Brian Kemple, PhD
Examining Thomas Aquinas’ fourth way of proving the existence of God, readers may discover a number of difficulties as to how it should be interpreted:
The fourth way is taken from the grades of being which are found in things. For we find in things something which is more or less good, and true, and noble, and other such things of this kind. But the more and the less are said of diverse things insofar as they approach diversely to something which is the maximum, as something is more hot which more closely approaches the maximum of heat. Therefore there is something that is most true, and best, and most noble, and consequently something which is maximally being, for those things which are maximally true are maximally beings, as Aristotle says in Book II of the Metaphysics. Now that which is said to be maximally such in any genus, is the cause of all things which belong to that genus: as fire, which is the maximum of heat, is the cause of every heat, as is said in the same book of the Metaphysics. Therefore there is something that to all beings is the cause of existence, and goodness, and every perfection, and this we call God.
How does truth admit of more or less? The same applies to nobility and being. One might also ask, just what does Thomas means by nobility? How is it distinct from goodness? How is the maximum in any genus supposed to cause everything else in that genus? Is this formal or efficient causality? Is this a Neo-Platonic argument from participation? Is the argument undermined by the Angelic Doctor’s outdated and erroneous example of fire as cause of all heat? Each issue is likely deserving of its own paper. Yet behind all these questions (except, perhaps, the meaning of nobility), lies a more primary difficulty: the meanings of “more or less” and the “maximum” by which they are denominated. How we answer the above questions depends on our understanding of these two key features of the argument.
Christopher Martin has argued that in all of the five ways, Thomas identifies some particular feature exhibited by created things, requiring an explanation from something uncreated, which is common to all created things. Martin claims that this “feature X” in the fourth way is “possessing existence to an (intensively) limited degree.”
This notion of the “feature X”, generally applicable to things in the world and necessitating some explanation outside of the world, seems a fair way to approach the five ways and the formulation of the feature X of the fourth way—possessing existence to an intensively limited degree—is mostly accurate. It would seem, however, that the difficulty many have in understanding the fourth way follows from interpreting what this “intensive limitation” means and why it requires explanation by something unlimited.
The More, Less, and Most
One issue seldom explored thoroughly in discussing the fourth way is what Thomas means by saying that having a “more or less” necessitates something that is “most”. We can designate something as being the “most” or the “maximum” in one of two ways: in either a relative or de facto sense, where something happens to exceed all others in a given genus, or in an absolute sense, wherein something is wholly identical with that act. Despite lacking investigation into this underlying issue, it is a common criticism of the fourth way that Thomas jumps invalidly from one kind of maximum to the other: that is, from a relative “most” to an absolute maximum, by moving from the example of there being a thing that is hottest to the claim that there is something maximally being, maximally true, and maximally noble in an absolute sense. That Aquinas appeals to an absolute maximum is clear in the case of the transcendentals; for although we could understand a relative “maximum” of truth, goodness, nobility, heat, brightness, and so on, it really makes no sense to say there is a relative “maximum” of being (since the maxime ens is God). It is, however, taken for granted by most interpreters that the maxime calido is a relative “most.”
If this is the case—that the maxime calido is a relative most—then Thomas is guilty of equivocating on the meanings of “the most,” and we would either have to ignore the example altogether and somehow reconstruct the argument or find a parallel text through which the wounds caused by this poorly-judged leap could be patched. Yet this assumption of a relative “most” mistreats the argument; since Thomas does not tell us whether fire is meant to be a relative or an absolute maximum, we owe it to the Angelic Doctor to investigate further and discern what he really means.
First, let us consider how it is Thomas says an entity is constituted as to have a “more or less” of something according to a degree. How can this comparison of intensity can be said of various things? Degrees of “more” and “less” are able to be considered, Thomas tells us, in three ways, and thus predicated in three ways:
- first, according to quantity and thus with a univocal meaning, such as whiteness is said to be more in the snow than it is in the wall;
- second, according to being essentially in one thing and participatively in another, such as goodness of itself is said to be better than some good thing; and
- third, according to a preeminence in one thing over the other, such as heat is said to be in the sun more than in fire.
The first, a simple comparison with no connection between the two things except that they each possess a common, univocally-meant attribute, does not apply here. The last two modes are how “more and less” are predicated of God and creatures; in other words, whatever is said of both God and creatures is said essentially and preeminently of God and of creatures by participation and in a mitigated sense.
In the case of hot things, with fire serving as a maximum, we can exclude the last sense—for if heat is preeminently in the sun, over and above fire, then it makes no sense to say that fire is the maximum of heat. If we understand the example Thomas gives of fire as being the most hot in an elemental sense, however, we will go a long ways towards obviating our difficulty of the apparent jump from the relative to the absolute; for in this sense, something is said to be the “most” not as having a quantitatively greater amount of something, but as being that something simpliciter. To make this more evident, we should proceed by considering how it is that things actually come to have some degree of the more or less.
Thomas says that whatever lies between two extremes, and thus in a range of more or less, does so by a composition of simple things which are contrary to one another while within the same genus. These simple contraries are not to be held as something equivalent, but rather one is something positive and the other is something negative. For example, fire is positive, being essentially hot, while cold is negative, being heat’s absence or negation. Again, Thomas says something is said to be more or less white inasmuch as it is a combination of white and black, and something is said to be more white the nearer it approaches white simpliciter. Similarly, we would say that the temperature of something is a medium between some ideal extreme amount of thermal energy and the total absence of thermal energy—that is, absolute zero—in one and the same thing.
This ideal positive extreme is merely an abstraction, for there would seem to be a very real impossibility of achieving any actual absolute extreme in anything material. For instance, regarding the actually existing positive extreme or contrary, while it is a de facto “most” in that particular genus, it can quite likely be exceeded in some way. Contrariwise, absolute zero is theoretically impossible to reach, and, although researchers at MIT have been able to come within trillionths of a degree F, it is still possible to come even closer and yet never all the way; and who knows what would happen were anyone to actually attain that temperature? At the opposite end of the thermal spectrum, by smashing gold ions together, a group of researchers produced temperatures around 7.2 trillion degrees F—hot enough, evidently, to break apart protons and neutrons.
The less directly or frequently we have experience of such things, and thus the more we are removed from them, the less certainty we have about the reality of their extremes and the variance we can observe between the two contraries. Nevertheless, no matter how much something is pushed towards one extreme, so long as it remains in some way in matter, it is still disposed to some degree by the opposite contrary. In other words, it is never purely simple.
Of course, this was quite unknown to Thomas. For him and his contemporaries, fire was an element: a mixture of the form of heat and matter which, in itself, did not admit admixture with any other form or matter. When mixed with other bodies, its substantial form was not preserved in its fullest actuality, but only virtually. In such a mixed case, fire would be present by participation but not essentially. Thus it would make perfect sense for Thomas to argue by example from an example of something essentially and simply hot as causing a limited heat in all other hot things, to an essentially and unqualifiedly simple being which causes a limited being in all others. However little he may have understood about the nature of fire, he certainly was not guilty of attempting to use an argument from a relative most to an absolute maximum.
A New Example
We could, were we so inclined, simply take the participative principle to which Thomas alludes in his example of the fire and explain the “feature X” of the fourth way by looking at the transcendentals. Yet to do this would be to jump immediately into a strictly immaterial, difficult to understand, and rather abstract sort of proof. While we do have direct experience of goodness and truth, we do not directly experience goodness itself nor truth itself; and thus arguing for the validity of Thomas’ argument simply on the basis of these transcendentals seems inadequate and even unfitting to the proper and proportionate objects of our knowledge. Rather, the argument cries out for another example found in nature which operates comparably to the mistaken example of fire. Since nothing is essentially and simply hot, the question becomes whether we can find a different but comparable instance of this principle to see if it holds true.
Looking to the level of elementary particles, I would argue that we do find a suitable analogate for Aquinas’ example of fire: the photon. Theorized to be a gauge boson, the photon is the basic unit of electromagnetic radiation. All radio and micro waves are produced by photons oscillating at different frequencies, or wavelengths; so are infrared lights, visible lights, and ultraviolet lights, as well as X-rays and Gamma rays. Considered explicitly as the principle of the visible spectrum of light, the photon appears to function in the same way Thomas mistakenly thought fire functioned.
All visible light, i.e. the wave-like oscillation of photons which occurs roughly between wavelengths of 420nm (714THz – violet), and 700nm (428THz – red), is differentiated by the manner in which it interacts with things, both those illumined by the light and the interposed medium between observer and that thing; or rather, by the manner in which things interact with it. White light contains photons oscillating at a wide range of wavelengths. If these photons can be diverted one from another by refraction, so that the wavelengths are differentiated, the white light will be differentiated into a variety of different colors displayed over a wider area than that covered by the source light. This is what Newton famously demonstrated with his prism experiment.
When we discuss the illumination of a thing, however, we can speak of a number of different photon interactions, by means of which the wavelength can be changed and the brightness of the light altered: specular reflection, diffuse reflection, transmission, diffuse transmission, iridescence, refraction, and a variety of modes of absorption. Of reflections, there are two kinds with which we are presently concerned: specular reflection, being the manner in which a bright metal, for instance, reflects, and diffuse (or scattered) reflection, being the manner in which a matte surface reflects. The brightness of each surface is determined by the manner and degree to which the photons are absorbed into the atoms of the object which they are illuminating: generally speaking, white bodies are those that diffusively absorb the least (diffusive absorption being that in which the photons which are absorbed are converted into other forms of energy and spread throughout the body), and black bodies those that diffusively absorb the most. The energy is not lost in either case, but either redirected, as in the case of white bodies, or converted into some other form, such as thermal energy in the case of black. That a black surface can nevertheless be reflective is due to the composition of the body itself; for instance, metal absorbs energy much better than most surfaces and, being a conductor of electricity, immediately re-emits some of the photons and thus gives a strong specular reflection, provided that there is not something added to the metal so as to give it a matte finish. Likewise a glossy black paint would have a high rate of absorption and a lower rate of diffusion than a matte black paint.
Thus we can see that the photon, when emitted within a certain range of wavelengths and although principally existing within electrons, is of itself the principle which produces visible light; when it contacts other atomically-composed things, they become visible because of the manner in which they receive the photons. Both their brightness and their color are said to be “more or less,” although the color (that is, the hue) is determined with respect to a relative most (the variability of electromagnetic wavelengths being indeterminate), whereas the brightness is determined with respect to an absolute—that is, the relative brightest thing is that which diffusively absorbs the fewest photons and re-emits the most which it receives. Regardless, the photons themselves are light, and the light which can be ascribed to things is dependent upon their reception of the photons. Granted, in order to produce directly visible effects, there must be a great many photons, and they must be generated by something causing them to oscillate within a certain range of frequencies, but simpliciter loquendo, they are the energy out of which all visible light is constituted and all things made visible by them are as participants in something participated. Unlike thermal energy, which exists only within an atomically-composed system, the energy carried by the photon can be entirely independent of any system; it is something which can be isolated in a single, simple instance, utterly uncomposed with other mass or particles. Thus, while the example of electromagnetic radiation within the visible spectrum, when compared to Thomas’ example of fire being composed with other substances to make them hot inverts the manner in which something is said to be more or less (i.e., that it is “more” by not becoming composed with the active principle, which composition entails a variety of possible consequences), nevertheless seems to prove the Angelic Doctor’s point, and coheres with the medieval notion of an element as being a material thing not divided by a plurality of forms. Furthermore, if we convert our terminology just a little, it aligns much more neatly: if we instead say that the illumined thing participates the illuminating of the photon which is illumination itself, there seems to be no significant difference at all.
Now we ask, how does this help our understanding of Thomas’ fourth way?
That there is a Neo-Platonic cast to the fourth way is generally agreed upon—or at least entertained and discussed—by most interpreters. For it appears to be an argument from participation, inasmuch as it is similar to many of the things Thomas says on participation; discussing things in terms of “more or less” in respect of their approximation to a maximum certainly appears Platonic. Unfortunately, this evident Platonism adds an altogether new difficulty: namely, that Thomas never wrote a treatise exclusively on participation. While much has been done in the last 70 years to attempt reconstruction of Thomas’ doctrine of participation, it remains rather unclear and has been thoroughly confused by some writers, particularly those who go too far in emphasizing its Platonic heritage. Nevertheless, we may hopefully erect an accurate if skeletal presentation of a genuinely Thomistic doctrine of participation.
In Thomas’ early exposition of the De hebdomadibus of Boethius, he identifies first a generic meaning of participation from its etymology (partem capere, to take part) and subsequently identifies three distinct ways in which something can be said to participate: 1) the way in which something more particular participates in something more universal, as man is said to participate in animal because “man” does not correspond with everything meant by “animal,” but rather takes only a part; 2) the way in which a substance is said to participate in an accident or matter in form, in that what is indifferent to what it receives is differentiated by receiving something specific; 3) the way in which an effect is said to participate in its cause. It makes no sense to claim that the fourth way uses participation in the first sense, for the “more universal” in this sense refers to something abstract, and anything which is said to participate as the concrete in the abstract does not signify a real cause. Nor would the fourth way entail the kind of participation by which a substance participates in an accident, for the substance is prior to the accident, while the created are not prior to the creator. It would seem then, that if the fourth way is an argument from participation, it is of that sort of participation by which an effect participates in its cause, and that cause is not anything of its essence; thus, an efficient cause. But what does it mean to say that an effect participates in an efficient cause?
To answer, we must make a qualification. The effect, or power, of a cause can be present in the thing effected in two ways: either univocally, as when the effect is formally identical to the cause, or equivocally, as when there is a deficiency in the effect in proportion to the cause. The former would be, for example, when human beings reproduces another human being or when fire produces more fire; the latter would be as when a horse and donkey produce a sterile mule, or as when fire produces a lesser heat (i.e., the power of fire composed with other elements, such as in heating water) but not more fire; or in Thomas’ example, as when the sun begets fire. The second of these two modes, the equivocal, parallels one of the ways in which a thing can be said to be “more or less;” that is, when something can be called “more” because it is had essentially and preeminently in one thing, which it seems safe to say is also the cause, and what is called “less” is had participatively and restrictedly in the other, which we can identify as the effect. Whereas something had essentially is in some way unlimited, whatever is had participatively, as an effect participating in its cause, is necessarily limited, insofar as something participated is something received, and something received results in the received and receiver constituting something composed. Thus when a cause acts upon something—and its effect becomes composed with that upon which it acts, which must be in potency to that which acts upon it—the presence of that which is in the effect is somehow deficient compared to its the presence in the cause.
This should clarify for us an interpretation of the fourth way. Things are said to be “more or less” good, true, noble and the like with respect to a maximum of goodness, truth, and nobility—which three designate different considerations of one and the same thing—by a mode of participation. To participate in the transcendentals would have to be to participate as an effect in its cause, and as something not had essentially by the participants, were it to work in the context of the fourth way. Therefore, that in which they participate must be, ultimately, that which is the participated character essentially (or else there would be an infinite regress). For no ens aside from that which is maxime ens essentially has existence; nor is any other ens essentially good, but only insofar as it has an act of existence making it to be actually some way, bringing it to some perfection. Thus we can say “more” and “less” good, true, noble, and the like of entities, inasmuch as each has a different degree of participation in what is essentially good, true, and noble, by each entity having an act of existence which makes actual within it some perfection; and such a participation is also a degree of reception, of limiting in the effect what is essentially unlimited, in the sense of purity or simplicity, in the cause. So long as we understand being not as the simple fact of existence, but as the perfection of all perfections, this limitation of esse is not difficult to understand.
For instance, we can easily discern that between a starfish, whose eyes are little more than light detectors, and an eagle, whose eyes are many times more sophisticated than man’s, the eagle has the better eyesight. Moreover, the “more” of visual capability in the eagle is not simply a greater ability to sense; it is a greater perfection of being. Given that essentially unlimited being is restricted to the form of a sensitive creature, it is better that it be less limited by being received into something that can perform more or better sensory operations, such as the essence of an eagle, than by being received into something that can perform fewer, like the essence of a starfish. Were we to compare exclusively the being of human vision to the being of the eagle’s, we would have to say that the latter has the greater being. Yet in such comparisons of more and less in terms of being, we compare the whole entity; we compare things in the light of their outermost existential limits. As such, anything living will be greater than something non-living and something which has understanding will be greater than something merely living, inasmuch as each is determined to a greater or lesser participation in being. Moreover, that a subsisting being itself—ipsum esse subsistens—would have to be the cause of all these things is obvious, for no limited being is its own act of existence, and therefore must receive its esse from something that is its own being. Just as if there were not something per essentiam light, there could not be light in other things, so too were there not something per essentiam esse, there could not be esse in others.
Thus, were we to reformulate the previously given “feature X” of the fourth way “possessing existence to an (intensively) limited degree” to something perhaps more coherent with the interpretation we have proposed here, we could call it “existing in a limited degree by virtue of participation.” The mere fact that a thing possesses existence, or even that it possesses it to a limited degree, does not as such cry out for an outside-of-creation explanation. But given that such existence is not something which pertains to the thing by its nature—which seems all the more clear given that some have “more” existence, so to speak, through a greater act and a greater capability for act—does demand some cause beyond all creation. And this cause, as Thomas says, we call God.
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De Raeymaeker, Louis. 1954. The philosophy of being. Trans. Edmund Ziegelmeyer H. Binghamton, NY: B. Herder Book Co.
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 1266-68: ST Ia, q.2, a.3, c.: “Quarta via sumitur ex gradibus qui in rebus inveniuntur. Invenitur enim in rebus aliquid magis et minus bonum, et verum, et nobile, et sic de aliis huiusmodi. Sed magis et minus dicuntur de diversis secundum quod appropinquant diversimode ad aliquid quod maxime est, sicut magis calidum est, quod magis appropinquat maxime calido. Est igitur aliquid quod est verissimum, et optimum, et nobilissimum, et per consequens maxime ens, nam quae sunt maxime vera, sunt maxime entia, ut dicitur II Metaphys. Quod autem dicitur maxime tale in aliquo genere, est causa omnium quae sunt illius generis, sicut ignis, qui est maxime calidus, est causa omnium calidorum, ut in eodem libro dicitur. Ergo est aliquid quod omnibus entibus est causa esse, et bonitatis, et cuiuslibet perfectionis, et hoc dicimus Deum.”
 Martin, Christopher, God and Explanations (Edinburgh University Press, Edinburgh, UK: 1997) 126-7.
 Van Steenberghen, F., Ontology, trans. Flynn, Martin J. (Joseph F. Wagner, Inc., New York, NY: 1952) 150-51. One could similarly argue that although we may see many things that are de facto the “most” in any number of genera, none of them, even the good or being that we know directly, in any way indicate the presence of an absolute maximum.
 Cf. Gilson, The Christian Philosophy of St. Thomas Aquinas, trans. Shook, L.K.. (Random House, Inc, New York, NY: 1956) 71-73.
 Cf. Martin, 176; Van Steenberghen, op cit. Gilson, ibid. Wippel, The Metaphysical Thought of Thomas Aquinas (The Catholic University of America Press, Washington, D.C.: 2000) 469-479, outright ignores the example, despite it appearing in all three loci classici of versions of this argument (ST Ia, q.2, a.3, c.; SCG I.13, n.34; and De potentia q.3, a.5, c.)
 Such as Wippel tries to do, ibid. De Raeymaeker, Philosophy of Being, trans. Ziegelmeyer. (B. Herder Book Co., Binghamton, NY: 1954), 298-99 makes no mention of the example whatsoever.
 This is precisely what is done by F. Van Steenberghen, Ontology trans. Flynn. (Joseph F. Wagner, Inc, NY: 1952) 192-96: “In brief, the quarta via as expressed in the Summa Theologica fails to prove that there exists at the summit of the hierarchy of beings an absolute Maximum (the Infinite Being) and not merely a relative Maximum (the most perfect of finite beings). St. Thomas gives a much better expression to the argument elsewhere, as for example in De Potentia, III, 5.”
 De potentia., q.7, a.7, ad.3: “Ad tertium dicendum, quod magis et minus tripliciter potest considerari, et sic praedicari. Uno modo secundum solam quantitatem participati; sicut nix dicitur albior pariete, quia perfectior est albedo in nive quam in pariete, sed tamen unius rationis; unde talis diversitas secundum magis et minus non diversificat speciem. Alio modo secundum quod unum participatur, et aliud per essentiam dicitur; sicut diceremus, quod bonitas est melior quam bonum. Tertio modo secundum quod modo eminentiori competit idem aliquid uni quam alteri, sicut calor soli quam igni; et hi duo modi impediunt unitatem speciei et univocam praedicationem; et secundum hoc aliquid praedicatur magis et minus de Deo et creatura, ut ex dictis patet.” – “To the third it must be said that the more and the less may be considered in three ways, and thus predicated in three ways. The first way is according strictly to the quantity of that which is participated in, as snow is said to be whiter than the wall, because whiteness is more perfectly in the snow than it is in the way, but nevertheless there is only one intelligible rationale; and such a diversity according to more and less does not diversify the species. The second way is when one thing is said to be by participation and the other is said to be essentially; as if were to say that goodness is better than a good thing. The third way is when the same thing belongs to one in a more eminent manner than it does to the other, as the heat of the sun compared to that of fire. These last two modes impede a unity of species and univocal predication, and it is thus that the more and less are predicated of God and creatures, as is clear from the aforesaid.” There is an interesting sort of similarity here (we would not go so far as to say a parallel) to the threefold predication of analogy which Thomas posited much earlier In I Sent. d.19, q.5, a.2, ad.1. Of particular note is the example of preeminence of heat in the sun over fire and the differentiated sense of material in heavenly and earthly bodies.
 Cf. ibid., lib.11, lec.1, n.35: “…illud quod est magis calidum, est magis igneum: unde quod simpliciter est ignis, est calidum simpliciter” – “that which is most hot, is most fire; whence that which is fire simpliciter, is hot simpliciter”; De malo, q.3, a.2, c.: “Omne autem quod per participationem dicitur tale, derivatur ab eo quod est per essentiam; sicut omnia ignita derivantur ab eo quod est per essentiam ignis” – “Everything which is said to be such through participation, is derived form that which is through its essence; as everything on fire is derived from that which is fire through its essence.”
 Cf. In Meta., lib.10, lec.9, n.14 (n.2110).
 Cf. ibid, lib.10, lec.5, n.2 (n.2024).
 Maggie Fox, “Hottest temperature ever heads science to Big Bang,” http://www.reuters.com/article/2010/02/15/us-physics-temperature-idUSTRE61E3OB20100215 15 February 2010. Accessed 20 November 2011.
 Cf. De mixtione elementorum: “Sunt igitur forme elementorum in corporibus mixtis, non quidem actu sed uirtute. Et hoc est quod Aristotiles dicit in I De generatione: Non manent igitur -elementa scilicet in mixto- actu ut corpus et album, nec corrumpuntur nec alterum nec ambo: saluatur enim uirtus eorum” – “Therefore, the forms of the elements are in mixed bodies, not in act but by virtue. And this is what Aristotle says in book I of De generatione: ‘Therefore they do not remain—namely, the elements in a mixed thing—in act, as do body and white, nor are they corrupted, neither one nor both; for their virtue is preserved.” Also see Compendium Theologiae, lib.1, c.43.
 I.e., the quidditas rei materialis.
 I.e., an indivisible element, a primary particle.
 According to the predominant theory in contemporary quantum electrodynamics, light behaves according to a wave-particle duality; that is, at larger and more powerful scales the effects of light are consistent with a wave-like motion, but inexplicable at smaller and weaker scales unless considered as being composed by discrete quanta, i.e., the photon particle. For math-free explanations see Anton Zeilinger, Dance of the Photons (Farrar, Straus and Giroux, New York: 2010) 11-15, and James Kakalios, The Amazing Story of Quantum Mechanics (Gotham Books, New York: 2010) 8-26. This is, nevertheless, no more than a theory.
 nm = nanometers; THz = terahertz. Cf. Nassau, The Physics and Chemistry of Color (John Wiley & Sons, Inc., New York: 2001) fig. 1-14 (p.19), 1-18 (p.25), 2-1 (p.38), 2-2 (p.39), 2-3 (p.39).
 Ibid., 5-9.
 Cf. ibid., table 1-1, p.23. Whether these functions are due strictly to a pattern of absorption and re-emission or whether there is spontaneous emission of photons from atoms proportionate to the “radiation by an oscillating dipole moment,” i.e., photon emissions resulting from the “excited” character of an electron. Cf. Rodney Loudon, The Quantum Theory of Light (Oxford University Press Inc., New York: 2008), 1-4 and 168-173.
 Ibid., 25-31.
 Ibid., 26-29.
 Ibid., 7-29, especially 21, 23, and 27.
 De principiis naturae, c.3. Cf. In Meta, lib.12, lec.4, nn.11-13.
 We take particular issue with the interpretations of Cornelio Fabro and Louis Geiger, who each posited a double participation. While we do not have the length to discuss this here, the idea of a “participation by similitude or formal hierarchy,” advocated by Geiger, has no basis in Thomas. Where Geiger came up with this theory, and how it became generally accepted, we speculate to stem from a tendency to identify any theory of participation with the Platonic model. Feser, “Existential Inertia and the Five Ways” (American Catholic Philosophical Quarterly, 2011, Vol. 85, No.2) 249-250 has given a fairly nice and succinct interpretation of the fourth way; but like many others, has posited a formal participation as being authentically Thomistic.
 In de hebdomadibus, lec.2: “Secundam differentiam ponit ibi, quod est, participare, quae quidem differentia sumitur secundum rationem participationis. Est autem participare quasi partem capere; et ideo quando aliquid particulariter recipit id quod ad alterum pertinet, universaliter dicitur participare illud; sicut homo dicitur participare animal, quia non habet rationem animalis secundum totam communitatem; et eadem ratione Socrates participat hominem; similiter etiam subiectum participat accidens, et materia formam, quia forma substantialis vel accidentalis, quae de sui ratione communis est, determinatur ad hoc vel ad illud subiectum; et similiter effectus dicitur participare suam causam, et praecipue quando non adaequat virtutem suae causae; puta, si dicamus quod aer participat lucem solis, quia non recipit eam in ea claritate qua est in sole.” It has also been argued, and we concur, that there is a further, implicit division in this text: namely, between participation in the abstract and participation in the concrete, which we would further argue is actually a division of the third way (since whatever subject participates in an accident will participate it as in the abstract). See te Velde, Participation and Substantiality, (E.J. Brill, Leiden, NLD: 1995) 79.
 De potentia, q.7, a.5, c.: “Cum omne agens agat in quantum actu est, et per consequens agat aliqualiter simile, oportet formam facti aliquo modo esse in agente: diversimode tamen: quia quando effectus adaequat virtutem agentis, oportet quod secundum eamdem rationem sit illa forma in faciente et in facto; tunc enim faciens et factum coincidunt in idem specie, quod contingit in omnibus univocis: homo enim generat hominem, et ignis ignem. Quando vero effectus non adaequat virtutem agentis, forma non est secundum eamdem rationem in agente et facto, sed in agente eminentius; secundum enim quod est in agente habet agens virtutem ad producendum effectum. Unde si tota virtus agentis non exprimitur in facto, relinquitur quod modus quo forma est in agente excedit modum quo est in facto. Et hoc videmus in omnibus agentibus aequivocis, sicut cum sol generat ignem.” – “Since every agent acts insofar as it is in act, and consequently its acts are alike to it, it is necessary that the form produced exist in some mode in the agent, nevertheless, diversely. For when the effect is adequate to the power of the agent, it is necessary that this form is according to the same intelligible rationale in both the maker and in the made; thus the maker and the made coincide in the same species, which belongs in all univocals: for a human generates a human, and fire generates fire. But when the effect is not adequate to the power of the agent, the form is not according to the same intelligible rationale in the agent and in the made, but is in the agent in a preeminent way; for insofar as the form is in the agent, the agent has the power to produce the effect. Whence if the whole power of the agent is not expressed in the thing made, it follows that the mode by which the form is in the agent exceeds the mode by which it is in the made. And this we see in all equivocal agents, as when the sun generates fire.” Cf. Aertsen, Medieval Philosophy and the Transcendentals, 58-60.
 Note that in the first mode mentioned in De potentia q.7, a.5, even though there could be a difference of degree, scil., a quantitative difference, according to a generic conception there must be univocity. If a horse reproduces something other than a horse, e.g., if bred with a donkey and a mule is produced, there is something deficient in the product—there is a formal difference between the two.
 We will not dwell on this participating-receiving-composed trifecta here, for it is a complicated issue; suffice it to say that it is a principle used and defended by Thomas in many places. See for instance SCG II.53; for a larger exposition, see Wippel, “Thomas Aquinas and the Axiom That Unreceived Act Is Unlimited,” The Review of Metaphysics, Vol.51, No.3 (Mar. 1998), 533-564.
 This is a well known Thomistic doctrine, and can be found clearly stated at ST Ia, q.3, a.4, c., second proof, De potentia, q.7, a.2, ad.9, and most poignantly at ST Ia, q.4, a.1, ad.3: “Ad tertium dicendum quod ipsum esse est perfectissimum omnium, comparatur enim ad omnia ut actus. Nihil enim habet actualitatem, nisi inquantum est, unde ipsum esse est actualitas omnium rerum, et etiam ipsarum formarum. Unde non comparatur ad alia sicut recipiens ad receptum, sed magis sicut receptum ad recipiens. Cum enim dico esse hominis, vel equi, vel cuiuscumque alterius, ipsum esse consideratur ut formale et receptum, non autem ut illud cui competit esse.” – “To the third it must be said that ipsum esse [existence itself] is the most perfect as all and is compared to all things as their act. For nothing has actuality, except insofar as it is, when existence itself is the actuality of all things, and the forum of all things. Whence it is not compared to others as receiver to the received, but rather as the received to the receiver. When I say the existence of this man, or this horse, or of any other such thing, existence itself is considered as the formal principle and as the received, and not as that to which the existence belongs.” Here, Thomas is explicitly speaking of the esse of creatures as that which is the most perfect, and what makes them to be in act.