Four Books on Conics (Commandino 1566)

lines are drawn to another point, which is not on the circumference of the circle: their proportion is not the same as that which c has to d. For if it could be, let that be done to point m, which is taken outside the circumference (for if taken inside, the same absurdity would follow). And with m a, m b, and m f joined, let a m be to m b as c is to d. Therefore, as e d is to d, so is the square of e d to the square of c; and as the square of a m is to the square of m b. But as e d is to d, so a f is set to f b. Wherefore, as a f is to f b, so is the square of a m to the square of m b. And from those things which were just said, if a line is drawn from some point b equidistant to a m, it will be demonstrated as a f is to f b, so is the square of a f to the square of f m. But it has been demonstrated that as a f is to f b, so is the square of a f to the square of f h. Therefore, the line f h is equal to f m itself, which cannot happen.

C
D
E
9. of the 5th.

Plane loci, therefore, are of this kind. Solid loci, however, are so called because the lines by which their problems are solved have their generation from the section of solids, such as the sections of a cone and many others. There are also other loci called loci to a surface, to which a name is imposed from their property. Apollonius then attacks Euclid—not, as Pappus and some others think, because he did not find two mean proportionals (since Euclid correctly found one mean proportional, and not unsuccessfully, as he himself says); but Euclid did not attempt to investigate two mean proportionals at all in the Elements, and Apollonius seems to inquire nothing about two mean proportionals in his third book. But it is probable that Euclid wrote about loci in another book which has not reached our hands. What he adds next concerning the fourth book is clear. He says the fifth book deals for the most part with minima and maxima. For just as we learned in the Elements, if lines are drawn from some point into a circle, of those which pertain to its concave circumference, the one passing through the center is the maximum; but of those which pertain to the convex, the one that is intercepted between the said point and the diameter is the minimum: so also does he inquire about conic sections in the fifth book. The purpose of the sixth, seventh, and eighth books is clearly explained by Apollonius himself. And let these things be said concerning the letter.

Solid loci

FEDERICO COMMANDINO’S COMMENTARY

ON THE PROBLEM OF APOLLONIUS.

A
12. of the 5th.

And likewise g is a mean proportional between a f and f b: Because, since as d is to b f, so is c to g; by alternating, as d is to c, so is b f to g: again, because as e is to a b, so is d to b f from the 12th of the 5th, e d will be to a f as d is to b f. But as d is to b f, so is c to g. Therefore, e d is to a f as c is to g; and by alternating, e d is to c as a f is to g; and by converting, c is to d e as g is to a f. Now, d was to c as b f was to g; and as d is to c, so is c to d e. Wherefore, as b f is to g, so is g to a f; and therefore g is a mean proportional between a f and f b. This is what had to be demonstrated.

B

But as a f is to f h, so is e d to c: For we have just shown that e d is to c as a f is to g; that is, to f h, which is equal to g itself.

C
22. of the 6th.
20. of the 6th.

Therefore as e d is to d so is the square of e d to the square of c, and the square of a m to the square of m b: For as e d is to c, so is c to d; and as c is to d, so a m is set to m b. Wherefore, as e d is to c, so is a m to m b; and therefore as the square of e d is to the square of c, so is the square of a m to the square of m b. As, therefore, e d is to d, so is the square of e d to the square of c, and the square of a m to the square of m b.

D

As however e d is to d, so a f is set to f b: For it was demonstrated above that as e d is to a f, so d is to b f. Wherefore, also by alternating, as e d is to d, so is a f to f b.

E
4. of the 6th.
22. of the 6th.
29. of the 1st.

And from those things which were just said, if from point b a line is drawn equidistant to a m: as a f is to f b, so will it be demonstrated that the square of a f is to the square of f m: Let the line b n be drawn through b to m f, which is equidistant to a m. Because of the similarity of the triangles a m f and b n f, it will be as a f is to f b, so is a m to b n. Therefore, since as a f is to f b, so is the square of a m to the square of m b; and so is the square of a f to the square of f b: the square of a m will be to the square of m b as the square of a f is to the square of f b: and therefore the line a m is to m b as a f is to f b: and by converting, m b is to a m as f b is to a f. Now a m was to b n as a f was to f b. Wherefore, by equality, m b is to b n as b f is to f b. But a m is to m b as a f is to f b: and as a f is to f b, so is b f to f b. Therefore as a m is to m b, so is m b to b n. Since, therefore, around the equal angles a m b and m b n, the sides are proportional: the triangle a b m will be similar to the triangle m n b: and angle b a m is equal to angle n m b. But of the triangles a m f and m b f, the angle f a m is equal to the angle f m b: and the angle at f is common to both. Therefore, the remaining angle is equal to the remaining, and triangle will be similar to triangle. Wherefore, as a f is to f m, so is f m to f b. As, therefore, the first is to the third f b, so is the square to the square of f m.