Squaring curved figures: lunes, rings and circle proportions

Why an older river runs slower

Among straight rivers of the same terrain, water, breadth, depth and slope, the older one becomes slower. This is because a straight river grows more winding with age, and the winding course, acquiring greater length, slows the water — just as waters descending between equal heights run slower along the longer path.

A hatched circular ring equals the triangle a b c

From the square, curved surfaces are drawn so that the remainder stays squarable; the hatched circular ring is worth the triangle a b c. If the ring is too broad and exceeds triangle a b n, its cut a b is narrowed to half the value (c d), whereupon the ring is worth the whole square m b a n.

Transforming a square into curved figures and back

On transforming one and the same square surface into diverse curvilinear surfaces. And on the transmutation of diverse curvilinear surfaces into rectilinear ones. This heading states the programme the many diagrams work through.

A lune equals a square (a is worth b)

a is worth b, because from the upper semicircle four equal parts c f d e are removed and the like g K h i from the lower. Remove the two curved fronts from a and a square remains; remove as much from the lune and it too becomes squarable, because the curvatures are similar.

Circles in double and quadruple proportion

Of circles of double proportion, half the greater is worth the whole of the lesser; of circles of quadruple proportion, a quarter of the greatest is worth the whole of the least. The excesses at the overlap are equal, namely a b.

Angolo superiore sinistro:

Dalla destra. Prima colonna. Quattro corone circolari entro un quadrato, con:

Quadrato colle lettere:

Due varietà d'una stessa figura:

Quadrato con lunule:

Altro quadrato con lunule:

Seconda colonna. Quadrato con lunule:

Altro quadrato con lunule:

Cerchio e lunula entro cerchio, con:

Semicerchio e quadrante di cerchio sovrapposti, con:

Cerchio e semicerchio sovrapposti, con:

Cerchio e quadrante di cerchio sovrapposti:

Colonna terza.

Due figure con:

Cerchio con:

Due figure di quadrati con cerchi iscritti:

Tre figure colle lettere:

Doppia figura di cerchio:

Cerchio con 4 hisangoli e triangolo punteggiato:

Ultima colonna, dopo il primo paragrafo. Due semicerchi colle lettere:

Cerchio entro semicerchio, con:

Semicerchio colle lettere:

Cerchio sovrapposto a semicerchio, con:

Cerchio sovrapposto a semicerchio, con:

Cerchio sovrapposto a semicerchio, con:

Cerchio entro cerchio:

Upper-left corner:

From the right. First column. Four circular rings within a square, with:

Square with the letters:

Two variants of the same figure:

Square with lunes:

Another square with lunes:

Second column. Square with lunes:

Another square with lunes:

Circle and lune within a circle, with:

Semicircle and quadrant of a circle superimposed, with:

Circle and semicircle superimposed, with:

Circle and quadrant of a circle superimposed:

Third column.

Two figures with:

Circle with:

Two figures of squares with inscribed circles:

Three figures with the letters:

Double figure of a circle:

Circle with 4 biangles and a dotted triangle:

Last column, after the first paragraph. Two semicircles with the letters:

Circle within a semicircle, with:

Semicircle with the letters:

Circle superimposed on a semicircle, with:

Circle superimposed on a semicircle, with:

Circle superimposed on a semicircle, with:

Circle within a circle:

On this page

Why an older river runs slower

A hatched circular ring equals the triangle a b c

Transforming a square into curved figures and back

A lune equals a square (a is worth b)

Circles in double and quadruple proportion