Mathematical Collections (Commandino 1588)

ratio. Wherefore, as CM is to MA, so is the square from CM to the square from ML—that is, the square from LM to the square from MA.

C But the proportion composed of the proportion of the square of AM to the square of MG, and the proportion of the line AM to MG, is the same as that which the cube made from AM has to the cube from MG. For all prisms and pyramids have a proportion between them composed of the proportion of the bases and the proportion of the heights: which we demonstrated in the book On the Center of Gravity of Solids, proposition xxi. For a cube is a certain prism, the side of whose base is equal to its height.

To take three means in a semicircle.

A geometric diagram shows a semicircle with diameter AC and center E. A vertical line segment EB rises from the center to the arc. A point D lies on the diameter between E and C, with a vertical line segment DB rising to the arc. A line segment FD is drawn within the semicircle, and a line segment connects A to B.

Someone else said this, and exhibiting a semicircle ABC, whose center is E, and assuming some point D on the straight line AC, and drawing DB at right angles to EC, he joined EB. He asserted that if DF is drawn perpendicular to this from point D, three means are simply exhibited in the semicircle: EC itself as the arithmetic mean; DB as the geometric [mean], and BF as the harmonic. But it is clear that BD is the mean between AD and DC in geometric proportion, and EC is the mean between AD and DC in arithmetic mean. For as AD is to DB, so is DB to DC: and as AD is to itself, so is the excess of AD and AE—that is, AD and EC—to the excess of EC and CD. However, in what way FD is the mean in a harmonic mean, or of which straight lines, he did not say; he only affirmed that it is the third proportional of the straight lines EB, BD, ignoring that a harmonic mean is formed from EB, BD, BF themselves, which are in geometric proportion. For it will be shown by us below that two [units of] EB and three [units of] DB and one [unit of] BF piled together make the greater extreme of the harmonic mean; and two [units of] BD and one [unit of] BF make the mean; and one [unit of] BD and one [unit of] BF make the least. But first we must discourse about the three means; in the second place about those which are in the semicircle, then about the other three which are opposed to them according to the ancients: lastly about the four means which are among the moderns, according to their opinion, and in what way each of the ten means can be found through geometric proportion. So that we may also refute that which is proposed with more [arguments].

B

8 of Book VI
C
D

μεσότης medietas
mean
ἀναλογία analogia
proportion

A mean differs from a proportion, for if something is a proportion, it is also a mean; but not the reverse. For there are three means,