by Robert A. Robinson
The achievement of Nicholas Copernicus, whom Johannes Kepler so much admired, is often misrepresented in astronomy textbooks, as the “discovery of the heliocentric system.”
Copernicus never claimed to be the originator of the heliocentric system, that is, the system of placing the sun, rather than the earth, at the center of the universe. Copernicus himself stated that the idea of placing the sun at the center of the universe originated, as reported by Archimedes in his work, “The Sand Reckoner,” with the ancient Greek astronomer Aristarchus of Samos, around 200 B.C.! Nor was the rediscovery of the ancient heliocentric hypothesis, in itself, what Kepler appreciated in Copernicus’ work. Indeed, Kepler almost rejected Copernicus’ theory, because it assumed the stars to be infinitely distant from the sun, and therefore that the sun is the absolute center of the knowable universe. If, as Kepler instead maintained, the distance to the stars were immense, yet implicitly measurable, why might not the sun be but one of those luminous bodies we call stars, and therefore, not be the center of the whole universe, but only the center of local planetary motion?
What, then, did Kepler find so beautifully significant in the work of Copernicus? In a word, it was Copernicus’ discovery, based on the work of Aristarchus, of a wonderful harmony, or congruence of measurement, within the domain of the solar system itself. This is a subject of elementary, yet profound, importance for the future development of science, as Kepler clearly realized. It is a crime that it has been so obscured in so many astronomy textbooks, apart from those “textbooks,” like “Mysterium Cosmographicum,” and “The Epitome of Copernican Astronomy,” written by Johannes Kepler himself.
Copernicus discovered that the heliocentric hypothesis supplies the “One” to unite a “Many.” As Copernicus himself wrote in his posthumously published masterwork, “The Revolutions of the Heavenly Spheres,” “Therefore, in this (heliocentric-RAR) ordering, we find that the world has a wonderful commensurability and that there is a sure bond of harmony for the movement and magnitude of the orbital circles such as connot be found in any other way.” (Quote P. 528-529 in Great Books #16).
Let us divide Copernicus’ breakthrough into three parts.
First, look at Mercury and Venus, the planets which never, in the evening or morning sky, deviate far from the sun. They never appear “opposite” the sun in the sky, like all the other stars and planets do periodically (including even the moon, every time it is full.)
Venus is best to look at for our purposes of demonstration. Venus never deviates more than 45 degrees from the sun. If you track Venus each day when in its full glory as an evening or morning “star,” you will notice that it moves out to a position of maximum divergence, or elongation, from the sun, hovers around there for a few days, then starts back toward the sun, with what appears as variably accelerating motion. (See last week’s pedagogical.)
Consider how the heliocentric hypothesis provides us with a simple method to measure the distance of Venus from the sun, using the earth-sun distance as an “astronomical unit” of measurement. Construct a circle on a piece of paper, with center S, and place a point E some distance outside of and below the circle’s circumference. Draw the two straight lines from E, that are tangent to the circle’s circumference, meeting the circle at V on the left and V’ on the right. Now, in coherence with the heliocentric hypothesis, let S represent the sun, E the earth, the circle Venus’ approximate orbit, and V and V’ Venus at points of successive tangency (assuming counterclockwise motion) between its orbit and lines of sight from the earth.
Note that a right angle is subtended at those points, V and V’, between, respectively, lines VE and V’E, and lines VS and V’S. Now, just focus on the left side of the diagram, and the triangle VSE, that has a right angle subtended by VS and VE. V not only is the vertex of the right angle in VSE. V also forms a point on the left hand side of the circle (V’ being the corresponding one on the right hand side) of maximum divergence, or elongation, of Venus from the sun as seen from the earth, that is, divergence between a line of sight linking the earth and the sun, ES, and a line of sight from earth to Venus, VE. That angle of maximum divergence, as seen from earth, between the line of sight from the earth to the sun, and the line of sight from the earth to Venus, is measurable, with a sextant, to within a certain (not to be ignored) variability, at around 45 degrees. Consequently, we know one angle of triangle EVS is a right angle, another is around 45 degrees, and therefore, assuming the space between the earth, the sun, and Venus is as flat as our piece of paper, we know all 3 angles of the triangle, EVS, formed by the earth, sun, and Venus. Copernicus takes the average earth-sun distance (determined by the earth’s orbit around the sun, which Copernicus assumed was approximately circular) as a unit, or one. It therefore becomes an easy matter, applying basic Pythagorean rules of mesurement to right triangle EVS, to measure Venus’ distance from the sun, as a proportion of earth’s distance from the sun. That proportion turns out to be about one to the square root of two.
Aristarchus, in ancient times, had used a remarkably similar method of measurement to determine (to a first degree of approximation) the relative distances and sizes of the earth, moon, and sun, as the crux of his argument for heliocentricity. Aristarchus had based his measurement on the idea that, when there is a “half moon,” the moon forms the vertex of the right angle of a right triangle formed by the earth, the sun, and the moon. Indeed, if, in repeating Copernicus’ measurement, you happen, just for fun, to view Venus through a telescope when it is at its maximum extension from the sun, Venus will appear to you as a tiny “half moon”! Of course, Copernicus, in the 1500s, had no such telescope with which to view Venus. Yet, therefore, ironically enough, Copernicus did not suffer the observational drawback of Aristarchus, who had had the nasty job of attempting to determine just when a “half moon” occurs, to complete his measurement, and so ended up being off by an order of magnitude in his earth-sun distance. Though Copernicus had no immediate means of re-evaluating Aristarchus’ earth-sun distance, he did determine a remarkably accurate relative measurement for the earth-sun and Venus-sun distances. He was able to do that because his measurement in no way depended on the observation of a “half Venus,” because a “half Venus,” as viewed from earth, occurs axiomatically, as one may see from our diagram, at its point of maximum divergence from the sun. The latter, as we have said, is a magnitude not so difficult to measure with a sextant.
The point of maximum observable divergence of Venus from the sun will vary, depending on what we now know to be the (very slight) eccentricity of Venus’ elliptical orbit, as well as on the (somewhat larger) eccentricity of the earth’s own elliptical orbit, on the inclination of the plane of Venus’ orbit to the plane of the earth’s orbit, and on other parameters of orbital motion. These parameters must all be integrated in order to accurately predict (as Gauss finally did in 1801) future positions of planets, asteroids, and comets, but their future comprehension would not have been possible without Copernicus first finding a method to know, at least approximately, the Venus-sun distance, and the other planetary distances from the sun, relative to the earth-sun distance.
Similar methods can be used to determine the distance of Mercury from the sun, relative to the earth sun distance, if you are ever fortunate enough to see that fleet footed rascal!
The Outer Planets
Copernicus used precisely the same “Aristarchian” method, only “in reverse,” which we have just identified for determining the Venus-sun distance (and Mercury-sun distance), to determine the Mars-sun, Jupiter-sun, and Saturn-sun distances, all relative, just as in the case of the inner planets, to an earth-sun “astronomical unit” of distance.
The phenomenon of “retrograde motion,” that is, when planets appear to move backward in their orbits for a time, as seen against the starry background, had long been a stumbling block for astronomers. The ancient Greek astronomer and student of Plato, Eudoxus, had, for example, built an ingenious, geocentric, model of spherical rotation on top of spherical rotation to account for it.
Then later, around 100 A.D., someone less ingenious than Eudoxus, but also a believer in a geocentric universe, the famous Ptolemy, developed a different theory to account for retrograde motion of the outer planets. Basing himself on the Aristotelean dictum that “nature abhors a vacuum,” Ptolemy made planetary distances just big enough to “fit” retrograde motion, in the form of “epicycles,” in between planetary orbits.
Copernicus, on the other hand, thinking in terms of the heliocentric (sun centered) hypothesis, saw in retrograde motion, a reflection of the earth’s own motion, and therefore, saw a way to measure the distances from the sun (and the earth, for that matter) to the outer planets. To illustrate his method, we shall employ exactly the same diagram as before, only with different labelling!
Construct a circle on a piece of paper, again with center S, and place once again a point, but this time labelled C, under and at some distance from the circumference of the circle. From C draw two tangents to the circle, as before, but now label the point of tangency of the straight line from C to the left side of the circle E, and the one to the right side of the circle E’. Extend EC and E’C past their point of intersection at C, to point towards a general region outside our diagram which we shall label the “starry background.” These labels reflect just two differences between this diagram and the previous one. First, the circle now represents the orbit of the earth, not Venus, as in the previous diagram, and thus E and E’ represent successive positions, moving counterclockwise, of the earth in its orbit around the sun. Second, the outer planet, at C, which represents in this diagram any of the outer planets (such as Mars, Jupiter, or Saturn) which Copernicus might observe moving night to night against the “starry background,” takes the place of the earth at E in the previous diagram. The sun remains in the same position in both diagrams.
Notice, first, that the progress of the earth from E to E’ (and discounting for a moment the outer planet C’s own “real” — but slower — counterclockwise motion along its own orbital path) accounts for the apparent clockwise, or retrograde, motion of the outer planet C against the starry background, as the line of sight from the earth to the outer planet moves from EC to E’C. Second, note that the maximum angular extension of that apparent retrograde motion of C exactly reflects the progress of the earth’s motion around the sun, from E to E’, in the diagram.
Divide that angular span of retrograde motion in half, drawing a line from the sun, S, to the outer planet, C, and forming the triangles SEC and SE’C on the left and right hand sides of the diagram. Focus on the triangle SEC. We already know the angular (degree) measure of the total span of retrograde motion against the starry background, and we know angle SCE will be half that. But we also know that the earth’s own orbital motion must begin to create the appearance of retrograde motion in the outer planet when the earth, at E, forms a right angle between a line of sight from the earth to the sun, and a line of sight from the earth to the outer planet C. So we know angle SEC must be a right angle. Therefore, we can determine all 3 angles of right triangle SEC in the diagram. Taking the earth-sun distance as “one,” the distance of the outer planet becomes measurable by the Pythagorean relationships in triangle SEC.
Well, almost. We still have to account for the outer planet’s own motion.
Integrating Modular Motions
Leaving aside for a moment Kepler’s future discovery of elliptical motion, it is not difficult to integrate the earth’s and the outer planet’s approximate respective contributions to the apparent retrograde motion of the outer planet as seen from the earth.
Suppose a planet moves ahead 10 degrees, then back 5 degrees, then ahead 10 degrees, then back 5 degrees, etc., forming cycles with 2 parts, plus 10 and minus 5. Divide the difference between plus 10 degrees and minus 5 degrees in half, which turns out to be 7 and one half degrees. The contribution of the earth’s own motion in each half of the total cycle of motion is thus plus or minus 7 and one half degrees, to the total apparent motion of the outer planet as seen from earth. That leaves 2 and one half degrees for the outer planet’s own contribution to its apparent motion. In the first half of the cycle, (from E’ to E in our second diagram) add 7 and one half (the earths motion in a contrary direction to the outer planet’s motion ), and 2 and one half (the outer planet’s own real motion), to give 10 degrees of apparent forward motion, as seen from the earth, to the outer planet. In the second half of the cycle, (when the earth is moving from E to E’ in our diagram) subtract 7 and one half degrees of earth motion from the outer planet’s 2 and one half degrees of motion, to give a net minus 5 degrees for the outer planet’s motion as seen from earth. (In modular arithmetic terminology, 2 and one half is the residue of real motion left to the outer planet, congruent with its apparent motions of plus 10 and minus 5, in terms of a modulus supplied by the earth’s periodic motion of 7 and one half.) This is only approximate, because more time is spent by the earth, going counterclockwise from E’ to E, than from E to E’, but it is sufficient to determine relative distances, and thus to make further refinement of measurement of motion possible.
Now, finally, open Kepler’s “Mysterium Cosmographicum” to its opening pages. What do you see? You see Kepler’s direct juxtaposition of Ptolemy’s non-measurement or non-congruence with Copernicus’ beautifully congruent measurements of planetary distances!
From Copernicus, Kepler saw, first, that the planets in reality move more slowly the further they are from the sun, in some regular progression related to their distances. (By what law?) Second, their spacing is somehow harmonic, which Kepler found to be congruent with the Platonic Solids. Third, there are all sorts of “little anomalies,” as we have noted, like variations in the apparent distances of the planets from the sun as seen from earth, leading to later triumphs like Kepler’s determination of the elliptical form of planetary orbits, and Carl Gauss’ 1801 determination of the orbit of the first asteroid to be discovered, Ceres.
Copernicus’ genius “in the small” determined an entire “curvature” for future astronomy. Enough to inspire Kepler, or you or me!