The whys and wherefores of navigation

Every mariner who has reached a position in the profession where he is intrusted with the responsibilities of navigating a vessel is under obligation to the late Capt. Thomas H. Sumner, of Boston. This shipmaster discovered and developed the principle of the so-called Sumner or Position Lines, a principle which has proved of inestimable value and which, with its subsequent improvements, has well-nigh revolutionized the methods of navigation.

The discovery was purely accidental and for that reason is interesting. Here, in Capt. Sumner’s own words, is how it occurred: “Having sailed from Charleston, S. C., 25th November, 1837, bound for Greenock, a series of heavy gales from the westward promised a quick passage. After passing the Azores, the wind prevailed from the southward, with thick weather, after passing longitude 21° W., no observation was had until near the land, but soundings were had not far, as was supposed, from the edge of the bank. The weather was now more boisterous, and very thick, and the wind still southerly.

“Arriving about midnight, 17th December, within 40´ by dead reckoning, of Tuskar light, the wind hauled S.E. (true), making the Irish coast a lee shore. The ship was then kept close to the wind and several tacks made to preserve her position as nearly as possible until daylight, when, nothing being in sight, she was kept on E.N.E. under short sail, with heavy gales. At about 10 A.M. an altitude of the sun was observed, and chronometer time noted; but having run so far without any observation, it was evident that the latitude by dead reckoning was liable to error and could not be entirely relied upon.

“However, the longitude by chronometer was determined, using the uncertain D. R. latitude, and the ship’s position fixed in accordance. A second latitude was then assumed 10´ to the north of the last and working with this latitude a second position of the ship was obtained and again a third position by means of a third latitude still 10´ further north.

“On picking off these three positions on the chart it was discovered that the three points were all disposed in a straight line lying E.N.E. and W.S.W., and that when this line was produced on the first-named direction it also passed through the Smalls Light. The conclusion arrived at was that the observed altitude must have happened at all three points, at the Smalls Light, and at the ship at the same instant of time. The deduction followed that, though the absolute position of the ship was doubtful, yet the true bearing of the Smalls Light was certain, provided the chronometer was correct. The ship was therefore kept on her course, E.N.E. and in less than an hour the Smalls Light was made bearing E. by N. ½N. and close aboard. The latitude by D. R. turned out to be 8´ in error.”

If the captain had worked more time sights using different latitudes, he would have added new positions on the line to which he refers, each placed upon it according to the latitude used. Had he cared to pursue his experiments farther, and used latitudes very wide of his dead reckoning position, he would have discovered that the resulting positions instead of lying in a straight line, were in a curve and an arc of a circle.

The principle involved is very crudely illustrated in the following experiment: Let the reader consider himself aboard ship lying at anchor—say a full-rigged ship, so as to insure a foremast of good height. Lower the dinghy and take along a sextant.

We start with a series of measurements to determine the angle, as read from the sextant in the dinghy, between the truck and the waterline about the vessel. As a result of these measurements, we discover that this angle becomes smaller as the distance from the vessel increases.

Carrying our tests farther, suppose when the sextant shows the altitude of the fore truck above the waterline to be 70°, that the distance to the vessel be determined. With this distance as a radius and the foremast as the center, we row in a circle around the vessel, the sextant will continue to read 70° all around the circle.

It is thus demonstrated that a circle surrounds that foremast upon which the altitude of its truck is everywhere 70°—a circle of equal altitudes.

Not being quite sure of this interesting fact, perhaps, another angle is selected by moving a little farther from the ship. The sextant shows the fore truck to have an altitude of 50°; the distance to the vessel is established, whereupon the dinghy is rowed around the vessel with this distance as a radius. Again the sextant reveals no change from 50° and it is clearly shown that we have moved about on a circle of 50° elevation of the truck.

We can continue experimenting in this way until the distance from the ship becomes so great that some physical condition prevents our reading the angle of the truck’s altitude.

These investigations show that there is a system of concentric circles of equal altitude about every elevated object like the little undulations we have seen so many times produced by the splash of a stone thrown into a pool.

These circles of equal altitudes surround not only elevated terrestrial objects but also celestial bodies, as will now be shown. As the sun is the most convenient body for this illustration, let us substitute it for the fore truck of the foregoing experiments, while for the waterline of the vessel we will use the point on the earth touched by a plumb-bob suspended from the center of the sun.

This point will fall on the equator on the 21st of March or thereabouts, as the sun coming up from his southern declination crosses the equator into north declination at this time. The instant of the transit is the vernal equinox. Now this point will be found an excellent one from which to study this problem, but, as this takes some time and the sun is ever on the move, we will imagine ourselves endowed with the power of Joshua to command the sun and moon, which will enable us to study this phenomenon while free from the restlessness of the Universe.

First of all, it must be understood that the sun shines on one hemisphere of the earth at all times; it matters not how the earth is tipped in relation to him, one half of the world is always enjoying sunshine. The center of the lighted area is the spot directly beneath the sun where the plumb-bob touches and about this point lies the system of concentric circles of equal altitudes of the sun.

Under the conditions shown above, the sun is in the zenith of the terrestrial vernal equinox, shining on the earth for a distance of 90° in every direction; but its altitude diminishes in direct proportion with the distance of the observer from the point of the equinox. On the great circle everywhere 90° from the equinox the sun is in the horizon with an altitude of 0° (provided we disregard dip and refraction). Suppose the members of some intrepid expedition have reached the northern or southern pole; they would, at the time being considered, see the sun in the horizon and in the direction of the meridian passing through the vernal equinox.

Eastward along the equator 90° of longitude from the vernal equinox, the inhabitants are just resting from the toils of the day, for with them the sun is setting in their western horizon, while away to the westward 90° the people are showing signs of activity, for it is just sun-up in their eastern horizon.

So all around the world just 90° from this selected position and at this appointed time is a circle of equal altitudes, namely 0°, for is not the sun seen in the horizon at all points on this circle?

The altitude of the sun is 90° at the point of observation and 0° on its outer circle of altitude; these are the two extremes and between them lies an infinite number of concentric circles of equal altitude for navigators to utilize. The zenith distance, derived by subtracting the altitude from 90°, indicates the distance of each circle from the center or sun’s position. Thus if an observation was taken by some bewildered mariner in which the altitude was found to be 80°, the corresponding zenith distance of 10° multiplied by 60 would indicate that the altitude was taken 600 miles from the sun’s position, or to put it in another way, the circle of equal altitudes upon which the observer was located in this case had a radius of 600 miles.

What is true of the sun on the equator regarding the principle of the circles of equal altitudes holds good throughout its range of declination, the whole system moving north and south with the continuous change of declination and from east to west with its apparent diurnal motion.

In the quoted article, Capt. Sumner shows a method by which the position of a vessel may be established on some particular circle of equal altitude; it matters not where the observed body happens to be at the time, for with the Nautical Almanac and chronometer it can be located should we care to know. The navigator, however, cares to deal ordinarily only with a very small arc of the circle embraced within his immediate whereabouts. Should he be somewhat uncertain of these he would simply require the use of a longer line to extend beyond the limits of his possible position.

Except when in a latitude that differs but little from the declination of the observed body, the circle of equal altitude will be sufficiently large to allow the mariner to represent its arc in his vicinity by a straight line. Thus the lines of position used to plot a vessel’s position on the chart are in reality chords or tangents of the circle of equal altitude. In geometry it will be remembered that we used to study about circumscribed and inscribed polygons and here we have a practical application of their use. If we consider the line of position to be a tangent, it is one side of a great polygon with a vast number of sides circumscribed about the circle of equal altitude; and if we consider it to be a chord, it is likewise a side of a great polygon inscribed within the circle of equal altitude. It matters not, however, if the line or curve of position is considered a straight line, except in the ill-chosen condition of the body near the zenith when the radius of the circle will be proportionately small. If exactly in the zenith there will be no circle of equal altitude at all and the sextant will measure an altitude of 90°. It is comparatively rare, however, that such a condition will embarrass the use of this method.

Another point to be remembered in connection with the inscribed and circumscribed polygon propositions and one which has a practical application in the use of position lines, is that the tangent or chord of a circle is at right angles to the radius passing through the point of tangency or center of the chord. It follows that the sub-celestial or terrestrial position of the observed body, being at the center of the circle, is always at right angles to a line of position.

This important fact gives the navigator an opportunity to check his compass error each time he establishes a position line, by comparing a compass bearing of the body taken simultaneously with the measurement of the altitude, with the true bearing.

To establish a position line as Capt. Sumner did and as it was done for years afterwards, by assuming two latitudes usually 10´ each side of the dead reckoning latitude, and drawing the line through the two resulting longitudes, is known as the chord method. The two longitudes being positions on the circle a line drawn between them is a chord of the circle.

The work of computing a time sight is more or less laborious to everyone and with some seafarers forms their most arduous mental exercise. At any rate no one wants to work any more than is necessary to insure accurate results. So when establishing a position line it will often be found convenient to use the short cut known as the tangent method.

With the latitude by account work the observed altitude as in the ordinary time sight, instead of assuming two latitudes. Seek the true azimuth in the tables or on diagram, using the latitude and declination employed in the time sight and the local apparent time gained from it, as the arguments. The true azimuth, it will be remembered, always bears at right angles to the position line. Hence if the azimuth is laid down through the position furnished by the time sight, the position line may also be readily plotted at right angles to the line of azimuth at the time sight position.

The navigator now-a-days is expected to think in position lines when he is clear of the land, as a pilot thinks in shore bearings and marks. That is, he must see these imaginary lines of the different visible bodies, and keep track of their availibility for his particular use. It is easy to get into the habit of this, for they are simply astronomical bearings instead of bearings on distant terrestrial objects, with the distinction that the celestial bearing allows of a 90° correction to produce a position line.

The morning sun on the prime vertical with a sufficient altitude to avoid any dangerous refraction, will produce a north and south line of position. During the forenoon as the sun passes toward the meridian, the northern end of the position line will move in direct proportion with the body’s change in azimuth to the eastward and the southern end to the westward, until at noon with the sun on the meridian we have an east and west position line.

It will be seen that at one moment of the day it is a very easy matter to establish a line of position; the mere working of a meridian altitude does this. This simple expedient of finding a position line was utilized a great deal as a means of making a landfall in the days before chronometers were perfected. In those good old days, before the clipper ship era, time was not held at such a premium as in the present hustling period, and a few days more or less at sea mattered but little. The shrewd shipmasters then would keep well offshore until in the latitude of Boston or the Virginia capes, as the case might be, when they would haul due west and let her go, making, no doubt, first rate landfalls, if the old pig yoke was in good working order.

The value of a position line was demonstrated to the writer some years ago when bound in from the eastward and running into a heavy and very extensive fog bank somewhere southeastward of Halifax. During a break in the prevailing conditions the navigator succeeded in securing an ex-meridian sight and fortunately got a fairly good idea of the latitude. The vessel was under sail and making but slow progress, and as a result of the protracted period of overcast sky the longitude became considerably a matter of guess work. The vessel, however, was kept on a west course with a careful allowance made for the set. “Sir William Thompson” was kept going at regular intervals and it was surprising to see the soundings check up with the chart as the vessel approached, crossed, and left astern the Roseway Bank, southward of Cape Sable. One felt as sure of the position as did the old Nantucket sailor in crossing “Marm Hackett’s garden.”

In cases where the soundings do not check so precisely as in this instance, it will sometimes be found a great help to lay off to scale the depths obtained on the edge of a piece of draftman’s transparent linen. Place it on the chart in the line of the course, and, should the soundings fail to agree, move the scale forward and back or to either side, always preserving the direction of the course, until a position is found where the soundings on the scale agree with the depths given by the chart.

Progress has been made in the science of navigation as in all other sciences, and the modern shipmaster is not obliged to hold aloof from Nantucket Shoals and Georges Bank under ordinary conditions as our ancestors were compelled to do, for with a correct chronometer and a knowledge of the position line such outlying dangers have been robbed of many of their anxiety-producing elements. Before showing the method of working around such places another point of value of the position line is called to the reader’s attention.

A line of position extended until it reaches the land or some danger will indicate to the mariner the bearing of that particular point of the coast or danger. If it so happens that this point is not the place of destination, the navigator, not being able to lay a course direct for his objective port through inability to determine the vessel’s distance offshore, overcomes the difficulty by sailing a sufficient distance at right angles, then hauling on to a new position line parallel to the original one. This is similar to what our ancestors did in the simple way cited above. If the line lies in the direction of an off-lying or isolated shoal that is dangerously near the course, an offset like that shown above will allow a course parallel to the position line to be sailed in safety. Here is an example to show its useful application:

A steamer sailing from St. John, N. B., for New York proceeded but about 10 hours on her voyage, when she ran into a terrific gale. The master was soon forced to heave his vessel to and ride it out as best he could. The driving snow and mountainous seas occupied the attention of the officers in their efforts to save the steamer and in this way the dead reckoning position became a matter of mere guesswork. The wind after some 20 hours in the northeast quadrant hauled to the northward, at length blowing out in the northwest with clearing weather.

It was the master’s intention to pass through the South Channel, between Georges Bank and Nantucket Shoals, but as he had lost his reckoning to such an extent he hesitated about laying a course through such a danger-strewn locality.

In the late twilight immediately following the clearing sky, the master succeeded in catching the altitude of a star bearing 300° and established a line, the direction of which led close westward of Cultivator Shoal (a 6-foot spot on Georges Bank). So to be on the side of prudence and give this shoal a good berth, the master steamed 8 miles at right angles to this position line. The course or direction of the new position parallel to the first was found to lead directly into the range of visibility of Nantucket Lightship. So the master’s mind was put at rest as he laid his course along the second position line, knowing he would at length make the lightship.

It often happens that a distant mountain peak is visible and the sun is in a suitable position to establish a set of cross bearings, using the mountain for one object and the sun for the other. Now with what has previously been stated, it is hardly necessary to remind the reader that a “line of position” obtained from observations of the sun will be at right angles to the sun’s true bearing; therefore, in order to judge whether these objects are properly placed to give a good intersection, due consideration must be given to the relative bearings of the objects. It is evident that the sun must bear by compass nearly in the direction of the mountain or in the opposite direction to have the position line and the line of bearing of the mountain cut at nearly right angles. Of course, as with any set of cross bearings the angle of intersection may still be effectual if the lines cut at 50° to 60°, but the nearer a 90° cut the more accurate the resulting position.

A position line is liable to displacement through a variety of causes among which is an inaccurate altitude and through incorrect Greenwich mean time. In the former instance, an error of 1´ will displace the position one mile; if the altitude is 1´ too large, the correct position of the line will be 1 mile directly away from the bearing of the body and vice versa. The effect of an error in time upon a position line is to displace it bodily eastward or westward the amount of arc corresponding to the error in the chronometer; the direction of the line is, however, unaltered. The sun carries his system of circles of equal altitude with him from east to west as he travels along a certain parallel of latitude corresponding to his declination (neglecting the slight change in declination). It is quite evident that any arc of a selected circle, will, if its position is plotted on a small scale chart—say every 20 minutes—be found continuously parallel with itself. And the intervals between each two plotted positions of the arc will be 5° (of arc) the corresponding value of 20 minutes. Thus the displacement of the position line due to an error of time is explained. If the time was slow, the line was too far to the eastward, if fast, it was too far to the westward.

The value of a position line has been demonstrated, yet with all it does not positively establish the position of a vessel. The mariner in locating his vessel in a harbor does not usually stop after he has taken one bearing, but proceeds to find another object whose bearing will make a favorable “cut” with the first, and thus at their intersection determines his position. As a further check against possible error a third object may be chosen and, if the three bearings plot without forming a triangle at their intersection, a very reliable fix will be obtained.

What applies to terrestrial objects thus employed may be used as an illustration to be followed in taking celestial bearings. If the mariner establishes a position line and knows his vessel is located at a point somewhere along it, let him look about for another body so placed that the position line derived from it will make a good intersection with the first line; if all data are correct this point will indicate the position of the vessel.

When the sun is used this is seldom possible but in lieu of another body the sun can again be employed to establish the second position line after it has moved sufficiently in azimuth to make a good cut. The thought no doubt immediately arises as to the effect of the vessel’s change in position during the interval. This is easily taken care of by means of the course and distance run during the interval between the sights.

The first position line must be considered carried bodily by the vessel without change of bearing from its first position to the position of the second observation. That is, if at 9 A.M. a position line was established bearing in a 15°-195° direction, and the vessel then steamed and made good a 40°-course for 6 hours and 10 knots an hour, when another position line was established, the 15°-195° line of 9 A.M. would be moved bodily in a 40° direction 60 miles; where its intersection with the second line would indicate the position of the vessel at 3 P.M. The determination of position at sea by employing two position lines of a body with the run between sights is called Sumner’s double altitude problem.

It has already been shown that one body, notably the sun, can be used to get an intersection of two of its lines of position by waiting a sufficient time between observations for the body to change its bearing at least 30°, the nearer 90° the better. The relationship between the interval of time and the amount of change of bearing varies greatly, depending upon the latitude of the observer and declination of the body. For example, let us consider the two extreme cases: Suppose a mariner to be observing the sun on the equator on March 21st, he will note practically no change in azimuth during the whole forenoon. Yet another mariner in the Polar sea, whose latitude differs about 90° from that of the former, will have the sun encircling his horizon making the whole amount of the sun’s movement a corresponding change in azimuth.

Therefore it will be seen that with a low-riding sun (or other body) the change of azimuth is greater in a given time, and for this reason the position lines derived from the sun are more advantageously practiced in higher latitudes, especially in winter. This is a point of great value in view of the fact that the sun’s diurnal course is such that it is never on the prime vertical in northern latitudes during the winter months, making longitudes derived from chronometer sights very unreliable.

But to go back to the mariner on the equator whose latitude and sun’s declination so nearly agree. He is in a predicament should he persist in the plan to determine his whereabouts by position lines of the sun. In such an unusual case, it would be well to resort to some other method or wait until evening and determine the ship’s position by establishing the position line of some star or stars. It will be but a few days before the ship’s progress will cause the sun to leave his right course across the sky and take the hour circles at an angle. Take a case when the sun at noon has a zenith distance of 10°, the change of azimuth during the forenoon is still small, but suppose the bearing was noted 1 hour, or even less, before noon and again in similar amount after noon, a change will be found of perhaps 90°, the difference of moving from the southeast quadrant (if declination is south of latitude) to the southwest quadrant. In this way, a remarkably good cut may be had within a comparatively short time.

The foregoing will convince the reader that he must be governed by the change of bearing and not by time elapsed, in predicting the value of the cut of his position lines.

In the use of position lines, it is necessary to bear in mind, that, when the body’s altitude begins to approach the zenith, or, what is the same thing, when the ship is getting close to the body’s sub-celestial position, the circle is getting proportionately smaller. Under such conditions the arcs of the circle of equal altitudes can no longer be shown as a straight line. The double altitude as it is ordinarily practiced is here impracticable. And even outside this impracticable area, discretion must be shown. The dead reckoning position must be proportionately accurate, and the assumed latitudes must be brought correspondingly close together, in order to have a shorter line of position, because the curvature of the circle is getting sharper as the sub-celestial point is approached. To put it in another way, a smaller arc must be used in order to avoid the error due to excessive curvature.

Very good results can be obtained by noting the time of observation by chronometer (G. M. T.) and correcting it for equation of time in order to get Greenwich apparent time. This, if converted into arc, is the longitude of the sub-solar position. By using the Greenwich mean time to correct the declination taken from the Nautical Almanac for that day, the latitude of the sub-solar position may be obtained. Plot this position on the chart and use it as the center of a circle; then with the zenith distance (90° - altitude) as a radius, draw an arc in the probable position of the vessel. Somewhere along this arc is the ship’s position. The bearing of the sun (rather hard to get so nearly overhead) corrected for compass error, reversed, and laid off from the sub-solar position will give a fair idea of the position of the vessel. Now by waiting a sufficient time for the sun to change its azimuth enough to make a good cut and using its new sub-solar position as a center with the zenith distance of a second observation as a radius, an arc may be drawn which will intersect the first arc at the position of the vessel. The run between the sights will, of course, require the first arc to be carried forward as the first position line in the ordinary double altitude problem.

Johnson’s Method

It is not always found convenient to plot the position lines of a set of observations on a chart; perhaps for lack of a chart of proper scale or possibly for want of the chart itself. Again many navigators do not take kindly to the graphic method, but prefer to solve their latitudes or longitudes by computation. In any event Johnson’s Method comes as a relief to such persons, saving them from the arduous duty of establishing a set of position lines by the chord method of assuming two latitudes to get two longitudes.

Johnson’s Method can be practiced in both the double altitude problem of the sun, where the first sight, or position line, is brought forward to the second sight by correcting it for the intervening run, or where stars are used simultaneously.

Chief among its merits is the saving of figures. It is only necessary to compute two (instead of four) chronometer sights in order to find the ship’s position, thus obtaining a mathematically accurate result by a short cut. But also a great advantage in the Johnson Method is that the resulting longitude is obtained by calculation and it is not necessary to plot the lines upon the chart to secure the position.

In using Johnson’s method it is not absolutely necessary to observe two stars simultaneously as the quick work of a good man is sufficiently close for the practical purposes of navigation.

It becomes evident to anyone reading the foregoing pages that every ordinary time sight places the vessel on a circle of equal altitude, the longitude resulting from the computation, depending on the latitude, by dead reckoning, used. Now rather than work two sights employing two assumed latitudes on either side of the supposed position, make the calculation only once, using the latitude by account.

Suppose by way of explanation that the altitude of a star bearing S. 55° E. is observed simultaneously with that of another star bearing S. 25° E. The longitudes derived by working the time sight of each should be identical, provided the altitudes are the true altitudes, the Greenwich time is without error, and the latitude used is correct. A combination of accuracy, indeed, and one not likely to be experienced often in actual practice. However, a skillful navigator should find no great difficulty these days in always having the correct Greenwich time at hand. There is always, of course, an opportunity for the display of skill in measuring altitudes, refraction particularly being an illusive element and not always easy to detect. But if care has been taken to eliminate the errors as much as possible from the time and the altitude, it is safe to consider any discrepancy between the resulting longitudes as accountable to an error in the dead reckoning latitude.

The method of obtaining the ship’s position from the difference in the longitudes, derived from double or simultaneous observations, was originated by A. C. Johnson, R. N., and its many advantages have for years made it the most popular form among progressive shipmasters. The working of this problem involves the application of a correction to each calculated longitude in such a way as to bring them into agreement. The tables (Bowditch Tables 47 and 48) furnish this correction, which is known as the longitude factor and is symbolized by the letter F. It constitutes the change in longitude due to a change of 1´ in latitude. This quantity changes directly with the change of azimuth of the body; for example, the change in longitude is nil if the change in latitude is made on a due north or south line, and change in longitude increases as the change in latitude is made on lines bearing more and more eastward or westward. So it is necessary in order to obtain these corrections to have the true azimuths of the bodies at the moment of observation to use as an argument in the table of longitude factors. These are readily taken from the Azimuth Tables or diagram using the data furnished by the time sight.

The two longitudes obtained from time sights in which the same dead reckoning latitude is used, lie on the parallel of this latitude, but (unless the two longitudes happen to be coincident) the ship’s position is either north or south of this parallel according to the error existing in the dead reckoning latitude. If the observed azimuth of the body (or bodies) fall within the same quadrant or in opposite quadrants, the correct longitude will be found to the eastward or the westward of both calculated longitudes. This is clearly shown in Fig. 7; both azimuths are between south and east. If the observed azimuth of the body (or bodies) fall in adjacent quadrants say, one between south and east and the other between south and west, the ship’s position will be found between the two calculated or erroneous longitudes. The position of this true longitude is determined by means of the before-mentioned factors. The factor of a longitude is the distance of the true longitude east or west of the meridian passing through the calculated or erroneous longitude, assuming the latitude to be in error 1´. The moment of this factor, it will be seen, depends on the azimuth of the body, which in turn determines the direction of the position line.

The combination of the two factors, by adding if the bodies are in the same or opposite quadrants or vice versa, is the combined error in difference of longitude due to 1´ of error in latitude. It now becomes a matter of proportion by which to obtain the error in the dead-reckoning latitude. As the combined error in difference of longitude for 1´ of latitude, is to 1´ of latitude, so is the difference between the two calculated longitudes, to the error in latitude.

The longitude factors are based upon an error of 1´, so if the error is more than 1´ it becomes necessary to multiply the factor by the error in order to obtain the correction to the calculated or erroneous longitude.

An altitude may be taken of any body and after a suitable change in bearing has taken place (not less than 30°) a second altitude may be taken and the first longitude advanced for the run during the interval to the parallel of the latitude by dead reckoning at the time of second sight.

In the usual event of a disagreement in the calculated longitudes the rule of procedure is as follows: With the body’s true azimuth at each observation, the difference between the longitudes and the latitude by dead reckoning, used at second sight, enter Table 47, Bowditch, and take out the corresponding numbers. If the azimuths are in adjacent quadrants, these quantities should be added, but if in the same or opposite quadrants, they must be subtracted. The result in each case gives the combined error in difference of longitude for an error of 1´ in latitude.

It is now only necessary to divide the difference between the two longitudes by this combined error and we have the error between the correct latitude and the latitude by dead reckoning. Now multiply the error in latitude by the number taken from Table 47 corresponding to the first longitude, to obtain the correction to that longitude, and by multiplying the same error in latitude by the number corresponding to the second longitude we have the correction for that longitude. The application of these corrections should bring the two calculated longitudes into agreement at the position of the true longitude.

Some difficulty may be experienced in learning how to apply these corrections to the calculated longitudes, but it is always easy to make a rough diagram if at all in doubt. A horizontal line representing the parallel of latitude by dead reckoning at second sight may be drawn with the two longitudes plotted upon it; establish the position lines through these longitudes by drawing them at right angles to the sun’s (or star’s) azimuth. The intersection of the two position lines indicates the true longitude and a glance shows how to apply the corrections to each calculated longitude to get the true. In Fig. 8 the westerly longitude requires the correction to be applied to the east and the easterly longitude correction to the west in order to arrive at the true longitude.

Without the use of a diagram a rule easy to remember in deciding whether to apply the correction in longitude to the eastward or westward is here given: If the error in latitude is of the same name as the first letter of the bearing, the change in longitude is contrary in name to that of the second letter and vice versa. For example take the case just cited.

When a body’s azimuth is less than 45°, it is wiser, and insures more accurate results to work by New Navigation, or, if sufficiently close to the meridian, as an ex-meridian. In the case of the latter the corrections in such a case are taken from the table of latitude factors (Bowditch, Table 48), the problem being the same in principle and in solution as that described above. Good results are even obtained by using two ex-meridians, one on each side of the meridian. The corrections in the latitude may be applied according to the following rule, if it is preferred to the rough diagram method: if the error of longitude is of the same name as the second letter of the bearing, the change in latitude is of the contrary name to the first letter, and vice versa.

The New Navigation

In every branch of science and industry since time immemorial a continuous process of simplification and increased accuracy has been taking place, and amid this general evolution of working systems the science of navigation will not be found an exception. Even now there is a tendency to displace the time-honored chronometer sight, together with a long list of more or less bewildering ways of obtaining latitude and longitude.

The advance method is popularly known as the New Navigation, yet its principles were originally brought forward by Marcq St. Hilaire, a French admiral, nearly 40 years ago. It is not a new method of finding position, but rather an improved way of establishing a Sumner line. Like many innovations it has taken all these years for navigators to become reconciled to the change and break away from the more familiar forms.

In order to facilitate a simple explanation of New Navigation it will be brought to mind that every heavenly body has a corresponding point on the earth directly beneath it, which bears the same relation in latitude and longitude to the earth, that the body does in declination and right ascension to the celestial sphere. To an observer at such a sub-celestial point the body is in the zenith with an altitude of 90°; and about him lies a system of concentric circles of equal altitude, which extends over a hemisphere of the earth 90° in every direction from the point of origin. This point, through the apparent diurnal revolution of the body, carries this whole system of circles around the earth each day and northward and southward with the body’s change in declination. On the outer limit of this system of circles, the altitude of the body is 0°. Thus it is seen that the altitude of the body decreases and its zenith distance (90° - altitude) correspondingly increases in direct proportion as the observer departs from the sub-celestial point, and vice versa. If, for instance, an observer is 100 miles (nautical) from this point, the zenith distance is 100´ or 1° 40´ and the altitude of the body is 88° 20´; at 2700 miles 2700´/60 = 45° of zenith distance, and 90° - 45° of altitude.

A feature is now introduced that has a close bearing upon the principle under discussion, to serve as an opening view of the subject: A navigator fortunate enough to have a body reasonably near his zenith, say 5°, has at hand an extremely simple way of graphically finding his ship’s position. This situation has previously been described, but is repeated to make clear the principle of New Navigation. The sub-celestial position of the body at the moment of observation is readily ascertained by noting the time by chronometer and recourse to the Nautical Almanac for its declination. With the point thus established as a center, and the zenith distance derived from the observed altitude as a radius, swing a circle upon the chart. The ship’s position is somewhere on the circumference of this circle of equal altitudes. This circle is now carried forward the amount and direction of the run of the vessel between this observation and a subsequent one similarly taken. During this interval the bearing of the body should have changed sufficiently to make a good intersection of the circles. The ship being on both circles must be at one of the two intersections, between which the mariner can readily decide. The conditions cited are comparatively unusual but show the practical use of a circle of equal altitude in its simplest form.

The zenith distance is ordinarily too large to become a radius for such use on the chart. The circles of equal altitude are in practice so large that 10 to 40 mile arcs in the vicinity of the vessel are treated by her navigator as straight lines, known as Sumner or position lines. These lines are, theoretically, chords or tangents according to the method employed in establishing the line, but in practice the divergence from the circle is negligible, excepting always when the body is too close to the zenith. The establishment of the position line has been done in several ways for many years until the advent of this new and more expeditious method.

The altitude of a body at any selected time for an assumed position can be readily calculated. If this altitude does not agree (and it seldom does) with the altitude measured simultaneously with the sextant, corrected for the usual errors, the assumed position is not coincident with the actual position of the vessel. The navigator now proceeds to lay off from the assumed position, the line of azimuth of the body taken from the azimuth tables, Weir’s Azimuth Diagram, or determined by observation. On this line the distance between the observed and computed altitude, expressed in minutes of arc, is measured, towards the body if the observed altitude is greater, and away from it if less, than the computed altitude. The point thus indicated is a position on a circle of equal altitudes, the arc in the immediate vicinity of the computed point being, approximately, the position line. This line is at right angles to the azimuth for the reason that a tangent is at right angles to the radius of a circle at a given point.

It is now known that the ship is somewhere on this line of position, and it is necessary to cut it with another such line to determine definitely her position. If the sun is the body being observed, it becomes necessary, in order to provide a good angle of intersection, to wait until the azimuth changes at least 30°, when the observation is repeated, a second line established, and the first line brought forward in exact accordance with the ship’s run. The interval required naturally depends upon the latitude of the ship and the declination of the sun. The intersection of the lines will be the position of the ship at the time of the second sight.

The use of stars has a decided advantage in that there are always some of these bodies available for observation lying in various azimuths; it is practicable, with a well-defined horizon, to observe simultaneously two or more of these bodies whose bearings show that they would produce desirable position lines. From the resulting intersections the position of the ship is secured. This obviates waiting for the second line, a feature that is always inconvenient and sometimes, perhaps, dangerous.

The calculation of the altitude is accomplished by the solution of the spherical triangle in which we have given the co-latitude (90° - assumed latitude), the polar distance and the hour angle of the meridian of the assumed position. Thus with two sides and the included angle, the third side or the zenith distance (90° - altitude) is easily determined by either of several formulas.

With the use of this method all the formulas that formerly, and still, often puzzle the navigator to remember can be reduced to this one sight. One of the most important features it possesses is that it can be utilized regardless of the altitude of the body (except when very high), its azimuth, or its hour angle, all of which are elements that have to be used under certain favorable circumstances in order to get accurate results from the older forms. The navigator is now given a greater freedom in choosing bodies to observe than is found in any other method.

The mariner to-day has been almost entirely relieved from the labor of computing position at sea, should he care to avail himself of a set of altitude tables, several excellent ones have made their appearance on the market, among them Hydrographic Office Publication No. 200. From them the altitude can be selected corresponding to the conditions of any particular observation. With a set of these tables a navigator is no longer required to be a mathematician or to remember the forms of a half dozen sights. Thus in this wonderful age the mariner’s utopian dream of obtaining position at sea by inspection, is, in a way, realized.

In order to illustrate the practical working of a problem by this method, the following example is taken up point by point:

Early on the morning of May 21, 1899, while in the assumed position of latitude 55° 00´ N., longitude 112° 08´ E. observed the true altitude of the star Arcturus to be 37° 14´ 50´´, bearing west of the meridian. The chronometer carrying Greenwich mean time read 20 d. 6 h. 20 m. 03 s. The observer desired his position.

The problem by the St. Hilaire method resolves itself into the solution of the spherical triangle shown in Fig. 10, where two sides and an included angle are given:

Polar distance = 90° - declination (Nautical Almanac).

Co-latitude = 90° - latitude (by dead reckoning).

Hour angle of star. See figure and solution below.

The hour angle of the sun is more readily found than that of a star. It is accomplished by applying the longitude (in time) of the assumed position to the Greenwich time shown by the chronometer at the time of sight. This hour angle of the mean sun must be corrected by the equation of time to obtain the hour angle of the actual sun.