Mathematical Collections (Commandino 1588)

PAPPI MATH. COLL.
COM-

has a smaller ratio than E to F. I say that, from equality, A has a smaller ratio to C than D has to F.

For since A has a smaller ratio to B than D has to E, by permuting, A will have a smaller ratio to D than B has to E. And for the same reason, B has a smaller ratio to E than C has to F. Therefore, again by permuting, A has a smaller ratio to C than D has to F.

A geometric diagram displays six vertical line segments of varying lengths. The top row is labeled A, B, C and the bottom row is labeled D, E, F, illustrating the proportional relationships described.

Prop. 8

Prop. 12

These are the things which I considered necessary to preface. Therefore, omitting to explain to you, and to those who are practiced in geometry, the things which he wrote concerning the construction and the objections we raised, I judged it best to explain what the ancients thought concerning the stated problem. I shall begin by saying a few things about the problems that are considered in geometry.

Plane problems. Solid problems.

ἐπίπεδα, στερεὰ, γραμμικά plane, solid, linear

The ancients established that there are three types of geometric problems, and that some are called plane, others solid, and others linear. Those that can be solved by straight lines and the circumference of a circle are rightly called plane; for the lines through which such problems are solved originate in a plane. However, whatever problems are solved by assuming in the construction one or more conic sections are called solid, for it is necessary to use the surfaces of solid figures, namely cones, for their construction. There remains the third type, which is called linear. For lines other than those already mentioned are assumed for the construction, having a varied and transmutable origin, such as spirals and those which the Greeks call τετραγωνιζούσας quadratrices, which we can call quadratrices. Also the conchoids and cissoids, to which many admirable properties belong.

Conic sections are not easily drawn. Do not define the problem.

Prop. 8

Since such are the differences among problems, the ancient geometers could not construct the aforementioned problem regarding two straight lines—which is by nature solid—using a geometric method, since it is not easy to draw conic sections on a plane. However, they wonderfully translated the operation into manual, convenient, and suitable construction using instruments, which can be seen in their volumes that are in circulation, such as in the mesolabum mean-finder instrument of Eratosthenes, and in the mechanical and catapultic works of Philo and Hero. For these men, asserting that the problem is solid, performed its construction solely with instruments, consistently with Apollonius of Perga, who also performed its resolution via conic sections, while others did so via the solid loci of Aristaeus; but none did so through those which are properly called plane. But Nicomedes also performed it using the conchoid line, through which he also trisected an angle. We shall therefore set forth his four constructions along with a certain treatment of our own. Of these, the first is by Eratosthenes, the second by Nicomedes, the third by Hero—most suitable for manual operation for those who wish to be architects—and the last is the one invented by us. For with any solid given, another solid similar to the given one is constructed to a given ratio if, for two given straight lines, two mean proportionals are assumed in continuous analogy, as Hero says in his mechanical and catapultic works.