CATEGORIES OF RESEARCH IN THE FIELD OF RUSSIAN AXONOMETRY

V. A. VOSKRESENSKII

(Presented by the scientific seminar of the departments of descriptive geometry and graphics of the Tomsk higher educational institutions)

1.

Axonometry is a large subject. Various articles, books, textbooks, monographs, and so on can be written about it. And it must be said that up to the present time axonometric projections have been one of the richest “ore veins,” where ever new questions for scientific and methodological research are found.

As is known, axonometry did not arise all at once. Its appearance was preceded by a very long period of searching—more than 800 years in our country—a period of attempts to create a visual representation.

And this striving, at first toward a visual drawing, subsequently led to the discovery of axonometric projections, one of the chief conditions of which was the need to show an object as we are accustomed to seeing it in reality.

Of course, “the concept of visuality,” as Academician L. I. Mandelstam said, “is in general extremely conditional. It essentially includes two elements. First, in order for something to be visual, it must be familiar; in this lies the condition of this concept. Then it is necessary that the visual conception be connected with things that can be directly perceived by the senses” [35].

One cannot but agree with such a definition, proposed by L. I. Mandelstam.

And indeed, all ancient drawings and map-plans, in addition to ordinary orthogonal projections, were always explained by visual representations, executed at that time in a highly arbitrary manner, without observing definite rules, as a result of which the term of the so-called “free perspective” appeared.

The development of mechanical engineering required from merely visual representations the condition of measurability of the object. And thus, beginning with the first half of the nineteenth century, the terms appeared: first “isometry,” later “axonometry.”

Sometimes the word axonometry was explained as “parallel perspective” [30].

Consequently, the initial understanding of axonometric projections was reduced to three obligatory conditions: a) visuality, b) measurability, c) parallelism of projection.

Such a conception of axonometry has, unfortunately, survived down to our time.

If one takes any modern textbook on descriptive geometry, then almost everywhere one can encounter the above definitions of axonometric projections.

Let us take the programmed textbook by E. A. Aksenova with co-authors [1]. It says that “an axonometric image... possesses great visual clarity” (p. 34).

In N. L. Ruskevich [55]: “Axonometric projections are a visually clear representation of an object” (p. 23). In A. I. Dobryakov’s textbook [23] we read that “...the visual clarity of an axonometric image is only slightly inferior to the visual clarity of a perspective image” (p. 12). And in N. S. Kuznetsov’s book [29] it is said that “axonometry possesses greater visual clarity than images made in orthogonal projections, but less than perspective” (p. 5).

The fascination with the “visual clarity” of the image led some authors to an imprecise understanding of axonometric projections. In particular, Yu. K. Vakhtin [11] wrote that “the projections characteristic of representation before the sixteenth century are oblique axonometry” (p. 12), although at that time no methods for constructing axonometric projections existed.

Modern studies have shown that, depending on the choice of the direction of projection, we can obtain various axonometric images; moreover, if such a direction is chosen poorly, we shall obtain a very ugly, non-visual projection, which nevertheless is axonometric.

Let us note that in the nineteenth century Professor F. E. Orlov of Moscow University [41] expressed the idea that orthogonal projections of an object in engineering and construction drawings are nothing other than a special case of axonometric projections. The author concluded that “the method of projecting a body onto a parallel plane of projections is used predominantly in the art of construction engineering, and the projections of the body in all these cases bear different names..., namely: 1) If the picture plane is placed parallel to the coordinate plane \(XOZ\), then under orthogonal projection the axis \(OY\) degenerates into the point \(O'\), and the projections of the axes \(OX\) and \(OZ\) are obtained in true size. In this case the ‘projection of the body’ is called the ‘principal view or principal elevation.’ 2) If the plane of projections \(P\) is parallel to the coordinate plane \(YOZ\), then the axis \(OX\) degenerates into the point \(O'\), and the axes \(OY\) and \(OZ\) are represented in true size. Then the projection of the object is called the ‘side view or side elevation.’ 3) Finally, if the plane \(P \parallel \text{pl. } OXY\), then the axis \(OZ\) degenerates into the point \(O'\), and the axes \(OX\) and \(OY\) are represented in true size. Here we obtain a projection of the figure which is called a ‘plan’” (p. 91).

Consequently, visual clarity cannot be the principal condition inherent in axonometric projections. And if we touch upon questions of multidimensional axonometry, then it will hardly be appropriate here to speak of visual clarity in \(n\)-dimensional space.

On the basis of the studies carried out by E. Müller [65], N. M. Beskin [7], and N. F. Chetverukhin [20], axonometry should be understood as a parallel (central) representation of an object, referred to a definite coordinate system, constructed at a prescribed axonometric scale and according to the coordinate principle.

Those images from which it is impossible to determine the scale and the actual dimensions of the original we shall call, depending on the type of projection, images in parallel or central projections.

2.

As was said above, axonometric projections appeared in connection with the requirements of machine production, when the need arose not only to explain the external appearance of a given part, but also to know the exact dimensions by which it could be manufactured.

In Western Europe the beginning of the creation of such images is considered to be 1820, when the English professor at Cambridge University William Farish, at meetings of the Philosophical Society on February 21 and March 7, delivered a report on “isometrical perspective.” Later, in 1844, certain questions of the theory of axonometry were set forth by the professor of the Saxon Mining Academy, Julius Weisbach.

In our country the beginning of the appearance of the first axonometric projection should be considered the year 1855, when engineer-colonel A. Kh. Reder published in No. 1 of the Journal of the Main Administration of Communications and Public Buildings his article “On Isometric Projection.” This work appeared as a separate book in 1861 [53].

The further study of axonometry in prerevolutionary Russia proceeded along a line pursuing a narrowly practical aim: to teach future engineers to construct quickly clear, conveniently measurable projections of technical or construction details.

A major scholar in the field of the art of construction and descriptive geometry, V. I. Kurdyumov, created a fundamental work on the theory of parallel axonometry [30] and was one of the first to propose for practical use the so-called “isometric grid” [31], which aroused the interest of V. I. Kurdyumov’s contemporaries, expressed in a favorable review [25].

A thorough study of axonometry began in the Soviet period.

Analyzing many works on axonometric projections, we established that, in the main, they can be divided into three categories:

1) historical,
2) instructional-methodological,
3) theoretical-applied.

3.

The beginning of any investigation is the history of the development of the question, and in this connection we decided to characterize the state of historical works carried out on axonometry. Here, unfortunately, we have an insufficient number of fully exhaustive studies, and probably because these questions are very labor-intensive, requiring a great deal of time and a complex so-called “mental” (imagined) experiment. This is on the one hand. On the other hand, among engineering and scientific workers there are very few persons interested in questions of the history of science and technology. Yet Academician B. N. Yuriev [63] wrote: “The study of the history of science and technology helps every scientific worker and engineer not only in the study of the corresponding special discipline, but also makes it possible to develop a broader and more correct view of the development and place of the given discipline among other sciences.”

The well-known French physicist Louis de Broglie said that “... the history of science cannot fail to interest natural scientists: the scientist finds in it... numerous lessons and experience made wise by his own—”

he may perhaps be better than anyone else at interpreting these lessons with knowledge of the matter.” [34, dust jacket].

A similar thought is expressed by the prominent English scientist and public figure J. Bernal, who asserts that “... if the lessons of the past are well learned, progress in science will be faster and more reliable.” [34, dust jacket].

“History,” as the well-known French historian of science Paul Tannery (1843–1904) said, “does not at all have as its sole aim the satisfaction of useless curiosity: the study of the past must, in the end, illuminate the future” [50, p. 4].

This statement coincides in meaning with the opinion of the well-known Russian writer A. I. Herzen, who said that “in order to understand the present state of thought, the surest path is to recall how humanity arrived at it” [4, p. 5].

Let us note that the statements cited above are not new, since even earlier, in the works of K. Marx, F. Engels, and V. I. Lenin, “... on the basis of an analysis of an enormous body of material from the history of science and technology as well as from the history of art, the scientific method was discovered, the basic principles of research in this field were established, and a number of initial fundamental questions were deeply resolved” [42, p. 5].

From this one may conclude that works of a historical character should occupy one of the foremost places in contemporary science.

By Order No. 606 of the Minister of Higher and Secondary Specialized Education of the USSR, dated 29 September 1967, the “Instruction on the Procedure for Awarding Academic Degrees and Conferring Academic Titles” was put into effect as of 15 October 1967; in § 21 it states that “the defense of dissertations on topics in the history of science and technology is conducted in councils for the corresponding specialty...” [10, p. 5].

Although we consider only one question from the course of descriptive geometry—axonometry—we believe that the history of its development should be assigned to the first category of research, on which the remaining scientific directions must be based.

Attention to the history of axonometric projections appeared in the Soviet Union during the period of the beginning of the restoration of the national economy after the Civil War.

One of the first brief historical essays concerning specifically the development of axonometry we consider to be the essay by N. A. Rynin, included in his book [56].

Let us note at once that the author limited himself to listing the principal works (predominantly foreign ones), did not provide any scientific analysis of the reasons for their appearance, and did not examine the prehistory of this question.

Sixteen years later, i.e., in 1938, N. A. Rynin published a brochure [57] in which he repeated the above-mentioned essay, making the corresponding additions for the time that had elapsed (since 1922). In this work, the author himself proposed only a “historical canvas” for future researchers.

After the brochure mentioned, up to 1942 no one dealt with questions of the history of either engineering graphics in general or the history of axonometry in particular; at least, we cannot name a single published work in the field under consideration.

Only in 1942 did V. A. Baskarev write the dissertation “The Principal Stages in the History of Technical Graphics up to the Descriptive Geometry of Gaspard Monge.”

Finally, in the period from 1949 through 1955 inclusive, six candidate dissertations on historical topics appeared by the following auth—

A. A. Kuzin’s Essays on the History of Technical Documentary Materials (the Russian Drawing) in Pre-Revolutionary Russia, 1949; D. P. Volkov’s Essay on the History of the Development in the Nineteenth Century of the Course on Complex Projections in Russia, 1950; A. I. Aleksandrov’s Paths of Development of Engineering Graphics in the Urals and Siberia, 1951; V. N. Bulanova’s A Study of the Development of Engineering Graphics in Russia before the Nineteenth Century, 1954; V. A. Gusev’s A Study of the Formation and Development of Descriptive Geometry in the Nineteenth Century, 1955; and N. A. Levitskaya’s On the Development of Technical Graphics in the Manufactural and Initial Period of Machine Production in Russia, 1955. But the works mentioned dealt with questions of the creation of technical drawings or the formation of descriptive geometry in pre-revolutionary Russia.

Almost nothing was said about the development of axonometric projections.

Only beginning in 1959 did articles begin to appear in print dealing specifically with questions of the development of axonometry in the USSR [12, 13, 14, 15, 16, 17, 18, 19]. In addition, several works appeared on the history of graphic representations in general, where some information about axonometric projections may be found [2, 28, 47, 59].

4.

As was noted above, one of the first domestic books on axonometry published in the Soviet period was N. A. Rynin’s book [56]. After it, in 1924, the 8th edition of A. N. Pal’shau’s textbook was prepared, where axonometry was added as one of the new sections.

Then, in the 9th edition, written in 1927, the editor N. F. Chetverukhin noted that “this edition differs from the preceding one by a newly compiled and considerably expanded chapter on axonometric projections (Section II, Part II).”

In 1930 the second section of this same book was once again subjected to considerable revision, and “substantial additions were made, concerning chiefly orthogonal axonometric projections and their practical applications (§§ 348—365)” [46, pp. 3, 4].

Thereafter, for a number of years, up to the present time, one of the principal textbooks on descriptive geometry for mechanical-technological, machine-building, and instrument-building specialties in higher technical educational institutions has been the course by V. O. Gordon and M. A. Sementsov-Ogievsky, which has already gone through 20 editions. The section on axonometry was treated with sufficient completeness.

Axonometric projections in our country gained a firm place in courses of descriptive geometry and machine-building drawing, and in 1934 they were introduced into the state standard, which fixed the position of the axonometric axes for rectangular isometry and dimetry, as well as frontal oblique dimetry.

In 1956, in accordance with the methodology of teaching the course of descriptive geometry at the Moscow Aviation Institute, a book by N. F. Chetverukhin with coauthors [60] appeared. The section on axonometric projections (Chapter XII) was written by V. S. Levitsky, and the proof and development of the fundamental theorem of axonometry by N. F. Chetverukhin. Let us note that the content encompassed not only the theory of axonometric projections, but also practical examples. V. S. Levitsky set forth in considerable detail the methods for constructing axonometry by establishing a projective relationship between the fields of projections, for the first time

in the history of the teaching of methods for constructing this type of image in our country. In addition, the author gave a brief account of the existing mechanization of techniques for drawing axonometry and, together with N. A. Babulin, developed a diagram by which one can graphically determine the length of a segment for any standard axonometry when multiplying it by the corresponding distortion coefficient. As appendices, V. S. Levitskii set forth methods for constructing shadows both on complex drawings and on axonometric drawings. In total complexity, the material on axonometry comprises about 78 pages, which is almost twice the size of this section in the textbook by V. O. Gordon and M. A. Sementsov-Ogievskii [21], and can be used not only by students but also by graduate students.

The second edition of this manual, revised and expanded, appeared in 1963 [61].

The gradual reduction of lectures and practical classes in descriptive geometry, carried out by the Ministry of Higher and Secondary Specialized Education of the USSR, made it necessary to pay attention also to reducing the size of courses published in print.

One of the first textbooks compiled according to the shortened program, we consider to be the book by A. D. Posvyanskii [51]. The section on axonometric projections here amounted to only 32 pages.

Finally, an even briefer presentation of axonometry (on 10 pages) is found in the book by D. M. Orlov, K. S. Kolotov, and G. I. Puchko [40].

In 1970 the third edition of A. D. Posvyanskii’s textbook was published [52], where the author reduced the volume of axonometric projections to 24 pages.

The authors mentioned strove to present the material in such a way that, while preserving the basic laws of axonometry, it would make it possible to teach students to construct correctly the standard types of axonometric projections.

Teaching aids developed for the correspondence-study system are of great importance in the methodology of presenting any course. As is known, in recent years this form of training specialists with higher education has had a significant share in the educational preparation of higher education.

It should be noted that the existing methodological instructions and control assignments for courses in descriptive geometry and mechanical-engineering drawing are clearly insufficient.

The most widespread among them are the manuals by N. N. Ivanov [24] and V. S. Levitskii [33]. Despite the fact that these methodological instructions have already gone through more than one edition, one would like the methodology of correspondence instruction to develop both with respect to the creation of programmed manuals and with respect to increasing the variants of assignments.

Questions of programmed instruction at the present time, in our opinion, are still at the stage of searching.

Because of the excessively large scope of the present article, we do not wish to dwell on the textbooks and manuals of other authors which, to one degree or another, set forth axonometric projections; nevertheless, we must state the fact of the almost complete absence of presentations of new theoretical studies on axonometric projections obtained in individual articles and dissertation works.

The situation that has developed at present requires a fundamental revision of the program in descriptive geometry and mechanical-engineering drawing, especially in the field of axonometry. This is a large and important question, which now awaits an effective solution from our scholars, teachers, and methodologists.

The book by Yu. G. Kozlovsky [26], which appeared in 1971, is the first to examine methods of teaching axonometric projections. The author determines what the student should know as a result of study and what must be done to achieve this. Of course, the propositions expressed by Yu. G. Kozlovsky cannot be regarded as some absolute truth to be followed by all lecturers. Here, in many cases, the result of teaching depends to a large extent also on the preparedness of the audience. Nevertheless, this first monograph on methodology should serve the further development and improvement of the teaching of axonometric projections.

5.

To the third category of works we assign theoretical-applied works. The latter arose and reached their greatest development during the years of Soviet power.

In addition to studies concerning the Pohlke–Schwarz theorem, at present we have many works that consider multidimensional and central axonometry.

Studies of the fundamental theorem of central axonometry, both for three-dimensional and for multidimensional space, received significant development in the works of I. I. Kotov [27], E. A. Mchedlishvili [38], O. Ya. Ryonk [58], N. V. Paluver [43, 44, 45], V. A. Manevich [36], V. N. Pervikova [48], G. K. Nikolaevsky [39], and others.

In the field of parallel axonometry, mention should be made of the article by M. V. Burde [9], who proposed a simpler proof of Pohlke’s theorem without using affine correspondences.

The work of L. N. Lambin [32] deserves attention; in it the completeness of representations of multidimensional objects is established.

One of the first monographs on the classification of representations of multicomponent systems may be considered the book by F. M. Perelman [49].

The number of scientific research works on descriptive geometry in general, and on axonometry in particular, carried out in contact with mathematics, has noticeably increased; evidence of this is provided by the works of V. N. Pervikova [48], V. A. Manevich [36, 37], O. Ya. Ryonk [58], N. V. Paluver [44, 45], M. M. Yuditsky [62], E. A. Mchedlishvili [38], I. I. Kotov [37], N. M. Beskin [5—8], L. A. Yatskevich [64], G. K. Nikolaevsky [39], and others.

In the field of carrying out axonometric projections with the aid of instruments, a significant step forward was made, characterized by the creation of electro- and video-axonographs by such inventors as A. V. Dmitrevsky [22], N. L. Russkevich [54], G. N. Bagrationi [3], and others.

In the present article we do not set ourselves the task of reproducing the history of the creation and the practical significance of the works of the above-mentioned authors, but we recommend that those interested consult the articles [15, 18].

Finally, what can be said about the future problems of axonometry? In our opinion, axonometric projections must be investigated and developed in the field of central projection (of both three-dimensional and multidimensional space), using isotopic transformations. In addition, much remains to be done in questions of the methodology of teaching axonometry, the creation of programmed textbooks, and the revision of lecture material with the aim of introducing the modern achievements of science into this major branch of descriptive geometry.

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