Harmony and Proportion by John Boyd-Brent

About Scotland Arts pages

Squared Circle

The Square Root of 2

"We shall therefore borrow all our Rules for the Finishing our Proportions, from the Musicians, who are the greatest Masters of this Sort of Numbers, and from those Things wherein Nature shows herself most excellent and compleat." Leon Battista Alberti (1407-1472)

Anyone who while standing in a city street has looked up at the dome of the sky will recognise this figure as being the distillation of a fundamental condition of human life. On the one hand; order, sometimes man-made, sometimes natural, and on the other hand; boundless encircling space. On the one hand things as they appear to be, and on the other a space of unlimited possibility.



  • Ground
  • Earth
  • Finite
  • Manifest Things
  • Exterior Effect
  • Things Encompased
  • Four Directions


When early geometers came to try to understand mathematically the proportional relationship between the side of a square and its diagonal, they found that there was no straightforward way. Cut and divide as many times as you like, there can be no numerical relationship between the two.


Today our system of numbers is able to describe the relationship as 1:1.414...(etc., a decimal without end...), but the two sides of the relationship remain incommensurable. We refer to the relationship as "irrational", not meaning "without reason", but "without numerical ratio".

Root Two

Because the irrational relationship between the square and its diagonal seems to give a glimpse of a kind of order which is not resultant from (or dependent on) manifest number, but appears to indicate a "meta-physical" cause, it has in the past been held by architects and artists to have special significance.
Its appearance in the underlying geometry of the Squared Circle echoes or symbolises the "dialogue" between the Heavens and the Earth, the Infinite and the Finite.


Musica Mundana , the Music of the Spheres, the exquisite and rarified interior sound of the Cosmos, is reflected in the harmonious relationships of certain sounds and geometrical forms within the space of our universe...
The relationship between the side and diagonal of a square is not of this same order, but nevertheless fulfills the same function as a symbol (analogous to the squared circle) of the harmonious relationship between Form and Essence, Finite and Infinite.


Leon Battista ALBERTI, The Ten Books of Architecture, 1775 Leoni Edt. (Dover)
Andrea PALLADIO, The Four Books of Architecture (Dover)
Keith CRITCHLOW, Order in Space, 1979
PLATO, The Timaeus
Rudolf WITTKOWER, Architectural Principles in the Age of Humanism 1949/1988
Very interesting pages at: Sacred Geometry by Bruce Rawles.
The following links used to be hypertext, but the pages have moved. Use Google:
Barbara Hero's Lambdoma Harmonic Keyboard
"River Nymph" by John Leicmon, based on the Square Root of 2