Solar rays and the shadow cast by a sphere
Rays intersecting at the top and axis of a ball, and how a circle's shadow becomes a semicircle
Written on brick-red prepared paper and inscribed upside-down, this leaf continues Leonardo's study of solar rays striking a sphere and the shadows they cast. He distinguishes rays that intersect at the upper end of the ball from those that intersect at its axis, and traces how the broken rays terminate the derivative shadow. A striking passage describes rolling a ruler along the edge of the ball's shadow to see a neighbouring circle's shadow swell and shrink as a semicircle, and notes that a sphere's derivative shadow weakens sharply where it meets the shadow of a touching sphere. Several cancelled lines and pen figures of rays, discs and a large tinted disc fill the sheet.
On this page
Rays intersecting at the ball's top and at its axis
The solar rays make one intersection at the upper end of the ball and one at the lower end of the axis. Those that break are the ones intersecting at the axis, broken afterwards by the upper part of the ball; the nearer a point is to the intersection, the more m behaves as circles.
A circle's shadow read off with a rolling ruler
The shadow of circle n grows or shrinks according to its nearness to the ball's shadow, always appearing as a semicircle outside it. Leonardo tells the reader to hold the ball's shadow near the eye and move a ruler slowly toward the sun without leaving contact with the ball to observe the effect.
Weakening of a sphere's derivative shadow
The derivative shadow of a sphere lit by a light equal to itself diminishes strongly when its striking falls at the boundary of the shadow of another sphere touching the first. Point a is identified as the lower end of the shadow of the axis.