Leonardo Da Vinci and Gears

Often, when we think of gears, we simply think of the mechanisms, physical capabilities, and qualities of different gears. This was my thought process, too, before coming to Italy and realizing the difference in paradigms between the Medieval period and the Renaissance. Leonardo Da Vinci's use of gears was as a language of temporal logic - constructing mechanical narratives that redefined motion as a means of power, not just force, contrasting the medieval view of time as divine and cyclical.

In Leonardo's studies, gears weren't just used as tools; instead, they were instruments of control, design, and meaning. Every rotation and transmission of energy is represented as a vector, a purposeful and controlled motion that was knowable, predictable, in harmony, and directed at a predetermined outcome. Through this lens, gears proved that time and logic could be engineered - that knowledge is eternal and produced through science. Leonardo's gears embodied the humanist view, where man can master these motions perfectly, and universal law reigns. As A. R. E. Oliveira notes, "Leonardo's great attraction to machines was due to his enormous interest in studying motion. By designing and building a machine, he could reproduce existing motions that appeared in nature" (232). This insight affirms that Leonardo's mechanical studies weren't just for technical pursuit, but also a philosophical one. Da Vinci's machines became models of cause and effect and analyzed systems. The gear can be concluded as a symbol of understanding the shape and flow of time, as was done in the Renaissance.

In class, we discussed how time was understood as a divine rhythm in the Medieval paradigm, but we began transitioning to an idea that humans could measure, manipulate, and ultimately control in the Renaissance paradigm. The concept of gears exemplifies this shift in ideology, where medieval thought was in place, while Da Vinci's gears made time functional - a resource to be harnessed rather than to be controlled, and something to sequence events and direction. Paul Virilio described the concept as dromology, which he describes as the study of speed as a political and strategic force. In his book Speed and Politics, Virilio states, "...the strategic value of the non-place of speed has definitively supplanted that of place, and the question of possession of Time has revived that of territorial appropriation" (149). In this view, time itself becomes a resource to gain power, contrasting the Medieval idea of placement and understanding through divine authority and control. Leonardo's machines offer this form of temporal control because of the use of gears in systems to create direct and purposeful sequences.

In fact, Walter Isaacson in Leonardo Da Vinci states that "[Leonardo] was interested in a part-by-part analysis of the transfer of motion. Rendering each of the moving parts—ratchets, springs, gears, levers, axles, and so on - was a method to help him understand their functions and engineering principles" (192). The idea of thinking through motion is clearly seen as his intent, and allowed him to model causality - and drive the profound new thought that the world wasn't a static place subject to God, but an idea of dynamic movement. Therefore, to a scientist such as Leonardo, the world becomes a system of flows and sequences to map and harness. This idea parallels what Hayden White discusses in The Content of Form about narrative - that humans try to develop the meaning of events not from the events themselves, but in attempting to construct a story through a beginning, middle, and end. In fact, he suggests that "this value attached to narrativity in the representation of real events arises out of a desire to have real events display the coherence, integrity, fullness, and closure of an image of life that is and can only be imaginary. The notion that sequences of real events possess the formal attributes of the stories we tell about imaginary events could only have its origin in wishes, daydreams, reveries" (23). Leonardo thought the same - he wanted to create a "narrative" or motion through gears, and find meaning in motion. Just as White argued that narrative gives form to reality, it can be argued that Leonardo tried to give meaning to time and space.

But how did Da Vinci go about implementing these ideas? Through notebooks such as the Codex Madrid I & II, it can be seen that he carefully studied and implemented spur gears, bevel gears, worm gears, rack and pinions, worm gears, planetary gears, gear trains, and even cams, though not specifically a gear, but still a concept of change in motion. In the sketches, it was evident that through his sketches and studies, he was able to break down complex motion into its elements, which revealed themselves through the gear elements.

In Codex Madrid I, we can see many of the systems: spur gears with proportioned tooth contact (see Fig. A1), rack-and-pinion conversions of rotational to linear motion (see Fig. A2), and bevel gear interactions for transmission of motion to different planes (see Fig. A3, A4). Most impressively, Da Vinci modeled a planetary gear in Folio 13v and 20v. He describes the motion with the inner planet gear n and outer planets a and m by stating, "When the big wheel revolves, pinions a and n will turn in motion contrary to each other. And pinions n and m will turn in the same direction, just as the big wheel and pinion a will turn in the same direction" (see Fig. A4). This was highly profound because while William Murdoch was credited with being the father of the planetary gear in the 1800s, Da Vinci had concepts of this concept hundreds of years before. This gear system is implemented in transmissions and gearboxes in modern automobiles, robotics, and aerospace systems, where efficiency and space constraints are in high demand. Not only was it innovative, but the planetary gear perfectly mirrored the Renaissance ideas of symmetry, order, and functional elegance. While Leonardo wasn't the inventor of the planetary system, his designs articulated the framework of motion that was modular, compact, repeatable, and intelligible. This was an absolute mastery of space and time, before many ideas of their relationship were even thought of. In fact, Einstein's theory of relativity is one of the first and most famous scientific frameworks to link space and time as interconnected dimensions, but it was published centuries later in 1931. The fact that Leonardo's machines unfolded across time in an ordered space showed his understanding of temporal logic in a spatial expression.

Leonardo's use of gears extended to complex mechanical systems as well. A notable example of this was a design for a mechanical calculator. According to Oliveira, the machine was "capable of performing the operations of addition and subtraction, manipulating numbers which can reach 13 digits" (224). According to the figure, gear trains with different gear ratios were used, which was an early implementation of a carry mechanism, which I was able to build as well in my ECE class (though with electronics). This and the planetary gear system highlight his understanding of modular gear systems, which allowed him to conceptualize his most famous machines, such as the self-propelled cart or the mechanical knight, discussed in class lectures. The mechanical knight in particular "was based on da Vinci's extensive anatomical studies. He had a deep understanding of the human musculoskeletal system and how muscles and joints work together to produce movement" (Ren, 22). These gear systems allowed him to extend his studies on anatomy and other fields, and embody the vision of a world governed by observable and reproducible laws of nature.

One other interesting aspect when looking at the Codex Madrid I and Leonardo's design was the use of bearings to reduce friction. While bearings are not directly related to the topic of gears, I believe they represent an extension of Leonardo's philosophy of motion, especially in continuity. Leonardo created ball bearings to reduce surface contact between moving parts (see Fig. A5), thrust bearings to support axial loads and vertical force (see Fig. A6), bearing supports to stabilize rotating shafts (see Fig. A7), roller bearings to distribute weight (see Fig. A8), and more inventions. Ladislao Reti discusses in Leonardo on Bearings and Gears how "[Da Vinci] studies the most frequent cause of their failure—frictional wear—and analyzes it quantitatively. He also proposes the substitution of rolling for sliding friction. In this respect, his invention of roller and ball bearings is the greatest merit. These were not occasional sketches, but serious studies, carried out with continuity over a long period of time and with full understanding of the working principles" (107). This statement reinforces that Leonardo's approach to motion and how much it aligns with Renaissance thought, how time is linear and constantly in motion.

Leonardo da Vinci's mechanical work proves to be more than just a simple mechanical function. His gears show an understanding of motion and vectors and the thought process of Renaissance thinkers, contrasting with the paradigm of the Medieval Ages. Leonardo saw time as engineerable, rather than the governable and fixed view of the church. Da Vinci's use of gears expressed causality and proposed a worldview of order, motion, and meaning. As Virilio states, "history progresses at the speed of its weapons systems" (67). Da Vinci wasn't just making machines, but also the logic to drive the whole world. The ideas of temporal logic implied an early understanding that control of movement was control over history and its narratives.

Fig. A1 Leonardo da Vinci. Codex Madrid I, Folio 5r. Biblioteca Nacional de España. Accessed via RWTH Aachen University:

https://www.codex-madrid.rwth-aachen.de/madrid1/f005r/index.html

Fig. A2 Leonardo da Vinci. Codex Madrid I, Folio 2r. Biblioteca Nacional de España. Accessed via RWTH Aachen University:

https://www.codex-madrid.rwth-aachen.de/madrid1/f002r/index.html

Fig. A3 Leonardo da Vinci. Codex Madrid I, Folio 96r. Biblioteca Nacional de España. Accessed via RWTH Aachen University:

https://www.codex-madrid.rwth-aachen.de/madrid1/f002r/index.html

Fig. A4 Leonardo da Vinci. Codex Madrid I, Folio 13v. Biblioteca Nacional de España. Accessed via RWTH Aachen University:

https://www.codex-madrid.rwth-aachen.de/madrid1/f013v/index.html

Fig. A5 Leonardo da Vinci. Codex Madrid I, Folio 20v. Biblioteca Nacional de España. Accessed via RWTH Aachen University:

https://www.codex-madrid.rwth-aachen.de/madrid1/f020v/index.html

Fig. A6 Leonardo da Vinci. Codex Madrid I, Folio 20v. Biblioteca Nacional de España. Accessed via RWTH Aachen University:

https://www.codex-madrid.rwth-aachen.de/madrid1/f101v/index.html

Fig. A7 Leonardo da Vinci. Codex Madrid I, Folio 101r. Biblioteca Nacional de España. Accessed via RWTH Aachen University:

https://www.codex-madrid.rwth-aachen.de/madrid1/f101r/index.html

Fig. A8 Leonardo da Vinci. Codex Madrid I, Folio 26r. Biblioteca Nacional de España. Accessed via RWTH Aachen University:

https://www.codex-madrid.rwth-aachen.de/madrid1/f026r/index.html