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    Re: My Haversine Table...
    From: Frank Reed
    Date: 2014 Apr 16, 10:55 -0700

    Hi Greg,

    Regarding haversines, you wrote:
    "Please let me know if I made a mistake. Also; I can not tell if this formula is pre-1840?"

    Haversines, in this fashion, would be anachronistic. May I suggest that you download the second (1807), sixth (1826), or tenth (1837) edition of Bowditch's "New American Practical Navigator". There are links here:
    http://www.fer3.com/arc/navbooks2.html
    Take a look at the articles that actually explain how to do these calculations "to find the true time" (they would not have called it LHA back then). You could even use the historical tables in those books directly. The principal method that you will find in Bowditch for this calculation was overwhelmingly the most common version of the logarithmic calculation, and almost identical recipes could be found in most other navigation manuals in this period.

    You could also look at a passage from an odd little book called "The Romance of Yachting". I've posted this before, but I think it's worth seeing again... And Greg, in particular, read the few paragraphs that begin with Hart writing, "I will now proceed, as plainly as possible, to show my work, corrections and all, by which I ascertained my longitude". Hart is describing calculations made in 1846. That's after your chosen time period, but this is also just how it would have been done thirty or forty years earlier.

    The book is like a modern "blog". The author rambles and spins yarns and pronounces strong opinions on a laundry list of subjects quite unrelated to the nominal topic, like any modern blogger might, but all these thoughts and ramblings are set down on paper. In this little book, in a relatively rare moment when he is on-topic, he offers up what we might call an "interested amateur's guide to nautical astronomy". He spells out in rather good detail how to determine latitude at noon, how to estimate longitude at noon, and how to determine longitude more exactly by time sight in the afternoon or morning. If you had nothing else, you could literally navigate using this book and the appropriate tables. But rather than describing Hart's instructions, I'm going to post here the transcription, nearly in full, as rendered by the Google Books archive. It's plain and simple navigation, just the way it was really done back then with no pretenses about solving spherical triangles. This is not rocket science. The calculations were "recipes" pre-cooked by mathematicians and astronomers on shore, decades earlier. A good navigator merely needed to know how to follow the steps. Here's Hart laying it all out:

    --------------------------------------------------
    "The Romance of Yachting: Voyage the First"
    by Joseph C. Hart, New York, 1848.

    CHAPTER XVII.
    SHOWING HOW TO NAVIGATE A YACHT ACROSS THE ATLANTIC, OR ELSEWHERE.

    The young sailor, or Amateur Yachter wishing to become a sailor, and to manage his own craft, will find this chapter, and the one following, more worthy of his attention, for practical uses, than any other in the work. The general reader, on the contrary, having no especial regard for "tar, pitch, and turpentine," nor any desire to box the compass secundem artem, nor to keep his reckoning at sea in good set phrase, nor to "take a pull at the main brace" when the occasion justifies that nautical ceremony, will do well, with but a slight glance, to let them pass for the present. Even he, however, will be sure to recur to them, after being a short time at sea, for a fund of rational amusement, as well as instruction in the occult science of navigation; and also for the means of illustrating much that he will find constantly going on around him, which otherwise would be a dead-letter or a sealed book to his comprehension.

    They are somewhat full of technical and practical navigation, and in fact contain more theoretical information, in a few words, and in the simplest form, than many a good sailor thinks it worth his while to acquire on shore in the schools. If I have been obliged to knock the rust off my own seamanship, in order to be able to present the information in this form, I am a great gainer, and have reaped an advantage already, which I hope to share with many of my young and aspiring countrymen, who are determined, like the daring mariners of ancient Tyre, to "go down to the sea in ships;" although those same men of that great Phoenician city, "whose merchants were princes," had none of the modern advantages here presented for practical navigation.

    Far be it from me to undertake to write an extended treatise upon navigation, while such excellent books as Bowditch's "Practical Navigator," and Brady's "Kedge Anchor" are so easily obtained in all our sea-ports. No professional seaman, of any pretension to nautical science, will go to sea without them. But I think I may venture to predict, that, in addition thereto, no amateur sailor will hereafter trust himself upon the deep waters, without tucking this, "The Romance of Yachting" into his berth or hammock, to solace him in the leisure hours of his voyage : And the more especially do I think thus, because I design to enlighten him in that which will be all-sufficient, (with some other slight aids), to enable him to keep a reckoning for himself, and daily to find the place of his ship upon the chart, so that he may "stick a pin there," and say to himself, "thus far have we gone."

    With the high example of Queen Victoria before the world, who is often on the sea a-yachting, why should not the intelligent women of this country, (who, if the truth were told, are as fond of yachting as the men,) also dip into the mysteries of navigation ? The rules for ascertaining their latitude and longitude, or their position at sea, are so few and simple, that when they come to know they will not be obliged to wade through ponderous tomes to acquire them, we may well expect to see some of them as good sailors as her little majesty, who has bothered her cousin, "Lord Adolphus," and studied Hamilton Moore, to good purpose in that behalf. In short she is said to be setting her naval marine a bright and particular example in seamanship, and can "box the compass" with the best of them. [...]

    TO ASCERTAIN THE LATITUDE.
    A good quadrant may be obtained for a few dollars : The plainer the better, so that it is a well-made and accurate instrument. A practical nautical man will select one for you at a glance, possessing every necessary requisite. He will also purchase for you a nautical almanac, which you may find indispensable ; but for the purpose of ascertaining the latitude solely, any almanac will answer in which you can find the sun's correct declination ; and the generality of common almanacs have it correctly stated, or within a few seconds perhaps, which are not very material for the amateur. I procured a sixpenny one just as I was going on board, which, happening to prove quite correct in the declination columns, saved me the trouble of daily making corrections from the nautical almanac.
    See that your quadrant is adjusted before using it, by ascertaining whether the index and horizon glasses are perpendicular to the plane of the instrument, which is managed by a simple screw on the back, and is easily understood by your being once shown how to adjust it. Once regulated, it will probably remain permanently so, unless the instrument is misused.

    You are now prepared for ascertaining your latitude ; which, you know, is distance from the Equator, north or south. It is measured by an arc of the meridian contained between the zenith and the equator. If the zenith distance of any heavenly body, when on the meridian, is ascertained, and the declination of the object is given, the latitude is easily found. When the sun arrives at his meridian for the day, at the place where you then are, it is noon at that place. You will proceed as follows for your latitude : Ascertain from your almanac, or from any other source, the declination of the sun for the day of the month ; note it; and then take your quadrant on deck a few minutes before noon.

    If the sun is bright, turn down a dark glass ; hold the instrument in a vertical position; apply the eye to the hole in the sight vane ; direct the sight to that part of the horizon beneath the sun; move the index till you bring the sun to touch the horizon; by a slight vibratory motion of the quadrant, right and left, the sun will appear to sweep the horizon, which, by observing the lower part of the arc it appears to describe, will enable you to determine whether you have brought it down sufficiently, or not enough. The degrees and minutes shown by the index on the arc of your quadrant, where your right hand has been moving, will give you the sun's altitude at that moment. Continue your observation, and move the index, keeping the sun close upon the edge of the horizon, until he ceases to rise and appears to stand still on the horizon, without either rising or dipping below. It is then on the meridian. A moment afterwards he will dip, or commence descending below the horizon, and it is then noon ; or rather just before he begins to descend and is entirely stationary, it is noon, and the sun is at his meridian altitude, which you will find noted in degrees and minutes on the quadrant.

    You scarcely need any other instruction to be able to use your quadrant correctly; nevertheless a practical man will show you in five minutes all about its use, which is so simple and easy that you will require but once showing.

    Then the following figuring takes place, according to the rule, which is as follows :
    Subtract the sun's meridian altitude, as found in degrees and minutes on your quadrant, from 90° (first deducting 12' generally allowed for corrections.) The remainder is the zenith distance. To this remainder add your declination, if both be named the same, that is north, or south ; but if of different names, or one north and the other south, subtract; and the [...] remainder is your latitude.
    Thus, on the 25th of May, 1846, my latitude was obtained by the following figures; few indeed and easy of comprehension.
    90°, less 12', equal to..................... 89° 48'
    Meridian Altitude of the Sun (quadrant)..... 71 05

    Zenith Distance, ........................... 18 43 N.
    Sun's Declination (almanac)................. 20 58 N.
    Latitude of the Ship ....................... 39 41 N.
    The zenith distance is north, if the sun is south of the place you are in, and vice versa.

    TO ASCERTAIN THE LONGITUDE.
    I take it for granted that every body has a watch of such accuracy as to be considered a correct time keeper, and therefore it is a chronometer or time measurer. The dial should have a second hand.

    Take care to have this watch regulated to the time of the meridian of Greenwich, before you leave the shore, which is easily done by ascertaining the longitude of the place you are leaving and turning it into time, at the rate of 15° for every hour of time. Longitude is time, and "nothing else." Thus suppose New-York to be in longitude 75° West, (it is 74° exactly at Sandy Hook light,) the division of 15 into 75 -will give you a result of 5, which may be assumed as the difference of time between Greenwich and New-York, namely 5 hours. This is supposing the longitude of the city as above given; and being a convenient round number serves the better for the illustration. But the real longitude of New-York is 74° 01', which, by the same divisor and the same rule of division, will produce a difference of time answering it, exactly, of 4 hours, 56 minutes, and 4 seconds. And thus also you will ascertain that for every degree of longitude you have 4 minutes of time.

    Put your watch forward, therefore, to answer the time of Greenwich. In New-York, then, supposing it to be 12 o'clock at noon, your watch must be instantly changed to 4 o'clock 56 minutes and 4 seconds—that time having elapsed since the sun passed the meridian of Greenwich ; but it will serve your purpose, with greater certainty, to leave your watch with a chronometer-maker, to be set and regulated for Greenwich time. If your watch has a daily variation of some few seconds, he will tell you the rate of loss or gain, which, during your voyage, you must allow in your time and calculation. You may see how important this will prove by supposing your watch either gains or loses half a minute per day. This would produce a difference of 5° in your longitude in crossing the Atlantic, which might plump you ashore some dark night, if you are not careful to allow for the variation of jour time-piece.

    You must never suffer your watch or chronometer to run down, nor alter its time, until the end of the voyage. It would be well, if convenient, to take a second watch with you, in order to keep the time at tlie skip ; and this watch should be regulated daily at noon.
    You have now nearly the whole secret of finding or keeping your longitude at sea. To be sure, there are some little allowances to be made, of which practice and an inspection of Bowditch will keep you informed; but for all practical purposes, you have the general principle already.

    Thus, suppose yourself at sea, the time at the ship being noon exactly, as ascertained by the quadrant, which is an unfailing instrument. Upon the announcement of 12 o'clock, look instantly at your watch set for Greenwich, and note the time. Turn that time into longitude by multiplying it by 15 (or by the more convenient numbers 3 and 5, which is the same thing.) If your watch is a correct time-keeper, and noon was announced at the moment the sun was on the meridian, you have thus obtained your longitude; at any rate near enough for your general satisfaction.

    I will now proceed, as plainly as possible, to show my work, corrections and all, by which I ascertained my longitude by the chronometer in the afternoon of the same day on which I obtained the foregoing latitude—namely, the 25th of May, 1846, at about 4 o'clock, ship's time, when a "sight" was taken with the quadrant, and the time instantly noted by the chronometer. I had already found my longitude at noon tolerably near, in the way I have above indicated; but the following is the minute and sure way of finding it at any hour of the day. The hours generally selected are about 8 o'clock in the morning and 4 o'clock in the afternoon. You will resort to the tables in Bowditch and the Nautical almanac for the necessary corrections. By patiently running several times over the figures I shall give below, and referring to the tables, the learner will easily make himself perfect.

    The rule for finding the time at sea, and thence the longitude, may be best illustrated by the work itself at large, as given below. It may be thus shortly stated, the necessary corrections being supposed to be previously made, as far as possible. "First catch your fish," said Mrs. Glass; and therefore—

    First, in order to find the polar distance, (figures that must enter into your calculation,) ascertain your declination and make your corrections; then subtract your corrected declination from 90°, and the remainder will be your polar distance. The altitude of the sun, obtained by the quadrant at 4 P.M. (corrected by deducting the usual 12 minutes,) is of course previously attended to, and the time noted by the ohronometer. The following work shows the polar distance:—
    20°56'00" Dec. Naut. Aim.
    ....1 44 Correction, Table 5. Bowditch.
    20 57 44
    ....1 53 Correc. 2nd, for 4h. from noon, Tab. 5.
    --------
    20°59'37" True Dec. at 4 P. M.
    90
    --------
    69°00'23" Polar Distance.

    If your latitude and declination are both North or both South subtract the declination from 90° ; if one is North and the other South, add, for the polar distance.

    These matters being ascertained, the rule for the body of the work is as follows :
    Add together the sun's altitude, latitude and polar distance ; take half the sum of all; from that half sum subtract the sun's altitude. Then by Table 27 of Bowditch, or by some book of Logarithms, ascertain the secant of the latitude, the co-secant of the polar distance, (rejecting 10 in each index,) the co-sine of the above-mentioned half sum, and the sine of the above remainder. Add them together and take half the sum of that addition; which half sum, found in the line of sines, will answer to the time at the ship, also found in the same table. Subtract the time at the ship, thus found, from the time given by the chronometer (which you have already noted) and the difference, turned into longitude by Table 21, will be the longitude your ship is in. Or, to obtain the last result, multiply the difference of time by 15, (or 3 x 5=15.) The figures for this day will then stand thus :—(See following page.)

    Sun's Alt...37°06'...Log.
    Lat.........40 .......11575 Sec. Tab. 27.
    Polar Dist..69 .......02985 Co-Sec. Tab. 27.
    Sum........146 06
    Half-sum....73°03'...9.46469 Co-Sine. Tab. 27.
    Sun's Alt...37 06
    Remainder...35°57'...9.76870 Sine. Tab. 27.
    ....................19.37899 Sum
    H.M.S.
    3 54 16 Eq.to Sine...9.68849 Half-Sum
    ...3 28 Equation of Time, (Tab. 4, A.)
    -------
    3 50 48 Time at Ship.
    7 35 58 Time by Chronometer.
    -------
    3 45 10 Equal to 56°17'30" Long, of Ship. Tab. 21.

    A companionable and urbane commander, (they are nearly all such in America now,) 'will set you right in these matters at sea, and take pleasure in doing so ; and therefore I forbear going more learnedly into the mysteries of navigation, believing that reference would be best had to some elementary work for more numerous examples. If you have time for preparation, a resort to a skillful teacher on shore is recommended, who, in a few days, will qualify you in all the ordinary rules of theoretical navigation. At sea you will soon get the practice. A long preparation for yachting, however, takes away much of its interest and freshness [...]"
    --------------------------------------------------

    And I'll mention again that I teach all of this in detail in my "Celestial Navigation: 19th Century Methods" class which will be offered again next weekend (that's ten days from now) at Mystic Seaport.

    -FER

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