Objection 1. It would seem that in creatures there is not necessarily found a trace of the Trinity. For anything can be traced through its traces. But the trinity of persons cannot be traced from the creatures, as was above stated (32, 1). Therefore there is no trace of the Trinity in creatures.
Objection 2. Further, whatever is in creatures is created. Therefore if the trace of the Trinity is found in creatures according to some of their properties, and if everything created has a trace of the Trinity, it follows that we can find a trace of the Trinity in each of these (properties): and so on to infinitude.
Objection 3. Further, the effect represents only its own cause. But the causality of creatures belongs to the common nature, and not to the relations whereby the Persons are distinguished and numbered. Therefore in the creature is to be found a trace not of the Trinity but of the unity of essence.
On the contrary, Augustine says (De Trin. vi, 10), that "the trace of the Trinity appears in creatures."
I answer that, Every effect in some degree represents its cause, but diversely. For some effects represent only the causality of the cause, but not its form; as smoke represents fire. Such a representation is called a "trace": for a trace shows that someone has passed by but not who it is. Other effects represent the cause as regards the similitude of its form, as fire generated represents fire generating; and a statue of Mercury represents Mercury; and this is called the representation of "image." Now the processions of the divine Persons are referred to the acts of intellect and will, as was said above (Article 27). For the Son proceeds as the word of the intellect; and the Holy Ghost proceeds as love of the will. Therefore in rational creatures, possessing intellect and will, there is found the representation of the Trinity by way of image, inasmuch as there is found in them the word conceived, and the love proceeding.
But in all creatures there is found the trace of the Trinity, inasmuch as in every creature are found some things which are necessarily reduced to the divine Persons as to their cause. For every creature subsists in its own being, and has a form, whereby it is determined to a species, and has relation to something else. Therefore as it is a created substance, it represents the cause and principle; and so in that manner it shows the Person of the Father, Who is the "principle from no principle." According as it has a form and species, it represents the Word as the form of the thing made by art is from the conception of the craftsman. According as it has relation of order, it represents the Holy Ghost, inasmuch as He is love, because the order of the effect to something else is from the will of the Creator. And therefore Augustine says (De Trin. vi 10) that the trace of the Trinity is found in every creature, according "as it is one individual," and according "as it is formed by a species," and according as it "has a certain relation of order." And to these also are reduced those three, "number," "weight," and "measure," mentioned in the Book of Wisdom (9:21). For "measure" refers to the substance of the thing limited by its principles, "number" refers to the species, "weight" refers to the order. And to these three are reduced the other three mentioned by Augustine (De Nat. Boni iii), "mode," "species," and "order," and also those he mentions (QQ. 83, qu. 18): "that which exists; whereby it is distinguished; whereby it agrees." For a thing exists by its substance, is distinct by its form, and agrees by its order. Other similar expressions may be easily reduced to the above.
Reply to Objection 1. The representation of the trace is to be referred to the appropriations: in which manner we are able to arrive at a knowledge of the trinity of the divine persons from creatures, as we have said (32, 1).
Reply to Objection 2. A creature properly speaking is a thing self-subsisting; and in such are the three above-mentioned things to be found. Nor is it necessary that these three things should be found in all that exists in the creature; but only to a subsisting being is the trace ascribed in regard to those three things.
Reply to Objection 3. The processions of the persons are also in some way the cause and type of creation; as appears from the above (Article 6).
Source: Summa Theologica (New Advent)