Angle of incidence in a concave mirror and compass constructions

Locating the angle of incidence on a concave mirror

To find the angle of incidence in the concave mirror n a m: it lies at the point a, because the portions n a and m a are equal by the definition of the larger circle, that is, because the chords are equal and the arcs of equal curvature.

The unique circle through three given points

Three points taken at random are always equidistant from a fourth point, on which the compass foot is set to draw a circle receiving them. Two points, by contrast, admit infinitely many circles. Finding the point equidistant from the three intersections S r d gives the curve whose crossing with the given circle marks the required angle of incidence.

A set of elementary compass constructions

A run of short problems: erect a line at the end of another making two right angles; find the centre of any circle from a part of its circumference; bisect any straight or curved line; draw a line parallel to a given one; and raise a perpendicular at a given point. Each is stated as the 'briefest way'.

Perpendicular at the end of a line, two right angles

Give a line across the end of another line so that it forms two right angles, the construction Leonardo flags as the briefest way. It supports the mirror work above, where the incidence is fixed by equal arcs about a right-angled setting.

Columns of calculation among the figures

Two clusters of arithmetic sit among the diagrams: one near the centre (42; 1 3 7; 2 x 4 = 8; 32) and a longer set of multiplications lower right (28 x 12, 56, 28, 8, 336). They read as working figures beside the geometric constructions.

A letter's closing: 'Farewell. Your Leonardo'

The three-point-circle passage ends epistolary in tone: 'Of the proof I will spare the already-blamed effort. Farewell. Your Leonardo.' The signed close treats the geometric result as if addressed to a correspondent.

Angolo superiore destro. Cerchi e archi di cerchi, con:

Analoga figura, con:

Sopra la figura:

Verso sinistra, figura con:

Figura con:

Linea verticale all'estremità d'una orizzontale:

Due corde e due semicerchi entro un cerchio:

Margine inferiore, dalla metà verso sinistra.

Due linee reciprocamente perpendicolari tra tre archi:

Due semicerchi fra due parallele:

Perpendicolare innalzata su una linea:

Operazioni aritmetiche. Al centro.

In basso dal centro verso destra.

Upper right corner. Circles and arcs of circles, with:

Analogous figure, with:

Above the figure:

Toward the left, figure with:

Figure with:

Vertical line at the end of a horizontal one:

Two chords and two semicircles within a circle:

Lower margin, from the middle toward the left.

Two mutually perpendicular lines between three arcs:

Two semicircles between two parallels:

Perpendicular raised upon a line:

Arithmetic operations. At the centre.

Below, from the centre toward the right.