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Re: Martelli's Time-Sight Tables
From: Frank Reed
Date: 2012 Dec 25, 14:10 -0800

Greg, you wrote:
"What would be the best way to modify Martelli's table V formula to generate local hour angle rather than hours minutes and seconds ?"

That's actually easy. But if we start re-engineering these tables, you will discover that there is really nothing to them after you sweep away the "special" constants and the odd sexagesimal tabulation involved in Martelli's Tables II-IV. That had me stumped, by the way, and my thanks, too, to Lars for posting the details. Let's take a look at Lars's analysis, and see what each table is "really" giving us. What is tabulated fundamentally?

--Table I. 10^4*(0.5+log(cos(x)))
That's "really" just a table of log cosines as you would find in any navigation manual. Generally navigation manuals added 10 to log trig functions. Martelli adds 0.5. This limits the tables to sub-arctic latitudes --probably not a problem in practice, but an un-necessary limitation. Might as well use a standard log cosine table here.
--Table II. 10^3*(0.2+cos(x))
That's "really" just a tables of natural cosines with 0.2 added. For positive values, this limits us to a maximum value of x of about 78.5 degrees. For most time sights, that wouldn't be a practical problem, but it's un-necessary. A table of natural trig functions is always positive if we add 1 (that is, list 1+cos(x)). Might as well use that table. Also note that Martelli tabulates these values as sexagesimals (minutes and seconds) which is pointless. This is just a little bit of that "mystery" that Cotter was talking about.
--Table III. 10^3*(1-sin(h))
That's "really" just a table of natural sines. same points as for Table II and it could be combined with II.
--Table IV. 10^4*[log(21600/(x-1200)) - 1]
That's "really" just a table of common logarithms, or if you prefer, it's a close cousin of a table or "proportional logarithms". Notice that you can re-write the expression Lars has given in many ways (of course --that's a feature of logarithms). For example, it's the same as -10^4*[1 + log((x/60 - 20)/360)]. If we drop the "mysterious" sexagesimal output from Tables II and III, then we can replace this with a more obvious table of common logarithms.
--Table V. log(21.6/(1-cos(t))
This is "really" just a log "versine" table. Apart from a constant, it's identical to -log(1-cos(t)). And if you want output in degrees rather than time units, then you can combine this with Table I again.

To sum up, Tables I,II,II, and V can be combined into one table of natural and log trig functions and Table IV is just a table of common logarithms (which, like Martelli's tables or the earlier "proportional logarithms" can be modified to remove one brief step in the calculation if desired). So you can re-build Martelli's method for modern use and remove all the mystery.

Certainly, one could do this. You could build a "better Martelli" but would it sell at all (sell, in the general sense of finding users) today and would it have sold any better than Martelli's a hundred years ago if we could stick it in our time machine and take it back into the past? Martelli's product power was not its mathematical science but rather its inscrutability. If this reconstruction is for pleasure and mental challenge, that's fine of course! But if you want an efficient time sight method that occupies relatively few printed pages and was almost universally used historically, why not use the standard method from Bowditch and many other navigation manuals? There was no navigational science "magic" in Martelli's Tables. But they surely had marketing "magic". It's interesting to contemplate how and why he might have made his choices, some clever and some peculiar, but in the end it's solving the exact same problem with roughly the same amount of work.

-FER

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