1871.] Ship's Place from Observations of Altitude. 



261 



from P O, we have Z O ; and this, with S in the triangle S Z O, gives 

 the zenith distance, S Z, and the azimuth, S Z O, of the body observed. 



Suppose, now, that the solution of the right-angled spherical triangle 

 S P O for P O and S O to the nearest integral numbers of degrees could 

 suffice. Further, suppose P Z to be the integral number of degrees closest 

 to the estimated co-latitude, then Z O will be also an integral number of 

 degrees. Thus the two right-angled spherical triangles S P O and S Z O 

 have each arcs of integral numbers of degrees for legs. Now I find that 

 the two steps which I have just indicated can be so managed as to give, 

 with all attainable accuracy, the whole information deducible from them- 

 regarding the ship's place. Thus the necessity for calculating the solutions 

 of spherical triangles in the ordinary day's work at sea is altogether done 

 away with, provided a convenient Table of the solutions of the 8100 triangles 

 is available. I have accordingly, with the cooperation of Mr. E. Roberts, 

 of the ' Nautical Almanac' Office, put the calculation in hand ; and I hope 

 soon to be able to publish a Table of solutions of right-angled spherical 

 triangles, showing co-hypotenuse* and one angle, to the nearest minute, 

 for every pair of values of the legs from 0° to 90°. The rule to be pre- 

 sently given for using the Tables will be readily understood when it is con- 

 sidered that the data for the two triangles are their co-hypotenuses, the 

 difference between a leg of one and a leg of the other, and the condition 

 that the other leg is common to the two triangles. The Table is arranged 

 with all the 90 values for one leg (b) in a vertical column, at the head of 

 which is written the value of the other leg (a). Although this value is 

 really not wanted for the particular nautical problem in question, there are 

 other applications of the Table for which it may be useful. On the same 

 level with the value of b, in the column corresponding to a, the Table shows 

 the value of the co-hypotenuse and of the angle A opposite to the leg a. 

 I take first the case in which latitude and declination are of the same name, 

 the latitude is greater than the declination, and the azimuth (reckoned from 

 south or north, according as the sun crosses the meridian to the south or 

 north of the zenith of the ship's place) is less than 90°. The hypotenuses, 

 legs, and angles P and Z of the two right-angled triangles of the preceding 

 diagram are each of them positive and less than 90°, and the two co-hypo- 

 tenuses are the sun's declination and altitude respectively. We have then 

 the following rule : — 



( 1 ) Estimate the latitude to the nearest integral number of degrees by 

 dead reckoning. 



(2) Look from one vertical column to another, until one is found in 

 which co-hypotenuses approximately agreeing with the declination and 

 altitude are found opposite to values of b which differ by the complement 

 of the assumed latitude. 



(3) The exact values of the co-hypotenuse and the angle A corresponding 



* It is more convenient that the complements of the hypotenuses should be shown 

 than the hypotenuses, as the trouble of taking the complements of the declination and 

 the observed altitude is so saved. 



