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LX. On finding the Latitude, $c. from three Altitudes of the 
Sun and the elapsed Times. By James Burns, Esq. 
To the Editor of the Philosophical Magazine and Journal. 
Sir, 
HESITATED some time whether I should again address 
you on any subject which may cause a controversy; but 
as I believe the following solution of another problem cannot 
be in that predicament, you are at liberty to make whatever 
use you please of it. You have already devoted more of your 
pages to a controversy (of which I have taken my leave) than 
its importance perhaps deserved ; and therefore your last cor- 
respondent Mr. Beverley will excuse me, if I pass him in si- 
lence, merely reminding him: ; 
TO YAR FEPlcTe TEATTLIY Bx EXsE VEY sd<va.— Soph. Antig. 
A controversy of that kind seldom ends in the conviction of 
either party, especially when there are misunderstandings on 
both sides; and I will readily own that my fist method was 
not so free from objection as I considered it. But to come to 
our present purpose. Three altitudes of the sun and the in- 
tervening intervals of time afford a ready and accurate me- 
thod of determining the true time from noon, the sun’s decli- 
nation, and the latitude. We have not however, so far as I 
have seen, an analytical solution of that problem, which can 
be considered short and practical*; or in other words, the 
formulz of solution are not embodied in the smallest possible 
compass. Mackay in his very useful work on the longitude, 
has given us a geometrical solution, which is very ingenious, 
and which, I believe, was first given by D. Bernouilli. But 
the usual objection to most geometrical solutions may be made 
to this,—that it breaks the formulze into too many equations, 
which render it more complicated. ‘To the following solu- 
tion, I believe this objection cannot be made. 
Let ZA =a 
ZB = 0b 
ZC =c 
if) Lee 
PC:= PBi= PAj= 2 
ZPA:= 2 
APB =y 
APC =z 
* Our correspondent will find an analytical solution of this problem, by 
: Euler, 
