472 PHILOSOPHICAL TRANSACTIONS. [ANNO 179Q. 



from the meridian; and consequently the area gc, which is proportional to it, must 

 likewise remain constant. If therefore we can ascertain this area when the hour- 

 angle is supposed to vanish, we may employ it when the sun is at any distance from 

 noon. Let us now conceive the declin. to be equal to nothing; then will our ex- 

 pression for the area gc become jj> ; and consequently, (since the tangents of the 

 azimuth and hour-angle vanish in the ratio of their sines, or of the sines of the 

 opposite sides in the triangle alluded to before), we shall have the area gc = —, 

 when the sun is on the meridian. But this area is always the same when the lati- 

 tude is given, whatever be the sun's declination, and therefore may always be repre- 

 sented by ^ = - X 7 = -7-; ana * tne area g c W M De generally expressed by ~ x 

 cos. o men a tit. ^en the hour-angle does not exceed the limits which have been 



cos. of declin. ° 



recommended. Hence, if we add together the constant log. 3.1015, the log. 

 radius, and the log. cosine of the merid. altitude, and from their sum subtract the 

 log. cosines of the latit. and declin. we shall obtain the log. value of gc. 



It will be necessary perhaps to meet an objection, which some may be inclined 

 to urge, against the method of deducing the hour-angle in terms of the cosine, 

 when this angle is very small. But it should be recollected, that with the angle 

 itself we have no immediate concern, the accuracy of our conclusion depending 

 entirely on the accuracy with which the area corresponding to any particular incre- 

 ment of time can be determined. Now this area, whatever be the sun's distance 

 from the meridian, will be nearly proportional to the increment of the latit. and 

 consequently its magnitude is totally unconnected with that of the hour-angle. A 

 given error in the quantity which expresses this area will equally affect our conclu- 

 sion, whether the angle be 2, or whether it be 20 degrees. But let us inquire what 

 effect will actually be produced, by admitting an error of half an unit in each of 

 the log. cosines whose difference is equal to the area gc; and of course in some in- 

 stances, an error of an unit in the area itself, on any particular supposition of latit. 

 and declin. We have only to ascertain the ratio which this area bears to unity: for 

 the same ratio will the correction of the latit. bear to the error in our result. If 

 the latit. for instance, be 50°, and the declin. 10°, on the same side of the equator 

 with the latit. then, radius being unity, i, the increment of the hour-angle, will 



equal — = - = (g being the sine of the hour-angle, and « the cos. of 



* T cos. 50° x 4- 



the altit.) r^ X ~ L— - =11' nearly, when z is 5°; and we have seen that, 



' cos. 50° go 



at any other distance from the meridian, the incremental area will be of the same 

 magnitude. Hence, subtracting the log. cosine of 5° from that of 5° ll', we get 

 the difference equal to 1238; and consequently the error in our approximation will 

 be to the error in the assumed latit. as 1 : 1238, when the log. cosines are carried 

 to 7 places of decimals. But, when the zenith distance of the sun, at his greatest 



