300 PHILOSOPHICAL TRANSACTIONS. [aNNO 1772. 



back observation ; in which case it will be best to use the 2d method, or else to 

 hold the quadrant perpendicular by judgment ; which will be much facilitated by 

 using a telescope containing wires in its focus parallel to the plane of the 

 quadrant, as described before : for, in this case, the perpendicular position of the 

 quadrant cannot be attained so near, by the method of turning the quadrant on 

 a line going to the sun as an axis, as it can by the other method. 



It remains to treat of the errors which may arise from a defect of parallelism 

 in the two surfaces of the index glass, and to point out the means of obviating 

 them in the celestial observations. It is well known, that if a pencil of parallel 

 rays fall on a glass, whose two surfaces are inclined to each other, and some of 

 the rays be reflected at the fore surface, and others passing into the glass and 

 suffering a reflection at the back surface, and two refractions at the fore surface, 

 emerge again from the glass ; these latter rays will not be parallel to those 

 reflected at the fore surface, as they would have been if the surfaces of the glass 

 had been parallel, but will be inclined to the same. I find that the angle of their 

 mutual inclination, which may be called the deviation of the rays reflected from 

 the back surface, will be to double the inclination of the surfaces of the glass, 

 which is here supposed to be but small, as the tangent of the angle of incidence 

 out of air into glass, is to the tangent of the angle of refraction. Hence, 

 in rays falling near the perpendicular, the deviation will be about 3 times the 

 inclination of the surfaces ; and if the angles of incidence be 50", 6o°, 70°, 80°, 

 or 85°, the deviations of the reflected rays will be about 4, 5, 7, 13, or 26 times 

 the inclination of the surfaces respectively. Had the deviation been the same at 

 all incidences of the rays on the index glass, no error would have been produced 

 in the obser\'ation ; because the course of the ray would have been equally 

 affected in the adjustment of the instrument, as in the observation. But, from 

 what has been just laid down, this is far from being the case, the deviation 

 Increasing according to the obliquity with which the rays fall on the index glass ; 

 so that in very oblique incidences of the rays, such as happen in measuring a 

 large angle by the fore observation, or a small angle by the back observation, the 

 least defect in the parallelism of the planes of the two surfaces of the index glass, 

 may produce a sensible error in the observation. 



What is here said, only takes place in the fullest extent, if the thickest or 

 thinnest edge of the index glass, or, to express the same thing in other words, 

 the common section of the planes of the surfaces of the index glass, stand 

 perpendicular to the plane of the quadrant ; but, if the common section of the 

 plane be inclined to the plane of the quadrant, the error arising from the defect 

 of the parallelism of the surfaces, will be lessened, in the proportion of the sine 

 of the inclination to the radius ; so that at last, when the common section 

 becomes parallel to the plane of the quadrant, the error entirely vanishes. For 



