PREFACE. VU every case likely to occur. The shape in which they are presented has been chosen with the view of shortening, as far as possible, the arithmetical operations they involve in their application. X think it due to Professor Miller to state, that nearly the whole of the assistance which I have been able to render to the composition of this volume consists in such information relating to many of the minerals described as a long acquaint ance with them has enabled me to afford, and in having sup plied such specimens and crystals as it appeared desirable again to examine and measure. In every other respect the treatise has been composed and arranged entirely by Pro fessor Miller. II. J. B.