8 INTRODUCTION. 
without measurement, while for the second class (cleavage angles, extinction 
angles, optic axial angle, refractive indices, and birefringence) numerical values 
obtained by actual measurement are required. The properties included in 
the first group are used as a general rule only in an essentially qualitative 
way in microscopic work and can usually be recognized at a glance. For 
these the ordinary petrographic microscope suffices. But in the second 
group, the quantitative element predominates, and, as such, demands accu- 
rate, precise measurements. In ordinary petrographic work, however, these 
properties are only very roughly measured and are then expressed in general 
terms, as optic axial angle, large; birefringence, strong, etc., actual numerical 
data being rarely given. But with the increased knowledge of rocks and 
minerals, the demand for data which are precise and quantitative in char- 
acter, rather than qualitative, has become more imperative, with the result 
that at present a thorough petrographic investigation should contain accu- 
rately determined optical constants of each of the rock-forming minerals ex- 
amined and, in critical instances, the probable error of each determination. 
This passage from qualitative to quantitative work implies consequences 
of profound importance; an additional burden is placed on the working 
geologist and the time and energy required for the investigation of a par- 
ticular problem are much greater under the present regime than formerly ; 
but at the same time this transition indicates that in one phase of geology, at 
least, the step from reconnaissance work to the higher level of precise and 
detailed work is being taken. 
During the past decade the results of comparative and critical studies on 
the relative merits and accuracy of a number of these methods have been 
published at different times by the writer, but there is still lacking a con- 
nected presentation of the entire subject, so far as it has been carried, in 
which the different methods are coordinated and the significance and use- 
fulness of each particular method are made to appear in their proper rela- 
tions to other available methods. 
In microscopic work with fine-grained preparations, where the average 
diameters of component grains are expressed in hundredths and often 
thousandths of millimeters, the different methods are frequently taxed to 
the limit, and all available resources must be called into play to obtain 
quantitative results of even a fair order of accuracy. In such investigations 
however, the quantitative element is essential and methods must be applied 
which are capable of furnishing quantitative data of a known degree of 
accuracy. It may be stated, as a result of experience, that on clear, indi- 
vidual grains measuring o.oi to 0.03 mm. in diameter, all the optic proper- 
ties ordinarily employed in the petrographic-microscopic investigation of 
minerals in the thin section can be determined with a satisfactory degree of 
accuracy. 
In the following pages the different methods best adapted for the micro- 
scopic examination of fine-grained and artificial preparations will be con- 
sidered with special reference to their degree of accuracy and range of 
general application. 
Experience has shown that in the determination of any one of the optical 
constants, a particular method or device may be satisfactory under certain 
conditions, but less so and even worthless under other conditions. For this 
