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    Re: Latitude from meridian observation
    From: Frank Reed
    Date: 2018 Sep 20, 16:42 -0700

    Paul Hirose wrote:
    "Clearly, Wylie doesn't have the traditional aversion to signed arithmetic in navigation."

    Starts right about then, too. Post-war... By this date, education is changing. Today, that change is the better part of a century in the past. And this is a big advantage we have in celestial navigation education today: everybody studies algebra. I'm not necessarily talking about solving algebraic equations, though nearly every adult has been exposed to that at least at a basic level, but the key algebraic concepts of addition and subtraction of positive and negative numbers are familiar to all but a very few students who take up celestial navigation. We can dispense with all those special "same name" and "opposite name" rules in most cases.

    And wrote:
    "I believe that in practice the great majority of navigators could ignore the case of lower transit."

    Absolutely. Except for the exams, this is nearly irrelevant to practical navigation, and in any case, we can simply define it as a separate problem with a separate methodology. There's no reason to force both cases into one shoebox.

    And wrote:
    "My own preference is to take the complement of altitude"

    Which we safely refer to as zenith distance! "Complement of altitude" is neither more nor less "jargony" than "zenith distance" so we may as well use the latter term. It's more common and has a long historical legacy. The zenith distance is 90°-(corrected altitude) or for a quick approximation (common in the 19th century), it's 89.6°-(raw sextant altitude). 

    And wrote:
    "Then apply that to declination, in the common sense way according to whether you're facing north or south."

    There's a marginally easier route which I have taught for many years which requires no application of "common sense" and employs only one equation for all cases (because we are using +/- algebraic signs which manage the cases for us implicitly). Latitude and Declination have signs that correspond directly to common practice and are familiar to many users today from computing applications, Google Maps, and so on: North is +, South is -. We can keep this same North positive, South negative rule by adding a simple rule for the sign of the zenith distance. The zenith distance as computed (90-corrected altitude) is necessarily positive, but we apply an algebraic sign to it by asking which way the observer's shadow is pointing. And shadow direction determines sign by the same N pos./S neg. standard: if your shadow points North, then the Z.Dist. gets a positive sign (it already has one, so no action required), if your shadow points South, then the Z.Dist gets a negative sign. This covers all cases and requires just one equation:
      Lat = Z.Dist + Dec.
    So easy! Deciding shadow direction can be done by actually looking at your shadow on the deck while you're taking the sight and thinking which way the bow is pointed, or it can be done in a textbook case or an exam by thinking through which way the observer's shadow would be pointing from information about the DR Latitude or other details. If the body is the Sun or the Moon, then the shadow is real. You can look at your shadow. For stars and planets, the shadow is "virtual" --ask yourself which way your shadow would point if that star were very bright. 

    Meridian transits below the pole (for example, in mid-northern latitudes, measuring the low point altitude of Kochab as circles beneath Polaris) is a separate case (Lat = CorrectedAlt + 90-Dec), and here we ignore signs on Lat and Dec. But again, this is a largely pointless circumstance. Most navigators will never shoot such a sight unless they go out of their way to plan for one. With the exception of some summer sights in the high Arctic, below the pole meridian altitudes are impractical textbook problems. You learn it for the exam, and then throw it away.

    Frank Reed
    Clockwork Mapping / ReedNavigation.com
    Conanicut Island USA

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