“Eadem sunt quorum unum potest substitui alteri salva veritate. Si sint *A* et *B*, et *A* ingrediatur aliquam propositionem veram, et ibi in aliquo loco ipsius *A* pro ipso substituendo *B* fiat nova propositio æque itidem vera, idque semper succedat in quacunque tali propositione, *A* et *B* dicuntur esse eadem; et contra, si eadem sint *A* et *B*, procedet substitutio quam dixi.” [2] Leibnitii *Opera Philosophica quæ extant*. Erdmann, Pars I. Berolini, 1840, p. 94. Leibnitz, then, explicitly adopts the principle of substitution, but he puts it in the form of a definition, saying that those things are the same which can be substituted one for the other, without affecting the truth of the proposition. It is only after having thus tested the sameness of things that we can turn round and say that *A* and *B*, being the same, may be substituted one for the other. It would seem as if we were here in a vicious circle; for we are not allowed to substitute *A* for *B*, unless we have ascertained by trial that the result is a true proposition. The difficulty does not seem to be removed by Leibnitz’ proviso, “idque semper succedat in quacunque tali propositione.” How can we learn that because *A* and *B* may be mutually substituted in some propositions, they may therefore be substituted in others; and what is the criterion of likeness of propositions expressed in the word “tali”? Whether the principle of substitution is to be regarded as a postulate, an axiom, or a definition, is just one of those fundamental questions which it seems impossible to settle in the present position of philosophy, but this uncertainty will not prevent our making a considerable step in logical science. Leibnitz proceeds to establish in the form of a theorem what is usually taken as an axiom, thus (*Opera*, p. 95): “Theorema I. Quæ sunt eadem uni tertio, eadem sunt inter se. Si *A* ∝ *B* et *B* ∝ *C*, erit *A* ∝ *C*. Nam si in propositione *A* ∝ *B* (vera ea hypothesi) substituitur *C* in locum *B* (quod facere licet per Def. I. quia *B* ∝ *C* ex hypothesi) fiet *A* ∝ *C*. Q. E. Dem.” Thus Leibnitz precisely anticipates the mode of treating inference with two simple identities described at p. 51 of this work. Even the mathematical axiom that ‘equals added to equals make equals,’ is deduced from the principle of substitution. At p. 95 of Erdmann’s edition, we find: “Si eidem addantur coincidentia fiunt coincidentia. Si *A* ∝ *B*, erit *A* + *C* ∝ *B* + *C*. Nam si in propositione *A* + *C* ∝ *A* + *C* (quæ est vera per se) pro *A* semel substituas *B* (quod facere licet per Def. I. quia *A* ∝ *B*) fiet *A* + *C* ∝ *B* + *C* Q. E. Dem.” This is unquestionably the mode of deducing the several axioms of mathematical reasoning from the higher axiom of substitution, which is explained in the section on mathematical inference (p. 162) in this work, and which had been previously stated in my *Substitution of Similars*, p. 16.