Circle-square proportions and construction of square roots

Circles are in proportion as the squares on their diameters

Leonardo states that the proportion between one circle and another is the same as that between the squares formed by the multiplication of the circle's diameter. It is his way of tying the areas of circles to a rectilinear measure.

Geometric construction of the roots of all numbers

In an isosceles right triangle a n h the base a n is divided into equal parts, and perpendiculars are raised to the hypotenuse. The first line o b is the square root of 1, d i the root of 2, and so on up to 6, though the rule extends to infinity. A 'dotted root' at the right angle is disregarded because it does not obey the rule.

Quadrant built from parts of seven nested circles

On the inverted sheet a quadrant labelled with fractions (quarter a b c, eighth a c n, sixteenth a n m) is used to find a 'half-portion.' The half-portion sought is drawn from the greatest half-portion a c n, divided as many times as there are circles above it. Leonardo adds that in circles of quadruple proportion a quarter of one equals the whole of the other.

Doubling squares: diagonal of one equals side of the next

For squares each double the other, incorporated one within another in succession, the diagonal of the preceding square is always equal to the side of the following one. The figure shows five such nested squares.

Circular 'parallels' change shape but not quantity

In a quarter-circle figure the half-portion o is removed from the curvilinear parallel and its value restored by the portion m. Leonardo concludes that the parallel has changed its figure but not its quantity.

In alto due cerchi concentrici, di cui uno tratteggiato, colle lettere:

Sopra i due cerchi:

Entro il triangolo sottostante:

A sinistra delle suddette figure:

Figura di quadrato con 5 quadrati " incorporati "

Margine destro, quadrato colle lettere:

A sinistra della precedente figura colle lettere:

A foglio capovolto. In alto a destra. Figura di quadrante comprendente parti di sette cerchi, colle lettere e i numeri:

Sotto la figura:

Nella figura, entro il settore che è 1/64:

Entro un uguale settore disegnato a sinistra della figura:

In alto, gruppo di figure circolari colle lettere:

Sotto la figura:

Colonna centrale, dall'alto in basso. Figura colle lettere:

Sotto la precedente, figura colle lettere:

Unito alla figura precedente con un tratto di penna:

Sopra lo scritto precedente, figura circolare:

Figura di quadrante di cerchio colle lettere:

Accanto alla figura di 5 quadrati e 4 cerchi reciprocamente iscritti e circoscritti:

A sinistra della precedente figura e sopra piccole figure circolari:

Sul margine inferiore:

Top: two concentric circles, one hatched, with letters:

Above the two circles:

Within the triangle below:

To the left of the said figures:

Figure of a square with five squares 'incorporated':

Right margin, a square with letters:

To the left of the previous figure, with letters:

With the sheet turned upside down. Upper right. A quadrant figure comprising parts of seven circles, with letters and numbers:

Below the figure:

In the figure, within the sector that is 1/64:

Within an equal sector drawn to the left of the figure:

Top: a group of circular figures with letters:

Below the figure:

Central column, from top to bottom. Figure with letters:

Below the previous, figure with letters:

Joined to the previous figure by a pen-stroke:

Above the previous writing, a circular figure:

A quarter-circle figure with letters:

Beside the figure of 5 squares and 4 circles mutually inscribed and circumscribed:

To the left of the previous figure and above small circular figures:

On the lower margin:

On this page

Circles are in proportion as the squares on their diameters

Geometric construction of the roots of all numbers

Quadrant built from parts of seven nested circles

Doubling squares: diagonal of one equals side of the next

Circular 'parallels' change shape but not quantity