THE ANGULAR MICROMETER 
395 
lucida and measuring the figure drawn. This method is open to many 
objections; tediousness and difficulty of operation would prevent the 
majority of workers from adopting it. 
The angular micrometer was designed to eliminate or greatly reduce 
the errors inherent in the instruments and methods referred to above. 
The mechanism for magnifying the movement of the pointer-line is 
rigid and of a nature to reduce mechanical errors of construction to a 
minimum. These errors are either negligible or such as to admit 
of ready detection and calculation. The parts subject to wear are 
only those whose abrasion with ordinary use will not affect the accuracy 
of the instrument. The pointer line moves so that the datum point 
selected can always be easily identified and the graduations are 
accurate beyond the severest demands. 
The principle upon which the micrometer is based is the trigo- 
nometrical calculation of the chord of a circle. The equation for this 
calculation is x = 2r-sin(^/2), in which r represents the radius of the 
circle and 6 the angle subtended by the chord, x, whose length is to be 
calculated. The apparatus by which d is determined for the chord 
in question (the distance between two points on a microscopic object) 
is shown in figure i. In A the micrometer is shown from above, in 
place on the microscope; in B, from the side. The instrument con- 
sists of three essential parts, a, b, and c. The plane of the arm a is 
perpendicular to the axis of the microscope. At the outer edge of 
this arm is a double vernier, d, whose center is on the axis of the 
microscope. Rigidity is secured by the brace, g, and the wide collar, 
k, which is split lengthwise and secured by two sets of screws attached 
similarly to those for the arm b {h-h, figure 2, A). This arm, b, is 
attached by a split collar, secured with screws, h-h, to the ocular of 
the microscope and bears at its outer edge a protractor, /, which is 
graduated to half-degrees and may be read to minutes of arc with 
the aid of the vernier. In the instrument which has been constructed 
the radius of the protractor circle is 17.5 cm., so that readings may be 
readily made with the naked eye. A glass plate, c, is placed in the 
ocular on the interior focal plane and bears a short photographed line 
which is tangent to some one of the imaginary circles whose centers 
are on the axis of the microscope. The point at the tip of this fine 
moves, as the ocular is turned in its sleeve, around the circumference 
of a circle whose center is on a line which passes through the centers 
of the protractor and vernier and is perpendicular to their planes. 
