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    Re: Scrap paper with time sights c.1870
    From: Frank Reed
    Date: 2013 Feb 26, 11:21 -0800

    Lars, you wrote:
    "I have made some further research into the "scrap paper" voyage presented by Frank. In the enclosed excel file is my final (I hope) summary. I am quite sure the voyage was made 1855-1856, or at least the NAs for those years were used. The voyage itself can of course have been made later. Can anyone find another period of time with as good match as this?"

    Excellent analysis, Lars. I had to dig through my notes to find out just which of my old Bowditches had these bits of scrap in the back originally. It was an 1847 edition, so that's no conflict with your 1855/56 results. As you note, it could have been some later year, since every navigator knew that the Sun tables could be re-used four years later and eight years later... even twelve years later without much error. In any case, the exact year doesn't actually tell us anything particularly useful, though it's fun to know.

    For your comparison data, you mentioned that you looked at the American Ephemeris and Nautical Almanac, which was just starting at this time. The first volume was printed with data for 1855. This was a rather "formal" volume trying hard to compete with the British Nautical Almanac (the Nautical Almanac and Astronomical Ephemeris) which had already evolved into a reference aimed more at astronomers than navigators. Meanwhile, there were numerous inexpensive private almanacs available. Hart alludes to them in his instructions in that quirky book I described in another post. Some of the private almanacs were no better than "farmer's almanacs" but even these usually had very good tables of the Sun's Declination and the Equation of Time. There were also privately published "Nautical Almanacs" which were more obviously intended to serve the needs of mariners. These were usually slimmed down extracts from the British Nautical Almanac, often with other articles added and maybe tide tables, too, and in many ways these practical volumes are the real precursors of the modern "Nautical Almanac". Nearly all of these ceased publication in the US during the 1850s, replaced by the paper-bound extract from the American Ephemeris. Of course from the "scrap paper" there would be no real way to tell any difference. It is possible, however, that a small difference in original source might be detected. The American Ephemeris was re-calculated by reconstructing the tables of the Sun, Moon, planets and stars from first principles. It was quite an undertaking (and controversially expensive in that bygone era when the US federal government was a small entity). The lunar tables, when published, were slightly superior to the British tables for a couple of years leading to some embarrassment in Greenwich. The solar tables were, of course, more nearly identical. But if our nameless navigator was using a private almanac, and presuming that its data was taken from the British Nautical Almanac, then that might be responsible for some of the few remaining differences in the data in the sights. Just a possibility.

    You wrote:
    "The equation of time is always interpolated to the time of observation, but for the chronometer correction the same value is used during the day, although it changes by 1s every five hours."

    Interesting point! I don't think I've ever seen instructions anywhere proposing that the chronometer correction should be interpolated for hours. An intelligent navigator might have done so, but most practical navigators simply followed the recipe as given. Tables that appear to be cumulative chronometer corrections are quite common in the margins of old logbooks and navigation manuals. Also, even though they worked everything out to the exact second of time, navigators were certainly aware that one second only shifted the longitude by a quarter of a minute of arc. The back cover of the same Bowditch that held this scrap paper has hand-written notes, apparently in the same hand, that note specifically "15 miles = one minute, 1 mile = 4 seconds" (that should be a navigator's bumper sticker).

    For another example of "rote" navigation, when interpolating for the Sun's Declination, navigators followed the instruction in Bowditch which described a two-step process of interpolation where one first interpolated by the DR difference in longitude and then by the local time (this is equivalent to estimating the Greenwich time). Of course, this instruction was originally written c.1800 when chronometers were rare. The obvious process when a chronometer is available is to interpolate between the values directly, based on the GMT read from the chronometer. Even at the end of the 19th century, navigators were still, blindly, following rules written down almost a hundred years earlier.

    You also wrote:
    "The altitude corrections show just a few irregularities, indicating that index error seems to be stable. A 4’ correction for index error and dip seems to be most usual."

    That's probably the standard 4' for dip alone. It seems to have been fairly common practice in the era to zero out the index error manually (certainly the case with wooden quadrants/octants) or to apply the index correction before working the calculation. You do see "Corr'n" sometimes, and from context, it's what we call "index correction" today, but it's less common than in modern practice. Also, while 4' for dip corresponds to a height of eye of 16 feet, this does not necessarily imply that our navigator measured this. It was considered a reasonable compromise value for any "normal" sailing vessel in the era. At least we can deduce that the sights were not being taken from the crow's nest!

    You concluded:
    "I don't think that much more information can be squeezed out of that paper."

    Yes, I think you've pulled nearly everything that can be calculated out of the available data! Nice work. We can now step back a bit and consider some more general points.

    Here's one to start: every navigator who learned in the late 20th or early 21st centuries has seen sight reduction forms. In fact, I would say that many navigators get "hooked" on them ...can't work a sight without a form! Yet these tiny, scribbled calculations with a common workflow but no fixed form were normal in the 19th century. I've seen books of "formulae" for clearing lunars, but even for lunars they weren't popular (the word "formula" has shifted in meaning: it meant "form" back then and was not synonymous with "equation", as it nearly is today). Paper was expensive and printed forms were a luxury item.

    And what of the mathematical method of solution? They worked time sights by this method (which we name a "haversine" solution today, if we name it anything at all), not because it had any superiority in the quality of solution. In fact, there are many different ways to solve the PZX spherical triangle for the local hour angle or, as they called it back then, the "true time". Most navigators today who've studied enough of the math to work spherical trig solutions directly would do it using the standard "law of cosines" either on a calculator or in a spreadsheet or some other software. The haversine solution (or also the equation underlying, for example, Martelli's tables) can be recovered from the law of cosines by the simple application of ordinary "trig identities", still taught to many secondary school or high school students. The law of cosines solution and the haversine solution are "identical" in that sense. They take EXACTLY the same inputs and yield EXACTLY the same results. From the point of view of modern computation, there is no difference whatsoever. But in the logarithmic calculations that were the norm for some 250 years, the haversine version was easier to set up and importantly required no verbal rules and no "cases" based on the signs or "names" (north or south) of the various inputs. The haversine method was overwhelmingly, far and away, the most popular method of solving this problem. You can find examples of time sights cleared this way throughout the history of navigation as long as time sights were still being worked. I've written up details in NavList posts on some from as early as 1827. Dave Walden once posted on one of the examples from the navigation notebook of the Charles W. Morgan from the 1890s that we also study in the "19th Century Methods" class that I teach. Brad Morris worked up some examples from Worsley's logbook of the harrowing open boat voyage with Shackleton to South Georgia. And let's not forget the examples that Henry Halboth posted on NavList just last month from his very own navigation notebooks from the 1940s which still continued that long tradition of working time sights by the haversine solution. It was one of the most stable aspects of celestial navigation during its entire history.

    -FER
    PS: I am "pressing" that other piece of scrap so that it will scan properly. I should have it for you to investigate in a day or two.

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