The Poetry of Logarithms

by Ted Andromidas

Note: For this pedagogical discussion, you will need Appendices I and X to {The Science of Christian Economy}, {So You wish to Learn All About Economics,}, and the April 12, 2002 issue of {Executive Intelligence Review}.

“You have no idea how much poetry there is in a table of logarithms.” — Karl Friedrich Gauss to His Students

Developing a function for the distribution of the prime numbers has been one of the great challenges of mathematics. An exact solution to this problem, of how many numbers generated between 1 and any given number, N, are actually prime, has not yet been discovered, though there is a general notion of a succession of manifolds as determining to any solution.

One of the most stunning demonstrations of the generation of number by an orderable succession of multiply-connected manifolds, is Karl Friederich Gauss’ discovery of the “Prime Number Theorem.” The wonderfully paradoxical nature of Gauss’ approach, in contradistinction to that of Euler, is that we must move to geometries associated with the physics of higher-order forms of curvature, such as the non- constant curvature of catenary functions, and those forms of physical action associated with living processes, for a first approximation solution.

To understand the importance, and the elegance, of this discovery, we must first investigate a class of numbers called logarithms. Hopefully, it will all so demonstrate the inherent differences between a “constructive” approach to the questions of the generation of such numbers as logarithms as over, and against, the formalisms of the textbook. I have included as an addendum at the end of this discussion, a short rendering on the subject of logarithms, modeled on that of a typical textbook , so the reader might more appreciate the conceptual gulf separating the constructive approach from that of classroom formalisms.

“It is more or less known that the scientific work of Cusa, Pacioli, Leonardo, Kepler, Leibniz, Monge, Gauss, and Riemann, among others, is situated within the methods of what is called synthetic geometry, as opposed to the axiomatic- deductive methods commonly popular among professionals today. The method of Gauss and Riemann, in which elementary physical least action is represented by the conic form of self-similar- spiral action, is merely a further perfection of the synthetic method based upon circular least action, employed by Cusa, Leonardo, Keller, and so forth. [fn. 1]

It is in this domain, physical least action associated with the self-similar spiral characteristic of living processes, that we search for a solution to the ordering principle which, in fact, might generate the prime numbers. Gauss approach involves understanding the idea behind the notion of a logarithm.

Logarithms are numbers which are intimately involved in the algebraic representations of self-similar conic action. In previous discussions, we saw that number measures more than just position or quantity; number can also measure action. We discovered that numbers in one manifold measure distinctly different qualities, than numbers in another manifold, and that what and how you count can sometimes leave “footprints” of a succession of higher ordered manifolds.

All descriptions of logarithmic spiral action, and the rotational action associated with them, are of two types of projection:

1) The 3 dimensional spiral on the of cone; we understand that each increase in the radial length of the 2 dimensional, self-similar spiral on the plane, is a projection from the 3 dimensional manifold of the conic spiral. The projection of the line along the side of a cone, which intersects and divides the spiral is called “the ray” of the cone. [See {The Science of Christian Economy}, APPENDIX I]

2) The 3 dimensional helical spiral action from the cylinder; the rotation of the three dimensional manifold of the cylindrical spiral (helix) projects on to the two dimensional plane as a circle. Nonetheless, some action is taking place, and that action is represented, therefore, by a “circle of rotation”, as simple cyclical action, i.e. we “count” the cycles of each completed, or partially, completed cycle of rotation of the spiral.

Turn to the April 12, 2002 issue of EIR, page 16, (See figure), “The Principle of Squaring”; review the caption associated with that figure [“The general principle of ‘squaring’ can be carried out on a circle. z^2 is produced from z by doubling the angle x and squaring the distance from the center of the circle to z.”] and construct the relevant diagonal to a unit square. The side of the square is one, the diagonal that square equals the square root of two. Use that diagonal, the square root of two, as the side of a new square; the diagonal to that square, whose area 2, will also be a length equal to two. We are generating a series of diagonals, each, in this case, a distinct power of the square root of two. In this case, it is a spiral which increases from 1 to 16 after the first complete rotation; 16 to 64 after the second rotation, etc. As we will soon see, each of the successive diagonal beginning with the first square 1, is also part of a set of “roots” of 16.

Each diagonal is 45 degrees of rotation from the previous diagonal; this should be obvious, since the diagonal divides the 90 degree right angle of the square in half. Therefore, each time we create a new diagonal and a new square, in turn generating another diagonal and another square, we generate a series diagonals, each 45 degrees apart. It should also be obvious that 45 degrees is equivalent to 1/8 of 360 degrees of rotation or 1/8 of a completed rotation of the spiral.

Let us now review a few fundamental elements of this action: we can now associate, in our spiral of squares, a distinct amount of rotation with a distinct diagonal value. In this case the diagonal values are powers of the square root of two or some geometric mean between these powers.

Table 1 
Rotation  Diagonal Value 
0        1 or ?20 
1/8      ?2 or ?21 
2/8       2 or ?22 
3/8      ?8 or ?23 
4/8       4 or ?24 
5/8     ?32 or ?25 
6/8       8 or ?26 
7/8    ?128 or ?27 
8/8      16 or ?28 
9/8    ?512 or ?29 
10/8      32 or ?210

The diagonals of this “spiral of squares” function much like the rays [fn2] (or radii) of a logarithmic, self similar spiral. We can imagine an infinite number of self-similar spirals increasing from 1 to any number N, after one complete rotation. Each successive complete, whole rotation will then function as a power of N[table 2]:

Table 2 
Rotation   Power 
0       N0 or 1 
1       N1 or N 
2       N2 
3       N3 
4       N4

Since each rotation of the logarithmic spiral increases the length of the ray (or growth of the spiral) by some factor that we can identify as the “base” of the spiral. In other words the base of the spiral which increases from 1 to 2 in the first rotation ( and doubles each successive rotation), is identified as base 2; the base of the spiral which increases from 1 to 3, as base 3; from 1 to 4, as base 4;…. 1 to N, as base N, etc.. The spiral, base N, will after one complete rotation beginning with ray length 1, generate a ray whose length is N^1; after 2 rotations, the spiral will generate a ray whose length is N^2; after 3 complete rotations the ray length will equal N^3, etc.

To measure or count rotation, we now define a “unit circle of rotation”. We can map a point of intersection with a spiral, and a ray spiral whose length is equal to or greater than one, on to a point on a unit circle. In this way it seems that a point on our circle of rotation can map on to, potentially, an unlimited number of successive points of intersection of a spiral and any given ray. But, when we look at our circle of rotation, we are looking at the projection of a cylindrical spiral. We can therefore “count”, as cycles or partial cycles, the amount of rotation required to reach the point on the unit circle which a ray maps onto the unit circle and the spiral at the same time.

Look again at the musical spiral of the equal tempered scale. (see figure 1, page 50, {So You wish to Learn All About Economics}). Here, I am looking, not at successive ROTATIONS of the spiral, but DIVISIONS, in this case one rotation of the octave or base 2 spiral.

When I divide the rotation of the spiral by half (6/12ths), I get F# or the square root of 2.[see chart 2]. When I divide the rotation of the spiral by 3 (4/12ths) the first division is the G# or the cube root 2. So each successive rotation is a power of N, i.e. N^1, N^2, N^3, etc. Each successive DIVISION represents a root of N, i.e. ?N, 3?N, 4?N, 5?N, etc.

Chart 2
Division    Root of Two   Musical Note 
0           0            C 
1/12       12th          B 
2/12        6th          A# 
3/12        4th          A 
4/12        3rd          G# 
5/12     5/12th          G 
6/12     square root     F# 
1         2            C

As we have now discovered, given any spiral base N, we can associate a distinct amount of rotation with a distinct power or root of N. Each successive complete rotation can be associated with a power of N; each division or partial rotation can be associated with some root of N, or a mean between N and another number. This distinct amount of rotation to a point on the “circle of rotation”, which can then be associated with a distinct rotation of a self-similar cylindrical spiral, is the logarithm of the number generated as a ray intersecting the spiral at a particular point.

For example, take our spiral of the squares; that spiral is base 16. The logarithm of 16 is one, written as Logv16(16) = 1[footnote 3]. Using our table 1, we can create a short “Table of Logarithms” for base 16. Turn once again to the April 12, 2002 issue of EIR, pages 16 and 17; as Bruce indicates, if I double the rotation, I square the length. Let us try various operations with the table of logarithms below. Table of Logarithms, Base 16 Logarithms unit value of diagonal or “ray” 0 1 or ?2^0 1/8 ?2 or ?2^1 2/8 2 or ?2^2 3/8 ?8 or ?2^3 4/8 4 or ?2^4 5/8 ?32 or ?2^5 6/8 8 or ?2^6 7/8 ?128 or ?2^7 8/8 16 or ?2^8 9/8 ?512 or ?2^9 10/8 32 or ?2^10

Add the logarithm of 2 to the logarithm of 4, base 16. What is the result? (2/8 + 4/8 = 6/8 or the logarithm of 8, base 16.) If I add the logarithm of 2, base 16 to the logarithm of 4, base 16, the two ADDED rotations give my the logarithm of 8, base 16, which is the product of 2 x 4.

Now subtract the logarithm of 4, base 16, i.e. 4/8 from the logarithm of 8, base 16, i.e. 6/8 and the remainder will be the logarithm of 2, base 16 or 2/8. Now take any of the logarithms from our table, base 16; add or subtract the logarithms of any number of numbers and see if they correlate with the division or multiplication of those same numbers. In other words: adding or subtracting the logarithms of numbers (i.e. the amount of rotation) correlates with multiplication or division of those numbers,

When I am looking at the number we call a logarithm, I am actually looking at the measure of two distinct forms of action in the complex domain of triply extended magnitudes, i.e. the cyclical nature of helical action, with the continuous manifold of the logarithmic spiral. Which is precisely why Gauss understood “…how much poetry there is in a table of logarithms.” We will look at this relationship in another way next time when we investigate why: “It’s Really Primarily Work.”



2) The ray of a cone is a line perpendicular to the axis of the cone, intersecting the spiral arm [It can also be constructed as a straight line from the apex of the cone to an intersection with the spiral. Both project onto the plane as the same length. When we project from the 3 dimensional cone to the two dimensions of the plane we assume that the incidental angle of the cone is 45 ray of the cone and the axis are of equal length.

3) LogvN(N) = 1 is the equivalent of saying “the logarithm (Log) in base N (vN) of N (N) equals 1. In the above case we’re saying the Logarithm of 2 in base 2 is 1

ADDENDUM I: “What is a logarithm?” according to the book.

“… a logarithm is number associated with a positive number, being the power to which a third number, called the base, must be raised in order to obtain the given positive number.”

Presuming we understand the concept of “the power to which a number is raised”, then a definition for “exponent” and a “base” might be necessary at this time. An exponent “…is a symbol written above and to the right of a mathematical expression to indicate the operation of raising to a power. In other words, in the simple function of 2^2 = 4, ^2 is the exponent, in the function 2^3 = 8, ^3 is the exponent, etc. The definition of a “base” is a little more complicated.

When we write our numbers we use the digits: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. Since we use these 10 digits and each digit in the number stands for that digit times a power of 10, this is called “base ten”. For example, 6325 means:

6 thousands + 3 hundreds + 2 tens + 5 ones.

Each place in the number represents a power of ten:

(6 x 10^3) + (3 x 10^2) + (2 x 10^1) + (5 x 10^0), or 6325

We could also use base 2, 3, 5, or any other that would seem most appropriate to our requirements.

Let us look at base 2, the mathematics of the computer. There are 2 digits in base 2, 0 and 1; as with base ten, each digit represents a power of the base number, in this case 2. For example the number 1101, base 2, is: (1 x 2^3) + (1 x 2^2) + (0 x 2^1) + (1 x 2^0) or 13, base 10.

Base 10 is called “the common base” and was most widely used in developing the Logarithmic tables. Let us take an example: the logarithm of 100 in base 10, which is 2. To say it in another way, in base 10, 10 ^2 (^ denotes exponent or power) = 100, and the exponent, in this case, is 2. We will note this relationship in the following way: v denotes the subscript followed by the base number, such that, in mathematical shorthand, the logarithm of 100 in base 10 will be written Logv10 (100) = 2.

The logarithm of 10 base 10 or Logv10(10) = 1, Logv10(100) = 2, Log v10(1000) = 3, etc. Therefore, if I add:

Logv10(10) + Logv10(100) = 3

I get a logarithm of 1000 in base 10, which is also the exponent of 10^3, or 1000.

If I subtract:

Logv10(10,000) – Logv10(100) = 2

I get 2, which is the logarithm of 100 base 10, which is also the exponent of 10^2, or 100.

In other words, adding logarithm of any number, N, to the logarithm of any other number of that base number system, N1, generates the logarithm of the product of those numbers:

Log(N) + Log(N1) = Log(N x N1)

Subtracting logarithm of N from N1 generates the logarithm which is the quotient of those numbers:

Log(N1) – Log(N) = Log(N1/ N)

Consequently, tedious calculations, such as multiplication and division, especially of large numbers, can be replaced by the simpler processes of adding or subtracting the corresponding logarithms. Before the age of computers and rapid calculating machines, books of the tables of logarithms of numbers were for engineers or astronomers or anyone else who needed to calculate large numbers.

I think the preceding discussion has been a relatively accurate one page “textbook” introduction to logarithms and their use. If it seems somewhat confusing, one solution is that described by a typical professor of mathematics identified as “Dr. Ken”, who, using the Pavlov/Thorndike approach to arithmetical learning, suggests that:

“The way you think about it is this: the log to the base x of y is the number you can raise x to get y. The log is the exponent. That’s how I remembered logs the first time I saw them. I just kept repeating ‘the log is the exponent, the log is the exponent, the log is the exponent, the log is the exponent,…’ “

A singular problem arises when we use the Pavlov/Thorndike approach, replacing the name of one number with that of another, “x is y” or “the log is the exponent”, and then simply memorizing it. If we don’t know the characteristic of action generating the exponent, then what the heck is the logarithm anyway; if this simple equivalency were all there was to the matter, then we have no concept of the characteristic action corresponds to this class of numbers.