In Mr. Midshipman Hornblower, the tyrannical senior midshipman, Mr. Simpson, given a navigation problem by the sailing master, computes the ship’s position as being in Central Africa. The captain acidly praises him for discovering the source of the Nile. Poor Hornblower, the most junior midshipman, is the only one with the correct answer; “everybody else had added the correction for refraction instead of subtracting it, or had worked out the multiplication wrongly, or had (like Simpson) botched the whole problem.”

The errors made by Hornblower’s peers differ only in degree from the real-life errors that were made by countless navigators, sometimes with the result that the ship ran aground, or sank. This is the first of two articles which will examine the art of navigation in the early seventeenth century, and what a bunch of landlubbers from Grantville (over two hundred miles from the ocean) can do to improve it. This article will focus on navigational instruments in the broad sense, while the sequel will address the celestial navigation methods by which longitude and latitude are determined.

Guides to Navigation

For down-timers, the leading English practical textbooks are William Bourne’s Regiment for the Sea (1573, 1631), which was based on Martin Cortes’ Arte de Navigar (1551);Thomas Blundeville’s Exercises (1594, 1597, 1606, 1613); John Davis’ Seamans Secrets (1594); Edward Wright’s Certain Errors in Navigation (1599); Richard Polter’s The Pathway to Perfect Sailing (1605, 1613); and Thomas Addison’s Arithmetical Navigation (1625). For those more mathematically inclined, there was Robert Tanner’s Brief Treatise of the Use of the Globe Celestrial and Terrestrial (1620), in eight volumes. And if you were more interested in navigational instruments, you could consult Anthony Ashley’s Mariner’s Mirrour (1588), William Barlow’s The Navigator’s Supply (1597), and various books by Edmund Gunter (1632, 1624, 1630 and 1636). (Swanick 57-67).

There are also various astronomical almanacs, but those will be discussed in the second article.

Grantville being far from the sea, it is not a likely repository for nautical texts. However, the consensus of the Editorial Board is that there will be at least a couple of editions of Bowditch’s American Practical Navigator around (several having been found at a used bookstore in Parkersburg, West Virginia, and the Up-timer’s Grid listing both Jack Clements, a retired Coast Guard boat pilot, and of course the aviator Jesse Wood). Even if we ignore APN, there are a surprisingly large number of useful entries in the 1911 Encyclopedia Britannica, and there are books on astronomy and mathematics in the school libraries. And the atlases and National Geographics maps will no doubt come in handy.

Navigation by Terrestrial Signs

Landmarks . The simplest form of navigation is to take note of prominent landscape features, and their bearings. Ideally, you take cross-bearings (simultaneous bearings of two different landmarks, lying in different directions), because that fixes your position. Lighthouses and buoys, of course, can be considered artificial landmarks.

Knowledge of landmarks was initially confined to local sailors. However, it became customary for long-distance mariners to draw “profiles” of the coasts they visited in their logbooks (Taylor 168). These sketches could be passed on to allied captains.

The “knowhow” of sailors was distilled into guidebooks known in ancient times as periplous, and, later as portolans, rutters and waggoners. They could include charts, profiles, and logbook summaries. The most famous of them all was Lucas Wagenaer’s Mariner’s Mirror (1584). Sailing directions are primarily directed to coastal navigation.

Soundings . In shallow waters, including the North Sea and Baltic Sea, it was common to navigate by soundings. This involved dropping a sounding line, a knotted rope with a lead weight, to the bottom. The number of knots passing over the side gave the depth. The weight could be “armed” with tallow to pick up sediment from the sea floor. Sailing instructions would tell mariners what to expect. For example, it might say that you have reached the shallow region between Cape Clear and the Isles of Scilly, when “at 72 fathoms [you] find fair gray sand” (Aczel, 12-3, 134-5).

The modern lead line is distinctively marked so that the leadsman can recognized the marks by feel even in the dark: two strips of leather at two fathoms, three at three, a white cotton rag at five, red woolen bunting at seven, and so on. The standard lines are twenty and one hundred fathoms long. (Mixter 11).

A sounding machine was invented by Lord Kelvin in the nineteenth century, and such machines are described in 1911EB. Soundings can be taken day or night, under harsh sea and wind conditions, and up to depths of several thousand fathoms, using galvanized steel wires which are reeled back in by an engine.

A twentieth century alternative to the sounding line is the fathometer, a SONAR-based echo sounder (Togholt 37). Unfortunately, I don’t know enough about electronics to venture a guess as to when these can be built, post-RoF.

Several up-time fishing fathometers made it through the RoF. The first reference in canon is in 1633 chapter 38: “At least I could send them out ahead with Al’s fishing fathometer to look for the really shallow spots.” In chapter 46, Eddie, in the powerboat Outlaw, says, “‘We ought to have enough water, and we’ll keep an eye on the fathometer.’ He tapped the digital depth display, and Larry nodded again.” 1634: The Baltic War, Chap. 34 reveals the existence of at least six fishing fathometers, two on the “up-time power boats leading the ponderous line of gunboats,” and the remainder on Simpson’s four ironclads.

The immediate contribution which the up-timers can make to navigation by soundings is to provide copies of up-time maps with depth markings. There is typically some sounding information on National Geographic maps. Of course, it is unlikely that anyone in landlocked Grantville has the detailed marine charts of coastal waters and, even if they did, they probably don’t correspond too well to seventeenth century reality; coasts and bottoms change over time.

Other signs . Those who live on the sea (and die there if they are unobservant) tend to notice subtle cues as to where they are. These include the color of the sea, the typical currents and winds, bird and fish movements, and clouds which hover over islands and perhaps even reflect the color of the land below. (Taylor, 59-60, Calahan 82).

One modern contribution might be the use of the thermometer. By sampling water temperature, you can map out currents.

Terrestial Latitude and Longitude

While we may not have thought so in high school, one of the great intellectual inventions of mankind is the coordinate system. For example, if a city is laid out as a grid, we can send someone to the intersection of, say, North Tenth Street and East Third Avenue.

Latitude and longitude are the dimensions of a gridded spherical surface coordinate system first devised by the ancient Greeks. The earth is not a perfect sphere, but for our present purposes, it is close enough. Imagine the Earth as a hollow, see-through globe, with you hovering somehow at the center. If your body were aligned with the earth’s axis, you could identify any point on the earth’s surface by two angles, one measuring “up-and-down” relative to “level” (latitude) and the other “left-and-right” relative to “front” (longitude).

For each of these angles, we need a reference, a “zero.” For latitude, it is the earth’s equator, the intersection of the earth’s surface with an imaginary plane perpendicular to the axis. Any point on the equator is zero degrees latitude. Angles are traditionally measured in degrees; by ancient convention, a circle is divided into 360 degrees (each degree, symbol d*, is divided into sixty arc minutes, symbol ‘, and each minute into sixty arc seconds, symbol “). Above your head would be the north pole, defined as 90 degrees north latitude. Below your feet, the south pole, at 90 degrees south latitude, sometimes represented as -90 degrees.

Except at the poles, the points on the earth’s surface which have a particular value of the latitude form a circle on the earth’s surface; the circles are parallel to each other (that is, they maintain a constant distance), and hence are also known as “parallels” (e.g., the 49th parallel, part of the border between Canada and the western United States).

The “lines” (really, half-circles) of constant longitude are called meridians. For longitude, we have to pick an arbitrary zero. Hipparchus proposed using a meridian which passed through the city of Rhodes. Currently, the zero longitude (prime) meridian is one established by an 1884 international treaty, and passes through the Royal Observatory at Greenwich, England. Longitude is measured as being so many degrees (up to 180) east or west of the prime meridian.

On a globe, the “lines” (circles) of latitude will always cross the “lines” (circles) of longitude at right angles. (A map may distort this relationship.)

If two points are on the same meridian (constant longitude), but one degree of latitude apart, that’s a distance of about 69 miles. It would be the same distance, regardless of where you were, if the earth was a perfect sphere. So an error of one degree latitude corresponds to 69 miles. An error of one arc-minute (‘), 1.15 miles. An error of one arc-second (“), 100 feet.

If two points are on the same parallel (constant latitude), but one degree of longitude apart, the distance between them would be a maximum of 69 miles (at the equator). The further away they are from the equator, the shorter the distance would be.

In 1632, the down-timers did not know the true length of a degree of latitude. However, it was measured with high precision (error <1%) in 1637 (EB11/Navigation). They did know the relative length of a degree of longitude, given the latitude, having published (1599) tables of “meridional parts.”

Thanks to land observations, the down-timers know the latitudes of many ports. Even those given in the Regiment of the Astrolabe (1509) are accurate to 30′, sometimes even 10′ (Taylor 166).

Globes and Maps

Globes, like the earth, are spherical. Maps are flat. As you can verify by trying to flatten out the skin of an orange while keeping it as a single piece, some creativity is required to flatten out a spherical surface.

The technical term for the mathematical manipulation by which points on a spherical surface are converted to points on a flat surface is “projection.” Any map projection is going to distort certain properties of the earth’s surface, and, hopefully, preserve others. Projections can preserve direction from a central point (azimuthal projection), distance from a central point (equidistant), local shape (conformal), area (equiareal), etc. You need to use the right map projection for a particular purpose.

It should be noted that the down-time mathematicians know quite a bit about map projections. For example, Oronce Fine (1494-1555) invented a heart-shaped projection. The empirically developed Mercator (1512-1594) projection, given proper mathematical form by Wright (1599), is still used for navigation.

Great Circles and Rhumb Lines

A great circle (orthodrome) is a circle on a sphere which has the same diameter of the sphere, and thus divides the sphere into two hemispheres. The equator (zero latitude) is a great circle, and the meridians are portions of great circles (with constant longitude). However, these are special cases, and great circles can connect points which differ in both latitude and longitude.

If a map uses a gnomonic map projection, great circles are shown as straight lines. On a mercator projection, they are curves.

The shortest distance between any two points on the surface of a sphere is a portion of the great circle which connects the two points. Unfortunately, traveling on a great circle path requires continual correction of one’s compass heading. Great circle sailing can also carry one to a higher latitude than is desirable (too much ice and fog).

A rhumb line (loxodrome) is a path on the spherical earth which corresponds to following a constant true compass bearing (azimuth), or, to put it another way, to crossing every meridian at the same angle. If a map uses a mercator projection, rhumb lines are straight lines. Parallel sailing is a special case of rhumb line sailing in which one sails along a parallel (line of latitude), thereby crossing every meridian at right angles.

As a compromise between minimizing the distance (great circle route) and facilitating steering (rhumb line), a great circle route may be approximated by a series of short rhumb lines connecting waypoints which lie on the great circle.

Composite sailing is a combination of great circle sailing to and from some limiting parallel, and parallel sailing in-between.

Dead Reckoning

In dead reckoning, the navigator plots the last known location on a chart, and extrapolates the present location based on the ship’s subsequent heading(s), speed(s), and time elapsed.

The Spanish called dead reckoning, navegacion de fantasia (Gurney 19), and Edward Wright (1599) referred to the estimated position as “the point of imagination.” (Williams) DR estimates of longitude were sometimes over 400 miles astray (Wakefield 165).

Surface currents usually exceed ten miles per day (mpd) and in many places are 40-50 mpd. If currents are ignored, the dead reckoning will accumulate error at a rate of 10-50 mpd. Even in the late eighteenth century, long-distance journeys typically accrued longitude errors of 5-15d* (Parr 68-9).

The Traverse Board was a device used to keep track of the courses steered. Every half-hour, a peg would be placed in one of 32 holes, each representing one point of the compass. There were eight such concentric circles of holes, thus recording an entire four hour watch. (Phillip-Birt 191).

Of course, steering a particular course didn’t mean that the ship necessarily moved in the expected direction. The helmsman could be lax, the ship’s steering arrangement could be inaccurate, and the ship could be forced off course by powerful winds and currents.

The prudent navigator attempted to estimate “leeway” (the extent to which the ship was forced off course) by looking at the angle between the wake and the heading. (Williams 22)

Moreover, even if the ship was placed on the desired compass bearing, that bearing might not be the desired true bearing, by reason of errors in correcting for magnetic variation and deviation, or of determining true north from the sky.

Logging Speed. For measuring speed, the sailor used a log. The common log was a piece of wood tied to a knotted line. The log was thrown out behind the ship, and the line allowed to run out. One sailor counted the knots as they passed over the rail, while another watched a sand glass. The count continued until the sand glass emptied. The first written description of this method was in William Bourne’s A Regiment for the Sea (1574)(Williams 39 n. 3), and the log was in general use, at least by the English and Dutch, in the 1620s (Swanick 100).

The sailing term “knots” refers to the fact that sailors estimated their speed, in nautical miles per hour, as the number of knots run out per “glass.” A knot is one nautical mile (6,076 feet, about one arc-minute of latitude.) per hour. Earlier schemes overestimated speed (perhaps deliberately), but the late eighteenth century, sailors used a knot spacing of 47.25 feet and a 28 second glass. (Gurney 25; Phillip-Birt 196)

There are some other obvious problems with this method. The log might be caught in the ship’s wake, and the line not pay out properly. There might be little delays in calling out the end of the time interval. The knot counter might miscount, or have trouble estimating an intermediate value. The speed of the ship might change, after the fact, as a result of shifts in wind and current.

An alternative form of the common log was the “Dutchman’s log”: throw a chip off the bow and time how long it takes to reach it. (Mixter 12)

The common log was ultimately replaced by the patent log. This was a towed rotator, with spiral fins (Togholt 36). The passing water caused it to spin, and the rotations were mechanically communicated to a mechanical counting device. The patent log had to be calibrated by testing it on a run of a known length. Preferably you carried out two runs in opposite directions, so as to reduce the effect of any local current.

A steamship engineer could construct a power curve relating ship speed to engine speed (RPM) by carrying out similar runs at each of several engine speeds. Then the engine tachometer could be used as a log. (Mixter 13-15).

To get the distance run, the navigator multiplied the speed (presumed constant) by the time elapsed. Measuring shipboard time in the early seventeenth century was a rather chancy proposition, typically involving sandglasses.

Plotting. When dead reckoning is figured as if the earth is flat, that is called “plane sailing.” For a DR plot to be accurate over long distances, you need to use a Mercator projection chart, or correct your eastings and westings for the changing length of a degree of longitude. The corrections are carried out with a table of meridional parts, which were first published in Wright’s Certaine Errors in Navigation (1599). But in the late seventeenth century, Sir John Narborough said, “I could wish all seamen would give over sailing by the false plane charts and sail by the Mercator’s chart . . . but it is a hard matter to convince any of the old navigators.” (Williams 43-6).

Navigational Use of the Compass

The compass has two purposes: determining which course is being steered, and providing a reference point for the measurement of azimuth (horizontal direction) in celestial navigation. An error of 3d* in setting the course of the ship results in a positional error of one mile for every twenty miles run (Mixter 48).

Magnetic Compass . The standard magnetic compass has a magnetized needle which only swings horizontally. However, there are also “dip” compasses which can pivot vertically, too.

The marine compass typically has a rotatable compass card, marked with the compass directions. At least one magnetized needle is attached to the underside of the card. (Unlike the boy scout compass, in which the needle turns, and the card is stationary.) The earth’s magnetic field causes the needle, and with it the compass card, to turn on its axis until the needle is properly aligned with the local magnetic field.

Needles were magnetized by stroking them with an artificial or natural magnet (lodestone). The up-timers can teach how to magnetize steel rods by inserting them into a current carrying coil.

Increasing the number of needles makes the compass more sensitive, and it thus performs better when the sea is quiet, but then it oscillates too much when the waters are rough (Walker 72).

The compass used by the down-timers is “dry,” the card pivots on a vertical pin, inside an empty bowl (Gurney 25). The epitome of the dry compass is, perhaps, the Admiralty Standard Compass, introduced in 1840 and still in use a century later (Gurney 208-10). It, together with the temperamental 1876 Thomson patent compass (240-64), are discussed in 1911EB.

In the wet form, the compass card is still attached to the needle, but they are floating on some kind of liquid, preferably a viscous one. The dry compass was favored during the sailing ship era, but steamship engine vibrations forced the eventual adoption of the wet version (Williams 136-7; Gurney 264-72), like the Ritchie model described by 1911EB.

Down-time, you had to be careful where you bought your compass. For example, in northern Europe, compasses frequently had hidden offsets (needle at angle away from north on card) of 6-11d*, to compensate for magnetic variation. On the other hand, Italian-made compasses lacked these offsets. (Gurney 63). An unsuspecting soul who bought a northern compass and then tried to use it in the Mediterranean could get an unpleasant surprise.

Curiously, compasses weren’t routinely tested until the nineteenth century. After the 1707 Scillies disaster, the Navy inspected its compass inventory, and found that only three out of 145 were working properly (Wakefield 45).

The magnetic compass is subject to a number of inherent errors (earth’s variation and ship’s deviation), so mariners speak of three different kinds of directions: compass, magnetic (compass direction adjusted for deviation), and true (magnetic direction adjusted for variation). A surveyor, such as Grantville’s Mason Chaffin, should be quite familiar with the phenomena of magnetic deviation and variation.

Magnetic Variation . The magnetic compass works, ultimately, because 1) the earth has a liquid iron outer core, 2) the molten iron is in constant motion, and 3) at least some of that motion is attributable to the rotation of the earth. The result is that a magnetic field is generated which, very loosely speaking, has one pole (place where a “dip” compass would point straight down) near the earth’s true North Pole, and the other near the true South Pole. However, the earth’s magnetic field is not a simple field, with two geometrically opposite poles, like the one generated by a bar magnet. Hence, the compass needles don’t necessarily point exactly toward the true poles.

The difference, expressed as so many degrees to the east or west of true north (or south), is called variation (or declination), and differs depending on where on the earth the compass is situated. Variation is unaffected by heading, and compensation with counter-magnets is not possible. But it varies with location (and time). It is thus essential, especially when sailing great distances, to keep track of the magnetic variation so that the correct course can be steered.

The down-timers are well aware of the existence of magnetic variation. According to Williams (26), magnetic variation was first indicated on a European chart in about 1504. Cape of Good Hope is called Cape Aguilhas (“Needles”) by Portuguese because of the way the compass misbehaves in its vicinity (Walker 1).Mercator tried to explain variation by postulating first one (1546) and then two (1569) north magnetic poles (NRC).

Nonetheless, one of the reasons for the loss of the English fleet off the Scillies in 1707 was that their navigators didn’t make allowance for the magnetic variation in the region (7.5d*W at the time)(Gurney 95-6).

Determining a compass’ variation requires taking the compass bearing of an object whose true bearing is known:

* Celestial object—The most commonly used celestial objects are Polaris, and the rising or setting Sun. While Polaris is always very close to true North, the Sun moves about, so you need to compute or look up its azimuth for a particular day and time.

* Landmark—If you have an accurate chart, and your ship’s position is known, take the bearing of a landmark shown on the chart.

* Place Line—If your position is not known, sail so that two landmarks shown on the chart line up. Preferably, the landmarks are far apart.

An example of calculating magnetic variation was given by Hariot in 1595. The azimuth of sunrise was measured with the meridian compass, the simultaneous solar declination was estimated from successive noon values in the Book of the Sun’s Regiment, and that was used as an entry, together with the ship’s latitude, into Hariot’s “Table of Amplitudes,” arriving at the true azimuth of the sun. The variation was the difference between the true and observed azimuths. (Taylor 221).

Determining the variation at a particular location is a bit tricky. Both daily and annual fluctuations occur. At Cheltenham, West Virginia, the westernmost declination is at 2 p.m., and the easternmost at 8 or 9 a.m. If time of year is considered, the range is from 6d*E on a summer morning, to 4.8d*W on an equinoctial afternoon (Sipe 77).

The Chief Pilot of the Portuguese India Fleet, De Castro, made numerous measurements of variation around 1540 and asserted that it could be measured with an accuracy of 0.5d* on smooth water and 2d* when the ship was rolling (Taylor 183).

The first map of magnetic declinations was made by Edmund Halley in 1699. I don’t think a copy of that map made it through the RoF, but the 1911EB has a world map showing the magnetic variation (declination) as of 1907. The contour lines connect points at which the variation is the same, that is, so many degrees to the east or west of north.

Unfortunately, the 1907 map is virtually useless in the 1630s (and the same would be true of Halley’s), because the magnetic variation changes dramatically over time.

The conventional wisdom in 1600 was that the variation was fixed (as taught by Gilbert in De Magnete). But by the time of the RoF, the down-timers already had collected evidence that Gilbert was mistaken. For example, Borough found that the declination at London in 1580 was 11d*4’E, while in 1622, Gunter said that it was only 6d*13’E. The discrepancy was at first ascribed to experimental errors. Sometime in OTL 1633, Henry Gillebrand began to suspect, based on new observations, that the declination had continued to trend westward, and he became sure of this in midsummer 1634 (and published his findings in 1635). This is explained at length in 1911EB “Magnetism”, which offers numerous tables showing the change in declination in different parts of the world.

This “secular change” is just as geographically diverse as magnetic variation itself. Even outside the polar regions, it can be as fast as a 20d* shift in one year.

One silver lining is that, for a specific location, the change is fairly close to constant (Bloxham). Hence, local maps (like the USGS quadrangle maps) can be published which state both the current variation, and the annual rate of change, and they are then useable for a few decades for local compass correction.

The other is that, if archaeomagnetic data is fitted to a standard geomagnetic model (Van Gent; Pickering), it appears that the early seventeenth century might have been a relatively good time to rely on a magnetic compass. Van Gent’s 1600 map suggests that for Atlantic voyages between 60d*N and 30d*N, the declination was usually not more than 10d* (the exceptions were between Newfoundland and Greenland, and in the SW Atlantic). Declinations were also less than 10d* in the waters lying in the Australia-SE Asia-Japan triangle.

If you are writing a story and you need to know the magnetic variation in a particular part of the world in the seventeenth century, I suggest taking a look at the tenth order CALS3K model (Pickering) and its successors.

Magnetic Deviation. The errors in magnetic compass bearings which are attributable to the ship and its contents are called deviations. They can vary depending on where the compass is located, and the direction of the ship’s heading.

The earth’s magnetic field induces transient magnetism in soft iron, and the resulting deviation is greatest when the ship is on an easterly or westerly course. Even in a wooden ship, there are iron items. João de Castro’s 1538 observation of variation were “troubled by the proximity of artillery pieces, anchors and other iron.” (Gurney 139)

These “soft iron” deviations change as the ship moves north or south (changes magnetic latitude). The force induced in “horizontal iron” (such as a beam) is greatest at the equator, least at the poles. The reverse is true for vertical iron, and its direction reverses when the ship crosses the magnetic equator. Vertical soft iron in early 19C sailing ships included “hanging knees, nails, and bolts in the deck, the capstan spindle, anchor flukes, stanchions, chain plates, belaying pins, rudder stock.” (180).

In wooden ships, the deviation is greatest when the ship is on an easterly or westerly course (Walker 67); this is the result of asymmetrical vertical soft iron, forward or aft of the compass (NGIA 13). Bear in mind that the compass is by the helmsman, at the rear of the ship.

Downie, master in HMS Glory, 1790, wrote: “I am convinced that the quantity and vicinity of iron, in most ships, has an effect in attracting the needle . . . the needle will not always point in the same direction, when placed in different parts of a ship . . . [T]wo ships, steering the same course by their respective compasses, will not go exactly parallel to each other yet when their compasses are on board the same ship, they will agree exactly.” (Walker 11)

A small amount of iron close to the compass can be as disturbing as a large mass further away. A belt buckle, moved closer than twelve inches, can cause a deviation. So can a ballpoint pen at five inches, or a wristwatch with a metal band a foot away, a knife at two feet, or a metal handle axe at four (Sipe 84-5). With some qualifications, the magnetic field strength is inversely proportional to the cube of the distance (83).

As you might expect, deviation became a greater concern in the nineteenth century when iron hulls were introduced. The deviations experienced on an iron ship can exceed 50d*! (Gurney 189, 200, 217). When the steel is hammered, bent, riveted or welded, the earth’s magnetic field imprints it, converting it into a “subpermanent” magnet which records the direction the ship was “headed” when built. (Mixter 60-1).This “semicircular” deviation can be observed on any heading, as it is maximized when the subpermanent dipole is at right angles to the compass needle.

Deviation is measured by “swinging the ship”; placing the ship on each standard heading and comparing the compass bearing to the true one. The known variation (unaffected by heading) is taken into account, and the residual error is the deviation.

There are two basic approaches to dealing with deviation. The 19C British Navy method was to never assume that the compass was correct; rather, routinely swing the ship. The Merchant Marine approach was to judiciously place counter-magnets so as to counterbalance the deviation. This can be tricky, especially until the underlying theory is rediscovered. (Gurney, 255-6; Togholt, 24-5; Williams 131-6). Some correction, at least, is desirable on iron ships, since large deviations can cause the compass needle to become sluggish or erratic.

Magnetic Dip. The first compasses had a needle which could only pivot horizontally. In 1581, Robert Norman discovered that if the needle were permitted to move vertically, it would dip (Walker 9-10). This magnetic “inclination” of the needle varies across the world. The needle will point straight down at the magnetic poles, and is flat at the magnetic equator (a wavy line ranging perhaps 10d* north and south of the true equator). The unreliability of magnetic compasses in the polar regions is a consequence of dip; the magnetic force on the needle is then primarily vertical, and the needle may be more responsive to ship movement than to the tiny horizontal magnetic force.

In 1602, Gilbert and colleagues suggested that dip could be used to determine latitude when the sky was overcast (Taylor 247). This was a forlorn hope; points of equal dip are not uncommonly 20-30d* latitude apart.

Looking at the NOAA 2005 World Magnetic Model, it appears that the lines of equal inclination are fairly shallowly sloped between around 60d*N and 30d*S (except near the Cape of Good Hope), so that if dip were regularly measured along with latitude, it might be possible, in a pinch, to estimate the change in latitude based on the change of dip. However, this probably wouldn’t be known to anyone in the 1632 Universe until there was a systematic study of magnetic dip.

Dipping needles are nonetheless useful in correcting the deviation observed when a ship heels over (tilts).

Gyrocompass . A gyroscope is a device designed so that it can spin rapidly, and mounted so that its axis can point in any direction. The axis will continue to point in the spin-up direction unless it is disturbed by an external force. The combinational of the earth’s gravitational force, and the centrifugal force imparted by earth’s rotation, causes it to precess so it points to true north (and south). Changes in the ship’s heading don’t change the forces acting on the gyroscope so it will continue to point that way. The principle of the gyroscope was enunciated by Foucault in 1852, and the first gyrocompass was installed by Sperry in 1910 (Mixter 73).

The gyrocompass has the advantage of pointing to True North; it is not subject to magnetic variation or deviation. And a properly adjusted gyrocompass is usually accurate to one degree or better. However, it requires constant electrical power (to keep the gyro spinning), and its accuracy decreases as the ship moves above 75 degrees latitude. (Dutton 171). Of course, the latter problem is also experienced with magnetic compasses, which tend to go haywire in polar regions. I don’t know when gyrocompasses will become available.

Altitude and Azimuth

For celestial navigation (see second article), we will need to be able to describe the positions of celestial objects. From the observer’s standpoint, the easiest system is to measure the angle between the object and the horizon (altitude) and between the horizontal direction to the object and true north (azimuth). While both altitude and azimuth are needed to completely define the position of an object in the sky, many celestial navigation problems are solved just by use of multiple altitude readings. Azimuth is used mostly for correction of compasses.

Measuring Altitude

To measure altitude, you need a reference, either “down” (established by a plumb line) or “level” (the true horizon). Unfortunately, at sea, plumb lines sway, and horizons are obscured by mist, waves, and spindrift (Callaghan 154).

Down-time Instruments . The down-timers had several devices to use for measuring altitude. The first was the mariner’s astrolabe. This had a circular scale, and a radially mounted, rotatable arm (alidade) holding a pointer and a sight at each end. The astrolabe was suspended from a cord so it would be vertical. You eyed the star through the two sights and then read off the altitude from where the pointer crossed the scale. For a sun sight, you let the sun shine through one hole and illuminate the other.

One person steadied it, a second took the sighting, and a third read off the altitude. The larger the astrolabe, the more accurate were its measurements. A seven inch diameter astrolabe might be read to thirty arc-minutes, and a two-footer to ten (Graham). However, large astrolabes were cumbersome to use. Sometimes, the navigator made a landing just so the astrolabe could be used more easily. (Miller)

Then there was the “sea ring.” In one version, the sun shone, through a hole, onto a scale engraved on the inner surface. In another, the sun cast a shadow onto the scale. Like the astrolabe, the “sea ring” was a hung device which didn’t need a horizon. (Swanick 87-88).

Another device was the simple quadrant, not to be confused with the later Davis quadrant. It was first used at sea in 1461. It was a quarter-circle arc, with sights along one straight edge. A plumb line hung from the vertex. One sailor sighted through the two holes while another read off where the plumb line cross the scale on the rim of the arc. (Williams 35)

According to John Davis (1595), the astrolabe and quadrant were difficult to use on shipboard, except when the sea was calm. (Phillips-Birt 139) You can imagine the astrolabe, or the plumb bob of the quadrant, careening wildly as the ship was buffeted by waves.

By the 1630s, these devices had been largely superseded by the cross-staff (forestaff), consisting of a staff and a sliding transom (the cross-piece).You put your eye at one end of the staff, and slid the transom toward or away from you along the staff, until the top of the transom was aligned with the star and the other end with the horizon. Since you looked somewhat like an archer, this was called shooting the stars. The staff was graduated so you could read off the location of the cross-piece. While the cross-staff could be used by one person, it was awkward to keep both the horizon and the celestial object aligned, simultaneously.

The sixteenth century cross-staff came with three or four cross-pieces, of different lengths. Typically, the staff was about 2.5-3 feet, and the transoms were 15, 10 and 6 inches long. The largest would be used when the star was high in the sky, and the smallest when it was close to the horizon (which would be the case for the North Star when a ship was near the equator). For each, there was a table for converting the location of the transom to the observed angle.

The length of the staff and cross-pieces dictated its region of usefulness; Swanick (76) says that it could only be used to sight objects between twenty and sixty degrees altitude, which would be degrees of latitude in the case of the Polestar. The cross-staff was best used when the altitude was substantially less than sixty degrees, since for higher elevations, the corresponding graduations on the staff were small. (Phillips-Birt 128, 145)

Digges’ Prognostications also warned the down-timers that there is a parallax problem with the cross-staff; it would yield the correct altitude only if the eye were at the center of the staff (Taylor 206). Since the staff wasn’t transparent, that was impractical, so the user had to make a downward correction to the nominal altitude. Hariot actually calculated the necessary individual correction for Raleigh and certain other English explorers, but on average it was about 1.5d* (Taylor 220).

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)


—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, the 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 celestial 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.

The “Mariner’s Creed” warned them that if they neglected any of the four L’s, they would “some day surely perish” at sea.

Thanks to the books of Grantville, the sounding machine will one day replace the hand-thrown lead; the towed “patent log,” the chip thrown overboard; and the sextant, the cross-staff and its ilk. For that matter, chronometers will replace nocturnals, magnetic compasses will be properly corrected, and twentieth century maps will be republished. But even with all of these improvements in navigational instruments, there will still be a need for a sharp-eyed sailor in the crow’s nest.

To be continued in Grantville Gazette, Volume 15