VOL. LXXXIX.] PHILOSOPHICAL TRANSACTIONS. 40Q 



whatever be the magnitude of the hour-angles, or however nearly the latitude and 

 declination may approach towards each other, we can always secure, with very little 

 additional trouble, an exact conclusion. 



We may remark that the latitude, determined in this manner, will be nearly 

 equivalent, in point of accuracy, to the mean result of 2 meridian altitudes. For we 

 know that the increment of latitude : increment of altitude :: radius : cosine of 

 azimuth; and since the cosine of a small angle differs so little from the radius, 

 this may be considered, within the limits which have been prescribed, as a ratio of 

 equality. If therefore one altitude of the sun were taken, and we could ascertain 

 the error in time arising from an error in the assumed latitude, without the aid of 

 a 2d observation, the latitude would be discovered with nearly the same precision 

 as if it had been deduced from the meridian altitude. But, by means of a 2d ob- 

 servation made on a different side of noon, we obtain a 2d error in time of the 

 same kind; and this being added to the former, and their sum divided in a just 

 proportion between the 2 observations, the same effect will be produced, with 

 respect to the accuracy of the result, as if 2 latitudes had been deduced from meri- 

 dian altitudes, and a mean between them had been taken. 



I might perhaps be allowed to say more; for I am satisfied, from experience, 

 that I can take an altitude of the sun with greater exactness, when he is in any 

 other situation, than when he is on the meridian. If we could ascertain, within a 

 few seconds, or even within a minute, the time when he attains his greatest alti- 

 tude, there would then be no reason why an observation should not be made with 

 the same degree of certainty in this, as in other cases; but we are generally 

 obliged to keep our eye stedfastly fixed, for several minutes, on the 2 images, and 

 it is well known that, in such circumstances, the best eyes are apt to be deceived. 

 Besides, it is impossible to preserve the contact of the limbs by perpetually moving 

 the index, while the sun continues to ascend so very slowly. We are compelled to 

 wait till they are evidently separated, and then, by one turn of the screw, to bring 

 them into contact again, which must necessarily be a source of some inaccuracy. 

 It is for the first of these reasons that, in taking an altitude of the sun, when he 

 is near the meridian, I have found it advisable, not, in the usual manner, to bring 

 the images almost to touch each other, and then to wait till they actually do so, 

 but to bring them at once into contact, with such a degree of velocity as would 

 make them sensibly overlap, or separate while the clock beats a second. 



But I consider it as one of the principal advantages of this method, that we can 

 avail ourselves of any number of altitudes, and of course approximate as near as 

 we please to a true conclusion with so little additional labour. If there be an equal 

 number of observations made on each side of the meridian, we must combine them 

 together by pairs, according to the preceding instructions, and thus determine the 

 different logarithmic values of gb. Having then added them all together, and taken 

 a mean among them, we have only to compute a single incremental area gc with 

 any of the altitudes and the lat. varied 1 minute, and subtracting its log. from the 



