# NavList:

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

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Re: Clearing Lunar Distances by trigonometry
From: Dan Allen
Date: 2004 May 15, 15:30 -0700

```I have posted a web page that contains this formatted correctly.

It is located at:

http://danallen46.home.comcast.net/Halboth.htm

Dan Allen

-----Original Message-----
Halboth
Sent: Wednesday, May 12, 2004 8:11 PM
Subject: Clearing Lunar Distances by trigonometry

I make one more effort this evening to get this format across straight -
hope it works this time.

As a matter of potential interest, the following is the haversine format
I employed  in clearing Lunar Distances by spherical trigonometry, before
the advent of calculators and, wherein ...

Hs = sextant altitude..............     Ds = sextant distance.............S =
sun
Ha = apparent altitude............      Da = apparent distance..........
M =
moon
Hc = true altitude....................  Dc = calculated distance........
Z
= angle at zenith

1)  hav Z = sin (s - Ha M) x sin (s - Ha S) x sec Ha M x  sec Ha S
in which ...  s = ? (Ha M + Ha S + Da)
2)  hav Dc = hav (Hc M ~ Hc S) + cos Hc M x cos Hc S x  hav Z,

therefore ...

Ha M    75-07-00        l sec   0.590318
Ha S    25-45-03        l sec   0.045424
Da      74-46-17
2s          175-38-20
s       87-49-10
s - Ha M        12-42-10        l sin   9.342213
s - Ha S        62-04-07        l sin   9.946211
Z               l hav   9.924166

l hav   9.924166
Hc M    75-22-00        l cos   9.402489
Hc S    25-43-12        l cos   9.954689
l hav   9.281344

n hav   0.191137
Hc M~S  49-38-48        n hav   0.176241
Dc      74-37-07        n hav   0.367378

To afford a comparison, the altitudes used, both apparent and true, are
as employed in an example of Borda's method,  set forth on page 417 of
Norie's 1889 edition where the cleared distance is found to be 74-37-10 -
all of 3" greater than that found by the haversine formulae stated above.

```
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