# Was the second originally defined in terms of the size of the Earth and c?

1. Aug 31, 2011

I posted this first in Astronomy but it belongs in General Physics.

With a little research, anyone can convince himself or herself that the following statements are true:

a) that in one second light travels a distance equal to 30.0 times the distance between the north pole and the equator and;
b) that one second is the half-period of a one-metre pendulum.

Either the world is a truly mysterious creation or, much more likely, men defined the second so as to make these statements true. In the paper, I show how men could have done so, with technology that we would consider almost medieval.

Anyone who has ever noticed that c is uncannily close to 300,000 km per second in the SI system should read the paper. I use the word uncanny advisedly, given that the metre dates from 1799 and the second is prehistoric in origin.

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2. Aug 31, 2011

### Astronuc

Staff Emeritus
3. Aug 31, 2011

### Ryan_m_b

Staff Emeritus
Eh? With a little research I convinced myself that this was not true.

Distance from north pole to equator - 10001.965km
x 30 - 300058.95km

1 light second - 299792.458km

http://en.wikipedia.org/wiki/Earth

4. Aug 31, 2011

Staff Emeritus
The second as a fixed unit of time (as opposed to something that varied with the seasons) goes back to about 1550. As Astronuc alluded to, the first measurement of the speed of light was more than a century later, in 1676.

Second, as Ryan points out, your statement is not true. If you want to say, "yeah, but it's close", it's off by about a part in 1125. Why say 30.0 and not 29 & 74/75ths? The second number is no less arbitrary than the first, and it's a heckuva lot closer.

PS When posting a new topic, please select the forum that best relates to the subject matter of your topic. If you are unsure, contact a mentor or the admin. Posting the same topic (or homework question) across multiple forums or multiple threads is considered spamming and is not allowed.

5. Aug 31, 2011

### olivermsun

Okay, but when did we first define a standard meter and then when were we actually able to measure the length from the north pole to equator with sufficient accuracy to determine that it was not 10000 but 10001.965km?

Anyhow, if we'd had the benefit of hindsight, maybe we should have just defined the meter such that c would be exactly 3e8 m/s! :tongue:

Last edited: Aug 31, 2011
6. Aug 31, 2011

### JeffKoch

More than that, at the meter level I'm not even sure what the distance means - the earth is not a sphere, it is not even an oblate spheroid, at the meter level it is a bumpy mess of mountains, valleys, land and water. Does the distance here mean the walking distance along a particular azimuth? Do I need to walk along the bottom of lakes and oceans or can I take a boat? Etc.

7. Aug 31, 2011

### D H

Staff Emeritus
Neither are true. Both are happenstance values: Numerology at work.

Had you done any research you would have found that the second was originally defined to be 1/60 of a minute, which in turn is 1/60 of an hour, which in turn is 1/24 of a day. These fractions go back to the Babylonians. If you want a remarkable similarity, why didn't you question why time is divided into intervals in a manner very similar to how a circle is divided into degrees, arc minutes, and arc seconds? These divisions and the similarities between these time and angle divisions are very intentional.

Your numerological nonsense is just that: Nonsense.

And reopened. Do beware of our rules, RadioTech. This is not a site for unjustified speculation.

Last edited: Aug 31, 2011
8. Aug 31, 2011

Staff Emeritus
There is a whole science of measurements of the earth, one that should properly be called "geometry", but since that was taken, is called "geodesy". I believe that particular distance is calculated using something called the "reference ellipsoid", which is an idealized geometry of something called the "geoid", which is the extension of sea level for equilibrium oceans at rest and allowed to penetrate into the continents (imagine a gridwork of tiny canals).

9. Aug 31, 2011

### D H

Staff Emeritus
Just as the concept of a second predates our knowledge of the hyperfine transition frequency of cesium 133 atoms, the concept of a "reference ellipsoid" predates our knowledge of the "geoid" by quite a bit. Most of our detailed knowledge of the shape is from 1990 and later. The reference ellipsoid has not been updated; it is WGS 1984.

One reason that it hasn't been updated is GPS. All of the GPS receivers out there in public use have the WGS84 reference ellipsoid built into their software.

10. Aug 31, 2011

### olivermsun

Well I do believe the original concept was "along sea level" which at some point was thought to mean "along a meridian" on the surface of a sphere, or at worse, an oblate spheroid. I don't think anyone ever seriously suggested that the bottoms of oceans should be traced to do this reference measurement. :tongue2:

Clearly, on practical grounds alone, the length of a meridian is now understood to be a less than stellar (ha) choice for a standard length. But it doesn't mean that the geometers who were envisioning such a concept were crazy.

11. Sep 1, 2011

I take note of all the comments, and will abide by the forum rules. I will just point out that my paper has been read by Professor Brown, the Astronomer Royal for Scotland. He did not reject it out of hand, although he did point out that my hypothesis was unconventional in terms of the accepted history of technology. He has passed it to a competent astronomer to review. The paper is not numerological fantasy, but it does rely on mathematical argumentation. Hence, my decision to post it on this web-site. The paper is only four sides of A4 long, and does not take long to read.

12. Sep 1, 2011

### JeffKoch

Yes, but you see the point - there is nothing profound about the distance between the pole and the equator of the earth, which varies by path and definition. There is a precise single distance between the pole and equator of a mathematical surface that approximates the earth, but the numbers will all vary by far more than a meter or even a kilometer.

13. Sep 1, 2011

### olivermsun

From my perspective, the distance between the pole and the equator is a pretty profound distance! :tongue:

Anyway, why would a well-defined single distance defined on a mathematical surface vary by a meter or even a kilometer?

14. Sep 1, 2011

### JeffKoch

Vary from actual distances on the actual earth.

15. Sep 2, 2011

I should answer some of the posts I guess - or leave the impression that I am innumerate, that I don't do any research, and that I don't care about wasting the time of other forum members with nonsense.

I used the number 30.0, rather than 30, because I knew a mathematician would understand this to be a quantity expressed with 3-significant-figure precision. If I say the distance from the north pole to the equator along any meridian is 10,002,000 metres, then I have to explicitly add 'correct to 5 significant figures' to make the statement true.

I should not have used the approximation 300,000 km/sec for c without adding 'correct to 3 significant figures'.

I should not have baldly stated that one second is the half-period of a simple pendulum of length one metre. I should have said something like, at a place on the Earth's surface where g=9.80 metres per second squared, the half-period period of a simple pendulum of length 1.0000 metres is 1.0035 seconds. I can express this latter figure, somewhat pedantically, as 1.00 x 10^0 s in scientific notation.

The Earth was believed to be spherical well before Aristotle wrote 'On the Heavens' in the 4th century BCE. He reported that the circumference of the Earth had been calculated to be 400,000 'stadia', much as we might say it is 40,000 km. (NB. I mean 40,000 correct to 3 significant figures. If I were expressing the quantity correct to 4 significant figures I would have to write 40,010).

Isaac Newton, in the 17th century, proposed that the shape of the Earth was closer to an oblate spheroid than a sphere. Otherwise, as he wrote, the waters in the oceans would build up near the equator owing to the Earth's rotation, and flood the land.

By the 18th century, the French knew the Earth was not even an oblate spheroid accurately, and despatched survey expeditions to Peru and and Lapland to better understand its shape. They used this information along with their survey of a meridian from a position at sea-level near Dunkirk to a position at sea-level near Barcelona, to calculate what the length of metre-rod should be, before constructing the first prototype in 1799.

Of course, the reason the astronomers wanted to know the dimensions of the Earth accurately was that they were using the Earth as a base-line to calculate distances within the solar system.

The first astronomer able to calculate the ratio of c to the mean speed of the Earth in its orbit round the Sun to 3-significant-figure precision was James Bradley, who announced his discovery of the aberration of light to the Royal Society in 1729. This was more than two centuries before that famous son of the city of Edinburgh, James Clerk Maxwell, did his ground-breaking work in electromagnetism.

The Babylonians, following the Sumerians, used a sexagesimal place-value system, leaving a space where we would put a '0', because they had not invented a symbol for nothing. Their system shows traces of a pre-existing purely decimal system, as they used a symbol for 10 to construct their numerals.

It was a more sophisticated system than the Greeks and Romans were using more than 2,000 years later. Compare, for example, MMXI to 2011. (I mean 2011 to be understood as a decimal number. In Babylonian I would write [33][31], or the equivalent in Babylonian numerals if these were to be found on my keyboard).

Mathematicians who have studied the Babylonian number system conclude that it is a system designed to divide a 'something', rather than one designed to count 'somethings'. This is likely to be true. We still use the Babylonian system to divide the day of 86,400 seconds. But we use the decimal system when we count things.

Last edited: Sep 2, 2011
16. Sep 2, 2011

### D H

Staff Emeritus
You are still guilty of numerological spelunking and of historical revisionism.

The concept of minutes and seconds as the first and second 1/60 subdivisions of a degree predates the concept of a meter, our knowledge of the figure of the Earth, and our knowledge of the speed of light by several hundred years. Ptolemy developed our current concept of degrees, minutes, and seconds as the angular subdivisions of a circle nearly 2000 years ago. As the Earth rotates once per day about an axis, analogously dividing the hour into minutes and seconds is an obvious extension of Ptolemy's angular subdivisions. The idea of minutes and seconds (of time) arose in the late 1200s / early 1300s. Clocks that could display subdivisions of the day into hours, minutes, and seconds were developed in the late 1500s.

What about the shape of the Earth? While Newton did derive that a rotating body such as the Earth must be in the form of an the oblate spheroid, measurements at Newton's time indicated that the Earth was a prolate spheroid. This discrepancy between theory and observation was not resolved for another hundred years.

What about the meter? The French Academy of Sciences were torn between two options regarding their new unit of length. They wanted something that was about as long as a human stride or a human arm. The two contenders were the length of a seconds pendulum and one ten-millionth of the length of the earth's meridian along a quadrant. The Academy chose the latter because the Earth has a non-uniform gravity field. That these two contenders are nearly equal to one another or to the length of a human stride or the length of a human arm is just coincidence.

Those French scientists had an inkling of the distance between the equator and the North Pole and developed a prototype meter bar based on this preliminary value. That prototype was just supposed to be a stand in until the precise value was known. The commissioned study finished their work years later. Rather than revise the meter to agree with this detailed study, the academy simply made the prototype meter the standard.

What about the speed of light? The concept that light had a finite speed postdates the concept of a second by several hundred years and postdates the development of a clock that could measure time down to the second by about a hundred years. So once again you have history backwards. That the speed of light is approximately 3×108 is just coincidence, a coincidence that was not known until well after the concept of a second was introduced or clocks that could measure seconds were made.

17. Sep 2, 2011

### uart

BTW. The second of the two "coincidences" is equivalent to $g = \pi^2$. As pointed out previously, that fact that these two numbers are even close is just happenstance.

Also the sea level value of g varies depending on your position on the globe, but as far as I know it's not exactly pi^2 anywhere.

18. Sep 2, 2011

### D H

Staff Emeritus
Not quite happenstance. The French seriously considered defining the meter (not the second) so that the numerical value of g would have been exactly equal to $\pi^2$ at the Paris Observatory.

On the surface of the Earth, that is correct. Inside the Earth, that is incorrect. g increases with increasing depth to over pi^2, then decreases with increasing depth below the transition zone (but does not drop below pi^2), then increases again at some point within the lower mantle. It reaches a global maximum at the mantle/core boundary, dropping toward zero at the center of the Earth. There are two surfaces inside the Earth where $g=\pi^2$, exactly.

Last edited: Sep 2, 2011
19. Sep 2, 2011

### uart

Yep, that's why I was talking about at sea level.

I didn't know that. You've got me googling "meter history" right now. :)

Last edited: Sep 2, 2011
20. Sep 2, 2011

### uart

Quote from wikipedia.

Code (Text):
Timeline of definition

1790 May 8 – The French National Assembly decides that the length of the new metre would be equal to the length of a pendulum with a half-period of one second.
1791 March 30 – The French National Assembly accepts the proposal by the French Academy of Sciences that the new definition for the metre be equal to one ten-millionth of the length of the Earth's meridian along a quadrant through Paris, that is the distance from the equator to the north pole.
1795 – Provisional metre bar constructed of brass.
...
So there goes that "coincidence" out the window.

21. Sep 2, 2011

### D H

Staff Emeritus
Not really out the window.

That the length of a human stride and the length of a human arm are more or less the same is pretty much coincidental. What if intelligent velociraptors rather than intelligent apes had become the intelligent species on this planet?

That the length of a seconds pendulum is roughly equal to those human-based standards is even more coincidental. What if the length of a day was something different than it currently is? What if the Earth's composition was something different than it is? After all, the length of a day has varied by quite a bit since the formation of the Earth, and the value of g depends heavily on the makeup of the Earth.

That 1/10,000,000 of 1/4 of the polar circumference of the Earth is roughly equal to any of the above is yet more coincidental. What if the average human hand had four or six digits? Our infatuation with powers of 10 is just a consequence of humans having 10 fingers. What if the Earth had formed slightly differently? That this is anything but coincidence means that life/intelligence can only arise on a planet that is almost exactly the same size as the Earth, and that such life will have the same infatuation with powers of 10 as do we.

22. Sep 2, 2011

### uart

Numerical coincidences are sometimes fascinating though. There was a guy here once who was posing a whole alternate theory of physics based on the fact that the number of seconds in a Lunar year (approx 354.37 days) times "g" was "exactly" (his words) equal to the speed of light.

The interesting thing about that one is that the units actually agree, m/s = m/s^2 * sec, and numerically it was only out by about 0.1%. Too bad his alternate theories didn't stand up to any scrutiny at all.

23. Sep 2, 2011

### Ryan_m_b

Staff Emeritus
That kind of reminds me of the (bad) film 23. For a long time after seeing that film as a teen me and my friends would parody it by point out "amazing" coincidences e.g.

"Hey guys! The house we just passed was number 63! If you add up our ages, times them by the number of eyes I have, divide it by the fraction if us that is currently speaking and add the number of us there are it comes to EXACTLY 63!!!11!1!!111!"

More realistically numbered coincidences are a brilliant example of conformational bias. We ignore the vast majority of relations we see and attribute arbitrary meanings to patterns we judge to be important. The arm span/height fallacy is a great one because no one ever points out the the circumference of a head =/= the length of a leg or that the size of the stomach is unrelated to the number of fingers on one hand etc

24. Sep 3, 2011

Speculation on the seconds-pendulum and the eventual choice to define the metre/meter as one ten-millionth part of the quarter-meridian is fine, but there is a good paper on the topic on the 'Istituto Nazionale di Fisica Nucleare' web-site:- 'Why does the meter beat the second?' by Paolo Agnoli and Giulio d'Agostini (ref: arXiv:physics/0412078 v2 25 Jan 2005).

It is perhaps unfortunate that the French revolutionary state did not impose the decimal-second with the same vigour it imposed the metre. If it had the 'coincidences' and 'happenstances' referred to in this thread would not exist.

However, the introduction of decimalisation throughout the new system of weights and measures was only one of the design criteria set for the system, as Agnoli and d'Agostini's paper describes.

Given the other design criteria, and the level of scientific and technological development attained by the 1790s, it is at least an interesting academic exercise to hypothesise how the French scientific rationalists of the period might have gone about designing a new unit of time.

Last edited: Sep 3, 2011
25. Sep 5, 2011

### D H

Staff Emeritus
Had the French Academy mucked with the definition of time, they would have made the day the base unit rather than the second.

In fact, the French Academy initially did muck with the definition of time (http://www.gefrance.com/calrep/decrtxt.htm): Le jour, de minuit à minuit, est divisé en dix parties, chaque partie en dix autres, ainsi de suite jusqu’à la plus petite portion commensurable de la durée.. (The day, from midnight to midnight, is divided into ten parts, each part into ten others, so on until the smallest measurable portion of duration.) This proposal took off like a lead trial balloon; the French abandoned the concept of decimal time less than two years after this mandate.

Unlike standards for length and mass, which varied incredibly from place to place at the time of the French Revolution, time and angle had pretty much the same representation across all of western Europe and beyond. While those representations were not decimal, they were very standard.

Some other coincidences and happenstances would have existed instead.

However, the following observations are matters of fact:
1. that one second is virtually identical to one half the period beat by a simple pendulum of length one metre and;
2. that one second is virtually identical to 30 times the period that light takes to travel the length of the quarter meridian.
These are surprising coincidences that are not explicable by reference to the laws of physics alone.​
Your paper is going along the path of numerology from the onset. This is in general a bad path to follow. Let's look at those coincidences:

That one second is virtually identical to one half the period beat by a simple pendulum of length one metre
This coincidence is rooted in two widely-used, human-based standards for a unit of length are coincidentally nearly equal to one another and that both are coincidentally nearly equal to the length of a seconds pendulum. That the number 360 holds a special place in pre-scientific numerological thinking is one of the key reasons we still subdivide angle and time the way we do.

That one second is virtually identical to 30 times the period that light takes to travel the length of the quarter meridian.
This coincidence is rooted in the coincidence that the speed of light in metric units happens to be close to a nice, round number and that the distance between the equator and the pole happens to be about ten million meters. What if the Indians (the source of our number system) had counted their fingers and toes rather than just fingers? Our ten million would be something like 32A000 in base 20, which isn't near as nice a number as 10,000,000. That part of the coincidence would have just vanished. The remaining coincidence is just that, coincidence.

I gave you a link showing exactly what they did: They chose the day as the base unit for time. If those French rationalists had had their way the 1/86,400 day second would have been history. That there are 86,400 seconds in a day is anything but an arbitrary choice. It is deeply rooted in pre-scientific Egyptian and Babylonian mythology, which placed undo importance on the numbers 6, 12, 60 and 360.

The French concept of a day as the unit of time lasted less than two years. The metric system did not have an official unit for time for another 150 years. The cgs system of course had a unit of time, as did the MKS system. That unit, the second, did not become a part of the official, treaty-bound metric system until 1960.