by Eric Flint
A modern test of a cross-staff replica, carried out at Lumberton, Mississippi (latitude 31.6d) resulted in readings of 32, 31.5, and 33d* latitude (calculated as altitude of Polaris) from one scale and 31.5, 32 and 31.5 for the other. This was done, remember, on land. It looks like the angular accuracy of a cross-staff was 0.5-1.5d*. (Cookman)
A similar device, the back-staff, was used for solar observation. Instead of the transom it had an element which cast a shadow back along the staff. With his back to the sun, the mariner slid the element until the shadow just touched the far end of the staff, and then read off the location of the slider. A table gave the corresponding angle.
The "state-of-the-art" on the eve of RoF was the Davis Quadrant, first described in John Davis' Seaman's Secrets (1595). It was called a quadrant, because it could measure angles up to 90 degrees (one quarter circle) It was also called a back staff, because, like the first back staff, it was a back sight instrument. That is, you observed not the sun itself, but the shadow it cast (which saved your eyesight, but also meant that you couldn't sight the stars)(Callaghan 157).
The Davis Quadrant had two arcs (vanes), one atop the other, with the same pivot point. The upper and lower vanes each have an attached slit, and there is a third slit next to the pivot point. You look at the horizon through the lower vane and pivot slits, and you adjust the upper vane so the sunlight passes through the upper vane slit onto the pivot slit.
The scale of the Davis quadrant might have 0.5d* divisions, and quadrants could be read to perhaps 0.25d* (Taylor 215). However, in 1631, (Miller), Pierre Vernier described the Vernier scale, by which a scale could be read to the nearest arc minute. The even more accurate micrometer was invented at the end of that decade (Gascoigne, 1639) but ignored until much later.
The Double Reflection Octant and Sextant . The first double reflection instrument was the Hadley octant (1731). It had a markedly different operating principle than the prior instruments. There are two mirrors. The first, the fixed horizon mirror, is only partially silvered. You sight the horizon through the unsilvered portion. There is also a rotatable index mirror, which is attached to a pointer. You rotate the index mirror until you can see the celestial object's second reflection in the silvered half of the horizon mirror, then check the pointer against the scale. The Hadley octant had a magnified scale, giving it an accuracy of 1-2' in Admiralty tests. (Taylor 257; Callaghan 158, 164).
The term "octant" arose because the frame was a one-eighth slice of a circle. Because of the double reflection, it could still measure an angle of 90d*. In 1757, Campbell suggested that if you wanted to measure the angle between two celestial objects, such as the moon and the sun, it was desirable to have an instrument with a greater angle of action (Williams 98). That led to the creation of the sextant, whose frame was one-sixth of a circle, and which thus could measure an angle of 120d*.
Subsequent improvements to the sextant included:
— silvered glass mirrors (the original ones were of speculum metal, and tarnished)
—larger mirrors (larger field of view)
—the tangent screw (to adjust the index arm)
—sun-shades
—vernier scales (and associated magnifiers)(EB11/Navigation)
—micrometer (for even finer adjustments)
—low-expansion frame material
—mountable monocular (for light amplification)
—spirit levels
—mountable artificial horizon
Altitude Measurements on Land
On land, the horizon may be hidden by mountains. Hence it is necessary to provide an artificial horizon. This took the form of a pool of mercury. Mercury, being a liquid, would naturally flow to form a horizontal surface. The observer would simultaneously sight on both the celestial object and its reflection; its altitude would be half the angle between them.
Use of mercury was not without its problems. Mercury is highly toxic and therefore had to be handled with care. Also, the artificial horizon could be disturbed by wind, smoke (from a fire 300 feet away), or ground vibrations (e.g., a horse galloping five hundred yards away). Topographers would dig a trench around the pan holding the mercury, to isolate it, and the observer would stand outside the trench. (Shafer)
Altitude Measurements in the Air
Pilots had a different set of problems. Because of the height of the aircraft, the observable horizon was far away (therefore often indistinct) and way below the geometric one (big dip correction needed). This also made it awkward to measure the angle between the horizon and the celestial object.
The mercury type artificial horizon was impractical because the surface would be continually perturbed by aircraft motion. Nor was it practical to use a damped pendulum to provide a reference vertical.
The basic solution, a bubble telescope attachment for a sextant, was developed in 1918 by R.W. Wilson. This was a shortened-T device. You looked, through a collimating lens, down the main tube, and thus at the horizon glass of the sextant. At the T-junction, there was a diagonal mirror, and this provided the observer with a view of a bubble in a bubble chamber at the end of the side tube. The bubble was confined by a spherical surface with a radius matching the focal length of the lens. The apparent position of the bubble told the navigator where the horizon was located.
The movement of the aircraft was still a problem, as the bubble took time to settle down. The bubble sextant was therefore refined by combining it with an averaging device of some kind. The averaging might be for a set number of observations, for a set time, or just between the "start" and "stop" of a trigger.
Navigators belatedly realized that the bubble position should be corrected for Coriolis acceleration; correction tables appeared in 1942.
Williams (119) says that "in large stable jet aircraft flying on auto-pilot above the weather," the residual error after correction was about 2'.
I doubt that these bubble sextants are described in any of the books in Grantville. Jesse Wood may remember what they look like, and he may even own one, but bear in mind that celestial navigation for aerial use declined as early as the Sixties.
Altitude Observational Errors
The apparent position of a celestial object may differ from its true (geometric position) for a variety of reasons, some dependent on the observer and the instruments used, and others on atmospheric and astronomical phenomena.
Sextant Construction Errors. To have accurate measurements, you need accurate scales. Scales were initially made by hand and eye. Later, "dividing engines" were devised for accurately dividing a scale into its units. The Ramsden apparatus allowed the sextant to be halved in size, without loss of accuracy (Gurney 112).
Sextant Calibration Errors . The sextant must be re-calibrated on at least a daily basis, to check for and remove index error (the two mirror faces aren't parallel), side error (the horizon mirror not perpendicular to the plane of the sextant), and perpendicularity error (the index mirror not perpendicular to the plane of the sextant). Recalibration is necessary because the sextant is affected by changes in temperature and, of course, accidental knocks.
The sextant must of course be designed to allow these errors to be corrected. According to Togholt (30-1), the only error which can be tolerated (and taken into account in calculations) is index error, and then only if the error is less than 5'.
Sextant Reading Errors . The observer must take the reading when the image of the object is just "touching" the image of the horizon (the moon and stars can be difficult to "land" on the horizon properly) and the sextant is absolutely vertical. (Toghold 33, 91). A 2d* error in "verticality" results in a 1.1' error in altitude (Manzari).
If the horizon is ill-defined, the altitude in turn is fuzzy. Usually, the stars and planets are observed during "civil twilight," when they aren't lost in solar glare but there is still a horizon. The best time is perhaps twenty minutes before sunrise, or after sunset (Schlereth 100-1).
Dip . Because the eye is elevated, and the earth is curved, t
he natural horizon (where the sea and sky meet) is lower than the celestial horizon. All sextant altitudes must be corrected for dip, or they will be over-estimated by several arc-minutes. The dip (') is about 1.06 times the square root of the eye level (feet).
The first dip correction table was constructed by Thomas Hariot (1560-1621), for eye levels of 5-40 feet; he figured the poop deck was at 20-25 feet, requiring a correction of 5-6'. His calculations are consistently about 1' too high. Wright's dip table was published in 1599. (Taylor 219-20).
Refraction . Light doesn't move in a straight line, but rather in the path which takes the least time. Since light moves more quickly through warm air than cold air, and the air nearest the earth's surface is warmest, the light you see, unless it is from directly overhead, has taken a curved path, favoring the warm air, in order to reach your eye. As a result, it comes from an apparent direction which is lower than the true direction of the object which you are observing.
This effect is greatest when the object is low in the sky; indeed, you can see the sun even after it has set below the celestial horizon. In the Nautical Almanac, for a star on the horizon, the refraction correction is -34.5'; at 10d* apparent altitude, -5.3'; at 45d*, -1'.
The Nautical Almanac assumes an air temperature of 50d*F and pressure of 29.83"Hg. The refractivity of air changes if either temperature or pressure changes, but this needs to be taken into account only for low-altitude observations. In the subsidiary table, the maximum correction for unusual temperature or pressure, for apparent altitude on horizon, is -6.9'. Above 8d* apparent altitude, it's less than 1' (Dutton 409-10).
The astronomer Tycho Brahe (1546-1601) published the first table of atmospheric refraction, determined by observation. He reported stellar refraction to be 30' at the horizon, 10' at 5d*, 3' at 15d*, and nonexistent from 20d* up. For the Sun, he was able to detect refraction only up to 45d*. (Heilbron 128)
Nonetheless, refraction tables didn't appear in nautical almanacs and weren't used by sailors. Hence, all low-altitude quadrant measurements at sea were subject to a systematic error.
The first theoretical model of atmospheric refraction was advanced by Cassini in 1666. It assumed that the atmosphere had a constant density up to a particular height, and then came to an abrupt halt. Cassini's model predicted that there was 1' refraction of a star at an altitude of 45d*, contrary to Brahe's teachings. Cassini was right.
1911EB, "Refraction" teaches the refractive power of air is (1) nearly proportional to density (and thus varies with temperature and pressure) and (2) proportional, at moderate altitudes, to the tangent of the zenith distance (90d*-altitude). The tangent law is the very one predicted by Cassini's model. (The 1911EB also says that near the zenith, the refraction is about 1" for each degree of zenith distance, and, at the horizon, it is about 34".)
Cassini's model doesn't do a good job of predicting refraction at low elevations. For that, we will need to either develop a better model of the atmosphere (one taking into account how the atmosphere thins out), or simply determine refraction by observation.
Aberration . This phenomenon was identified by Bradley in 1729, so it isn't known to the down-timers. If the observer is moving away from the true line of sight to the star, the latter will appear to be displaced is the direction of the orbital motion. The principal source of aberration is the motion of the earth around the sun. The maximum displacement is 20.5" (Pasachoff, 499).(The rotation of the earth can also cause aberration, but only, at most, 0.33".)(Williams 95).
Parallax . We use geocentric equatorial coordinates to describe the positions of celestial objects because they simplify calculations. However, a person on the earth's surface would see the sky from a slightly different angle than that of an imaginary observer at the center of a transparent earth. The angular separation of their lines-of-sight is called parallax. Lunar parallax was first measured by Hipparchus and is well known to seventeenth century astronomers.
The maximum lunar parallax occurs when the Moon is on the horizon, and it disappears when the Moon is at zenith. The parallax also varies with the distance of the Moon from the Earth, so that in the horizon case it is 54-61'. Hence, if you are using the Moon for celestial navigation, you have to take parallax into account. The Sun is much further away, so its maximum parallax is 0.15'. (Mixter 240).
Semidiameter . The stars and planets can be treated as point sources, but the Sun and Moon have discernible disks. To use the astronomical tables for the moon or sun, you need to know the altitude of the center of the body. However, you are actually measuring the altitude of the lower or upper limb. They are both about 15.7' from the center, and vary as the distance to the moon or sun changes (by 2' for the moon and 0.6' for the sun).(Mixter 240). Hariot (1595) told Raleigh to use a correction factor of 16' (Taylor 221).
Augmentation. The Moon is closer to the observer (by slightly more than the radius of the earth) when it is at zenith than when it is at the horizon, and hence looks larger, altering the semidiameter correction (Mixter, 242). At most, it is about 0.29'.
Measuring Azimuth
To measure azimuth (bearing), you use an azimuth compass. This instrument was first described in a 1514 Portuguese manual (Wakefield 40), and it combines a standard compass with an azimuth circle.
The azimuth circle, in its simplest form, is a ring with opposed sights, such as a peephole on one vane and a vertical wire on the other. The ring is turned until, looking through the peephole, the wire is directly in front of the object, and then you read off the orientation of the ring relative to the compass arrow.
That version only allows the navigator to take the bearing of an object close to the horizon, such as a landmark. However, there are more sophisticated forms in which a dark glass reflector is attached to the far vane, and is pivotable so that at an object at any altitude can be "brought down" to the horizon. (Dutton 177). A well designed azimuth circle will have leveling screws and "bubbles" so it can be made perfectly horizontal. Also, the near vane can be equipped with a telescopic sight.
The use of a simple "peep" system to observe the Sun would be hard on the eyes, and so the modern bearing circle comes with a second pair of "sights," a slit and a mirror on one end, and a prism on the other. The sunlight passes through the slit, and the prism creates a band of light on the compass card.
Celestial navigation usually makes more use of altitude than azimuth. That is probably because of issues of accuracy. Mixter (48) says that azimuths can be measured only to 0.5d* in quiet water, 1d* with the slightest roll, and 2d* or more at sea.
Measuring Time
On shipboard, short time intervals were measured with a sandglass. A 28 second glass was used for logging speed and a half hour one for governing the ship's daily schedule (a bell was sounded every half hour).
The nocturnal, which looked something like a ping pong paddle with an extra moveable arm, was used to determine the orientation of the "Guard Stars" relative to the Pole Star, and thus (given the day and month) to find the local sidereal time. While it could only be used at night, and then only if the stars in question were visible (i.e., not in the Southern Hemisphere), it had an accuracy of perhaps fifteen minutes. (Swanick 108; Navigation/EB11).
There was also the planisphere, an example of which is depicted in Gunter's The Description and Use of the Sector (1623). The basic principle was that you set the date, matched the planisphere to what was observed in the sky, and read off the time. In the simplest form, the "sky" was represented rather abstractly by radial lines corresponding to various bright stars. The volvelle was rotated so the line of the star then on the meridian (due south if in the northern hemisphere) matched the date, then the observer looked up the time. There was also a pictorial type, with simplified constellations. (Turner 67).
The nocturnal and the planisphere could be combined into a single device. A "planispheric nocturnal" was taken from the wreck of the LaBelle (1686). It includes a planisphere with 27 constellations inscribed, some located in the southern c
elestial hemisphere. (Swanick 155-67).
Chronometers are used to determine the time at a point of known longitude (where the time was set), and the difference between local time and chronometer time is indicative of the local longitude. In Jules Verne's Mysterious Island, Harding reports when it is local noon (based on the length of a stick's shadow) and Gideon Spilett reads off the time on his watch (set to standard time in Washington). (Conveniently, the date of the observation was April 16, when standard and true time were identical.) I will discuss the use of chronometers in more detail in Part 2.
Conclusion to Part 1
Prudent seventeenth century sailors were mindful of the "four L's": Lead, Log, Latitude and Lookout. The Lead was the sounding line, which not only warned whether the ship was in danger of running aground, but also gave a clue as to its location if it was roaming familiar coastal waters. The common Log gave the ship's speed, and hence was essential for "dead reckoning" the movement since the last celestial observation. The Latitude, computed from observations with astrolabe, cross-staff, etc., helped fix the position. Finally, the Lookout was needed to spot hazards which either were not shown on the maps, or which were unsuspected because of faulty navigation.
