Geometry, a mirror problem, accounts, and an upright bombard
A dense working sheet of constructions and sums, with a bombard drawn below.
The upper fragment is a crowded working page mixing geometry, arithmetic and household accounts: constructions on the semicircle and on circles crossed by arcs, a figure of rays striking a spherical mirror, columns of calculation, and expense notes in lire and soldi down the left margin. Among them is Leonardo's often-cited reckoning with his assistant Salaì — 93 lire 6 soldi given, 67 received back, 26 lire 6 soldi still owing. The lower fragment carries a large drawing of a bombard set upright on a curved timber base, which the catalogue note explains allows only a limited angle of fire. The geometrical notes assert that the two sides of a right angle inscribed in a semicircle always sum equally, and that no general geometric rule can fix the angle of incidence on a convex mirror.
On this page
The right angle inscribed in a semicircle
Beside a figure lettered a b c d e Leonardo states that the sums of the two sides of the right angle formed within a semicircle are always equal to one another. He repeats the rule for the several right angles so formed and begins to explain why this happens.
Why the angle of incidence on a convex mirror has no general rule
A figure of rays striking a spherical surface, lettered a b c and c b a, accompanies the claim that no general geometric rule can be made to find the angle of incidence on convex mirrors. Because the eye and the object stand at varying distances from the mirror, Leonardo says one must instead proceed case by case, as shown.
Finding the central line with needles and a thread
A construction directs the reader to set a needle at b and another at a and stretch the thread a b p e until it touches at p, equally distant from the central line h L. That central line is found by drawing equal the lines running from the centre of the circle to b f.
Account with Salai: 93 lire, 6 soldi
In the lower right corner Leonardo records that he gave Salai 93 lire and 6 soldi, has had 67 lire back, and that 26 lire 6 soldi remain to be paid. Further expense notes for fodder, irons, a saddle and journeys, tallied in lire and soldi, run down the left margin.
A bombard mounted upright on a curved timber base
The lower fragment shows a bombard standing vertically on a heavy timber frame. The catalogue note explains that the curve of the base permits only a limited angle of fire, and that a lost fragment of the sheet removed some lines of the drawing, since redrawn on the patch.