260 



Sir W. Thomson on the Determination of a [Jan. 19, 



finding a second angle (Z) of the one triangle (P Z S) ordinarily solved (P 

 being the earth's pole, Z the ship's zenith, and S the sun or star). The 

 angle ordinarily calculated is P, the hour-angle. By calculating Z, the 

 sun's azimuth also, from the same triangle, the locus on which the ship must 

 be is of course found by drawing on the chart, through the point which 

 would be the ship's place were the assumed latitude exactly correct, a line 

 inclined to the east and west at an angle equal to Z. But, as Captain Mo- 

 riarty pointed out to me, the calculation of the second angle would involve 

 about as much work as solving for P a second triangle with a slightly dif- 

 ferent latitude ; and Capt. Sumner's own method has practical advantages 

 in affording a check on the accuracy of the calculation by repetition with 

 varied data. 



A little experience at sea suggests that it would be very desirable to 

 dispense with the morning and evening spherical triangles altogether, 

 and to abolish calculation as far as possible in the ordinary day's work. 

 When we consider the thousands of triangles daily calculated among all the 

 ships at sea, we might be led for a moment to imagine that every one has 

 been already solved, and that each new calculation is merely a repetition of 

 one already made ; but this would be a prodigious error ; for nothing short 

 of accuracy to the nearest minute in the use of the data would thoroughly 

 suffice for practical purposes. Now, there are 5400 minutes in 90°, and 

 therefore there are 54 00 3 or 157,464,000,000 triangles to be solved each for 

 a single angle. This, at 1000 fresh triangles per day, would occupy above 

 400,000 years. Even with an artifice, such as that to be described below, 

 for utilizing solutions of triangles with their sides integral numbers of de- 

 grees, the number to be solved (being 90 3 or 729,000) would be too great, 

 and the tabulation of the solutions would be too complicated (on account 

 of the trouble of entering for the three sides) to be convenient for prac- 

 tice ; and Tables of this kind which have been actually calculated and 

 published (as, for instance, Lynn's Horary Tables *) have not come into ge- 

 neral use. 



It has occurred to me, however, that by dividing the 

 problem into the solution of two right-angled triangles, 

 it may be practically worked out so as to give the ship's 

 place as accurately as it can be deduced from the ob- 

 servations, without any calculation at all, by aid of a table 

 of the solution of the 8100 right-angled spherical triangles 

 of which the legs are integral numbers of degrees. 



Let O be the point in which the arc of a great circle s ' 

 less than 90° through S, perpendicular to P Z, meets P Z 

 or P Z produced. 



If the data were S P, P Z, and the hour-angle P, the solution of the 

 right-angled triangle S P O would give P O and S O. Subtracting P Z 



* Horary Tables for finding the time by inspection &c, by Thomas Lynn, late Com- 

 mander in the eea-service of the East- India Company. London, 1827, 4to. 



