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Re: Bowditch Table 15
From: Jim Thompson
Date: 2005 Jan 27, 16:04 -0400

```> -----Original Message-----
> From: George Huxtable
>
> Jim Thompson writes-
>
> >I searched through the rest of Bowditch 2002 for more references to the
> >Table 15 procedure, and found this odd paraphraph in Chapter 22, Article
> >2202:
> >
> >"Distance by vertical angle between the waterline and the top of
> an object
> >is computed by solving the right triangle formed between the
> observer, the
> >top of the object, and the waterline of the object by simple
> trigonometry.
> >This assumes that the observer is at sea level, the Earth is flat between
> >observer and object, there is no refraction, and the object and its
> >waterline form a right angle. For most cases of practical significance,
> >these assumptions produce no large errors.
> >D = sqrt[(tan^2a/.0002419^2) + ((H-h)/0.7349)] - (tan a/.002419)

I did transcribe wrong.  It is:
D = sqrt[(tan^2a/.0002419^2) + ((H-h)/0.7349)] - (tan a/0.0002419)
With the correct number of leading zeroes in the second instance of
0.0002419.

> >where D is the distance in nautical miles, a is the corrected vertical
> >angle, H is the height of the top of the object above sea level, and h is
> >the observer's height of eye in feet. The constants (0.0002419
> and 0.7349)
> >account for refraction.".
> >
> >I don't see why the Table 15 equation is given in this
> paragraph.  Looks to
> >me like a chunk of text is missing between the end of
> "...produce no large
> >errors" and the equation, and that in fact the second half,
> which contains
> >the equation, belongs to a missing title that should describe
> "Distance by
> >Vertical Angle Measured Between Sea Horizon and Top of Object Beyond Sea
> >Horizon".  Can someone check an older Bowditch?
>
> ================
>
> Response from George.
>
> Quite astounding! Bowditch is compounding his errors even further!
>
> My edition, which is 1977 for vol 1, and 1981 for vol 2, has quite
> different text, at section 503 in vol 2.
>
> It is indeed quite reasonable to state, as Jim quotes from section 2202 of
> the 2002 edition:
>
> >"Distance by vertical angle between the waterline and the top of
> an object
> >is computed by solving the right triangle formed between the
> observer, the
> >top of the object, and the waterline of the object by simple
> trigonometry.
> >This assumes that the observer is at sea level, the Earth is flat between
> >observer and object, there is no refraction, and the object and its
> >waterline form a right angle. For most cases of practical significance,
> >these assumptions produce no large errors."
>
> =============
> In fact, working on those simplifying assumptions, the "simple
> trigonometry" would be to use
>
> distance in feet = height in feet / tan angle
>
> or distance in miles = height in feet / (6080 tan angle)
>
> or (as near as dammit) for 106 ft height at a mile distant you will see an
> angle of 1 degree. Smaller angle, then proportionately bigger distance.

Table 16 in Bowditch 2002 yields 0.99 Miles for Angle 1.00d, Object Height
105 ft.

So the wording of the title of Table 16, "Distance by Vertical Angle
Measured Between Waterline at Object and Top of Object" translates into
plain English as, "Distance between the observer and the object, using the
vertical angle measured between the waterline at the object and the top of
the object, for an object within the visible horizon of the observer".

> In my edition, such problems are assisted by using table 41, "distance by
> vertical angle; measured between waterline at object and top of object".
> Does the newer edition carry that table, perhaps with a different number?

Bowditch 2002, in "Explanation of Navigational Tables": "Table 16. Distance
by Vertical Angle Measured Between Waterline at Object and Top of Object -
This table tabulates the angle subtended by an object of known height lying
at a particular distance within the observer's visible horizon or vice
versa. The table provides the solution of a plane right triangle having its
right angle at the base of the observed object and its altitude coincident
with the vertical dimension of the observed object. The solutions are based
upon the following simplifying assumptions: (1) the eye of the observer is
at sea level, (2) the sea surface between the observer and the object is
flat, (3) atmospheric refraction is  negligible, and (4) the waterline at
the object is vertically below the peak of the object. The error due to the
height of eye of the observer does not exceed 3 percent of the distance-off
for sextant angles less than 20? and heights of eye less than one-third of
the object height. The error due to the waterline not being below the peak
of the object does not exceed 3 percent of the distance-off when the height
of eye is less than one-third of the object height and the offset of the
waterline from the base of the object is less than one-tenth of the
distance-off. Errors due to earth's curvature and atmospheric refraction are
negligible for cases of practical interest."

> ==============
>
> But then, to follow that text directly with the formula for Table 15,
> without further comment, is just crazy. Because that formula DOES
> allow for
> the fact that the Earth isn't flat, and it DOES allow for refraction!
> That's its whole point.
>
> And notice the formula as Jim transcribed it. It's wrong, but differently
> wrong compared to everything we've seen before.
>
> D = sqrt[(tan^2a/.0002419^2) + ((H-h)/0.7349)] - (tan a/.002419)
>
> In the third term, the constant divisor, of .002419, should have
> been .0002419.

I did transcribe wrong.  In article 2202 of Bowditch 2002 it is:
D = sqrt[(tan^2a/.0002419^2) + ((H-h)/0.7349)] - (tan a/0.0002419),
with the correct number of leading zeroes in the second instance of
0.0002419, not
D = sqrt[(tan^2a/.0002419^2) + ((H-h)/0.7349)] - (tan a/.002419)
as I wrongly trascribed.

> For the first time, however, we see Bowditch putting the square-root
> brackets in the right place, so that the square root embraces the
> first two
> terms, and not the third.
>
> In the earlier edition, in which it was Table 9, the explanation wrongly
> put all three terms within the square root, and also made the
> same error as
> above in the denominator of the last term.
>
> In the explanation of Table 15 in the later edition, the denominator error
> was corrected, but the square-root wrongly embraced just the first term of
> the three.
>
> So, in three tries, Bowditch has provided three different formulae for his
> own table, NOT ONE of which is correct.

My mistake: Bowditch did use the correct formula in article 2202.

> Assuming, that is, that the consensus Nav-l expression is indeed the
> correct one, being-
>
> d = sqr{(tan A / .0002419 )^2 + ((H-h) / .7349)} - (tan A / .0002419)

Yes, assuming that.

Jim

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