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## A Community Devoted to the Preservation and Practice of Celestial Navigation and Other Methods of Traditional Wayfinding

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Re: Multiplication by addition of logs
From: Grant Nixon
Date: 2019 Sep 4, 13:34 +0000
The negatives you cite apply only to the characteristics, not the mantissas.
That is,
Log10(Hav(322))=Log10(Sin(322/2)^2)=-0.974716
Log10(Cos(lat))=Log10(Cos(35))=-0.086635
Log10(Cos(DEC))=Log10(Cos(8+49.3/60))=-0.005168

So, the sum is -1.066519

Of course, making the mantissas positive (and characteristics negative), we have:
-0.974716= 0.025284 - 1
-0.086635 = 0.913365 - 1
-0.005168=0.994832 - 1

so, adding the now positive mantissas. we get 1.933481 and adding the negative characteristics we get -3 so their sum becomes -1.066519, as before.

This part looks correct to me. But you should note that we are still missing a term in the Haversine formula. See below:

Grant

Grant Nixon, Ph.D., P.Phys.

On Wednesday, September 4, 2019, 01:18:04 a.m. EDT, Ashley Betts <NoReply_AshleyBetts@fer3.com> wrote:

Hi All,

I'm finding it a bit difficult to find a source of information on how to correctly process logs for multiplication. I'm using the haversine method to reduce an artificial horizon sighting I took in my backyard. My goal it to only use Nories tables for the reduction. I followed the approach documented at the start of the tables (and as mentioned on here) but the aspect I'm struggling with is how the characteristic of the log is adjusted during the additions for the multiplication aspect of the formula (hav(LHA)*cos(Lat)*cos(Dec)). I have looked at a few example reductions on this site but I can not see how the characteristic of the log is adjusted in these examples. I ended up solving using an excel spreadsheet to see the actual values which then lead me to understand what characteric I needed for the resulting log, in this example. I however want to understand how I derive this value without the assistance of spreadsheet or other source going forward.

The example that I used involved:

LHA = 322deg

Lat = 35S

Dec = 8deg 49.3min

The values I got from the table were:

log.hav(LHA) = -1.02528

log.cos(Lat) = -1.91336

log.cos(Dec) = -1.99483

If I add the mantissa's the result is 1.93347

The resulting log I need is -2.93347. I can see no association between the original characteristics (all -1) or the carry over (1) and the resulting -2 that is needed.

Can someone point me in the right direction here with either an explanation of how to go about it or a source that explains this?

Regards,

Ashley

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