276 American Quarterly Microscopical Journal. 



about ten feet distance and illuminated by monochromatic 

 (blue) sunlight. This appears, when the adjustments are rightly- 

 made, as an extremely minute blue star in the center of the field. 

 The radial arm is then swung until the star comes to the extreme 

 margin of the field, the adjustment is made as exact as possible 

 with the tangent screw, and the vernier read. The radial arm 

 is then swung till the star comes to the opposite margin of the 

 field, where the same process is repeated. The difference between 

 the two readings is the aperture of the objective for any medium 

 of the same index of refraction as the crown-glass semicircle em- 

 ployed. My apparatus reads to half-minutes, which is closer than 

 is at all necessary, and, indeed, closer than the observations can 

 be accurately made. In fact, after the star comes to the edge of 

 the field it usually begins to fade just before it entirely disappears, 

 and a motion of several minutes is necessary to effect the change. 

 My plan is to adjust the instrument as exactly as possible at the 

 point at which the star begins to fade, and then read to the next 

 lowest sixth of a degree, neglecting the small fractional remainder. 

 The same instrument answers very well to measure the glass angle 

 corresponding to the actual air angle of dry objectives of any 

 power ; or, the semicircle of glass being removed, and the mi- 

 croscope still used as a telescope to view the blue illuminated slit, 

 as before, air angles may be directly read with a degree of pre- 

 cision not attainable when the sector is used in the ordinary way. 



From the angles of aperture measured in the semicircle of 

 crown-glass, it is quite as easy to compute air angles, water angles, 

 glycerin angles, balsam angles, etc., as from the numerical scale 

 of Abbe. It is only necessary to subtract the logarithm of the 

 index of refraction of the rarer medium, in which the aperture is 

 to be expressed, from that of the index of the glass semicircle, 

 and to preserve the difference as a constant, for use whenever the 

 aperture in the selected medium is to be computed from the angle 

 observed with the semicircle. Then, to perform the computation, 

 it will only be necessary to add this constant to the logarithmic 

 sine of half the observed angle, and take from the table of loga- 

 rithmic sines the angle corresponding to the sum, which will be 

 half the angle required. 



It will readily be understood that if the crown-glass semicircle 

 of the apparatus is of precisely the same index of refraction as 

 the crown-glass front of the objective, the rays of light passing 

 into the objective from the semicircle will, after more or less 



