Mathematical Collections (Commandino 1588)

Arithmetic mean

they are the arithmetic, the geometric, and the harmonic. The arithmetic mean is said to exist when, there being three terms, the middle term exceeds one of the extremes by an equal excess in quantity and is exceeded by the other, as 6 is to 9 and to 3, or when it is so that the first term is to itself as the first excess is to the second. One must understand the exceedings as the first [items].

Geometric.

The geometric mean, which is properly called a proportion, is when it is so that the middle term is to one of the extremes as the other is to the middle, as 12 is to 6 and to 3; and otherwise when it is so that the first term is to the second as the first excess is to the second.

Harmonic.

The harmonic mean is when the middle term exceeds one of the extremes and is exceeded by the other by the same part, as 3 is to 2 and to 6; or when it is so that the first term is to the third as the first excess is to the second, as 6, 3, 2 have it.

With these things posited, we will find three means together in five minimal straight lines.

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COMMENTARY.

A "EC itself as the arithmetic mean": The Greek codex reads τὴν μενε μὲ σην κρικμητικὴν the mean arithmetic, but this is erroneous; it should be read as τὴν μὲνε γ μὲ σην ἀριθμητικὴν the mean arithmetic.

B "And EC is the mean between AD and DC in arithmetic mean, it is clear": In this place, the Greek codex is faulty, in which it is read ἡ δὲ ε̄ τῆς ᾱ δ̄ λ̄ γ̄. Correct it to ἡ δὲ ε̄ γ̄ τῆς ᾱ δ̄ δ̄ γ̄.

C "And as AD is to itself, so is the excess of AD, AE—that is, AD, EC—to the excess of EC, CD": He shows that EC is the mean between AD and DC in an arithmetic mean from its definition, for an arithmetic mean is, as he himself writes below, "when there being three terms, the middle exceeds one of the extremes by an equal excess and is exceeded by the other. As 6 is to 9 and to 3, or when it is so that the first is to itself as the first excess is to the second." Therefore, it is as AD is to itself, so is the excess of the first and second (AD, EC—that is, AD, AE, which is ED) to the excess of the second and third (EC, CD, which is the same ED), for AE and EC are equal to one another, since E is posited as the center of the circle.

D "For it will be shown by us below that two EE and three DB etc": In book 20 of this [work].

E "The harmonic mean is when the middle term exceeds one of the extremes and is exceeded by the other by the same part, as 3 is to 2 and to 6": For 3 exceeds 2 by the half-part of 2, and is exceeded by 6 by the same half of 6.

F "With these things posited, we will find three means together in five minimal straight lines": About these, later in the 15th of this [work].