Against an Incorrect Assessment of the Scientific Legacy of Leonardo da Vinci*

D. G. Stan’ko

Numerous researchers, both foreign and some Soviet ones, fail to perceive in the works of Leonardo da Vinci the development of questions of mathematics. At the same time, F. Engels counted Leonardo da Vinci among the great mathematicians.

The question arises: was this a slip of the tongue on Engels’s part, or did he have some other grounds for assertions of this kind? Engels lived in an epoch when mathematics had already attained a high level of development and operated with concepts of high degrees of abstraction.

Interpreting mathematics from the materialist point of view, Engels writes the following:

“The concept of figure, like the concept of number, is borrowed exclusively from the external world, and did not arise at all in the head out of pure thought. Before people could arrive at the concept of figure, there had to exist things that had form, and whose forms were compared. Pure mathematics has as its object the spatial forms and quantitative relations of the real world, that is, a very real content. The fact that this content appears in an extremely abstract form can only faintly obscure its origin from the external world. In order to study these forms and relations in their pure form, one must separate them completely from their content, eliminate it as something irrelevant to the matter. Thus are obtained points without extension, lines without thickness and breadth, $a$ and $b$, $x$ and $y$, constants and variables; only at the very end do we arrive at the real ‘free creations and fantasies’ of the reason, namely at imaginary quantities. Likewise, the derivation of mathematical quantities as if from one another proves not their aprioristic origin, but only their rational connection…”

“…Like the other sciences, mathematics arose from the needs of man: from the measurement of land and the capacity of vessels, from the computation of time and mechanics. But, as in all fields of thought, at a certain stage of development the laws abstracted from the real world become detached from the real world, are opposed to it as something independent, as laws that have appeared from outside and according to which the world must be directed. So it was with society and the state; so, and not otherwise, pure mathematics is subsequently applied to the world, although it is borrowed from this world and represents only a part of its composite forms, and, in fact, only for this reason is it at all applicable to it” [1, 33].

Here there is an entire program of views on mathematics from the materialist point of view. And in this light the posing of the question whether Leonardo da Vinci was a mathematician acquires a principled character—namely: what and how should be included in questions of mathematics.

* Editor’s note. The article is published for discussion.

In the present work, proceeding from Engels’s positions, an analysis is made of the work of Leonardo da Vinci in connection with questions of mathematics.

Leonardo da Vinci (1452–1519) was one of the outstanding representatives of the Renaissance. A great artist, sculptor, architect, and scientist, he touched upon and gave to humanity brilliant ideas in all fields of science: physics, mathematics, astronomy, geology, and chemistry. His creations and ideas outlived by many centuries the epoch in which he lived. Only in our time is it possible to assess the whole many-sided activity of Leonardo da Vinci.

As regards the wealth of factual data in the field of Leonardo da Vinci’s scientific work, one may note the detailed works of L. Olshki [3] and M. A. Gukovsky [4].

By critically analyzing the conclusions of the above-mentioned authors, and by interpreting the factual material from materialist positions, we are able to establish why Engels ranked Leonardo da Vinci among the great mathematicians.

Leonardo da Vinci formulated his thoughts as follows: “He who loves practice without science is like a sailor who boards a ship without rudder and compass.” On the other hand, sciences without practical benefit are not viable and “die just as quickly as they live” [3, 174].

In questions of cognition Leonardo da Vinci “held that one must begin with the simplest things and pass on to the second in order only when one has well remembered and learned the first” [3, 193].

As an artist, he gives preference to optics. Seeing in the latter one of the important physico-mathematical sciences, Leonardo da Vinci writes that, if painting “extends over the surface, the colors, and the form of each object created by nature, then philosophy penetrates inside the latter and considers its essence, but it is not in a position to attain that truth which the painter can know when he perceives the first evident truth of things with the eye, which is less deceived” [3, 175]. Moreover, he regarded all kinds of syllogistic proofs as dubious chatter. At first glance, all these statements would seem to speak of Leonardo da Vinci’s limitedness. But this is far from the case. One must not forget that Leonardo da Vinci’s attitude toward philosophy was quite natural and regular. He could not have had any other attitude toward that philosophy which was capable of “profoundly” considering such questions as how many angels can fit on the point of a needle? can God know more than He knows? did Adam have a navel? and so on and so forth. Leonardo da Vinci’s merit lies in the fact that he broke free from the chains of medieval scholasticism and theology.

It is noted that throughout his life Leonardo da Vinci treated the theological explanation of the world skeptically or indifferently. Nor is it accidental that Leonardo da Vinci valued optics highly. He considered perspective the guide of painting, and in its domain “the light ray is treated by means of proof, so that in optics one may see the glory not only of mathematics but also of physics, for it is adorned with the flowers of both the first and the second” (3, 175). Optics enriched mathematics through geometrical images, as the laws of perspective. The latter, in turn, made it possible to reveal the regularities inherent in the optical in the physical sense.

Not only painting but also technology was for Leonardo da Vinci a “paradise of the mathematical sciences, for here the fruit of mathematics ripens” [3, 175].

Leonardo da Vinci began very early to concern himself with technology, carrying out difficult projects of various kinds. He touched on almost all questions of mechanics. Thousands of drawings, projects, sketches, have survived to our time,

encountered accompanied by a few cursory indications and descriptions in his manuscripts. All this outlines the indefatigable and colossal activity of Leonardo da Vinci. From his early years he set himself the task of writing an encyclopedia of technology, theory and practice, for himself and for others.

Leonardo da Vinci critically mastered earlier works on technology, conducted experiments, investigated, and, in the light of accumulated facts, rejected or accepted old propositions, or else revealed new regularities.

On the basis of Duhem’s research, Olyshki indicates that “indeed, Leonardo da Vinci came close to the concept of the moment of a force; he guessed at various kinds of equilibrium of weights; he tried to determine motive force as the product of a raised weight by the height attained; he was even on the way to pointing out that the moment of the resultant of two forces is equal to the sum of the moments of the components, and so on; but not once was he able to express, in a general or definitive formulation, laws which, as a result of repeated calculations and experiments, would have presented themselves to a less gifted but more systematically working mechanic. This fell to the lot of Galileo’s immediate predecessors—Benedetti, Tartaglia, Cardano, and Baldi—who all drew from the very same scholarly sources as Leonardo da Vinci, or else, as Duhem showed in a number of cases, ‘drew directly from his manuscripts’” (3, 188).

Olyshki has doubts concerning the fact that Leonardo da Vinci, having failed to formulate laws, did not understand the character of the universality of the questions under consideration. But this is far from being the case. In fact, suppose we have the image of a rectangle and, within it, diagonals. This image can be read in different ways: namely, as a series of intersecting segments; as the boundary of a surface connected in a definite way with segments; as an area expressing the product of two numbers; as a figure composed of two right triangles; as an expression of the fact that the sum of the areas of the squares constructed on the sides of the rectangle is equal to the area of the square constructed on the diagonal, using the Pythagorean theorem as a means by which mathematical relations are expressed. One could continue the reading, but even this is enough to state that a meaningful geometrical image—the rectangle—is an object of investigation and a means of reflecting regularities in the mathematical sense, as a measure-number. The regularities inherent in a unit rectangle retain their force for all similar rectangles, thereby encompassing the diversity of measure-number, and through the mediation of the latter connect regularities with quantitative number. The unit rectangle, as a geometrical image, also reflects a series of regularities inherent in the rectangle in general—if only the sum of two rectangles in connection with the Pythagorean theorem.

All this was known even before Leonardo da Vinci. But in his investigations he developed this method, applying the geometrical image to express the relations of magnitudes in observed physical phenomena. Leonardo da Vinci obtained geometrical images as abstract impressions from actually observed objects. Interconnected lines and points, forming geometrical figures, served as a means of abstract expression in a general form of the questions under consideration. It is characteristic, too, that by means of the geometrical image Leonardo da Vinci operated in the mathematical sense, while changing the relations between circles, lines, and points. This can be seen directly from Leonardo da Vinci’s manuscripts, photographs of which are given in M. A. Gukovsky’s work The Mechanics of Leonardo da Vinci. This work contains 236 drawings, of which 14 are photographs of sheets from Leonardo da Vinci’s manuscript. Most of the drawings are geometrical images, accompanied by a scant description of the questions under consideration. Gukovsky does not pay proper attention to the geometrical im-

at times as the abstract expression of certain regularities; therefore his investigations and conclusions often have a limited character. One must in no case forget that Leonardo da Vinci freely read geometric images. It is also obvious that a drawing in general, and in the mathematical sense in particular, will be perceived differently by persons able to read a drawing and by those unable to read it. What form to give to what has been read in this sense is another question; in most cases Leonardo da Vinci limited himself to brief remarks on the geometric image, remarks at least sufficient for him. At the same time it must be noted that Leonardo da Vinci’s remarks do not fully exhaust the essence of the geometric image. If we now turn to concrete geometric images, one may point to the photograph of fig. 174, where we have an investigation of the equilibrium of a beam for various positions of it relative to the horizon, where the character of the change in the arm of action of the force is shown. The photograph of fig. 28 gives geometric images of an investigation of the equilibrium of a horizontal beam on two supports, with various positions of the supports. Photograph 227 (in the text, fig. 127, as a result of a misprint) gives geometric images connected with rolling friction. In implicit form geometric images are found in a number of the other photographs. Most of the drawings taken from Leonardo da Vinci’s manuscripts and reproduced by Gukovsky are not merely illustrations, but express the form of operations and of the solution of problems in the mathematical sense, by means of geometric images.

It is characteristic that Gukovsky, citing Leonardo da Vinci’s statement that “There is no certainty where one cannot apply one of the mathematical sciences, or where there is no connection with this mathematics,” and further, “Let no one read me in my foundations who is not a mathematician,” draws a peculiar conclusion: “it would be quite wrong to represent the matter as though Leonardo recognized only abstract mathematics in our sense of the word and as though all his science were strictly mathematical.”

Such an assertion would be incorrect in two respects: first, by the word “mathematics” Leonardo, like his whole epoch, understood something very different from what we understand at the present time. For him mathematics is a rather vague and unsettled concept, embracing first of all, evidently, everything that has to do with calculation and measurement. Second, and this is most important, Leonardo always firmly understood that true science is based on observation of reality, on experiment, while mathematics is only a means of processing the results of these observations and experiments” (4, 430—431).

In connection with Gukovsky’s statement, questions involuntarily arise: why should Leonardo da Vinci have had to understand mathematics in “our” sense of the word, and what is the meaning of such a way of posing the question for revealing Leonardo da Vinci’s scientific creativity?

One asks: what is vague and unscientific in the understanding of abstract mathematics as a reflection of counting and measurement? Of course, the essence of mathematics is not exhausted by the latter. But at the same time we have no grounds for denying that mathematics is connected with counting and measurement; to deny this means not to recognize the experiential origin of mathematics, that is, not to recognize that in mathematics we have the material, transplanted into our head and transformed by our consciousness.

Gukovsky believes that, for Leonardo da Vinci, mathematics is only a means of processing the results of observations of reality. Such a conclusion is one-sided and does not reflect Leonardo da Vinci’s views, if only because he used geometric images not only for processing observations, but also for revealing regularities in the mathematical sense. This follows from the fact that Leonardo da Vinci, in the phenomenon under study, singled out and abstracted the geometric image, thereby showing

saw how abstract mathematics is born. Moreover, Leonardo da Vinci operated with geometric images directly, that is, in the form of abstract mathematics. On this point Olski remarks: “In a number of the artist’s manuscripts, especially (1497), we find a long series of notes on elementary geometry, containing the usual Euclidean definitions of the point, the line, the angle, the triangle, and so on. No one would venture to say that Leonardo, having passed beyond 45 years of age—that is, at the time when he was engaged with the most difficult problems of mechanics—was memorizing the rudiments of geometry” [3, 206].

Olski believes that Leonardo da Vinci was not engaged in geometrical investigations without reason. Gukovsky, however, simply fails to understand the significance of the geometrical image. This is evident at least from the fact that he assesses as geometrical amusement the drawings—the geometrical images—studied by Leonardo da Vinci outside their connection with a physical phenomenon.

Gukovsky underestimates Leonardo da Vinci’s mathematical creativity in connection with geometrical images. But if the question is put thus: what would remain of Leonardo da Vinci’s legacy if the geometrical images were cast out of his scientific work?

Then, perhaps, it would be almost impossible to reconstruct and explain the creative path and those achievements that Leonardo da Vinci possesses.

Approximately the same assessment of Leonardo da Vinci’s mathematical creativity as Gukovsky’s, but in a sharper form, is given by Olski. He asserts: “Anyone who is capable of understanding and calling mathematics an experimental science thereby proves that he is completely devoid of a correct, or at least general, conception of its essence and aims. He, in general, cannot be a mathematician” [3, 181].

Such a position is plainly idealistic, since in mathematics the primacy of the material is denied. Mathematics is not reduced to experience, but is derived from experience; consequently, experience is not indifferent to mathematics.

Geometrical images are an intermediate stage of cognition between experience and subsequent generalizations. But an already comprehended geometrical image, as an expression of the relation between counted and measured objects and quantities, acquires relative independence, appears in the form of an object of investigation and in the form of expressing the result in a general form. All this as a whole is indeed the domain of mathematics, regardless of the opinions of both Olski and Gukovsky.

Along with this, Olski correctly noted the characteristic features of Leonardo da Vinci’s creativity when he writes: “The result of this is that the so-called universality of the reality he considered, and not of the spirit considering it, for him is all equally interesting and knowable, once only his eye has come to rest on something” [3, 181].

Of course, such a materialist conception does not satisfy Olski, which is why he assigns Leonardo da Vinci to the category of those who do not understand mathematics, although here the opposite is precisely the case. Olski’s remark concerning the eye must be understood in a relative sense, since cognition for Leonardo da Vinci is not simple contemplation. But it is also true that the eyes of the brilliant artist and thinker noticed much that escaped and passed unnoticed by the ordinary person. Olski understands this and gives a correct assessment of Leonardo da Vinci’s views on cognition, observing that “...practice meant for him (Leonardo), above all, the disclosure of all the secrets of nature, and also that he expected to find in it a means of comparison with theory, as well as a criterion of scientific truth. This view was supported by the circumstance that such a method, applied within proper limits, had already rendered good service to art and technology, and directly connected science with life and reality” [3, 195].

Olschki’s idealism leads him to erroneous conclusions. He believes that Leonardo da Vinci’s method had a negative effect on his scientific work, especially in the area of generalizing the questions he studied. Olschki’s conclusion follows from an incorrect understanding of the essence of cognition in general, and in mathematics in particular. Olschki cannot understand or explain why Leonardo da Vinci, in his manuscripts, repeatedly returns to a drawing in connection with the study of an inclined plane; why, in studying questions of mechanics, problems of infinity in the mathematical sense arose for him, seeing in all this the unsystematic nature of the artist’s work. Leonardo da Vinci himself gives a clear explanation of all Olschki’s perplexing questions. Leonardo da Vinci writes that “no human investigation can be called true science if it has not passed through mathematical demonstrations” [3, 221], (emphasis by D. Stanko).

Here, with complete obviousness, is revealed the significance that Leonardo da Vinci attached to his drawings. He attributed to the drawing not only the form of visual representation, but also the form of expressing universality through a geometric image. Leonardo da Vinci’s aphorism contains elements of a general theory of cognition, since only the unity of the quality and quantity of the phenomenon under investigation makes it possible to penetrate more deeply into the essence of the question being studied, to encompass it more broadly, and to draw correct generalizing conclusions.

Olschki denies the presence of any method in Leonardo da Vinci’s investigations, seeing in his works only the disclosure of individual scattered facts, often in the form of conjectures, in the form of aphorisms.

Olschki believes that Leonardo da Vinci’s style of work is determined by his individualism and by the absence of systematic education. If one can agree with the first point, understanding by individualism the individual traits inherent in Leonardo da Vinci as an artist, thinker, and scientist, then the second assertion is doubtful. It is difficult to imagine that scholastic education, even if systematic, could have had a positive influence on Leonardo da Vinci. More likely, the opposite could have been expected: that such education would have crippled him and turned him into an ordinary person of the era contemporary to him.

Olschki’s idealistic conception leads him to incorrect conclusions. He writes that “therefore we shall not call Leonardo da Vinci a ‘universal genius,’ but shall call him a brilliant eclectic and shall regard his work as the most direct and obvious expression of his personality and of his individual peculiarities in this sense. From this it follows that we must investigate Leonardo da Vinci’s activity not as a certain whole, but consider it as the sum of independent separate cases, and hence that we must speak not of Leonardo da Vinci’s method of investigation, but of his scientific style” [3, 202].

Thus, according to Olschki, in Leonardo da Vinci’s work the decisive significance remains with style, while method is absent.

If we speak of the external aspect of Leonardo da Vinci’s style, then it is a long series of aphorisms on various questions, often not connected with one another; it is a series of manuscripts that can be read in mirror writing; it is a series of illustrations—drawings and diagrams, very poorly supplied with text; it is, on the whole, preliminary sketches, notes in the form of draft manuscripts. At the same time, a series of questions arises: why, then, has Leonardo da Vinci’s work been preserved and survived down to the present time?

Why has his work attracted and continued to attract numerous researchers, who gradually reveal the whole many-sided nature of Leonardo da Vinci, proceeding from the various kinds of his activity?

Why does his work, as a sum of independent, separately scattered cases, retain its significance for the present time?

If style is to be considered the main thing in Leonardo da Vinci’s work, then it is impossible to give any reasonably adequate answer to the questions posed, because his style has lost its significance for the present time. One could justify the enciphering of the manuscripts by the conditions of the epoch, when it was necessary to “keep one’s tongue behind one’s teeth,” and also by fear of plagiarism. The style of exposition in the form of aphorisms cannot be indicative, since this style did not become a model for the presentation of scientific research later on. It cannot be said that the style of investigating individual, disconnected questions was a model for the subsequent development of science.

Nor are the experimental results of Leonardo da Vinci’s investigations valuable—at least, for our time his experimental data have been replaced by new, more precise ones.

All this indicates that in the scientific work of Leonardo da Vinci there is something that has withstood the test of time and has preserved its freshness despite being 500 years old.

It is indisputable that Leonardo da Vinci was a brilliant artist. This is confirmed by the surviving legacy of his pictorial art. But Leonardo da Vinci was no less brilliant as a scientific investigator. This follows from his worldview and creative method.

According to Leonardo da Vinci’s worldview, practice was the criterion of truth; the universality of Leonardo da Vinci is connected with the universality of the reality he examined. It is also obvious that in his investigations and remarks the material is primary. This follows from a number of statements by Leonardo da Vinci cited above. In addition, let us cite a number of other thoughts of Leonardo da Vinci that confirm the propositions we have advanced.

On the question of the structure of the universe in connection with the problem of gravity, he drew the correct conclusion, believing that “the Earth is not at the center of the world, but only at the center of the elements kindred to it and joined with it” [3, 201].

At the same time, understanding by “element” everything connected with experiments carried out and observed within the bounds of the earth, Leonardo da Vinci is not at all satisfied with the biblical interpretation of the flood as to the height reached by the water and the duration of the flood in time. He remarks with wicked irony that “who measured it and calculated the time?” [3, 253].

In the same way he mocks those who explain the origin of fossil shells, fish bones, and the like by the flood and by “heavenly influence,” when he writes that the heavenly influences also took care to feed them [3, 253].

It cannot be said that Leonardo da Vinci’s views were accidental in character. His views followed organically from his scientific work, in which the material was primary.

Let us now dwell on Leonardo da Vinci’s method of investigation. Let us cite his remarks on friction. We read that “Since the friction of dense bodies is always equal to one fourth of the bodies rubbing... But in complex movements, i.e., with the aid of levers, which are of various kinds, namely: simple ones, in which only one lever is used; compound ones, in which many levers move one another; but here I shall speak of simple ones.”

“Hard moving body with a hard surface.”

“Soft moving body with a soft surface.”

“Soft moving body with a hard surface.”

“Rough moving body with a smooth surface.”

“Smooth moving body with a rough surface.”

“Rough moving body with a rough surface.”

“Smooth moving body with a smooth surface.”

“Rough and hard moving body with a soft surface.”

“Rough hard with rough hard—motion of extreme difficulty.”

“Rough soft with rough soft—of medium difficulty.”

“This is motion with the action of intermediate bodies—as when sand is placed between the movable body and the surface; and if the intermediate grains have a round shape, the motion will be easy; and when the intermediate grains are larger, the motion will be easier, if they are round; and this property of motion is not a feature peculiar to rubbing motions, but also occurs in motions through liquids and friction...”

There follows a series of generalizations: “On the material that offers the greatest resistance in friction. Friction occurs in three ways, namely: hard with hard, soft with soft, and soft with hard. The contact of these three varieties is distinguished into two kinds, of which the first is the simple contact of rubbing bodies, the second—composite contact. In simple contact no substance enters (between the rubbing surfaces), but only these surfaces themselves come into contact with one another. In composite contact, however, different substances enter between the surfaces of the rubbing bodies, and these substances abrade the above-mentioned bodies the more, the rougher these surfaces are. But if the rubbing bodies have different hardness, then the softer abrades the harder, and this occurs as a result of the roughness of the substance located between the surface, which eats into the substance of lesser hardness and, having become fixed in it, works like a file, with which it then abrades the hard substance.

But if the substances possess equal hardness, then the substance placed between the surfaces grinds itself down if it is less hard than the rubbing things; but if it is harder, then it absorbs this friction, just as two files of equal strength abrade one another” [4, 790—791].

In his investigations Leonardo da Vinci strives to embrace the entire aggregate of causal connections that condition sliding friction, and to single out from all of them the principal connections. Such a method of investigation is by no means metaphysical, but includes elements of the dialectical method on a materialist basis. And it must be said that dry sliding friction appears in Leonardo da Vinci in all its diversity. Moreover, if one does not count the quantitative side of the question, then not so very much was added to his explanation of dry friction in subsequent works. The expression of the form of universality of the phenomenon under study leads to the discovery of laws. In this respect there are data showing that Leonardo da Vinci came close to formulating the laws of friction. At least he singled out the point that the coefficient of friction is equal to 0.25. And subsequently this quantity was only refined and subdivided qualitatively and quantitatively, while in essence preserving within itself those general propositions that were formulated by Leonardo da Vinci on questions of friction. No less thoroughly does he reveal causal connections in the study of rolling friction.

Observing the flight of birds and making sketches, Leonardo da Vinci set himself the task of uncovering the laws of flight in order to build a flying machine. In fact, researchers of a later period began to solve this problem in a similar way, when observation of the soaring of birds led to the creation of the glider, and after it the airplane.

Analysis and synthesis of any phenomenon and object under study—this is a characteristic feature of Leonardo da Vinci’s method of scientific creativity. Olschki cites an excerpt from Richter stating that “the synthetic concision, modeled on the object of nature and on mathematical proofs, which he (Leonardo) set himself as the ideal of exposition” [3, 232], draws incorrect conclusions,

considering that Leonardo da Vinci did not succeed in doing this. Rather, the opposite is true here, if one does not follow Olschki in his understanding of cognition, which reduces cognition only to formally logical propositions.

Science can develop only where abstract concepts are present. By processing the concrete in our consciousness, by singling out the characteristic traits and features inherent in the manifoldness of the concrete, we arrive at abstract concepts.

Abstract concepts possess a great power of universality and a relative indifference to the concrete. In cognition, abstract concepts have their own relative stages of development, transitions from a lower stage to a higher one.

At the highest stage of abstraction the concrete is so veiled that it makes possible an incorrect, illusory conclusion, turning the abstract into a kind of absolute. This is especially observed in mathematics and in those sciences where the concrete is connected with the abstract in an indirect way.

The objectivity of scientific abstraction consists in the fact that abstraction finds its place in the concrete, for it has been singled out from the concrete. But at the same time any abstraction cannot exhaust all the richness of the concrete, since it reflects sides and features of the concrete. From all that has been set forth it follows that abstraction cannot be identical with the concrete, despite all the depth of its penetration into the concrete, the material. A deeper knowledge of an object requires the singling out of a number of abstract concepts characterizing various aspects of the manifoldness of the concrete. The highest stage of abstraction is attained in the philosophy of dialectical materialism, as the objective reflection of knowledge of the world and of its laws in their most general form.

Leonardo da Vinci’s scientific legacy is connected with a number of generalizations, with abstract concepts. Leonardo da Vinci’s sketch of a plan for mechanics reads: “Speak first of motion, then of weight, why it arises from motion, then of the force that is born of weight and motion, then of impact, which is born of movable weight and often of force.”

“Speak of motion and of impeto.”

“Speak of wheels rotating in one direction.”

“Speak of wheels going and returning.”

“Speak of wheels of increase.”

“Speak of infinite screws.”

“Speak, first, of the lever and the scales.”

“Speak of mills and other devices for motion and throwing.”

“Of teeth.”

“First you will speak in detail of axles and of where they are placed, then of wheels and pulleys without teeth, then of wheels and pulleys with a thread, and then with teeth. And you will say which teeth are more suitable by nature, according to motion and the rules” [4, 445].

Here we see reflections of the method of investigation and the basis for the subsequent generalizations that Leonardo da Vinci mentions, such as “On motion,” “On weights,” “On machine elements.”

The program is carried out and recorded by Leonardo da Vinci in manuscripts, where a very brief deciphering is given of what is being considered in the questions touched upon, and how.

If one seeks formal logical rigor in definitions and concepts in Leonardo da Vinci’s notes, then in this respect he is not beyond reproach. But it would be wrong to reduce the essence of abstract concepts to formal logical rigor, because formal logic, owing to its limitations and metaphysical character, cannot encompass the connection of the abstract and the concrete in their totality. It does not follow from this that this or that definition,

a concept must have the character of a collection of unrelated words and sentences. There exists an objective dialectic, and subjective dialectic is a reflection of the former. It is in this light that Leonardo da Vinci’s investigations must be considered. Concepts such as the inclined plane are regarded by him not only as the motion of a real body, but are also studied from the standpoint of mechanics, and therefore of mathematics, through the medium of a geometric image.

The inclined plane in the form of a geometric image, the right triangle, and the slider on the hypotenuse, already as something abstracted from the concrete, has retained its significance down to the present day, if only in the study of the screw pair, the wedge, the worm gear, and so on. Consequently, the inclined plane has a universal character, since it finds its place in the manifoldness of the concrete. At the same time, we cannot identify the screw pair, the wedge, or the worm pair with the inclined plane, because this would mean confusing the concrete with the abstract. The inclined plane was studied by Leonardo da Vinci in connection with weight; but this in no way diminishes his achievements for the subsequent period, when a more general case was investigated, since his method of inquiry retained its force.

In a similar way Leonardo da Vinci developed the doctrine of the lever as applied to a system of pulleys and to gear transmissions. Here again we have the separation of geometric schemes from real bodies. The lever itself, as a physical body, is used in lifting weights and in weighing. The lever and scales were already known in antiquity. Archimedes studied the condition of equilibrium of the lever. In his investigations Leonardo da Vinci considerably broadened the scope of the problem of the lever, extending it to other mechanisms. Applying the rule of the lever, he in essence operated with moments of forces—or, more precisely, with the conditions of equilibrium of the moment of forces for an instantaneous position of the mechanism. This operation, at a higher stage of cognition, became the basis of the kinetostatic investigation of mechanisms of the present day. His geometric schemes played the role of kinematic schemes of the mechanism and were the prototypes of the kinematic pairs on which the general theory of machines and mechanisms is built.

In investigating the equilibria of an inclined lever under the action of a vertical and a horizontal force, Leonardo da Vinci introduced the concept of the potential lever, representing, on the one hand, the arm of the action of a force and, on the other, the components of the resultant acting along the sides of a rectangle. In this analysis, Leonardo da Vinci uses the geometric scheme of a rectangle, in which the diagonal is the distinguished axis of the lever and the sides express the relative magnitude of the weight; that is, we have the composition of forces according to the rule of the rectangle, later developed into the rule of the parallelogram. It is characteristic that, with the aid of a geometric image, Leonardo da Vinci reveals the laws governing the change of the components of forces for a constant resultant [4, 28].

We have cited certain characteristic features of Leonardo da Vinci’s creative method, which may be expressed as follows:

1. In him, various concepts appear in the form of causal connections. The concepts are deeply connected with the content of the questions under study.

2. His abstractions find their place in the concrete, since he mentally transformed the material; moreover, Leonardo da Vinci shows how this transformation is carried out.

3. The form in which the result is expressed, in the mathematical sense, finds its place in him through geometric images—as number-measures.

4. Practice and theory in Leonardo da Vinci appear in unity, in mutual connection.

The points enumerated above are precisely what retain their force and freshness

for the present time. By virtue of their objectivity, these points retain their force for the future as well.

Of course, Leonardo da Vinci cannot be regarded as a consistent dialectical materialist. In his work there are also some erroneous conclusions, which are explained to a considerable degree by the insufficient amount of accumulated knowledge, and also, to a certain extent, by the influence of the epoch in which he lived.

Leonardo da Vinci appeared as a revolutionary in many fields of science. He did enormous work in the critical assimilation of the old and in the creation of the new in the science and technology of that time. His works have survived to our time in the form of various kinds of drawings and rough notes. The works show, in their immediacy, the laboratory of Leonardo da Vinci’s scientific creativity; in this lies their considerable value for the present time.

But for Leonardo da Vinci his rough sketches constituted material for subsequent processing. From his early years he had already set himself the task of writing an encyclopedia of technology. To the end of his life Leonardo da Vinci intended to systematize his fragments by specialty. His studies of the Latin and Italian languages were not accidental. Most likely these studies were connected, to a considerable degree, with the need for the final shaping of the accumulated material, when the vocabulary of the Italian language was insufficient for scientific exposition. It must be assumed that Leonardo da Vinci realized that his form of presenting the fragments was adequate for him, but not for others. Of course, as in any new, revolutionizing scientific discovery, significant difficulties arise in giving it form. It turns out as follows: form lags somewhat behind content; the old form cannot correspond to the new content. The problem of the relationship between form and content confronted Leonardo da Vinci in all its acuteness.

It is necessary to note that this problem was not resolved by Leonardo da Vinci. Its resolution was also greatly impeded by the fact that, by the make-up of his character, Leonardo da Vinci gave preference to direct practical activity, which captivated him in all its diversity. In the final analysis, although his generalizations were fragmentary in form, they were profound and far-reaching in content. All this as a whole in no way diminishes the achievements in the scientific creativity of Leonardo da Vinci.

In his work The Mechanics of Leonardo da Vinci, Gukovsky considers Leonardo da Vinci as a man and as a scholar. It would seem that Gukovsky had every opportunity to give such a characterization, in which what was decisive, principal, and of interest for our time would be objectively brought out. If Olschki recognizes Leonardo da Vinci as a brilliant eclectic, then Gukovsky, by his characterization, simply dissolves the image of Leonardo da Vinci.

Gukovsky’s characterization of Leonardo da Vinci reads as follows:

“A worldly dandy and an eccentric, but at the same time a profound thinker and experimenter, a salon chatterbox and an infinitely secretive creator, an artist and a scholar of heroically bold ventures and childishly weak perseverance, a resolute innovator, always and in everything striving to solve matters himself, in a new way, in his own way, not like others, and at the same time never completing anything; a man passionately loving and pitying animals and with cold cynicism sketching executed people—such is Leonardo, one of the creators of modern science” [4, 418].

Such is the characterization of Leonardo da Vinci as a man. No better is the characterization of him as a scholar, which reads: “However, a number of defects in Leonardo da Vinci’s mechanics, its fragmentariness and mathematical imperfec-

…perfection in no way diminishes its general significance. Despite them, and to some extent precisely because of them, it is the first creation of a new science that closely linked its fate with practice. This science not only explains the world, but also makes the first, still timid and awkward, attempt to remake it. And it is precisely this quality of Leonardo da Vinci—the Florentine artist, technician, and scholar, the brilliant eccentric and failure—that raises it above the level of a separate, particular, though very important phenomenon; it gives it a special principled significance and symptomatic character” [4, 310].

In both of Gukovsky’s characterizations there is nothing that would make it possible to include scholars among the ranks of geniuses. Such a characterization is superficial and not profound. It follows from Gukovsky’s incorrect understanding of Leonardo da Vinci’s work.

Engels gives an entirely different characterization—an objective one, in keeping with Leonardo da Vinci’s work—when he writes:

“Leonardo da Vinci was not only a great painter, but also a great mathematician, mechanic, and engineer, to whom the most diverse branches of physics owe important discoveries…” [2, 87].

This clear characterization retains its force even in our own day. Gukovsky also cites the characterization given by Engels. It remains only unclear how Gukovsky contrived to reconcile his own characterizations with the characterization given by Engels.

In conclusion, let us note that, proceeding from the historical development of mathematics in its primary material—that which has been transplanted into our heads and transformed in them—we have the following in Leonardo da Vinci’s mathematical legacy:

1. He singled out abstractions from the concrete in the form of geometric images, thereby showing how mathematical relations arise in abstract form.

2. He went somewhat beyond Euclid, since he studied relations among the geometric elements of moving geometric images. His attempts to comprehend infinity on the basis of geometric images are quite natural, if only because he encountered the question of the continuity of motion. As an artist, Leonardo da Vinci freely “read” geometric images.

3. His mechanics was mathematically formulated by means of a series of geometric images. The elements of the latter—segments—expressed directed quantities: forces associated with moments of forces; there were operations of geometric addition according to the parallelogram rule, etc. In this sense we find ourselves at the sources of the emergence of vectors and vector calculus, when the abstract had not yet been obscured.

Such is the conclusion, if one abandons the understanding of mathematics merely as the development of logical forms, assuming that the latter can flow from objective content only as something secondary.

LITERATURE

1. F. Engels. Anti-Dühring. Gosizdat. Moscow—Leningrad, 1930.

2. F. Engels. Dialectics of Nature. Partizdat, Moscow, 1934.

3. L. Olshki. History of Scientific Literature in the New Languages. Vol. 1, GTTI, Moscow—Leningrad, 1933.

4. M. A. Gukovsky. The Mechanics of Leonardo da Vinci. Publishing House of the Academy of Sciences of the USSR, Moscow—Leningrad, 1947.