Means In Pythagorean Arithmetic

I am reading the fascinating Wooden Book titled The Diagram, by Adam Tetlow.

It demonstrates how through a simple Diagram all of the common, and many other divisions, can be found, claiming that this tool has existed for 7000 years as an aid to builders.

However I have some questions over the definitions of the Means as given on page 4 of the book. Not that I wish to declare them as invalid, but simply as not primary in the consideration of these Means. They are not the Principles of the Means, but their result.

According to the author “The Arithmetic Mean is half the sum of the extremes. In the Geometric Mean, the square of the Mean equals the product of the extremes. The Harmonic Mean is twice the product of the extremes divided by their sum.” Simples, as they say.

However these differ from the traditional definitions of those terms following Pythagoras, with a resulting obscuration of inherent elegance. For the Ancients whole numbers only were considered, since these hold a specific relationship to one another, and to The One from which all is derived.

The definitions are given by Thomas Taylor in his exposition of Pythagorean Arithmetic. Writing in the 18th century, Taylor tells us

“an arithmetical arrangement of terms, divides the quantities by equal differences only; but a geometrical arrangement conjoins the terms by an equal ratio. The harmonic arrangement, however, being more amply conversant with relation than the other middles, neither speculates ratio solely in the terms, nor solely in differences, but is in common conversant with both. For it shows that as the extreme terms are to each other, so is the difference between the greater and middle term, to the difference between the middle and last term.”

An arithmetical arrangement, according to Taylor, separates the terms of the series by the same interval. In the arithmetic series 3, 4, 5, the unit, 1, separates each from the succeeding number. Similarly we may find an arithmetical series among the numbers 2, 4, 6, where the separation is by 2 units, progressively. The arithmetical series follows simply with an equal separation between the figures considered. It is not that the sum of the extremes is the significant factor, but that the separation is equal between them. The other, the Mean, or middle as Taylor calls it, being half the sum of the extremes is interesting but incidental to this.

By Mr Tetlow’s definition with the Geometrical Mean the square of the Mean is equal to the product of the extremes. For example 1, 2, 4, is a geometrical series, and sure enough we find that 2 x 2, the square of the Mean, is equal to the product of the extremes, since 1 x 4 = 2 x 2. Similarly 3, 9, 27 is a geometrical progression, with 3 x 27 = 9 x 9.

In comparison Taylor defines this series as separated by a similar proportion between the terms that is reflected in the relationship between the intervals.

In the case of 1, 2, 4, the intervals between them are 1 and 2. The entire series is bound by the relationship, 1:2. Let us take a different example from the same series. 2, 4, 8. We see that 2 is to 4 as 4 is to 8, or, as it is written, 2:4::4:8, and that the intervals maintain this ratio since 4-2 = 2 and 8-4 = 4 and 2:4 = 1:2. Similarly in our second example of 3, 9, 27, we find that 3:9::9:27. A relationship of 1:3. While the intervals between them also reflect this, since 9-3 = 6 and 27 – 9 = 18 while 6 :18 = 1:3, the bounding relationship of this series. This is elegant and evident.

Mr Tetlow’s definition serves as a corollary to this first and apparent expression. I would argue that the definitions of Means, given in The Diagram, serve as proofs of the condition of three numbers. But they do not deal with the purpose of identifying those types. It is a case of observing Results rather than Causes. Ratio, as Mr Tetlow observes, is the relationship between two numbers, where ‘proportion’ is a continued sequence of a ratio. In this way Proportion may be seen as the progressive generation of those relationships, the manner in which these ratios generate further numbers, limiting, as it were, the series of numbers by their chains, links and meshes, beyond the limits of their initial expression, which are found to be the seeds of each series.

In defining the Harmonic Mean, Taylor defines the series as sharing in the properties of both the Arithmetical and Geometrical Means since the relationship between the extremes is reflected in the relationship of the intervals. Again a simple example will demonstrate this.

In the series 3, 4, 6; 3:6 = 1:2, where the intervals between the numbers in the series are 1 and 2 respectively, which also hold that same relationship. It is these properties of elegance and harmony which bring Joy to the Hearts of Pythagoreans everywhere.

That the measure within them reflects in other outer expressions is incidental to these and a result of the basic laws each of the Means possesses, and demonstrates.

Beauty and elegance, for the modern mathematician, appear to be in obscuration and the manipulation of numbers, rather than in the relationships between them. It is not through manipulation that we find the wisdom of numbers, but in their unique and stable properties.

Consider Mr Tetlow’s definition that ‘the harmonic mean is twice the product of the extremes divided by their sum’. Taking our example of 3, 4,6, by this definition

(3 x 6) x 2 / (3+6) = 4.

But why the need for Algebra when the simple definition offered by Taylor shows us all we need to see? Not only within its own identity but also in relation to the other types of Means considered by the ancients.

Later followers began to play with further distinctions in relationship to bring the full complement to 10, such as the Inharmonic Mean in which the series 3, 5, 6, demonstrates that 3:6 holds an inverse relationship to the intervals, since 5-3 = 2 and 6 – 5 = 1. But let us leave these aside since we are concerned with simplicity and its application to our lives in our return to The One, simplest of all Things, and beyond definition.

Author: Keith Armstrong

Dance teacher, writer, film-maker, educationalist, enthusiast.