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.