From Nicolaus Of Cusa To Leonardo Da Vinci: The “Divine Proportion” As A Principle Of Machine-Tool Design, Part I

Can You Solve This Paradox?

by Jonathan Tennenbaum

The following two-part discussion is intended to prompt a richer reflection on what was presented earlier, concerning Analysis Situs, the paradox of “incommensurability” in Euclidean geometry, and Nicolaus of Cusa’s discovery of a higher geometry based on “circular action.” At the same time, I will set the stage for a new series of pedagogical demonstrations, to be developed in coming weeks.

When you have encountered a new physical principle, you cannot just put it in your pocket and walk away. The new principle, if validated, implies a more or less revolutionary change in the entirety of existing knowledge. We have the task of integrating the new principle (“new dimensionality”) into a new, comprehensive hypothesis-system, incorporating the results of all pre-existing valid demonstrations of principle (i.e., the valid side of existing knowledge), as well as the new demonstration, as a new manifold of “dimensionality N|+|1.” What is the measure of the change in the per capita productive power of society, associated with the “impulse ratio” (N|+|1)/(N)? And how can we push the new manifold “to its limits,” uncovering new experimental anomalies which will provide us the stepping-stones on the way to future manifolds N|+|2, N|+|3|,|…?

From Cusa to Leonardo and Beyond

Would this sort of process be a fair way to characterize what happened during the 50-year period from Nicolaus of Cusa’s <cf2>“De docta ignorantia,”<cf1> to the collaboration of Leonardo da Vinci and Luca Pacioli on the “Divine Proportion”? Is it valid to conceptualize the scientific developments of the European Renaissance, from the Council of Florence through Leonardo and beyond, as a process of “integrating” Nicolaus of Cusa’s crucial discovery, with the best previous accomplishments of Classical Greek, Arab, and other European civilization?

Before entertaining the possible merits of such a working hypothesis, we should first make sure to reject any temptation to impose “linearized” misinterpretations on what Nicolaus of Cusa actually discovered. Here, as always, there is no substitute for “re-experiencing” the {process} of discovery, which at the same time constitutes its real {content.}

Among the most “tempting” and commonplace misinterpretations, for present-day readers, is to substitute naive visual imagination’s image of circular motion in empty space, in place of the radically different ontological conception of “circular action,” which Nicolaus actually adduced in his discovery. The promotion of this error by the Venetian agent Paolo Sarpi and his successors, as a willful fallacy, was key to the Enlightenment assault on the European Renaissance. Among other things, it provided the basis, via Galileo, Newton, D’Alembert, Lagrange, Euler, et al., for the elaboration of a so-called “analytical mechanics” as the model for an “Establishment science,” thoroughly “sterilized” against the seeds of discovery.

Circular Motion and Circular Action

Yes, there is a connection between the visible phenomena of rotation or circular motion, and Cusa’s principle of circular action. But the connection is that of a shadow to the real object, whose existence it lawfully reflects.

Two brief quotes from Nicolaus of Cusa himself might be helpful in this context. Both are taken from his mathematical essays on the quadrature of the circle and related topics. The first emphasizes the <cf2>Analysis Situs<cf1> principle of “relationship of species” as crucial to his discovery:

“Since polygons are not magnitudes of the same species as the circle, it is still the case, even though we can always find a polygon which comes closer to the circle than any given polygon, that among things, which can be made smaller or greater, the absolutely largest can never be attained in existence or possibility. In fact, the area of the circle is the absolute maximum relative to the areas of the [inscribed] polygons, which are capable of being more or less and therefore cannot reach the circular area, just as no number can ever attain the encompassing power of the Unity, nor the Composite the power of the Simple.”

Full Scope of Circular Action

Another essay ends with a magnificent stretto, in which Nicolaus reveals the full scope of his conception of “circular action,” encompassing the relationship between hypothesis, higher hypothesis, the hypothesis of the higher hypothesis, and “the Good”:

“We assert, therefore, that there exist beings of the nature of the circle, which could not be their own origin, since they are not like the absolutely greatest circle which alone is eternity. The other circles, which, indeed, seem not to have a beginning and an end, since they are conceived through abstraction from the visible circle, nevertheless, since they are not infinite Eternity itself, are circles whose being derives from the first, infinite and eternal circle. And these circles are, in a certain way, Eternity and complete Unity relative to the polygons inscribed in them. They possess a surface which incommensurably exceeds the surfaces of all the polygons, and they are the first images of the first, infinite circle, even though they cannot be compared with the latter on account of its infinity. And there are beings having an unending circular motion around the being of the Infinite Circle. These contain within themselves the power of all the other species, and from their enveloping power they develop, in imitation, all the other species; and, beholding everything within themselves, and beholding themselves as the image of the Infinite Circle, and through beholding this image–themselves–they raise themselves up to the eternal Truth or to the very Origin. These are the beings endowed with Reason, who comprehend everything by the power of their minds.”

Machine-Tool Design Prototype

By what mode of action do we expand the “enveloping power” of the human race, exercising increasing dominion over the Universe, and knowing Reason in the mirror of its own active participation in developing the Universe? What could be more fruitful, to deepen our understanding at this point, than to follow the track of Nicolaus’s discovery into the busy workshops and “design bureaus” of Leonardo da Vinci and his Renaissance friends! Here is the prototype of the “strategic machine-tool design sector,” which has been key to the emergence and survival of the modern nation-state up to the present.

Much oligarchical effort has been expended, over the centuries, to mystify and conceal the “machine-tool principle” underlying Leonardo’s work in all fields. For example, Leonardo is often portrayed as a “speculative genius” whose designs were wildly impractical in his day. As a matter of fact, much of Leonardo’s time was spent in direct collaboration with machine-building workshops and factories, as well as with construction teams involved in infrastructure and other projects, developing solutions to problems as they came up. Thus many, if not most, of Leonardo’s actual designs were implemented in his day.

Another malicious piece of gossip, spread by Joseph Needham among others, was that Leonardo made “no fundamental breakthrough” in the principles of machine-design. That assertion is commonly coupled with the assertion, that Leonardo was not a scientist, and that the real breakthrough, leading to the Industrial Revolution, came with the formal mathematical physics of Galileo, Newton, et al. For example, a book on Leonardo’s engineering work, published by one L. Olschki in 1949, claims: “The technical principles employed by Leonardo were hardly different from those handed down from antiquity and the Middle Ages…. He never attempted to frame new theoretical approaches or theories of mechanics.”

Leonardo’s Breakthrough

Leaving aside such malicious nonsense, get out a good collection of Leonardo’s sketches. Concentrate particularly on his designs for machines and mechanical devices of machines. Looking over those sketches, ask yourself: What was Leonardo’s crucial breakthrough in these matters? What is stunning, revolutionary, about Leonardo’s approach to the design of machines, and related matters, which went decisively beyond what had existed before? I am not talking about individual “inventions,” so often played up as isolated entities; I am asking for a “One.”

Whoever tends to read Nicolaus of Cusa’s principle of circular action as merely a form of “motion,” in the manner indicated above, will be plunged into a rather profound paradox at this point.

Looking at Leonardo’s designs, what do you see except mere mechanical linkages–assemblies of gears, pulleys, and levers, which transmit motion from one place and direction to another, without “adding” any new motion? Didn’t Archimedes already describe the basic mechanical principles involved, as typified by the action of the lever or pulley? Or is there something more than just “mechanics” in Leonardo’s machine-designs, something absolutely banned from the textbooks of “analytical mechanics,” but which is a key to the unprecedented rate of increase in the productive powers of labor, unleashed by the Renaissance?

To be continued.

From Nicolaus of Cusa to Leonardo da Vinci:

The “Divine Proportion” as a Principle of Machine-Tool Design Part II


by Jonathan Tennenbaum

Lyndon LaRouche’s discoveries in physical economy provide the key to unlocking the secrets of Leonardo da Vinci and the Italian Golden Renaissance, to a degree which would have been impossible at any earlier time, before LaRouche’s work.

Observe that the leading features of Leonardo’s designs for machine tools and other machines–most emphatically including the method of “non-linear perspective” employed in his drawings–all cohere with one central conception:

The emergence of {nation-state physical economy} as a {living process} based on development of the cognitive powers of individual members of society, imposes a unique “curvature of space-time” upon the Universe, such that each and every particular must be conceptualized and measured by reference to the “horizon” defined by that curvature.

That central conception subsumes the following features and consequences@s1:

1. A physical economy is a special type of living process, whose maintenance and growth depends on development of the cognitive powers of the individual members of society.

2. The action of human Reason upon the Universe, occurs {solely} through the instrumentality of living processes. That is, through the activity of sovereign human individuals, working in and through society, upon the expanding domain of Man-altered Nature which constitutes the “substrate” of physical economy as a living process.@s2

Reason’s Dominion Over the Universe

3. Hence, Leonardo da Vinci’s conception of non-linear-perspective curvature is based on a relationship of Nicolaus of Cusa’s “species”: Living processes exercise increasing dominion over inorganic processes, and human Reason exercises increasing dominion over the entire Universe via its dominion over living processes (i.e., human individuals, the physical economy, and an expanding biosphere).

4. In particular, the required notion of “technology,” appropriate to the maintenance and development of physical economy, {cannot} be derived from inorganic physics. No mere physical laws, of the sort suitable to “inorganic physics,” could ever account for the impact of a new machine or other invention on increasing the productive powers of labor.@s3 Although it is possible to design a machine on the basis of a simple hypothesis, we cannot measure its economic {effect} that way. The survival of human society, therefore, depends on shifting attention from the mere “engineering approach” of simple hypothesis, to encompass the “horizon” defined by higher hypotheses. Leonardo’s drawings have the included purpose, to communicate exactly that conception.

5. For these and related reasons, Leonardo’s studies of anatomy, and his collaboration with Luca Pacioli on the “Divine Proportion,” were decisive inputs to his approach to machine-tool design. Leonardo sought to apply to the design of machines, a reflection on the principles and means by which living organisms exercise dominion over the inorganic domain.

6. When a living organism incorporates non-living material into its active domain, it {imposes} its own characteristic {ordering} upon that material. (One day soon, the environmentalists might turn against plants and trees, denouncing them for imposing their “authoritarian values” upon poor, defenseless dirt!)

Harmonic Proportions

7. Leonardo, Pacioli, and others demonstrated how the peculiar space-time ordering of living processes finds its lawful {visible} expression in self-similar elaborations of the harmonic {proportions} derived from the division of the circle and sphere. The latter all belong to the dominion of the circle’s “Golden Section.”

8. This sort of approach points to a principle of {harmonic composition of motion} for the evolution of machine-tool designs integrating an increasing number and density of degrees of freedom. The harmonic principle of the “Golden Mean” will be reflected, not necessarily in the individual machine per se, but rather in the context of the evolutionary series of species of technology. The latter constitutes, on the one side, a central functional feature of the growth of physical economy as a living process, while at the same time embodying an ordering of mutually inconsistent theorem-lattices of increasing “power” under the principle of “higher hypothesis.”

9. In the continuation of this process, with the increase in energy-flux density and precision of machine-tool design, discoveries in microphysics oblige us to replace the concept of “motion” by a generalized notion of “harmonically ordered physical action.” The approach of Leonardo (and later Kepler) received preliminary, but brilliant confirmation in the domain of atomic and nuclear physics.

Beauty of Leonardo’s Drawings

10. Hence, the stunning beauty of Leonardo’s drawings! He communicates not merely a set of “specifications” for a machine, but a {conception}–a conception of that invention as seen in the perspective defined by the creative principle of the Universe as a whole. By this method of “non-linear perspective,” Leonardo is able to communicate the creative process itself, and not merely a particular product. Thus, Leonardo’s designs and machines are vehicles for the communication of higher ideas, for the generation of higher qualities of labor power. Like great Classical music, they embody Reason’s ironic reflection on the principle of life.


1. Besides Lyndon LaRouche’s writings, I would especially recommend juxtaposing to our discussion, the relevant articles by Dino de Paoli on Leonardo and related matters.

2. Have you stopped to consider the significance of the fact, that we need a brain in order to think? Actually, we need more than that: To develop, individual creative reason must continually expand and intensify its “active domain.” By the term “active domain,” I mean, roughly, the region of the Universe which is directly subject to the deliberate actions of a given individual. The growth of the active domains of members of society, is obviously correlated to increase in per capita and per hectare consumption of energy and other components of the market baskets, as it is to increase of the productive powers of labor. To the extent that the creative contributions of individuals are communicated and realized by society, their active domains may encompass the entire physical economy, and more. Would it be justified to conside, that growth of the “active domains,” in some respects represents an enlargement of the physiological processes of the brain, as an instrument for the development and realization of valid ideas?

3. Some might reject such a categorical proposition as preposterous. Don’t we know countless examples of inventions, whose labor-saving effects can easily be explained by any physics student? For example:

@sb|Levers, pulleys, and similar devices permit a single man to lift a weight which would otherwise require the muscle power of many men.

@sb|Ball-bearings and similar devices improve the performance of an existing machine by reducing friction and wear (a major focus of Leonardo’s work, by the way).

@sb|Steam engines and other power-generating devices multiply the amount of useful power at the disposal of an industrial operative, etc.

What could be more obvious, than the increase in productivity, caused by the above-mentioned inventions? Yet, such a casual affirmation overlooks at least one decisive point: What about the direct and indirect {costs} (in real terms) of developing, producing, and maintaining a given machine or technical improvement? How can we be {sure,} in any given case, that that additional cost will not actually exceed the saving in labor, or other benefits provided?

Observe, for example, the vastly greater complexity and intensity of motion of Leonardo’s machines, compared to the rudimentary gadgets of pre-Renaissance Europe. Even the simple act of introducing ball-bearings and related devices into machine design, adds new degrees of freedom to the system as a whole, raising the demands on the {quality of labor} required for the manufacture and maintenance of the machine. Actually, the purpose of the machines themselves, as means for urban-centered development of the nation-state, is not to “economize labor” per se, but to rather to {uplift its cognitive quality}. (Thus, while industrialization subsumes as a necessary aspect the reduction and final elimination of manual labor, a healthy industrial society actually increases the “work load” which must and can be borne by the average member of society.)

Reflecting upon such matters, we realize that the increase in the productive powers of labor, associated with the introduction of a new machine into the productive process, can hardly be determined from a mere analysis of the machine itself. It requires that we carry out a measurement of the entire economic process within which a proposed new machine design is to be “inserted.” Since the insertion of a new technology changes the characteristics of the economic process, that measurement must take into account, not only the present, but also its projected development in the future. In the last analysis, there is no adequate answer which does not center on the rate of improvement of the cognitive powers of labor, associated with any given “pathway” of economic development. Herein lies the cause of the essential “incommensurability” of real economic growth, relative to any linear sort of engineering or “systems analysis” standards of measurement. The significance of the Golden Section (“Divine Proportion”) comes once more to the fore.