Notices respecting New Boohs. 309 



b may also have any value from 0° to 90°. The Tables are calcu- 

 lated for the successive values of a and b, increasing by 1° at a 

 time. Instead of the values of the hypotenuses (S P and S Z) being 

 tabulated, the complements of their values are tabulated, so as to 

 give directly the declination and altitude instead of the co-declina- 

 tion and co-altitude. These quantities are given in the columns 

 headed ' co-hyp.' The values of the angles opposite a are given in 

 the columns headed ' A.' 



" When we have made an observation and estimated our latitude 

 to the nearest degree, we do not know the value of b for either triangle ; 

 but we know what the difference between the values of b for the 

 two triangles is, it being equal to the estimated co-latitude. We 

 do not know the value of a ; but we know the values of the two co- 

 hypotenuses, the complements of S P and S Z, since they are equal 

 to the declination and altitude respectively. We therefore take 

 the Tables and search from column to column until we find two 

 places in one of the columns headed ' 5,' distant from each other 

 by an amount equal to the estimated co-latitude, and which are 

 such that opposite to one of them in the adjoining column headed 

 ' co-hyp ' is a number approximately agreeing with the declination, 

 while opposite to the other (and also in the ' co-hyp ' column) is a 

 number approximately agreeing with the altitude. We then take 

 from the adjoining column headed ' A ' the number level with the 

 declination for the hour-angle, and that level with the altitude for 

 the azimuth. In fact we have two level sets of numbers in the 

 three columns \ 6,' ' co-hyp,' and ' A,' one set distant from the other 

 by a length of the column ' b ' equal to the estimated co-latitude, 

 and which give the solutions of the two triangles S PZ and S Z 0. 

 ♦ . . . The search from column to column until the proper places 

 are found, is, with the help of a pair of compasses, a very simple 

 matter. It will almost always happen that the numbers found in 

 the ' co -hyp ' column agree only approximately with the actual 

 declination and altitude. It is then necessary to get a second set, 

 such that the actual value of the declination will lie between the 

 two tabular values, and then to interpolate intermediate values for 

 the altitude, hour-angle, and azimuth corresponding to the actual 

 declination." 



In cases where the point O comes between P and Z : — 

 " The two triangles S P O and S Z O will, as before, have the 

 common side a ; their co-hypotenuses will, as 

 before, be the declination and altitude respec- 

 tively ; and their angles A (S P O and S Z O) 

 will be the hour-angle and azimuth respectively. 

 The side b of the one will, however, no longer 

 differ from the side b of the other by an amount 

 equal to the estimated co-latitude PZ; in place 

 of this relation the following one will exist. 

 Let P Z be bisected in Q ; then P O (the side b 

 of the one triangle) = PQ + OQ = |PZ + OQ, 

 while OZ (the side b of the other triangle) 

 =Q.Z— OQ=-VPZ-OQ. In other words, the 

 side b in the one triangle is just as much greater 



