Similar Portions of Circles in Sixteenfold Proportion
Geometry of curved figures: portions of two circles compared, with sector-to-semicircle equalities
This inverted sheet works through the geometry of curved figures: given two circles whose areas stand in sixteenfold proportion, portions cut from a quarter of each circumference are shown to be similar in shape and likewise sixteenfold, since circle is to circle as diameter squared is to diameter squared. A left column manipulates sectors and segments (labelled a, b, c; e, g, f, d; h, i; L, m, o; n, m), removing a diminished sector and adding its completion, then matching a perfected sector to an equal semicircle. A lettered semicircle marked 1/2 and a sector marked as 'an eighth of a quadruple' accompany the demonstration, alongside a large circle with its inscribed smaller circle and diameter at lower left. Much additional faint text across the top and right of the sheet is not transcribed here.
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Similar portions of two sixteenfold circles
Of similar surfaces the proportion in the parts equals that in the wholes. So for two circles in sixteenfold proportion, the portions taken from a quarter of each circumference are similar in figure and stand in the same sixteenfold ratio.
Circle-to-circle ratio equals diameter squared
The sixteenth part of the larger circle equals the whole smaller circle, so each similar portion of the larger is worth sixteen of the smaller. The proportion of circle to circle is that of diameter to diameter multiplied into itself.
Sector, segment and equal semicircle (a b c, g e f, L m o, n m)
The diminished sector a b c is removed from the larger portion and completed as g e f, the parts cut off with a straight line h i. A semicircle L m o equal to the perfected sector is then given; subtracting the added part a from both leaves n m equal to f.