Astronomical and Nautical Collections. 173 



communicated a similar improvement to the respectable veteran 

 Von Zach, who has inserted it in the number of his valuable 

 Correspondence, bearing the nominal date of March, 1820. 

 But Mr. Querret's formulae, though more geometrically accu- 

 rate than Mr. Delambre's, possess no practical advantage over 

 them, and Mr.Dubourguet's method appears to be almost exactly 

 the same as one of those which Mr. Delambre has employed. 



Rule for double Altitudes. 



1 . Having corrected one of the observations for the change 

 of the ship's place during the intervfil, take the logarithmic sine 

 of the mean polar distance ^ (PA+PB), and the sine of half 

 the interval converted into space, that is, | f ; add them toge- 

 ther, and the sum will be the sine of half the distance AB. 



2. Then as the sine of AB is to the sine of the opposite angle 

 APB, so is the sine of one of the polar distances ; for instance, 

 the second PB to the opposite angle PAB. 



3. Having thus the three sides ZA, ZB, and AB, we have 

 next to find the angle BAZ. For this purpose, add together the 

 two polar distances ZA, ZB, and the distance AB, and from 

 the half sum subtract the^rs^ polar distance ZA and the distance 

 AB : add together the sines of the remainders, and the arithme- 

 tical complements of the sines of the sides last mentioned, half 

 the sum will be the sine of half the required angle BAZ. 



4. The difference of BAZ and PAB, or sometimes the sum, 

 between the tropics, will be the angle PAZ, subtended by the 

 colatitude PZ from the sun's place A. To find this colatitude, 

 take out the logarithmic cosines and sines of the first polar dis- 

 tance and zenith distance, and with the sines set down the co- 

 sine of the included angle PAZ : add them separately together, 

 and find the corresponding natural numbers, the sum of which 

 will be the natural sine of the latitude, and its logarithm of 

 course the logarithmic sine. But if the angle PAB lies without 

 BAZ, and their sum exceeds a right angle, the cosine becomes 

 negative, and the diflTerence of the natural numbers must be 

 taken : and if there is any doubt in the computer's mind, it will 

 be easy to try both suppositions. 



