The lunar distance method was proposed by Regiomontanus (1475) and Werner (1514), but it wasn't seriously pursued until the 1700s. There are several methods of calculating the "lunar distance", with different tradeoffs between accuracy and speed; Bowditch's 1802 method was the first one considered practical by mariners. While we expect to find copies of "Bowditch" in Grantville, it is unlikely that we will find an edition which contains either the original lunar distance methods (dropped in 1880) or the replacement method (dropped from the Appendix in 1914).
Although the rapid movement of the Moon makes it a potentially useable celestial clock, it was a somewhat frustrating one in practice. All errors in observation, prediction and computation are multiplied by the ratio of the earth's rate of rotation (15°/hour) to the rate of change in lunar longitude (~0.5°/hour): 29.5.
Lunars were difficult from an observational standpoint. Normally, a sextant is held vertically, and used to measure altitude above the horizon. For lunars, it had to be held obliquely, depending on where the Moon and the reference object were located.
The observed angle would be affected, like any other sighting, by dip and refraction. However, to correct the lunar distance, you needed to know the altitudes of the Moon and its "partner." So that meant, ideally, taking three simultaneous sextant measurements: the two altitudes, and the lunar distance. That was usually impractical, so what was done instead was to 1) measure the altitudes of both objects, 2) then the lunar distance, and 3) the altitudes again. The "before" and "after" altitudes for each object were averaged together to estimate the altitudes at the time of the lunar distance measurement.
Then you had to apply the tables. Their accuracy depended on the astronomers' understanding of lunar movements. Predicting lunar position is complicated because the moon's orbit is strongly perturbed by the Sun, so it can't be calculated purely by Keplerian methods. In the 1783 Nautical Almanac, the average error was 14" in ecliptic latitude and 30" in longitude. In 1817, the average latitude and longitude errors were 5". (Williams 96)
Then the ship's longitude had to be computed correctly. In practice, "longitude by chronometer" (see below) was perhaps ten times more accurate than "longitude by lunar distance," because the lunar observations and calculations were so complex and prone to error (Sobel 162). Preston (180) says that in the early nineteenth century, it was not unusual for lunars to yield a 30' error in the longitude. Lewis and Clark used the lunar distance method, and their errors in longitude were as great as 185' for moon-star and 76' for moon-sun measurements (185).
Chronometers. In 1530, Gemma Frisius pointed out that if one had a good clock, one could set it according to the time at a location of known longitude. Multiplying the time difference in hours between the reference clock time when the sun peaked (local noon), and noon, by 15 then gave the longitude difference in degrees from the known longitude.
An alternative to reading the chronometer time at local noon is to take two readings, one before noon and one after, when the sun is at the same altitude. The time of local noon is then the average of the two equal altitude times (Schlereth 96; Preston 172).
Practical use of this method had to await the development of a ship-friendly clock.
The down-time clocks have a cumulative error of 10-15 minutes/day. "After a few weeks at sea, clock error could correspond to a longitude error as wide as the ocean." (William 78; Mixter, 271; Wakefield 136).
The first reliable marine chronometer was designed and built by Harrison in the late eighteenth century. His H-4 (1760), lost only five seconds after 81 days at sea. After another two months, its temperature-adjusted total error was still under two minutes (Sobel 120-1).
The Harrison chronometer's principal weakness was the time and expense necessary to build it. The copy of H-4 made by Kendall (1770) cost 500 pounds, and took two years to construct. Kendall told the Board of Longitude, "I am of the opinion that it would be many years (if ever) before a watch of the same kind . . . could be afforded for 200 pounds." (200 pounds in 1776 was equivalent in buying power to 150 pounds in 1632; enough to buy a large yacht of 25-35 tons burden.) In contrast, a sextant and a set of lunar distance tables would cost a mere 20 pounds (153).
Later clockmakers nonetheless brought costs down by having the less critical parts made by lesser craftsmen. by the 1780s, you could buy an Arnold box chronometer for 80 pounds, or an Earnshaw for 65. There was also the Arnold pocket chronometer, which only gained or lost three seconds a day. (Sobel 156-63)
The 1632 Tech Dead Horses page comments, "Yes, we can make a lot of $ using up-time clocks, and no, no one is going to recreate Harrison's clock, it's way simpler to copy something newer. And people will." And that's where I am going to leave it. This is an article on navigation, not horology.
It is not necessary that the clock keep perfect time. What is necessary is that its error, if any, be known and predictable. A modern chronometer is set, in port, to approximate GMT, and certified as to its initial departure from GMT it is, the average rate at which it gains or loses time, and the date of the time check. (Mixter 272)
Most of my sources emphasized how much more difficult it was to determine longitude at sea than on land. However, maritime travel did have the advantage of being fast; an Atlantic crossing was something like two months by sail, less by steam. Hence, the chronometer's time error—especially the unpredictable error—wouldn't have the chance to accumulate to unbearable levels before you reached a port of known longitude, and could re-calibrate it. However, the Lewis and Clark expedition (1803-6) took years, and its rigors, in some respects, were greater than those of a sea passage. Consequently, L&C periodically re-calibrated their chronometer by the lunar distance method. This revealed considerable rate variation; it lost 15.6 sec/day in the summer 1803, 36 in late November, and 46.44 in mid-December. (Huxtable 1.4)
"Lunars" were only rarely used after 1850, in view of the convenience and accuracy of calculations based on improved chronometers. One use was to verify that the chronometers were still in working order. The Nautical Almanac stopped publishing lunar distances in 1907.
Modern Celestial Cross-Fix
Lines of Position. Earlier celestial navigation methods required making an observation at a special time (e.g., local noon) which simplified the calculations. Latitude and longitude were determined separately. But you can determine both ship's latitude and longitude simultaneously if you know the reference time (per chronometer) and take two (at most three) sextant readings ("time sights") on a celestial object whose position is tabulated in a nautical almanac.
From the almanac and the reference time, you know the GP of the celestial object. The sextant reading defines a circle about the GP, whose radius equals 90°-altitude. If a second mariner simultaneously takes a sextant reading of a second object, you get a second circle, and the ship must be at one of the two intersection points. Usually your dead reckoning from the last known position will tell you which of the two is right, but if it doesn't you can take a reading of a third object to be sure.
If the sextant readings aren't simultaneous, the older reading is "advanced" (moved, based on ship's course and speed) so that they are effectively simultaneous. This is called a "running fix."
It is inconvenient to plot these circles, which are usually very large, on a globe or a chart. Fortunately, a small enough arc of a large circle can be approximated by a straight line, and the position is then where the two "lines of position" (LOPs, Sumner lines) cross. These can be plotted on a large-scale map (EB11/Navigation).
Graphic methods work best when the two LOPs meet at a substantial angle, and the objects can be picked to ensure this. Sun and moon will generate LOPs crossing each other at an angle of 45° or better perhaps ten days a month (Schlereth 77). Or you can pick stars which are in different quadrants of the sky.
As described in EB11, Saint Hilaire suggested (1875) a major improvement in the 1847 Sumner method. This involved using an assumed position to compute an expected altitude and azimuth f
or the GP, then plotting the LOP perpendicular to the computed azimuth line, moving the LOP toward or away from the assumed position to account for the difference between the expected altitude and the observed (after correction for observational errors) altitude.
The expected altitude and azimuth can be obtained by spherical trigonometry, by solving the navigational triangle (two sightings yields two linked triangles). This is formed by the assumed position (AP), the GP of a sighted celestial object, and the nearest pole (P). For each triangle, we have three sides, whose lengths are:
GP-AP: 90-expected altitude
GP-P: 90-object declination (looked up in almanac)
AP-P: 90-assumed latitude
The angle with vertex at P is the expected difference in longitude (meridian angle) between AP (assumed longitude) and GP (looked up). The angle with vertex at AP is the expected azimuth (or 360°-azimuth) of the GP, if the ship is at the assumed position. The sides AP-P and GP-P, and the meridian angle, are used to calculate the expected altitude and azimuth.
At local noon, the meridian angle becomes zero and the triangle degenerates into a straight line, vastly simplifying computations.
Log and Trig tables. Sight reduction (the conversion of observations to positions) is heavily dependent on knowledge of spherical geometry and trigonometry. Many different formulae were known by the early seventeenth century. Typically, these required multiplication of trigonometric functions. In the sixteenth century, the trigonometric functions of angles had been calculated to fifteen decimal places. Logarithms were important because instead of multiplying trigonometric functions you could just add their logarithms. Napier published logarithmic tables in 1614, and by 1624 they had been computed to fourteen decimal places (Williams 47-54).
Sight Reduction Tables. So sailors don't have to do trigonometric calculations, sight reduction tables have been prepared. They compile precomputed navigation triangles, typically covering each possible whole degree value of the meridian angle, latitude, and declination. Unlike almanacs, these are always valid; the math doesn't change.
To use the tables, you assume a ship position which is at a whole degree latitude and longitude within 30' of the dead reckoning position, and calculate the meridian angle from the longitude and the almanac listing of the "hour angle" of the object. The declination is in the almanac. Together with the assumed latitude, you use the SRTs to find the expected altitude (nearest 0.1') and azimuth (0.1°) of the object. You then compute the altitude difference and draw the LOP accordingly.
There is no assurance that any sight reduction tables will exist in Grantville. If anyone has them, it is Jesse Wood, since they are also used in aerial navigation. If they don't exist, they can be generated by computer, and printed in volume on dot matrix printers. In fact, that would probably be better than trying to typeset Jesse's copy, since it would avoid typesetting errors.
Conclusion
Jack London, sailing in the Pacific, noted that every step of the navigational art must be performed correctly, or else the captain will hear, " 'Breakers ahead!' some pleasant night, receive a nice sea-bath, and be given the delightful diversion of fighting the way to shore through a horde of man-eating sharks." The up-timer's mathematical knowledge and mechanical skills will make navigation a bit safer than it was before the Ring of Fire.
The bibliography is included in the Navigation Addendum posted to www.1632.org .
The Geared Locomotive or What Wood You Shay To?
Written by Kevin H. Evans
Geared locomotives were developed to handle rough track industrial applications. Most notable were logging short lines, and mining short lines. The traditional steam locomotive has cylinders parallel to the ground with the effort of those cylinders transferred to the drive wheels via reciprocating side rods. A geared locomotive transfers the power to the wheels by a shaft connected to the wheels through a gear meshed to another gear on the wheel or axle of the locomotive. Said wheels are usually about the same size as the wheels found on the load carrying cars. Some of these locomotives used power transferred to the center of the axle sets, but the most common type had all the power train (engine, shafts. and gears) mounted on the side of the locomotive.
This type the "Shay" (named after it's inventor, a logger of the same name from Minnesota) had the great advantage that all the working parts were easily accessible and mounted where the Engineer could see them. Additionally the short wheel base of the driven wheel sets (or trucks as they are called) allowed the locomotive to use extremely rough track. In fact the small wheels gave the geared locomotive a lot of adhesion and allowed the movement of large loads for relatively low horse powers. It was commonly said that you could "Draw two lines on the ground and a 'Shay' would follow them." This made it a supremely good locomotive for industrial use. Temporary track, tight turns, and difficult grades were handled with ease. The greatest failing of the geared locomotive was it's slow speed. Because of the gearing a Shay sounds like it is going a hundred miles an hour when it is really making about fifteen miles per hour.
The geared locomotive would also be attractive to the residents of Grantville because a large well run tourist railroad, the "Cass Valley Scenic RR" is only three hours drive from the pre ROF location of Grantville. This operation, formerly a logging road, has what is probably the best collection of geared locomotives in existence. Railroad fans, large machinery enthusiasts, machinery restorationists, and industrial historians would be frequent visitors and would be well aware of the advantages of the geared locomotives in use there.
Building a geared locomotive first would be very attractive because: first, all the "works" are where you can get at them. Second, you can get a lot of work for a lower horsepower prime mover. Third, the locomotive can handle really rough track. Fourth, many of the structural components can be made from wood. Lastly the suspension and equalization of the locomotive is much simpler.
So how they would build it? Well, first it would not look like a product of the 1920s industrial age. It would be coal fired, wood framed, and steam driven. The locomotive would look a lot like a flat car with an engine hanging over the side. Set a boiler on the flat car, wrap the boiler with a saddle type water tank, and add a cab with a fuel bunker on the back.. Lights, bells and whistles would be nice, but secondary to getting the mover on the track..
The first thing to build would be the actual engine. In view of the need for a strong locomotive as quickly as possible, converting a big block "V-eight" engine to steam would have to do. This conversion has problems as the block would have small cylinders compared to a purpose-built engine. It would only have power on the down stroke of those cylinders, and need extensive refitting to run on steam. Luckily, by using an engine block from a big ICE (internal combustion engine) you would have a reciprocating cylinder/piston set without having to do all the design work and machining from scratch.
Converting an engine block to steam can be done in a number of ways, but the easiest is to make it into what is commonly called a bash valve engine. A bash valve engine, also known to the model aircraft crowd as a CO2 engine, uses a rod mounted to the top of each piston to lift a ball off of it's seat when the piston reaches "Top dead center". This allows steam to enter the cylinder and pushes the piston down. When the piston reaches "Bottom Dead Center" it reveals an exhaust opening that allows the steam to escape. The steam is collected and used to power the draft of the boiler or alternatively condensed to add to the power of the engine and extend the amount of time between water stops.
In order to fit the new valving, the block must be stripped. The valve heads, timing belt, front pulley, and water pumps would all be removed. New valve heads would be fabricated to hold the ball valves and return the lubrication galleries of the block. The block must have an exhaust port cut into each cylinder and have the port fitted with a steam recovery device. Also each piston must be fitted with a push rod for opening the ball valve as needed. Provision must be made so that the engine always rotates the same
direction and the existing transmission can be used for forward and reverse. Note that only the forward and reverse gears are needed so second and third gears can be removed if needed or practical.
As for numbers, I am going to fiat the cylinder diameter as 5 inches (I haven't torn down a big block for a long time). Therefore each cylinder will have an area of about 19.6 inches or about 157 square inches total. This, multiplied by the working pressure of the boiler, would give the foot pounds produced by the engine, or about 23,562 foot pounds at 150 psi. This translates into roughly forty-two "steam" horsepower delivered to the transmission.
The use of an automotive transmission would have an effect on the power delivered to the shaft and gearing. While low gears would deliver a lot of power, it would also serve to further limit the top speed of the locomotive. It may be more desirable to keep the forward reverse gearing to a one to one ratio.
The power would be transferred to the wheels via a shaft and bevel gear set. This gear set serves to rotate the motion of the shaft by 90 degrees, and is accomplished by having a pinion gear meshed to a ring gear around the face of the wheel. Each wheel would be driven, and the shafts are usually square tube with telescoping joints to allow motion to the trucks.
The trucks can be made with wooden frames and wheels. The construction would need to be fairly massive, and steel strap reinforcement will be desirable. The wooden wheels would need to be fitted with steel tires and the ring gears could be bolted directly to the wheels. Each truck would also have the shaft and pinion gear mounted to one side. The frames of the trucks would hold the journal boxes, (the bearings that the axles ride on) and would be sprung to a transverse bolster by leaf or coil springs. Also the wheels must be rigidly connected to the axles so as to transfer power to both sides of the truck.
The frame of the locomotive would be made from heavy timber, probably bigger than eight by twelve inch beams fastened by bolts and plates. Cross stringers would be added as needed and it all would be further strengthened by fish plates and tie rods.