Four Books on Conics (Commandino 1566)

meaning what is sought. The other, however, acts as a universal proposition, declaring when, and by what method, and in how many ways that which is proposed can be done. As in the twenty-second theorem of the first book of the Elements of Euclid: From three straight lines, which are equal to three other given lines, to construct a triangle; it is necessary, however, that two such lines be greater than the remaining one, however they are taken: for it has been demonstrated that in every triangle, any two sides, however taken, are greater than the remaining one. He says the third book of Conics contains many admirable theorems that will be useful for the composition of solid loci. Ancient geometers were accustomed to call them plane loci when a problem is produced not from one point alone, but from many. For instance, if someone proposes: Given a bounded straight line, find a point from which a perpendicular line drawn to the given line is a mean proportional between the parts of that same line. Geometers call a locus of this kind a plane locus, because there is not just one point that satisfies the problem, but a whole locus, which the circumference of a circle described around the given straight line, as if around a diameter, possesses. For if a semicircle is described on the given straight line, whatever point you take on the circumference, and from it draw a perpendicular to the diameter, it will satisfy the proposal. Similarly, however, given a straight line, if someone proposes to find a point outside of it from which lines drawn to its endpoints are equal to each other: in this case too, there is not just one point that satisfies the problem, but a locus which the line contains, drawn from the midpoint of the given line at right angles. For if the given line is bisected, and from that point a line is drawn at right angles, whatever point you take on it will do that which was proposed. Apollonius himself writes something similar in the anaioymeno topo analyzed place/locus.

Plane loci.

Let the given points be a and b; let the given proportion be that which c has to d: and let c be greater: and it is required to do that which is proposed. Let a b be joined: and produced towards the parts of b: and let it be done so that as d is to c, so c is to another line, which will be greater than d: let it be e d. Again, let it be done so that as e is to a b, so d is to b f, and c is to g. It is clear, therefore, that the line c is a mean proportional between d and e d: and likewise g is a mean proportional between a f and f b. Wherefore if from the center f, and with the interval g, a circle k h is described, the circumference k h will cut the line a b. Let any point h be taken on the circumference: and let h a, h b, and h f be joined; h f will be equal to g itself: and therefore, as a f is to f h, so h f is to f b. Now, the sides are proportional around the same angle h f b. Therefore, triangle a f h is similar to triangle h f b: and angle f h b is equal to angle h a b. Let b l be drawn through b equidistant to a h. And because as a f is to f h, so is h f to f b; the first f a will be to the third f b as the square of a f is to the square of f h. But as a f is to f b, so is a h to b l. Therefore, as the square of a f is to the square of f h, so is a h to b l. Again, because angle b h f is equal to angle h a b: and angle a h b is equal to angle h b l, for they are alternate: and the remaining angle will be equal to the remaining: and triangle a h b is similar to triangle b h l. Wherefore the sides which are around the equal angles are proportional: namely, as a h is to h b, so h b is to b l: and as the square of a h is to the square of h b, so a h is to b l. But it was as a h is to b l, so is the square of a f to the square of f h. As, therefore, the square of a f is to the square of f h, so is the square of a h to the square of h b. And for this reason, as a f is to f h, so is a h to h b. But as a f is to f h, so is e d to c, and c to d. Therefore, as c is to d, so is a h to h b. Similarly, we will show that all other lines which are inclined from the points a b to the circumference of the circle have the same proportion that c has to d. And so I say if from points a b...

6. of the 6th.

cor. 20. of the 6th.

4. of the 6th.

29. of the 1st.

4. of the 6th.

22. of the 6th.

A geometrical diagram displays a circle with points a, k, b, and f arranged along a horizontal diameter line. A triangle is constructed with its vertex h on the circumference and its base on the diameter. To the right, two vertical line segments, labeled c and d, illustrate the given geometric proportion.