Leonardo da Vinci:Body as Machine and Machine as Body

The elements of engineering

Do not forget that the book on the elements of machines with its beneficial functions should precede proofs relating to the power of man and of other animals; on their basis you will be able to verify your propositions.

Faced with the divine fittingness of form to function, what is the human engineer to do? Clearly the standard had been set, and no-one was going to be able to rival what god had accomplished in the totality of the human body. At one level, the human engineer could not but piously follow in nature’s wake, inventing devices that obey the principle of “Necessity”, ensuring that all designed forms perform their functions with no redundancy and no insufficiency. Yet it was evident that human inventions had gone beyond merely aping nature and had resulted in mechanisms that nature herself had not made. There was, to take one obvious example, no crossbow to be found in nature. Thus he could claim as an engineer, as he had claimed as a painter, that “the constituent parts of nature are finite, but the works which the eye commands of the hands are infinite”. The engineer learnt how nature designed its forms to fit functions, respecting her principles and the absolute sovereignty of her natural law, in order to become what he called a “second nature” in the world. In this, the artist and the engineer are at one. They make new things on the basis of the inner workings of nature rather than simply imitating what nature has already done.

We have already seen that there is good evidence of Leonardo working on specific engineering projects for his patrons, both as a roving consultant and as the designer of utilitarian items like tongs, locks and jacks within hisworkshop. It is exactly this kind of utilitarian item that has not survived. A drawing of a jack (fig.11) gives some idea of what he may have accomplished for those who employed his services. In order to minimise what we call frictional forces (which are particularly severe with wooden machines and in the absence of modern lubricants), he has combined a worm gear with a circular roller bearing, a ring bearing in modern terms. A drawing of the mechanism separated from its containing frame and a section of the bearing demonstrate his concern to separate the rolling balls, so that their counter-rotating faces should not encounter each other. In another design he showed how intervening rollers, shaped like concave cotton reels and revolving independently on their axles, could achieve the desired separation.

11. A Screw Jack with a Roller Bearing, Madrid, Biblioteca Nacional, Codex Madrid I, 26r

Lifting devices were of notable importance in any operations that require large loads, such as blocks of stone, to be lifted from the ground – particularly for loading on to transportation. Leonardo’s senior colleague, Francesco di Giorgio Martini of Siena, one of whose manuscripts on architecture and engineering Leonardo owned and annotated, .devoted considerable efforts to lifting devices, particularly ones that could raise large columns into upright positions. The two masters were in Pavia together in 1490, advising the Cathedral authorities on structural matters. In more ways than one, Francesco provides a close precedent for Leonardo. He was a painter, sculptor, architect and engineer, particularly renowned for his mastery of fortifications. He etched out a very substantial career for himself, much in demand from leading military and civil patrons, like Federigo da Montefeltro, Duke of Urbino. His native city of Siena went to considerable expense in an effort to attract and retain his services. In the hierarchies of court and civic employment, engineers were far bigger fish than painters – unless the painter was both. When Leonardo was listed alongside Bramamte as one of the top engineers in the Sforza court, he had moved into a league above that available to a maker of pictures.

Francesco’s ingenious disposition of walls and bastions in fortresses, relying upon an intricacy of geometry rarely applicable to other forms of architectural design, was taken up and developed by Leonardo, who characteristically re-thought the configuration of every part of a fortress in response to the “percussive” blows dealt out by new-style artillery, The design of a fortification becomes an exercise in impacts, deflections and rebounds, so that the bombardment can be neutralised. His drawings assume the character of dynamic trigonometry (fig.12), as he strives to maximise the firelines from the bastions – so that there should be no dead ground in which the enemy can manoeuvre – at the same time as creating angles such that incoming projectiles strike only glancing blows. Where there is necessarily a vulnerable opening in the walls to emit fire from the defenders’ armaments, the apertures are designed so that anything other than the most direct hit expends is impetus impotently in successive rebounds. Canted gun ports had long been in use, but Leonardo works with a series of curved profiles, so that the majority of the blows might glance tangentially off the surfaces, much like the “percussion” of light on the chin of the man drawn in profile (fig.13).

12.Three Alternative Plans for a Fortified Wall Demonstrating the Fire Lines, Milan, Biblioteca Ambrosiana, Codice atlantico, 767r.

13. Demonstration of the Brightness of Light on a Face According to Angles of Impact, Windsor, Royal Library, 12604r.

Where Leonardo is fundamentally innovatory is not so much in the details of his design as in developing the principle that military architecture, like every other form of invention, should be founded on principles drawn from the relevant rules of nature. Even Francesco, who was notably interested in Aristotelian ideas, had not conceived the marriage of theory and practice in quite this way. Looking into the future, we can see how the abutting of bookish physics and practical operations, which was to stand at the heart of Scientific Revolution effected by Galileo and others, was already present in what Leonardo was striving to accomplish. What he was not able to achieve was to revise in a fundamental way the laws of dynamics he inherited from Aristotelian natural philosophy. That reform was to await Galileo and levels of mathematical analysis to which he could not aspire.

An excellent example of how he thought about military devices is his design for a giant cross-bow (fig.14). Although repelled by the effects of war, which he called “beastly madness”, he was drawn deeply into the design of ever more destructive weapons – drawn both by the wishes of his patrons and his own fascination with the challenge of systems that could amplify human power a thousand-fold. This highly-finished design looks at first glance like one of the kinds of unrealistic “treatise” designs, intended to impress patrons rather than to be actually constructed. On such a scale, the problems become formidable, not least the thickness of the arms of the bow, for which he has conceived sliding laminations to overcome their propensity to crack when drawn back with the enormous power required. When the giant bow was actually built for a television programme, the limbs cracked alarmingly - though the builders mistakenly set off on a track dictated by their confidence that they knew more about laminations than Leonardo, who was working in a context where wood technology had been developed to levels no longer available. In contrast to what looks like the unrealistic aspirations of the design, a series of preparatory sketches, not least for the release mechanism, indicate that he was taking the practical aspects of the design very seriously indeed. In theory, the increased scale should (according to the proportional laws of dynamics) allow a missile to be hurled immense distances or, as is more probable here, permit the firing of extremely heavy projectiles to breach stout defences. Elsewhere he was concerned with the design of explosive shells, which could make such a missile launcher even more penetrative.

14. Design for a Giant Crossbow and its Components, Milan, Biblioteca Ambrosiana, Codice atlantico, 149rb.

Being Leonardo, the issue of the ratio of the force exerted by the released bow string and the depth of its withdrawal is an issue to be confronted. The obvious formula to apply would be to take the force as directly proportional to the depth of pull; that is to say, if the string is withdrawn to double the distance of the first pull, it would be doubly as effective. However, this will not work. The act of drawing back the ends of the arms upsets the geometry. The points of attachment of the bow string will be drawn towards the direction of the pull, and the distance between them will become less. The simple proportional triangles beloved of Leonardo become inoperable. However, there must be a definable proportional ratio, otherwise Leonardo’s principles would have collapsed in a way that he considered unthinkable. He settles on the power of the withdrawn bow string as being inversely proportional to the angle it makes at its point of withdrawal (i.e. where the base of the projectile is lodged). Thus, if the string inscribes an angle of 45o the force which thrusts the projectile towards its target will be double that generated when it inscribes a 90o angle. Since he had no trigonometry, he cannot take the mathematics of this hypothesis any further, but he is content that he has the right answer because it feels right within his system of thought. We might say, anachronistically, that the rule of the angle of the bow string harmonises properly with the aesthetics of dynamic law.

Every device – a lever, a pulley system, the gearing required to equalise the power of an unwinding spring (fig.15), a jack, a pair of tongs – acted in similar accordance with the underlying structure of natural law. Thus a volute gear is designed to operate in accordance with the “pyramidal law” that inexorably governs how force diminishes over time.The action of biceps in the arm, the biting power of our jaws, the levers operated by muscles attached to the ribs to accomplish breathing, all conformed to the same “Necessity” of mechanical law. He explained that the muscles in the neck that were necessary to sustain our heavy head in an upright position worked according to exactly the same set of principles as applied when the designer of the mast of a sailing boat, supported by stays, was trying to give it maximum stability.

15. Volute Gear for a Barrel Spring, Madrid, Biblioteca Nacional, Codex Madrid I, 45r.

Given this insistence on the rational anatomy of machines, it makes sense that Leonardo should attempt to design sets of those stock components that might be permutated to create any variety of new “body”. Previous engineers had concentrated on complete machines. Leonardo turned his attention to the “elements of machines”, components that potentially exercised their utility beyond that of a single context. A properly designed component, in which all the mathematics was right, should be of universal applicability. This we find wonderfully succinct drawings for ring bearings, axles, hinges, gears, ratchets, cams, couplings, springs, and so on. Any kind of new body could potentially be assembled to undertake any designated function. At their best his designs have that air of compact inevitability that makes one wonder why no-one else had thought of it. A patron who received a pair of tongs made by a German technician at Leonardo’s behest was, in effect, being presented with an implicit lesson in theoretical mechanics and the proper practice of engineering design.