“It was Pythagoras who discovered that the 5th and the octave of a note could be produced on the same string by stopping at 2⁄3 and ½ of its length respectively. Harmony therefore depends on a numerical proportion. It was this discovery, according to Hankel, which led Pythagoras to his philosophy of number. It is probable at least that the name harmonical proportion was due to it, since1:½ :: (1-½):(2⁄3-½).Iamblichus says that this proportion was called ύπ eναντία originally and that Archytas and Hippasus first called it harmonic.”

— James Gow (scholar)

Nicomachus gives another reason for the name, viz. that a cube being of 3 equal dimensions, was the pattern άρμονία: and having 12 edges, 8 corners, 6 faces, it gave its name to harmonic proportion, since:<center>12:6 :: 12-8:8-6</center>
Footnote, citing Vide Cantor, Vorles [Vorlesüngen über Geschichte der Mathematik ?] p 152. Nesselmann p. 214 n. Hankel. p. 105 sqq.
A Short History of Greek Mathematics (1884)

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James Gow (scholar) 22

scholar 1854–1923

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“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)

“The Pythagoreans discovered the existence of incommensurable lines, or of irrationals. This was, doubtless, first discovered with reference to the diagonal of a square which is incommensurable with the side, being in the ratio to it of √2 to 1. The Pythagorean proof of this particular case survives in Aristotle and in a proposition interpolated in Euclid's Book X.; it is by a reductio ad absurdum proving that, if the diagonal is commensurable with the side, the same number must be both odd and even. This discovery of the incommensurable… showed that the theory of proportion invented by Pythagoras was not of universal application and therefore that propositions proved by means of it were not really established…. The fatal flaw thus revealed in the body of geometry was not removed till Eudoxus discovered the great theory of proportion (expounded in Euclid's Book V.), which is applicable to incommensurable as well as to commensurable magnitudes.”

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

p, 125
Achimedes (1920)

“With a note of music, one strikes the fundamental, and, in addition to the root note, other notes are generated: these are called the harmonic series…As one fundamental note contains within it other notes in the octave, two fundamentals produce a remarkable array of harmonics, and the number of possible combinations between all the notes increases phenomenally. With a triad, affairs stand a good chance of getting severely out of hand…”

Robert Fripp (1946) English guitarist, composer and record producer

[Denyer, Ralph, The Guitar Handbook, 2002, 114, 0-679-74275-1]
Elsewhere

“The Plus Factor makes its appearance in a person's life in proportion as that person is in harmony with God and His universal laws.”

Norman Vincent Peale (1898–1993) American writer

Power Of The Plus Factor (1987)

“On symbolic use of equalities and proportions. Chapter II.
The analytical method accepts as proven the most famous [ as known from Euclid ] symbolic use of equalities and proportions that are found in items such as:
1. The whole is equal to the sum of its parts.
2. Quantities being equal to the same quantity have equality between themselves. [a = c & b = c => a = b]
3. If equal quantities are added to equal quantities the resulting sums are equal.
4. If equals are subtracted from equal quantities the remains are equal.
5. If equal equal amounts are multiplied by equal amounts the products are equal.
6. If equal amounts are divided by equal amounts, the quotients are equal.
7. If the quantities are in direct proportion so also are they are in inverse and alternate proportion. [a:b::c:d=>b:a::d:c & a:c::b:d]
8. If the quantities in the same proportion are added likewise to amounts in the same proportion, the sums are in proportion. [a:b::c:d => (a+c):(b+d)::c:d]
9. If the quantities in the same proportion are subtracted likewise from amounts in the same proportion, the differences are in proportion. [a:b::c:d => (a-c):(b-d)::c:d]
10. If proportional quantities are multiplied by proportional quantities the products are in proportion. [a:b::c:d & e:f::g:h => ae:bf::cg:dh]
11. If proportional quantities are divided by proportional quantities the quotients are in proportion. [a:b::c:d & e:f::g:h => a/e:b/f::c/g:d/h]
12. A common multiplier or divisor does not change an equality nor a proportion. [a:b::ka:kb & a:b::(a/k):(b/k)]
13. The product of different parts of the same number is equal to the product of the sum of these parts by the same number. [ka + kb = k(a+b)]
14. The result of successive multiplications or divisions of a magnitude by several others is the same regardless of the sequential order of quantities multiplied times or divided into that magnitude.
But the masterful symbolic use of equalities and proportions which the analyst may apply any time is the following:
15. If we have three or four magnitudes and the product of the extremes is equal to the product means, they are in proportion. [ad=bc => a:b::c:d OR ac=b2 => a:b::b:c]
And conversely
10. If we have three or four magnitudes and the first is to the second as the second or the third is to the last, the product of the extremes is equal to that of means. [a:b::c:d => ad=bc OR a:b::b:c => ac=b2]
We can call a proportion the establishment of an equality [equation] and an equality [equation] the resolution of a proportion.”

François Viète (1540–1603) French mathematician

From Frédéric Louis Ritter's French Tr. Introduction à l'art Analytique (1868) utilizing Google translate with reference to English translation in Jacob Klein, Greek Mathematical Thought and the Origin of Algebra (1968) Appendix
In artem analyticem Isagoge (1591)

“A wise man, therefore, proportions his belief to the evidence.”

David Hume book An Enquiry Concerning Human Understanding

Section X: Of Miracles; Part I. 87
An Enquiry Concerning Human Understanding (1748)
Context: In our reasonings concerning matter of fact, there are all imaginable degrees of assurance, from the highest certainty to the lowest species of moral evidence. A wise man, therefore, proportions his belief to the evidence.

“I. The subjects of every state ought to contribute towards the support of the government, as nearly as possible, in proportion to their respective abilities, that is, in proportion to the revenue which they respectively enjoy under the protection of the state.”

Adam Smith (1723–1790) Scottish moral philosopher and political economist

Source: (1776), Book V, Chapter II, Part II, p. 892.

“The proportions of good and evil in any society depend partly upon the proportion of consent to that of refusal and partly upon the distribution of power between those who consent and those who refuse.”

Simone Weil (1909–1943) French philosopher, Christian mystic, and social activist

Draft for a Statement of Human Obligation (1943)
Context: The proportions of good and evil in any society depend partly upon the proportion of consent to that of refusal and partly upon the distribution of power between those who consent and those who refuse.
If any power of any kind is in the hands of a man who has not given total, sincere, and enlightened consent to this obligation such power is misplaced.
If a man has willfully refused to consent, then it is in itself a criminal activity for him to exercise any function, major or minor, public or private, which gives him control over people's lives. All those who, with knowledge of his mind, have acquiesced in his exercise of the function are accessories to the crime.
Any State whose whole official doctrine constitutes an incitement to this crime is itself wholly criminal. It can retain no trace of legitimacy.
Any State whose official doctrine is not primarily directed against this crime in all its forms is lacking in full legitimacy.
Any legal system which contains no provisions against this crime is without the essence of legality. Any legal system which provides against some forms of this crime but not others is without the full character of legality.
Any government whose members commit this crime, or authorize it in their subordinates, has betrayed its function.

“To the Greek harmony was of supreme importance, and if the triglyphs represented the harmonic scale… their meaning and use is abundantly explained. …The universal admiration which Greek proportions have always excited proves that the method of obtaining them was correct.”

Ernest Flagg (1857–1947) American architect

Small Houses: Their Economic Design and Construction (1922)

“Every man is rich or poor according to the proportion between his desires and his enjoyments”

Samuel Johnson The Rambler

No. 163 (8 October 1751)
The Rambler (1750–1752)
Context: Every man is rich or poor according to the proportion between his desires and his enjoyments; any enlargement of wishes is therefore equally destructive to happiness with the diminution of possession, and he that teaches another to long for what he never shall obtain is no less an enemy to his quiet than if he had robbed him of part of his patrimony.

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James Gow (scholar) 22

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