OF THE t'NITED STATKS. 318 



Expressina; tlicroforc the nidiusn/ I)y tlie seconds and de- 

 cimals whicli it subtends at the station a, the nitiltiplication 

 ot" this by the cosine of hall" the angle observed gave the ap- 

 propriate correction or corr. {cd)" co%. kcas. 



To construct this correction with great case. 1 divided a 

 quadrant on pasteboard of one foot radius, numliered with 

 tlie doul)le angles as in a reflecting instrument, beginning to 

 count as if Jroni the point of the circle perpendicular lo va, 

 whicli represents olcoursc the jiosition of tlie sun in the pro- 

 traction of ac behind the sioinal. This radius was divided 

 into ten parts, and liiiKs drawn perpendicular to ii, cutting 

 the circumference in the corresponding points, indicatiui? the 

 angles to whicli they coircspoml. Upon a smaller piece, 

 cut at right angles, one of these decimals was divided into 

 ten parts again ; this piece being slided along the line near- 

 est to the angle observed between the sun and the signal, 

 the subdivisions of it being perpendicular to the same, until 

 these intersected the circumference at the actual observed 

 angle, the tenths and hundredths of the radius correspond- 

 ing to this angle, were indicated, these being midti|)lied in 

 the seconds, tenths, and hunilredths subtended by the radius 

 of the signal, gave the correction corresponding to the ob- 

 served angle. 



This operation was of course of sufTicirnt accuracy, and 

 nuich shorter than the calculation either by natural cosines 

 of the half angles or their logarithms. 



Opcratina; in the same manner for both signals, between 

 which the ani!:lc is to be corrected, the total correction of the 

 angle is obtained according to the following easy princi|)les, 

 which will be evident, without demonstration, from a mere 

 inspection of the figure: 



1. When the sun is seen between the two signals, the sum 

 of the two corrections is added to the angle. 



2. When the sun is behind the station a in the vertical 

 ancle of the signals, as it) liak, the sum of the two correc- 

 tions is to be subtracted from the angle. 



