The whys and wherefores of navigation

by Gershom Bradford

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.