Christmas Trilogy 2018 Part 1: The Harmonic Isaac

Isaac Newton is often referred to, as the ‘father’ of modern science but then again so is Galileo Galilei. In reality modern science has many fathers and some mothers as well. Those who use this accolade tend to want to sweep his theological studies and his alchemy under the carpet and pretend it doesn’t really count. Another weird aspect of Newton’s intellectual universe was his belief in prisca theology. This was the belief that in the period following the creation humankind had perfect knowledge of the natural world that got somehow lost over the centuries. This meant for Isaac that in his own scientific work he wasn’t making discoveries but rediscovering once lost knowledge. Amongst, what we would now regard as his occult beliefs, Isaac also subscribed to the Pythagorean belief in Harmonia (harmony), as a unifying concept in the cosmos.

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Robert Fludd’s Pythagorean Monocord

Although he was anything but a fan of music, he was a dedicated student of Harmonia, the mathematical theory of proportions that was part of the quadrivium. According to the legend Pythagoras was the first to discover that musical interval can be expressed as simple ratios of whole numbers related to a taut string: 1:1 (unison), 2:1 (octave), 3:2 (perfect fifth), 4:3 (perfect fourth), 5:4 (major third), 6:5 (minor third). Unfortunately, anybody who has studied the theory of music knows these ratios don’t quite work. If you start on a given tone and move up in steps of a perfect fifth you don’t actually arrive back at the original tone seven octaves higher after twelve fifths but slightly off. This difference is known as the Pythagorean comma. This disharmony was well known and in the sixteenth and seventeenth centuries a major debate developed on how to ‘correctly’ divide up musical scale to avoid this problem. The original adversaries were Gioseffo Zarlino (1570–1590) and Vincenzo Galilei (1520–1591) (Galileo’s father) and Kepler made a contribution in his Harmonice Mundi; perhaps the most important contribution being made by Marin Mersenne (1588–1648) in his Harmonie universelle, contenant la théorie et la pratique de la musique.

Marin_Mersenne_-_Harmonie_universelle_1636_(page_de_titre)

Harmonie Universelle title page

Here he elucidated Mersenne’s Laws:

Frequency is:

  1. Inversely proportional to the length of the string (this was known to the ancients; it is usually credited toPythagoras)
  2. Proportional to the square root of the stretching force, and
  3. Inversely proportional to the square root of the mass per unit length.
mersenne001

Source: Gouk p. 115

As a student Newton took up the challenge in one of his notebooks and we don’t need to go into his contribution to that debate here, however it is the first indication of his interest in this mathematics, which he would go on to apply to his two major scientific works, his optics and his theory of gravity.

After he graduated at Cambridge Newton’s first serious original research was into various aspects of optics. This led to his first published paper:

A Letter of Mr. Isaac Newton, Professor of the Mathematicks in the University of Cambridge; Containing His New Theory about Light and Colors: Sent by the Author to the Publishee from Cambridge, Febr. 6. 1671/72; In Order to be Communicated to the R. Society

In which he described his experiments with a prism that showed that white light consists of blended coloured light and that the spectrum that one produces with a prism is the splitting up of the white light into its coloured components. Previous theories had claimed that the spectrum was produced by the dimming or dirtying of the white light by the prism. Newton wrote an extensive paper expanding on his optical research, An hypothesis explaining the properties of light, but due to the harsh criticism his first paper received he withheld it from publication. This expanded work only appeared in 1704 in his book, Opticks: A Treatise of the Reflections, Refractions, Inflections & Colours of Light. Here we can read:

In the Experiments of the fourth Proposition of the first Part of this first Book, when I had separated the heterogeneous Rays from one another, the Spectrum ptformed by the separated Rays, did in the Progress from its End p, on which the most refrangible Rays fell, unto its other End t, on which the most refrangible Rays fell, appear tinged with this Series of Colours, violet, indigo, blue, green, yellow, orange, red, together with all their intermediate Degrees in a continual Succession perpetually varying . So that there appeared as many Degrees of Colours, as there were sorts of Rays differing in Refrangibility.

This is of course the list of seven colours that we associate with the rainbow today. Before Newton researchers writing about the spectrum listed only three, four or at most five colours, so why did he raise the number to seven by dividing the blue end of the spectrum into violet, indigo and blue? He did so in order to align the number of colours of the spectrum with the notes on the musical scales. In the Queries that were added at the end of the Opticks over the years and the different editions we find the following:

Qu. 13. Do not several sorts of Rays make Vibrations of several bigness, which according to their bignesses excite Sensations of several Colours, much after the manner that the Vibrations of the Air, according to their several bignesses excite Sensations of several Sounds? And particularly do not Vibrations for making a Sensation of deep violet, the least refrangible the largest for making a Sensation of deep red, and several intermediate sorts of Rays, Vibrations of several intermediate bignesses to make Sensations of the several intermediate Colours?

Qu. 14. May not the harmony and discord of Colours arise from the proportions of the Vibrations propagated through the Fibres of the optick Nerves into the Brain, as the harmony and discord of Sounds arise from the proportions of the Vibrations of the Air? And some Colours, if they be view’d together, are agreeable to one another, as those of Gold and Indigo and other disagree.

In the An Hypothesis, Newton published a diagram illustrated the connection he believed to exist between the colours of the spectrum and the notes of the scale.

mersenne003

Source: Gouk p. 118

Interestingly Voltaire presented Newton’s theory in his Elemens de la philosophie de Newton (1738), again as a diagram.

mersenne004

Source: Gouk p. 119

Turning now to Newton’s magnum opus we find the even more extraordinary association between his theory of gravity and the Pythagorean theory of harmony. Newton’s Law of Gravity is probably the last place one would expect to meet with Pythagorean harmony but against all expectations one does. In unpublished scholia on Proposition VIII of Book III of the Principia(the law of gravity) Newton claimed that Pythagoras had known the inverse square law. He argued that Pythagoras had discovered the inverse-square relationship in the vibration of strings (see Mersenne above) and had applied the same principle to the heavens.

…consequently by comparing those weights with the weights of the planets , and the lengths of the strings with the distances of the planets, he understood by means of the harmony of the heavens that the weights of the planets towards the Sun were reciprocally as the squares of their distances from the Sun.[1]

Although Newton never published this theory David Gregory (1661–1708) did. David Gregory was a nephew of the physicist James Gregory who in 1684 became professor of mathematics at the University of Edinburgh, where he became “the first to openly teach the doctrines of the Principia, in a public seminary…in those days this was a daring innovation.”[2]

David_gregory_mathematician

Davis Gregory bust Source: Wikimedia Commons

In 1691, with Newton’s assistance, he was appointed Savilian Professor of Astronomy at Oxford going on to become an important mathematician, physicist and astronomer. He worked together with Newton on the planned second edition of the Principia, although he did not edit it, dying in 1708; the second edition appearing first in 1713 edited by Richard Bentley. In his Astronomiae physicae et geometricae elementa, a semi-popular presentation of Newton’s theories first published in Latin in 1702

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Gregory wrote the following:

The Elements of Astronomy, Physical and Geometrical By David Gregory M.D. SavilianProfessor of Astronomy at Oxfordand Fellow of the Royal Society (1615)

The Author’sPreface

As it is manifest that the Ancients were apprized of, and had discover’d the Gravity of all Bodies towards one another, so also they were not unacquainted with the Law and Proportion which the action of Gravity observ’d according to the different Masses and Distances. For that Gravity is proportional to the Quantity of Matter in the heavy Body, Lucretiusdoes sufficiently declare, as also that what we call light Bodies, don’t ascend of their own accord, but by action of a force underneath them, impelling them upwards, just as a piece of Wood is in Water; and further, that all Bodies, as well the heavy as the light, do descend in vacuo, with an equal celerity. It will be plain likewise, from what I shall presently observe, that the famous Theorem about the proportion whereby Gravity decreases in receding from the Sun, was not unknown at least to Pythagoras. This indeed seems to be that which he and his followers would signify to us by the Harmony of the Spheres: That is, they feign’d Apolloplaying on a Harp of seven Strings, by which Symbol, as it is abundantly evident from Pliny, Macrobiusand Censorinus, they meant the Sun in Conjunction with the seven planets, for they made him the leader of that Septenary Chorus, and Moderator of Nature; and thought that by his Attractive force he acted upon the Planets (and called it Jupiter’s Prison, because it is by this Force that he retains and keeps them in their Orbits, from flying off in Right Lines) in the Harmonical ration of their Distances. For the forces, whereby equal Tensions act upon Strings of different lengths (being equal in other respects) are reciprocally as the Squares of the lengths of the Strings.

I first came across this theory, as elucidated by Gregory, years ago in a book, which book I have in the meantime forgotten, where it was summarised as follows:

Gravity is the strings upon which the celestial harmony is played.

 

 

 

 

 

 

 

 

[1]Quoted from Penelope Gouk, The harmonic roots of Newtonian science, in John Fauvel, Raymond Flood, Michael Shortland & Robin Wilson eds., Let Newton Be: A new perspective on his life and works, OUP, Oxford, New York, Tokyo, ppb. 1989 The inspiration and principle source for this blog post.

[2]Quoted from Significant Scots: David Gregory

https://www.electricscotland.com/history/other/gregory_david.htm

 

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5 Comments

Filed under History of Astronomy, History of Mathematics, History of Optics, History of science, Newton

5 responses to “Christmas Trilogy 2018 Part 1: The Harmonic Isaac

  1. Pingback: Christmas Trilogy 2018 Part 1: The Harmonic Isaac — The Renaissance Mathematicus | Die Goldene Landschaft

  2. Another fascinating blog. I was intrigued by the colour/sound harmony scale to wonder if synaesthetes see those colours for those sounds. I found this website but am not sure it answers that question: https://synesthesia.com/blog/sound-synesthesia/

    • Very probably not, since Newton’s inventing 7 colours to correspond to 7 notes has no basis in fact.

    • Laurence Cox

      Newton may not have had synthesia, but another great scientist, Dick Feynman certainly did. In What do you care what other people think? and quoted in John and Mary Gribbin’s biography of Feynman he wrote:

      “When I see equations, I see the letters in colours – I don’t know why. As I’m talking, I see vague pictures of Bessel functions from Jahnke and Emde’s book, with light tan j’s, slightly violet-bluish n’s, and dark brown x’s flying around. And I wonder what the hell it must look like to the students.”

  3. I’m a little puzzled by Gouk’s having (diameter)2. Assuming uniform density, isn’t the squared diameter of the string proportional to the mass per unit length? Since we want the square root of the mass/length, shouldn’t that be just the diameter? Maybe Gouk’s typo?

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