MICROSCOPICAL PETROGRA PHY 485 
in which the quantitative element enters. It is a highly developed 
system and impressive to most of us but ‘‘Huxley warned us that 
the perfection of our mathematical mill is no guaranty of the 
quality of the grist.”* If we put into our mill loose data with large 
probable errors we cannot expect our final results to be more than 
first approximations. The more accurate the initial data the more 
accurate and satisfactory the final result. A statement cast in 
mathematical form does not prove that it is correct even though 
the mathematics be rigidly true. Mathematics, because it is 
logical and concise, is often used to express, in general form, rela- 
tions the exact numerical values of which are not definitely known— 
such values being then represented by appropriate symbols. For 
purposes of generalization and the framing of a theory by the logical 
grouping of observed facts into a simple co-ordinated system, 
mathematics is invaluable because it serves to express in a single 
sentence the results and essentials of a whole course of reasoning. 
The more accurate the facts and results thus used, the greater 
the degree of probability for correct generalization and the easier 
the process of such generalization. The non-mathematical reader 
may examine the premises on which any mathematical argument 
is based and then use his common-sense in testing the conclusions. 
Quantitative work requires more time than qualitative work 
but petrography has now reached a stage where quantitative work 
is required. To the observer accustomed only to rapid qualitative 
methods, quantitative methods necessarily seem slow and irksome 
and not yielding of immediate results, and he may even be tempted 
to question whether such methods are really worth the while and 
repay the energy and time which must be put into them. But in 
petrography the qualitative reconnaissance period has passed 
and it is no longer permissible in good work to ignore the quanti- 
tative element altogether. It is only by the accumulation of 
precise data that many of the large problems of petrology will be 
solved, and until then the solutions will remain matters of opinion 
supported more or less by a slender foundation of fact. Never 
before has the need for exact evidence, both from the field and 
t Extract from address by R. S. Woodward, ‘‘On the Mathematical Theories of 
the Earth,” Proc. Amer. Ass. Adv. Sci., 1889, p. 62. 
