Continuous proportion and the doubling of the cube
The tripled ratio 1:3:9:27 explained, with cones and circles; faded memoranda dated 5 July 1458
The page is filled with Leonardo's mirror-script working through continuous proportion: when four quantities stand in continued proportion, the ratio of the first to the fourth equals the ratio of the first to the second "tripled," illustrated throughout with the series 1, 3, 9, 27. Marginal diagrams show cones inscribed in circles and, at mid-page, three circles proportioned as quarter, half and whole. The Codex Atlanticus tags connect this to the duplication of the cube (the Delian problem) and to Euclid's Elements. Turned about, the sheet also carries faded fragments and a stained memorandum bearing the date 5 July 1458 and the name Antonio; further faint mirror-script beyond the transcribed blocks is present.
On this page
The tripled ratio: 1, 3, 9, 27
When four quantities are continuous and proportional, the proportion from the first to the fourth equals that from the first to the second tripled — composed of three such ratios. With 1, 3, 9, 27: one times 3 makes 3, 3 times 3 makes 9, 3 times 9 makes 27, so 27 is built of three triples and is multiplied cubically.
Cones in circles and three proportioned circles
Along the left margin are cones and triangles inscribed in circles, working geometric means. At mid-page three circles are drawn proportioned to one another as quarter, half and whole, tying the numeric proportion to figures.
Stained memorandum dated 5 July 1458
In a stained corner, partly illegible, stands a memorandum with the numbers 5 5 7 and the words "year ... given ... on the 5th of July, 1458," with a further fragment naming Antonio. It sits apart from the mathematical work on the sheet.
