Vol 4 Iss. 5

Difficulties in crystallographic practice include difficulties in compiling tables of crystal chemical analysis and the actual research techniques. The author formulates the course of operations necessary for a complete geometric study of crystals of a single substance, assuming that the goniometer has been calibrated and satisfies the conditions stated above (see the article). Before each individual series of measurements, the zero setting should be established. The first measurement, leading to the construction of a stereographic projection diagram, serves for an initial examination of the crystal and for the selection of the principal faces. If the substance has already been described, then often this first measurement makes it possible to derive the correct orientation and the corresponding symbol of the complex, and therefore toidentify the substance from the tables.

It has long been obsered that certain physical and chemical processes involving sulfur-containing substances produce colors of violet, indigo-blue, blue and green, some of which are remarkable in their beauty. These color reactions are historically closely linked to hypothetical modifications of elemental sulfur; black sulfur, which is translucent blue in thin layers, Magnus's sulfur and Wöhler's blue sulfur. The presence of these modifications in ultramarine was explained by a number of scientists as the reason for its diverse colors.

If we set a type of symmetry and, in accordance with it, place equal spheres on a given sphere in layers according to the distance of their centers from the center of the given sphere and, in such a way that these spheres fit into the recesses between the previous spheres and form a regular set, then the number of spheres in the layer will be quite definite, namely, it will be equal to the multiplicity of symmetry in the general case, when the direction of the radius vector of each such sphere (starting from the center of the initial sphere) will be common (that is, it will neither coincides with the axes of symmetry, nor lie in the planes of symmetry), and will be a definite divisor of this number in particular cases. I shall consider three sets of spheres of the hexakisoctahedral type of symmetry, corresponding respectively to the three systems of parallelohedra possible in this case: triparallelohedra, hexaparallelohedra and heptaparallelohedr, as well as a set of dihexonal-bipyramidal type of symmetry (and a the system of tetraparallelohedra).

Imaginary circles play a very important role in solving geometric and crystallographic problems. Imaginary spheres also play a significant role. However, in solving certain problems related to some of the simplest sets of imaginary circles and spheres, as problems of an elementary geometric nature, difficulties arise that are easily resolved precisely due to the simple properties of these sets. The article discusses possible ways to solve these problems.

If, taking the vertex of the cone as the center of the sphere, we intersect it with the surface of this sphere, then the cone is replaced by a spherical conoprima, which is why the the stated problem is reduced to recognizing the classes of conoprima on the sphere. An analogous problem of determining the classes of planar conoprimes—is solved by determining the type of involution of the points of the conoprima on the straight line at infinity or the involution of rays at its center. The article derives new classes of conoprimas and a method for recognizing them.

The numerous and useful results obtained from the construction of the diagram of spherical conoprimas prompted me to undertake, as a simpler case, the construction of the diagram ofplanar conoprimas. Of course, in both cases the difference is enormous. There we are dealing with the second(-order) of conoprimas; here, only with a prima, since the totality of all similar conoprimas must be regarded as a single entity. In that case, each conoprima is characterized by the angular magnitude of its two axes, which are always real; here, only the principal (major) axis is always real, while the minor axis in the case of hyperbolas is an imaginary axis. The diagram is based on uniting all similar conoprimes into one. But in the composition of hyperbolas there is a striking exception with regard to similarity, namely the limiting case of hyperbolas with equal angles between the asymptotes, that is, the pair of asymptotes themselves, considered as a hyperbola, cannot be said to be similar to all the others. For this reason, the diagram does not include the special hyperbolas consisting of a pair of rays.

The author has constructed the appended diagram, using, as for the diagram of spherical conoprimas, a stereographic net, excluding small circles from the latter. The diagram of conosecunds, as well as the diagram of planar conoprimas, is based on the principle of similarities, that is, all similar conosecunds are taken as one. The author regards the main purpose of the diagram is to determine, from the ratio of the three principal axes of a conosecund, the three conoprimas which are formed in the conosecund in the three planes of its symmetry.

The formulas of spherical tetragonometry are considered, which are also applicable to plane tetragonometry. For the practical purposes of crystal-chemical analysis, the graphical techniques used are quite sufficient, despite the inaccuracies associated with them. But over time, as the body of material expands, the need for replacement of roughly obtained firures with more precise ones will be felt more and more, in many cases this will reduce the increasingly complex labor of searching tables for a substance identified by its symbol of the complex. A closer examination of the problem at hand shows that it is not just a matter of solving spherical triangles from three given angles, which is precisely what spherical trigonometry covers, but rather that here we have the opportunity to calculate spherical elements, obtained in an indefinite number by constructing from four given points, and to find for each such element the corresponding formula, expressing it even when the positions of the four fundamental points are arbitrarily varied.

Experience has shown that, at present, the approximate numbers that are obtained using relatively crude graphical techniques are quite sufficient for the individual characterisation of each substance, that is, for crystal-chemical analysis. But as the number of crystallographically described new substances accumulates, and such accumulation is proceeding at an accelerating pace a time must eventually come when greater precision will be required in expressing the results of measurements.

It is precisely to this problem that Steiner devoted his famous treatise on linear primas of conoprimes (Kegelschnittbüschel), and it is precisely there that he presented it with such exhaustive completeness that absolutely nothing could be added to it, had he not from the outset, restricted his problem real conoprimas; among the given data there may also be imaginary conoprimas, even if in essence they are only ellipses, since imaginary hyperbolas are equivalent to real hyperbolas with the same asymptotes, the so-called conjugate ones.

At the very foundations of the new geometry lies the concept of involution, and two cases are distinguished: involution with a pair of real (hyperbolic) and a pair of imaginary (elliptical) double elements. In particular, just as for points on a line we can pass from one type of involution to another, if we invert one of the the systems of points constituting the involution so that the points that were self-conjugate (double) become conjugate to one another, so in the involution in the plane (a polar system), the conoprima that determines the involution becomes imaginary (see the article). The distinction between a real and an imaginary conoprima, as well as between a real and an imaginary conosecund, is elucidated. The example illustrating polar relations for all imaginary conoprimas and conosecunds is presented.

By a strange coincidence, not a single geometer, as far as I know, has ever considered the system of spheroprimas of rays, whereas the system of spheroprimas of points was one of the first geometric systems to be established, except for the system of spheroprimas, both of points and of rays, which was so far ahead of its time, and was derived by Mr. Steiner. At the foundation of the construction of the system lies a linear prima undertake the construction of a linear prima of spheroprimas of rays.