Dr. Everett on a new Proportion -Table. 355 



niently available for use with the Proportion-Table. For ex- 

 ample, when the sides of a spherical triangle are given, an angle 



, j, t , ,, „ t . eos a — cos b cose , , 



can be found by the formula cos A = : — ; — : , and that 



sm b sin c 



with more ease and expedition than by the ordinary logarithmic 

 methods. The hour-angle can thus be calculated from the ob- 

 served altitude of sun or star ; and if the calculation is carefully 

 performed with the aid of a Table of natural sines carried to four 

 places of decimals, the result will be correct to I or at most 2 

 minutes of arc — that is, to 4 or 8 seconds of time. 



Another problem to which the Proportion -Table is well adapted 

 is the clearing of lunar distances. Thus, 

 if A B C be the spherical triangle formed 

 by zenith, moon, and star, as observed, the 

 sides of this triangle are known, and the 

 cosines of the angles B and C at moon 

 and star respectively can be calculated as 

 above. Then, if accuracy be required, the 

 correction a of the observed distance can 

 be computed from the corrections {3, y of 

 the apparent zenith-distances of moon and 

 star respectively, by the formula 



a=/3cosB + YCOsC + (^tt; ) (1 — cos 2 B)cot«, 



the small arcs a, /5, y being expressed in seconds ; and if the 

 first term in this formula be worked to 4 figures, the second to 

 3, and the third to 2 figures, the result will be as correct as if 

 6-figure logarithms had been used in any of the ordinary me- 

 thods. When no great accuracy is required, it suffices to use the 

 first and second terms only, and to carry the work to only three 

 figures. The average value of the third term is only 4" or 5", 

 and its sign is the same as that of cot a ; so that it can be roughly 

 allowed for without calculation. 



The Proportion-Table is in fact sufficient and convenient for 

 all the calculations which are usually made at sea ; and its general 

 use in navigation would go far towards enabling seamen to per- 

 form intelligently those processes which, from the very artificial 

 form of the rules at present in use, they are now compelled to 

 perform empirically. " Proportional logarithms," which, as at 

 present used, are a source of perpetual mistakes even to expert 

 computers (see note to Boper's ' Navigation/ p. 419), and have 

 the effect of entirely concealing from a beginner every vestige of 

 analogy in processes which are essentially identical, would no 



2 A2 



