The vortex The paradise of mathematics
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Leonardo famously said that
mechanics is the “paradise of mathematics”. Today, few might agree with
him in light of the complexities of the subject! In accordance with
Aristotle’s concept of science as a systematic inventory of the world,
he set out to classify vortices just as a botanist might classify
plants. Ultimately the task defeated him.
During his last years, the natural forces manifest in the gentle curls of the lamb’s coat in the Madonna, Child, St Anne and a Lamb,
and in the beautiful twisting motion of the leaves of the Star of
Bethlehem plant took on a destructive force in Leonardo’s artistic
imagination. In Ms A he wrote of “hollows in which the water, whirling
around in various eddies consumes and excavates and enlarges
chasms…consuming and devouring whatever stands in its path, changing
its course in the midst of the ruin.”
In the late Deluge Drawing
this prophetic vision of the destructive powers of nature is given full
visual expression. The turbulent forces that condition man’s existence
take the form of swirling vortices of air and water which have now
become synonymous with the uncontrollable and unknowable forces of
nature in Leonardo’s mind.
Movement in water and air was Leonardo’s greatest preoccupation. As the
most tangible evidence of natural force, spiraling forms and vortices
held a particular fascination. Their forms pervade the entire spectrum
of Leonardo’s work in the realms of hydraulics, anatomy, mechanics and
art.
- Size Water
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As a “Master of water”, Leonardo needed to manage
the movement of water and control its destructive, erosive power. He
studied and recorded the movements of water under variable conditions
repeatedly, adopting a scientific approach to the subject, as
illustrated on Ms F Fol18v.
In the upper diagram, Leonardo
illustrates the turbulent flow of water around a rectangular obstacle
labeled “f”, noting that such “envelopings of courses” are caused by
“the impetus of water under other water”. The lower water strikes the
bottom first and immediately rises to the surface. The upper water does
not descend, but when it encounters the rising water it collides with
it and the two waters “unite and turn about together in their contact.”
In the second diagram, water cascades from the gate of a
lock, falling into other water. That which is closest to the middle is
“most oblique” and that which is closest to the extremities “most
upright”.
Leonardo’s visualizations of water in motion in
these drawings are not unlike the spiraling curls of hair. The visual
appearance of natural phenomena was a natural consequence of its
dynamic force. He remarks that “the motion of the surface of water
resembles the behaviour of hair, which has two motions, of which one
depends on the weight of the strands, the other on the line of its
revolving; thus water makes revolving eddies, one part of which depends
upon the impetus of the principle current, and the other depends on the
incident and reflected motions.”
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An accidental force
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The framework for Leonardo’s
understanding of the vortex was drawn from late medieval natural
philosophy, derived originally from classical sources. The four
elements – earth, water, air and fire were arranged in concentric
circles. Each desired to find its own level in nature. An “accidental
force” was required to move one of them from its assigned position. An
object above its natural level would exercise weight and desire to
descend according to the rule of gravity. One below its natural level,
such as a submerged bubble, would desire to ascend by the shortest
route possible.
In late medieval theory, the quality impressed
in a moving body by force was termed “impetus”. The concept of impetus
differed from the Newtonian law of the conservation of motion in that
impetus “drains” from an object in accordance with the natural rule of
the pyramidal diminution of power, rather than persisting until opposed
by contrary forces. Nonetheless, impetus was just as remorseless as
Newtonian inertia. On encountering an obstruction, a moving object
would necessarily strive to complete its assigned motion, creating
patterns of eddies or vortices in the process.
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