let's say a timer shows a elapsed time of 3 hours and 40 minutes, which is written as "3:40". In our decimal system (i.e. positional base 10), that would be 3.667 hours or 220 minutes. In sexagesimal, 3:40 represents either 3x60^0+40x60^-1 hours or 3x60^1+40x60^0 minutes but the bolded numbers do not change and the colon remains where it's at.

But what if there were more than 59 hours involved, for example three days, 13 hours 43 minutes and 22 seconds? "Easy!", you say, "just write 85:43:22" - but sorry, that is not a true sexagesimal number because 85 is more than 59! To be truly "sexy", it would have to be written 1:25:43:22 which is 1x60^1+25x60^0+43x60^-1+22x60^-2 and which is also thoroughly confusing! Indeed, this is why we still have a DMS function (angular Degrees, Minutes and Seconds) on our engineering calculators due to the mixed bases of decimal degrees with sexagesimal minutes and seconds.

Best regards,

xpatUSA