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As is well known to us all, "Analogy of Being" is a very important term created by the Scholasticism of St. Thomas Aquinas. While this term cannot be found in Aristotle's original work, the similar notion of Aristotle is the "analogia proportionalitatis" in his book Nicomachean Ethics. How did Aquinas understand Aristotle's notion? There is a basic opinion of mine when my dealing with those questions: the substance of God and the substance of other things, like animals/humans，is univocal or of the same essence in Aristotle's mind; but, in Aquinas' view the substance of God must be higher or superior to other things.

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Thomas Aquinas's short work De Principiis Naturæ concisely summarizes the three types of predication:

  1. […] "Being" [ens], however, is not a genus because it is not predicated univocally, but only analogically.

  2. In order to understand this last we must notice something is predicated of many things in three ways: univocally, equivocally and analogically.

    1. Something is predicated univocally according to the same name and the same nature, i.e., definition, as animal is predicated of man and of ass, because each is called animal and each is a sensible, animated substance, which is the definition of animal.
    2. That is predicated equivocally which is predicated of some things according to the same name but according to a different nature, as dog is said of the thing that barks and of the star in the heavens, which two agree in the name but not in the definition or in signification, because that which is signified by the name is the definition, as is said in the fourth book of the Metaphysics.
    3. That is said to be predicated analogically which is predicated of many whose natures are diverse but which are attributed to one same thing, as health is said of the animal body, or urine and of food, but it does not signify entirely the same thing in all three; it is said of urine as a sign of health, of body as of a subject and of food as of a cause. But all these natures are attributed to one end, namely to health.

  3. Sometimes those things which agree according to analogy, i.e., in proportion, comparison or agreement, are attributed to one end, as was plain in the preceding example of health. Sometimes they are attributed to one agent, as medical is said of one who acts with art, of one who acts without art, as a midwife, and even of the instruments; but it is said of all by attribution to one agent which is medicine. Sometimes it is said by attribution to one subject, as "being" is said of substance, quantity, quality and the other predicaments, because it is not entirely for the same reason that substance is being, and quantity and the others. Rather, all are called being in so far as they are attributed to substance which is the subject of the others.

  4. Therefore being is said primarily of substance and secondarily of the others. Therefore "being" is not a genus of substance and quantity because no genus is predicated of its species according to prior and posterior; rather, "being" is predicated analogically. This is what we mean when we say that substance and quantity differ generically but are the same analogically.

The related text of Aristotle is Metaphysics IV ch. 2, 1003b22 (cf. Aquinas's commentary on this):

The term being [ὂν] is used in many senses, but with reference to one thing and to some one nature and not equivocally. Thus everything healthy is related to health, one thing because it preserves health, another because it causes it, another because it is a sign of it (as urine) and still another because it is receptive of it. The term medical is related in a similar way to the art of medicine; for one thing is called medical because it possesses the art of medicine, another because it is receptive of it, and still another because it is the act of those who have the art of medicine. We can take other words which are used in a way similar to these. And similarly there are many senses in which the term being is used, but each is referred to a first principle. For some things are called beings because they are substances; others because they are affections of substances; others because they are a process toward substance, or corruptions or privations or qualities of substance, or because they are productive or generative principles of substance, or of things which are related to substance, or the negation of some of these or of substance. For this reason too we say that non-being is non-being.

Réginald Garrigou-Lagrange's Reality: A Sythesis of Thomistic Thought ch. 5 art. 3, § "The Idea of Being," explains "analogy of being":

Being, for St. Thomas,194 is a notion, not univocal but analogous, since otherwise it could not be divided and diversified. A univocal idea (e.g.: genus) is diversified by differences extrinsic to genus (animality, e.g.: by specific animal differences). Now, nothing is extrinsic to being (ens). Here Parmenides enters. Being, he says, cannot be something other than being, and the only other thing than being is nothing, is non-being, and non-being is not. St. Thomas replies: "Parmenides and his followers were deceived in this: They used the word being (ens) as if it were univocal, one in idea and nature, as if it were a genus. This is an impossible position. Being (ens) is not a genus, since it is found in things generically diversified."195

Duns Scotus196 returns in a manner to the position of Parmenides, that being is a univocal notion. Suarez,197 seeking a middle way between Aquinas and Scotus, maintains that the objective concept of being (ens) is simply one (simpliciter unus): and that consequently everything that is in any manner (e.g.: matter and essence) is being in act (ens in actu). This viewpoint granted, we can no longer conceive pure potency. It would be extra ens, hence, simply nothing. The Aristotelian notion of real potency (medium between actuality and nothing) disappears, and the argument of Parmenides is insoluble.

Footnotes
194. Ens non est univocum, sed analogum, alioquin diversificari non posset
195. In Metaph.: Bk. 1, chap. 5, lect. 9. See the fourth of the twenty-four Thomistic theses
196. Opus Oxon.: Bk. 1, dist 3, q. 2, nos. 5 ff. ;dist. V, q. 1;dist. 8, q. 3; IV Met.: q. 1.
197. Disp. met.: II, sect. 2, no. 34; XV, sect. 9; XXX and XXXI

See also Part II of his Essence & Topicality of Thomism for more on Parmenides vs. Heraclitus and analogy of being.

A good introductory book on Aquinas's doctrine of analogy of being is:

For an advanced text, see the 4 vol. De Analogia by Jacobus M. Ramírez.

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