The Table of Proportionality and the Squaring of Portions
Building 84 small portions from six, with common notions on parts and number
A dense demonstration Leonardo calls the 'table of proportionality': a circle cut into six sectors of equilateral triangles, each holding a square divided into 14 equal parallels, so that six times fourteen makes 84 small portions. A long left column walks through squaring one parallel with the last proposition of Euclid's second book, transmuting it into a square, building an equilateral triangle on its side, and describing a new circle to be subdivided in turn. Framing 'common notions' state that equal parts grow as their number lessens and shrink as their number grows, while the whole keeps its natural quantity. Three worked figures — a squared circle, a 14-portion sector, and a double-circle rosette of 84 portions — anchor the argument; further untranscribed notes fill the sheet.
On this page
The table of proportionality built from square and compass
The circle is divided into six sectors facing six equilateral triangles, each enclosing a square divided into 14 equal parallels — this is the table of proportionality. Squaring one parallel (with the help of the last proposition of Euclid's second book), it is turned into a square a b L K; an equilateral triangle a b d is built on its upper side, and the fixed foot of the compass set at d describes a new circle to be cut into six, then 84, similar portions.
Common notions: as number grows, equal parts shrink
The equal parts grow just as much as their number diminishes, and diminish just as much as their number grows; yet the whole of a quantity divided into parts neither grows nor diminishes in its natural quantity. The aliquot part here is 6: six times the 14 marginal portions yields 84 portions in the figure at the foot of the page.
The hexagon as a circle lacking its six portions
The hexagon is a circle that lacks its six portions, which together compose the whole circumference of the circle. In an allied note Leonardo poses the classic problem: let a curvilinear surface be given equal to a rectilinear surface.
Fourteen portions worth one greatest portion
A sector of a circle lettered m, n, f, r, h, o carries the note that 14 portions are worth one of the greatest portions, with n m o marked as the aliquot part. The white field within the greatest circle is declared worth the greatest hexagon that can be inscribed in that circle.