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.