I will be glad if someone who has seen Peirce's paper could summarily describe here Peirce's axioms and describe their relation to Peano's.
C.S.Peirce, On the logic of number (1881).
Peirce's system is equivalent to Peano axiomatization.
A discrete system is one in which every quantity greater than another is next greater than some quantity (that is, greater than without being greater than something greater than).
An infinite system is one in which any quantity greater than x can be reaclhed from x by successive steps to the next greater (or less) quantity than the one already arrived at.
we may say that an infinite class is one in which if it is true that every quantity next greater than a quantity of a given class itself belongs to that class, then it is true that every quantity greater than a quantity of that class belongs to that class.
The minimum number is called one.
The language is cumbersome but the basic "ingredients" are there: an initial number; the successor function; induction.
Then the standard recursive definitions of $+$ and $\times$ follow:
By $x + y$ is meant, in case $x = 1$, the number next greater than $y$ [i.e. $s(y)$, the successor of $y$]; and in other cases, the number next greater than $x' + y$, where $x'$ is the number next smaller than $x$ [i.e. $s(x)+y=s(x+y)$].
Induction is explicitly used in the proof of theorems, starting from the associative principle of addition:
The proof in each case will consist in showing, first, that the proposition is true of the number one, and second, that if true of the number $n$ it is true of the number $1 + n$, next larger than $n$.