Four Books on Conics (Commandino 1566)

to d k, so is the rectangle which is contained by both d g, e f, and d e to that contained by both d g, e f, and d k. Now the triangle a b c is half of the rectangle contained by a h and b c: A and the trapezoid d e f g is half of that which is contained by both d g, e f and d k. Therefore, as B the rectangle a b c is to the rectangle contained by both d g, e f, and d e, so is the triangle a b c to the trapezoid d e f g. But if a b c is a triangle, and d f a parallelogram; by the same reasoning it will result that as the triangle a b c is to the parallelogram d f, so is the rectangle a b c to double the rectangle d e f.

From which it is evident that the rectangle a b c, if indeed d f is a parallelogram, C is equal to double the rectangle d e f: but if it is a trapezoid, it is equal to that which is contained by both d g, e f and d e itself.

COMMENTARY.

A BUT the triangle a b c is half of the rectangle contained by a h, b c, and the trapezoid d e f g is half of that which is contained by both d g, e f and d k.] For, having joined d f, the triangle e d f will be half of the rectangle contained by e f and d k: and the triangle d f g likewise half of that which is contained by d g and d k. Therefore the whole trapezoid d e f g is half of the rectangle which is contained by both e f, d g, and d k itself.

B Therefore as the rectangle a b c is to the rectangle contained by both d g, e f and d e, so is the triangle a b c to the trapezoid d e f g.] For it is gathered from the aforementioned that as the rectangle a b c is to the rectangle from a h and b c, so is the rectangle from d g, e f and d e to the rectangle from d g, e f and d k. Therefore, by alternating, as the rectangle a b c is to the rectangle from d g, e f, and d e, so is the rectangle from a h and b c to the rectangle from d g, e f and d k; and so are their halves, that is, the triangle a b c to the trapezoid d e f g.

C From which it is evident that the rectangle a b c, if d f is a parallelogram, etc.] This follows when the triangle a b c is equal to the parallelogram or trapezoid d e f g. Which is also demonstrated by Eutocius in his commentaries on the 49th proposition of the first book of Apollonius. Therefore, it is likely that something is missing in Pappus's words here.

LEMMA IX.

Let there be a triangle a b c, and let c a be produced to d, and let a straight line d h e be drawn as it may happen; to which let a g be drawn parallel: and a f parallel to b c. I say that as the square of a g is to the rectangle b g c, so is the rectangle d f h to the square of f a.

A LET a rectangle a g k be placed equal to the rectangle b g c: and a rectangle a f l B equal to the rectangle d f h: and let b k and h l be joined. Since therefore the angle at c C is equal to the angle b k g: and the angle d a l in the circle is equal to the angle f h l: the D angle g k b will also be equal to the angle f h l. Therefore as b g is to g k, so is l f to f h. But it is as d a g to g b, so is h e to e b: and as h e is to e b, so is h f to f a. As therefore a g is to g b, so is h f to f a.

lemma in 22 of Book 10 But as b g is to g k, so is some other line l f to the antecedent f h. Therefore, by equality in perturbed ratio, as a g is to g k, so is l f to f a. But as a g is to g k, so is the square of a g to the rectangle a g k, that is, to the rectangle b g c. And as l f is to f a, so is the rectangle l f a, that is d f h, to the square of f a. Therefore as the square of a g is to the rectangle b g c, so is the rectangle d f h to the square of f a. But it is permitted to demonstrate that same thing also by composition of ratios. For since the ratio of a g to g b is the same as that of h e to e b; that is h f to f a: and the ratio of a g to g c is the same as that of d e to e c; that is d f to f a: the ratio compounded of the ratio of a g to g b, and of the ratio of a g to g c, which is that of the square of a g to the rectangle b g c, is the same as that which is compounded of the ratio of h f to f a: and of the ratio of d f to f a. This, however, is the ratio of the rectangle d f h to the square of f a.

Geometric diagram showing a triangle abc with extended lines and parallel constructions. Points labeled include a, b, c, d, e, f, g, h, k, l. A line d-h-e intersects the triangle's extensions, with parallel lines a-g and a-f forming various geometric relationships.