How Birds Turn Without Rising or Falling
The double wing-beat, and steering flat with the wingtip rudder (alula)
Folio 45r opens by calling the wing's action 'double': part of each stroke pushes down toward the earth, checking the bird's descent, while the backward beat drives it forward. Leonardo then asks what lets a bird curve its path without climbing or diving, and answers that it uses the small rudder (alula) at the leading edge of one wing. Taking the bird as a p o g, he shows the straight course m p a bent into the curve a b by the left-wing rudder t, invoking his rule that an equal-sided body keeps a straight course in air but curves the moment one side gains or loses area. Small marginal figures show birds and a kite turning in flight.
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The double motion of the wing: downstroke checks descent, backstroke drives forward
The wing-motion of birds is double, since one part descends toward the earth and one part goes toward the place from which the bird flees. The part made toward the earth prevents the bird's descent, and the beating backward drives the bird forward.
Turning flat with the wingtip rudder (alula)
The bird curves its straight motion, made in the site of equality, without raising or lowering itself, by means of the right or left rudder at the fronts of the wings. Let the bird be a p o g, moving through the site of equality; it bends the straight motion m p a into the curved motion a b by means of the rudder t placed at the front of the left wing.
Equal-sided bodies keep straight; unequal sides curve the path
By the ninth proposition, bodies of equal sides about the central line of their gravity always keep their motion in a straight line when in the air. If one side gains or diminishes its quantity, the straight motion curves, showing the concave part of the curvature toward the part of greater inequality of the moving thing.
Marginal figures of turning birds and a kite
The right margin carries small pen figures of birds banking and turning, including a kite curving its course. They accompany the analysis of how the wingtip rudder bends the flight path without a change of height.