200 Mr. T. T. Wilkinson on Mathematics and Mathematicians : 



rises perpendicular in ©, S©Z is a right-angled spherical tri- 

 angle ; whence the following rules may be deduced for finding 

 the proper time of making the observation ; the Sun's altitude 

 at that time and the azimuth angle SZ © . 



* Now, tan. ZS : tan. SO : : r : cos. ZS©. 



" Or, r : cot. SZ : : tan. S© : cos. ZS©. 



u Again, cos. S© : cos SZ : : r : cos. Z©. 



" Or, r : sec. S© : : cos SZ : cos. Z©. 



" Similarly, sin. ZS : r : : sin. S© : sin. SZ©. 



" Or, r : cosec. ZS : : sin. Z© : sin. SZ©. 



" Hence the following rules : — 



" 1. To find the time. 



" Add the tangent of the Latitude to the cotangent of the 

 Declination; the sum is the cosine of the hour-angle from 

 noon. 



* 2. To find the Azimuth Angle. 



" Add the secant of the Latitude to the cosine of the Declina- 

 tion; the sum is the cosine of the Azimuth. 



"3. To find the Altitude. 



" Add the cosecant of the Declination to the sine of the Lati- 

 tude ; the sum is the sine of the Altitude. 



" General Limitation. 



" It is evident that the polar distance Sn must be less than 

 SZ the colatitude, otherwise the vertical circle Z©i> cannot 

 touch the parallel of Declination, and hence the Latitude and 

 Declination must be of the same name, and the Latitude less than 

 the Declination. 



" This rule saves a great deal of trouble when it can be used ; 

 for a number of observations may be taken a little before and a 

 little after the proper time found above, and there is no trouble 

 of calculation whatever ; for the difference between the constant 

 Azimuth angle found above and the angle observed gives the Varia- 

 tion. A Table might be readily made with the Declination at 

 the top and the Latitude at the side ; whence the proper time 

 and constant angle might be taken out. I think it would be 

 proper to put such a table in the Nautical Almanack, not be- 

 cause it is anything either new or difficult, but because sailors 

 are such idle scoundrels that they neglect thousands of oppor- 

 tunities of finding the Variation because they Will not be at the 

 trouble of working the Azimuths, and such a Table would save 

 them that labour." 



The object of the preceding problem evidently is a to find the 

 time when the apparent motion of a celestial body is perpendi- 

 cular to the horizon, from having its declination and the latitude 

 of the place given ; " — this being the most favourable time for 

 making the observations required. Under this enunciation the 



