“The problem of doubling the cube was henceforth tried exclusively in the form of the problem of the two mean proportionals.”

p, 125
Achimedes (1920)

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Thomas Little Heath46

British civil servant and academic 1861–1940

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“Hippocrates also attacked the problem of doubling the cube. …Hippocrates did not, indeed, solve the problem, but he succeeded in reducing it to another, namely, the problem of finding two mean proportionals in continued proportion between two given straight lines, i. e. finding x, y such that a:x=x:y=y:b, where a, b are the two given straight lines. It is easy to see that, if a:x=x:y=y:b, then b/a = (x/a)3, and, as a particular case, if b=2a, x3=2a3, so that the side of the cube which is double of the cube of side a is found.”

Thomas Little Heath (1861–1940) British civil servant and academic

Achimedes (1920)

“Take the case of a famous problem which plays a great part in the history of Greek geometry, the doubling of the cube, or its equivalent, the finding of two mean proportionals in continued proportion between two given straight lines. …if all the recorded solutions are collected together, it is much easier to see the relations, amounting in some cases to substantial identity, between them, and to get a comprehensive view of the history of the problem. I have therefore dealt with this problem in a separate section of the chapter devoted to 'Special Problems,' and I have followed the same course with the other famous problems of squaring the circle and trisecting any angle.”

Thomas Little Heath (1861–1940) British civil servant and academic

A History of Greek Mathematics (1921) Vol. 1. From Thales to Euclid

“A problem shared is a problem doubled.”

David Gemmell book Legend

Source: Drenai series, Legend, Pt 1: Against the Horde, Ch. 1

“While then for a long time everyone was at a loss, Hippocrates of Chios was the first to observe that, if between two straight lines of which the greater is double of the less it were discovered how to find two mean proportionals in continued proportion, the cube would be doubled; and thus he turned the difficulty in the original problem into another difficulty no less than the former. Afterwards, they say, some Delians attempting, in accordance with an oracle, to double one of the altars fell into the same difficulty. And they sent and begged the geometers who were with Plato in the Academy to find for them the required solution. And while they set themselves energetically to work and sought to find two means between two given straight lines, Archytas of Tarentum is said to have discovered them by means of half-cylinders, and Eudoxus by means of the so-called curved lines. It is, however, characteristic of them all that they indeed gave demonstrations, but were unable to make the actual construction or to reach the point of practical application, except to a small extent Menaechmus and that with difficulty.”

Thomas Little Heath (1861–1940) British civil servant and academic

Apollonius of Perga (1896)

“Empirical problems of a nonlinguistic sort are not solved by linguistic considerations, but it is important that the two kinds of problems not be confused and that nonlinguistic problems be expressed in such a form as aids their empirical solution.”

Charles W. Morris (1903–1979) American philosopher

Source: "Foundations of the Theory of Signs," 1938, p. 57

“Some… agreeing with Philolaus, believe that the proportion is called harmonic because it attends upon all geometric harmony, and they say that 'geometric harmony' is the cube because it is harmonized in all three dimensions, being the product of a number thrice multiplied together. For in every cube this proportion is mirrored; there are in every cube 12 sides, 8 angles and 6 faces; hence 8, the [harmonic] mean between 6 and 12, is according to harmonic proportion…”

Nicomachus (60–120) Ancient Greek mathematician

this harmonic proportion may be expressed as <math>\frac{12}{6}=\frac{12-8}{8-6}</math> or inversely.
Nicomachus of Gerasa: Introduction to Arithmetic (1926)