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    Sign conventions and units.
    From: George Huxtable
    Date: 2001 Apr 21, 4:08 PM

    MUSINGS about sign conventions and units used in navigation.
    >From the earliest days of their art, navigators have been reluctant to give
    a mathematical sign (plus or minus) to the various quantities involved.
    Instead, "Names" have been used (North or South, East or West). And a
    complex series of rules-of-thumb has evolved for handling these Names, such
    as "take the difference between A and B if the Names are the same, add them
    if Names differ". In the days when logs were used, trig functions of angles
    were expressed as Versines, which never go negative.
    Not surprising, really, when many mariners were unlearned, and got their
    results by rote rather than understanding. The manipulations of signed
    arithmetic, such as subtracting -5 from -2, and getting the right answer
    (+3) could have them out of the depth. Nowadays, every schoolchild is
    taught how to handle such problems (even if some of them may fail to learn
    Anyone who has tried writing a program for astro-nav computation quickly
    meets the problem that the mathematics handles quantities with signs rather
    than names. The first thing he has to do is to convert from the names, such
    as North or South, into a sign, plus or minus, and then perhaps convert
    back at the end. Or he can ask the user to fall in with a simple sign
    convention when inputting data, such as North = Positive, South = Negative.
    That convention, for Latitude, is one that everyone can accept. Everyone
    draws maps with North upwards, even Australians. So, for Latitude, there's
    no argument.
    Not so for Longitude, however. There isn't one accepted convention, but
    two! The worst possible state of affairs.
    There are good reasons for choosing the West-is-positive convention. The
    Geographical positions of all heavenly bodies travel Westwards as the Earth
    turns, so it's convenient to have the longitudes of those positions
    increase with time: therefore GHA of Aries and star are measured Westwards.
    And in practice, many instruments and programs originate from the US, who
    have found it convenient to presume a positive longitude for their own
    However, there are equally valid reasons for measuring longitude positively
    to the East. It fits in with the conventional way that we draw a graph,
    with the X-axis increasing to the right, and the Y-axis increasing upwards.
    It fits in with the way astronomers have always measured celestial
    positions, in terms of the sidereal time at which a star will cross the
    Greenwich meridian, and is measured in hours, increasing in an Easterly
    direction. Certainly some instruments, such as certain Decca Navigators,
    presumed that East was positive.
    If the navigators of this mailing list were in a position to choose their
    preferred direction for a convention for positive longitude for the world
    to adopt, which would it be, East or West? I wonder.
    Of course, whichever direction was adopted as positive, longitudes could be
    measured either 0 to 360 degrees from Greenwich in that direction, or as 0
    to +180 in that direction and 0 to -180 in the other, with no ambiguity.
    There's a similar difficulty when specifying azimuths and courses.
    Navigators finally broke clear of the complex and non-numerical "points"
    system of "NE by N three-quarters N" (and so on), and it's rarer to see
    azimuths measured quadrantally, 0 to 90 degrees E or W of N or S. Instead
    me now use (mostly) the simple convention, 0 to 360 measured clockwise from
    N, as marked on modern compasses. Surely there's no argument about that,
    one might hope. But indeed there is!. Jean Meeus, in his invaluable book
    "Astronomical Algorithms", chooses to define azimuths as measured clockwise
    from the South, not from the North. He claims that this is a convention
    commonly used by astronomers, but accepts that there's a choice. So again
    we have the worst possible situation, tmo contradictory standards. If the
    world could settle on one, which should it be? And would the astronomers
    Angles such as lat or long are measured in degrees, we all accept. But what
    about subdivisions of the degree? Do we prefer to use decimal parts of a
    degree? Or minutes? And if minutes, should we state parts of a minute as
    decimal minutes, or as seconds? Instruments, and charts, are calibrated in
    many different ways. Do we navigators know what we would prefer to be the
    standard for the future?
    We are accustomed to using nautical miles, which fits in (almost) to the
    length of a minute of latitude. But this is an awkward unit in that it's
    incompatible with the length of the mile as used on land, and with the
    kilometer. As the kilometer is becoming the standard of length measurement
    over much of the world, excluding the US, are we willing to accept a change
    to the kilometer as a unit of navigational length, with its implications.
    The text above is a serious suggestion that we navigators should think
    about, and decide, what conventions we wish to see adopted. What follows is
    rather more frivolous.
    Much of the calculation in navigation stems from converting quantities from
    one awkward and incompatible unit to another. The revolutionary French had
    a good shot at reforming the system of measurements, but from a navigator's
    point of view they left much of this work undone, or made unsatisfactory
    choices. Wouldn't a navigator's work become simple if we could make the
    following changes? I know it's only a pipe-dream, but just think about it.
    1. Measure angles, not in degrees, but in Turns, corresponding to one
    complete rotation. 1000 milliTurns would bring you back to where you
    started. A right-angle would be 250 mT. For many calculations, one could
    simply cast off whole integer turns, and use the fractional part.
    2. Measure time in Days and milliDays, so that time in milliDays and
    longitude in milliTurns become instantly interchangeable (rather than 15
    degrees per hour). Breakfast would be around 200mD and midday at 500mD. If
    dates were registered by counting up whole days from the year start, 0 to
    364 or 365, ignoring months altogether, then time intervals between two
    events become a simple subtraction process, except over a year-end.
    3. Measure distances, not in miles or in kilometres, but in units equal to
    the Earth's circumference (the Circ?), so that one milliCirc corresponds
    (almost) to a milliTurn of latitude. The French approached this idea by
    making the kilometer one 10,000th of the distance between equator and pole,
    but they really got it wrong. So one Circ would correspond to 40,000 km,
    and a microCirc would be worth 40 metres. One would need to coin a name for
    a smaller decimal subdivision of this unit.
    You will be aware that I have ignored all the practical implications and
    difficulties of making such far-reaching changes, such as recalibrating the
    World's clocks and chronometers, sextants, compasses, logs, charts. That's
    because these suggestions are really.nothing more than an exercise in
    wishing. But wouldn't it be nice if the French had done the right thing in
    the 18th century?
    George Huxtable.
    George Huxtable, 1 Sandy Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.
    Tel. 01865 820222 or (int.) +44 1865 820222.

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