It is well known that the class of all special Jordan algebras does not form a variety of algebras, but it is not difficult to see that this class forms a quasivariety of algebras. The natural question then arises whether this quasivariety can be defined by a finite number of quasi- identities. The answer to this question turns out to be negative. Moreover, it is proved that any quasivariety of special Jordan algebras which contains all special nilpotent algebras of index n (\(n\geq 16)\) cannot be defined by any number of quasi-identities depending on a finite number of variables. In other words, any such quasivariety has an infinite axiomatic rank. To prove this theorem, the author constructs very interesting series of nonspecial Jordan algebras \(J_ n\), \(n\in N\), all of whose n-generated subalgebras are special; besides, the algebras \(J_ n\) are homomorphs of a special Jordan algebra.