48 Mr. Burns on the Double Altitude Problem. 
that should induce him to think so. I have there asserted that 
Mr. R. misunderstood the foundation of my method; and I 
think he has. proved that he knew nothing further of the de- 
monstration, which he attempted to advocate, than the mecha- 
nical computations derived from it. I shall now, therefore, 
notice more in detail those solutions of the problem to which I 
objected, than I had originally intended. The equations which 
I first gave are, 
__ cos. 4 (A +a) .sin. 3 (A — a) 
C085 hFS Tawe (+ + ka). sin. Fi. cos. 9 (1) 
ae cos.4(A + a).sin.} (A — a) (2) 
sin. 3(T + 7) sin. 4 (T— 7). cos. 3 
Now the second of these is identical with 
cos. 3 (A +a) .sin. $(A — a) (3) 
sin. $i. sin. (Zi — +) . cos. 3 
since, T+r=2; 4(T +.7) = 42; also, T —7t.=i —2r, 
&e. The only unknown quantity in the equations (1) and (3) 
is tr, * or the time nearest noon: if that could be ¢rzly deter- 
mined, the question could be rigorously and easily solved. 
But it is plain that + cannot be so determined by means of 
the middle time (as in Douwe’s method), which is itself deter- 
mined by means of the latitude by account,—a quantity that 
maybe very far from the truth. Hence the method of Douwe’s 
is no other than a pure paralogism. ‘The more probable 
way, therefore, of arriving near the truth, would be to take 
from the * Horary Tables” the angle corresponding to the 
latitude by account, the greater altitude, and the declination, 
and substitute it, in one of the above formule according to the 
case; and if the greater altitude were near the meridian, the 
probable error would be diminished. Mr. R. will now pro- 
bably understand what is meant by * all that is necessary to 
be known is, the time, the interval, and the altitudes,” which 
before appeared to him so inexplicable.—Now, Dr, Brinkley’s 
method is professedly a correction of the latitude computed by 
Douwe’s method, which, by the by, will be often further from 
the true latitude, than that by account. Let us now see how 
this correction is derived. ‘lhe Doctor first deduces the fun- 
damental equation (see Nautical Almanac) 
cos.A = 
dl (vers.t — sin.¢. tan. m) 
~"T Stan, D.cot.2 
_— _1— tan. D.cot./ 
ei vers. ¢ — sin. ¢ .tan. m 
dc= ; or, dl: dce::n: 1, making 
. The quantity x, therefore (on which 
* The time 7 could be rigorously deduced by means of a third altitude 
and preceding interval; but that being a distinct problem, may be consider- 
ed on another occasion. 
the 
