Page 81 RECENT MATHEMATICAL TABLES  
     
  295 [U].—Hamburg, Deutsche Seewarte, publication no. 2154, F-Tafel.
Tafel zur vereinfachten Berechnung von Höhenstandlinien. 3 Auflage.  Hamburg, August, 1941. xxiii, 88 p. 19.6 X 29.2 cm. In the third edition there were extensions and corrections of the introductory material, and of 8 of the 11 tables.		  
     
          The method and principal table of this volume are similar in many respects to those of H. O. 208 (Dreisonstok, see MTAC, v. 1, p. 79f). The astronomical triangle is divided into two right spherical triangles by a perpendicular from the zenith upon the hour circle of the star; U is the co-declination of the foot of the perpendicular, and V is log cos B, where B is the angle subtended at the zenith by U. By Napier's rules,  
 
tan U = cos t cot L
  
  and  
 
sin B = sin t cos L,
  
  where t, L, and d are the local hour angle, latitude and declination respectively. By applying another of Napier's rules to the right triangle of which the star is one vertex, the altitude, h, may be found by  
 
sin h = cos B sin (d + U)
  
  or  
 
log sin h = V + log sin (d + U).
  
     
          For the determination of azimuth, Z, two more auxiliary quantities are introduced, P which is the great circle distance from the star to the east- or west-point of the horizon, and Gr. δ which is the declination of the intersection of the hour circle of the star with the prime vertical. Thus, sin t cosd = cos P and sin Z = cos P sec h. Also, tan Gr. δ = tan L cost.  
     
     

 

     
 
 
  Page 82 RECENT MATHEMATICAL TABLES  
     
          In Table F I, with vertical argument, latitude 0(1°)70°, and horizontal argument, local hour angle 0(4m)6h, three values per page, there are tabulated four quantities, U to the nearest 0°.1, V to 5D, Gr. δ and P, each to the nearest 0’1. In the second part of Table F I, the vertical argument is latitude, 70°(1°)90°, and the horizontal argument is local hour angle 0(4m)6h, nine values per page, three in each horizontal section.  
     
          At the bottom of the vertical columns in Table F I are azimuths; entering the left hand column with altitude as argument, and moving across the pages horizontally until one finds under P, the value already copied out, one can drop to the bottom of the column and read off the azimuth angle. Since the tabulated values of the azimuth angle go up to 90° only, it is necessary to have another device to determine the quadrant. When the hour angle is greater than 6h, the azimuth is measured from the elevated pole; when the local hour angle is less than 6h and L and d are of opposite name, the azimuth is measured from the depressed pole. If the local hour angle is less than 6h and L and d are of the same name, the azimuth is measured from the elevated or depressed pole according as the declination is greater or less than the quantity Gr. δ.  
     
          In case the altitude is great, or the azimuth near 90°, the value of the azimuth may be poorly determined by the use of Table F I. In such a case, it will be noted that the value ofP lies below a dotted line running across the page. One must then use instead Table F XI, which gives P to the nearest minute of arc and the variation in P corresponding to 1' change in d or h.  
     
          Table F II is a table of log sin x, x = [0(0'·1)6°(1')90°; 5D], with generous tables of proportional parts.  
     
          Tables F III and F IV represent the principal advantages this volume possesses over other similar tables; they permit one to determine the corrections (to the nearest 0’.1) to the computed altitude corresponding to slight changes in time (up to 2m by 10s steps) or latitude (up to 30' by 1' steps) respectively. In both cases, one can interpolate very easily by a shift of the decimal point. Table F III is a well-designed triple-entry table occupying only five pages; one starts down the column at the left headed by the value nearest the assumed latitude, stops at the value nearest the computed azimuth and moves to the right to the column headed by the number of seconds change in time.  
     
          Table F IV is a small double entry table on a single page; the vertical argument is azimuth 0(5°)20°(2°)90°, and the horizontal argument is change in latitude, 1'(1')10'(10')30'. These two tables allow one to work either with an assumed position or with a dead reckoning position.  
     
          Table F V is for changing time into angular measure and conversely.  Table F VIgives the corrections for refraction, semi-diameter and parallax to be applied to the altitude (3°-90°) of the lower- or upper-limb of the moon; there is a supplementary table for height of eye. Table F VII gives the combined correction for refraction and height of eye (0-30 meters) to be applied to the altitudes (3°-90°) of fixed stars or planets.  Table F VIII yields the correction for refraction, semi-diameter and height of eye (0-30 meters) to be applied to altitudes (3°-90°) of the sun's lower limb; there are also two auxiliary tables to provide corrections to the altitudes to take care of the varying semi-diameter of the sun through the year, and for the case where the sun's upper limb was observed. The latter takes only a very small amount of space and would seem to be quite worthwhile. Tables F IX and F Xprovide similar corrections for use with the bubble sextant.  
     
          The tables are well printed on a good grade of paper. In a number of cases, the rules needed to make decisions as to quadrants, etc. are printed on each page. As for the accuracy of the tabulated values, only a few rounding off errors of a unit in the last place were discovered in a brief examination.  
     
 
Charles H. Smiley
  
  Brown University