308 Notices respecting JS T ew Books. 



spending to each. If the two points thus found be joined, the lino 

 joining them will be the Sumner line. Or we may calculate one 

 longitude (and so obtain one point through which the line passes), 

 and also the true azimuth of the body observed, which gives us 

 the inclination of the Sumner line to the parallels of latitude. 

 The latter plan is the one adopted by Sir W. Thomson. The former 

 is the plan most commonly used, although neither can properly be 

 said to be in common use at all ; for, as Sir W. Thomson says, 



" Sumner's method, although undoubtedly the clearest and com- 

 pletest mode of interpreting observations, has never found very 

 general favour with navigators. A few of the most skilful no 

 doubt use it habitually ; and many more use it occasionally in criti- 

 cal times ; but its tediousness has hitherto prevented it from coming 

 into every-day use. I have prepared the Tables which follow with 

 a view to remove this drawback to the use of a method which I 

 have long felt convinced should be the rule and not the exception 

 of practice at sea." 



When the latitude is estimated, the problem of how to find the 

 hour-angle and azimuth is simply the solution of the spherical 

 triangle PSZ, in which P is the earth's pole, 

 Z the shij)'s zenith, and S the sun or star. 

 The data are P Z (the estimated co-latitude), 

 S Z (the zenith distance or co-altitude), and 

 S P (the polar distance or co- declination). It 

 is required to find the angle S P Z, which is 

 the hour-angle from the ship's meridian, and 

 S Z O, which is the azimuth. Sir W. Thomson 

 shows that to tabulate the values of the angle 

 P corresponding to all possible values (differ- 

 ing by not more than 1') of the three sides of 

 the triangle would be impracticable, as it 

 would necessitate the tabulation of the solu- 

 tion of 157,464,000,000 triangles, which at the rate of 1000 

 triangles per day would take 400,000 years. He has overcome this 

 formidable difficulty by dividing the problem into the solution of 

 two right-angled spherical triangles, and by aid of the tabulated 

 solutions of 8100 triangles whose sides are integral numbers of 

 degrees, has made it possible and easy to work out a ship's place 

 as accurately as the observations admit of, with hardly any cal- 

 culation, and without the use of logarithms. The account of how 

 this has been done is given in the preface with a conciseness which 

 hardly admits of abbreviation : — ■ 



" Let O be the point in which the arc of a great circle less than 

 90°, through S, perpendicular to P Z, meets P Z, or P Z produced. 

 We have now the two right-angled spherical triangles S P O and 

 S Z O, which have the common side S O or «, whose angles A 

 opposite to a are in the one the hour-angle, and in the other the 

 azimuth, whose sides b (O P and O Z) differ by an amount equal 

 to the estimated co-latitude P Z, and whose hypotenuses (S P and 

 S Z) are the co-declination and co-altitude respectively. 



" The side a may have any value from 0° up to 90°. The side 



