XI. An enquiry into the variation of angles observed in crystals; especially of potassium-alum and ammonium-alum

All who have been practically engaged in goniometric measurements are well aware that in crystals of one and the same substance small variations of angle are usually encountered even when the crystals are taken at the same time from the same solution. The differences are sometimes so small that they are only appreciable by a goniometer which will read accurately to 1 minute or less, and very often the crystal faces themselves are too irregular or too imperfect to yield sharp images of the collimator slit such that they can be adjusted within, say, half a minute. But even if account be taken only of those faces which are truly plane and yield a perfectly sharp reflection of the collimator slit, the individual variations may amount to 5 or 10 minutes, and the measurements of the same angle made upon, say, 20 good crystals of the same substance, may differ to this extent. I have studied an example from the mineral kingdom in the case of the very beautifully crystallised Proustite, where the angle between perfectly smooth and plane rhombohedron faces varied between 42° 39' 50" and 42° 48' 27", even upon the same crystal.