Constructing the angle of incidence within a circle
A compass construction for the equal angles that flank the angle of incidence
A single large diagram fills the top of f. 19v: a circle e f d with an inscribed triangle, two external points a and b, a tangent p t, and a great arc swung below. The text below gives the construction: from the two given points a and b draw lines meeting at the circle's centre, extend b a until it cuts a line f n, draw the tangent p t there, then drop a perpendicular r c to obtain a semidiameter whose point of contact r fixes the angle of incidence, set midway between two equal angles. Leonardo notes this rule is not general, since the flanking angles vary with the distance of the two points, and says he has set down these three rules hoping to make of them all a single general one, achieving what his predecessors never did.
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Fixing the angle of incidence at the point r
From the two given points a and b draw lines a c and b c to the centre of the circle e f d; where they cut the circle at e and f, draw the line f n; extend b a until it cuts f n, and from that intersection draw the tangent p t. Dropping a perpendicular r c to the centre gives a semidiameter whose point of contact r is the angle of incidence a r b, standing midway between two equal angles o r e and i f r.
Toward one general rule from three
The equal angles are proved by the semicircle e o i f, which cuts the sides of the angles with equal length, since the arcs o e and i f are equal, lying between the curvilinear parallels g s and K h. Leonardo warns the rule is not general, for it varies the flanking angles more as the two given points differ in distance; he sets down these three rules hoping to make of them all a single general one.