## The Metric System for Angles and Time

June 5, 2016

Roscoe, N.Y.

“I often say that when you can measure what you are speaking about and express it in numbers you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind: it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of *science*, whatever the matter may be.” — William Thomson (Lord Kelvin), 1883.

For this reason, a major contribution to science was the development of the metric system during the French First Republic in the late 18^{th} century. The metric system had two big advantages over earlier systems of weights and measures. First, all units were based on powers of ten, and secondly, the units were interrelated in various ways: The liter is a cubic decimeter, and a liter of water weighs one kilogram. (The original names for some of these units were different, and the relationships have evolved somewhat.)

The major flaw in the metric system is that it is based on decimal arithmetic, the time-honored number system deriving from the number of fingers on humans and other primates. While ten is divisible by both two and five, it is not divisible by three and four, and that’s often inconvenient for simple calculations. During the early development of the metric system during the French Revolution, Pierre-Simon Laplace (1745 – 1827) recommended that the new system of weights and measures be based on a duodecimal (base 12) system but that was rejected.

I’ve recently been reading Laplace’s *The System of the World* (volume 1 and volume 2), an 1809 English translation of his *Exposition du système du monde*, first published in 1796.

Unlike Laplace's mammoth and mathematically impenetrable *Traitè de mécanique céleste* (1799–1825), *The System of the World* was intended for a popular audience. It has no math, but it has no diagrams either and they are sorely missed. Aside from its discussion of planets and comets, *The System of the World* also includes several insights into the development of the metric system, as well as discussions of two aspects of the metric system that did not catch on: decimal angles and time.

In Book I, Chapter 12 (pages 113-164 of the first volume) Laplace discusses the intention of basing the standard unit of length on something in the natural world — either the length of a pendulum with a half period of one second, or a fraction of the circumference of the Earth. They went with the latter, and it was decided that the meter would be a 10,000,000^{th} of the distance between the equator and the North Pole.

A major problem, however, was that this was not an easy distance to measure. It would be much easier to derive if the Earth were a sphere, but that is not the case. Due to centrifugal force resulting from its rotation on its axis, the Earth has assumed a figure of an oblate spheroid, which means that a cross section of the Earth through the poles is an ellipse bulging at the equator and flattened at the poles. But in Laplace's day, even this did not seem to agree with the measurements that had been made.

The meter was actually based on the distance between Dunkirk in north France to Montjuïc (spelled Montjoui in Laplace’s book) near Barcelona. These two locations were both on the same meridian that passed through Paris, and about 1/10^{th} of the total distance between the equator and the North Pole.

But the original metric system went a step further with the decimalization of angles.

We're all familiar with a circle being 360 degrees, but the original metric system gave the circle 400 degrees based on a right angle of 100 degrees. Instead of dividing the degree into 60 minutes, and the minute into 60 seconds, the metric degree was divided into 100 minutes, and each minute was 100 seconds. This required new tables of trigonometric functions that were created by interpolating values from existing tables. (Roger Hahn, *Pierre Simon Laplace , 1749–1827: A Determined Scientist*, p. 107)

Now here's where it clicks together:

The right angle is also the difference in latitude between the equator and the North pole. That terrestrial distance had been established as 10,000,000 meters or 10,000 kilometers, from which emerges an incredibly beautiful and easy equivalence between degrees of latitude and distance, what Laplace terms “the advantage of making nautical and celestial measures correspond.” (page 155):

1 degree of latitude ≈ 100 kilometers

1 minute of latitude ≈ 1 kilometer

1 second of latitude ≈ 10 meters

These equivalences also apply to degrees of longitude at the equator. They aren’t exact because degrees of latitude are slightly longer near the equator than at the poles.

As you know, decimal angles didn't catch on, so now we're stuck with 1 degree of latitude being about 111 kilometers, 1 minute of latitude being about 1.85 kilometers, and 1 second of latitude being about 31 meters, which are not nearly as easy to memorize or derive.

Time was also decimalized in the early metric system. The day was divided into ten hours, the hour into 100 minutes, and the minute into 100 seconds. This means that each day contains 100,000 decimal seconds, compared with the 86,400 seconds of our conventional clocks, so the metric second is somewnhat shorter than the second we know.

Of decimal time, Laplace confesses “This division of the day which will become necessary to astronomers is less advantageous to civil life, where there is little occasion to employ time as a multiplier or divisor. The difficulty of adapting it to clocks and watches, and our commercial connexions with foreigners in the sale of watches have suspended its use indefinitely.” (p. 162)

But that doesn’t prevent Laplace from using decimal angles and time throughout *The System of the World*, forcing the English translator to provide footnotes for the converted values.

On page 8 Laplace indicates the axial tilt of the Earth to be 26° 0796 (which we would tend to write as 26.0796 degrees), which decomposes easily into 26 degrees, 7 minutes, and 96 seconds. But the English translator has to convert that to 23° 28′ 18″.

On page 76 Laplace refers to the period of the revolution of Mars around the Sun as 686^{d} 979579, which we would write as 686.979579 days, and the decimal part easily decomposes into 9 hours, 79 minutes and 57.9 seconds. The footnote converts this to the ungainly 1^{y} 321^{d} 23^{h} 30′ 35″6.

As I continued to read Laplace's *System of the World*, I also began remembering how clumsy it was to work with minutes and seconds of both time and degrees in astronomical calculations. The elegance of decimal angles and decimal time began appealing to me more and more, and I'm now disappointed that these components of the metric system didn’t catch on.

Not that we'd be using them in the United States anyway, but it would be nice to know they existed elsewhere.