If we divide a cube by three mutually perpendicular central planes into eight small cubes, we obtain a periodic operation that we can continue infinitely in both directions. In this process, the volume of the cube decreases (or increases) by a factor of eight, that is, two cubed. But this period is very simply divided into three smaller periods with a decrease (or increase) in volume by a factor of two. In this periodicity, a special role is played by A) the centers of the faces of the original cube and B) the centers of the cubes of the lower period.

In the article "Results of the First Stage of an Experimental Investigation of the Structure of Crystals," the term *basic parallelohedron* denotes that smallest parallelohedron with marked arrangements of atoms within it, from which the entire system of atoms is derived by means of the symmetry elements of the bond. In the system of atoms, therefore, there are several different orientations of the basic parallelohedra with their atoms, namely according to the degree of symmetry of the bond (see the article).

Direct experience shows that the phenomenon of crystallization is a highly complex one, in which various factors play a decisive role. Although, according to Steno's law, the angles between corresponding faces in crystals of the same substance are constant, the appearance of the forms that condition crystallization is far from being completely constant. At first glance, something even contrary seems to be the case. From one and the same substance, one can obtain crystals with a very limited number of pairs of faces, in exceptional cases not even reaching three, and one can also obtain crystals with a very rich combination. The forms that make up the combination also vary greatly in their development, and those very forms which in one case are predominant in terms of their size, in other cases recede into the background or are even completely suppressed by other forms.

In establishing the structure, we were guided even earlier not by the complete sum of combinations that a given crystal exhibits, but only by a small number of the most important pairs of faces. In this respect, as we see, no change has occurred. Even now, we must first of all note these most important forms, but previously we calculated the density of these forms, since, after all, the sequence of faces by importance was still the basis. Now this calculation is no longer necessary, and the matter is significantly simplified, because it only remains to determine, based on these forms, which of the three belts these forms fall into.

The initial material was combustible shale collected during prospecting work by N. F. Pogrebov. After the distillation of illuminating gas, a tar remains, which served as the material for this study. During the distillation of this material in a water bath, crystalline, very thin plates of the distilled organic substance deposit in the tube, with inclusions of another substance having a significantly higher refraction and apparently capable of very good crystallization. This distillation was carried out by Mr. Valchis.

One theorem states that the distance from the gnomonostereographic projection to the linear projection in a certain plane is equal to the distance from the latter to the vanishing point of the rays. The proof reduces to the fact that the vanishing point of the rays Z, the gnomonostereographic projection P, and the midpoint of the linear projection of the plane O constitute the vertices of an isosceles triangle, having the first two points at its base, and this, in turn, reduces to proving the equality of the angles at the base.

Since on a plane the analogous problem is solved very simply on the basis of Pascal's theorem, it is quite natural that the thoughts of geometers have been persistently directed toward finding a simple solution to the problem for space, and the lack of success has elevated this question to the status of a difficult problem. Pascal's theorem can be formulated in various ways, but, especially from the point of view of modern geometry, for which it served as one of the first foundations, this formulation must be connected with collineation, specifically one that transforms the conoprima determined by five points into itself. The construction based on a theorem analogous to that of Pascal is carried over into all geometric systems, particularly into the system of planes. However, the latter theorem, being of a non-positional character, is carried over only to related systems; for example, it is not carried over to the system of planes, just as the theorem of a non-positional character just cited is not applicable to the system of lines in a plane.

Let us turn to analogous constructions in space, which are a consequence of a theorem analogous to Pascal's theorem. Since the construction based on this theorem involves the construction of two hyperboloids of a linear prima, to which the required conosecund also belongs, and for this it is necessary to construct two hexaprimas, it is clear that the given data may consist of such tangents, together with their points of contact, as are sufficient for the construction of the hexaprimas.To understand why Pascal's theorem, and consequently its analogue, is of fundamental importance, it suffices to point out that these theorems are merely particular expressions of the deepest and most important fundamental theorem of new geometry, according to which, in two projective systems, linear aggregates correspond to linearaggregates, quadratic to quadratic, and, in general, aggregates of the n-th order correspond to aggregates of the same order. Moreover, intersections correspond to intersections, tangencies to tangencies, and involutions to involutions.

In the issue of the journal "The Mineralogical Magazine" (No. 81, September 1915), a note by G. Césaro is published, in which the author derives a formula concerning the numerical property of the set of axes of symmetry located in the planes of symmetry of a polyhedron. From the very title of the article, it is evident that the author was interested not in the application of the formula for determining planes of symmetry—which is quite limited, as it applies only to cases where the planes of symmetry pass through all axes of symmetry (i.e., only to mirror types of symmetry, when symmetric figures can be reproduced in zonohedral mirrors)—but rather in the formula itself with its numerical relationships. However, the author was completely unaware that this very formula had been derived by the author [i.e., the writer of the present text] for a very wide application to zonohedra (and in this sense was called zonohedral) and to paired polyhedra (in which case it is called polar zonohedral).

The goal of these preparations is to enable, in all cases where laboratory crystals are obtained from supersaturated solutions, an optical study to be carried out in parallel with goniometric measurement, with the most precise possible orientation of the axes of the optical ellipsoid. Until recent years, I used homemade preparations, assembling them from offcuts of cover glasses, cut and glued onto a slide in such a way as to create the required wedge-shaped space for crystallization. I have now settled on a relatively simple and, for research purposes, very convenient type, which is fully clarified by the attached plan and section of the microscopic preparation (see the article).

I mean to confine myself only to the simpler conclusions that I use in my lectures (see the article). From this triplicity we deduce the necessity of limitations for the principal numbers of the symbol of a complex, namely: for an octahedral structure, one cannot accept a number greater than 63 1/2°; for a hexahedral structure, only numbers within the range of 45° to 63 1/2° can be accepted; and for a dodecahedral structure, only numbers greater than 45°.

These few examples (see the article) provide highly important indications of the dependence of crystallization on the arrangement of atoms in particles of complex structure. In such particles, it is therefore necessary to distinguish between the central and peripheral parts (in simple particles, e.g., ClNa, this is impossible), and only the latter, by the density of the arrangement of atoms, determines the position of possible faces.

According to the essence of the fundamental law of crystal chemistry, the position of each individual atom is expressed in coordinates (see the article), where both the numerators and the denominators are integers, and moreover, the denominators are necessarily numbers greater than the numerators if the points are located inside the elementary parallelohedron or on its surface. Using the results, we can easily and directly obtain, from the symbols of three points, the symbol of the face passing through them.

(Regarding Bowen's book "The later stages of the evolution of the igneous rocks"). The author of this note has long distinguished, among igneous rocks, between normal ones with more or less strict signs of chemical equilibrium and anomalous ones, which are not subject to any laws of equilibrium or any strict scientific classification, but bear clear signs of a sequential course of phenomena, the sequence of which, thanks to them, can be elucidated. Such rocks can only be described, and from the description, historical and geological dates can be deduced. Now, after the publication of Bowen's important work, one involuntarily inclines to the thought of the very weak representation of normal rocks, and perhaps even their absence; in nature, only approximations to them are represented, which is why they should rather be regarded not as normal, but as ideal.

In the article "Results of the First Stage of Experimental Research on the Structure of Crystals" (in the footnote on p. 361), a principle for determining the structure based on the parallelhedron of the smallest volume is established, which is equivalent to the principle of establishing a system of parallelhedra of the highest order permissible for a given regular system of points. A criterion for the correctness of constructing the parallelhedron can be the test of the possibility of a different, more specific, arrangement of one of the atoms represented in the chemical formula in the smallest number, or, even better, if there are no single atoms, the placement of that special point which occupies a unique position in the parallelhedron.

The first conclusion to be drawn from the observations made is the undeniable connection between the formation of actinolite and the decomposition of plagioclase. This connection is so inseparable that, as mentioned, augite decidedly never comes into contact with plagioclase pseudomorphs anywhere. We are, of course, unaware of the immediate primary cause of the decomposition of plagioclase and augite, and it undoubtedly stems from the still-unknown composition of the mineralized solution that penetrated the rock, which, however, must have contained both CaO (from the decomposition of plagioclase) and K₂O (otherwise muscovite would not have formed), with the first component diffusing from the plagioclase and the second diffusing toward it. And since at the points where the diffusing components exited the boundaries of the plagioclase, we see the transformation of augite into actinolite, and since CaO is the only component originating from the plagioclase whose content in actinolite is greater than in augite, we must conclude that it was precisely this component that was halted and absorbed by the augite, which was also affected by the overall chemical alteration of the rock—more pronounced in the plagioclase than in the augite.

According to Steiner's famous theory, two given involutions of pairs of points on lines in a plane determine an involution on any line in the plane, that is, the complete secund of involution. The determining factor of all these involutions is a linear prima of curves, namely a pencil of conics (Kegelschnittbüschel, according to Steiner), having two pairs of points in common, of which not only one, but both pairs can be imaginary. Each line intersects each curve of the pencil in a pair of points belonging to its involution. Specifically, we can define a collineation by two axes without any involutions. If we call the axes, whose points are the real double points of all involutions, real axes, and the axes of isotropic involutions imaginary axes, then we obtain that every axial collineation can be defined by a pair of axes, real or imaginary (see the article).

If through any point on a quadratic cylinder we draw a secant plane (defining a conoprima k), a tangent plane, and in it some skew line d) (not in the plane of the curve k) and not a generator of the cylinder), and then from each point of this line draw two rays in the diametral plane of the cylinder through the points of the curve k), we obtain a ruled surface of the third order.

Systems of points and rays in the plane are, as is known, not related. However, related to a system of points are the secunds of conoprimas of points and rays having three fixed elements, and among those having two fixed elements, only the conoprimas of points (tetraprimas) are related. Now we will show that the secunds of parabolas of rays with two fixed rays are related to the system of rays in the plane.

The projective relations elucidated in the preceding note would achieve a high degree of clarity if it were possible to establish, by a simple construction, such a correlation between the conoprimas of a secund and the points of the plane that the extra-elements of the one correspond to the extra-points of the other. In the secund of pencils of rays, there is contained a prima of those which are represented by a pair of points (more precisely, by a pair of linear primas of rays). Since in each linear prima there are three such special conoprimas represented (of which one pair may be imaginary), it is perfectly obvious that the prima of special primas of rays is correlative to a curve of the third order.

In the preceding note, we solved the problem of transferring all kinds of constructions carried out in the plane to the system of conoprimas of points and rays; and conversely, we reduced the solutions of all kinds of problems in the secunds of conoprimas of points and rays having three fixed elements to ordinary problems in the plane. The thought naturally arises to extend the solution also to those simplest cases where in the secunds of conoprimas there are only two or only one common element.

In any given conoprima, we can arbitrarily take two groups of points, four in each, and establish a general collinearity based on them (see the article). A cycle can consist of a different number of points, up to infinity. If, for example, a point is self-homologous in the collineation, then the entire cycle consists of a single point; if we have a double homology of points A and A', then the entire cycle reduces to two points, and so on. In the general case, the cycle encompasses a significant number of points, or even an infinite number, and it may happen that all points of the conoprima form part of a single cycle. If the collineation, established in some way from the given points, makes a conosecund self-homologous, then the problem of constructing the points of the latter reduces to a simple problem of collinear constructions.

In the note "Polar Relations of Imaginary Triangles and Tetrahedra," we showed that these relations are identical to those defined by a known imaginary ellipse or ellipsoid, with neither passing through the given points. However, from the principles set forth in that note, it follows that definite polar relations of real triangles and tetrahedra may exist. This work constitutes a natural continuation of the preceding note, but pertains to triangles and tetrahedra taken as real.

Having derived a series of principal aggregates, both positional and non-positional, we can now more clearly define the very concept of such aggregates. We call an aggregate principal if it is completely and uniquely derived from a given number of elements, with all these elements playing an equal role in the construction. If we denote these elements by letters and derive the construction of the aggregate, first introducing elements marked by some letters and then by others, then, if the derived aggregate is principal, we can rearrange the letters with respect to the elements differently, and the construction remains valid provided that we retain the original letters in its procedural order.

Although the preceding article represents something complete, having carried the principal aggregates up to those determined by six elements, it immediately strikes the eye that the highest of them — hexaseconds — do not represent aggregates of the most general character — conosecunds — but only their special varieties, capable of being generated by straight lines, that is, only linear conosecunds. Only such aggregates are completely and uniquely determined by no more than six points and planes. Since it is precisely conosecunds of a general character that establish a polar correlation between points and planes in space, the proof of the theorem in question reduces to the fact that such a correlation can be established in ∞⁹ ways.

A large number of singular points are known, the positions of which are strictly derived for each given triangle. These points are discovered through the study of various properties of triangles, the number of which is very considerable. Despite all the simplicity of such a figure as an ordinary triangle, its study cannot yet be considered exhausted, and over time, although rarely, new properties are discovered. The problem arises to find the point that serves as the center of the linear prima of rays — the polars of points of the circumscribed circle. To solve this problem, it suffices to find the polars of any two points of the circle with respect to the triangle; the point of their intersection is the desired one. It is precisely this that constitutes the new singular point of the triangle referred to in the title.

As is known, these surfaces of the fourth order can be generated by two projective quadratic primas of planes. The theory of these surfaces is expounded in detailed manuals. In the general case, the surface possesses a hexaprima of points at which two rays of the surface intersect. Planes passing through such a pair of rays can be considered tangent planes, and since the plane section of the surface is in the general case a curve of the fourth order, it is clear that if a plane passes through one of the rays of the surface, this curve decomposes into this ray and a curve of the third order, and if a plane passes through a pair of rays, the same curve decomposes already into this pair and a conoprima; hence we see that tangent planes passing through pairs of rays of the surface intersect this surface further in a conoprima.

In the article on linear aggregates of rays, I showed that the system of rays is not an independent system, but that for it one must adopt a parameter in the form of an extra-ray, necessarily entering into the composition of linear aggregates. Then a linear prima is completely and uniquely determined by two, a linear second by three, and a linear tertia by four arbitrary rays. If three arbitrary rays are given, then together with the constant fourth extra-ray we obtain the necessary and sufficient data for determining the rays of a complete linear secund. Since four arbitrary rays are in the general case intersected by a pair of transversals, real or imaginary, it is clear that we can define a linear second only as the aggregate of rays intersecting a given pair of lines a and b. If, consequently, we intersect the lines a and b with three mutually perpendicular rays, then by these are determined a linear straight tetraprima.

By definition, this surface is generated by two homologous quadratic primas of planes. In the general case, this surface possesses a hexaprima of double points, at each of which two rays of the surface intersect. A correlative transformation yields the same surface, since to each point with two intersecting rays, by which a tangent plane is determined, there corresponds correlatively a tangent plane with two rays in it, intersecting at the point of tangency. Consequently, we can also generate such a surface by a correlative path, that is, determine it by two conoprimas with an established projectivity of points.

Tertias of rays, which represent the so-called null systems and are completely and uniquely determined by five arbitrary rays, are usually called linear in view of the fact that in each plane and from each point of space there exists a prima of rays that intersect at a single point and are contained in a single plane; such a prima of rays in a plane or emanating from a single center is called linear. But for a system of rays determined, as a parameter, by a special extra-ray, such primas are no longer linear, since the latter must necessarily contain this extra-ray. Therefore, the null systems themselves do not represent linear tertias in this system, but are quadratic tertias.

A series of successive improvements in the precise representation of the chemical composition of rocks has shown how to find the figurative point of the chemical composition based on given four ratios of oxides. However, the spatial position of this point is determined not by one, but by two projections onto mutually perpendicular planes, following the method of descriptive geometry. Although the highest achievable simplicity was attained in all other respects, the clarity of the representation still suffered somewhat precisely because of the depiction in two projections. In this article, I aim to systematically present the sequence of all operations necessary for graphical representations so that they become clear even to persons who have no conception of either the tetrahedral scheme or the system of vectorial circles.

Yu. V. Wulff kindly drew my attention to a very small but exemplary textbook by Miller, "Tract on Crystallography," published in Cambridge in 1863. This little book is only 86 pages long, but it not only lists and depicts the most important forms in crystallography, but, what is especially characteristic of it, it presents and derives the most important formulas for calculation, and moreover, according to that system original to the author, in which double (anharmonic) ratios predominate. Miller's formulas were the first to introduce the principles of the new geometry into the practice of computational crystallography, although their derivation still relies entirely on the formulas of plane and spherical trigonometry.

P. P. von Weymarn kindly sent me a small bottle containing large crystals, along with a letter: "At the bottom of the bottle, where reagents were discarded when washing glassware, crystals of green and yellow color formed; I am sending you these crystals; perhaps they will be of interest for your crystal-chemical analysis." The crystals of different colors turned out to be of different sizes and different shapes. The yellow crystals are distinctly tabular; the thickness of the plates is nearly half a centimeter, and their largest dimension exceeds two centimeters. The greenish crystals are presented in a more isometric form and are at least four times smaller in linear dimensions.

The general law referred to here is that a crystal precipitating from a solution tends to assume the smallest surface area. This law, which has a simple and well-known theoretical basis, is usually demonstrated by examples of crystallization, or rather recrystallization, requiring a long time, even months, or at least days. I came across a preparation in which this demonstration can last a few seconds; this preparation is Chilean saltpeter (sodium nitrate), the microscopic crystals of which dissolve from breathing in a few seconds and recrystallize in approximately the same time due to evaporation. Thanks to this rapidity, the aforementioned law is, of course, demonstrated just as quickly.

Involutory-collinear transformations belong to the most elementary operations of the new geometry. However, this always refers to the transformation of real geometric figures. The problem of transforming imaginary figures, e.g., imaginary circles, appears not to have been posed and seems incomprehensible. In the study of a system of pairs of rays, this problem presented itself in all its reality in the following form. If two pairs of rays are given, we take them as two pairs of tangents to a parabola, which we easily construct, and thus we find the linear prima of pairs of rays, the centers of which constitute a line, and the pairs of rays themselves are pairs of tangents to the parabola.

One of the very first principles of the new geometry is the theorem according to which projectivity on lines (linear and quadratic) is established by the correspondence of three elements. Therefore, if four arbitrary lines are given on a plane, then each of them, at its intersection with the other three, gives three points, and this is sufficient to establish the projectivity of points on all these lines, because on each of them we have three corresponding points. If spherical aggregates are given partly by real, partly by imaginary conoprimas, then from them it is necessary to construct two linear aggregates of the same degree, for one of which the value of the class of conoprimas must be changed: the real taken as imaginary and vice versa.

The principal classes of hexaprimas, or what are commonly called space curves of the 3rd order, were derived by Seidewitz and are given in the well-known manual by Reye under the names 1) space hyperbola, 2) space ellipse, 3) parabolic hyperbola, and 4) space parabola. This note is the result of the question: is it possible to construct a hexaprima possessing symmetry? The term hexaprima denotes a prima of points that is completely and uniquely determined by six points, and a space curve of the 3rd order is precisely such a curve. We obtain three constructions leading to hexaprimas of three types of symmetry (see the article).

In my previous works, I examined a number of geometric systems whose elements consist of pairs of points. The simplest and most important of these is the system of parallel vectors, then the systems of harmonic segments and vectors, and finally the system of midpoints of harmonic pairs. However, in all of these systems considered, a certain restriction is introduced, either in the form of vectoriality or in the form of a special parameter of the system. Here I intend to consider a system of such elements given without any restriction; that is, I conceive that an element of the system on the plane can be an arbitrary pair of its points, which simultaneously constitutes a segment.

This theorem very simply resolves a problem which can be formulated as follows as a problem of elementary geometry (see the article). Despite all the simplicity of its solution as a problem within the system of circles, it would hardly be solvable on the basis of the theorems of elementary geometry. It is clear that this theorem can be directly transferred to the system of spheres, replacing in it the words "circle" with the words "sphere". For the proof, it suffices to take the center line Q R as the axis of rotation. Thus, in its most general form, the problem of finding the centers of spherothertions of spheres is also solved, a problem which was previously solved by means of formulas.

In previous works, I have developed in detail the system of vectors. But since vectors essentially represent pairs of points, albeit non-equivalent ones (the initial and the terminal), and since rays can be correlated to each point, it is clear that the systems of vectors can be correlative to systems of pairs of rays, which can hardly be called otherwise than vectorial. However, if it is somewhat difficult to reason about vectorial pairs of rays in the plane, then, for the sake of complete clarity, I prefer to establish such a sequence of systems that are included in the composition of all linear primas of this system.

We have come to the conclusion that a system of rays can be introduced into the general series of geometric systems, and we can bring it into a general correlative connection. However, this system no longer belongs to the class of independent systems, but rather to the class of systems restricted by a certain parameter, which we must recognize as some extra-ray, constant for all linear primas, just as we have an analogous system of points with a point as a parameter, included in the composition of all linear primas of that system.

Any aggregate of circles whatsoever will not differ from the collection of circles of the previous system, but will constitute only half of the collection of that system, with the linear primas and seconds of ordinary circles remaining such also for this system; but the linear collections of vectorial circles of the previous system will no longer be such for this system, because the tangent linear primas of the previous system are no longer linear primas of this system. It is easy to prove that in this system aggregates of vectorial circles are altogether absent; indeed, one cannot even postulate them. In fact, if I postulate, for example, a right vectorial circle, then the diametrically opposite one is already a left vectorial circle; we obtain essentially two vectorial circles, by which their linear prima on the sphere is completely and uniquely determined; it is clear that in its presence one cannot postulate a third, arbitrary circle; generally, it would no longer be part of a defined linear prima.

In the article "A Simple and Accurate Representation of Points of Four-Dimensional Space on a Plane by Means of Vectors," not only is the system of vectors on a plane examined in detail, but also the foundations for constructing a linear prima of vectors in space from two given ones are indicated, namely, that this linear prima consists of segments generating a hyperbolic paraboloid, enclosed between two directrices, one of which is the line of initial points and the other the line of terminal points, with the construction itself being achievable by decomposing the two given vectors into components along three coordinate axes and by constructing, from the components, linear primas of parallel vectors; three vectors with a common initial point, but parallel to the coordinate axes, whatever directions we may choose for the latter, are precisely the component vectors of the linear prima; the terminal point of the latter lies on the line of terminal points.

The application of X-rays gave W. L. Bragg (and his father) means which have led to conclusions extremely important for the theory of crystal structure. In part, these conclusions are unexpected, at least in the respect that it was anticipated that the points of regular systems would be seen as the centers of chemical particles, whereas the experiments of the said scientist have led to the conclusion that these are the centers of atoms. Thanks to this, in substances of the simplest chemical composition, special regular systems of points are also obtained, with the centers of symmetry occupied by individual atoms, as if the atoms themselves also possess high symmetry.

This note is prompted primarily by the desire to present Bragg's final conclusion in a more graphic form; and then, in view of the complete originality of this conclusion and its rather sharp divergence from our previous ideas about the structure of particles, a desire arose to determine whether it is possible to reconcile it with them. Reflecting on the arrangement of the atoms, we can easily understand that it is of two kinds. Some atoms occupy the positions of the centers of rhombic dodecahedra, others occupy the position of such four trigonal vertices of a dodecahedron that together belong to a tetrahedron. It is precisely this arrangement that determines the hexakis-tetrahedral type of symmetry, and, although the arrangement of the centers of the some particles corresponds to a dodecahedral structure, the situation is altered by the arrangement of the other atoms.

Strictly speaking, as many projections can be constructed as there are geometric systems of the second degree — that is, an unlimited number — and if I now wish to mention such projections as new crystallographic ones, it is solely because they offer peculiar conveniences for solving certain crystallographic problems, conveniences not provided by other projections. Here I have in mind those projections which are obtained from the linear and gnomonic projections if they are subjected to transformation by inverse radii, which is why they can be called grammacyclic and gnomocyclic, respectively.

In previous works, methods for such a determination were given with the aid of the tables of Sokolov and Artemyev, with the exception of the cases listed in the title, unless the pole corresponding to the twofold axis of symmetry is simultaneously the pole of the (1000) face. In all these cases, the performance of specific shears was assumed in order to determine the density of the faces of the principal zone. In this note, I will show that even for these cases one can manage without shear (see the article).

Even from the preceding note, one can see the importance, in the initial study of crystallography, of deriving special elementary formulas for determining the density of isotropic complexes, both cubic and hypohexagonal. Beginners become acquainted with the technique of determining densities from tables best and most easily precisely on examples of isotropic complexes, since the simple formulas provide an ideal check on the determinations made and immediately acquaint them practically with the degree of accuracy—or rather inaccuracy—of graphical operations.

The known properties of gnomonic projections of trigonaloid crystals prompted me about the presence of the relations mentioned in the title, which seemed to me paradoxical. For the case under consideration, the theory of poles and polars unfolds in its usual form: two points are the poles of two polars and, in turn, determine a line — the polar of the intersection point of these polars. To each vertex of a triangle, the opposite side is polar, etc., and in no case is there a point through which its polar passes, as is the case for imaginary conoprimas of projectivity (see the article).

Pascal's theorem lies at the foundation of the theory of conoprimas, expressing their fundamental property of being completely and uniquely determined by five elements. In its modern generalized form, it can be expressed as follows: if six elements ABCDEF of one linear second belong to one conoprim, then, finding the intersection elements of the linear primas AB with DE, BC with EF, and CD with FA, we obtain that all three belong to one linear prima. (see the article). This expression clearly testifies to the deep organic connection of each sixth element with the five remaining ones that determine the conoprima. A simpler analogue of this theorem can be found in the well-known theorems expressing the fundamental properties of spheroprims and spheroseconds.

The content of this note is a direct consequence of the preceding one. It presents a theorem that makes it possible to construct a hexasecond from seven arbitrary points. Since a hexasecond is a positional figure and is correlatively transferred to all geometric systems, the construction of a hexasecond of planes from seven given ones is implied by itself. But if only six planes are given, then the extra-plane is always at our disposal as the seventh and is not counted, being unique in its kind.

If we consider only the plane on one side and the homological hyperboloid on the other, we can still recognize the kinship of these linear secunds of points, because the points at infinity of the first system, and consequently their entire linear prima, are homologous to the points of the line of intersection of the hyperboloid with the plane of involution, and consequently to this entire line as the linear prima of extra-elements. From this, in particular, it follows that if three arbitrary points are given in the second system, the spheroprima determined by them is easily obtained as follows: we project these three points through center Z onto the plane, construct a circle through them, and transfer the points of the latter by inverse projection onto the parabolic hyperboloid. The center of this spheroprima is projected in the same way. It is clear that this center on the hyperboloid, with respect to the spheroprima, is the pole of the aforementioned line on the hyperboloid.

When beginning the goniometric study of a crystal, the researcher cannot yet foresee how to appropriately orient the crystal on the goniometer so that, upon completion of the work, the symbol of the complex—the primary goal of any goniometric study—can be derived in the simplest way from the obtained diagram. From this it is clear that, in general, when the question of the correct orientation of a crystallographic complex is resolved, it is finally resolved only after completing a series of measurements, deriving the most significant faces, and calculating the planes—it becomes necessary to transform the projection plane, taking the main zone as the circumference of the projection.

I take this opportunity to note, for the thousandth time, the advantages of the method of new geometry, the theorems of which know no exceptions, but always possess perfect generality. This author has already presented the proof of this theorem, which I communicated to him, using the method of new geometry. It seems to me far more expedient and simpler to formulate A.K. Boldyrev’s theoremas follows (see the article). A.K. Boldyrev's theorem in its generalized form, according to the method of new geometry, reveals one of the interesting properties of spheroprimes of vectorial circles.

With regard to the theory of confocal aggregates, the conclusion drawn shows that the aggregate of surfaces derived from an imaginary hyperbola taken as the focal curve does not represent anything new, and is included among those derived on the basis of a real hyperbola. If we take into account that, in the general case, we have, linked by the principal axis, two focal curves in two mutually perpendicular planes of symmetry, one of which is an ellipse and the other one a hyperbola, and that in the third plane of symmetry the focal curve can be neither an ellipse nor a hyperbola, and, as it now turns out, an imaginary hyperbola, then the only remaining possibility is to admit an imaginary ellipse, thereby completing the derivation of focal curves. In conclusion, we note that involutions can also be derived on the plane at infinity; since from any point three normally conjugate rays are projected onto it, the corresponding projectivity curve is an imaginary circle, and this is the case for any confocal aggregates in space.

In this article, I present, as complete a list as possible of crystals of the cubic system obtained to date. Here we are indeed dealing with a series of substances exceptional in their properties, as the forms of the cubic system are exceptional among all others. Once the list is compiled, this exceptionality in the chemical composition of the substances is striking, if only the possibility of dividing them into the few categories that form the basis of my exposition. Other features of the chemical composition of substances in this series will be discussed at the end of the article. It was necessary for me to compile this list already for the purpose of singling out,from among those described, the crystals that are not amenable to determination by the method of crystal-chemical analysis.

The article raises the question of constructing edges from symbols in crystals for complexes of hypohexonal type — a question that has not yet been raised by anyone. It is resolved, of course, extremely simply and moreover in a manner completely analogous to its resolution for the cubic type. I have shown that in complexes of the hypohexagonal type, the symbols of edges are such that, in the particular case of a hypohexagonal-isotropic complex, the indices of the edges and perpendicular faces are the same, as required by the theory of crystal systems (because in this case, the ellipsoid of the crystal system is a sphere).

In a grammastereographic projection, any plane is projected as an arc of a great circle, that is, an arc passing through two diametrically opposite points of the projection circle. This circle represents one circular section of the cone having its center at the point of convergence of the rays; the other circular section of the same cone is the diametral circle of the sphere in the plane being projected. Apparently, no crystallographer has yet noted that these projecting cones are not cones of a general nature, but are special cones, known as the cones of Pappus, who first noted their simple construction. The two special axes of the projecting cone are perpendiculars to both circular sections, that is, perpendiculars to both the given plane and the projection plane (see the article).

In these Notes (III 287), I devoted a brief article to this subject, in which I confined myself to a complete derivation of the geometric images pertaining thereto. The same derivation, of course, could be carried out by other means, which would necessarity lead to identical results. All conoprimas, for example, can be derived from circles by means of a collinear transformation, but also by the intersection of two projective primas of rays, and it is evident from elementary manuals that the latter example, if not more accurate, is at least more intuitive, and in this sense simpler (see the article). From the brief article mentioned at the beginning, it is clear that the existence of special circles and spheres introduces great disruption into the concepts of the circle that have become established, not merely over centuries, but over millennia. As proven herein, the notions of center and of equal radii must be completely eliminated from this concept.

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.

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.

In the article "Parolleloëder in kanonischer Form und deren eindentige Beziehung zu Raumgittern” I developed the concept of parallelohedra in canonical form or simply canonical paradelohedra. I based this primarily on the property that their derivation from spatial lattices should be unambiguous. However, in this is article I examined only one aspect side of the question, related to the angular relationships of crystalline complexes, which, in accordance with the crystallographic law of limits, bring crystalline complexes in general closer to ideal types. These relations characterize the distribution of angles, which determines the assignment of these types to certain types of syngony, with right angles playing a primary role (see the article).

In the article “Paralleloëder in kanonischer Form und deren eindeutige Beziebung zu Raumgittern” I showed that by performing a monoclinic shift, you can always obtain an identical spatial lattice, and in doing so sometimes you can reduce nonorthogonality, that is why the expression for the probability of correct orientation will increase, even though this causes the form symbols to become more complex. But I did not dwell in detail on the criterion for exactly at which shift the lattice remains identical. This is the explanation I want to make in this note.

The more processed material is accumulated according to the calculation of the correct orientation, the more pronounced is the need to limit this calculation to the minimum number of the most important faces. Laboratory crystals are most often exhibit a minimal number of developed forms. This was confirmed in the case of barite. Despite their appearance, which is almost no different from the usual appearance of many natural barite crystals, their combination is minimal and through this the first most important faces are particularly emphasized.

This note contains neither any significant innovations nor any systematic solution to graphical issues. However, given the widespread development of graphical solutions that crystallography has received recently, and especially with the introduction of crystal chemical analysis, even the most insignificant simplification or reduction in techniques takes on considerable practical significance. Finally, some rules lead to a reduction in graphic operations in a particular field of applications only in certain, although numerous, cases, it is inconvenient to include them in elementary courses, where only the most general rules should be presented systematically, so that students receive a valid ability to solve problems of all kinds, even if not always in the simplest way.

In view that the tables for this analysis have now been compiled, it was possible to begin applying this scientific discipline. A total of 5 tables have been compiled, namely, all tetragonal crystals are divided into 3 tables according to structures (hexahedral, dodecahedral and octahedral), and in addition, there is one table for hypohexagonal and trigonal crystals (for the latter, structures are marked only for ideal crystals due to their significant accumulation). In all tables, ideal crystals are highlighted in a special column, and it is for them that we have the densest arrangement of points, which is why, bearing in mind the inevitable inaccuracies in the graphically obtained constants, for such crystals in particular we will have to compare the largest series of crystals.

Enigmatic faces can practically be viewed as irrational, not allowed by the fundamental laws of crystallography. The assumption of their irrationality is further confirmed by their overgrowth upon contact with an important face of the complex, wetted by a saturated solution of the substance, whereby a part of the latter face is subject to dissolution. In addition to this characteristic, enigmatic faces are distinguished by their singularity and non-repetitiveness.

This amendment is made by me to the note “An interesting crystal of apatite, a companion of neptunite from California.” (Zapiski G.I. II 253) based on a written indication from Dr. Slavik of Prague, who noted my error based on data from my own description, primarily pleochroism. Having tested the hardness, which turned out to be slightly higher than that of orthoclase, I can now state with full conviction that an error occurred.

In the last article “Chemical relations of rocks and their graphical representation” I focused on the method of tetrahedral representation , which I had proposed earlier, as the most perfect and simple method. At this moment, with the intentionof further simplifying the same method, I once again focus on it as the most perfect. Thus, the purpose of this note is neither to revisit the theoretical issues examined using this method in the aforementioned article nor to introduce any changes to its applications. Everything elaborated in this article remains equally valid to me at present. Now, I only mean to demonstrate that the method proposed then can be applied in a wide variety of forms and to select exactly the one associated with the simplest operations.

If we are given a triangle ABC and define its orthocenter D in it (that is, the common point of intersection of the perpendiculars from its vertices to the opposite sides), then ABCD can be taken as a complete quadrangle with pairs of opposite sides AB and CD, BC and DA and CA and BD . Drawing a circle through the bases (a₁, b₁, c₁) of the perpendiculars on the sides of the triangle, we obtain a Feuerbach circle, which, in addition to these three points, will pass through six more midpoints of the just listed sides of the complete quadrilateral, that is, points a₁, b₁, c₁, a'₁, b'₁, c'₁.

In my article on the system of spheres, I outlined their linear and spherical aggregates, the collinear and reciprocal transformations of these aggregates, but did not address at all the special circles present in each of their linear primas. It was only implied that among all the circles of such a prima there is a circle of infinitely large radius, and such a circle is a straight line constituting the radical axis of the prima. As if in contrast to this, in the scientific literature, starting with Shteiner, it is implied that in the linear prima of circles, the special element is not a single straight line, but a pair of straight lines, one of which is the radical axis, and the other is the line at infinity; but I am not aware of any work that specifically analyzes the question of special circles. With this short article I intend to fill this gap.

Let us consider systems of second-order curves (conoprimas). In a system of conoprimas of points, circles can be taken as extra elements, because these elements themselves constitute a special system, and, at the same time, any curve together with a circle defines a linear prima. But in general, such an extra element does not exist in a linear prima, only in a linear second. However, one can form a linear second from the linear prima of ordinary (not vectorial) circles and some other conoprime. Such a linear second, however, will already be a special one, and must therefore be regarded as a particular system, and such a system will be related to a system of points on a plane, with the points at infinity of the latter being projectively correlated in a special way with the circles of the former. Also, if we compose a linear third from any linear second of circles and some other conoprima, then such a system will be related to a system of points in space. But all these will be particular , special systems of conoprimas of points.

In the "Annual of Geology and Mineralogy of Russia", in the article on “Crystallization in a solid medium,” I have already described an experiment on the transformation of a highly hydrated magnesium sulfate, which forms directly upon evaporation of the solution, into heptahydrate. In that account, I noted that the rapidly growing needles and fibers of the heptahydrate propagate at a seemingly equal rate both in the free solution and when penetrating the crystals of the higher hydrate (namely, MgSO₄ 12 aq). It must be assumed that such an extreme slowdown in the process in extra-thin layers occurs under the influence of partial capillary attractive forces of between the walls of the wedge-shaped space and its contents.

It is clear that the complete aggregate, that is, the quint of conoprimas, possesses the highest possible symmetry, that is, circular symmetry. The symmetry of quarts is completely determined by the symmetry of one conoprima, because the symmetry is derived from it is completely and unambiguously. Therefore, in the general case, such a aggregate has a twofoldaxis of symmetry and two perpendicular planes of symmetry (orthorhombic type of symmetry in the plane). In the particular case of the parabola, only one plane of symmetry remains (the hemiorthorhombic type of symmetry). The circle possesses absolutely exceptional symmetry, and therefore there exist linear quarts that exhibit circular symmetry. From this we conclude that if one takes an arbitrary conoprima and a pentad axis of symmetry, from which five equal elements are derived to define a linear quart, the resulting a quart will possess circular symmetry. All curves contained in it, are in every orientation, arranged in continuous circles of equal elements.

If it turned out that by choosing one linear prima in one diametral second and then arbitrarily another linear prima in an arbitrary other second, and thus constructing an infinite multitude of hyperboloids, we obtain that the entire aggregate of such hyperboloids is contained within a single third, which itself lies within a single linear quart, then we would be dealing with an entity representing a generalization of the concept of a hyperboloid; such a hyperboloid we could call a hyperboloid of the 4th degree system. The special third, possessing circular symmetry, which was just indicated in the article (“Symmetry of linear sets of conoprimas”) is precisely such a generalized hyperboloid in the system of conoprimes. Since in this system, which addresses an entirely different topic, it would be inappropriate to dwell on the consideration of this issue in all its details, this note has been dedicated specifically for this purpose.

As is well known, the French mathematician Dupin used the name "cyclides" to refer to certain curious surfaces which can be defined as surfaces enveloped by the totality of all spheres tangent to three given spheres. These surfaces are extraordinarily are distinguished by many simple properties inherent to them, studied both by the author himself and by several other mathematicians. They possess two special axes, and if a plane is rotated about these axes, it will intersect the surface in a continuous series of circles, consequently, this surface can be conceived as the trace of a circle moving according to a specific law, being at every point perpendicular to all circles of another such system. All the properties of cyclides are set forth in my manual “New Geometry as the Basis for Drawing” . Therein, however, a special cyclide possessing extremely interesting properties was also derived.

There is absolutely no indication that the former mining operations have ceased and that the ore-bearing conditions, which are evident here in any case, have disappeared. Admittedly, there is no reason to claim that we will certainly encounter rich ore deposits here; but it must be said that everywhere and even in other places, no matter how favorable the observed conditions may be, it is risky to make positive statements, but given the conditions depicted on the geological map, it is more likely to assume favorable rather than unfavorable results. The Nikolo-Podgornyi mine occupies a very special position. From a geological point of view, it is one of the most interesting points in the area, both in terms of the distinctiveness in the development of the rocks, their complete exclusivity among others, and in terms of the unexpectedness and novelty of the rocks themselves and the geological conditions in which they were formed.

Taking into account the immeasurably greater simplicity of the method of new geometry as a method of mental construction (without the help of any auxiliary complex adjustments), one does not need to be a prophet to foresee that modern geometric analysis, in pursuit of its goal, will supplant algebraic analysis, and the role of the latter will be reduced to such symbolic a expression of the conclusions of geometrical analysis (which is necessary in order to replace the essentially imprecise implementation of geometric constructions in practical applications with precise calculations and calculations), which makes it possible to express results in exact numbers.

A collineation with an imaginary involution can also be characterized by two such self-collinear rays, of which one is infinitely distant in the horizontal plane, and the other is vertical. Although these two rays are indeed self-collinear, and are in no way axes of collineation with a real involution, as two special rays, characterizing the symmetry of the system, we could conditionally call them the axes of imaginary collineation (a conventional abbreviation for collineation with an imaginary involution). We see that these systems possess a center, three double axes of symmetry passing through it, and three planes of symmetry, each passing through a pair of these axes.

This includes, on the one hand, the substance obtained by Anschutz u. Beckerhoff as the benzoyl derivative of amylphenol, and the benzoyl derivative from tertiary amylphenol, the crystals of which were described by Hartmann, and on the other hand, the substance obtained by the same chemists, benzoyl ester of tertiary-amylphenol, the crystals of which were described by Schwanke.

The question posed is so elementary that, it would seem, its solution should be found in the most elementary textbooks. However, this has not happened, and in the most comprehensive manual available - Reye, 'Geometrie der Lage', in Chapter 3 of Volume II, which specifically treats the perspective position of linear seconds, only the conditions under which two linear primas are considered (as usual, only two systems are considered: a system of points and a system of planes) are in perspective position. Therefore, I deem it useful to consider this issue in its general form.

If it is impossible to uniquely determine an infinite set of rays from two arbitrarily given rays, then this can be achieved based on three arbitrarily given rays. It is well known from elementary textbooks that three arbitrarily given, non-intersecting straight lines, can completely and uniquely determine a certain one-sheeted hyperboloid. Since this curved surface of the second order does not consist of one, but two systems of non-intersecting lines, it is clear that only from one of them, which includes the three given lines, can be determined directly by the three lines, and then it is logically inevitable to also accept the other set which occupies a position in space identical to the first system, i.e., the surface of a hyperboloid of one sheet.

The plane passing through the polar line a and the point, has as its zero point the one at which the polar intersects with the zero plane of the point A. The straight line connecting this point B with point A, like a polar, has a point on the polar a as its pole, and these two points form a conjugate pair on this polar. Each plane, simultaneously tangent to two conosecunds of such a prima, has as its polar a straight line connecting the two points of tangency. If the plane is simultaneously tangent to more than two conosecunds, then it is tangent to all conosecunds of the linear prima, which in this case have one common point of tangency with it and with each other. The zero system is polar with respect to linear primas of conosecunds, just as an ordinary polar system follows from a single conosecund.

Recently I noted the likelihood of the identity of two substances obtained by Anschütz and described, respectively, one as Benzoylderivat des Amylphenols and the other as Benzoylparatertiaramylphenol. After the publication of that note, I received a kind letter from Prof. Groth, who had inquired about this matter in writing from Prof. Anschutz and had received an answer confirming the actual identity of these two substances, despite the enormous differences in the crystallographic constants attributed to their crystals by Messrs. Hartmann and Schwantke. Now, I have found in my old notes an indication of the same ratio between two substances, which have been assigned very different chemical formulas, but which in crystallographic terms turn out to be very close.

The geometric constants, even for a triclinic crystal, are five angles, the values of which can be easily determined by direct measurement on a universal goniometer, and then no preliminary calculations are needed; using these same five angular values with the help of a fundamental formula, the determination of any face, not only measured, but any possible one, given by the indices of the symbol, is produced, as mentioned, by simple additions and subtractions, but with the exception of the faces located in the original belt itself, that is, the belt of faces a and b. And in the textbook (“A Brief Course in Crystallography”) no formula is provided for calculating the angles between the faces in this belt. Here I will present the derivation of this remarkable and extremely simple formula.

The usefulness of compiling such detailed tables becomes clear, which, of course, will be compiled when it is necessary to perform more accurate calculations of lattice face densities or edge densities. In all cases in general, except for triclinic crystals, such calculations will be distinguished by significant simplicity, or more precisely, they not at all will be needed once the tables are compiled. Accuracy can also be enhanced when using the graphical method, if we take as the basis for calculations not the gnomostereographic (or gramma-stereographic) projection, but the gnomonic (or linear) projection.

Reflecting on the state of crystallography as I found it 40 years ago, and comparing it with the present, I find that its transformation during this period is almost deeper than of any other science. The closeness of crystallography and chemistry seems natural: both belong to sciences of an intermediate nature, situated between the exact sciences, the study of which is entirely reduced to the application of mathematical methods, and the descriptive sciences, where mathematical methods are not applied at all. If we compare the role of mathematics in chemistry and in crystallography 40 years ago and now, we will, of course, see that the latter has undergone a much greater transformation.

When dualism was recognized in the New Geometry, only two geometric systems were adopted the system of points and the correlative to it system of planes. This note aims to show the special significance of such a particular case, which is expressed by the following theorem, if we attribute as cognate two such correlative systems, in which the spheroprims of one are correlative to the spheroprims of the other. "Absolutely all graphic constructions, and therefore all theorems, of one cognate system are transferable to the other." In addition, I mean to show that in general case it is possible to establish such systems where for each of its linear seconds, it is possible to reproduce a cognate system of points on the plane.

In the article “The ultimate simplification of zonal calculations and crystallographic calculations in general,” I noted the extreme simplicity that calculations of spherical bipolar coordinates obtain if we take Miller’s formula as the basis. Now I will supplement the formulas derived then with those that relate to the calculation of the cotangents of angles formed with an arbitrarily chosen initial face of the zone (see the note). It can be can concluded that the system of zonal calculations with bipolar coordinates, in its particular simplicity, is applicable without the use of a sequential, recursive procedure of calculation, and directly to faces with arbitrary, complex indices.

Here I mean only what is meant by the word “twin crystals”, and not those regular intergrowths that are caused by mechanical shifts - a case already discussed in my previous works. It seems to me that the experiments of the regular overgrowths of heterogeneous crystals on each other and the conclusions drawn from them by F. Barker completely establish the physical cause of the formation of twins. The most general conclusion from the principle is that the possibility of twinning overgrowth along any plane of a complex, no matter how complex the symbol expressing it may be, for cases of twin faces with a complex symbol, is used in nature under exceptional circumstances.

Vl. Meyer addressed me in a letter, in which he reported on a graphical method for dividing a circle into equal parts using a simple technique (see the article). Of course, theoretically this is not correct; it is impossible even establish the perspectivity of points on a circle and on any straight line in general otherwise than by placing the center of perspectivity on the circle itself. Mr. Meyer's method has no theoretical basis and can only be approximate in relation to a semicircle. This method can indeed be used at least in construction work, for example, when placing pillars and columns around the circumference of a circle.

When solving problems about the correct orientation of crystals (as a basis for crystal-chemical analysis), one constantly has to resort to symbol transformation, while the verification of this solution is associated with the second and further transformations. The task arises to express the final symbols in terms of the original ones. Instead of the well-known transformation formula, in practice we use an abbreviated expression in the form of a determinant (see the article).

The structure of parts of the earth’s crust, such as the modern Ural Range is something so immensely complex that no human imagination is able to comprehend it in all its details, and any attempt in this direction reduces it to a more or less detailed schematic representation. But especially grandiose mining areas conceal not only this complexity of structure, but also so many subsequent alterations and transformations that even a schematic representation of the processes that occurred there encounters barely surmountable difficulties. It is enough to point to the intense activity of metamorphism and weathering to highlight these distinctive features. See the results of the study in the article.

Since, in general, to define a chrystal form four faces are required and sufficient, it is clear that with three given points, no matter what triangle their poles constitute, further development of the form is impossible, and it is absolutely necessary to know the position of some fourth face. In general, the choice of such faces depends on our preference, and I will consider the case when these four points constitute the vertices of two adjacent triangles that share a common side.

A network for the hypohexagonal type was established and it was shown that the numerical law of the development of forms is essentially the same for this network as well, although now the faces are expressed by symbols not of three, but of four numbers; from these four numbers, you can always choose three, including necessarily the first number that will be completely identical to the numbers of the first network. But even if we do not make such a selection, but limit ourselves to three numbers, of which one is in the first place, and we choose the other two arbitrarily (that is, the second with the third, or the second with the fourth, or finally the third with the fourth), the law in question here will still remain valid. Of particular importance is the distribution of even and odd numbers. This law states that of the seven symbols relating to any elementary triangle of the network, one certainly encloses three (that is, all) odd numbers, three enclose two, and three enclose one odd number; all other numbers in the symbols are even.

Nothing is more natural than to generalize the conclusions of the previous note relating to three-digit numbers or the triangular geometric network, to numbers of higher digit counts and, first, to four-digit numbers, and the result is a tetrahedral network. Such a network of numbers has found its application for the chemical tetrahedron in petrography. Reflecting on the mathematical foundations of the construction of a triangular network, we will find that the fundamental theorems remain valid for this network with a corresponding complication of the constructions themselves. This complication lies in the fact that the total number of points associated with an elementary tetrahedron (more precisely, a sphenoid) of any given period is no longer 7 (3 + 3 + 1), but 15: four at the vertices, six midpoints of the edges, four centroids of the faces, and one at the centroid of the tetrahedron itself.

In this article, I severely narrow my task and do not even mention the various geometric systems and their combinations that offer a means to solve it; I focus exclusively on considering a single system of the 4th order on the plane, which most directly corresponds to the essence of the matter. This system is a direct extension of the system of parallel vectors, namely, extended in the sense that vectors, as elements of the system, may also be non-parallel. It is clear that this condition elevates the system by one order, making it precisely a system of the 4th order (corresponding to the geometry of 4-dimensional space).

The crystals of this mineral from the Museum of the Mining Institute had already been systematically described by me in a special article. Recently, the museum has received two new interesting small crystals of this mineral. What is most striking is their extraordinary thinness, reaching 0.1 mm with a planar dimension larger than a square centimetre. With such extreme differences, it is especially instructive to raise the question of the existence of a relationship between form and combination.

A. E. Kupffer, while crushing the rock from San Zenito in California, which contains almost black and rather large neptunite crystals in abundance, isolated, among other things, an excellently formed crystal of a rather deep blue color, developed in combinations typical of a rhombic crystal with four very sharp pyramids (see the article).

Using the hemispheres of potassium alum I had and the corresponding cavities in a large crystal, I wished to test how impurities in the solution, which do not exert a decomposing effect on the solute, influence the process. A likely conclusion is the formation of faces, although of poor quality, with more complex symbols on the spherical surfaces, but of such small size that reflections are generally undetectable, and only some of them begin to become detectable due to impurities that improve crystallization.

Vicinaloids, or vicinal surfaces, are those surfaces that, while forming the true faces of crystals and are very close to planes, are actually not planes, but highly complex and varied curved surfaces. But if, in general, during crystal growth, clustering, that is, the non-parallel staking of particles occurs chaotically, that is, equally in all directions (complete disorder of clustering), then one cannot deny the possibility of the existence of causes that disrupt this complete disorder and produce partial disorder.

The basis of the modern concept of the regularity in the formation of faces lies in the principle of correspondence between the order of importance of faces (manifested both in their very appearance and in their size) and the order of their reticular density. This principle is derived from experience as a statistical law, that is, not as an exact law that manifests always and unconditionally, but as a regularity, which manifests itself in a significant majority of cases. The exceptions that we generally find in experience by no means exclude the idea of the absolute significance of the order of the density of faces, but they indicate that the formation of faces, in addition to this absolute factor, is influenced by other factors, the significance of which has not yet been expressed numerically; and these factors may be quite numerous, since the degree of formation of particular faces is also influenced by various external, partly difficult-to-detect conditions.

I would like to note not only the diversity evident from these tables, important for crystallochemical analysis, but also the importance of the intermediate minimum that is observed for the principle numbers. It should be noted that in isotropic complexes, the distinction between tetragonal-like and trigonal-like disappears and, in general, crystals could be classified as pseudo-cubic. However, if we were to add genuine cubic crystals to them, we would obtain a special cluster of crystals for this particular gap, which would be an unfavorable factor for the analysis, and it turns out that just in this gap, some rarefaction in the distribution naturally occurs.

Amphibole. Pyroxene. Aegirine. Enstatite. Hyperstene. Babingtonite. Lievrit (ilvait). Beryl. Phenakite. Troostite. Willemite. Sphene (titanite). Parisite. Gold. Cerussite. Aragonite. Quartz. Olivine. Neptunite. Cinnabar. Celestine. Barite. For a detailed description of the crystals, see the article.

In the Zapiski G.I. on page 259, regarding the compilation of tables for crystallochemical analysis, some statistical data on this subject were already presented, although these data were based on less material than is currently available to me. A major step has now been taken to compile these tables, namely 3,730 diagrams have been compiled based on the material contained in 42 volumes of the Zeiteschrift für Krystallographie, including both its original articles and abstracts (several hundred of these diagrams have already been verified by the staff and thus their orientations , and therefore their places in the tables, have been confirmed).

Currently, crystallography widely uses graphical techniques to solve its problems. Increasing accuracy is also one of the goals set in the development of graphical operations. As for the design of the new device, we mainly note the much larger dimensions of the device, of which the black hemisphere, as the main working part of the device, has a diameter of about one arshin plus. The second device is a universal three‑axis contact goniometer. It is used for routine measurements by the universal method of crystals so large that they can no longer be mounted on the crystal holder of ordinary goniometers. The third device is designed to facilitate the crystallization process. The principle of its operation consists in the rhythmic heating and cooling of the vessel containing the solution in which crystallization occurs.

We now know that geometric systems can be very numerous and varied, since very diverse geometric images can be taken as elements of systems. To establish any such system, it is necessary to determine the complete set of its elements and prove that from any two elements arbitrarily chosen from it it one can uniquely construct an infinite set of them, such that by replacing the two chosen ones with any two arbitrary other elements included in its composition, we would also uiquely derive the same set, which would constitute the linear prima of the system.

The author concludes that there exists an infinite number of geometric systems of the same order, derived from each given one. Since the conclusion about the possibility of reproducing from any given system another, paired one, is not limited by any conditions and is determined solely by the possibility of the same positional constructions as for all systems, it is clear that it is equally applicable to paired systems. In other words, we can reproduce a new, paired system not only from any geometric systems in general, but on absolutely the same grounds and from each paired system.

Among the remarkable examples of potassium feldspar in the Museum of the Mining Institute there is a very large adularia quadruple from Gotthard St. (Fibia), reproduced in Fig. 1 of the attached table (up to two decimeters in length).

The author came across clear indications (Fig. 3) of the formation of striations in microcline in a thin section from the shores of the White Sea (No. 8, that is, from Gorely Island in the Keretsky Roadstead). I felt obliged to present such an image, which is perfectly clear at 120x magnification.

Since the plane P is only a special case of a second-order curved surface (a conosecund) K, then from this and any other arbitrarily given conosecund their linear prime is uniquely determined. Such will be the aggregate for the determination of which these data are sufficient and can be replaced by any two conosecunds of the same aggregate.

Correlativity is established not only between a system of points and a system of planes, but also between transformations of these systems. Precisely because of this correlative naure this theorem has a dual meaning, so that in its formulation an ordinary lattice can be replaced by a polar one and vice versa. The author considers it necessary to publish this theorem in view of the fact that in crystallography, to determine the symbol of a complex, we perform precisely the operation of shifting the polar lattice using the gnomosteographic projection, while the essence of the change that the ordinary lattice undergoes in this case remained unknown.

I began experiments of this kind back in 1901, when I cut out discs or small rings from thin sections of rock salt and alum, which separated the inner convex from the outer concave spherical line; I introduced a drop of an unsaturated solution into this annular space and covered it with a cover glass, which I sealed with Canada balsam.

The article “Precise representation of points in space on a plane”, solves the problem of such representation in three different elements: vector and ordinary circles, and parallel vectors. It also provides a practical application of representations in parallel vectors of a system of mines. I will now show an essential application of the theory for the representations using vectorial circles of spatial lattices of each crystal studied in structural terms.

We know that from two given points eand e' and a conic section K in the plane, we can generate a second‑order curved surface, if we take one of these points, e,as the center of a second of rays, and the other point, e', as the center of a second of planes, and bring these two seconds in a correlative relationship such that to the ray ea (where a is a point on the conic section plane) will be considered correlative to the plane e'A, where A is the polar of point a with respect to the conic section K. It is known that in such a surface a set of rays and their correlative planes intersect.

No matter how elegant the construction of conic sections using Pascal's theorem may be, it lacks sufficient generality, since it is applicable only for five real points of a curve, and in practical application it is more complex than some other methods.

I consider it useful to note one property of stereographic projection, which, although it does not open up new ways for solving problems, still can contribute to greater accuracy in solving some of them.

Among the deviations from the laws characteristic of real (ideal), that is, completely crystalline-homogeneous individuals, small deviations are often noticed both in the orientation of faces and, in general, in the intergrowth of subindividuals.

Under this general term we mean all those curves and surfaces that are represented in the geometry of harmonic segments. Indeed, in this geometry, to every middle point of a segment there uniquely correspond both endpoints of this segment; consequently, whatever curve is represented in this geometry as a set of endpoints of segments, its points are always grouped in pairs that are harmonic with respect to the main orthogonal sphere O, with the endpoints located on a radius of this sphere.

One of these systems or spherical geometries can be considered generally known, although I am not aware that anyone has categorically noted its complete parallelism or equivalence with the geometry of points on a plane. This geometry on the sphere introduces one limiting condition: it deals only with points on a sphere. Therefore, although its points occupy all three dimensions of space, it is essentially the same geometry of two dimensions (that is, of the second order), as is the corresponding geometry on a plane, in which the limitation consists in considering only points lying on a single plane.

Despite a long line of first-class researchers of this mineral, starting with Brooke who studied this mineral in 1824, its crystallization, specifically, the correct orientation of its crystals, has hitherto encountered significant difficulties. I can now make use for this purpose of an excellent specimen from Alston Moor, available in the Museum of the Mining Institute, from which hundreds of crystals suitable for this purpose could be extracted. I extracted 15 crystals and subjected them to comprehensive measurement on a universal goniometer.

During the fundamental revision and reorganization of the minerals in the Museum of the Mining Institute, currently being carried out by A.E. Kupffer, the opportunity arose to get acquainted in some detail with the figures mentioned in the title, which are sometimes developed with great clarity on a fairly large number of specimens from various deposits of this interesting mineral, which is so abundantly represented in the museum's collection.

Natural crystals are the most challenging object for the method of crystal-chemical analysis, and I must admit that my attempt to assign the proper orientation to all minerals proved to be imperfect in many respects. But it was precisely this imperfection that prompted the further development of criteria for correct orientation, and pointed to the need to re-examine and individually study the crystals of many minerals.

Having specially studied pyroxenes, I came across, among other things, intergrown crystals resembling twin intergrowths. Let me present here the result of one such study, based on whose numerical data the attached diagram was compiled.

The criterion used so far contained a fundamental, principled drawback, which consisted in calculating the reticular density of any given complex as if it were an isotropic complex. Although th criterion with this drawback was used quite consciously, for the sake of simplicity, in view of the complexity of the operation of calculating reticular densities, but, of course, there always was the desire to eliminate it, if only a method could be found to determine this density quite correctly, without resorting to simplifying, but still an erroneous assumption.

I intend to present additional theorems concerning linear aggregates in the geometry of vector spheres and quadratic aggregates of ordinary spheres. For a detailed description, as well as a comparison of megaspheres of vector and ordinary spheres and an overview of the system of parallel vectors, please see the article.