THE ANGULAR MICROMETER 
401 
III. Failure to Graduate Accurately the Vernier and 
Protractor 
Case I : Spacing on either scale unequal. If the protractor and 
vernier are used at different points on their scales and the ratio be- 
tween the length of the spaces on the two scales is not constant, the 
error is readily detected by the failure of the end lines of the vernier 
to cover invariably the same distance on the protractor. Suppose 
the vernier to be so graduated that thirty half-degree spaces on the 
protractor equal twenty-nine spaces on the vernier. A slight dis- 
placement of the twenty-ninth line of the vernier from the prolongation 
of the thirtieth on the protractor would be evident and hence a varia- 
tion of even one minute of arc would be immediately perceived. 
Case 2: Spaces on the protractor not accurately half-degrees. 
This error is impossible of easy detection, but it in no way alters the 
accuracy of the instrument for it is necessary only that each interval 
be equal to every other interval; their absolute magnitude is a matter 
of complete indifference. This is true because all the constants used 
in the calculations depend upon a known magnitude, which is used for 
the calibration of the instrument. 
IV. Failure to Fix the Plane of the Vernier Perpendicular 
TO THE Axis of the Microscope 
(It is only necessary to consider the vernier, since its position 
determines that of the protractor.) 
Case I : The angle between any radius drawn on the surface of 
the vernier-carrier and the axis of the microscope is the same for all 
radii. This is equivalent to shortening the radius of the protractor, 
because the common center of both circles is still on the axis of the 
microscope although in a lower plane. The results of this error are 
the same as for source II. 
Case 2: The angle referred to under case i is not the same: the 
plane of the vernier is tilted from the horizontal. If the tilting is 
great enough to bring about even 0.5 mm. difference in level between 
the highest and the lowest portions of the vernier, it may be readily 
detected by laying a straight edge upon the vernier-arm, when in 
position flush with the open end of the microscope tube. If the 
straight edge is then moved over the surface of the vernier-carrier a 
very slight space between the two will be plainly evident. Suppose, 
