Four Books on Conics (Commandino 1566) (1566) - Page 18

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is that of the rectangle d f h to the square of f a.

COMMENTARY.

A LET a rectangle a g κ be placed equal to the rectangle b g c: and a rectangle a f l equal to the rectangle d f h.] Almost all of these are missing in the Greek codex, which we have supplied; it is to be understood in this way, that a g is produced to κ; and the rectangle a g κ is made equal to the rectangle b g c; and again, a f being produced to l, the rectangle a f l is made equal to the rectangle d f h.

B Since therefore the angle a d c is equal to the angle b κ g: and the angle d a l in the circle is equal to the angle f h l.] From the twenty-first proposition of the third book of the Elements: for the points a b κ c are on the circumference of the same circle, since the rectangle a g κ is equal to the rectangle b g c, from the converse of the thirty-fifth of the same: and for the same reason, the points a d l h will fall on the circumference of another circle.

29. Book 1.

C The angle g κ b will also be equal to the angle f h l.] For the angle a d c is equal to the angle d a l, because b c and f a are parallel.

D Therefore as b g is to g κ, so is l f to f h.] For it follows from what has been said that the triangle l f h is similar to the triangle b g κ, since the angle a d κ is equal to the angle f h l; as was demonstrated; and the angle l f h is equal to the angle l a g, that is, to b g κ itself. Therefore the remaining angle will also be equal to the remaining one.
29. Book 1.

E This, however, is the ratio of the rectangle d f h to the square of f a.] From which it results that the rectangle d f h to the square of f a has the same ratio as the square of a g to the rectangle b g c. Which indeed it was required to demonstrate.

A geometric diagram showing two intersecting triangles or a complex polygonal figure with labeled vertices (a, b, c, d, f, g, h, κ, l). The lines form various angles and ratios discussed in the text, illustrating the relationships between rectangles and squares mentioned in the commentary.