THE VARIATION OF ANGLES OBSERVED IN CRYSTALS. 
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deviations; the crystals of lead nitrate showed a certain regularity in tlie angles; 
they had all grown lying on an octahedron face (ITT); now if tiie obtuse angles made 
by the upper face (111) with the three adjacent faces be compared with the acute 
angle which it makes with tlie three other faces, it is found that the average 
value of the former is less than 70° 31' 44", and that of the latter is greater tlian 
109° 28' IG". Brauns drew the conclusion that the action of gravity is tlie 
distuihin^ cause ; tliis is not perceptible in spinel and alum, Imt does come into 
evidence in the denser substance, lead nitrate. 
The investigations of Brauns are the only precise and direct attempts to determine 
and explain the angular variations of culiic crystals. Tliey lead to the conclusion 
that the planes of the crystal are octahedron faces, distorted by some cause which in 
the case of lead nitrate appears to be the action of gravity. The faces were all 
sufficiently good to yield measurements reliable to 1 minute, and the author makes 
no mention of multiple images, so that they were all presumably single plane 
surfaces, and were, therefore, regarded as not vicinal forms Imt true octahedron 
faces. 
(2.) Ob,tect of the Present Investigation. 
One serious difficulty underlies all attempts to study the problem by the ordinary 
methods of crystal measurement. Where several octahedron crystals, for example, 
are taken from a solution and found to give angles which are not tlie theoretical 
angles of the octahedron, it is impossible to ascertain whetlier any face of the one 
crystal corresponds exactly to any face of another. It occurred to me that the only 
way of studying changes in angle, and of investigating kindred problems, is to 
ascertain how far the variations are constant for one and tlie same crystal during its 
growth. For this purpose it will not suffice to take a crystal out of the solution and 
measure it, and then to re-immerse tlie crystal, allow it to grow, and measure it 
again, for one is ignorant whether any faces of the nucleus are parallel to those 
of the shell by which it becomes enveloped; it is necessary to devise a plan by 
which one and the same crystal can be measured at different times durina; its oTowth 
in the solution. 
I his would further render it possilile to trace the changes in the position of any 
given face as the crystal grows, and, if the changes are due to variations in the 
conditions of equilibrium between crystal and solution, might conceivalily afford 
means for. studying these conditions. Mucli has been wiitten concernino’ internal 
O 
structure of crystalline media, but it must not be forgotten that every point within a 
crystal has at some period been a point on its surface, and that our knowledge of tlie 
structure is very incomplete until the life-history of the crystal has been studied by 
observing the changes that take place at its surface and determining tlie conditions 
of equilibrium which obtain there, 
