An Astronomical Gear-Train Calculation
Forty geared wheels multiplied out to a number of forty digits, reduced to miles
With the sheet turned upside down, Leonardo fills the page with a towering multiplication: a train of 40 wheels, each carrying a pinion of 10 spindles and a wheel of 200 teeth, so that a single turn of the first wheel yields a number of turns of the last running to some forty digits. He builds the powers of 20 down a long column, divides by 3, and reduces the outcome to an equally colossal figure of miles. Beneath the calculations, faint metalpoint sketches - including what appears to be an animal and small human figures - remain untranscribed.
On this page
A single turn multiplied through forty geared wheels
If you drive the gearing with 40 wheels, each with 10 spindles on its pinion and 200 teeth, then one turn of the first wheel makes the last turn 536,316,837,888 followed by more than thirty further zeros of times. Leonardo names the colossal figure aloud - '536 million of million of million...' - and reduces it, at a braccio and one-seventh per turn, to 1,787,722,792,960,255,588,974,300 miles.
The compound gear train that produces the number
The calculation describes a real mechanism: forty wheels in series, each meshing a 10-spindle pinion (rocca) against a 200-tooth wheel, giving a step-up of 20 at each stage. The powers of 20 counted down the column are the successive turn-ratios of this train.
Faint metalpoint sketches beneath the figures
Under the number columns, lightly drawn metalpoint studies survive that the transcription does not record - among them what appears to be a saddled animal and small standing figures at the lower left. They are too faint to read with confidence and are not translated here.