"The universe of Riemann and Einstein, for example, is a dynamic system, of a type best described… as finite and self-bounded. That means, for example, that gravity, as discovered uniquely by Johannes Kepler (but not the modern sophists Galileo and Newton) is an efficiently universal physical principle. This means, in other words, a principle of action as extensive as the universe, in a universe which extends no further than is reached by the universal principle of gravitation. Our universe is therefore self-bounded, and finite in that sense. Its bounds are expressed in mankind's expanding accumulation of discoveries and applications of universal physical principles."
Lyndon LaRouche, Jr. EIR volume 33, number 29, July 21, 2006
The Personality of the Sun
Picture taken by Chris Landry
The sun, a choirmaster, conducts the system of the harmonies; he generates the motions of the planets, and receives the harmonic effect of these motions as the apparent angles the various planets make from him as they orbit him. He conducts and is the beneficiary of his conducting:
“Not only does light go out to the whole world from the Sun, as from the focus or eye of the world, as all life and heat does from the heart, all motion from the ruler and mover; but in return there are collected at the Sun from the whole cosmic province, by royal right, these, so to speak, repayments of the most desirable harmony, or rather images of the pairs of motions flowing to it are linked together into a single harmony by the working of some mind, and so to speak stamped into coin out of rough silver and gold."
Harmony of the World
What would this mind be that is repaid by the pleasure of the harmonies that it has created? Without delving into sublunary nature, let us say that the Sun is a creative personality, as Kepler’s entire conception of the orbits of the Earth and Mars in the New Astronomy rely on his perception that the Sun is the cause of their motion. So it is in how these motions appear from the sun that Kepler’s harmonies are created.
This idea, that the Sun should be more than a fiery ball sending its heat and light over millions of miles, that it should be more than its matter, that, in fact, it should be like a perceptive entity, sounds terribly strange, even mystical. The idea is even more troubling as Kepler clearly conceives of the “heavenly harmonies” from this standpoint.
Pictures from NASA website
Why should this be troubling? Because the very notion is entirely unscientific and verges on animism? But what do we mean by science that we might call this idea “unscientific?” Is science all that which is free of human emotion, of joy, of sorrow, of happiness? Is it the slow death of a human calculator, endlessly tabulating statistics in a dark hole somewhere? If science were this, what would the universe that it claimed to analyze be?
Is the Sun creative? If so, what is his relationship to the solar system?
A Three Body Problem
If the Sun sends out a species whose affect lessens as this species moves further from the Sun, if its affect on the planets varies as the inverse of their distances from the sun, or if its apparent affect, as viewed from the Sun, varies as the inverse of the square of these distances:
“Why should the relationship between speed and distance be different within a planetary orbit than it is between orbits? Why should this be so? If the proportion of two apparent daily arcs, taken as an infinitesimal measure of angular speed from the sun at two “moments” of the planet’s orbit have a proportion which is close to the inverse proportion of their corresponding distances squared, then why does this relationship not hold across two orbits? Why is the proportion of two apparent daily arcs of two different planets, like Mars and Jupiter, not close to the inverse of the ratio of their two corresponding distances squared?” The question erupted from me, for suddenly the paradox had gripped me. Calculator in hand, papers spilling over the top of an armful of notebooks, attempted drawings, and calculations, I demanded that a friend assist me. “At a single point on any planet’s orbit it is its mean distance from the sun. At that point the planet’s distance from the sun is associated with its mean speed, with the speed it would be traveling if it swept out its orbit at a constant speed over the entire period of its orbit:
In the first animation, Mars is shown as it travels around the sun. In the second, it is shown as it would travel if it always traveled at its mean speed. In the third, they are shown together.
"If I compare that particular moment with another on that planet’s orbit, the mean distance compared to the second distance will be close to the inverse square of the second apparent motion compared with the mean motion. That holds for the single orbit. But if I were to compare one planet at that moment to a second planet at the same moment in its orbit, the proportion of their apparent motions would not be close to the inverse square proportion of their corresponding distances. The proportion of the distances would be equal to the cube root of the square of the ratio of their mean speeds.
"I see that this would be true at this moment because at that moment the mean speed reflects the total periodic time of the entire orbit, and it is this which is the real subject of Kepler’s so called third law. Yet at the same time, I wonder why should the planets not have the same inverse relationship of speed and distance to each other that each has internally? Here, I made this animation to demonstrate my point:
The red orbit represents Mars and the green orbit represents Jupiter. The "fake" planet is purple. As you can see from the fake planet's orientation, its aphelial distance is the same as Jupiter's mean distance, while the fake planet's perihelial distance is the same as Mars's mean distance. If you watch for a moment, you should be able to see that the fake planet travels slower at its perihelial distance than Mar's does at its mean distance, and it travels faster at its aphelial distance than Jupiter does at its mean distance.
"I made a fake planet whose aphelial distance is Jupiter’s mean distance, and its perihelial distance is Mars’s mean distance. I used Kepler’s principles to create it: finding its mean distance as the arithmetic mean of the aphelion and perihelion I gave it, I then set up a ratio of Jupiter’s mean distance to the fake planet’s mean distance. I took the square root of the cube of that and multiplied it by Jupiter’s mean speed, to get the mean speed of the fake planet. I then did the same thing for the fake planet and Mars. For the first calculation, using the relationship of its orbit with that of Jupiter, I got a mean speed of 575s for the fake planet. The second time, using Mars, I got 569s. These numbers are off by a hair of a hair’s breadth, by not more than 1%. So, for the purpose of calculation, I used 575s for its mean speed. I then set up two proportions. First I took the fake planet’s mean distance over its aphelial distance; I squared this proportion and set it equal to the unknown apparent aphelial speed over the fake planet’s mean speed.For the other proportion, I used its perihelial distance over its mean distance. As you can see from the table, its aphelial speed is quite a bit faster than Jupiter’s mean speed (about 20% faster); and its perihelial speed is quite a bit slower than Mars’s mean speed (about 30% slower.)
If we compare the fake planet's apparent aphelial motion with Jupiter's mean motion, they create a harmonic interval which is close to a major third. Right click on the animation and select play. When the two planet's pass the point where they are an equal distance from the sun, their respective apparent speeds will flash on the screen, accompanied by the sounding of the interval that these speeds make with each other.
"You can even see this in the animation.
In this animation, the fake planet's apparent perihelial speed is compared with Mar's mean speed in the same way. Again, you must right click on the animation and select play to activate it.
"In fact, you can hear it. If I take the interval between Jupiter's mean speed and the fake planet's apparent aphelial speed, they create an interval that is close to 4/5, a major third.
"So Kepler’s so called third law has implications which are obviously more far reaching than merely the relationship between the various planets’ orbital periods and mean distances. The third law is, in a sense, concerned with the quantization of space.”
My friend laughed. “Didn’t we have this conversation before and weren’t you on the other side then?” I remembered arguing in the car months earlier about this very question. At that time it had seemed so clear to me. Of course the planets would have a relationship of speed and distance internally that was different than the relation of speed and distance which they had to each other. It was a simple question of the difference between a simple relation and a relation of relations. Like triangles: Each has a certain internal proportionality which is different then the proportional relations between two individual triangles. (Image) But then, on returning to the problem again, it no longer made sense. My mind had grabbed hold of a solution to the apparent paradox too quickly and had not allowed the problem to work on me, to creep in and confuse the seemingly well defined order of my mind.
This is the crux: The harmonic nature of the relationship of the individual planets and the sun is reflected in the total orbital period of each planet, the total area of the orbit swept out as equal areas in equal times, or better, as Kepler views it, the area swept out by the planet is the time it has traveled. This is echoed in the fact that within an individual orbit, at two moments, the proportion of the apparent (from the sun) speeds has an inverse relationship to the proportion of the squares of the distances of the planet from the sun at those moments. But this relationship does not hold between planets. If the area a planet sweeps out is the time it has traveled, this time is unique to this individual planet. 100 units of Mars’s orbit are not equal to 100 units of Jupiter’s orbit. If we were to evaluate these two portions from the standpoint of how we think of time on the earth, according to the earth’s rotation about its axis, the number of days Mars took to travel 100 units would be different than the number of days Jupiter took to travel 100 units. If you are engaged with this problem, this either seems obvious to you, or it seems absurd. Those who see clearly will forgive those of us on a slower orbit, and allow a small demonstration:
Archimedes proved that the area of the ellipse has to the area of the circle which circumscribes it (the circle with the ellipse’s semi- major axis as radius,) the same proportion as the ellipse’s semi-minor axis to its semi-major axis. In other words, if you multiply the area of the circumscribing circle by the proportion of the ellipse’s semi-minor to its semi-major axis you will have the area of the ellipse.
From this it follows that the area of the ellipse is simply pi times the semi-major axis times the semi-minor axis. (I was going to include Archimedes’s proof and an animation, but it is more involved and requires more time then I have to give to it, so I leave it for you to demonstrate. It is proposition 4 of On Conoids and Spheroids. Have fun!) So using the distance data I have for Jupiter and Mars in the table below, Jupiter’s orbital area would be: pi(3413.19 X 3409.21) = pi(11636286.78). And Mars’s orbital area would be: pi(1000 X 995.68) = pi(995680). Their ratio would be: (pi(11636286.78)/ pi(995680)) = 11.69. But if we were to look at the ratio of their orbital periods, we would have Jupiter’s 4332.62 days divided by Mars’s 686.98 days. * 4332.62/ 686.98 = 6.31. Obviously 6.31 does not equal 11.69. So the ratios of areas and orbital periods are not the same.
Likewise, there is no absolute relationship between speed and distance. In fact, I have no direct way of comparing two seemingly equal quantities among the orbits. I can only directly compare the two orbits as wholes. I want to say that at a given distance from the sun, any object would travel at a certain speed, but it is not so. Perhaps here you may interject: "but isn't this simply a question of each planet’s mass?" But, I took no consideration of mass to create my fake planet. I simply used the harmonic considerations Kepler lists in chapter 3 of book 5 of the Harmony of the World. At that time, Kepler was only able to have the vaguest idea about the real size of the planets. He conceived of the planets' masses and volumes as reflective of the higher harmonics of the universe, but he did not conceive of these as causal. Mass is not the reason the planets orbit as they do.
But it is important to reiterate that Kepler did conceive of the planets as bodies in a direct break with Aristotle:
“the planetary bodies moving or revolving around the sun must be considered, not as mathematical points, but obviously as material bodies having, as it were, a certain weight… that is, insofar as they are endowed, in proportion to the bulk of their bodies and the density of their substance, with the capacity to resist motion imparted to them from without.”
From Gesam. Werke, VIII, 94:9-14, as quoted in Kepler’s Somnium
So, in the universe, there is no absolute time or space. There are regions of action defined by the relationship of the sun and the individual planets.
“The virtue (gravity) which lays hold (of the planet) does not overcome every whit: for the resistance of matter in the planetary body stands up against it and restricts it: hence the planet does not follow exactly the forward movement of the prehensive force, but is left behind and abandoned by it and in that mutual struggle there is place for time.”
J. Kepler, Epitome of Copernican Astronomy pg 62
Perhaps, in a way, time is the result of this struggle.
In chapters 38 and 39 of the New Astronomy Kepler begins to develop his concept of the cause for the eccentricities of the individual planets, the reason why the planets do not travel in the same or even in proportional orbits. “The approach and recession of a planet to and from the sun arises from that power which is proper to the planet.” (pg 407 NA) The planet and the sun play in their own peculiar way, forming time through the union of their two personalities, and the personality of each planet contends with the sun in its own way. Nature plays. As Kepler quips in his “On the Six-Cornered Snowflake,” as he speculates why the snow-flake has six sides: “Formative reason does not strive to fashion only natural bodies, but is in the habit of playing with the passing moment… I transpose the meaning of all such from playfulness (in that we say that Nature plays) to this serious intention. I believe that the heat, which till then was protecting its matter, is now conquered by the surrounding cold; but just as previously, animated as it is by a formative principle, it had acted and fought in an orderly fashion, so now it displays an order of its own in preparing for retreat and withdrawal, and holds out longer in the selected branches, or outposts, that are distributed in good order over the line of battle (in the star-like branches of the snowflake) than in all the rest of the matter.” Kepler plays, as nature does. Or he and nature play like Schiller describes the “instinct of play” as the key component of Man’s aesthetic sense: “the instinct of play would have as its object to suppress time in time, to conciliate the state of transition or becoming with the absolute being, change with identity.” (AL letter 14) This is the harmony of contending powers, the timeless unity in time, the marriage of the unchanging personality and a universe where nothing is constant but change. Paradoxically, it seems that it is here that time and space exist, as defined by this play. For instance, take this famous duet, “Un di, Felice, Eterea” from Verdi’s La Traviata as an example.
13th
The thirteenth of Keplers principles of astronomy necessary to investigate the harmony of the Universe in chapter 3 of book 5, derives from this reality. Kepler, as was stated previously, wants to compare the motions and distances of the planets not just in totality, but particularly at the singular moments of aphelion and perihelion. From this relationship, Kepler can test his hypothesis for the reason for the individual planets’ eccentricities. The convolution of Keplers treatment is the result of the indirect path he must take to analyze the relationship of these particular moments of different planetary orbits.
We begin with the statement of the 3/2s law, as we have come to call it. That is, that the proportion of the mean motions is the inverse of the square root of the cube of the proportion of the mean distances, that is the 3/2s power of that ratio. The question is, if this relationship is precise, what is the relationship of the ratio of the extreme motions of the two planets, to the inverse of their corresponding distances? There is a definite relationship, but it is not direct.
There is no less circuitous path I have found then to simply plunge ahead on Kepler’s track. It is confusing, but I have tried to make the path clearer. If you are following the map Kepler left at the end of chapter 3, I will show how I have changed the markers in his example. I provide the data for Mars and Jupiter, so you can compare actual data to Kepler’s numerical examples, and I include the numerical examples Kepler creates, which are straightforward enough if you can figure out what Kepler is doing otherwise.
Here is a table of how my notation corresponds to Kepler’s:
(For this example, Kepler analyses the relationship of distance and apparent motion for the converging motions of two planets, so there are only two distances and two speeds associated with either planet. For the outer planet it is its perihelial distance and speed and its mean distance and speed. For the inner planet it is its aphelial distance and speed and its mean distance and speed.)
What we will finally come to is that the proportion of the converging motions has the same proportional relationship to the inverse proportion of the corresponding distances that that proportion has to the square root of the proportion of the mean distances. “Or, which comes to the same thing, the proportion of the two converging distances is the mean between the square root of the proportion of the orbits and the inverse proportion of the corresponding motions.” (416) Instead one simple relationship, which equates one quantity to another quantity, like the relationship between the mean distances and mean motions of two planets, we have a constant relation among the relations of the two planets, which is reflected in this consistent proportional relationship just stated, of which Kepler will make frequent use.
Part of the difficulty with unlucky 13th is that Kepler is dealing with proportions and thinks of them in a way different than we are used to (for Kepler the simple proportion 2/3 = 3/2 because either way the “space” between the two numbers is the same. This is fine, but if you are going to use that proportion in relation to another, for instance if you were going to multiply or divide by it, then it would matter whether it were 2/3 or 3/2.) (Example) Secondly, where we would tend to think of “space” in terms of arithmetic proportion, Kepler often thinks of it as a geometric proportion. For instance, Kepler states that the proportion of the two apparent converging motions is either less or more than the proportion of their corresponding distances by the (proportional) amount, that the proportions of the two corresponding distances multiplied with the proportion of the two mean distance comes to less or more than the square root of the proportion of the orbits. This sounds terrible, I know. But first realize that Kepler is, in a certain way, taking the proportion of the mean distance of the outer planet to its perihelial distance and multiplying it by the ratio of the inner planet’s aphelial distance to its mean distance. So he is thinking of the planet’s eccentricity, its relationship of aphelion or perihelion to its mean distance, in a geometric way, as a ratio, where we would tend to think of this as a simple arithmetic quantity. If a planet’s mean distance from the sun were 9 units and its perihelial distance were 8 units, we would think of that planet as having an eccentricity of 1. But Kepler looks at that eccentricity as a relationship between 9 and 8 of 9/8.
The next thing to remember is that that with which 13th works through derives entirely from the two proportional relations about which I have been speaking: the proportion of the mean motions of the two planets (which is the proportion of their orbital periods) is equal to the 3/2 power of the inverse of the proportion of their mean distances and the proportion of the apparent motions at any two moments of an individual planet’s orbit is equal to the inverse ratio of the corresponding distances squared. That is all. In 13th, Kepler merely sets out to show how these two different proportional relations are themselves related.
Here is a chart for diverging motions. Kepler only gives the example of the converging and infers the relationship for the diverging apparent motions and distances from that. I am going to state both “algebraically,” so I include this other table for your use.
What can we deduce from this? Given that √(D1)/ √(D2) is to (P1/A2) as (P1/A2) is to (MA2/MP1), we know that, because they are in geometric progression, the amount by which (P1/A2) exceeds or is exceeded by √(D1)/ √(D2) is the same as the amount by which (P1/A2) exceeds or is exceeded by (MA2/MP1). We say “exceeds or is exceeded by,” because simply knowing that they are in geometric progression does not tell us the direction of that progression. 3 : 9 :: 9 : 27, but so is 27 : 9 :: 9 : 3. In the first case, the factor of excess, so to speak, is 3, in the second case it is 1/3. For Kepler, the relationship “3” and “1/3” are equivalent in terms of quantity, but not in terms of directionality. It is through this crack that a great deal of confusion seeps in.
The amount by which (P1/A2) exceeds (or is exceeded by) √(D1)/ √(D2) is simply the former quantity divided by (which is the same as multiplying by the inverse of) the latter: (P1/A2) x √(D2)/ √(D1) (Which is the same idea as saying that 5/8 is greater (for Kepler) than 2/3 by 5/8 x 3/2 or 15/16.) (If (P1/A2) were exceeded by √(D1)/ √(D2) instead of exceeding it, the proportion you got as a result would have the larger quantity in the numerator. For instance, if you were to divide 2/3 by 5/8 instead of the other way around, you would get 2/3 x 8/5 or 16/15.)
If the direction of the geometric progression is such that (MA2/MP1) is less than (P1/A2) then it is obvious that it will also be less than (P1/A2) 2/3 because this quantity is (P1/A2) increased by its own root. For instance 5/4 < 9/4 and it is even lesser than (9/4 X 3/2,) or 27/8. But it rarely happens that the proportion of the converging motions is less than the proportion of the inverse of the corresponding distances. For the motions to come quantitatively close to each other, the eccentricities of the planets must be very large, and ironically, the corresponding ratio of their converging distances must become very small. Yet, at a certain point, even as it constantly diminishes, the ratio of the distances will still exceed the ratio of the corresponding motions. In the animation above, this point is crossed when the eccentricity of the changing planet’s orbit becomes absurdly large, in the second to last example!
The second case, in which (MA2/MP1) is greater than (P1/A2), which is the only case that actually occurs in our solar system, is not as easily demonstrated. The first indication is that because (P1/A2) 2 = (MA2/MP1) x √(D1)/ √(D2) then (P1/A2) 2 X (A2/P1) 1/2 = (MA2/MP1) x √(D1)/ √(D2) X (A2/P1) 1/2 which means that (P1/A2) 3/2
is equal to (MA2/MP1) multiplied by the quantity (A2/ D2 X D1/ P1.) 1/2 So the question is, does that quantity increase or decrease the size of the ratio (MA2/MP1?) If it increases it, than (P1/A2) 3/2 is obviously larger than (MA2/MP1) by that quantity.
The second clue is that if (MA2/MP1) is greater than (P1/A2), it is greater by the quantity (MA2/MP1) X (A2/P1). This quantity is itself equal to the quantity (P1/A2) X √(D2)/ √(D1) because since these three proportions are in geometric progression, the amount which (P1/A2) exceeds √(D1)/ √(D2) is the same as the amount which (MA2/MP1) exceeds (P1/A2). Just as 27 exceeds 9 by the same amount 9 exceeds 3. Or 27/9 = 9/3. So for (MA2/MP1) to be larger than (P1/A2), but smaller than (P1/A2) 3/2, then the amount by which (MA2/MP1) exceeds (P1/A2) must be less than (P1/A2) 1/2. For instance, 3456/2025 is greater than 8/5 but less than (8/5) 3/2, because 3456/2025 X 5/8 = 432/405 is less than (8/5,) 1/2 meaning that the amount by which 3456/2025 exceeds 8/5 is less than the square root of 8/5. So, for (MA2/MP1) to be greater than (P1/A2) and less than (P1/A2) 3/2, (P1/A2) X (D2/D1)1/2 – the quantity by which (MA2/MP1) exceeds (P1/A2) – must be less than (P1/A2) 1/2, so (P1/A2) X (D2/D1)1/2 < (P1/A2)1/2. The ratio of the converging distances will always be less than the ratio of the mean distances, because the converging distances represent the point where the two orbits come closest to each other. Therefore, the square root of the ratio of the mean distances will always be greater than the square root of the ratio of the converging distances. (P1/A2) X (A2/P1) 1/2 = (P1/A2)1/2. (A2/P1) is less than 1 because the inner planet’s aphelial distance will always be smaller than the outer planet’s perihelial distance (unless they cross orbits!) Likewise, (D2/D1) will also be less than 1 because the inner planet’s mean distance will always be smaller than the outer planet’s mean distance. So (D2/D1)1/2 will diminish the proportion by an even greater amount than (P1/A2)1/2 would . Therefore, (P1/A2) X (D2/D1)1/2 < (P1/A2)1/2 will always be true. For instance 8/5 X (4/9)1/2 < (8/5)1/2 because 9/4 > 8/5, so the square root of 9/4 is greater than the square root of 8/5 so dividing 8/5 by the square root of 9/4 will diminish it more than if it were divided by the square root of 8/5. (Like: 4/4 < 4/2.)
Kepler uses this proportion to establish a simple measure for the amount the ratio of the converging motions, while always less than the 3/2 power of their corresponding distances, will be greater (or less) then the simple ratio of their corresponding distances. From our geometric progression, if (P1/A2) is greater than (D1/D2)1/2 then if I multiply both sides by (D2/D1), I will get: (P1/A2) X (D2/D1) is greater than (D1/D2)1/2 X (D2/D1). And (D1/D2)1/2 X (D2/D1) is simply equal to (D2/D1.)1/2 So this proportionality that Kepler states at the beginning, and uses as a first indicator of what harmonies were possible between the extreme motions of two planets, is really just a restatement of the geometric progression we found to exist among the square root of the mean distances, the extreme distances and the extreme motions.
This relationship between the relative distances of the orbits implies what their potential converging and diverging harmonies could be. Within this region of possibility, the best possible harmonies with respect to the entire system of planets then dictate the planets’ eccentricities, which in turn, can violate the dictate of the distances to a certain degree. From the proportional relation of the planets’ mean distances to their converging extreme distances, Kepler can quickly check what harmonic relationships were possible for these planets. For instance, if the “proportion of the spheres” of two planets were 1000 to 795, how large would the proportion of their eccentricities need to be for these planets to make a major or minor third?
Conclusion
The nested solids define the large question of how the planets are ordered, “the number of the six primary spheres has properly been taken from the five solids alone, their proportion principally from the five geometrical solids; but it has conceded very small amounts all around to the motions, as it was the final cause which was accepted for the idea of the operation right from the start. And this is to be understood of the motions of each planet, its slowest on the one hand and its fastest on the other, that is of the motions considered as the cause of its particular properties. Indeed the periodic motions, that is to say, the number of days assigned to the revolutions of each individual planet, have both on account of the proportion of the orbits and on account of the eccentricities (which have been established from the harmonies) regressed further from the five solids.” (Mysterium Cosmographicum). Both the motions and the relative distances are coherent in one universe. “The bodies which make up the universe are not parts of one continuous quantity… the bodies of the universe are allotted spaces which are solid, or of three dimensions, to traverse.” (Mysterium Cosmographicum) The “space” between the planets is a function of distinct regions delimited by the solids, whose “nobility depends on their simplicity… for just as God is the model and rule for living creatures, so the sphere is for the solids… (and the sphere) is extremely simple, because it is enclosed by a single boundary, namely itself.” But the motions of the individual planets, their voices, were “the final cause which was accepted for the idea of the operation right from the start.” The harmony of the individual voices is the primary cause, yet the range of action which is possible for these harmonies is delimited by the boundaries of the solids, which in turn must give way for the harmonies. The relative distances and relative motions of the planets are subject to different domains, and are yet unified in a single idea.
Thus, the personality of the Sun acts through the entire solar system, like a conductor conveying an idea in music through the medium of the activity of the individual members of the orchestra. And each of those members is himself a sovereign in his own right, not a featureless automaton, but a unique individual with his own voice. The conductor then unifies these multifarious voices into a one.
This thirteenth proposition of chapter 3 is therefore concerned with potential. What were possible in this self-bounded universe of dynamic play?
*(If you are wondering why my fractional parts of a day do not match Kepler’s, it is because he records fractions as parts of 60 like the hexagesimal fractions used by the Greek astronomers. I have converted these to decimal fractions (parts of 10,) which were just beginning to enter the language Western Europe during Kepler’s life.)
