by Martin Kemp
He formulated the rule that “where the angles made by the line of incidence are most equal [i.e., 90 degrees] there will be greater light, and where they are most unequal [i.e. oblique] it will be darkest.” Leonardo was in effect proposing a system of ray tracing, akin to that used in computer rendering of light on surfaces. The difference is that modern ray tracing uses a cosine law named after Johann Heinrich Lambert, the eighteenth-century Swiss-German mathematician. Leonardo’s intuition that the solution was geometrical was justified. This is not to say that he expected any painter to conduct actual calculations in painting a head, but that the grading of light on a face should be informed by an awareness of the rule, given a specific light source.
The smaller diagram to the left shows that if three spheres are illuminated from a square window, the angle of their shadows traced back to the window will converge on the point n at the center of the window.
43. A Truncated and Fenestrated Dodecahedron (Icosidodecahedron) from Luca Pacioli, De divina proportione
1496, Milan, Biblioteca Ambrosiana
Among the luminaries brought to the Milanese court was the mathematician Luca Pacioli, who in 1494 had published his important encyclopedia, Summa de arithmetica, geometria, proportioni et proportionalita (Compendium of Arithmetic, Geometry, Proportions, and Proportionality). Within two years he had joined the court and established a close rapport with Leonardo, who benefited from Luca’s knowledge of mathematical texts, particularly Euclid’s Elements. In 1498, Leonardo illustrated his friend’s treatise, De divina proportione (On Divine Proportion), printed in 1509, which centered on the five regular or “Platonic” solids.
We need some mathematics to understand Leonardo’s illustrations. The five regular bodies are the only ones whose faces are composed from identical polygons: 4 triangles for the tetrahedron, 6 squares for the cube, 8 triangles for the octahedron, 12 pentagons for the dodecahedron, and 20 triangles for the icosahedron. They were exalted by Plato as the fundamental building blocks of the universe. For Leonardo they were the highest exemplars of the mathematical perfection that underlies the design of nature.
Pacioli’s treatise related the construction of the solids to the divine or mean proportion, later known as the “golden section” (which results from a line’s being divided unequally in such a way that the ratio of the longer part to the shorter part is the same as that of the whole to the longer part). Pacioli explored what happens when the solids are truncated (cut symmetrically at their corners), transforming them into semiregular bodies. Additionally, he stellated both types, constructing symmetrical, pointed figures on their faces.
The solids had been represented before, but Leonardo devised a more “concrete” way of visualizing them. They are portrayed on vellum in pen, ink, and wash as material objects, hanging in space and modeled systematically in light and shade. Each is also portrayed in its “fenestrated” form so that we can view its full spatial array.
The illustrated example shows a solid and a hollow dodecahedron (icosidodecahedron) that have been truncated to create semiregular bodies, composed of pentagons and equilateral triangles. The illustrations in De divina proportione were very influential, not least on the great German astronomer Johannes Kepler (1571–1630).
44. Number Square, with Figures in Motion and Other Studies
c. 1501, London, British Library, Codex Arundel, 153r
Leonardo stated that geometry deals with “continuous quantities”—lines and surfaces that could be artificially divided but run continuously. On the other hand, arithmetic deals with “discontinuous quantities”—separate numbers that progress step by discrete step. On the whole he preferred the former, since they related to space and form in nature. He particularly valued geometrical proportions that could not be reduced to numbers, such as √2, which is the diagonal of a square with sides one unit long.
Arithmetical proportions were crucial to Leonardo in one respect. They served as the foundations of harmony in music in the system established by the sixth-century BCE Greek philosopher Pythagoras. As a performer on the lira da braccio (a Renaissance string instrument), Leonardo would have been well versed in the basics of music, and he would have been able to draw upon the wisdom of his colleague Franchino Gaffurio (1451–1522) in Milan, who was the foremost theorist of music in Italy.
Leonardo’s collaboration with Luca Pacioli after 1496 extended his understanding of mathematical proportions. He recorded that he had purchased Luca’s Summa de arithmetica, geometria, proportioni et proportionalita (see page 81), which treated the mathematics of proportions in some detail.
Leonardo twice transcribed a 10 × 10 number square from Pacioli’s book, clearly attracted by its unalterable sets of sequences and multiples. In the page shown here from the Codex Arundel, he has with minute care transcribed the 100 numbers in left-to-right script and begun to note the names of the multiples: “double / triple / quadruple / quintuple.” He then broke off, probably because it was becoming too crowded. In the number square in Codex Madrid II (1503–5) he started his listing down the left side of the square, but again stopped at “quintuple.”
Just below the square here he wrote in tiny script a variation of his recurrent refrain of frustration: “tell me if ever anything was done.” This appears across his manuscripts in a number of variant and abbreviated forms, often triggered by the mysteries of math and its endless sets of conundrums.
45. “Vitruvian Man”
c. 1497, Venice, Gallerie dell’ Accademia
In a note at the top of “Vitruvian Man”—the world’s most famous drawing—Leonardo tells us that the work is based on a statement by the first-century Roman architect Marcus Vitruvius Pollio. In his treatise De architectura, Vitrurius wrote that “if a man lies on his back with hands and feet outspread, and the center of a circle is placed on his navel, his figure and toes will be touched by the circumference. Also . . . a square can be discovered as described by the figure. For if we measure from the sole of the foot to the top of the head, and apply the measure to the outstretched hands, the breadth will be found equal to the height.”
Earlier illustrators assumed that the circle and square should both be centered on the navel. The only convincing way to make the formula work is to assume that the square is centered on the man’s genitals. Leonardo additionally wrote, “If you open your legs so much as to decrease your height by 1/14 and spread and raise your arms so that your middle finger is on a level with the top of your head, you must know that the navel will be the center of a circle of which the outspread limbs touch the circumference; and the space between the legs will form an equilateral triangle.” Leonardo’s positioning of the fingers and toes is the only arrangement that works with the main circle.
The overall schema is geometrical, but the internal proportions are numerical. For example, “the face from the chin to the top of the forehead and the roots of the hair is a tenth part [of the body]; also the palm of the hand from the wrist to the top of the middle finger is as much; the head from the chin to the crown, an eighth part.” The many compass marks show that the internal music of the body is composed from measured intervals, not the pentagons or other geometrical figures that are often imposed on the image.
“Leonardo’s positioning of the fingers and toes is the only arrangement that works with the main circle. . . . The many compass marks show that the internal music of the body is composed from measured intervals.”
46. Centralized Temple, Design for a Small Dome, and a Lifting Machine (?)
c. 1488, Bibliothèque de l’Institut de France, MS Ashburnham I 5v
Using a series of simple geometrical units, Leonardo worked inexhaustible variations on centralized “temples” (as he called them). The centralized form was beloved of Renaissance theorists and designers, but it was less well regarded by the commissioners of actual buildings, since it was not readily adapted to the functional demands of church rituals. The impression is that Leonardo’s “temples” would have been the si
ze of baptisteries rather than cathedrals.
His careful demonstration of the multiple symmetries of the plan is combined with a perspectival view of the exterior as a solid form in space. This was a novel way of presenting the plastic form of architecture. In other drawings, he sectioned the “temples,” much as he did the human skull. When he wrote to the authorities of Milan Cathedral to present his model for a domed crossing tower, he specifically drew an analogy between a building and the human body.
The compact, modular design consists of a central octagon surrounded by eight cruciform spaces, each like a miniature temple in its own right. The geometrical units cluster symmetrically on the exterior, interrupted only by the necessary atrium at the entrance. At the top left Leonardo illustrated an alternative for the small domes, adding oculus windows to admit more light. The ribbed central dome is notably like that built earlier at Florence Cathedral by Filippo Brunelleschi.
In the note to the lower left, Leonardo expressed his hope of avoiding an attached bell tower, which would violate his cherished symmetry: “Here a campanile neither can nor should be made. Rather they should stand separately. . . . If you nevertheless want it to be with the church, make the lantern serve as the campanile.”
Although Leonardo himself did not construct such a church, his manner of visualizing its form in space and his modular way of combining geometrical shapes had a massive impact, particularly on Donato Bramante, his colleague in Milan, who was later to embark on the construction of St. Peter’s in the Vatican.
47. Design for a Roller Bearing, Pinions Turned by a Wheel, and a Conical Screw with a Shaft and Rod
c. 1499, Madrid, Biblioteca Nacional de España, Codex Madrid, I 20v
Leonardo was well versed in the extensive achievements of his fellow engineers and their predecessors. In Florence he could draw on the tradition of Filippo Brunelleschi, the builder of the dome of Florence Cathedral. Leonardo expanded on the study of engineering in a number of ways, not least in his depiction of the “elements of machines” as discrete components that could function in a wide variety of devices. Other engineers designed and drew complete machines as functioning entities. There is a series of such “elements” in the first of the Madrid Codices.
Friction and wear were major problems in load-bearing machines, particularly those constructed in wood and lacking effective lubricants. The design on the top right of the opposite page is for a new kind of bearing that uses balls. A series of balls could be placed in a ring to facilitate rotary motion, but the rear of one rolling ball would rotate in the opposite direction to the front of the one that follows. Leonardo devised concave reels as separators for the balls. Through each reel runs an axle attached to rotating wheels on the outside and inside of the ring.
He explained in the accompanying note that he was using “wooden balls in place of rollers to move a weight. . . . The wheels with their axles . . . keep the wheels in order, so that they perform their rotating function and cannot escape.” The three moving elements—the wheels with their axles, the concave reels, and the round balls—act entirely in concert with the minimum of friction. Like most great designs, it looks obvious once it is conceived.
The drawing on the left examines the related problem of the direction of rotation of four pinions driven by a large-toothed wheel. In the note on the right of the page, Leonardo argued that “the pinions that are found engaging one of the semicircles of the wheel that moves them will turn in a contrary direction to that which occurs with the pinions turned by the other semicircle.” On either side of the vertical axis of the device he wrote “semicircle.” At the bottom right he illustrated a compact conical screw of unknown size, resembling a gimlet or auger that is rotated by a horizontal bar.
Contemporary wooden model of Leonardo’s Design for a Roller Bearing.
48. Gearing for a Clockwork Mechanism and Wheels without Axles
c. 1499, Madrid, Biblioteca Nacional de España, Codex Madrid, I 14r
In the age before the pendulum, it was difficult to measure small increments of time in a regular manner. None of the sources of power to drive a clock—including hanging weights and springs—delivered power entirely consistently over time.
Leonardo was much attracted to the potential of barrel springs, since they offered compact sources of power. The problem was that a wound-up spring delivered a lot of power at the beginning, and progressively less as it wound down. Various devices had been invented to counteract the problem. Leonardo’s favored solutions were pyramidal and spiral gears. These were designed according to his pyramidal theory of the diminution of power, which applied to all dynamic systems in which motion gradually slowed to a complete halt (see page 77). Thus he looked to gears that compensated mathematically, in effect negating the pyramidal decline. Other engineers were not using mathematics in the context of the theory of dynamics.
In this wonderfully precise pen drawing of a mechanism for translating the power of a barrel spring into a steady rotational motion, the teeth of the gear wheel at the top of the barrel engage with progressively wider whorls in a pyramidal gear to compensate proportionally for the winding down of power. To accommodate the changing lateral position of the shaft, a worm gear at its base turns a vertical gear wheel that transports the platform supporting the shaft away from the spring. The wheel at the top of the pyramidal gear turns a vertical shaft and a large upper wheel with what is intended to be a steady degree of power.
The other studies at the bottom of the page show wheels in the form of rings without central axles. They are supported by their small pinion wheels, either three or four in number. Leonardo wrote that he did not see “any difference between them, other than that the three pinions that support the wheel a [on the far right] are better located and separated.” He noted that it is often necessary to use a wheel that has no axle.
49. Designs for a Giant Crossbow
c. 1488, Milan, Biblioteca Ambrosiana, Codice atlantico, 149br
Many Renaissance military engineers advertised their abilities in illustrated treatises that promised superiority. Leonardo’s letter of introduction to Ludovico Sforza, boasting of his mastery of a wide range of military inventions (see page 33), was well positioned to attract the attention of a patron whose father had risen to power as commander and whose territories were under regular threat.
Among Leonardo’s armaments that went beyond those in “common usage” were monstrous versions of known devices. The giant crossbow is one of the most striking, particularly because it is drawn in such a way as to facilitate its construction. It is portrayed in nonconvergent perspective so that the more distant parts of the construction are not diminished in size. On the drawing, Leonardo wrote that it had a wingspan of almost 100 feet (30.5 meters), and that it could hurl 100 pounds (45.4 kilograms) of stone.
The arms of the bow are made from laminated wooden strips so that they might bend without cracking. The mighty shaft is borne on six canted wheels with linked axles. The train that tightens the ropes is pulled backward by a pair of windlasses, as detailed in the drawing at the lower right. The drawings on the left extract the “element” of the trigger block. Leonardo explains that the upper mechanism releases the rope when the hammer strikes the nut, while the lower one uses a lever.
Characteristically, he considered the mathematics of crossbows. We might think that the power with which the projectile is thrown would be proportional to the distance that the rope of the bow is withdrawn. However, the configuration of the arms changes as the rope is pulled back, complicating the calculation. Leonardo decided that the power is inversely proportional to the interior angle of the rope at its point of withdrawal.
The giant crossbow seems to promise the destruction of castle walls. When the ponderous weapon was reconstructed for a television program in 2006, its impact proved to be more visual than vicious.
“Among Leonardo’s armaments that went beyond those in ‘common usage’ were monstrous versions of known devices. The giant crossb
ow is one of the most striking, particularly because it is drawn in such a way as to facilitate its construction.”
50. Men Struggling to Move a Huge Cannon
c. 1485, Windsor, Royal Library, 12647
Leonardo was fascinated by the technology of war machines but he was repelled by war, which he called “beastly madness.” In one of his riddles, he played on the hideous destructive power of major weapons: “Emerging from the ground with terrible noise it will stun all those standing nearby and with its breath it will kill men and ruin castles and cities.” The answer to the riddle is a cannon cast in a pit.
In the work here, Lilliputian men within an armory dedicate all their manic efforts to serving their massive metal master. Some are straining at the axles of wheels far larger than themselves. Other squads of subjugated men haul on the levers of windlasses to wind ropes over pulleys attached to the pyramidal frame in order to raise the huge body of the bombard onto the carriage.
Stored in the background are cannons of various sizes, while on the far left, approximately one-third of the way down the page, we can see a single mortar. The massive barrel of the cannon closely resembles one of the two sections of the Dardanelles Gun, cast in Turkey in 1464. Each section of the Turkish bombard was screwed onto the other using levers inserted in the rings of rectangular apertures. Leonardo was much interested in Turkey, and made a design for a Bosporus bridge. Knowledge of the most advanced technologies traveled fast. The overall length of the Dardanelles cannon was 17 feet (5.2 meters). Leonardo’s monster looks to be even larger than the Turkish supersize bombard.
The drawing is full of closely observed technology, including in the foreground a half-mold on rollers for the casting of a smaller cannon. The hordes of little “pen-men” resemble those he was intending to use in his planned Treatise on Painting (see page viii) to illustrate human motions of pushing and pulling. Yet these rationally observed details are placed in the service of the visionary fantasy that falls outside the known types of drawing produced by Renaissance artists. For whom was the drawing intended? It seems to be the personal exorcising of a military nightmare—perhaps a nightmare with a Turkish dimension.