From the History of Axonometric Images

V. A. Voskresenskii

(Presented by the scientific seminar of the Department of Descriptive Geometry and Graphics)

Questions concerning the construction of axonometric images and their development occupy a large place in the theory and practice of engineering graphics. Earlier, in 1959–1960, the author published articles concerning the development of this type of projection \[1, 2\].

The present work sets itself the task of giving a more detailed account of the path of the successive improvement of axonometry and of showing examples of its practical use, chiefly in the pre-revolutionary period.

The brief historical surveys contained in courses on descriptive geometry and technical drawing, as well as in certain dissertations on engineering graphics, speak of axonometry in insufficient detail and do not investigate the particular features of its development.

The information on the history of axonometric images that is found in works \[3, 4, 5, 6, 7\] cannot claim to be a profound scientific study of the present question, since it represents a sequential listing of the principal works on axonometry and does not give a proper idea of the successes that have been achieved in this field.

In this connection, the present work aims to fill the existing gap in this question and to show the development, distinctive features, and significance of axonometric projections.

Before speaking about the axonometric method, which arose with the development of machine production and required a combination of visual clarity with convenience in measuring the object being drawn, one should dwell on visual graphic images that served as prerequisites for theoretically grounded, geometrically correct axonometric drawings.

Pre-axonometric images appeared as early as remote antiquity, when people wished to express their thoughts by means of drawings executed in a very primitive manner.

The appearance of the first drawings used in construction is attributed to the seventeenth century; moreover, as I. E. Zabelin \[8\] indicates, they, “although made by hand, by eye, were at that time compiled as occasion arose for each construction.”

N. N. Voronin \[9\] notes that “earliest of all, the drawing as an abstract and scale-reduced image transferred to paper appeared for recording river and overland routes, as well as ‘boundaries,’ ‘frontiers’—the typical geographical map.”

A distinctive feature of the first plans and maps is their visual clarity in depicting terrain, cities, and individual buildings, often from several viewpoints and by different methods of projection, both central and parallel. This mode of representation was transferred from icon-painting art. In [9] it is noted that, in the search for qualified draftsmen, icon painters were approached, apparently on the assumption that drawing up plans of a building or a city meant, above all, giving a visual representation of one object or another. But these images could in no sense be regarded as axonometric in the sense in which we understand them today.

The beginning of the eighteenth century—the beginning of the epoch of the so-called “Petrine transformations”—is remarkable in that Russia, for the first time in its entire history, made an attempt to take its place in trade and industry among the Western European countries that had outstripped it.

The strengthening of the country’s defensive capacity and the construction of a navy required the intensified development of the mining and metallurgical industry, which during this period was being created in the Urals and represented, as V. I. Lenin noted [10], “an original system of industry based on serfdom, which in the eighteenth century served as the basis for the prosperity of the Urals and its predominance not only in Russia, but partly also in Europe... in the epoch of the embryonic development of European capitalism.”

Detailed descriptions, supplied with drawings and illustrations, had already been compiled in the eighteenth century concerning the planning of the Ural factories, their equipment, and technological processes. The most complete among them are: Description of the Siberian Mining Works (1735) [11], and Description with Drawings of Mining Works and Mines in Russia [12], produced in the 1760s. These descriptions contain series of graphic images representing views of factories, workshops, and their equipment. Examining them, we obtain a complete idea of the individual sections of production, its technology, and so forth.

In the series of drawings of 1735 we see that, for almost every drawing made in orthogonal projections, an explanation was made in the form of a visual image, which in most cases, because of its non-scale construction, cannot be classified as axonometric. Figure 2 shows an excerpt from the drawing of a “rolling mill” (Fig. 2), representing a bearing device drawn in parallel projection, the overall dimensions of which were taken from the orthogonal drawing.

Fig. 1

If axonometric axes are constructed, we obtain correct oblique frontal isometry. The angle of inclination of the axis \(OY\) to the axis \(OX\) will be \(40^\circ\). Above this image are drawn a detail of the bearing frame (Fig. 1, c) and the upper liner in central projection (Fig. 1, b), although the latter (as is evident from the schematic construction) is executed incorrectly. It should be noted that, in drawing the bearing device in parallel projection, the exact ratio of the constructive dimensions was not observed. Thus, for example, in Fig. 1 the dimension \(f > n\) (although they should apparently be equal). The same

one may also say this with respect to the orthogonal projection. It is evident that the indicated scale was used for constructing the principal, overall parts of the details, while small sections were taken “by eye,” without definite accuracy.

The same can also be observed in the second “description,” the so-called “Schlatter series,” preserved in only one copy in the archive of the State Geographical Society of the Academy of Sciences of the USSR in Leningrad. However, in character of execution, the quality of the drawings in the latter is higher than in the preceding series (the “Gennin” series).

Fig. 2

Schlatter’s visual representations are for the most part executed in parallel oblique projection, and some of them may be assigned to a parallel oblique frontal projection of the isometric type, since the dimensions along three mutually perpendicular directions correspond to the dimensions on the plan drawn to scale. A shortcoming in individual drawings is the change in the direction of projection when depicting shop pipes. The upper cut of a pipe is shown as the artist was accustomed to seeing it, looking at it from below (whereas the shop is represented in a view from above). If the draftsman had not made such a gross error, the drawing could have been considered entirely axonometric, as, for example, the well-known “perspective view” of the Barnaul plant, executed by hand ...

by the military inventor I. I. Polzunov in 1751 [13] and representing an isometric frontal projection.

Abroad, in the second half of the eighteenth century, the first attempts appeared to give a theoretical justification for the method of parallel projection. It must be said that the practical method of parallel projection for the purpose of obtaining pictorial images found wide application at that time in matters of fortification.

As an example one may cite the work of N. Kurganov [14], published in 1777 on the basis of foreign sources. In it the author gives an explanation of “military perspective (cavaliere),” the rules of which “... are used for taking and drawing views of fortresses” when considering it from an infinite distance. Such perspective is considerably simpler than central perspective, but “quite well represents fortifications, drawings” (p. 176), since it does not distort the size of the objects depicted.

As the study of old drawings and literary sources of that time has shown, “military perspective” was not used in depicting technical details, apparently because individual machines and mechanisms appeared in an unnatural form in relation to the viewer, who was accustomed to viewing usually small objects from the height of normal human stature, whereas large areas are customarily observed from a more elevated point of view, in order to take in the whole site of a fortress, city, or other large engineering structure under construction.

In 1817 a book by Engineer-Colonel A. Maiorov [15] appeared, in which, expounding analytic geometry and the defilading of fortresses, he used oblique frontal images for explanation, in particular for \(1/8\) of an ellipsoid (Fig. 3).

Fig. 3

Fig. 4

In considering questions of the defilading of fortress works, A. Maiorov proposed, in contrast to foreign authors, using the theory of oblique surfaces. Fig. 4 shows a pictorial drawing in frontal oblique projection. The angle formed by the \(y\)-axis with the \(X\)-axis is \(52^\circ\). Here it can be seen that the oblique surface is obtained “from the motion of a straight line passing through all points of the lines \(SS\) and \(AX\) and perpendicular to the \(X\)-axis” (p. 217). This surface is formed by the direction of “enemy shots from the chain of mountains directed at the line \(X\),” which is considered the limit up to which “enemy shots cause damage.” Having constructed such a surface bounding the space struck by fire, one may, relative to it, build the proposed fortification, finding at any place the corresponding “height of the crest of the breastworks.”

It must be said that the pictorial representations in parallel projection used in technical literature could not always be regarded as axonometric, and should be classified as drawings.

The first work that laid the foundation for the theory of axonometry in Russia was an article by Engineer-Colonel A. Kh. Reder, published in the Journal of the Main Administration of Ways of Communication and Public Buildings (1855, No. 1). Against the background of works that already existed abroad, the article mentioned could not have served as something new and original in the study of isometry, especially since, as Professor V. I. Kurdyumov noted, in preparing the article the author used material from Möllinger’s book (1840). Nevertheless, A. Kh. Reder’s merit lies in the fact that he was the first in Russia to publish the necessary rules for constructing projections most convenient for technicians. In 1861, with some additions, A. Kh. Reder’s work appeared as a separate book [16]. The author did not confine himself to an exact translation of Möllinger’s book, but independently considered a number of problems concerning the practical application of isometry, such as the construction of the octahedron, shadows, and curved surfaces. In addition, he compiled a syllabus on isometric projections, later included in courses of descriptive geometry as one of its applications, although A. Kh. Reder himself regarded isometry as a special method of representation. His book was the most complete printed manual on isometric projections until 1885, when similar works by N. I. Makarov and V. I. Kurdyumov appeared.

As for other kinds of axonometry—dimetry and trimetry—the basic formulas determining the values of the distortion coefficients and the angles between the axes were derived for them. The author of these derivations was Professor F. E. Orlov of Moscow University, who taught a course in descriptive geometry at the end of the 1870s. Considering various types of rectangular axonometry, Professor F. E. Orlov expressed the idea that orthogonal projections of an object in engineering and construction drawings are nothing other than a special case of axonometric projections; moreover, if the axonometric plane \(P\) is parallel to the frontal plane of projections, the principal view, or front elevation, is obtained; in the other two cases the side view, or side elevation, and finally the plan are obtained.

A major shortcoming of the first works on isometry is the absence of practical examples of the construction of technical parts, assemblies, or structures. The problems proposed by A. Kh. Reder, N. I. Makarov, F. E. Orlov, and others were reduced to the construction of geometric figures.

As was already mentioned above, even before the appearance of the first theoretical works in Russia on rectangular isometry, parallel frontal and horizontal oblique projections were used practically, although questions of their theory were not considered in technical and instructional literature until the end of the nineteenth century.

In 1896 Professor V. I. Kurdyumov wrote that “on free perspective, or oblique projection, in the Russian language we have only one work: Section III of Issue 2 of A. Makkaveev’s Course of Geometrical Drawing, published in 1890. In this work only one practical aspect of the question was considered—the construction of representations—while the theory of oblique axonometric projections was passed over in silence” [17] (p. XXXVI).

We have found that, even before the appearance of A. Makkaveev’s work, an attempt to give a theoretical justification of the method of parallel oblique projection was made in a short article by A. Frolov [18] (pp. 12–18) in 1884, where he noted that the wide use of oblique projections in practice and in the study of stereometry caused—

calls for the necessity of obligatory knowledge of the scientific foundations of “this kind of design.” A. Frolov emphasizes that “the plane \(P\) of an oblique projection must always be assumed to be vertical.” In this case, straight lines situated parallel to this plane are projected without distortion, while segments of lines perpendicular to the plane \(P\) will be inclined to the horizon. Having proved the dependence of the magnitude of the projection of a segment perpendicular to the plane \(P\) on the direction of projection, the author recommends depicting such straight lines at half their length, inclined to the horizon at an angle of \(30^\circ\). In Fig. 5 the orthogonal projections of the straight line \(AB\) \((ab,\ a'b')\) are given. It is required to construct its oblique projection. \(rs,\ r's'\) is the direction of projection.

Fig. 5

\[ mn = np \cdot \operatorname{ctg}\beta = ab \cdot \operatorname{ctg}\beta. \]

From \(\triangle ABC\) we have

\[ AB=\frac{AC}{\cos\alpha} =\frac{mn}{\cos\alpha} =\frac{ab\cdot \operatorname{ctg}\beta}{\cos\alpha} =ab\,\frac{\operatorname{ctg}\beta}{\cos\alpha} =ab\,\frac{\cos\alpha}{\sin\alpha\cdot\cos\alpha} = \]

\[ =\frac{ab}{\sin\alpha}. \]

Putting

\[ \frac{\operatorname{ctg}\beta}{\cos\alpha}=\frac12 \]

and taking \(\angle \alpha = 30^\circ\), we find

\[ \cos\alpha=\frac12\sqrt{3}, \qquad \operatorname{ctg}\beta=\frac12\cdot\frac{\sqrt{3}}{2} =\frac14\sqrt{3}, \]

\[ \angle \beta = 66^\circ 35'12.4''. \]

In the subsequent drawings A. Frolov assumes that \(\alpha\) and \(\beta\) have these values, and therefore recommends depicting the oblique projection of a segment of a straight line perpendicular to the plane of projections by a segment half as long, inclined to the horizon at an angle of \(30^\circ\).

Further, in § 6, problems are given for constructing oblique projections of various geometric figures: a rectangle, a polygon, parallelepipeds, a pyramid, a building, a circle, cylinders, a sphere, and cones. Owing to its brevity, A. Frolov’s article did not receive wide circulation and was not noted by contemporaries (in particular, by Prof. V. I. Kurdyumov).

The problems of constructing correct, visually clear, and fully measurable images, in connection with the growth of technology and industry in the 1880s–1890s, compelled scholars to turn their attention to theoretical work in the field of axonometry.

Thirty years after the publication of A. H. Reder’s article, a work by V. I. Kurdyumov appeared [19]. In the preface, dated January 6, 1885, it is said that “the purpose of the present article is to fill a gap in the latest technical literature with a brief exposition of the method of isometric projections” (p. 3). A characteristic feature of V. I. Kurdyumov’s work is

the use of isometry for practical purposes (which cannot be said of A. H. Reder).

Working as a communications engineer in the Caucasus, V. I. Kurdyumov was compelled, when explaining working drawings to workers, to resort to isometric projections; moreover, for more rapid construction he used paper ruled in advance, the so-called “isometric grid,” a sample of which he published in the above-mentioned work.

His book received due appraisal from his contemporaries. In No. 12 of the journal Engineer for 1885, on p. 531, there appeared a review by L. L., who wrote: “...for greater clarity, when making images of more complex objects, it is very often necessary to resort to a pictorial perspective drawing or an isometric drawing, which combines the visual clarity of a perspective drawing with the property of a drawing in orthogonal projections, namely, to give the direct dimensions of the object, although, to be sure, only in three definite directions; ...the brevity of the exposition, as well as the distinct drawings, make this book accessible to anyone who is not a stranger to the foundations of elementary mathematics, while the isometric grid devised by Mr. Kurdyumov considerably facilitates the preparation of isometric drawings.”

Simultaneously with the appearance of V. I. Kurdyumov’s work, on January 23, 1885, there appeared a work by Prof. N. I. Makarov [20], in which the second section contained a method of isometric projections compiled from lithographed notes. In contrast to the projection method of V. I. Kurdyumov, who placed the picture plane perpendicular to the diagonal of a cube, Professor N. I. Makarov proposed projecting the system of coordinate axes, together with the object referred to them, orthogonally onto a plane that intersects all three coordinate axes and is equally inclined to them.

A major shortcoming of his work is the absence of examples for constructing technical parts and assemblies. Despite this, N. I. Makarov’s course of lectures on isometric projections was a fairly detailed guide for constructing pictorial images of objects specified in orthogonal projections and, along with the work of V. I. Kurdyumov, was used as a textbook in many institutes of that time.

The largest and most complete work on axonometry in Russia was considered to be the work of Professor V. I. Kurdyumov [21], printed in 1892. It should be noted that this work was published 4 years earlier than his course in descriptive geometry [17], since the pressing need to develop an appropriate theory of parallel projections and to investigate its various types (besides isometry) compelled the author, first of all, to write this important section.

V. I. Kurdyumov’s work is remarkable for the character of its exposition and for its profound analysis of the application of various kinds of axonometric projections. Throughout the entire course he sought to demonstrate the possibilities of using axonometric projections for “narrowly practical purposes” and proposed, when explaining individual machine assemblies and building structures, to use axonometric sketches.

Among the shortcomings of the work should be counted an underestimation of general theoretical questions of axonometry and the absence of a description of its state in foreign works. In particular, there is no mention of the fundamental proposition of axonometry—the Pohlke theorem—which, as is known, abroad...

...an extensive literature. Having focused all his attention on the practical side of axonometry, V. I. Kurdyumov did not provide a sufficiently scientific-theoretical basis and did not show the prospects for the further development and use of axonometric projections.

V. I. Kurdyumov’s work, despite the shortcomings noted, for a long time served as the only and most complete manual for many generations of engineers and, owing to the simplicity and convenience with which the material was presented, even at the present time is a textbook in the study of the theory and practice of axonometric projections.

Professor V. I. Kurdyumov, who taught at the Petersburg Institute of Engineers of Ways of Communication, in addition to a course in descriptive geometry and a course in construction works, showed the application of axonometric projections for interpolating earthwork profiles.

“Interpolation of profiles,” he wrote, “can be carried out by using orthogonal projections or axonometric oblique-angle projections. The use of oblique-angle projections is more convenient, in the sense of the greater clarity of the images of the forms of the individual parts of the earth body under consideration” [22] (p. 144).

For this purpose he solves the problem in oblique frontal isometry, taking two adjacent cross profiles in a system of projections onto a plane \(Q\) parallel to these profiles and the projection of the track axis \(O_1O_2\), parallel in turn to the axis \(Oy\), making an angle of \(20^\circ\) with the axis \(OX\), with a distortion coefficient equal to 1 (Fig. 6).

Fig. 6

First of all he finds the projections of the points \(O_1\) and \(O_2\) on the surface of the ground (\(o_1\) and \(o_2\)). Connecting \(O_1\) and \(O_2\) with one another, we obtain the point of intersection \(O_1O_2 \times o_1o_2 = P\), representing the “projection of the point of transition of the axis.” The distances \(O_1P\) and \(O_2P\) determine the line marking the points “\(P\)” relative to the complex profiles. Through this point, too, the plane of the theoretical profile must pass, with \(AB \parallel A_1B_1\), since the plane of the theoretical profile is parallel to the plane of the given profiles. Then he determines the points of intersection of the profiles \(D\) and \(G\), drawing cutting planes through these points parallel to the plane \(O_1o_1Po_2O_2\). The points \(D\) and \(G\), \((dD \times D_1D_2 = D,\ gG \times G_1G_2 = G)\), are points of break in the theoretical profile.

Having constructed the slope lines \(AK\) and \(GL\) to their intersection with \(DK\) and \(BT\), we obtain the theoretical profile of the half-embankment \(PACD\) and the half-cut \(PBTG\).

The points $M$ and $N$ at which the edges of the roadbed $A_1A_2$ and $B_1B_2$ pass are obtained by intersecting the lines $a_1a_2$ and $b_1b_2$ with $A_1A_2$ and $B_1B_2$. At point $M$ an additional excavation of earth must be completed, and at point $N$—an additional embankment.

The works of Professor V. I. Kurdyumov on the theory of axonometry and on applying the latter to engineering and construction served as the beginning for the further use of axonometric projections in various fields of science and technology.

A pupil of V. I. Kurdyumov, later Professor N. A. Rynin, devoted a number of his works, in addition to setting forth the theory of axonometry, to its wide application in practice. Analyzing axonometric projections, he concluded that, with respect to visual clarity, they stand in second place after perspective; however, this in no way diminishes their merits, owing to the preserved convenience of measurement.

N. A. Rynin’s works in descriptive geometry, especially on questions concerning its significance and methods of application in science and technology, served as the beginning of further research in this field.

In 1911 Professor P. M. Leontovsky [23] noted that, for a detailed representation of part of a mining working, “one may recommend depicting it in axonometric projections.”

A major contribution to the development of the theory of axonometry was made by Professor E. S. Fedorov (although he did not use the word “axonometry”), who investigated oblique and vectorial projections in works [24, 25], published respectively in 1907 and 1917. However, his ideas on the study of such projections found wide dissemination only in the years of Soviet power.

While in Russia, in the second half of the nineteenth century and at the beginning of the twentieth century, works on axonometry appeared that pursued “narrowly practical aims,” abroad the theoretical questions of axonometric images continued to be perfected; these led to a new proof of the fundamental theorem of axonometry by Schwarz, Schur, and others, to the discovery by E. Kruppa [26] in 1910 of the theorem of central axonometry, which was not mentioned in the works of our scholars, and also to the study of the axonometry of multidimensional space [27].

Having considered the state of the theory and practice of axonometry in the pre-Soviet period, the following conclusions may be drawn:

1. The first graphic images, which appeared in remote antiquity, were for the most part very primitive drawings reflecting various events from people’s lives, such as scenes of hunting, battles, views of implements of labor, weapons, and the like.

2. With the growth of science and culture, methods of representation also improved; in the Western countries considerable successes were achieved in this direction, culminating by the sixteenth century in the creation of the theory for constructing visual images, which received the name of perspective.

3. In our country, owing to the special conditions of its socio-economic development, graphic art followed its own distinctive path, which later affected not only Russian painting but also the engineering graphics of the eighteenth century, when Russia entered upon the path of development of industrial production.

4. In the drawing-pictures of the sixteenth–eighteenth centuries, alongside orthogonal drawings, visual images were used as explanations, often in oblique parallel projection; moreover, depending on the performers, some of them were made with observance of scale, which allows them to be classified as axonometric and thereby to regard the practical use of the latter as occurring considerably earlier than...

the theoretical foundations of this method of constructing projections arose. In this connection one cannot agree with the statement made by A. A. Kuzin [6] (p. 70) that axonometry began to be applied only at the end of the nineteenth and the beginning of the twentieth centuries. More correctly, from our point of view, it should have been noted that at that time it became more widespread, owing to the growth of the machine-building industry.

1. Beginning in the second half of the nineteenth century, in connection with the successive development of machine production, which required the use of visual-to-scale images, works appeared on axonometric projections and, above all, on isometry.

2. Interested in the techniques for constructing such images, on the one hand, and in their use in practice, on the other, the authors of the works mentioned did not provide profound theoretical investigations in this field (in comparison with those abroad).

LITERATURE

1. V. A. Voskresenskii. On the main directions in the development of axonometric projections in the USSR. Izvestiya TPI, vol. 104, 1959.

2. V. A. Voskresenskii. A brief outline of the development of axonometry in the Soviet Union. In: Questions of the Theory, Applications, and Methods of Teaching Descriptive Geometry. Riga, 1960.

3. A. I. Aleksandrov. From the history of engineering graphics of the Urals and Siberia. Sverdlovsk, 1959.

4. N. M. Beskin. The fundamental proposition of axonometry. (Questions of Modern Descriptive Geometry, collection of articles.) Moscow, 1947.

5. Yu. K. Vakhtin. The Russian School of Engineering Graphics. Moscow, 1952.

6. A. A. Kuzin. A Brief Essay on the History of Russian Drawing. Uchpedgiz, 1956.

7. N. A. Rynin. Materials on the history of descriptive geometry. Leningrad, 1938.

8. I. E. Zabelin. The Domestic Life of the Moscow Tsars. Moscow, 1862.

9. N. N. Voronin. Essays on the history of Russian architecture. Moscow–Leningrad, 1934.

10. V. I. Lenin. The Development of Capitalism in Russia (Works, vol. III, 4th ed.). 1941.

11. V. Gennin. Description of the Siberian mining works. Manuscript, Archive of the State Historical Museum, 1735.

12. Schlatter. Description, with drawings, of mining works and mines in Russia (manuscript, Archive of the State Geographical Society of the Academy of Sciences of the USSR, section 48, inventory No. 36).

13. V. V. Danilevskii. I. I. Polzunov (Works and Life). Moscow, 1941.

14. N. Kurganov. The Book Naval Engineer. St. Petersburg, 1777.

15. A. Maiurov. Higher Geometry in Spaces, or the Application of Analysis to Descriptive Geometry, with an Exposition of the Theory and Defilading of Fortification Structures. St. Petersburg, 1817.

16. A. Kh. Reder. On isometric projection. St. Petersburg, 1861.

17. V. I. Kurdyumov. Course of Descriptive Geometry. Orthogonal Projections. St. Petersburg, 1896 (Collected Papers of the Institute of Engineers of Ways of Communication, issue XXXII).

18. A. Frolov. Supplements to Elementary Descriptive Geometry (in the book Elementary Descriptive Geometry with Drawings in the Text and with Problems for Independent Work by Students. For real schools). St. Petersburg, 1884.

19. V. I. Kurdyumov. The Method of Isometric Projections. St. Petersburg, 1885.

20. N. I. Makarov. Appendix to Descriptive Geometry. 1. A Method of Projections with Additional Numbers. 2. A Method of Isometric Projections. St. Petersburg, 1885.

21. V. I. Kurdyumov. Axonometry in rectangular and oblique projections, or parallel perspective. Collected Papers of the Institute of Engineers of Ways of Communication, issue XXIII, 1892.

22. V. I. Kurdyumov. Materials for a Course of Construction Works. Issue 2, Earthworks, 2nd ed., St. Petersburg, 1898.

23. P. M. Leontovskii. On the application of orthogonal, vectorial, and axonometric projections in mine-surveying plans (Izvestiya of the Ekaterinoslav Higher Mining School), issue II, 1911.

24. E. S. Fedorov. Exact representation of points of space on a plane (Notes of the Mining Institute, vol. 1, issue 1, St. Petersburg, 1907).

25. E. S. Fedorov. New descriptive geometry (Izvestiya of the Russian Academy of Sciences, 4th series, vol. XI, Petrograd, 1917).

26. E. Kruppa. Zur axonometrische Methode der darstellende Geometrie (Sitz. ber. der K. Ak. der Wiss. Mathematisch naturwissenschaftliche Classe. Abt. II a, Bd. 119, Heft 4, S. 487–506, Wien 1910).

27. P. H. Schoute. Mehrdimensionale Geometrie. I Teil. Leipzig. 1902.

Corrections and Noticed Misprints