Transmuting curved surfaces into rectilinear ones (lunes)

Transmutation of curved surfaces into rectilinear ones

The governing heading of the sheet announces the programme: transforming curved surfaces into straight-sided ones. Every figure below is an instance of squaring a lune or reducing a ring or sector to a rectilinear equivalent.

Quadruple circle: semicircle worth the lune

In a circle subquadruple within a semicircle, the semicircle a b c is worth the lune; removing b and c leaves the triangle a. Removing from the lune n m, which equal b c, leaves r S t equal to the triangle a.

Two equal circles reduced to a rectilinear figure

Two circles equal to one another each lose a fourth part — one inside as a circle, the other outside as a sector. Then three sectors are removed from each, leaving a rectilinear figure equal to a curvilinear one.

Sector made equal to a circular ring

Make the sector n worth the circular ring a n; remove the half-portion from the ring and an equal value from the sector (b), leaving n squarable and equal to the ring. Removing the rectilinear part above a from each, the remainder of the sector equals the remainder of the curvilinear ring; b may be equal but not similar to a, which does not matter.

Proportional divisions and a column of numbers

The left margin carries a portion divided as 1/8, 1/4, 1/2, 1, geared to the doubling and quadrupling of circles used throughout. At the foot of the sheet stands a column of numbers (27, 47, 36, 37, 48, 38, 50, 39).

Metà destra, dall'alto in basso.

Cerchio subquadruplo entro semicerchio biffato, con:

Due figure: corona circolare con quadrato iscritto e settore di cerchio, con:

Due figure: corona circolare e semicerchio, con:

Due figure biffate: corona circolare e cerchio con quadrato iscritto:

A destra del precedente, corona circolare, con:

In basso, corona circolare e cerchio, con:

Metà sinistra. Figura analoga alla prima della metà destra:

Tre piccole figure con:

A sinistra sotto la didascalia: corona circolare e settore di cerchio, con:

Due figure: cerchio e corona circolare, con:

Settore di cerchio con:

Cerchio e corona circolare con quadrato iscritto, con:

Margine sinistro, porzione divisa in parti:

In basso:

Right half, from top to bottom.

Subquadruple circle within a biffed semicircle, with:

Two figures: circular ring with inscribed square, and circular sector, with:

Two figures: circular ring and semicircle, with:

Two biffed figures: circular ring, and circle with inscribed square:

To the right of the preceding, circular ring, with:

Below, circular ring and circle, with:

Left half. Figure analogous to the first of the right half:

Three small figures with:

At left below the caption: circular ring and circular sector, with:

Two figures: circle and circular ring, with:

Circular sector with:

Circle and circular ring with inscribed square, with:

Left margin, a portion divided into parts:

Below: