See Aristotle's Mechanica (or Mechanical Problems; Greek: Μηχανικά), traditionally attributed to Aristotle, though his authorship of it is disputed, 847b16:
The original cause of all such phenomena [lever, balance] is the circle. It is quite natural that this should be so; for there is nothing strange in a lesser marvel being caused by a greater marvel, and it is a very great marvel that contraries should be present together, and the circle is made up of contraries. For to begin with, it is formed by motion and rest, things which are by nature opposed to one another. Hence in examining the circle we need not be much astonished at the contradictions which occur in connexion with it.
The "marvel" discussed here is specifically the power of a less weight to rise a greater one through the balance (with uequal arms), see 848a11:
Therefore, as has already been remarked, there is nothing strange in the circle being the origin of any and every marvel. The phenomena observed in the balance
can be referred to the circle, and those observed in the lever to the balance; while practically all the other phenomena of mechanical motion are connected with the lever. Furthermore, since no two points on one and the same radius travel with the same rapidity, but of two points that which is further from the fixed centre travels more quickly, many marvellous phenomena occur in the motions of circles, which will be demonstrated in the following problems. [...] Mechanicians seizing on this inherent peculiarity of the circle, and hiding the principle, construct an instrument so as to exhibit the marvellous character of the device, while they obscure the cause of it.
Thus, in the end, the "magical power of the circle" is nothing more that the ability of the skileld mechanician to master the "law of the circle" that explain the functioning of the device: the balance.
See also: The Mechanical Problems in the Corpus of Aristotle by Thomas Nelson Winter.
Also useful: Edoardo Benvenuto, An Introduction to the History of Structural Mechanics. Part I, Springer (1991), §1.8 The "Mechanical Problems": The Peripatetic Explanation of the Law of the Lever and the Parallelogram Rule, page 34.