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    Using logs in nautical tables
    From: George Huxtable
    Date: 2003 Jun 5, 17:52 +0100

    Using logs in nautical tables.
    I was introduced to logs in maths classes in school, over 50 years ago now.
    I wonder if logs still appear on the modern syllabus, in these days of
    calculators and computers. It was a bit of a surprise when I first met
    nautical tables which also used logs, but on a rather different basis. It
    may have surprised some readers of this list in the same way. Let's discuss
    Take the number 2 as a simple case. log 2.000 is 0.3010, log 20.00 is
    1.3010, log 200.0 is 2.3010
    You can consider the log as having two parts. There's the part after the
    decimal point (which is called the mantissa: sorry about these fancy names,
    but it has to be called something...) which remains the same whenever we
    are dealing with an original number (in this case 2), multplied by some
    power of ten. And there's the part before the decimal point, an integer
    called the characteristic, which defines where the decimal point was in the
    original number (it's actually one less than the number of significant
    figures to the left of the decimal point).
    So far, we have discussed logs of numbers that are greater than one.
    Log 1 is 0.0000. Numbers smaller than 1 will have a negative log. For
    example what about the number 0.2000? This is a factor of 10 smaller than
    2.000 that we considered before, so its log must be smaller by 1. So
    instead of 0.3010, its log must be (0.3010 - 1), or -0.6990. Similarly, log
    of 0.02000 is -1.6990, and so on. This is perfectly valid and correct, but
    it's awkward when you start to add and subtract logs.
    There are two alternative tricks to make things easier.
    Trick 1. How we did it at school. (skip this if you didn't come across the
    "bar" notation in school, as it will only confuse you)
    Instead of writing log 0.2000 as -0.6990, we write it instead as <1.3010.
    At school, the "characteristic", the quantity to the left of the decimal
    point in the log, was written here as a 1 with a bar over the top, and
    referred to as "bar one". However, email typography doesn't allow me to do
    that, so in its place I have invented here a notation of preceding the
    characteristic with the "<" character: still using the expression "bar one"
    when putting it into words,and meaning the same thing.
    "<1.3010" is different in its meaning from "-1.3010" (which would be
    wrong). The meaning of <1.3010  is that it represents (-1 +.3010), which
    has exactly the same value as -0.6990. The < symbol means "treat the
    characteristic as a negative integer, but leave the mantissa as a positive
    quantity". Similarly, log of 0.02000 can be expressed as <2.3010.
    Remember here that "<" has the meaning, when you are adding and subtracting
    logs, "always treat the mantissa as a positive quantity, but if the
    characteristic is preceded by <, take the characteristic as a negative
    This standard trick makes manipulating logs a lot easier, believe me.
    Trick 2. How (presumably) it's taught in nautical colleges and done in
    nautical tables.
    In this case, use of negative logs, for quantities less than 1, is got
    round by a different trick. If any quantity in a calculation is less than
    1, multiply it by an enormous number (actually 10 to the power 10, or
    10,000,000,000) to make it greater than 1, but so much greater that it must
    become obvious in the answer, and confusion can't result (one hopes). Then
    allow for that factor at the end of the calculation.
    This is done very simply by adding 10 to the log, when it would otherwise
    become negative. For example, to find log 0.2000, first multiply it by
    10-to-the-10, which gives
    0.2000 x 10,000,000,000, or 2,000,000,000.0 and find its log by the rule as
    before, which will give a characteristic of 9 (10 significant figures to
    the left of the decimal point, so characteristic is one less, i.e, 9) and a
    mantissa of .3010, so log = 9.3010
    Feel free to add 10 to the characteristic whenever it helps in this way. A
    number of terms in a complex calculation may have been so modified. At the
    end of the calculation, before going back from logs to ordinary numbers,
    strip off or add on 10, or 20, or whatever is appropriate, to the
    characteristic, to bring the result into the working range that corresponds
    to the real world. There may be a set of rules defining how exactly to go
    about this in a logical way, but I haven't met it.
    There's a weakness in this second method, which I presume is a reason why
    it wasn't used in my maths lessons at school. If you wish to raise a number
    to a power, or take a root, using logs is an easy way to do it. But if the
    characteristic has had an extra 10 (say) added on, and you want to take a
    cube-root (say), that involves dividing the whole thing by 3, which would
    involve tacking on an extra 3.3333 to the result, so completely upsetting
    the mantissa as well as the characteristic. It simply wouldn't work.
    Fortunately, such problems of raising-to-a-power just don't occur in
    nautical astronomy, as far as I know. The worst case that really does occur
    is in finding a square-root, and then one has to be aware that an odd 5 may
    turn up in the characteristic.
    Handling negative quantities, using logs.
    Above, we considered the question of handling negative logs, of quantities
    less than 1. Now we come on to the handling, in a calculation using logs,
    of quantities less than zero, negative quantities.
    The problem is that logs are defined only for positive quantities: the log
    of a negative number has no meaning whatsoever.
    Standard trig formulae have been manipulated to death, often involving
    considerable extra complication, to rewrite them in terms where no quantity
    ever goes negative. Complex rules have been developed to avoid negatives,
    on the lines of "subtract the lesser from the greater, and name it
    according to the nearer Pole", or some such thing. It's one reason why
    "names" such as North or South, get used instead of quantities with signs
    (such as North latitude is +, South lat is -).
    And yet, I've found from reading Chauvenet ("Spherical and Practical
    Astronomy", 1853), astronomers had a simple way of handling such negative
    quantities in log calculations. As far as I can tell, it's done this way.
    If any quantity in a log calculation is negative, it is instead taken to be
    the corresponding positive quantity (as if it had been multiplied by -1)
    and the log of that quantity written down. Alongside that log, the fact
    that the sign of the quantity has been changed is noted (Chauvenet's
    notation is to precede it with a letter n). At the end of the calculation,
    when all the logs have been totted up and the result in real quantities
    determined, the number of "n" notes is counted up, and if it's odd, a
    negative sign is given to the final result. Simple, but effective.
    I should add that Bruce Stark's Lunar Distance Tables have their own clever
    way of overcoming the difficulties described above.
    Of course, with computers and simple hand-held calculators able to perform
    all these calculations directly, without using logs, the above
    considerations are of interest only to those following up the historical
    techniques of calculation by tables.
    contact George Huxtable by email at george@huxtable.u-net.com, by phone at
    01865 820222 (from outside UK, +44 1865 820222), or by mail at 1 Sandy
    Lane, Southmoor, Abingdon, Oxon OX13 5HX, UK.

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