As for the rational soul, or mind...it is not only a mirror of the universe of created things, but also an image of the divinity. The mind not only has a perception of God's works, but it is even capable of producing something that resembles them, although on a small scale. For to say nothing of the wonders of dreams, in which we effortlessly (but also involuntarily) invent things which we would have to ponder long to come upon when awake, our soul is also like an architect in its voluntary actions; and in discovering the sciences according to which God has regulated things (by weight, measure, number, etc...), it imitates in its realm and in the small world in which it is allowed to work, what God does in the large world.
Leibniz, “Principles of Nature and Grace, Based on Reason”
For since shape is demarcated by several limits, it comes about that on account of their being plural shape partakes of proportions. However what proportion is without the action of the mind is something which cannot be understood in any way. Hence by the same reasoning, one who gives limits to quantities as their essential basis supposes that quantities which have shape have an intellectual essence.
Kepler, Book I
The task to which Kepler has set us is to unveil which proportions were used by the Creator in the ordering of the solar system. Seems like a pretty tall order, considering man's physical limitations, doesn't it? However, as is demonstrated in Book I, we can begin investigating this celestial construction with the investigation of the proportions which were made available to man in his construction of things in the created world, for their laws are the same. Let's begin by asking this question: if the planets are moved about the sun in particular orbital pathways, why do they exist at the intervals they do, and not in some other arrangement? Why would the Creator choose certain proportions over others?
Well, what is a proportion? What does one look like? Perhaps a better question is: what perceives proportion? And how? For example, could you tell what the relationship is between the side of a square and its diagonal just by looking at it? By what procedure would you judge it? As one may quickly realize, the sense of sight is rather poorly equipped to make such judgments, and can only come up with an approximation, at best. As Kepler tells us: “...proportions are entities of Reason, perceptible by reason alone, not by sense, and that to distinguish proportions, as form, from that which is proportioned, as matter, is the work of Mind.” (Book III, page 150).
So how would the mind come to know a proportion which is ordering what we see? Well, if the relationship between two lines were known, I should be able to generate one from the other, right? Beginning with a simple line, we find that once we discover the right method, we could divide any line into two parts, three, ten, or ten-thousand, all in the same way. It seems, then, that I could generate any whole number relationship between two lines that I wish. Why, then, would the orbits in the solar system lie in one particular proportional relationship versus another, if all proportions are created equal?
Lets return to our geometrical example of the square, because here we encounter something different. Both the side of a square and its diagonal are lines, but lines which are the sides of squares in a double relationship (this can easily be shown). That would make the relationship of the lines themselves 1:√2. But wait, that's an “irrational” proportion! A number which is too big for my calculator to handle! No matter how hard I try, there doesn't seem to be a common unit of measure between them. However, when we look not just at the lines, but instead compare what generated them, their squares, we find a very commensurable 1:2. Though commensurability cannot be found in the shadow, the lines, it is found in the higher domain of their cause, and it is through this relationship that the relationship of the lines can be known.
Now, we uncover a much higher degree of complexity existing in the proportion of 1:4 when we use a square to divide the circumference of a circle into this proportion, since it involves exactly that “irrational” relationship, which I didn't encounter when I was just dividing lines. So what would it take to construct any proportion in the circle? As Kepler points out, in order to do this, it's necessary to begin with knowledge of a particular proportion: that between the diameter of the circle, and the side of the figure I wish to construct.
In this animation, the "particular proportion" is a cut of the radius into the Golden Section Proportion.
This is the unique relationship which yields the side of the decagon.
The yellow dot which appears represents
a cut of the line into the Golden Section.
What's so difficult about this? Look back to the square inscribed in a circle. While the arc which it cuts off is ¼ of the circle's circumference, the chord which does the job has an “irrational” relationship to the diameter, its diagonal. How could we begin our construction, then, if we only have the diameter to start from? Unlike some of his contemporaries like Petrus Ramus, Kepler doesn't discard these “irrational” quantities as unknowable and ridiculous, but shows that a proportion can be known if it can be constructed from some sequence of steps, “however long”, from the diameter, just as our “irrational” square side could be constructed from its area. As you'll see when working through Book I, we begin to encounter stranger and stranger relationships between the diameter and the sides of our polygons as the number of sides increases, but as Kepler says: “...there are many lines which, although inexpressible, can be defined by the best computations.” In his investigation, Kepler uncovers a certain ordering principle of the proportions (which lay hidden from us in the simple domain of lines or counting numbers) by demonstrating an order of nobility through knowability and contractibility of the polygons which produce the proportional divisions of a circle. It is in the classes of knowability of the constructible figures that our first step is taken toward uncovering the ordering principle which the Creator found to be the best in his crafting of the heavens.
In the first place, then, as is evident to all, fire and earth and water and air are bodies. And every sort of body possesses solidity, and every solid must necessarily be contained in planes; and every plane rectilinear figure is composed of triangles; and all triangles are originally of two kinds, both of which are made up of one right and two acute angles; one of them has at either end of the base the half of a divided right angle, having equal sides, while in the other the right angle is divided into unequal parts, having unequal sides. These, then, proceeding by a combination of probability with demonstration, we assume to be the original elements of fire and the other bodies; but the principles which are prior to these God only knows, and He of men who is the friend of God.
The function of Book I in the Harmony of the World, was to arm the reader with the most important aspect of Harmony; Constructive Physical Geometry. Kepler sites the fact that not one follower of the Platonic school was fortunate enough to write a commentary on Book X of Euclid's elements. By commentary, I mean that no one understood the cause or the foundation of the geometric figures. Euclid simply asserts the construction based on the axiom that to build a geometric system one must begin with a point. A point being that which is without breadth width or height.
If you were to base science off of the purely Aristotelian "Blank Slate, quantitative" view of the world, discovery would be impossible fn1. It is the Quality which defines Quantity. That is the essence of Knowability.
The understanding that Kepler defines is a rigorous work through of the origins of the plane figures and their relationship to the human mind's ability to knowably construct them.
Kepler states in the Introduction to Book I;
we must seek the causes of the harmonic proportions. That is, divisions of a circle into equal aliquot parts, which are made geometrically and knowably, that is, from the constructable regular plane figures.
Kepler, introduction to book I
Plato says in the republic, "God is a geometer and employs geometry in all his works." This couldn't be more scientifically true through the work of Kepler.
He begins with the definition;
To describe a Figure is to determine by geometrical means the ratio of the lines subtended by the angle to the lines round the angle, and, from what we have determined, construct Elementary triangles of the figure, and fit the triangles together to complete the figure.
book I , Definition V
See if you can do it. Try to construct as many regular figures as possible to independent of the circle. The animations is an example of the Pentagon.
From here we move to the degrees of knowability. Once a line has been determined by the elementary triangle, the question becomes, how does the mind come to know it?
“In geometrical matters, to know is to measure by a known measure, which known measure in our present concern, is the diameter of the circle, the inscription of figures in a circle, is the diameter of the circle.”
Book I, Definition VII
Degrees of knowability are defined by "how removed" a line or area is from the diameter, either by length or area. In other words, what series of actions must be taken by the mind in order to bring a quantity into relation with the known.
We have as the First Degree of knowledge, the length of the diameter of the circle, and its area. Anything that can be compared to this, like the square drawn, is knowable by the first degree. Both the line and the area are knowable in this way.
The Second Degree deals with any "expressible" portion of the diameter. "Expressible" is what we term the so called "rational numbers." In this case, we have the example of the hexagon. The side of the hexagon is of the second degree because its side is 1/2 the diameter.
Now, the Third Degree deals with a line which is "inexpressible" in relation to our diameter, but it is commensurable to the area of the diameter. Here we have the example of a square, whose side is in the relation of √2:1, but whose area is 1/2 that of the area of our original diameter. So the area of the square is commensurable with the area of the diameter, as in the case of the second degree, but its side is not. This line is even further removed.
Kepler states here that all quantities beyond this point are of the quality "inexpressible" (what we might call irrational). Here he makes an important point; he suggests that we bury the term irrational, because it makes perfectly intelligible numbers seem crazy. Kepler says that the quantities that follow, albeit inexpressible, "can be defined by the best computations." This is where the idea of number becomes interesting.
Thus the Fourth Degree, and the first of inexpressible quantities, is when neither the line nor its square is expressible; but, the square can be transformed into a rectangle such that its sides are expressible, at least in square. In the octagon we find an example of the fourth degree. The first animation below shows the construction of the octagon, from the bi-section of the side of a square.
The area of the octagon, Kepler says, is an area called medial. Medial, meaning that the area of the octagon cannot be expressed, but can be known. In other words the medial quantities need "ambassadors", or something in between, to relate it to the expressible length, the diameter, which can speak for itself (so to speak).
In the animation, the inexpressible lines QT and QS form the yellow rectangle. The area of the rectangle formed by QT and QS is also inexpressible. The new rectangle formed with the diameter as one side, and the length QM as its other side, which are knowable lengths, is equal to the area of the yellow rectangle, whose sides were found to be inexpressible, even in square. The animation below shows geometrically that the squares of the sides of the new rectangle are expressible, and therefore commensurable. (Click Here For A Short Pedagogy on Medial Quantity).
Next, you can see how the area of the octagon is comprised of two medial rectangles. The green square formed in the middle is produced by the overlap of the two medial rectangles, and then is distributed to the remaining empty spaces as triangles. Therefore, the area of the octagon is medial.
The square of a medial line is also called medial, so we have this other type of area, following the expressible area. So the following types of areas can be distinguished into these two types, the expressible and medial.
Pairs of Lines
What do we do when neither our line, nor our area can be measured by the diameter, and we can't even find an "ambassador" to deliver us a medial line or area to create a relationship for us? What would possibly seem like completely unknowable, or ignoble quantities, Kepler shows can be coupled into pairs, such that “what is lacking in one square, making it less expressible, is exactly compensated by the other square that is associated with it.” (Book I, page 24). There are three distinct types of these classes of quantities.
Lines of the Fifth Degree of knowledge are such that while individually they may not both be knowable even in square, and therefore hold a rather low rank, in combination, that is by adding the areas of their squares, and by taking the rectangle formed from the two as sides, two new, mutual areas are created which can be known and measured by the diameter. In fact, both new areas are expressible.
The side of the dodecagon, HL, and its star, HP, are an example of the fifth degree. HL² is equal to 2-√3, and HP² is equal to 2+√3. For as can be seen in the figure below, the sum of their squares add up to exactly the diameter squared, and as can be seen from the animation above, the yellow rectangle HPDL, is equal to rectangle HOLP, and is exactly ¼ of the area of the diameter squared.
The Sixth Degree of knowledge is a continuation of what came before in the fifth, but this time only one of their mutual areas (not both) is expressible, the other is only medial and requires another rectangle of the same area which can be constructed through the diameter.
There are two types of the sixth degree: the first when the sum of the squares is expressible and their rectangle is medial, and the other when the sum of the squares is medial and the rectangle is expressible. The octagon gives an example of the first case.
In the diagram it can be seen that the squares, each of which is an inexpressible area, add together to equal the square of the diameter, an expressible area, each making up for what is lacking in the other.
In the animation, the yellow rectangle made by the two lines (QT and QS) is medial (see fourth degree).
See if you can find an example of the second type of sixth degree lines.
The Seventh Degree of knowledge is when not both, not one, but neither of the mutual areas are expressible. Both are medial, and need an ambassador to be known by the diameter.
What happens when both of the mutual areas are inexpressible, yet neither is even medial? This falls into the final class of the Eighth Degree of knowledge. For example, the area of a pentagon can be divided into five triangles with a base FH the side of the pentagon, and a height AN (see diagram).
As can be seen in the animation (above), the area of the rectangle formed by AN and the pentagon side FH is 2/5 of the entire pentagonal area. So what is this rectangular area?
If the radius is 1, it can be proven (and the reader should indeed prove it) that the side of the pentagon, which can form a right triangle with the radius and the side of the decagon, is √((5-√5)/2). The height can be found as the square root of the difference between the radius of 1 squared, and half the side of the pentagon squared, which would be √((3+√5)/8). So the area which is two fifths of the pentagon's area is (√((10+2√5))/4), and the whole area would be (5√((10+2√5))/8). What a strange quantity! As Kepler points out, these arise when we must combine different types of lines, such as subtracting an inexpressible from an expressible line, or a medial from an expressible, in order to know a quantity.
As you work through Book I, you'll encounter many lines, all quite different, which fall within the eighth degree. All, however, bear the distinguishing feature of being known through the combination of different types of quantities.
At a certain point, as you will soon find, a figure whose elementary triangles cannot be found, is considered unknowable. That point is reserved for the Heptagon pedagogy.
* Since it is the mind in which knowledge of proportions exists, and which is shaped by them, not the senses, the reader is strongly encouraged to take these examples as a guide, but to work through each example from the book for themselves.
fn1 For an elaboration of this idea see Book IV page 296-299. In this section, Kepler heavily defends Plato's Doctrine of Recollection found in the Meno Dialog.