Excerpt from Lectures on the Principle of Symmetry and Its Applications in All Natural Sciences After seriously testing the methods of argument hitherto elaborated, the writer has in many places finally adopted that of Schoenflies, chiefly because in his opinion it offers, from a teaching standpoint, undeniable advantages over the often not less happy and concise ways of treating the problem employed by authors such as Von Fedorow, Wulff, ViolaBarlow, Boldyrew, and others. However, many alterations and extensions have been occasionally made, chiefly with the intention of keeping the deduction as general as possible, even for cases which are of no special crystallographical interest, though doubtless important for biologists. The author is convinced that Mobius definition of symmetrical figures has some logical advantages above the somewhat dualistic definition of Von Fedorow and Schoenflies adopted here; and also, that from a mathematical standpoint, the methods of demonstration of Wulff and Viola, and more especially that of Boldyrew, may perhaps be considered more homogeneous. But he is convinced also that confusion is more readily created in the mind of students of these subjects, when all symmetry-properties are reduced to mere reflections in planes of different functions, than when the axial symmetry is considered as well. Attention is, however, occasionally drawn to several theorems demonstrated by the authors last named and their importance for certain purposes has in such cases been indicated as clearly as possible. About the Publisher Forgotten Books publishes hundreds of thousands of rare and classic books. Find more at www.forgottenbooks.com This book is a reproduction of an important historical work. Forgotten Books uses state-of-the-art technology to digitally reconstruct the work, preserving the original format whilst repairing imperfections present in the aged copy. In rare cases, an imperfection in the original, such as a blemish or missing page, may be replicated in our edition. We do, however, repair the vast majority of imperfections successfully; any imperfections that remain are intentionally left to preserve the state of such historical works.