The end of the De Principiis Naturae seems to explicitly state that analogy is a strictly logical notion, and not a metaphysical one. After arguing that analogy involves using the same name according to related notions {rationes}, he says:

The principles of those things which agree [in name] according to mere analogy are the same according to analogy or proportion alone.  For matter, form and privation, or potency and act are principles of substance and of the other genera. Nevertheless, the matter of substance and consequently and similarly the form and privation differ in genus, but agree only according to this proportion: as the matter of substance has the notion of matter with respect to substance, so the matter of quantity [has the notion of matter] with respect to quantity. Neverthless, just as substance is the cause of the others, so the principles of substance are principles of all the other [genera].
[E]orum quae conveniunt secundum analogiam tantum, principia sunt eadem secundum analogiam tantum, sive proportionem. Materia enim et forma et privatio, sive potentia et actus, sunt principia substantiae et aliorum generum. Tamen materia substantiae et quantitatis, et similiter forma et privatio differunt genere, sed conveniunt solum secundum proportionem in hoc quod, sicut se habet materia substantiae ad substantiam in ratione materiae, ita se habet materia quantitatis ad quantitatem. Sicut tamen substantia est causa ceterorum, ita principia substantiae sunt principia omnium aliorum.

In this text he says that the principles are the same “according to analogy alone”: that is, they are only the same insofar as (a) they are given the same name and (b) the things signified stand in a certain way to other things. For example,

(α) prime matter : (β) substance :: (γ) the matter of quantity : (δ) quantity;

because prime matter and the matter of quantity are both called “matter” and because each of them stands similarly to substance and quantity respectively, matter is said analogously of each.  There is no more to the notion of analogy than this.

The relation St. Thomas adds at the end “substance is the cause of the others and thus its principle are the principles of the others.” Is not a case of analogy, nor is any metaphysical relation between the matter of substance and that of quantity: such relations may be real and may even be the basis for analogous naming, but they are not the reason a name is imposed analogously.

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