Assuming then, that the true Greenwich time is calculated; that the observer is competent and the instrumental errors are known, the accuracy of a sea position, obtained by the observation of heavenly bodies, depends mainly on the accuracy of the sea hori zon, On account of abnormal refraction occasioned by special conditions in the atmosphere, the sea horizon, as seen by an observer, is often displaced from the position which the tables of normal dip of the horizon would assign to it. This refraction is most common in localities where the temperature of the air and water differ widely. By combining the results obtained from observation of bodies situated in opposite quarters of the horizon, the errors in finding the true position of the horizon are largely counteracted. The conditions at dawn and twilight, when the horizon is visible and suitably placed stars can be selected, lend themselves to this purpose. And, more over, it is in these periods that opportunities are present for another important purpose, that is, the selection, for simul taneous observation, of bodies whose relative bearings are such that the lines of position determined from the observations shall afford a well defined intersection, and thus serve. to accurately find the latitude and longitude at approximately the same instant. The only information that is obtained from one observation of a heavenly body is that the ship is somewhere on the circumfer ence of a small circle, on the earth's surface, the centre of which is the geographical position of the body and the radius of which has the same arc-measure as the zenith distance at the instant of ob servation. At this instant the heavenly body observed must be
vertically over some one point on the surface of the earth. The latitude of this point is equal to the declination of the body and the longitude is equal to the hour angle of the body from the meridian of Greenwich.
Owing to the size of the globe which would be required, it is, of course, not practicable to plot this small circle on a globe and plotting it on a chart is inconvenient unless the radius is very short and the ship in low latitude. A small circle becomes very greatly distorted when plotted on a Mercator's chart, unless near the Equator. The usual method of plotting, on a chart, is to represent a portion of the small circle as a straight line, a tangent to the circumference; that is, a straight line drawn at right angles to the bearing of the body observed. But, as a straight line represents accurately only a very small portion of the circumference of a circle, it is necessary to select some point through which to draw this line, as near the actual position of the ship as possible. This straight line is the "line of position" somewhere on which the ship must be. If another heavenly body, suitably placed, is now ob served or, after a reasonable interval, to allow of a change of bearing, a second observation of the same body is made, a second line of position is obtained. The intersection of these two lines of position gives the actual position, provided the first line of posi tion has been moved parallel to itself to allow for any run of the ship in the interval. The line of position, being a tangent to the small circle, is at right angles to the radius or bearing of the body observed ; it follows, therefore, that the angles at which the several lines of position will intersect one another must be the same as the angles at which the bearings intersect. Bodies should, therefore, be selected which will give a good "cut." Each line of position has precisely the same value, provided the altitude has been observed with equal accuracy. An error of I' in the observed altitude, and thence in the zenith distance, produces an error of one mile in the line of position.
Now as to the manner of obtaining the point through which to draw the line of position. Whatever method is used, provided the method is a correct one, the point found will be somewhere on the line of position. There can be only one line of position derived from the observation of one body. Different methods of calcula tion do not give different lines of position, but give different points through which the line of position can be drawn. A point can be found by assuming the latitude and calculating the resulting longi tude, which is a procedure applicable to observations of heavenly bodies which have a large azimuth ; or by assuming the longitude and calculating the resulting latitude, which is applicable to ob servations of bodies with a small azimuth. The best method, how ever, is to assume both latitude and longitude and, therefrom, to calculate the zenith distance of the body observed, or in other words, the radius of the circle of position. Having, at the same instant, observed the actual altitude, and thence the zenith dis tance, it is at once ascertained whether the ship is on a line which is directly nearer to or directly farther from, the geographical posi tion of the body than the assumed or dead reckoning position.