Making An English Foot

Making an English Foot

First we need to look at how to measure circles and time. We do this the way the ancients did in degrees, minutes, seconds (not in things like radians).

For time, a day will have 24 hours, each of 60 minutes, each of 60 seconds.

For arc of a circle: the earth will have 360 degrees of circumference, each of 60 minutes of arc, each of 60 seconds of arc.

Why do we use these numbers? Because the ancients did a lot of their math in fractions and ratios, and 60 was discovered by the ancients to be highly divisible by a lot of other smaller numbers: That is, it has a lot of factors. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60. And even things like dividing by 8 are not too bad in that the fraction rapidly reduces to a fairly easy number to calculate 60/8 =15/2 or 7 1/2. So if you are going to be doing a lot of calculations in your head or scratched on the dirt, having things that “divide easily” is a big feature. Compare with the base 10 metric system where your factors are 1, 2, 5 and 10. Not very useful for anything other than 10s letting you move a decimal point.

A Standard In Time

With our calculation system settled and our way of measuring time and the earth decided, we can proceed with making a time standard, and from that a length standard.

SideBar on Precision: I will be doing this demonstration using simple common definitions of “day” “year” and related things. I don’t want to get tied up in the minutia of sidereal, vs. tropical, vs. whatever year; precession of the equinoxes, etc. Yes, those are important out in the small digits of precision; and I expect the ancients put a lot of time into polishing the system for those details over the several thousand years it evolved. I just don’t expect folks to absorb all that in 10 minutes of modern impatience. So we keep it simple and direct. That “polish” can be added later by anyone who cares to go down that path. We’re taking the “KISS path” – Keep It Simple…

The Solstice In Time:

The Earth rotates. Not as consistently as the ancients believed, but far more consistently than most folks need for anything like day to day use. There are 360 degrees of rotation per day. That is 15 degrees per hour (360 / 24 = 30 / 2 = 15 You see, I told you those factors would come in handy…) So all we need is a way to mark off 15 degrees of arc of the sky and we have an hour of transit time for a star (or most other heavenly bodies – modulo the planets and sun that are slightly different, but not enough different for minor uses)

To draw a circle we need a rope and a pole. The pole is stuck in the ground and the rope is stretched out from it. Take the other end and swing it. That makes a circular arc. You don’t need to make the whole circle, just a bit more than 1/6 of a circle. But where to put that 1/6 ?

While it doesn’t matter too much, pointing it at the point where the sun, moon, planets et. al. rise from the horizon at the time of a major celestial turning point (the Solstice) has been the common way to do it. Many circular “monuments” from Stonehenge to Medicine Wheels have had an “alignment” to the summer solstice. It is also more likely that you will have a clear night sky in the summer, so this is a practical point too.

You can use a modern calendar to find the solstice date (much faster – about June 21st in the Northern Hemisphere) or you can do it the way the ancients did. Put a pole in the ground where you will observe, and another toward the horizon a ways away. Every morning from mid winter on, when the sun rises, it will rise a bit further to the left (north) of the pole, until one day it doesn’t. That is the Solstice. Where the sun stops its migration and starts migrating back the other way. That is the longest day of the year.

The Arc and a Time Covenant:

So now, from your observation pole (made plumb with a bit of string and a weight / plumb bob) to your Solstice pole (also made plumb) you can swing an arc off to the south.

Circles and Hexagons

Circles and Hexagons

There is an interesting property of circles. A string (or rope) one radius long (like the one used to draw your arc) will divide the circumference of a circle exactly 6 times. Another way to think of this is that the 60 – 60 – 60 degree triangle, the equilateral triangle, will exactly fit in a circle 6 times. We use this to make a 360 / 6 = 60 degree arc. Use your “radius line” but now measure from your solstice pole and swing an arc to the south. Where it crosses your circumference arc is exactly 1/6 of a circumference, or 60 degrees of arc. Put a marker or pole at that point (plumb, if a pole). Now use your rope to draw a straight line from one of the circumference poles to the other. The “chord”.

Now we need to cut the arc into smaller pieces. Pick a spot a little more than half way down your “radius line” rope. Swing one arc from the first circumference pole a bit past the midline of the chord. Make sure it crosses the circumference a bit more than 1/2 way from one pole to the other. Go to the other pole and do the same thing so that the two arc cross each other. Stretch your rope between the two points where these two arcs cross each other. Where it crosses the circumference is the halfway point, or 30 degrees. Put a marker there.

Starting from the 30 degree point, do it all again to get a 15 degree arc. That is your 1 hour transit arc. If you watch a star disappear behind one pole, it will disappear behind the next pole in one hour. Now, you can repeat this division process as many times as you like to get a workable arc. You will be timing transits between the two poles a few times to “tune” your clock, so if each time takes an hour, that might be a bit much. Also stars rise as they rotate, so you need a fairly tall pole for the 1 hour marker! In my opinion, the 7 1/2 degree or 3 3/4 degree arcs are easier to time (so divide in half two more times…).

The Clock:

We now have a celestial clock. There are 3600 seconds of time in 15 degrees of arc. For 7.5 that would be 1800 seconds or for 3 3/4 degrees it would be 900 seconds (and for the 1 7/8 degree arc 450 seconds etc.). This is your universal time standard.

Build a Pendulum

Take a bit of string somewhat near 4 feet long and tie a weight on the end. A large metal washer or stone doughnut works well. You want to be able to find the exact middle of the weight, since that is the centre of mass / centre of oscillation of the pendulum. A large used wheel bearing set works well too, as does any large metal or stone ring with a small hole in the centre.

For a formal standard, you could even go so far as to build a large grandfather clock mechanism with a long pendulum and room for a wide swing. You want a pendulum that makes one “swing” from one side to the other in exactly one second. “Second Clocks” with a one second pendulum were very popular at one time (I wonder why…) For your washer on a string, you need to calibrate it. So start it swinging. It needs to swing 3600 times (or 1800 “periods” of out and back, from one side, to the other, and back) in an hour, or as a star transits 15 degrees from entering the left side of the first pole to entering the left side of the second pole.

Now you know why I like the idea of a 3 3/4 or even a 1 7/8 degree arc! It is a lot easier to count 225 periods (450 swings out, then 450 swings back) than 3600. A longer pendulum swings slower. A shorter pendulum swings faster. If you get a number larger than 225 beats, your pendulum is swinging too fast, make your pendulum longer. If you get a smaller number, your pendulum is too slow, make the pendulum shorter. The “arc” that the pendulum swings in ought to be about 90 degrees when you start it swinging. When your count is right, your string, from centre of the weight to pivot point at the top is 1 yard long.

A Pendulum In Time

For small angles of swing, the time a pendulum takes does not change much. It is more or less driven by gravity and length. There is a small variation as the angle gets larger (exponentially with size of the angle). For an example of the amount of variation, you can visit this site and play with the numbers:

http://www.calctool.org/CALC/phys/newtonian/pendulum

You will find that 42.445 degrees of swing each side of straight down ( a bit more than 84 degrees in total) with a 3 foot long pendulum ( I used a 1 pound setting for the weight, but that doesn’t change the time, just the energy equation.) gives an exactly 2.00000 second period (or a 1 second “swing” from one side to the other, but not back). Play with the exact degrees of swing and you can see how many digits of precision you get from less accurate control of the arc of swing.

Just for fun, change the length of the pendulum to 2 cubits.
(Cue spooky music ;-)

As a pendulum becomes of lesser swing, it becomes more consistent, so modern pendulum clocks use a very short swing. It also becomes longer by a little bit. This is also why the commercial “seconds pendulum clocks” have a pendulum longer than a yard (a bit over 39 inches) and approaching a meter. To some extent, the move from a yard, to a seconds clock pendulum, to the meter can be seen as changing the length of the swing of the pendulum from an easy to observe by hand and eye rather large 84 degrees of arc, to something much much smaller, but more consistent and precise.

Oddly enough, if you narrow the swing to 5 degrees (2.5 alpha in the calculator) you get 2.00665 seconds from a 1 meter pendulum. While you can not get a time from a zero length swing, it does look like we are approaching a 1 meter pendulum at a near zero swing in the 3rd decimal point. Kind of makes me wonder if someone on the committee to make the meter standard knew something about the older yard and seconds pendulums and just looked at taking the pendulum to a limit point. Rampant speculation, yes, but there was that odd moment when the meter ended up being not quite 1/10,000,000 of the arc from pole to equator … “an error” by folks that didn’t make many errors. It still ended up a bit long. Then again, perhaps allowing for a non-point mass in a pendulum would tighten that up to 2.000…

An Alternative Pendulum

Now 84 degrees is a very large arc, and is not very accurate nor precise and repeatable. Is there another way? Say we made our pendulum very long, rather than a yard, and say we made it swing very slowly in a very small arc. Then it would be quite precise and not very sensitive to exact swing length. How about if we make it swing in a 4.6 degree arc, and had it be one “rod” in length. (A rod being 16.5 feet). What time would it measure then? 4 1/2 seconds to the period, exactly.

Gee, that looks useful… Our very small arc of 1 7/8 degrees took 450 seconds. So this pendulum would only take 100 swings to have a “match”, and it would be far less sensitive to the exact size of the swing, varying in the 1/1000 place if you increase the arc to 10 degrees. So if we ‘go large’ to increase our precision and accuracy, the “rod” works very well.

I could easily see our paleo-astronomer counting 450 a few times, and with less than stellar repeatability, wondering if maybe counting 100 would be easier, then finding that (with the same distance of swing, but a far smaller arc) his repeatability and ease of determination became much easier. Then it would just be a matter of calibrating his yard to the rod. Below we will see how this might be meaningful…

So now you have two very useful things.

1) An excellent time standard. The second.

2) A very good distance standard. The yard (or cubit). And a more precise and more easily timed / measured distance standard, the rod.

Your precision is limited mostly by the care you put into the construction, the size you make things (which is probably why Stonehenge and Medicine Wheels are so large) and a bit by the “fiddly bits’ we ignored about the minor variations in the day length from orbital mechanics, along with keeping the arc of your pendulum swing near 84 degrees (or using the “rod” to make it much easier to make more precise). If you need much more precision than that, you need to be taking a degree in astronomy…

For a set of cave man tools (a couple of straight poles, a rock, some cordage) and the sky to be able to make a decent time and distance standard that can be recreated by anyone anywhere is rather a neat trick! That it matches the English Yard and Rod is no accident.

But you said “Foot” not Yard!

OK, we need to divide that yard long string into 1/3 parts. We could just fold it back on itself twice to get three lengths, but that is a bit imprecise due to the radius of curvature in the ‘turns’. Good enough to build a house, but not for fine precision.

So lets think about this for a minute ( or second or degree ;-). A yard is 36 inches long… Golly, 36 inches is 360 tenths, where have I heard of 360 before…

We know how to divide a circle into 1/6 parts. We have several choices at this point. We could simply stake out our string in the shape of a hexagon (overlay it on the center of a circle, with diameter lines at each 60 degrees constructed as we did above for the arc of 60 degrees). Each segment is now 60 tenths of an inch long (or 1/2 foot). A line from one vertex to another through the centre is 1 foot long, or we could take a segment of line from 2 sides of the hexagon and that would be a foot too.

We could lay the line out as three 60 degree arc chords of a half circle. Then each segment would be a foot long (as would the radius of the circle). Or we could form the string into a circle and figure out how to divide it into 360 degrees, each degree segment 1/10 Th. of an inch long.

At this point, it is really just a matter of High School Geometry how you choose to divide up the string. I’m sure you can find many other interesting ways to make the same divisions… 2 rods are 11 yards or 33 feet long. A bit inconvenient to divide, but not impossible. I’ll leave dividing that one for another day…

Now it suddenly makes a lot more sense why a yard is 36 inches ( 360 1/10ths) and why there are 3 feet in a yard. Maybe this system of time and distance standards is a bit more rational than some folks think…

My inspiration for this exercise came from:

http://www.robertlomas.com/megyard/index.html

Wherein they recreate the “megalithic yard” that is a unit of measure widely found in very old stone works over much of the ancient world. It is based on a 366 “degree” circle and the number of sunrises in one orbit of the sun from solstice to solstice, not of the celestial sphere that gives our 365.x day year.

More detail on the kinds of “year” that are defined:

http://en.wikipedia.org/wiki/Year
and
http://en.wikipedia.org/wiki/Precession_(astronomy)

For even more details on the “fiddly bits” see:

http://scienceworld.wolfram.com/astronomy/TropicalYear.html

And if you think nobody would do something this complicated, here is an interesting circles and hexagons construction done for far less inherent value, but beautiful in the construction.

And what Mascheroni did because he felt using a straight edge and compass was too much technology for constructing geometric forms and wanted to prove all you really needed was a compass alone (our rope and pole for swinging arcs, though we treat the stretched rope as a straightedge of sorts when stretched to make a chord line).

Past Tense and Past Time

One can now see the change from the Megalithic Yard to the English Yard and some of the various “cubits” as the shift from a 366 degree circle standard to a 360 degree circle standard (for more factors and a bit easier fractional math) and a transition from a solstice sunrises year standard to a Tropical year. We can now see in the stones on the ground the discovery that the Solstice Sunrises Year was only one way of seeing the orbit of the earth (and in some ways a bit too sun centered and not in touch with the greater celestial sphere) along with a move from the “366” degrees, based on that Solstice Sunrises Year, to a circle divided for easier math (since 366 was no longer so “special” as to deserve preserving and 365.x is hard to use in dividing). In the units of measure we can see the step forward in understanding.

Interestingly enough, a “rod” is 16.5 feet, or 5.5 yards, or 11 cubits, or 6 megalithic yards. In that context we can see the “rod” as a unit that unifies the various systems of measurement. I can even envisage an alternate path of history where the megalithic yard came first, then the Rod as 6 megalithic yards, but using the “new” 360 degree circle and divisible by the hexagon method above; and then the English yard coming to replace the megalithic yard (with it’s own 360 divisions for fine work). Not wanting to fully replace the megalithic yard and rod, but augmenting it, only replacing it later. The exact path through history we will likely never know. Far fetched? Look at the present change from the yard to the meter…

Since a “chain” is 4 “rods” or 22 yards, and an acre is 1 chain x 10 chains of area, we carry with us in our 1/8 or 1/6 acre urban home lot the history of both the Megalithic Yard, The English Yard and Foot, and The Greek and Minoan foot. All neatly interoperable.

Just don’t try to convert these measurements into the less flexible metric system nor to do fractional math in your head with metric to the same accuracy and precision…

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About E.M.Smith

A technical managerial sort interested in things from Stonehenge to computer science. My present "hot buttons' are the mythology of Climate Change and ancient metrology; but things change...
This entry was posted in Earth Sciences, Metrology, Stonehenge and tagged , . Bookmark the permalink.

7 Responses to Making An English Foot

  1. davidc says:

    I’ve just dug out my copy of Ureil’s Machine by Lomas and Knight to refresh my memory on this stuff. I rember looking into the material on the Book of Enoch. Very strange but it looks like something happened.

  2. Ulric Lyons says:

    Greetings. I reckon the 360 degree division of the circle came from musical ratios and division of a string. The ratios of 16/15, 9/8, 6/5, 5/4, 4/3, 3/2 and their inverses cover all but the flat fifth in a natural temperament chromatic scale.
    With a root of 360Hz, every note in the scale would be a whole number of Hz. For the flat five (45/32), you need to start 2 octaves up, at 1440Hz, the number of minutes in a day.
    The lunar tides return close to 25hrs, so 24 is a useful division of a day for this reason, 1 day and 1 hour.
    The first whole number ratio between feet and Megalithic yards is 25:68, five times this is the diameter of the inner stone circle rings at Avebury (125:340). A very handy number of Meg. yards to measure the circumference when done in tenths of Meg. yards. 3927/1250 = 3.1416. This gives an error of 0.03 inch on the circumference from the real value of Pi.
    3927/17 = 231, a triangular number, 3927/7 = 561, also a triangular number, the ancients were known to divide the Earth and the the Heavens by 7 divisions.

  3. David says:

    Past history wisdom and science can be very fasinating. I thought our host and others may be interested in the following if he has never heard of it.

    Vedic Mathamatics

    Ancient math from time untold… In 1958 the Sri Sankaracarya Bharahti Krsna Tirtha of India paid a vist to the United States. He was the ecclesiastical head of the Gorvardhana monestary in Puri India, and was the apostolic sucessor of the first Sankaracarya. (ninth century; considerd by many to be India’s greatest Philosopher). It was the first time in the history of the thousand year old order that one of its leaders had visited the United States. He delivered many lectures at numerous universties, did radio shows, and at Washington and Lee University, Lexington, Virginia he engaged in a debate on religion and peace with distinguished historian, Arnold Toynbee, insisting that peace with honor is something worth fighting for.

    One of the most fasinataing aspects of his visit was his lectures on mathmatics from ancient India, at least 5,000 years old according to him. He has the academic credentials to at least lend some veracity to his claims. While still in his 16th year he was awarded the title of “Saraswati” for his mastery of Sanskrit. He appeared for the M.A. examinations of the American college of Sciences Rochester, N.Y. from Bombay Center in 1903; and in 1904 at the age of twenty passed M.A. examinations in further subjects simultaneously, securing the highest honors in all. his subjects included Sanskrit, Philosophy, English, Mathematics, History and Science. Quite an achivement for anyone, let alone a twenty year old.

    The history of Vedic Mathmatics he relates is a fasinating study. The entirely new twist of doing math with a completely different method, often easier and more efficient then the accepted comon methods. And all of this derived from the Indian sutras.

    Additionaly his concept of unity between variou religions, and religion and science you would I think find very stimulating.

    The title of the book is Vedic Metaphysics / DK Tirthaii
    Publisher was Montilal Banarsidass. It is out of print. I have a copy. If you would like to borrow it I would be happy to send it to you. Having thoughts one has never thought before is such fun, and should occur daily.

    Thanks.

    REPLY:[ I'd heard some of this before, but as different bits here and there; never so nicely packaged together. I read a bit about the vedic math and saw that it would do what was claimed, but also realized that at 50 something years old I was not going to be learning a whole new math AND Sanskrit too (it helps to know Sanskrit to really understand what the Vedas are saying...); so I let it pass me by. Were I a 20-something I'd probably have learned Sanskrit, then the rest of the Indo-European languages just become a subset with modernizations ;-) At any rate, the book sounds really interesting, and when I've caught up on the 3 or 4 years of things on my "must do" list, I'll take you up on the offer of a loan 8-{ One of the things I found fascinating what the size of a Bramahn year; roughly 3 Trillion years: http://www.sanskritmantra.com/brahma.htm and when you read of the 'cycle of creation' and that much of our universe is supposed to exist in other indistructible universes "above"... it really starts to sound like a big bang cosmology with multidimentional phase space... Some times I wonder if we are not just rediscovering what was known 12,000 years ago before the comet hit North America and melted all the ice causing "the great flood" and washing it all away ... Sigh. Cycle of creation... -E.M.Smith ]

  4. slick says:

    You forgot that 2 is also a factor of 10.

    It’s quite a useful factor of ten, actually–especially when trying to multiply a long number by 5.

    Just to be devil’s advocate, I find that the most common *demand* for conversions in daily life is between small, medium, and large distances, followed by converting between small, medium, and large areas.

    Converting cm to m to km is somewhat easier than converting in to yds to miles, but converting cm^2 to m^2 to km^2 is a heckuva lot easier than converting in^2 to yd^2 to miles^2.

  5. E.M.Smith says:

    @slick: OK, it was implied by the 5, but yeah, I left it out. Fixed.

  6. Verity Jones says:

    Knowing your interest in henges and the like:
    RTÉ.ie will be streaming the Winter Solstice at Newgrange from 8.55am on 21 December. This live feed will be available on RTÉ.ie/live
    http://www.rte.ie/news/2010/1220/newgrangefeature.html

  7. Pingback: Unifying The Cubits The Yard and The Rod « Musings from the Chiefio

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