Lune and circle quadrature, dated 7 July 1514 at the Belvedere

Dated colophon: finished 7 July 1514 at the Belvedere

The central proof closes with a personal note of completion, fixing the date, hour and place of the work: finished on the 7th day of July, at the 23rd hour, at the Belvedere, in the studio made for him by the Magnificent, 1514.

Reducing sectors to a rectilinear triangle

Adding the notion that equal parts removed from equal things leave equal remainders, Leonardo strips the half abc from portion abcS and the whole portion dbe from sector gac, leaving the rectilinear triangle gab. Working through the marginal figures L, v and r, t he reduces the remainder to a rectilinear surface equal to sector r, which is 1/8 of its circle, so as to construct a circle an eighth of the original.

Euclid on triangles of equal height and base

The second demonstration invokes Euclid: triangles of equal height on equal bases are equal, so triangles anf and nof on the equal bases an and no are equal. Since triangle h equals triangle i and triangle K equals triangle i, h equals K; the equal bases follow from portions of equal chords and sagittae, the triangle sides being semidiameters of one same circle.

Semi-annulus equal to semicircle and sectors

The third figure argues that the semi-annulus efichb is worth the semicircle ehi and the sectors abgc and bgf. Subtracting rectilateral triangles from equal-content sectors leaves rectilinear remainders, so sector eih equals the rectilinear fgh, and the remainder of triangle fhg equals aci, a quarter of a circle.

Transferring a lune and fractional labels

With the sheet inverted, a semicircle carries the instruction to take away the lune a and set it in b, one of many area-transfer operations rehearsed across the sheet. Neighbouring lunes are labelled by fraction, an eighth, a quarter, a third or a half, and by ratio, quadruple and double.

A destra, figura colle lettere:

e, nel margine, figure colle lettere:

Figura con:

Figura con:

Figura con:

Capovolto il foglio. Da destra a sinistra. Semicerchio con:

Dall'alto in basso. Falcata tripartita:

Figura analoga alla precedente:

Cinque semicerchi colle lettere:

Semicerchio colle lettere:

Figure di falcate con:

Semicerchio con due bisangoli:

Semicerchio con:

Semicerchio biffato:

Entro un altro semicerchio biffato:

Salendo lungo il margine sinistro, semicerchio con:

At right, a figure with the letters:

And, in the margin, figures with the letters:

A figure with:

A figure with:

A figure with:

The sheet turned upside down. From right to left. A semicircle with:

From top to bottom. A lune divided into three:

A figure like the preceding one:

Five semicircles with the letters:

A semicircle with the letters:

Figures of lunes with:

A semicircle with two bi-angles:

A semicircle with:

A semicircle struck through:

Within another semicircle struck through:

Rising along the left margin, a semicircle with:

Lune and circle quadrature, dated 7 July 1514 at the Belvedere

On this page

Dated colophon: finished 7 July 1514 at the Belvedere

Reducing sectors to a rectilinear triangle

Euclid on triangles of equal height and base

Semi-annulus equal to semicircle and sectors

Transferring a lune and fractional labels