The true rule for squaring any portion of a circle

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Bibliothèque de l'Institut de France, Paris

Upper margin. Circular segment, with:

first - L v. Squaring of the sector L v.

Figures along the right margin, separated by a pen stroke. Double curvilinear triangle, with:

a - b c.

Curvilinear triangle divided in three, with:

d - e f.

The same divided into 24 sectors, with:

g - h i.

Rectangle divided into triangles, with:

n p - m f.

Two triangles on the same base, with:

a - b - d c.

Circular segment, its decomposition and rectification, with:

n. The squaring of irregular portions, that is of varying curvature, requires that, once it is divided into sectors of unequal length ....

Column of four figures. Sector of a circle, rectified arc, with:

first a - b c - e d f.

The same divided into eight sectors, with:

second g - h K - i.

Rectification of the second figure, with:

third r S - t v.

Insertion of further sectors between the preceding spread-apart ones, with:

fourth n m - o p.

Main column.

Lend the triangle a b c to the portion b c d, and divide it into sectors, as is shown in the second figure g i h k. Then separate the vertices of these sectors one from another, so that the space set between these vertices equals the flattened bases of the sectors. Then lend to the sectors of the third figure r S t v as many sectors again, that is an equal amount, and you will have made a quadrilateral, which is n m o p. Once the quadrilateral of the fourth figure is made, take away half of it, and you will have removed the lent sectors, and there will remain a quantity equal to the second figure g h i K, which will remain squared. Beyond this, you will take away from that square as much as is equivalent to the triangle of the first figure a b c, and there will remain to you the squared portion of a circle, that is the portion b c d, whose curved side was straightened by the motion made upon the straight line e d f. And this is the sole and true rule for squaring every portion of a circle smaller than the semicircle, for which no science avails except by the lending and by the motion described above, etc.

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

Margine superiore. Segmento circolare, con:

Figure lungo il margine destro separate con un tratto di penna. Doppio triangolo curvilineo, con:

Triangolo curvilineo tripartito, con:

Lo stesso diviso in 24 settori, con:

Rettangolo diviso in triangoli, con:

Due triangoli sulla stessa base, con:

Segmento circolare, sua decomposizione e rettificazione, con:

Colonna di quattro figure. Settore di cerchio, arco rettificato, con:

Lo stesso diviso in otto settori, con:

Rettificazione della figura seconda, con:

Inserimento di altri settori tra i precedenti divaricati, con:

Colonna principale.