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Codex Atlanticus — 0771 v

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© Veneranda Biblioteca Ambrosiana, Milan

Quadrati di proporzionalità, porzioni e bisangoli:

Qui son false.

Qui stan bene.

Sempre ciascun lato del quadrato è semidiamitro del circolo che li tocca li sua quattro angoli1.

Tal proporzione è infra le parte simili, quale è infra li lor tutti simili.

Tal proporzione hanno infra lor le parti simili, quale è quella delli lor tutti. Seguita che essendo le porzioni de' cerchi la sesta parte delle circunferenze d'essi cerchi ...

Tre cerchi secanti, con:

c a – e b d – o n

6 sono il maggiore numero di bisangoli equilateri che nel circolo far si possino. Provasi essere vero per una delli Elementi geometrici, che mostra2: il circolo posato sopra la linia retta non tocca quella se none in punto. Ora direno che la linia d e riceva sopra di sé 'n contatto la parte della circunferenzia d'un circolo, e sia a b c, che tocca la linia retta d e in punto b, e 'l simi[le] fa la curva opposita n b o. Adunque la linia retta d e passa per il primo contatto di due circoli sanza tagliarne parte alcuna3.

Squares of proportionality, portions and bi-angles:

Here they are false. Here they stand well. Each side of the square is always the semi-diameter of the circle that touches its four angles. Such proportion is between the similar parts as it is between all their wholes. The similar parts have the same proportion among themselves as that of their wholes. It follows that, the portions of the circles being the sixth part of the circumferences of those circles ...

Three secant circles, with:

c a - e b d - o n. Six is the greatest number of equilateral bi-angles that can be made within the circle. It is proved true by one of the geometric Elements, which shows that a circle set upon a straight line touches it only at a point. Now we shall say that the line d e receives upon itself, in contact, the part of the circumference of a circle, and let it be a b c, which touches the straight line d e at the point b; and the opposite curve n b o does the like. Therefore the straight line d e passes through the first contact of two circles without cutting any part of them.

Circles, squares and the geometry of lune-figures

Rosette constructions and the greatest number of equilateral bi-angles in a circle

Geometry Arithmetic & Proportion

The fragment is filled with geometric figures: squares drawn over ruled grids, circles enclosing six-petalled rosettes, squares inscribed in circles, and a small stack of cylinders. The notes state a rule that each side of a square equals the semi-diameter of the circle touching its four corners, and reason that similar parts of circles share the same proportion as their wholes. A second passage argues, citing one of the geometric Elements, that six is the greatest number of equilateral bi-angles (lune shapes) that can be fitted into a circle, because a circle set on a straight line touches it at only a single point.

On this page

Square and circle in proportion; portions of circles

Some constructions are marked 'false' and others 'correct'. Each side of a square is stated always to be the semi-diameter of the circle that touches its four angles. The similar parts of circles are said to hold the same proportion among themselves as do their wholes, applied to circular portions equal to a sixth of the circumference.

Six equilateral bi-angles as the maximum within a circle

Six is claimed to be the greatest number of equilateral bi-angles that can be drawn within a circle. The proof rests on the point that a circle set upon a straight line d e touches it only at a single point b, so the opposing arcs a b c and n b o meet the line at their first contact without cutting it.

Reader's Edition — described with AI from the e-Leo transcription.