of a Place by Observations of the Pole-Star. 447 
largest circles and with the most perfect instruments. — Besides, 
we have always the means of eliminating this error: we have 
only to take the altitudes at an equal iaeaice on the opposite 
side of the meridian, and the error destroys itself.” 
“ We see therefore, in every view of’ the case, that in point 
of accuracy, zt is indifferent whether we observe the pole-star 
at the time of its culmination, or at any other point of its pa- 
rallel. But for the convenience of the observer, and the infinite 
advantage of being able to collect in a short time an immense 
number of good observations for the latitude of the place, this 
latter method seems preferable to all others.. The observer does 
not depend on a single instant, which presents itself only once 
in 12 or 24 hours, and of which he may be deprived by the 
weather, a passing cloud, or some other untoward circumstance. 
By the method here proposed he may at any time he thinks 
proper take the altitude of the star, by day or by night, if the 
weather permit, and if he have time and opportunity: and he 
can thus in 24 hours make as many observations as he pleases, 
and collect in this interval a great number of latitudes.” 
The advantages thus detailed by M. Littrow must be evident 
to every practical astronomer; and have since engaged the at- 
tention of several mathematicians. In Bode’s Astronomische 
Jahrbuch for 1823, Dr. Dirksen has also given a formula for de- 
ducing the latitude from such observations. And in M. Zach’s 
Correspondance Astronomique, vol. v. page 308, Mr. Horner © 
has deduced another formula for the same purpose. In vol, vi. 
page 71, of the same work, M. Littrow asserts that the formula 
of Dr. Dirksen is erroneous, and that Mr. Horner’s might be 
~ much sitnplified. He also states that it appears that M. Schu- 
macher has calculated his tables for finding the latitude by the 
pole-star, inserted in his Ephemeris of the planets, from a for- 
mula similar to that of Dr. Dirksen. Indeed, the argument not 
only of one of M. Schumacher’s tables, but also of Mr. Horner’s, 
is the latitude of the place ; which is, in fact, the very quantity 
sought, This remark will probably attract the attention of these 
astronomers, and induce them to revise their formule. The 
- formula, ultimately adopted by M. Littrow, is investigated as 
follows : 
“ J.et p be the apparent polar distance of the star, x the ob- 
served zenith distance of the star, ¢ the horary angle of the star, 
or the sidereal time elapsed since the moment of its upper cul- 
mination, and the height of the equator or the co-latitude of 
the place: and make x (= the correction) equal to ~ — x, 
Then, by spherical trigonometry, we shall have 
cos % — cos p, cos (%+X) — sin p, sin (z+7), cos t = 0, 
By 
