Mathematics and Metaphysics

in the

Premodern World



Assignment 2b

Due: Thursday,

February 7th


Do the following for Thursday...

Read the following:


Plato. Plato’s Cosmology: The Timaeus of Plato. Translation, notes, and commentary by Cornford.  New York: Liberal Arts Press, 1937. Excerpts. (ca. 30pp of English)

56-73: Bury's translation: Creation, World-Soul, means, Cosmic Blacksmith

57-61, 66-75: Cornford's translation of roughly the same section with commentary.  [page number coincidence]

Image: Chalcidius (Calcidius), and Plato. Commentarius in Platonis Timaeum. Chalcidius (Calcidius) - MS from Bodlian. 12th c.




Do the following:  (just thinking at this point)

1. As you read these excerpts from Timaeus, think about all the musical math.  Think about how the cosmos of Timaeus fits with the cosmos of "The Myth of Er" from the Republic.  Try to do some of the math.  Your assignment for next Tuesday will involve doing a lot of this type of math, so get ready.  Play around with the various types of means (see below).


Notes on Bury translation (pp. 56-73): Things get mathematically confusing starting on p. 68.  Don't get discouraged.  [The exercises that I will hand out on Thursday will help with this section.]  Plato gets back to being only sort-of-confusing by p. 71. 

            Immediately after p. 73 of Bury's translation I inserted Cornford's translation and commentary of the same material which have page numbers very close to that of Bury's.  This is a total coincidence.  (...or is it?)


Notes on Cornford's translation and commentary (start on p. 57, "The World Soul" and read to 61, skip pp. 62-65, then read 66-75)  This reading can get pretty mathy and it is rather cryptic.  Cornford's commentary can be useful and it can also be aggravating. If you rush the reading it will all be aggravating.  Savor it like a good single-malt.  Do what you can with it.  I'll try to fill in as much as I can in class.

Back to Quadrivium Main Page



Notes for in-class material.


If you sing Do-Sol and then Sol-do, you are going up a fifth (3/2) and then up a fourth (4/3) which combine to be an octave. It seems like you are adding a fifth to a fourth, but mathematically you are multiplying.

          which is an octave.


So... if going up in intervals is multiplication.  What do you think going down is?  Division, of course.

Go up a fifth, Do-Sol, then go down a fourth, Sol-Re.

The difference between a fifth and a fourth is

   This is the definition of a tonus, or tone.  It is not technically consonant.


In Timaeus (pp. 69-71 in Bury trans. and pp. 68-72 in Cornford) Plato divides all tetrachords, fourths, into two tones (toni) and a semitone, which is defined as follows....

A tetrachord is a fourth, 4/3,... Do-Fa.  [Here comes the bride.]

It is divided into tonus-tonus-semitonus.  Here's how...

       Solve for x.


 = the semitone.


Check it: 


So a full diatonic scale over an octave is this:

































fifth (3/2)

fourth (4/3)



fourth (4/3)

fifth (3/2)


Two tetrachords with a tone in-between.

...or a fifth and a fourth.

Plato also uses the harmonic and arithmetic means to fill up the intervals between his geometric sequence:

1, 2, 3, 4, 8, 9, 27. 

The resultant intervals all superimpose exactly onto a 4+ octave diatonic scale like the one above.


Arithmetic Mean


c is linearly equidistant

from both a and b.

The arithmetic mean is "equidistant between both extremes." Alberti’s example:

a : c : b = 4 : 6 : 8

The common average.

Geometric Mean


a x b is a rectangle,

c is the square with the same area.

"…the dimension of a side [c] that generates a square of equal [area]" to that of a rectangle with sides measuring a and b. Alberti’s example:

a : c : b = 4 : 6 : 9

Harmonic Mean



"The proportion between the shortest and the longest dimensions is the same [proportion] as that [quantity] between the shortest and the middle, and … that [quantity] between the middle one and the longest." The ratio between the outer numbers equals the ratio between the differences between the numbers. 

Alberti’s example:

a : c : b = 30 : 40 : 60

The quotations on the right are from Leon Battista Alberti's description of the means from 

De re aedificatoria (1452, On the Art of Building)