Doubled Circles and a Bird Steering in the Wind

A bird steering by the horn of its tail

Under 'compound obliquity': a bird above the wind uses the lower horn of its tail to bend from a straight course, and below the wind uses the upper horn. If the wind comes head-on and it wishes to turn to the left, it must lower the left wing and the left horn of the tail, and so it turns at once.

Doubled circles: whole in the half, half in the quarter

On double circles: the whole in the half, the half in the quarter, the quarter in the eighth, the eighth in the sixteenth, and so on to infinity, with four figures of superimposed circular parts illustrating the halving series.

Circles of quadruple proportion

On circles of quadruple proportion, the same halving series is repeated. In quadruple circles the whole of the smaller is worth a quarter of the larger, the half worth an eighth, and the like parts correspond to their like larger ones.

A lune equals a triangle (octuple proportion)

Among the circles of octuple proportion, the figure marked 'quarter and eighth' with points a c b shows that the lune a is worth the triangle b, one of the recurring equalities between curvilinear and rectilinear areas on the sheet.

Overlapping equal surfaces: touching parts equal in figure

If two surfaces differing in figure but equal in quantity are laid partly one over the other, what of them touches is equal in figure and quantity, and what does not touch is equal in quantity but not in figure — the principle underlying the segment-swapping proofs.

Rule for the value of curvilinear parts

A marginal heading announces a rule for knowing the value and proportion of many curvilinear parts of various circles, the programmatic statement governing the demonstrations below it.

Angolo superiore sinistro. Figura biffata di uccello in virata:

Lungo il margine superiore.

Angolo superiore destro.

Sotto le quattro figure di parti di cerchio sovrapposte:

Margine destro

Quattro figure analoghe alle precedenti; l'ultima colle lettere:

Quattro figure analoghe alle precedenti e una quinta con:

Figura con:

Figura con:

Figura con:

Semicolonna a sinistra delle precedenti figure. Semicerchio su settore, con:

Semicerchio, parzialmente biffato, su settore con:

Due settori di cerchio sovrapposti, con:

Semicolonna a sinistra della precedente.

Cerchio su quadrante di cerchio, con:

Figura simile alla precedente, con:

Lungo il margine sinistro, dopo il brano biffato

Figura di cerchio su semicerchio, con:

Semiottagono iscritto in semicerchio; settore di cerchio con falcata:

Upper left corner. Struck-through figure of a bird banking:

Along the upper margin.

Upper right corner.

Beneath the four figures of superimposed parts of a circle:

Right margin.

Four figures analogous to the preceding; the last with the letters:

Four figures analogous to the preceding, and a fifth with:

Figure with:

Figure with:

Figure with:

Half-column to the left of the preceding figures. Semicircle on a sector, with:

Semicircle, partly struck through, on a sector, with:

Two superimposed circular sectors, with:

Half-column to the left of the preceding.

Circle on a quadrant of a circle, with:

Figure similar to the preceding, with:

Along the left margin, after the struck-through passage.

Figure of a circle on a semicircle, with:

Semi-octagon inscribed in a semicircle; a circular sector with a lune:

On this page

A bird steering by the horn of its tail

Doubled circles: whole in the half, half in the quarter

Circles of quadruple proportion

A lune equals a triangle (octuple proportion)

Overlapping equal surfaces: touching parts equal in figure

Rule for the value of curvilinear parts