June i6, 1898] 



NATURE 



53 



The line of position by an altitude for time was first dis- 

 covered by Captain Sumner, who, being doubtful of what 

 latitude he was in, worked an observation with three 

 different latitudes. On projecting these positions on the 

 chart, he found that all three were in a straight line, 

 which produced, led to the Smalls light, whose bearing he 

 thus had, without knowing how far it was away. He 

 steered along the line till he found it. He did not ob- 

 serve, however, that this line was at right angles to the 

 sun's bearing, nor would it have shortened his problem if 

 he had, because it then took as many figures to calculate 

 one longitude and the azimuth as two longitudes with 

 different latitudes. In these days, when azimuths can 

 be taken out of tables by inspection, nearly half the 

 figures are saved by using the azimuth to obtain the line 

 of position. 



Thus, no matter what the bearing of a heavenly body, 

 if we can observe its altitude and the corresponding time 

 at Greenwich, it will afford us some information as to the 

 position of the ship. If it is on the meridian, with a 

 minimum of labour we get the latitude in the simplest 

 and most accurate way available to the navigator. If it 

 is not too far in azimuth from the meridian, there are 

 plenty of methods by which the observation can be re- 

 duced to the corresponding meridian altitude, and the 

 latitude obtained. If it is on the prime vertical, the line 

 of position will be a portion of a meridian. If it is on 

 any intermediate bearing, the line of position will be at 



right angles to the bearing of the body, through the lati- 

 tude by account, and the longitude deduced from it and 

 the observation. Any two lines of position, provided 

 they do not cross at such an oblique angle that the inter- 

 section is ill-defined, will fix the position of the vessel. 

 When the star is so far from the meridian, and the time 

 too uncertain to be favourable for working as an ex- 

 meridian, and yet too far from the prime vertical to give 

 an accurate hour angle, the new navigation, originated by 

 the French, and introduced into England by Captain 

 Brent and Messrs. Williams and Walter, R.N.,^ gives a 

 better line of position than the older methods. By it you 

 calculate the altitude for the position of the ship by dead 

 reckoning. If this agrees with the observed altitude 

 (corrected), the line of position is at right angles to the 

 bearing of the star, through the position by D.R. If, 

 however, the observed altitude is, say, lo' greater than 

 that calculated, the ship must be that much nearer the 

 spot on the earth where it was in the zenith at the 

 moment of observation ; so you lay off lo miles (i sea 

 mile being practically i' of a great circle) from the D.R. 

 position, in the direction of the star, and through this 

 point rule the line of position at right angles to the bear- 

 ing; or the corrections for the D.R. latitude and longitude 

 may be calculated by trigonometry (see Fig. 2). 



The triangle apz (see Figs, i, 3, 4 and 5) is the 

 most important in nautical astronomy. Up to this, I 

 have only referred to it as a means of finding hour 



1 " Exmeridian Altitude Tables 

 is an excellent work. 



NO. 1494, VOL. 



and other Problems," by these authors, 



58] 



angles (angles at p) ; but not only is it also used for 

 finding azimuths (angles at z), for if the time be ac- 

 curately known, we can utilise it for finding the latitude 

 by a star with a large hour angle. To make it clearer^ 

 and avoid complicating Fig. i, I give figures here on the 

 plane of the horizon. In these, let A represent three 

 different stars, and from A let fall a perpendicular on the 

 meridian. Then right-angled spherics can be utilised, 

 and the latitude obtained with fewer figures than by the 

 new navigation. Either before or after the >jc or >|cs 

 shown in the figure, are obtained for latitude, observe one 

 on or near the prime vertical, for longitude and time, 

 which will give accurately the hour angles of the latitude 

 5+cs, allowing, of course, for any easting or westing made 

 between the observations. Then 



Sin AB = sin -^ sin/, tan PB = co?,hX2iWp and cos ZB 

 = sin a sec. AB 



h being the hour angle, p the polar distance, and a the 

 true altitude. The sum or difference of PB and ZB 

 = the colatitude. This method is even shorter than it 

 appears at first sight (because the logs, can be taken out 

 in pairs), and is concise and accurate when the data is 

 trustworthy, and, even if the hour angle is doubtful, will 

 give a good line of position. 



Unfortunately, the navigator has often to work with 

 data that are more or less doubtful. In the triangle, apz, 

 he uses the three sides to find the hour angle (p). Of 



these the polar distance is accurate ; the latitude is often 

 doubtful enough to affect the hour angle, though not 

 generally the line of position, and the altitude may be 

 vitiated in various ways. It therefore behoves him to- 

 take his observations in a way that errors, that he can 

 neither detect nor avoid, will neutralise each other. Few 

 human eyes are optically perfect ; the best sextants,, 

 though beautiful instruments, are not absolutely fault- 

 less, and their errors are liable to alter by a knock or 

 jar ; the sea horizon is fickle, and refraction uncertain ; 

 but the whole of these errors may be minimised, if not 

 absolutely, eliminated, in the resulting latitude — for 

 example, by observing (Fig. i) the >|<:'s s^ and S^. With 

 about the same altitude, their refraction will probably be 

 similarly affected ; the horizon is generally subjected to- 

 the same influences all round ; the personal and instru- 

 mental errors may be taken as constant, for the same 

 observer and sextant, at any particular time and place 

 when the altitudes are somewhat similar. Suppose the 

 sum of these errors to be - 2', and unallowed for, the 

 effect would be, in each case, to make s' and s^ appear 

 nearer z than the truth ; and while each resulting lati- 

 tude would be 2' wrong, the mean would be correct. 



Again, in the single altitude problem (Figs. 3, 4 and 

 5), if the time had been calculated by two stars, one east 

 and the other west, the time and thence the hour angles 

 of the latitude >|<s would be less liable to the foregoing 

 errors ; and if the three stars were taken and worked 

 for latitude, each would be a check on the others, and 

 opposite bearings would tend to neutralise errors of 



