Hi Bryan,
The "9" in "9.11943" can't be right for any reasonable "log" of "tan",
so it's likely a misprint. The "log" than Waugh would likely be using is
the common log, LOG10, rather than the natural log we mostly use these days.
And indeed LOG10(TAN(7.5)) is −0.8805708975, which is "-1 + 0.1194291025".
In Logarithm books (for those of us who remember using them), when you
looked up the logarithm of a number you had to first express the number
in scientific notation as the lookup tables only had numbers in a
certain range. So in this case, the logarithm in base 10 of 7.5 degrees
is 0.1316524976, but you had to first express that as 1.316524976 x
10^(-1), then lookup the logarithm (base 10) of 1.316524976, which is
0.1194291025. The logarithm of the exponent is -1. However rather than
waste time subtracting 1 from 0.1194291025 immediately, the exponents
(whole numbers) and fractions were treated separately in further
calculations and only combined at the end. Hope that makes sense.
I suspect Waugh was just using logarithms as a means for making
calculations by paper easier. They would not be inherently part of the
equations he was trying to solve. He was really interested in the TAN,
and finding the LOG10, was just so he could add intermediate numbers
rather than multiply them.
Regards Hank
On 9/8/22 3:19 pm, Bryan Mumford wrote:
I’m working from Albert Waugh’s book “Sun dials, Their Theory and
Construction”. On page 45 he presents a method for computing hour lines. I lack
significant math skills, but I know how to work Excel. I don’t understand how
he is calculating these values.
He says, for example, that “log tan t” of 7°30’ is 9.11943.
In my simple-minded way I asked Excel to show me log(tan(7)) and got a very
different value.
I tried converting 7°30’ to radians and that didn’t get any closer.
How can I calculate "log tan t" or "log sin latitude” with Excel to get the
values he shows?
I anticipate further problems with the last two columns, but you have to start
somewhere….
- Bryan
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