For a long time, it has been known that the circularity is the overly demanding property for index numbers such as price and quantity indexes. When we insist on this property exactly, the resulting indexes turn out to have constant weights across time, which could lead to serious measurement bias. Recently, the attention to this property has re-emerged and the index that satisfies it was even dubbed as proper index by neglecting the possible bias induced by it. The main purpose of this paper is to highlight the problem of imposing the circularity by using simple examples. We also show that the Fisher index satisfies the weaker variant of circularity axiom and some proper indexes violate the circularity test in the axiomatic system of bilateral comparison.