Tapered Rollers, a Cord-Breaking Ratio, and a Mirror Template
Geometry of a cone's point of concurrence, applied to grinding a concave mirror
The sheet studies the geometry of a tapered roller (curro), whose sides, if extended, meet at a point of concurrence; the gentler the taper, the more distant that point. A branching table of numbers headed 39060 and a doubling column illustrate the ratios, with a note that a one-pound weight could break a cord able to resist 39060 pounds, the transcription marking where an arithmetic error begins. Leonardo then applies the same principle to grind a concave mirror (spechio), describing how to sight a target a mile off through a one-braccio aperture and shape a roller with a circular cutting edge to trace the mirror's template (centina). Additional untranscribed writing and faint underdrawing are visible across the sheet.
On this page
Section of a pyramid and where a roller's sides concur
The section of the pyramid n m c enters six times into the side n c that meets at two acute angles, and all the figures below do likewise. Every roller moves naturally about the point where its sides concur, and the smaller the proportion of its sides, the more distant that concurrence lies.
A number-tree and doubling series; breaking a 39060-pound cord
A branching table descends from 39060 through 19530, 6510, 2160, 720, 240, 80, 27, 3, 1, while a left-hand column doubles from 1 (a) to 129472 (t). A note claims a one-pound weight would break a cord resisting 39060 pounds, and the transcription flags where the arithmetic error begins.
Grinding a concave mirror to a distant focus
If a columnar body narrows by one braccio over a braccio, its sides concur at one braccio; if it tapers 1/3000, the concurrence lies a mile off. To make the roller that traces the mirror's template, divide its face in half on the diameter and continue for 19 divisions, and the concurrence falls at 46 miles.
Sighting through an aperture to shape the mirror template
To hollow the mirror to one mile, sight a target m a mile away through a one-braccio window a b, drawing back to e and marking c and d so that eye, window edges and target lie on one line. Then make the roller g to the lines a b c d marked on the ground, giving it a circular cutting edge on its thicker side to trace the template d n.