I am working my way through Astronomie by A.F. Möbius, a (historical) introductory astronomy book by the great mathematician and astronomer Möbius written during the 19th century. I’ve made my way up to p. 100 somewhat laboriously (because of the beginner level of my German).

There, in the section on Kepler’s three laws of planetary motion, I ran across the following correct sentence describing the third law:

Die Quadrate der Umlaufszeiten je zweier Planeten verhalten sich wie die Kuben ihrer mittleren Entfernungen von der Sonne

Below that is an example which is also correct but a little harder to interpret, partly because of the language.

… die Umlaufszeiten von Merkur und Mars sich genähert verhalten wie 1 zu 8; da 8 = 2×2×2 ist, verhalten die Umlaufszeiten sich also wie die Kuben von 1 und 2. Die mittleren Entfernungen sollten sich also verhalten wie 1 zu 2×2, d.h. wie 1 zu 4.

The example is saying that ((1/2)^3)^2 = ((1/2)^2)^3), that is, (1/8)^2 = (1/4)^3, which in turn is the square of the ratio of the orbit times is the cube of the ratio of the mean solar distances.

However, perhaps particularly for a German learner, the phrasing can mistakenly lead one down the path of thinking the example is incorrect! It is not, I now see, but, there is a language issue here, and that is what my question is about.

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    I am not sure if this is a language problem or just confusion due to lack of basic algebra knowledge. 'Die Kube von n' or in English 'the cube of n' is not 'the cube root of n', which you seem to believe. – jarnbjo May 17 '17 at 12:22
  • Actually I did not believe that but was attempting to express the relationship using my own words. – Circulwyrd May 17 '17 at 12:35
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    In your interpretation you are talking about the cube root of the orbital period and therefore claim that the German text should be referring to the cube of the orbital period. If you understand the difference between the cube of n and the cube root of n, why do you want to replace the expression in the German text? – jarnbjo May 17 '17 at 12:46
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    This is a nice question, but it has nothing what so ever to do with German as a language. Its proper place would be a maths or astronomy forum. – Christian Geiselmann May 17 '17 at 14:04
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    @jambjo Well, suppose someone told you that a + b = c and gave an example in which something wasn't clear and you found it easier to pose your question by referring to the inverse operation, subtraction. For example c - b does not appear to be equal to a. Would that mean that you do not know the difference between addition and subtraction? – Circulwyrd May 18 '17 at 15:39

All search results I found for "Keplersche Gesetze" claim the 3. law similar to german Wikipedia:

Die Quadrate der Umlaufzeiten zweier Planeten verhalten sich wie die Kuben (dritten Potenzen) der großen Bahnhalbachsen.

which is T1 ^ 2 / T2 ^ 2 = a1 ^ 3 / a2 ^ 3. So your book is totally right.

The english version of Wikipedia formulates this law for one planet:

The square of the orbital period of a planet is proportional to the cube of the semi-major axis of its orbit.

which is T1 ^ 2 / a1 ^ 3 = const. So your example is right, too.

It seems that the example in the book is wrong, but the facts are right.

  • I'm glad to get the German Wikipedia reference to show the law is correctly stated (I'd looked at the English Wikipedia and the slightly different way of framing it there caused difficulty). I continue to be unsure if the apparent mismatch of the example is a language problem (there is a second example and some data that are similar), but I'm afraid it's hard to continue following up on it unless someone else is able to look at the actual text of the book: I can't reproduce everything here. Just knowing the law is stated correctly is something though. – Circulwyrd May 18 '17 at 15:45
  • As a follow-up, the example is in fact correct (so the book is NOT in error). See my edits to the original question. I do see though why the wording of the example is challenging, and why it confused me when I first posted the question. – Circulwyrd May 19 '17 at 20:08

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