Tuesday, 10 May 2016

Degrees, Minutes And Seconds

Here's a printable 360 degree protractor:

If you print this on plastic film and cut it out, you will have a protractor.  This to me is a tiny example of how post-scarcity would work because although you could in theory go down the local stationer's and buy a protractor, this one's available for free plus the cost of the ink and plastic.  In an ideal world, oh never mind.

The reason I've put this here, apart from it being a resource for home edders, which in theory is what this blog's supposed to be about, is that it clearly shows the 360 degrees of a circle.  Most people know degrees well enough to be able to understand what's meant by a ninety or forty-five degree angle.  One's surroundings can also be thought of as a 360 degree circle, meaning that the traditional description of objects being at "two o'clock" or "six o'clock" can generally be supplanted by talk of degrees.  It also turns up as a way of describing positions in the sky and on the surface of this planet and other roughly spherical celestial bodies such as Mercury, turning up as latitude and longitude:

In the case of the sky, i.e. the celestial sphere, there's a slightly different system where right ascension and declination replace longitude and latitude respectively, and whereas declination also uses degrees, right ascension uses hours, minutes and seconds.  I tend to think of it as representing the time something rises above the horizon even though that can't be exactly what it is:

This last system, however, is confusing because the words "minutes" and "seconds" refer to different things in different directions.  There are of course 86 400 seconds in a day, so the smallest complete unit of right ascension, a second, is in fact fifteen seconds of arc (arc seconds).  This is the bit I didn't explain.

Minutes and seconds are not just units of time but also of angle.  Perhaps surprisingly, neither of these two systems are completely metricated although metric systems for time and angle do exist.  I'll get back to those.  A minute of arc, or arc minute, is a sixtieth of a degree, that is a relatively tiny shift in direction which would be microscopic if it was marked on the above protractor.  However, in terms of things such as the night sky, the circle of vision and the surface of Earth, arc minutes are fairly large.  An arc minute, coincidentally, is the size of the smallest object visible to the naked eye.  An object an arc minute across would be a tenth of a millimetre across from a distance of twenty-five centimetres.  In fact it's not strictly true that a minute of arc is the smallest part of the visual field discernible because luminous objects are still visible even when they're much smaller and the brighter they are, the smaller they can be.  This means, for example, that the immense distances to the stars still doesn't render them invisible to most people even though they are far smaller than one minute of arc across.

Both big lights in the sky, the Sun and Cynthia, are well over a minute of arc across.  They are in fact both around thirty minutes of arc in diameter from here, meaning that solar eclipses are possible.  This is such an unlikely coincidence that unless there's some reason why habitable planets need such a ratio, solar eclipses are not only a wonder of the world but of the entire Milky Way.  It's unlikely that there are any other planets in this whole Galaxy which have solar eclipses, and in some imaginary Galactic Empire where faster than light travel is possible, this could make Earth a top tourist destination.

Arc minutes also turn up in the form of nautical miles.  If you went a quarter of the way round the equator you would have travelled ninety degrees.   A single degree of longitude on the equator is around fifty miles.  Hence in this context a mile on the equator can be thought of as a measure of angle as well as distance, but it's not a convenient unit because the planet is not exactly 25 000 miles in circumference.  It ought to be exactly 40 000 kilometres in circumference because the kilometre was originally defined as a ten thousandth of the distance between the North Pole and the Equator on a line across the surface passing through Calais.  This is not how it works due to surveying inaccuracies and the fact that the planet is slightly tangerine-shaped rather than absolutely spherical.  However, a nautical mile is defined in a similar way as a minute of longitude, which is 1852 metres, slightly more than the 1609-metre mile.  Knots are then defined as nautical miles per hour, thereby combining two angular sexigesimal (sixty-based) systems in a very neat and appealing way.

Unsurprisingly, minutes themselves are divided into sixty seconds in this system too.  A second of arc or arc second is a sixtieth of an arc minute, which is such a tiny angle that there are 3 600 in a degree.  Again, this sounds fairly useless but again it has many functions.  I'll just mention two.  Measured six months apart, Earth's orbit is 300 million kilometres across, so objects in space shift very slightly against their background in a process referred to as parallax.  The nearest star apart from the Sun shifts by slightly under one arc second in position due to this.  The distance over which an object would shift by this angle is known as a parsec - parallax of one second.  A parsec is just over three and a quarter light years.  Another use of an arc second is to describe the size of distant objects in the sky, so for instance yesterday Mercury was twelve arc seconds across as it crossed the Sun.  Pluto is between four and seven seconds in diameter as seen from here.

As I mentioned before, there are metric and decimal versions of angles.  One of these is the radian:

A radian is easy to describe but for me very hard to understand the point of.  It's simply the radius of a circle moved onto its circumference, or 360 degrees divided by π.  This is around 57 degrees.  I don't know why this is useful although programming languages tend to use it, so I have to use it.

Another decimal angle system was used on the Peter's Projection map, though not on this one:

By Strebe - Own work, CC BY-SA 3.0, https://commons.wikimedia.org/w/index.php?curid=16115242
I'm not a fan of the Peter's projection but that's another story.  Some versions of the Peter's world map divide the "globe" not into degrees but gradians.  There are four hundred of these in a circle instead of three hundred and sixty and they were probably invented by someone French.  They're used in surveying.

I hope this makes more sense now.