Questions on the History of Axonometry

V. A. Voskresenskii

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

In December 1958, in the opening address at the conference of the Department of Descriptive Geometry of the Moscow Order of Lenin Aviation Institute named after S. Ordzhonikidze, Professor and Doctor N. F. Chetverukhin emphasized that “the methods of descriptive geometry possess a generality characteristic of all mathematical sciences. Thus, descriptive geometry may without error be assigned to the number of mathematical disciplines” [1].

From this one may conclude that, since descriptive geometry is a mathematical science, many general questions concerning mathematics must also belong to descriptive geometry; in particular, the question of its history, which should be assigned to the history of mathematics. However, taking into account that descriptive geometry is very widely applied in engineering, the process of its development therefore cannot be separated from the history of technology.

V. I. Lenin wrote: “The concept (cognition) in being (in immediate phenomena) reveals the essence (the law of causality, identity, difference, etc.)—such is the truly general course of all human cognition (of all science) in general. Such is the course also of natural science, and of political economy (and of history). Hegel’s dialectic is, consequently, the generalization of the history of thought. It seems an extraordinarily rewarding task to trace this more concretely, in greater detail, in the history of the separate sciences” [2].

Taking into account V. I. Lenin’s instructions, the Plenum of the Central Committee of our Party in 1929 proposed introducing the teaching of the history of technology in higher technical educational institutions. In connection with this, both methodological and scientific work in this direction was revived, as is proved by the publication of a number of monographs and articles.

The inter-institutional conference on the history of the physical and mathematical sciences, held in Moscow in 1960, noted in its resolution that “knowledge of the history of natural science and technology is necessary for every specialist, since the history of knowledge helps one to understand contemporary concepts and theories of science more correctly, makes it possible to gain deeper knowledge of the laws of development of science and technology and thereby to evaluate their state correctly and outline the prospects of their development” [3, p. 6].

But, despite such significance of the history of science and technology, attention was drawn at the conference to a noticeable decline both in organizational work and in research and methodological work on the history of the sciences and, in particular, on the history of the physical and mathematical sciences.

A matter of great concern was the fact that in our country too few monographs were being written in the field of the history of science, while at the same time the capitalist countries were showing corresponding activity in this direction.

The conference emphasized that in England, Italy, the Federal Republic of Germany, the United States, and other countries, bourgeois historians of science (Rosenberger, Whittaker, Rutherford, and others) had produced a number of relevant monographs in which, although a large body of factual material is presented, it is nevertheless offered to readers from the standpoint of bourgeois objectivism. From this follows the definite conclusion that Soviet scholars must not allow students and specialists to make use only of such literature. It is necessary to counter these works with our own studies, written from the standpoint of dialectical materialism.

At the conference 45 reports on the history of mathematics were delivered; among them of substantial interest is the report by I. V. Kuznetsov and L. E. Maistrov [3], which examined the significance of historical-mathematical investigations that, in the authors’ opinion, “contribute to the development of correct conceptions of the very history of mathematics as a science, of its tasks and methods” (p. 183).

In connection with this, the speakers recommend that contemporary researchers devote attention to “the development of the history of mathematics in the newest epoch of its development, so that historical investigations may lead directly to contemporary mathematical ideas and schools” (p. 186).

One must agree with this proposal, since in the study of the history of descriptive geometry as a mathematical science, very, very little has been done in our country.

In 1963 the Second Interuniversity Conference on the History of the Physical and Mathematical Sciences was held [4, pp. 3–7], where, along with successes, it was noted that “the implementation of Lenin’s directives and the need for a dialectical treatment of the history of science and technology are still being carried out at an impermissibly slow pace.” The reasons for this phenomenon are an underestimation of “the relevance of research both in the history of the physical and mathematical sciences and in other sciences.” The conference established that the number of postgraduate students, doctoral candidates, and applicants for Candidate and Doctoral dissertations investigating the history of the physical and mathematical sciences and other sciences had decreased.

In addition, the underestimation of the study of the history of the sciences in higher educational institutions has an adverse effect on the formation of a scientific dialectical-materialist worldview among students.

In order to lecture competently and effectively in higher education on various general technical and specialized courses, taking into account at least individual questions in the history of their development, it is necessary to devote attention to raising the scientific qualifications of instructors in the field of their historical level.

In this connection, the Higher Attestation Commission sanctioned the advisability of defending dissertations on the history of science and technology, approving on September 28, 1966, a new “Instruction on the procedure for awarding academic degrees and conferring academic titles,” where § 21 states that “the defense of dissertations on topics in the history of science and technology is conducted in councils for the corresponding specialty” [5].

Proceeding from all that has been said above, one may confidently consider that questions concerning research into the history of the development of descriptive geometry in general and, in particular, of one of its major branches—axonometry—are now acquiring substantial significance.

In works [6, 7, 8] the process of the historical development of axonometric projections up to 1955 inclusive has been analyzed in considerable detail.

…Ultimately, we established that in the brief historical surveys placed mainly in textbooks on descriptive geometry and in dissertations directly concerned with the history of engineering graphics, the distinctive features of the development of axonometry were not investigated, and everything was reduced merely to a listing (in chronological order) of published works, the beginning of whose appearance in Russia, as is known, dates to the second half of the nineteenth century.

After 1955, separate articles and books began to appear in print that set forth various questions in the history of engineering graphics. These include the books by A. A. Kuzin [9], A. I. Aleksandrov [10], and others, and the articles by D. P. Volkov [11], N. A. Levitskaya [12], V. A. Shulzhevich [13], and others.

In addition, in new textbooks on descriptive geometry by such authors as N. F. Chetverukhin, V. S. Levitskii, Z. A. Pryanishnikova, A. M. Tevlin, and G. I. Fedotov [14, 15], N. N. Krylov, P. I. Lobandievskii, and S. A. Men [16], R. S. Brilin [17], and others, brief historical information was given on the development of descriptive geometry in our country.

However, in all the literature mentioned, there were likewise no special studies on the history of axonometric projections.

In 1963, a book by V. A. Pakhomova [18] was published in Omsk. The author attempted to describe the development of modern methods of representation and the most interesting works and outstanding scholars in this field. In view of the book’s limited size (8 printer’s sheets), V. A. Pakhomova was not able to illuminate all questions in the history of descriptive geometry sufficiently fully. In our view, her work is a synoptic presentation of a rather large body of material that requires detailed elaboration by many researchers.

On the question of the history of axonometry, the author, considering the ancient pictorial representations of the Egyptians, Greeks, and Romans, concludes that many drawings in which artists used the device of “shifting” a similar figure by some amount in relation to its first representation “were the first step toward the creation of a specific type of representation—frontal axonometry” (p. 7).

We consider such an assertion insufficiently substantiated, since in those ancient times the concept of a metric connected with a coordinate system did not exist. In making a pictorial representation, artists approached the metric of its construction purely intuitively, without taking exact account of the dimensions of the original. In essence, they made a drawing in parallel projection, which, in our view, it is not entirely correct to call frontal axonometry.

The author also gives an incorrect definition with respect to certain drawings of structures and buildings made in the eighteenth century by S. Remezov, I. I. Polzunov, and others, allegedly in “frontal axonometry” (pp. 44, 47). When we carried out a graphic analysis of such drawings, we established that the lines drawn in the direction of the third dimension (the depth of the picture), when extended, intersect one another at various “vanishing” points, which is inadmissible for an oblique frontal projection. It should be noted that D. P. Volkov makes precisely the same error in his article [11], when he characterizes I. I. Polzunov’s pictorial drawings as axonometric.

Describing the development of descriptive geometry in Russia up to 1917 and in the Soviet Union after 1917, V. A. Pakhomova confines herself to a brief enumeration of works in this field, where works and theorems created by our scholars are mentioned. A detailed scientific analysis of…

because of the small size of the book, she, like other authors before her, does not carry it out.

We said above that descriptive geometry is a mathematical science and, consequently, the history of the development of this subject must be connected with the history of mathematics.

In special courses on the history of mathematics one can find a brief description of the origin and development of descriptive geometry. In K. A. Rybnikov [19] it is said that “the methods of descriptive geometry took shape in the area of technical applications of mathematics” (p. 117). Speaking of the appearance of axonometry, the author notes that Desargues may be considered the creator of this type of projection, having applied in 1636 “the method of coordinates for constructing perspective and the corresponding rudiments of axonometric projection” (p. 117). Nothing more detailed is said about the development of axonometric projections.

The German historian of mathematics G. Wieleitner [20] also says very briefly (in all, on two pages) about descriptive geometry, chiefly about its founder Gaspard Monge. Nothing is mentioned about axonometry.

Articles directly considering only questions of the history of the development of axonometric representations, written by the author of the present work, appeared in 1959 [21], 1960 [6], 1963 [22], 1965 [23], 1966 [7, 8]. In these works, concerning the gradual formation of the theory of axonometry in our country, both the stages (periods) in the creation of axonometric projections and the main directions of their development, which also exist at the present time, are defined.

In conclusion it is necessary to note the following:

1. The history of the development of axonometry, as one of the sections of descriptive geometry, must be connected with the history of mathematics and technology.

2. Individual articles, dissertations, and books on the history of engineering graphics, as well as historical surveys in textbooks that appeared by the fiftieth anniversary of the Soviet state, do not adequately reflect the special features of the development of axonometric projections.

3. The results of special studies on the history of axonometry began to appear in print only from 1959, in the articles [21, 6, 22, 23, 7, 8].

LITERATURE

1. N. F. Chetverukhin. Descriptive geometry as a mathematical science. Proceedings of the interuniversity seminar on descriptive geometry. Moscow, Publishing House of the All-Union Correspondence Power Engineering Institute, 1959.

2. V. I. Lenin. Philosophical Notebooks. Moscow, Publishing House of the Central Committee of the VKP(b), 1934.

3. Questions of the history of the physical and mathematical sciences. Collection of reports of interuniversity conferences, Moscow, “Vysshaya shkola,” 1963.

4. History and methodology of the natural sciences. Issue IV, pp. 3–7, Moscow, Moscow University Press, 1966.

5. Bulletin of the Ministry of Higher and Secondary Specialized Education of the USSR, No. 11, Moscow, “Vysshaya shkola,” 1967.

6. V. A. Voskresenskii. A brief outline of the development of axonometry in the Soviet Union. Collection “Questions of the theory, applications, and methodology of teaching descriptive geometry.” Riga, Publishing House of the Riga Institute of Civil Aviation Engineers, 1960.

7. V. A. Voskresenskii. From the history of axonometric representations. Proceedings of TPI, vol. 143, Tomsk University Press, 1966.

8. V. A. Voskresenskii. On theoretical studies in the field of axonometry. Proceedings of TPI, vol. 143, TSU Press, 1966.

1. A. A. Kuzin. A Brief Outline of the Development of Drawing in Russia. Moscow, Uchpedgiz, 1956.

2. A. I. Aleksandrov. From the History of Engineering Graphics in the Urals and Siberia. Sverdlovsk, “Uralskii Rabochii” Publishing House, 1959.

3. D. P. Volkov. Some materials on the question of the development of descriptive geometry in Russia. Proceedings of the Georgian Polytechnic Institute. Collection on Descriptive Geometry and Engineering Graphics, No. 1 (49), Tbilisi, Publishing House of the Georgian Polytechnic Institute, 1957.

4. N. A. Levitskaya. Questions of the history of technical graphics in the course of machine-building drawing. Proceedings of the Moscow Scientific-Methodological Seminar on Descriptive Geometry and Engineering Graphics. Issue II, Moscow, 1963.

5. V. A. Shulzhevich. On the question of the significance of descriptive geometry in the nineteenth century. Collection of Works of LIIZhT. Issue 240. Moscow–Leningrad, LIIZhT Publishing House, 1965.

6. N. F. Chetverukhin, V. S. Levitskii, Z. I. Pryanishnikova, A. M. Tevlin, G. I. Fedotov. Course in Descriptive Geometry. Moscow, Gostekhizdat, 1956.

7. N. F. Chetverukhin, B. S. Levitskii, et al. Descriptive Geometry. 2nd ed., Moscow, “Vysshaya Shkola,” 1963.

8. N. N. Krylov, P. I. Lobandievskii, S. A. Men. Descriptive Geometry. Moscow, Fizmatgiz, 1959.

9. R. S. Briling. Descriptive Geometry. Kharkov, Kharkov University Publishing House, 1962.

10. V. A. Pakhomova. The History of the Development of Methods of Representation. Omsk, Publishing House of the Omsk Institute of Railway Transport Engineers, 1963.

11. K. A. Rybnikov. History of Mathematics. Part II. Moscow, Moscow State University Publishing House, 1963.

12. G. Wieleitner. History of Mathematics from Descartes to the Middle of the Nineteenth Century. 2nd ed., translated from the German by A. P. Yushkevich, Moscow, “Nauka” Publishing House, 1966.

13. V. A. Voskresenskii. On the principal directions in the development of axonometric projections in the USSR. Proceedings of TPI, vol. 104, Tomsk University Publishing House, 1959.

14. V. A. Voskresenskii. Stages in the formation of axonometric projections. Proceedings of the Moscow Scientific-Methodological Seminar on Descriptive Geometry and Engineering Graphics. Issue II, Moscow, 1963.

15. V. A. Voskresenskii. Oblique axonometry. Abstracts of Papers of the Second Scientific-Methodological Conference on Descriptive Geometry and Engineering Graphics of Higher Educational Institutions of the Uzbek SSR, with the participation of representatives of higher educational institutions of the fraternal republics. Tashkent: Publishing House of the Tashkent Institute of Railway Transport Engineers, 1965.