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In the present document we will discuss the nature of  "quality", still [doing so] by largely following HOENEN, 1947.  The existence of true qualities -- i.e. the fact that not all alleged "qualities" can be completely reduced to quantity -- is very important in Aristotelian-Thomistic metaphysics and forms the basis for considering organisms, free molecules, free crystals, and free atoms as true substances in the metaphyscal sense :  They are heterogeneous -- as a result of the pattern of qualities in each one of them -- continua. Much of this was already discussed in  Fourth Part of Website ,  but in the present series of documents (Sixth Part of Website) the results are integrated in our latest development (also in Sixth Part of Website) of the theory of the Explicate-Implicate Orders.

Our world is a world of multitude and diversity. Bodies not only do differ in size and shape, but also show great diversity in qualities, color, sound, heat, weight, hardness, and tension, diversity also in intensity of a same quality. It is also a world of change. And again we find change not only of size or shape, and place, but also of these qualities and of their intensities. It even is a world of  "becoming and corruption" (of bodies) in the strictest sense, a world of birth and death. All this, daily experience seems to teach us in all clarity.
We know how the ancient Eleatic metaphysics came to the baffling conclusion that only one single  ens  (being) existed, and that this one ens must be unchangeable in all respects, and that all the multitude and change is just appearance, opinion, and not a reality of Nature. We know how mechanicism, that of the ancient atomists as well as that of Descartes, tried to save multitude and change, but not arriving at any result other than the possibility of multitude of extension and of shapes of the extensa (i.e. bodies differing in size and shape only). We know that they could not render intelligible any change other than that of place, local motion, and that they were forced to explain Nature from these data only. And then, a priori, all intrinsic, true qualitative, change was out of the question. A fortiori was excluded all real becoming and corruption, the origin of new, and the vanishing of existing substances, - birth and death.
But we know that the principles of Aristotle, the principles of the potentia-actus theory, do make intelligible the possibility of qualitative change, and of (radical) becoming and corruption, containing a complete (but still general) solution of the antinomy of the Eleates. And from this it follows that the metaphysical fundament of the mechanistic world-view is false. This fundament consists in the thesis that intrinsic change, qualitative as well as substantial change, is not intelligible and thus impossible. So if the analyses of Aristotle demonstrate the intelligibility of these changes, then the falsity of the mechanistic fundament is demonstrated.
But mechanistic natural philosophy rested on yet a second, a cognitive fundament, onto which especially Descartes, and those who came after him, the founders of "classical physics", erected their philosophical systems :  [having as central thesis] the theoretical conviction that in Nature only the geometric elements, extensionality and motion, and what is included in these two, are intelligible and thus may have reality, excluding qualities, i.e. secondary qualities as these are called since Locke, especially when these qualities were supposed to be changeable. We also know, from history, that the analysis of Descartes was incomplete, and that further analysis of the qualitative demonstrates intelligibility  ( for the human intellect :  as to its genus. That is, we can apprehend the possibilty of qualitative change, but cannot a priori forecast  what  quality will be the result )  also of it. And from this may rightfully be concluded -- according to the very principles of Descartes-Leibniz themselves -- that in matter also qualities, even variable qualities, are possible. We even can go one step further and conclude from the same principles, that in Nature such qualities must actually exist (i.e. not only be possible in principle), in no less a way as in the case of geometric elements. However, our knowledge of these qualities was restricted -- in contrast to the geometric -- to only the generic nature of them, and to the fact that there exist differences in qualities. In the beginning, the nature of these differences escaped from our insight.
With this first result, that consists in the  possibility  of intrinsic, qualitative, change, and, consequently, excluding the  necessity  of a mechanistic explanation of Nature, might go together that Nature  in fact  might be completely mechanistically explained (and thus had to [as long as qualities are not actually found] ).
The second result already brings with it that also in fact variable qualities, at least one or another, have been found in Nature. What qualities precisely they are, a general theory [as is the Aristotelian-Thomistic metaphysics] cannot determine. For this it has to be  specified,  and, in order to find these specifications, an extensive, experimental and theoretic or explanatory investigation is needed. This investigation is the proper task of physics, and therefore, we will, in what follows, make extensive use of the data, experimental and theoretic, collected by three centuries of mathematical physics.

These specifications are necessary in every philosophy of Nature. It may take its point of departure from mechanistic principles as those of the atomists or those of Descartes, or from the naturalistic principles of Aristotle or from those of the medieval philosphers. Surely there is a distinction between the three systems :
Descartes wanted to find these specifications purely a priori, i.e. by thought alone :  and experience was only consulted by him to determine, in complicated cases as they are encountered in living organisms,  which  of the a priori deduced specifications were actually realized in Nature. He was a genuine rationalist.
In the atomistic systems of philosophy, as they are worked out in classical physics, this aspiration is less apparent. Not so much in virtue of the system, but rather because the development of science has forced to attribute a greater value and influence to experience.
Finally, in a system of aristotelian origin one needs, to find the specifications of the general theory, in a much broader degree the help of experience. It follows from the nature of the system :  It not only assumes geometric elements which are totally transparent to the human intellect, and thus admit an a priori specification, but it also contains qualitative elements, which are known to us only generically, and thus urgently demand a specification to be found in experience only. And with this we are well aware that we say something which is precisely the opposite of what one usually ascribes to medieval philosophy as compared to more modern views of nature. The mistake is not ours. If we understand that the nature of Aristotle's system necessarily demands consultation of experience, we will not be astonished when we hear a St Thomas declare :  " Who, in the philosophy of Nature neglects experience (sensum), falls into error."  But isn't it so that experimentation was completely lacking in the Middle Ages? This is undoubtedly true. But in addition to experimentation yet another [type of] experience exists. And then :  the absence of experimentation did not derive from scholastic principles, surely demanding specification by experience. The lack of experiments doesn't urge us to suppose unfaithfulness to these principles, because it was the result of another cause :  Experimentation in the inorganic world demands, if it wants to be of some value, mathematical methods, and these were not yet developed. Now someone might hold that the theories of scholasticism formed a hindrance to the application of these mathematical methods, and so certainly to their invention. Later, we will see that also this view is erroneous. The theories of natural philosphy of scholasticism, as they are already developed in St Thomas, call for further specification by mathematical methods.
So what we need is specification of the general principles if we want to erect a philosphy of Nature, which, as it is its task, wants to go all the way to an explanation of the specific essences. Discovery of these specifications is the task of physics and demands physical and especially mathematical-physical methods in experiment and theory.

Here we admit a difference to exist between natural philosophy and physics in the modern sense of this word. We more or less acknowledge this difference, but are not prepared to assume a  s e p a r a t i o n  between the two. Then, physics (and chemistry) has the task to find the data, experimental and theoretic (providing already a first explanation). These data lead to specifications, which will be incorporated into the general principles by natural philosophy. This concise description would deserve further elaboration, which we cannot present here. This is the task of epistemology, which is, of course, not natural philosophy.

Classical physics has looked for these specifications and found many. But they always were sought for and interpreted, up to desperation, as further determinations of mechanical principles. At the end of the 19th century the task was even thought to be almost completed, until the facts, being irrationalities to mechanicism, piled up, and the 20th century, finally, clearly revealed the bankruptcy of the mechanical view of Nature. Earlier we already heard the words by which Bavink describes this bankruptcy and its recognition by "alle genaueren Kenner der modernen Physik mit wenigen Ausnahmen". But Bavink did not know by what [principles] the Eleatic-Cartesian principles had to be replaced. This will be clear to us :  They are the Aristotelian principles. So we will attempt to use these data of physics, chemistry, and crystallography for finding specifications of these [Aristotelian] principles, rendering intelligible also intrinsic changes, qualitative-accidental as well as substantial. Substantial changes we will deal with later on. Now we will discuss the qualitative-accidental ones [where quality appears as an entity further determining some given substance (in the metaphysical sense) but not necessarily deriving from the intrinsic whatness of that substance.]

In our analysis we must attend to the difference between  extension  and  intension  of a quality -- [the extension of a quality just refers to the greater or lesser surface area or volume (in a substance) in which the quality is manifest. This extension is itself, of course, not quality but quantity.], -- [attend to the difference] between the categories which Aristotle calls "quantity" (to poson) and "quality" (to poion). [Quantity is, of course, not limited to the extension of any given quality (if the latter has such an extension). As Aristotelian category it may be the size of any substance, or any number of parts of it.]. Later we will deal with it further. Here it is sufficient to present a simple description, of which every element is obtained by insight :  A quantity,  extensive  magnitude, may be larger or smaller. A quality,  intensive  magnitude, may, as to its intensity, be stronger or weaker (more, or less intensive), but not larger or smaller in the strict sense of the words. Two quantities of the same species may be placed adjacent to each other and added to each other :  A line of three with a line of two meters results in a line of five meters. On the other hand, two intensities, a temperature of three and one of two degrees cannot be placed adjacent to each other and thus not added to each other. The operation, in this case, doesn't even make sense. [water of three degrees mixed with water of two degrees does not result in water of five degrees.]. The difference (difference in the strict arithmetic sense) between two quantities of the same species, two lines, again is a quantity of the same species, a line. The difference between two intensities, two temperatures, is not an intensity, not a temperature. And although one may speak of a difference between two temperatures, the word does not have the same arithmetic meaning as it had above. A quantity may become smaller, by having cut off a section of it, and also this section may continue to exist on its own. An intensity may become weaker, i.e. smaller in a broader sense, but not by division leaving some section behind. From a line, say of 10 meters, one may, by division, take out a middle section, say between the second and eight' meter, and this may remain in existence as a line of five meters. But from a temperature of 10 degrees one cannot remove a section lying between the second and eigtht' degree. Not only because it is physically impossible, it even cannot take place in thought. Again, the words do not have any meaning.
All this cannot be derived by reasoning or argument, but is a case of immediate insight. It is a "Wesensschau" of the two categories, quantity and intensive quality, and, the fact that one has these immediate insights is a criterium of our knowledge of these two categories, extension and intensive quality. It is also a criterium of that particular knowledge saying that they cannot be reduced to one another. The fact that one can  measure  intensities nevertheless, and  compute  with the results, as does mathematical physics, will be discussed later, including how, precisely, one does the measuring  [While it is clear from the start  h o w  to directly measure the magnitude of  e x t e n s i o n  of something, such as measuring its length,  it is not immediately evident how to directly measure the magnitude of  i n t e n s i t y  of something. This is because the proper scale of intensity has its points (its grades) not distributed spatially, but distributed non-spatially, i.e.  w i t h i n  one another]. Meanwhile, what has been said is sufficient to conclude the existence of a quality there where we find something in Nature that becomes stronger and weaker. [Some such 'qualities' might in fact be emergent resultants of collective physical behavior which itself is merely quantitative or configurative.]
We must, as we've said, investigate four cases :  The principle of inertia, the collision, electromagnetic fields, and gravitation.

Our analysis of the principle of inertia (previous document) has teached us that, in order to render uniform rectilinear motion intelligible, we must assume an active cause, a cause accompanying the projectile in its motion. The medievals called this cause the  impetus.  Two aspects of it we find in modern mechanics, where it considers the (linear)  momentum (mv) and the  kinetic energy (1/2 mv²) [and these are measurable aspects of the impetus].  This impetus is not motion, which, afterall, is "becoming", i.e. something passive [The projectile is being moved]. It is the active cause of the inertial motion [uniform rectilinear motion]  [the impetus is the active cause, and its effect is the projectile being moved]. The impetus will, so we concluded, have its seat in the projectile itself, but necessarily has a relationship with the medium, the aether, in which it moves the projectile rectilinearly and with constant speed (as long as no "forces" are applied from without). It thus is an accidental determination of the substance of the projectile itself [generally, such a projectile is an aggregate of substances, and the determination is applied to each one of them]. It, the impetus, may be absent, its value then being zero, while the projectile itself remains what it is [therefore the impetus is an accidental determination]. It may (under the influence of "forces" from without) increase or decrease and with it the speed does. As to the nature of things [as they are in themselves], the impetus (the active) is that what is preceding, of which speed (a determination of the passive "becoming" which motion is) depends. To our knowledge, on the other hand, speed may be that what is preceding, the first known, from which we then derive the value of the ontologically-preceding, the impetus (by means of measuring its aspects, linear momentum and kinetic energy). [Motion of a body is caused by its impetus. The beginning of motion of a given body is caused by the for the first time implementation of impetus (i.e. the "handing-over of impetus (from, maybe, another body)) to the body. And this "applying-to" is the external force, intitiating motion. Also after cessation of that force, the body remains in possession of the impetus, and this causes the continuance of its motion. Newly applied external forces may change the intensity of the impetus of the body, causing it to change speed. After cessation of such newly applied external forces, the body continues its motion with a speed corresponding to the newest value of the intensity of its impetus. If, finally, all its impetus is transferred to another body (or bodies), the body will come to rest, its impetus is zero.
The impetus is an intensive quality of a body :  If the impetus is weak, then the resulting speed of the body is low. If the impetus is strong, then the resulting speed is high. The speed of the body, resulting from the intensity of the impetus, is itself an extensive magnitude (a species of quantity), because speed is the ratio of spatial distance and temporal duration, which both are extensive magnitudes.]

The impetus as [being an] intensive quality.
So we have found, from the analysis of the existential conditions of the experimentally known "inertia", in the projectile something active that may be, or may not be, present in the projectile, that, further, may become weaker or stonger, i.e. we've found a qualitative-intensive accident. So the material body is subjected to an intrinsic-accidental change ["intrinsic", because the change is not a mere change of size, or a mere addition or subtraction of material parts. It is a non-mechanical change. The change is "accidental" because it does not necessarily follow from the very nature of the projectile.], which [type of change] contradicts the first principle of mechanicism [i.e. contradicts the philosophical position that all change can be reduced to motion of material particles], which demands intrinsic unchangeableness. We find a quality which may become stronger or weaker, which contradicts the second principle [of mechanicism] which only admits geometric elements [geometric elements, that are only geometric elements, cannot become stronger or weaker]. So already in the first general principle of motion [the principle of inertia] itself we find contradiction with mechanicism. The mechanical view of Nature carried the seed of its demise already with it right from the beginning.
So for an explanation of intrinsic qualitative changes necessarily presupposed by the principle of inertia, we must apply the Aristotelian principles of changeability, potency, and act. Experience teaches us the actuality of the principle of inertia and in this way provides the data for specification of the general Aristotelian principles. We have found the first intensive quality, of which we know one effect, namely motion.

Heat as quality.
Sense experience shows us "heat" to be a quality, although it tells us nothing about the specific nature of it. Physics, has, as we already pointed to, teached us that heat is a regime of unordered motions, be it of molecules in a gas or liquid, or of material particles in a solid body. And to the average energy of these particles does correspond the temperature [of the gas, liquid, or body of which they are constituent particles]. Does this contradict that what immediate experience teaches us, namely that there is a quality "heat"? From the above it follows that there is complete agreement. The motions, each for themselves, bring with them their respective impetus [i.e. each particle has its impetus], which is an active and qualitative principle. As a result of this quality, not as a result of the motion itself, which, afterall, is just a passive "becoming", the moving particles can actively influence anything, especially also the sense organ. So indeed in "heat" there resides a quality which can be perceived by the sense organ in certain conditions (those of unordered microscopic motions). [We [JB], however, believe that in this perception an extensive effect of this quality is involved. In fact, the (extensive effect of the) quality is "measured" by the sense organ.]. The fact that the sense organ lets us be aware of only the generic, not the specific, nature of this quality does not need to be concluded from the doctrine of physics. Already analysis, or "Wesensschau", of what we perceive in "heat" teaches us this :  We cannot read off from it any specific propositions (judgements) concerning the properties of heat, but only generic ones concerning the properties of heat insofar as it is an intensive quality, propositions of which we had made use above. The fact that what is perceived turns out to be merely an average is a specification, the knowledge of which is provided by physics. [Here it is clear that one single particle may have an impetus of a certain degree (i.e. with a certain intensity), but not a temperature. We cannot say that the particle is "hot" when the particle is elementary in the sense of not containing a great many free-moving constituents. Also a macroscopic body in motion, cannot, in virtue of that motion and thus in virtue of its corresponding impetus, make us perceive a coressponding degree of heat. The quality "heat" is a collective quality, and only as such it is, in the present case, a true a quality. Although the impetus of every particle is a quality, it is not the quality "heat". So the only proof, as given by HOENEN, that "heat" (of a body) is a true quality (despite it being connected with motion and speed of constituent particles, which [motion and speed] are extensive magnitudes, and thus quantities) is the fact that it can be reduced to the impetus which itself is a quality [but, a different quality]. While the impetus may be the quality of some given substance (in the metaphysical sense), it can also be the quality of an aggregate of substances (such as a stone). But it is not simply the arithmetical sum of the impetuses of all constituent substances (a body does not move faster because it contains more constituent macroscopic or microscopic particles). The impetus is a true quality, and therefore an intensive magnitude, not allowing simple addition.

Sound as quality.

Something similar holds for "sound", which physics also tries to interpret as in fact [representing] motion, - here, it is true, not as unordered motion (if one has not to do with a mere noise [which is a mixture of all sorts of regular vibrations] ),  but, on the contrary, as a very regular one, which may be analyzed into simple vibrations. The simple tones are simple vibrations, and thus motions of a very determined nature. Now we can, in motions, just as in lines and surfaces, distinguish yet a special quality, which in geometric entites is called [geometric]  figure.

["Geometric figure" being a quality :  In this case (in contrast to the figure of vibrations) it then is not an  i n t e n s i v e  quality but an 'extensive quality'. This, however, seems to be a contradiction, because quality, in contrast to quantity, is an intrinsic, not an extrinsic property. So "geometric figure" belongs to the category of quantity. A geometric triangle, for instance, may be described and defined quantitatively by having the triangle located in a coordinate system, i.e. letting it be relative to such a system. The quantitative nature of the triangle then consists in the counting of units of distance from the origin of the coordinate system. But now we have smuggled in something extrinsic, extrinsic with respect to the triangle. How things should be in a genuine intrinsic description and definition of a triangle? Such a definition might be :  a triangle is a closed curve consisting of three straight lines. But such a definition appears to be wholly quantitative, except perhaps for the notion "closed".  Anyway, the categorical status of "geometric figure" is not completely clear.]

This quality [the geometric figure] we may, in the case of motions (which have extension in their own way), analogously call "figure of motion" [which is the trajectory in space of the entity which moves, if pure local motion is considered. If motion is depicted as a formal dependence of place upon time, then we do not get the trajectory in space, but a geometric figure expressing this dependency]. So a sinusoid as the graphic depiction of a simple [i.e. not compound] vibration [of something, and here not the depiction of its trajectory, because time is co-determining this graphic depiction] may rightly be called a "figure", but notice that "the figure of this motion" is not itself a sinusoide but consists in a period of changes of speed [along its trajectory]  [So here "figure of motion" is equated with the internally differentiated trajectory, differentiated as to its differents speeds in different sections of it. The sinusoide, on the other hand, graphically depicts the relationship between intensity and time, i.e. the formal dependence of intensity upon time.]. Now, such a form of period [i.e. the sinusoid linking amplitude (place) with time] we see not only in the motion [of the air-particle], but also (and even primarily so, as this follows from our expositions above) in the variation of the impetus, the active quality of the vibrating particle [This same sinusoid, but now that of the impetus, depicts with its amplitude, not place, but intensity of the impetus, and displays this intensity as it is formally dependent on time.]. So we must also have a very determined "figure" in the variation of this quality [the impetus], if we want to have a simple tone. This "figure" may be different in different cases :  Firstly, as a result of a different frequency, corresponding to the different pitches of tone [musical note], and secondly, with the same pitch, [different as a result] of different amplitudes, corresponding to different intensities ["volumes" of the tone]. So if our ear lets us perceive "sound" as a quality, in which we perceive pitch and volume as differences (without that perception teaching us in what these differences consist), then our ear indeed reveals what there, in this case, actually is in Nature :  a quality, the impetus, further determined by various qualitative properties - "figures",  a quality that moreover may have different intensity. And again, we know that our ear only reveals these  generic  properties, namely that there is a qualitative difference between tones of different pitches, and that in a same tone [musical note] different intensities [volumes] may occur. That the ear does not reveal to us the  specific  nature of tones as vibrations, physics does not have to tell us. Analysis of what we come to know by merely "hearing of tones alone" gives us already the same [generic] result. [As to sound, different musical notes, such as B and C, may be interpreted as different qualities (like we do with colors), but in fact they only differ on the basis of different formal dependencies of intensity upon time in one and the same quality, the impetus of air-particles.]

Naive idealism.
There exist certain naive forms of idealism which take their view from the position that physics teaches us that heat and sound are nothing else than motions, while our sense preception makes us think them to be qualities. So this perception, so it is said, is not reliable. It will then also not be reliable where it makes us assume extension in Nature. So a non-extensional outside-world remains. These forms of idealism we call naive because they are, apparently, not critical as to their own position. Detailed analysis lacks there in two respects.  from the side of the subject,  because it isn't checked what precisely it is what we think to know by attending sense data, and that [what we know from these data] is only the  generic  aspect in those qualities.  From the side of Nature  detailed analysis is lacking in naive idealism, because one assumes that in those "motions" [as they are found in physics] there do not reside qualities.
But also defenders of qualities are not always off the hook here. For they often argue such as if our mind would be able to read off that specific nature of qualities from sense experience of them, like it can do in the case of extension. They neglect the fact that the insight that our mind extracts from qualities given by sense-perception is, it is true, a genuine insight, but only a generic one. And in setting up a theory of knowledge one should not neglect this fact.

Collision as cause of change of motion.
To understand what precisely happens in collisions of bodies must be of fundamental importance to mechanicism. Certainly, study of collision in every system is needed, but strict mechanicism takes collision as the only and unique case in which bodies change each other's motion and exert forces on one another. And this strict system is in fact the only one that asserts to remain loyal to the principle not to assume anything in Nature that lacks the clarity of geometric relationships. In it, one did consider every attractive force as to be something that hasn't that clarity for our reason (and, indeed, the "how" of attraction partly escapes our understanding), and therefore the notion of true attractive force was rejected. Above, we already heard the reproaches of Descartes to Roberval. And Leibniz writes, even after Newton's discoveries, no different :

"comme nous soutenont que cela (the attraction) ne peut arriveé que d'une manière explicable, c'est-à-dire par une impulsion des corps plus subtiles, nous ne pouvons point admettre que l'attraction est une propriété primitive essentielle à la matière, comme ces Messieurs (the adherents of Newton) le prétendent."

Also repulsive forces, other than those in collisions, were rejected by the same reason. But in collision, as one held, everything was clear. Experience shows us the impenetrability of bodies, but this necessarily derived from extension [so one said]. If two parts of the same extensum are necessarily "outside each other", then (so one thought) even more so two bodies. But these cases are very different. In order to have impenetrability, a necessary "outside one another", of two bodies, extension certainly is needed, but not sufficient. In addition to extension an active principle is necessary, because extension is by itself not active. But because any two bodies were seen to be impenetrable to each other, one saw it as self-evident that two bodies, of which one collides to the other, and can, therefore, not proceed its motion any further without help, have to undergo, by reason of their extension, a change of their motion. Another cause of change of motion was not clear to the intellect and thus non-existent.

Is there perhaps still something else, a daily experience, an individual experience of human beings, that lies at the base of the assumption that only collision, only "push", may bring a body in motion, or influence its motion? We mean this :  We can, with our limbs, influence the motion of other bodies only by "pushing" (also "pulling", toward us, is in fact "pulling" at the other side (not facing us) of the body). Is this the origin of the collision-only theory of strict mechanicism? If it is, then the severe critique of the famous positivistic nineteenth century critic of the explicative physical theories J.B. Stallo is right. It says of the collision-only theory the following :  "What does the demand of the atomo-mechanical theory, to admit no interaction between bodies other than that of impact, imply? Nothing less than this, that the first rudimentary and unreasoned impressions of the untutored savage shall stand forever as the basis of all possible science." J.B. Stallo, The concepts and theories of modern physics, 2e ed. London, 1882, p. 181.

Descartes then attempted to derive the laws controlling collision, and he did this, according to the demands of his system,  a priori,  i.e. by thinking [about it] only. His general law affirms the principle of the conservation of the amount of motion (as that was conceived by him. He had not yet any idea of a vector [a magnitude with direction]. And also this principle he derived a priori, from the unchangeableness of God). Then he implicitly assumes that the true collision, the [mutual] hitting of the two bodies, is over in an indivisible fraction of a moment, and attempts to derive, solely from the constellation [as it was] before the collision, how, in different cases the motion of the two bodies will be. He then deduces seven rules for different cases, and the tragic fate of the philosopher of the  idée claire et distincte was :  His seven rules are largely false contradicting experience. The true laws of collision have been discovered by Christiaan Huygens.

Elasticity.
Descartes' main error is contained in the fact that he takes into consideration only the conditions as they were before and after the collision, while not attempting to analyse what exactly happens during the collision itself. Well, this analysis gives a result that contradicts mechanicism. One can find this analysis in every textbook of mechanics although no mention is made there about the catastrophic consequence of it for the mechanical view of Nature. It is sufficient to consider the simple case of central collision of two elastic balls [One, ball A, moving, the other, B, at rest]. At the moment of encounter, of the hitting, the one ball, A, has its full speed with respect to the second, B, resulting in a collision. This speed cannot instantly disappear (which was what Descartes in fact assumed), resulting in the fact that A is, after this moment, still moving into the direction of the midpoint of  B. Given the mutual impenetrability, a deformation must take place in both balls, a deformation increasing up to a maximum ["Deformation" here means the rearrangement of the constituent particles of the body that is being deformed]. This deformation generates elastic forces in both balls [trying to restore the deformation], acting on the balls in opposite directions, forces that decelerate A and accelerate B into the direction of the first motion [like we see it in the game of billiards]. The maximum of deformation (and also of the corresponding elastic forces) is reached when A and B have attained equal speed [equal speed, in the same direction, because then they do not press onto one another anymore, resulting in the subsequent decrease of the deformation]. Then the elastic forces still continue to act in the same way, they slow down A and speed up B, resulting in the decrease of the deformation (and the magnitudes of the eleastic forces themselves), until the balls, at the moment when they have regained their original shape, do not influence one another anymore, and then continue their motion with the speed they then have. [The restoring-force generated by the deformation of ball B slows down the movement of ball A toward the center of ball B, because the restoration of the shape of ball B works into the direction of where ball A is. The restoring-force generated by the deformation of ball A initiates and acceleates the motion of ball B (into the same direction of that of A), because the restoration of the shape of ball A works into the direction of where ball B is.]. What precisely happens in special cases can nowadays be easily computed from the laws of momentum and energy.
Above we assumed the balls to be perfectly elastic. If this assumption is not satisfied, it will not harm the generality of that what follows. Moreover :  In the case of imperfect elasticity heat is generated consisting in the motion of small particles. This, of course, will not hold anymore in the case of the collision of the  smallest  particles [i.e. simple, uncomposed particles]. There, the law of the conservation of energy demands elastic collision [Here the lost energy of the slowed-down particle (coliding with another particle) cannot be transformed into heat, i.e. cannot be dissipated among the constituents of the particle, because it by definition doesn't have any. So the lost energy must be found back in the elastic forces. And thus at least in the case of the smallest particles (the uncomposed particles) we have perfectly elastic collision.]
All this [i.e. the collision] takes place in a very short time, but still not in an indivisible moment. And it all stands in contradiction to the principles of mechanicism [because deformation is found even in uncomposed particles, meaning that this deformation cannot be the result of a rearrangement of the constituents of these particles, it must be a qualitative change.]. In this classic analysis we encounter forces [i.e. the restoring-forces in elastic deformation] not consisting of motion, increasing and decreasing in intensity, which, accordingly, represent a true intensive quality. As to this quality the bodies are intrinsically changeable, opposing the basic principle of mechanicism. This system of thought, accordingly, cannot produce an explanation of the collision. For this we'll have to assume Aristotelian principles, even in the case of collision of bodies, which is in the strictly mechanistic view of Nature the only means to effect change of motion.

Remark.
One might bring forward an objection :  The repulsive forces of the collision may be of an electric nature. In that case no deformation of colliding particles has to be assumed -- at least not with certainty. And then there will be even no contact between the particles. They already repell one another at a distance, by means of their electric fields. This will indeed be often the case, but it doesn't save mechanicism, because then we have an example of the third case, that, as we will see, also contradicts the principles of this system. By the way, this remark cannot not refer to the collision of neutrons [electrically neutral subatomic particles], where we have to suppose something that corresponds to the above analysis of the collision of macroscopic bodies. And especially :  In strict mechanicism collision is the only first-given factor influencing the change of speed. There our consideration is strictly valid :  Precisely there [in the collision of elastic bodies] not all qualitative change is expelled, but merely overlooked.

Fluida.
The doctrine of electricity and magnetism always has resisted to be fully incorporated into strict mechanicism, if one at least does not consider the attempts like those of Descartes himself. But such attempts should rather be viewed as a (unfulfilled and unfulfillable) promise to explain the phenomena than as themselves being such explanations. So apart from such constructions, one has always been forced to introduce a non-geometric-qualitative element, but one still did so always according to the demands of general mechanicism, i.e. one stuck to motion as the one and only possible  change  in Nature. And, accordingly, these qualities were attributed to  fluida,  a kind of imponderable substances, carriers of these  unchangeable  qualities. Above we had already mentioned such a fluidum, the heat-stuff, that should explain, i.e. reduce to some cause, insofar possible, thermic phenomena. Heating of ponderable bodies then consisted in supply of heat-stuff. And so there was, in line with the demands of general mechanicism, no question of intrinsic change in bodies. There only was a relocation of unchangeable particles, in casu the heat-stuff. But this was not strict mechanicism, because something qualitative was assumed that was not a mere geometric quality, not a mere figure. [Strict mechanicism acknowledges as the only true qualities the geometric shapes (figures) of unchangeable particles. Their shape thus is, ex hypothesi, unchangeable. These particles may arrange into all kinds of configurations as the result of their motions (changes of position). As to the supposed particles of the heat-stuff, assumed in general mechanicism :  alongside their geometric shape these particles possessed an extra quality which was responsible for heat :  More or less heat as a result of more or less of such particles.]
And in the same way one had introduced fluida also for the cases of electricity and magnetism, in which one did not agree how much [different] fluida were needed [to explain these phenomena].

Remark [JB] :
The chemical bond  (see  First Part of Website, "The Chemical Bond" ),  playing such a huge role in the constitution of material things, entirely lies within the context of electromagnetism, and, consequently, within the context of electromagnetic fields, now to be discussed.

In electricity talk about fluida included positive and negative, glass- and resin-electricity. Are they two fluida, or is the one only the absence of the other?
The heat-stuff has automatically vanished without philosophical critique as a result of the progress of science :  Magnetism is reduced to electrical factors. But the electric fluida have been preserved up to the present day in atomistic form, as electrons and positrons. A conductor is (negatively) charged if it carries a surplus of electrons. And so the mechanical view of Nature, at least insofar as it excluded intrinsic changes, seems to be confirmed here :  a higher "intensity" of negative electrical charge appears to exist merely by the presence of a larger amount of electrons, who before were elsewhere.
Nevertheless, this confirmation of mechanicism is merely appearance. Also electric phenomena bring with them intrinsic changeability in Nature. [So alongside the fact that a net charge of a given body is caused by the difference in numbers of charged elementary particles (which are, as to their charge, unchangeable), in (other) electric phenomena there can also take place intrinsic changes, namely those of the aether (field theory).]. This turned out to be the case when one learned to know better the behavior of the electric elements, the electrons and positrons. Coulomb was the first who investigated the attraction and repulsion of electrically charged bodies more precisely. Initially one interpreted this behavior as an "actio in distans", an influence that immediately, without actively involving the medium, or undergoung its influence, acts on the body-to-move (which for Descartes would be anathema). All these things took place according to the principles of general mechanicism in this sense :  That what is being caused is motion, and only motion, be it of "electricity" (the imponderable), be it of a ponderable body. There was no question of any intrinsically qualitative change.

"On account of the difficulties into which they [the mechanistic explanations of the states of the aether] lead us, there has of late years been a tendency to avoid them altogether and to establish the theory on a few assumptions of a more general nature. The first of these is, that in an electric field there is a certain state of things which gives rise to a force acting on an electrified body and which [field] may therefore be symbolically represented by the force acting on such a body per unit of charge [and these forces may be different in different regions of the field]. This is what we call the  electric force,  the symbol for a state in the medium about whose nature we shall not venture any further statement. The second assumption relates to a magnetic field. Without thinking of those hidden rotations [I, JB, take them to be the small circular currents] of which I have just spoken, we can define this [field] by the so-called  magnetic force,  i.e. the force acting on a pole of unit strength."

So in the field there is a certain condition, "a certain state of things", giving rise to the sharply defined "electric force". A further specification of that state is not presented ("we shall not venture any further statement"). And something similar is found in the magnetic field. This "state of the medium", as it is described, is  a  true quality.  It can increase and decrease in intensity, and, as a result, is the cause of stronger and weaker electric, respectively magnetic, force.
Lorentz then continues :

"After having introduced these two fundamental quantities, we try to express their mutual connexions by a set of equations which are then to be applied to a large variety of phenomena. The mathematical relations have thus come to take a very prominent place, so that Hertz even went so far as to say that, after all, the theory of Maxwell is best defined as the system of Maxwell's equations."

So the equations of Maxwell are, as a hypothesis, presupposed, to express the connection that there is between the changes of intensity of both forces [and with this describing the nature of light and all other electromagnetic phenomena]. A hypothesis that was then beautifully successful.

Field qualities.
So what physics in its latest development concerning electromagnetic fields gives us, is a further (not yet the last) specification, integrated into theory as a result of the rich and subtle experience, of the general Aristotelian principles of quality and intrinsic(ally)-qualitative change. After the downfall of the mechanical view of Nature in these things, science, automatically and without knowing it, has arrived at a specification of Aristotelian general principles. Carrier of this quality and subject of these changes is the aether. Further specification consists in the action and behavior of these qualitative states on experimentally known objects, electrified bodies and magnets (or paramagnetic and diamagnetic materials), and in the connections that are laid by Maxwell's equations between the changes of these states.
The fact that Lorentz speaks of "these two fundamental quantities" should not confuse us :  These quantities are the electric and magnetic force, true intensities, which, as we will see further down, also in aristotelism may and should be called "quantities in a broader sense". [Quantity in the strict sense coincides with extension and that what results from it by division, categoric multitude. Quantity in a broad sense is nothing else than the intensity-scale of a quality, i.e. the dimension of such a quality, making possible its determination as to how strong or weak. We may assume that the intensity-scale is in itself an intensive one-dimensional space, that may be mapped onto a corresponding extensive one-dimensional space being the scale measuring the quality's extensive effect. We may also assume that the intensity-scale, being a dimension, is not a space but just an ontological condition (a hartmannian category) in order for a quality to vary its strength.]

Vectors
In passing, we make yet a remark or two. These two forces, intensive factors, are no scalar magnitudes, but "vectors", they have a direction. In our case, at least for the electric force, this direction, and thus the vectorial nature of the qualitative, certainly is a secondary aspect, something resulting. The primarily-qualitative will certainly be a scalar magnitude [a magnitude wholly expressible by numbers], thus without direction, for instance the [electric] potential. The electric force, and thus the vectorial magnitude, then results from the  difference  of electrical potential, implying by itself, of course, a direction [the direction over which the difference spans, whereby difference merely as difference is just a scalar magnitude]. Are primary intensities possible which are of themselves vectorial magnitudes? As things now stand, we cannot give an answer to this question. We cannot demonstrate this possibility, but the impossibility we cannot see either.

Light and colors as qualities.
Maxwell and Hertz also have demonstrated that there is a connection between electromagnetic fields and light phenomena. Like tones consist of periodic motions, mechanical vibrations [of air particles], light consists in periodic changes of the electromagnetic field. Difference in frequency of these periodic changes corresponds to differences in color, as in sound in pitch. The fact that light consists in undulations, analogous to sound waves, was already known to Fresnel. It was demonstrated with diffraction, interference, and polarization phenomena. But Fresnel, still, in line with the presuppositions in classical physics, had taken those changes to be vibrations in the ordinary sense of the word, i.e. as periodic  motions,  displacements of aether particles, supposing an elastic medium :  It was the theory of the mechanical aether. The great success of the theory in explaining these interference and polarization phenomena meant an irrefutable demonstration of the fact that it must at least contain true elements. But there were difficulties speaking against this theory :  The light vibrations were not longitudinal but transversal, as followed from the polarization phenomena. Therefore, the substrate must be something like solid matter, for in liquids or gasses transversal vibrations do not occur. It had to be even a solid matter with an enormous coefficient of elasticity to warrant the high speed of propagation of light, a coefficient of elasticity much greater than that of steel for instance. On the other hand the substrate had to be like a very rarefied gas, such as not causing any detectable friction in the motion of the planets. Although this is not an absurdity, it was hard to apprehend. In addition to all this, the theory of the mechanical aether of Fresnel appeared to bring with it unsurmountable difficulties in the explanation of reflection and refraction of light. [The reader must realize that the "medium" which is discussed here is not the medium through which light travels (for instance air, or empty space for that matter), but the specific carrier of the electromagnetic vibrations, i.e. that what in fact vibrates (either extensionally or intensionally.]. All these difficulties vanished in the theory of Maxwell, making it possible, to introduce, instead of a mechanical vibration, an electromagnetic change of state.
And so the mechanical explanation of natural phenomena is pushed back also in the theory of light.  Light is not a periodic  change of place,  but a periodic  change of intensity  of electric and magnetic qualities [and the (ontological) substrate of these qualities is the aether of Lorentz. These qualities determine the  s t a t e  in which the aether finds itself.]. Furthermore, also here we must assume what we already saw in the consideration of sound :  Also here there is a "figure" or "shape" in such a period. A simple [i.e. elementary] color of the spectrum corrresponds to a periodic change which, we may, in a broad sense, call a simple [not-composed] vibration. The unity which this "figure" or "shape" brings into the period is in light even more accentuated. Light, after all, appears as  photons,  well-characterized units of every color from the spectrum. Yet another pecularity in all this may be mentioned :  Precisely what photon is created when an amount of energy is to be radiated, seems to depend only on the measure of this energy. Indeed, the frequency is equal to the energy divided by the constant of Planck. Here certainly there is still enough for further philosophical enquiry. It is also clear how far we now stand from mechanicism.

This fourth case, in which we investigate mechanicism, stands in connection with a second fundamental property of matter finding its expression in the factor "mass" or "gravitational mass". This case has, in analysis and result, much resemblance with the third case, that of electric properties, expressions of the first fundamental quality of matter.

Theory of Lesage.
Newton discovered the general attractive force of (ponderable) matter, and the law controlling it. We already above heard how Leibniz reproached the newtonians for taking the attractive force as a "propriété primitive essentielle à la matière" and how the german philosopher demanded a strictly mechanistic explanation of this attractive force, instead of seeing this force as a primitive property, a quality - he also spoke of a "qualité occulte". A strictly mechanistic explanation of the attractive force, he demanded, "par une impulsion des corps plus subtiles". One indeed has attempted to work out such [mechanistic] theories in order to avoid this quality. A certain degree of fame at the time received the theory, proposed by Lesage. He took the Universe as filled with swarms of very small particles flying around with high speed into all directions. If there is, in such an environment, a large body, for example the Earth, then countless particles will collide with it constantly and transfer to it impulses. However, no motion of the Earth will result from it, because the resultant of the enormous number of forces acting on the Earth will on average be zero. They cancel one another, because as a result of the large number of collisions the averages of each pair of opposite directions will be equal. All this will change as soon as there is a second body (for instance the sun) present at a certain distance from the first body. For then, this second body acts like a screen that stops the particles, encountering this screen, from moving further. And now the equilibrium is broken. A resultant force acting on the first body, the Earth, remains, that is a force acting into the direction of the second, the sun (and, the other way around, a force acting on the sun into the direction of the Earth). It is  as if  the sun attracts the Earth [and, to a lesser extent, the Earth attracts the sun]. In reality, though, there is (ex hypothesi) no attractive force, but only a  vis a tergo,  resulting from the collisions of the buzzing-around particles, as demanded by strict mechanicism.
Nevertheless, this success of the theory of Lesage is merely apparent. It collapses under the weight of precise mathematical analysis, like Darwin has demonstrated [not the Darwin of organic evolution, I presume]. If one, namely, supposes that the collisions are perfectly elastic, then the sun will not merely act as a screen to the Earth but also as a reflecting wall. As a result of reflection from the sun the particles will reach the Earth at its sunward side, particles who would have been flown further if the sun had not been there, flown further, that is, without reaching the Earth. And the exact calculation tells us that this increase of the number of collisions neutralizes the decrease caused by the screen. So in the case of elastic collisions the resultant apparent attraction is zero (after the kinetic gas-theory this result could have been foreseen). If, on the other hand, the collisions are totally unelastic, then an apparent attractive force can result, but then such collisions will increase the internal energy of the Earth, and with a catastrophic result :  According to the calculations of Darwin, in order to accomplish the actual attractive force, so much heat would be generated in the Earth as a result of the supply of energy, that its temperature would increase with 10²⁶ degrees per second. The Earth would receive 10²⁰ as much heat as the sun normally transmits to it by radiation!  If, finally, the collisions are semi-elastic, then the energy of the [colliding] particles is not entirely converted into heat, but then also the resulting attraction becomes proportionally smaller [because the impact of these particles on the Earth does not entirely result in a force pushing the Earth towards the sun, and the sun, although semi-effective, still works as a reflection wall]. In order to obtain the same attraction [as actually observed] one must suppose the number of collisions larger again, and, according to Poincaré (1908) the result of Darwin remains the same. By all this the theory of Lesage is refuted.

Gravity as elementary property.
So with the old newtonians one takes general gravitation to be a "propriété primitive essentielle à la matière", at least of those elements which are called neutrons. One had essentially done this already before Lesage's theory was refuted, except for the fact that also great scientists had attempted to derive the gravitational force from electrical forces, with which mechanicism is of course not helped. Initially this thesis still corresponded to the wider mechanical view of Nature, insofar as it claimed no changeability in this quality, precisely as we saw it with the electric charge [dependent only on the number of electrically charged particles with unit charge]. Indeed, mass -- thus the gravitational force exerted by a body -- appeared to be intrinsically unchangeable. A given body only obtains a larger mass when another body with its mass is added to it  ( The fact that according to the data of the theory of relativity mass itself is changeable [it may turn into energy] we will neglect here. This changeability is, if it is objective, of course immediately contradicting mechanicism). With this, one ascribed, just as in the case of electrical influences, also to gravity an immediate action at a distance. If this is true, the sun would immediately attract the Earth, without changing a medium, without needing time of propagation. Then the only effect of the force of gravity would be motion, then gravity would not demand a new intrinsic change.

Gravitational field.
But also this position did not hold out. After one had given up  actio in distans  for electromagnetic influences, and replaced it by the generation of a field, a field that consists in real changes in a real medium, the aether of Lorentz, one finally has done the same thing also with respect to gravitation, to which one attributes even the same speed of propagation as it is in the influence of electromagnetic fields, the speed of light. Definitive all this became not until the general theory of relativity, but before it these ideas were already held anyway. That a real gravitational field does exist in the light-aether follows from the influence it has on the direction of light-rays, which was, it is true, first derived from the theory of relativity, but also independently of it was found experimentally. So we find for gravity what we found for the electric field :  As a result of the "gravitational mass" of ponderable matter an intrinsic qualitative change in the aether is formed. Let us hear it from Lorentz, who also has described the qualitative electric and magnetic field so acutely. He says :

"Dans un champ gravifique il y a un changement d'état qui produit les effects qu'on attribue à la gravitation. Du reste, pour établir et appliquer la théorie, il n'est aucunement nécessaire de nous former une idée de la nature de ce changement".

We again have, as in electromagnetic fields, a change, which is known to us only generically, "un changement d'état". We know the effects, caused by this state, i.e. the motions given by the fields to the bodies. We do not need to further know the nature, i.e. the specific nature, of those changes (and in fact do not know them). Again, only the genus is known to us :  It is a quality, it may have greater or smaller intensity [strong field - weak field]. The intensity decreases with the square of the distance of the mass causing the field, i.e. according to Newton's law, or roughly according to it. So the final word of science in these matters is again a specification (not the last) of an Aristotelian general principle, the principle of the intrinsic changeability of matter as to qualities. And again mechanicism, the theory of intrinsic unchangeableness, fails to account for the facts.

The notion of mass.
In the theory of gravitation we met the notion of "mass". What precisely the essence of mass is nobody knows. We also will not attempt to answer this question, but only venture to make a moderate attempt to approach it more or less. And also this we do not without hesitating.
Anyone who makes an attempt to apprehend the nature of "mass" must realize that it is a notion of the science of mechanics, which is connected with the motion of bodies, and with the force which is taken to be the cause of acceleration -- of the increase of the impetus as we will say.

Inertial and gravitational mass.
One will further have to take into account the fact that in mechanics there are two concepts of mass, which mass, however, as it turns out experimentally, but for the time being just experimentally, is perfectly proportional, and thus, when measured with the same units, perfecty equal. They are the "inertial" and the "gravitational" mass, which, from the finding of their being identical onwards having become the foundation of the general theory of relativity, attracted attention. They, however, are not always properly described. Soon we'll see a reason.
And see here a very correct description of the difference between both concepts, and of the experimental data, that so clearly tell us their equality such that formerly one often took the concepts to be unqualifiedly identical. The description is Einstein's (1914). Because of its correctness we may be excused to insert this long quotation :

" Wenn wir von der  Masse  eines Körpers reden, so verbinden wir mit diesem Wort zwei Definitionen, die logisch gansz unabhängig von einander sind. We verstehen unter der  Masse  einesteils die dem Körper zukommende Konstante, welche den Widerstand des Körpers gegen eine Beschleunigung desselben miszt ("träge Masse"), andernteils diejenige Konstante des Körpers welche für die Grösze der Kraft massgebend ist, welche der Körper in einem Schwerefelde erfährt ("schwere Masse"). Es ist  a priori  durchaus nicht selbstverständlich, das die träge Masse und die schwere Masse eines Körpers übereinstimmen müssen. Wir sind lediglich daran  gewöhnt,  deren übereinstimmung vorauszusetzen. Die Ueberzeugung von dieser Uebereinstimmung stammt von der Erfahrung, dasz die Beschleunigung, welche verschiedene Körper im Schwerefelde erfahren, unabhängig is von deren Material. Eötvös hat gezeigt, dasz die träge und der schwere Masse jedenfalls mit sehr grosser Präzision übereinstimmen, indem er durch Versuche mit der Drehwage eine Existenz von relativen Abweichungen beider Massen voneinander von der Gröszenordnung 10^-8 ausschlosz."

English translation :
" When we speak of the  mass  of a body, we have with this word two definitions in mind which logically are totally independent of one another. We mean by the  mass,  on the one hand, the constant of the body measuring the resistance of the body against its acceleration ("inertial mass"), on the other hand [we mean with mass] that constant of the body which is indicative of the magnitude of the force experienced by the body in a gravitational field ("gravitational mass"). It is  a priori  not self-evident that the inertial mass and the gravitational mass of a given body have to coincide. The belief in such an equality comes from the experience that the acceleration, experienced by different bodies in the field of gravitation, is independent of their material. Eötvös has demonstrated that the inertial and gravitational mass at least with very large precision do coincide, excluding, on the basis of experiments with the balance [Drehwage], the existence of relative deviations of both masses from each other of the order of 10^-8."

A larger "inertial mass" (which is the "resistance against acceleration") demands for getting the same acceleration a greater force than a smaller inertial mass does. If in the same gravitational field there are two unequal masses, then they, nevertheless, get the same acceleration while one of them must experience a greater force than does te other [it is heavier]. But the forces, experienced by both bodies in the same gravitational field are proportional to their "gravitational mass" -- [So a greater force is needed for the heavier body to give it the same acceleration. And thus the resistance against this same acceleration is proportional to the degree of heaviness of the body, and thus when the body's weight increases the body's inertia increases.] -- from which immediately follows the proportionality between the gravitational and inertial mass of a same body, and thus, when measured with the same units, the equality of both masses. This result is obtained from experience that shows us the equality of acceleration of all bodies in the same gravitational field [i.e. in a field with the same intensity, that is, in fields in which there is the same distribution of the same intensities.]. An attempt to apprehend the notion of "mass" will have to explain the equality of gravitational and inertial mass. [The gravitational and the inertial mass are, before any further interpretation of them, nothing else than certain constants belonging to the body.]

Passive and active mass.
The description of Einstein is correct, yet not complete. For in each of the two masses we can distinguish yet another aspect  ( The mixing up of these aspects with those mentioned by Einstein, sometimes may lead to confusion).
What Einstein describes is a  passive  property. The "resistance against acceleration" evidently is nothing else than a passive capacity to receive an amount of motion, or of kinetic energy, of impetus. It is a "passive resistance". It is like the resistance offered by a container with a large diameter against the rising of the water with which one wants to fill it [for the wider the container is, the slower the water-level in it will rise]. But also the above mentioned "gravitational mass" is something passive, namely proportional to the force which the body  undergoes  in a given gravitational field.
But both masses do also possess an  active  aspect, which wasn't described above. This is immediately clear in the case of the gravitational mass. It not only is subjected to the influence of the gravitational field, but it itself causes one. And these passive and active masses are, as experience shows, equal again. But also the inertial mass possesses an active aspect. A larger inertial mass with the same speed as that of a smaller one possesses more impulse, more kinetic energy, it can have greater effects. [In  F = ma  (Newton's second law)  F  is the force causing the acceleration  a  of a body with mass  m.  One might also say that F is the force needed to give the body with mass  m  an acceleration  a.  The acceleration  a  can be written as  d/dt v,  where  v  is the velocity of the body and  d/dt v  the rate of change of  v.  So we can write the formula  F = ma  as  F = m d/dt v,  or, equivalently,  F = d/dt mv,  where  mv  is the impulse (momentum) and  1/2mv²  the kinetic energy of the moving body.]. And again there is equality between active and passive "inertial mass".
But now having four masses that seem to be identical, the problem about the nature of mass has not become easier to solve. Nevertheless, see here an attempt to approach the solution, which we do, we again say so, not without hesitation.

Trial of the solution.
[The idea of the solution is :  Mass = degree of bondage to the aether. If this idea is correct, then from it the four masses should derive.]
Mass is something that connects with the mobility of bodies. And motion is primarily and directly :  motion with respect to the aether. In order to be able to occupy a place in the aether, it is, as follows from the contact-theory of place, sufficient that the thing-to-be-placed is an  extensum,  This is also sufficient for the thing to be able to now occupy this place and later occupy another [i.e. in oder for it to be able to move]. If nothing else would play a role in these matters, then at the most volume might influence mobility. But bodies are not merely extensa -- this has become clear enough against mechanicism -- but in addition do possess qualitative aspects. Thus, equal extensa not always have equal mobilities, [and not always] equal masses. Let us now suppose that mass  primarily  is something through which the body is "bonded" to that part of the localization-medium, the aether, which it mathematically touches. The notion "bonded" is, necessarily a bit vague, but sharp enough to be able to say that the "bond" may be stronger or weaker. It is something intensive, in the same object perhaps not changeable, but nevertheles intensive.
This "being bonded" immediately expresses itself as follows :  In order to be moved, the stronger bonded body demands a greater active influence, which experimentally boils down to the formula :  For getting equal acceleration it demands a greater force. In this way we find the passive aspect of the inertial mass.
And now the active aspect is easy to derive. It is not primarily intrinsic to the inertial mass, but only a consequence of its passive nature of capacity. In order to be moved, the inertial body must take up impetus, and only the latter - we return to it later -- is the  direct  cause of motion, not the cause of acceleration, but of speed, or rather of the motion itself. A body with a greater inertial mass has, in the case of equal speed, taken up a stronger impetus than a body with smaller inertial mass has done. From this its equal speed together with stronger "bondage" to a place in the aether is clear. Then the active aspect of the inertial mass is only the activity which the impetus can develop. It is not a direct property of the inertial mass, but still proportional to it. And then the equality of active and passive (as the latter is the inertial mass proper) inertial mass is not surprising anymore.
Does, from that "bondage to the aether" also the "gravitational mass" become clear? We think it does. As to the "passive gravitational mass", we may, without becoming too speculative, say the following :  Mass consists, according to our supposition, in a property which brings with it "bondage to the aether" :  A larger mass is, by "qualitative contact", in a stronger degree bonded to the aether, in a stronger degree under the influence of the aether -- because mass is primarily passive -- than a smaller mass. Then we are not surprised that the  larger  mass in an area of aether in which there is a gravitational field, also in a  stronger degree  experiences the influence of it [of the gravitational field], i.e. experiences a  stronger  force, than does a smaller mass in the same area, as a result of that  more  intensive qualitative contact [We must keep in mind that a gravitational field -- as also the electromagnetic fields -- are supposed to be conditions of the aether.]. This cannot, of course, strictly be derived, but it is nevertheless very probable. Further it is also probable that there is proportionality between "bondage to the aether" and that active influence of the gravitational field, as this proportionality existed between the inertial mass and that bondage. Then the experimental equality between both masses, the inertial and the gravitational, must result, and the reason now is that the mass is in both cases the same :  Something in the body by which the latter is bonded to the aether.
What, on the other hand, can we say of the  active  gravitational mass? This active mass is necessary for the gravitational field which in fact results [i.e. is caused by] from the presence of mass in the aether [In order to be able to cause anything, in this case the gravitational field, the mass must be active.]. But is it necessary for this to suppose activity, as effective cause of the gravitational field, in the gravitational body itself? We don't think so :  the same passivity found above may be sufficient. In this sense :  Let the "bondage to the aether", and thus the consequence of mass, be something purely passive with the mass itself. If the body is then immersed into the aether, the aether will seize upon this passivity and bond the body. But that activity of the aether may very well happen at the expense of the aether itself and its qualities. Exerting of that activity, namely, may result in a change in the aether itself, not needing a new effective cause, because the sufficient reason is already there. This change in the qualitative state of the aether may propagate itself, and there we have the gravitional field without  activity  of the gravitational body, only as a result of its passive capacity, its passive mass, which again  is  the same as the passive inertial mass. The activity comes from the aether. Remarkable! in a much more fundamental setting we find again, without looking for it, the ancient thesis of Aristoteles, that "place" has [not only a passive, but] an active capacity as well.
In our suppositions also the changeability of mass, the appearance of longitudinal and transversal mass, can, we will say, not be derived, but they will be less perplexing [because mass is taken by HOENEN to be the degree of "bondage to the aether" of the body having that mass. And this may be more or less intense, making mass to be an intensive quality that may  c h a n g e  as to its intensivity.]

Remark.
So mass will be a qualitative passive property of a body [and with this we have found yet another genuine quality], which [quality] is essentially related to the aether. The body, even the [free] atom, does not necessarily have equal mass in all of its equal parts. It may, with respect to this property, as in all qualities, be heterogeneous. It will not be too speculative to suppose that these different masses -- (and thus densities [A given mass may be added to another mass, resulting in the sum of these two masses, but not so in the case of densities, i.e. a body may obtain more mass by addition of other bodies to it (i.e. by accretion), but its mass may also increase when the body's density increases.] ) -- of a same body do not need to always be connected to the same parts of it. Then the displacement of two masses  in  a body with respect to each other -- not of two parts of the body [because mass is a quality] -- will be possible. [So, two such masses in one and the same body both form part of the total mass of that body, they are not necessarily masses of two particles in that body or, said differently, masses of two individual material parts of that body (in such a case these masses are qualities of just these parts or particles, and not of the body itself). The two masses in one and the same body may just be two regions of it having densities differing from their surroundings in that same body, and the locations of these different densities in the body may change from time to time.]. But because then these masses move also with respect to the aether, the laws of mechanics will apply to them. From which follows :  That what is in classical physics described as displacement -- for example vibrations -- of parts of a body with respect to one another, also of atoms and molecules in a body, may very well be interpreted as displacement of merely masses  within  the one continuous body [i.e. of mere spatial shifts of qualities of such a body, instead of local displacements of constituent parts or particles in that body. And also the thermic movements of particles in a (warm) body may be so interpreted]. Motions in a continuum may be such shifts. That this result may have important philosophical consequences, also with respect to living beings, is perfectly clear. For the time being this remark should be sufficient. [As to the density in different parts of a body :  The relation of (gravitational) mass with volume makes the result -- the density -- intensive (instead of extensive). The relation of mass with the aether (here only the definitional involvement of the aether) also turns mass into an intensive magnitude.]

[If it is really true that  mass  is a quality, a quality of the body that has this mass, then indeed "moving individual material parts" in such a given material body, for example an organism, may (without sacrificing any result as to the physics or chemistry of that given body) be interpreted as a quality of that body, a quality unevenly distributed in it, unevenly distributed that is, as to the quality's intensity which is different in different regions of the body, a distribution, finally, that may change from time to time. This interpretation of mass makes possible to take a given individual organism (or a free atom, molecule, or crystal, for that matter) to be a true substance (in the metaphysical sense), i.e. to be a single material entity not consisting of separate material parts or particles, because that would turn it into a mere aggregate of substances instead of one single substance. Such a substance then is a "continuum", especially a "heterogeneous continuum" as HOENEN calls it. Its heterogeneity is not the result of it consisting of individual material parts or particles, but of it having variable qualities. Later we will extensively deal with this "holism of substances" and the part played in it by the Implicate Order.]

Summary.

Mechanicism has tried to explain Nature with purely geometric elements, to which also, since Descartes, is reckoned motion. All intensive qualities were banned.

Among these geometric elements there is, of course, one which is itself qualitative :  the figure or external shape. But this is not an intensive quality. In fact the speed of a motion is a concept that also  points  to something qualitative, namely to a genuine intensive quality [the impetus]. But this was overlooked by mechanicism.

In motion itself, then, the principle of inertia was taken to be something that was in no need of further explanation, that was a prime datum. We saw that this isn't correct. And the analysis of the conditions, under which the principle -- which in fact is given to us in experience only -- is valid already earlier has led us to the discovery of the  impetus,  which has all the properties of an intensively changeable quality.
When does the speed of a motion change? Strict mechanicism accepts only one influence that may cause acceleration [or deceleration] of bodies :  the mutual collision of bodies. Broader mechanicism also accepts other forces, among which also attractive forces. And therefore it had to be prepared to accept qualities -- that attractive force is certainly something qualitative -- but thought that it could exclude  change  in qualities, i.e. intrinsic change. So all change in Nature was supposed ultimately to be motion. This is the general thesis of mechanicism, to which also the broader system adhered.
These forces, causes of changes of motion, we have investigated. First the collision, accepted in all systems. Also here, mechanicism thought it could do it all with two primary data, which should be taken as self-evident :  with the conservation of the amount of motion, and with the demand that two extensa must necessarily (in virtue of the extension alone) stay "outside each other" [The amount of motion being conserved, motion has to reverse direction because the extensa (the colliding bodies) cannot penetrate each other.]. However, analysis demonstrates that these data -- apart from their truth -- are not sufficient. Also here we undoubtedly encounter a change which is not motion, but which points to intensive-qualitative changeability, expressing itself with forces of elasticity, the continually increasing restoring force. Here, mechanicism failed to exclude all intrinsic changeableness. [Elasticity of a body 'of course' in turn derives from motion of (smaller) particles with respect to one another inside that body. But between the two members of every pair of such particles (particles colliding with each other) there is still a restoring-force and this force is of an electrical nature (which is qualitative).].
The broader variant of mechanicism in addition introduced electromagnetic influences and gravitational force. And all this had led to "fields" which we have described. That what here, in the form of experimental data and theoretic results, as final result of analysis, and thus as first data for synthesis, was found, again has all the nature of intensively changeable qualities, does not consist of motion. If we here indeed have to do with first principles of Nature, then we find two fundamental, real, variable qualities. Are they once and for all last, and thus for synthesis first data? It is certainly out of the question that they can be reduced to pure motion, to mechanical principles. The probability that they may be reduced to still deeper qualitative principles seems to us low. And so the development of science has automatically called back qualities and qualitative changes, which had been banned by mechanicism in an a priori way and on the basis of incorrect arguments. And for us it has turned out that the crisis of the mechanical view of Nature, broken loose in the atomic theory of the 20st century, was there already long before, or, if one prefers, the elements of the crisis were there already, namely in the 19th century in the field theories (to which many physicists had already pointed to), and more, they were already present from the beginning of cartesian mechanicism, namely in the principle of inertia and the very collision itself.
Intrinsic change only finds its explanation in Aristotle's theory of  act  and  potency.  This then will have to be realized in Nature. But these general principles are not sufficient for a specific explanation of what takes place in Nature. So we need specifications of these general principles. And it is experience which will largely provide them. Aristotle's own experience and that of St Thomas and other medievals still was far from sufficient. No wonder that we cannot accept most of their specifications anymore. Thus their fundamental quality-pairs, warm-cold and dry-humid, having played an important role, must be rejected. Only the powerful mathematical method in experiment and theory (i.e. explanation with specified  near  causes of phenomena) was able to provide and process the necessary exact experience. These we then have to apply to specify the general Aristotelian principles of which we already know that they are found to be true (out of general experience and from the failure of the mechanical view of Nature). The specifications found in this way, lead to the following fundamental qualities :  impetus (cause of inertial motion), elastic forces, electromagnetic and gravitational qualities (electric charges and mass in ponderable bodies, field states in the aether). Are there in inorganic Nature still other fundamental qualities, or do they result from all those mentioned? [Today (end of 20th, beginning of 21st century) we in addition have discovered the so-called weak and strong interactional forces in atomic and subatomic dimensions as (to be) fundamental (or semi-fundamental) qualities. The book of HOENEN dates from before particle-physics, begun in the 1950's.]. A definite answer we cannot give. We expect that all other qualitative data may be reduced to these, unless impenetrability, which we learned to know as an active principle, refuses to be so reduced.
Finally, we may certainly repeat what we, in a study from 1928, already described :  Explicative physics will either be Aristotelian-Thomistic, or it will not. And then we may ask :  Wouldn't science have fared better when it, without having taken the detour through the failure of the mechanistic view of Nature, directly had applied its beautiful mathematical method -- according to us the credit of Descartes for most of it -- to specify the principles of Aristotle? The same conclusion we will find and the same question will automatically be asked in the last chapter where we will discuss atomic theory.

Mechanicism often is described as a system of thought that denies  activity  totally or partly in inorganic Nature. Partial denial, then, consists here in the fact that mechanists only accept causes that realize local motion, i.e. change of place, while holding that other activity is impossible. This splits them up into two groups. The first, adhering to strict mechanicism, limits this activity to collisions, while the second, the group of broader, more comprehensive, mechanicism, admits also other forces. This description of mechanicism is only partly correct. Sometimes one ascribes the  total  denial of all activity to Descartes. Also this is erroneous. Certainly, Descartes does not accept any activity other than that what develops during collision  as a consequence of the motion  of the colliding body. But this is for him a genuine active influence of that body, not only a passive transition of an amount of motion from this to the other body. The latter would mean the transition of an accident of a given substance to another, which also for the mind of Descartes would be absurd.

There certainly are cartesians, like Cordemoy, denying all activity in Nature.
The denial of active causes, not only in inorganic Nature, but also in all living creatures, is a different [different from mechanicism] philosophic theory, occasionalism.  According to this system of thought all what is happening is a direct consequence of the activity of God only, not of the creatures, lacking all activity. The name of this system derives from the following :  Where people usually assume that an event is to be ascribed to the active influence of a creature, mind or matter, there the presence of the alleged cause is nothing else than an occasion, which is used by God, to realize the effect only through his own activity. Such a system was already adhered to in [circles of] arabian philosophy, in the so-called Mutakallimun (in St Thomas Aquinas they were called "loquentes in lege Maurorum"). Later it was [further] developed by Geulincx and by Malebranche.

Certainly, Descartes does deny all other activity in inorganic Nature, and this is the only distinction between his system, strict mechanicism, and the system which has become the fundament of classical physics, broader mechanicism, which, as we saw, accepts also other forces. And this is the only difference between them. For all mechanicists agree that all forces of Nature can only produce motion, change of place. And moreover, they agree in the more fundamental thesis that matter cannot  experience  any other change than motion, change of place. This thesis is more fundamental, because, if it is true, also other agents, also a mind, can effect in matter nothing else than motion, and a true intrinsic unity (wholeness) between matter and mind, as we see it in human beings, and in general in all living creatures, becomes then impossible. Therefore, Descartes had to degrade the animals into automatons. More fundamentally [it is also], because this thesis is the  reason  why mechanicism has limited the activity of matter to activity of motion. Where there is no other "becoming" than "becoming moved" every other activity other than "to move" is superfluous. Mechanicism essentially consists in its denial of  passivity  of matter (except the possibility of being moved), while the denial or limitation of  activity  is only secondary.

Because now, as we've seen, matter is not only movable, but also intrinsic-qualitatively changeable, also the derived thesis of mechanicism, limiting activity to one or more moving forces, and excluding all other activity, is incorrect. In addition to moving forces there are also active causes, intrinsic-qualitatively changing the bodies by their influence. From the various types of change, found above, we can easily sort out what kinds of active causes correspond to them. In fact the result is remarkable.
Earlier we had found an active cause of  motion,  namely of inertial motion :  the  impetus.  And such that to a faster motion of a same mass does correspond a more intense impetus. Are there yet other causes of motion, i.e. activities  directly  causing motion, change of place? All other activities, influencing motion, "forces" in the technical-physical sense of the word, are causes of  acceleration (or deceleration), and thus of change of impetus, and so of change of quality. And  only by means of  the change of the impetus they can cause acceleration. Consequently, they are  no direct  causes of motion [causes of the initiation of motion, i.e. by giving impetus, and of the change of speed, by changing the impetus]. There is no place anymore for a direct influence of these forces upon motion, they only influence motion by means of qualitative change of impetus, which they directly cause [force-impetus-motion].
And even not all these forces directly influence the impetus. This may be so in the case of elastic forces, affecting the impetus through impenetrability, but the "gravitational mass" first causes a gravitational field, and only the tensions in this field are causes of the impetus in a body to be moved  [The "gravitational mass" of the Earth does not directly causes the fall of a body, but only through the mediation of the gravitational field (which is caused by the Earth) and then through the mediation of the impetus of that body :  gravitational mass ==> gravitational field (force) ==> impetus ==> fall of body]  [Here HOENEN speaks of "not all forces directly influence the impetus", but the example which he presents is about the "gravitational mass" not influencing directly the impetus. But the gravitational mass is not itself a force, it causes the gravitational field, and this field makes ponderable bodies feel a force which then causes or influences in them the impetus]. The same holds for positive and negative electrical charges, and for magnetic forces [So also here, through a field].
So we arrive at the conclusion, that the impetus -- which in mechanicism, and in fact in classical mechanics, was neglected, if not, denied, as active principle -- is in fact the only directly moving activity [the direct cause of motion and of its change]. The "forces" of mechanicism, and consequently those of classical mechanics, are activities, which directly do not effect motion, but only effect intrinsic-qualitative changes in Nature. This may rightfully be called a remarkable conclusion. But our analysis, we think, necessarily leads to this result.

The scholastics distinguished two groups of effective causes which were named :  causa in fieri  and  causa in esse,  what we here translate as :  cause of  "becoming" (cause of  "coming into being") and cause of  "being" (cause of  "to be", cause of  "existing", cause of  "sustained existence"). The meaning is this :  Some active causes have influence upon their effect only at its origin. After that, the effect is independent of the effective cause that has generated it. It keeps on existing without further effective influence from that cause. In this way, the newly generated living individual is independent of the effective cause which had generated it. Therefore, this cause was only a  causa in fieri.  But as it was in its generation also dependent on the effective influence of the First Cause, it remains so dependent on it in order to be able to keep on existing. This, then, is thus not only a  causa in fieri,  but at the same time a  causa in esse  as well. [Nowadays we would prefer in having this "First Cause" (God) being replaced by "life-sustaining conditions" prevailing on the Earth's surface, conditions which generate and sustain life. The new individual of some given organic species depended for its very generation on the parent organisms, but after that (apart from the highly sophisticated cases of parental care) it is, as to its further existing, independent of them. For sustained persistent existence it needs an appropriate environment.].
This distinction seems to hold for the activities which we had discovered. The forces, causing impetus, are, apparently, only  causae in fieri.  After cessation of their influence the impetus remains existing independently and unchanged.
The forces, first causing a field, be it electrostatic, be it magnetic, or be it gravitational, are genuine  causae in esse.  They are necessary to maintain the field, which disappears when these forces cease to work  [It is questionable whether a force may cause a field. It is the electric charge of a body, the magnetic poles of a body, or the gravitational mass of a body, that causes a field. And indeed these bodies are not merely causae in fieri,  but also causae in esse of the fields.]
The impetus is the cause of continuous motion. Now, motion is a "to be" of a very special sort, as we will further demonstrate later, namely a "to be" which consists in continuous "becoming". The impetus appears as the cause of this "to be" not only at the first moment of this remarkable "to be", but ongoing. Upon cessation of the impetus also the motion ceases, and so the impetus is the  causa in esse  of this "to be". A cause of a special nature, because it is "to be" of a special nature. [The force, applied to a body, is the causa in fieri of the impetus, while the impetus is the causa in esse of the motion of the body.]
These few indications may be sufficient in matters where so little is philosophized. Maybe further pondering will produce valuable results.

Earlier we spoke a few words about the history of the light-theory. It told us how, especially by the work of Fresnel, the phenomenon of light was taken to essentially consist of vibrations, more precisely, transversal vibrations of an elastic medium, the aether of Fresnel. The undulation theory of Fresnel was, in the course of years, confirmed by wonderful and convincing experiments. These experiments were carried out with high precision, and concerned diffraction, interference, and polarization phenomena. Complicated laws had first been theoretically derived before they were experimentally comfirmed. We only mention here the discovery of the totally unexpected phenomenon of conical refraction, in 1832, after Fresnel's death, calculated from the theory by the mathematician Hamilton, and thereafter, on Hamilton's request, experimentally demonstrated by Lloyd.
Nevertheless, there were dark spots in Fresnel's theory. Difficulties presented themselves, which, as we indicated earlier, in the end turned out to be unsurmountable, becoming the cause of the fact that towards the end of the 19th century the light-theory of Fresnel was replaced by the electromagnetic theory of Maxwell, further developed by Lorentz and completely cleared from mechanical elements. The aether of Fresnel gave way to that of Maxwell, and finally became the aether of Lorentz.

True and false elements.
We here have a striking example -- it is not the only one -- of a physical theory which, by its many and wonderful and precise results, is in such a degree confirmed that doubt as to its correctness seems to be out of the question, but nevertheless leading to unsurmountable difficulties, to contradiction with observation, so that it must be assessed as being false afterall. That we here have a problem is clear. A solution of this problem expressed in general terms is easy :  The theory must contain in addition to true, also false elements. The first account for the good results, the others are responsible for the demise of the theory. One should thus retain the first, the true, elements and reject the second group (the false elements), and, if possible replace by new elements. That this applies to Fresnel's theory is clear already by the fact that one could adopt a great many formulas of Fresnel from which observed phenomena could be deduced, adopt, that is, as to their  form  unchanged, into the electromagnetic theory of light.

It is, however, of philosophical importance to more precisely qualify the relationship between "true" elements that are taken over into the more correct theory, and "false" elements which have to be eliminated. We will do this in two stages.

First specification.
According to Fresnel, light consists of waves having resulted from "vibrations" in the ordinary mechanical sense of the word. In the Maxwell-Lorentz theory, light also consists in "vibrations", but in a somewhat other sense. They are not mechanical vibrations anymore, but electromagnetic vibrations. And in this way no "waves" in the usual sense (like we see them in water) do arise anymore, but electromagnetic undulations. Vibrations in the usual sense are periodic  motions (in the strict sense), i.e. periodic changes of  place  of the moving particles. In light, that change of place must be transversal, i.e. perpendicular to the direction of propagation of the light-ray. Here [in the mechanical aether of Fresnel] the vibrating particles are aether-particles. For this, the aether must be elastic (more precisely elastic as is a solid body, [because in gasses and liquids the waves are longitudinal or of a mixed type] ),  and then the difficulties arrive [The mechanical aether had to have a very high coefficient of elasticity (such as that of steel), but at the same time be very rarefied, because it does hardly or not at all cause any friction with bodies moving through it. These are the difficulties of the mechanical aether.]. In Maxwell's theory these electromagnetic vibrations are also periodic changes of state, in which along a light-ray and perpendicular to its direction periodically (with the periods appearing in Fresnel's formulae) the electric and magnetic forces in every point of the ray continually change continuously as to their direction and magnitude [i.e. change as to the direction and intensity of these forces]. Initially, one wanted, holding to the demands of the mechanical, anti-aristotelian view of Nature, to mechanically explain also these electromagnetic changes, i.e. to see in them hidden changes of place. But, as we saw already earlier, also this brand of mechanicism is, by Lorentz, removed from the theory. What remained was, as we heard it from Lorentz, this :  Along the light-ray, periodic  qualitative  changes take place, expressing themselves in periodic transversal changes of electic and magnetic force [themselves in turn expressions of field-strength].
So in Fresnel's theory we can distinguish and separate two elements, or groups of elements :  The first group includes the hypothesis that light consists in a change of  place [we do not here refer to light's propagation, but to its vibrations], while the second group includes the fact that this change is  periodic (and transversal). Well, all the wonderful confirmations of Fresnel's theory, which we summarize under the names diffraction, interference, and polarization, can be deduced from the second group of elements [of the theory]  alone,  in which it is assumed that in a light-ray there is one or another  periodic  change (to be sure, a transversal).  Exclusively  from this do follow all the wonderful results of Fresnel's theory. The assumption that these changes consist of motions, changes of place, is  completely superfluous  in deducing the results, has no influence whatsoever on the whole reasoning leading to  those particular  results. Lorentz (1922) describes, again with his usual acuteness, what is necessary and sufficient to account for the phenomena of interference :  "that at a same moment along every light-ray opposed states alternate in succession" and "at each place there must be a continuous alternation of mutually opposed states". There are changes and opposed  states.  That these changes of state are changes of place is totally irrelevant to the deduction of the interference phenomena. And it is precisely because of that, that  all those  formulas of Fresnel, mathematically expressing the periodic aspect in these changes, could be taken over by the electromagnetic theory unchanged, where the latter theory assumes changes of field-qualities, from wich the electric and magnetic forces originate. With all this, precisely all those particular elements of Fresnel's theory were automatically eliminated which had to be introduced to bmerely justify the mechanical element in the theory, and which had led to unsurmountable difficulties or contradictions.

Principle of elimination.
From these historical data we easily can read off a general principle valid for all explicative physical theories, a principle that in fact is just a simple demand of logic, although logicians keep silent about it. An explicative physical theory (explicative, in contrast to theories of the purely thermodynamic type) serves to explain phenomena summarized in experimental laws. In this way :  One erects a hypothesis about the nature of the causes of a series of phenomena. So, concerning to the phenomenon of light it was supposed that it might consist of transversal vibrations of aether-particles. Then, from this hypothesis, mathematical conclusions are derived, which are a description of phenomena which either have been already observed, or "predicted" by this derivation and then being observed. In this way, the undulation theory of Fresnel was able to derive an uncountable series of optical phenomena, known and new, and even completely unexpected phenomena (think of the conical refraction). That is the structure of every explicative physical theory. [And so, by the way, Newton's theory of gravitation explains the experimental laws (laws found by repeated observations) of Kepler concerning the trajectories of the moving planets.]
Is the hypothesis, lying at the basis of the derivation, proved by this? It is true that it is refuted when only already one of the conclusions to which it leads contradicts undoubted observation. Then the hypothesis is, at least partially, false and must be rejected or at least be amended. But if it does not run up against such a contradiction it still is not certain. As long as another hypothesis is possible, leading to the same conclusions, this other hypothesis retains it probability. This is sufficiently expounded by all logicians since Aristotle :  From the falsity of a conclusion (in a correct reasoning) the falsity of [at least one of] the premises follows (and on this the "proof from the absurd" is based). But from the truth of the conclusion the truth of the premises does not necessarily follow. The logicians also explain under what conditions such a premise -- thus in physics a hypothesis -- may gain probability and in the end even be accepted as a certain thesis. Then one may, with full legitimacy, call the hypothesis a "theory". All this is very well known, and we do not consider it further. For in the cases that we will investigate these conditions are, for everone who is rightly informed, sufficiently satisfied. Precisely so in the case of the theory of light, where nobody doubts the presence of undulations. But there is, as we said, still another logical demand, about which the logicians are silent, and it is precisely this one that is urged by the history of the theory of light. See here.
A physical hypothesis may consist of different elements or groups of elements, which usually are considered unseparated, while they are nonetheless well separable. So things are in the theory of Fresnel, the elements of which, according to the above, may be reduced to the following two :  In a light-ray there are periodic changes. And secondly :  these changes are changes of place. The one element in the theory is the periodic aspect, the other is the local nature of the changes. The marvellous confirmations of Fresnel's theory follow from the first element only. The second element, the local, does not influence these results at all. If, accordingly, the undulation-theory is in such a degree well established in virtue of the success of its results that one has to accept it as certain (and the same remains true if one would take this theory as to be merely probable in whatever degree), then this does not at all apply to the second element, the local. This hasn't gained any more probability at all from the confirmation of Fresnel's theory (later, it even turned out that this local element is unacceptable). For anybody who sees this, it is immediately evident that we here have to do with a general and purely logical principle, which we can formulate as follows :

If a physical theory consists of different elements (groups of elements) which can be separated from each other [i.e. elements that do not imply each other], and if the omission of one or more of these elements (groups) has no bearing on the conclusions which are confirmed by observation, then these elements (groups) do, globally taken, not gain any extra probability as a result of that confirmation of the theory.

Who grabs the meaning of this principle, immediately sees its obviousness. We will call it :  The principle of elimination of superfluous elements of a theory.  Or, shortly,  principle of elimination.  We now may also formulate it succinctly :  Superfluous elements of a theory do not gain probability as a result of any confirmation whatsoever of the theory.
Later we will make often use of it.
So we have now reached a first specification of the relationship between the true and the false elements, we now say :  between the necessary and superfluous elements of a theory like that of Fresnel. The first group is necessary. If one eliminates one of its elements, the deduction of the verified results is cancelled, totally or partially. The second group (that later was found to be false), upon appropriate critical separation of elements (or groups) and enquiry into their influence on the deduction, turns out  to be superfluous already from the very beginning.  Therefore it should have been eleminated right from the start if one had wanted to construct a strict logically-pure theory. And it would not surprise us when this principle finds its application also beyond physics and natural philosophy.

Neglect of the principle.
One will, considering the simplicity and clarity of this principle, ask the question :  Why, then, hasn't one applied this principle immediately to the theory of Fresnel, i.e. to its very construction? The answer is not difficult to give. It is a consequence of the fact that classical physics was set up on the presupposed fundament of the mechanical view of Nature. Indeed, according to this view every change in Nature is a change of place. Change or substitution of qualities is excluded. Now, Fresnel found in light a  periodic  change. And consequently it necessarily had to be a periodic change of place, a mechanical vibration.
But this forces us, first of all the philosopher of Nature, but also the physicist, to be cautious in accepting other physical theories. If classical physics a priori presupposes mechanicism as a fundamental explanation of all natural events, then that what we had found in the theory of Fresnel may also be present in other theories. Later we will see examples. Therefore, we must investigate such theories as to our principle of elimination.
How would the theory of light have looked like in the first half of the 19th century, if one had, right from the start, excluded the superfluous elements, and preserved those that had been confirmed by observatiom? In its initial stage it would have been like it was formulated by Lorentz  "(that) at a same moment opposed states alternate in succession along every light-ray" and "at every place there also must take place a continuing alternation of mutually opposed states". To which it should be added that vectors appear in these alternations being perpendicular to the direction of the light-ray's propagation. The theory should speak of  states  and changes of  states,  without determining whether these states and changes are [respectively] places and changes of place of vibrating particles, or whether they might be qualitative states and qualitative changes. Further development of the theory then had to determine whether the first or the second member of the alternative is demanded for further working out this [originally still] general theory. If classical physics would not have been bewitched by mechanical, and thus anti-aristotelian, preconceptions, it would have constructed the theory as described, and ... be spared from many disillusions and also from much useless trouble. By which we do not want to say that all effort spent for the working-out of the theory along mechanistic lines, was superfluous. On the contrary, the failure of the mechanistic attempts is necessary for reaching the positive result that in the phenomenon of light qualitative changes are found, and thus for the gaining of the insight that Nature can be explained along aristotelian lines, and not mechanistically.
But history also forces us to make yet another remark :  In modern mathematics one, since Weierstrasz, laid much stress on the logical "Sauberheit" [purity] in developments. Shouldn't the same be demanded of physics as a strict science? But then one should use, even already in elementary education, the principle of elimination. It isn't, afterall, a sound reasoning to say :  " The vibration-theory of Fresnel is sufficiently proved by the many and precise confirmations from experiments." So with the theory of light, so with other theories. It is of course very pedagogic to use this easy analogy, i.e. to use real local vibrations as illustration [i.e. to illustrate the real proces] [like one, in quantum mechanics, may say that an electron has "jumped" onto another "orbit" around the atomic nucleus.].
So if one, in the first half of the 19th century, had constructed the theory of light as described above :  " light is an undulation, but whether it is mechanical or qualitative undulation has not yet been decided upon", then the further development would have gradually resulted in the acceptance of the second member of the alternative (qualitative undulation). And that development also would then have identified these qualities as electric and magnetic ones, defined by experimental data, as we earlier heard from Lorentz.

Second specification.
We promised the reader yet a second specification of the relation between the two elements (or groups) of the theory of light. So let us consider them further. We have two theories of light. They have the following in common :  Light consists of undulations. They differ as to further determination. The one says :  They are waves of motion, the other says :  they are quality-waves. In this way we can formulate things succinctly. Now the element in which both differ from each other is more fundamental and more essential than the element which they commonly possess. The one [element] says :  All changes in Nature necessarily is motion (and it is attempted to prove it a priori). So if light is something periodic (an undulation), it necessarily is a periodic  motion.  The second, the aristotelian, says :  A change in Nature  may  also be qualitative (and a metaphysical justification is given). Were this the case in the theory of light, then it must be a  periodic  change of quality (a quality-undulation). The periodic aspect, discovered by experiment, thus is a further  specification  of general principles. If one adheres to the mechanical view of Nature, then the periodic nature, which light turned out to have on the basis of experiments, specifies the general mechanistic principle. An aristotelian  may  (but does not a priori  have to,  from which follows that Aristotelianism is a broader system of thought) consider this [periodic nature] as further specification of possible qualitative changes. Here we find, in a very clear example, confirmed what we have said earlier :  Philosophical consideration provides the general principles, but these general principles are not sufficient to also explain the more specific features. For this they themselves need to be specified, and observation and experiment -- preferably experience collected and processed by mathematical physics -- must provide that specification [In our case, changes of place or of quality]. That precisely is in our case -- and in others -- the relation of element-groups which we have learnt to separate above.
There is yet another pecularity to which we must turn our attention. An analogous specification -- and, viewed purely mathematically, even an identical one, for the formulae remain unchanged, despite the partly change of their meaning -- may be applied to two philosophically opposed systems :  the one, indeed, denies, the other proves the possibility of intrinsic-qualitative changes. We encounter this as a fact. Explanation is not difficult. Both systems, after all, commonly admit changeability in Nature. The one system [does admit] only a change of place, the other also a change of quality. And it is precisely this changeability [of place or of quality] that is further specified by the second element, the periodicity [and this is possible in the mechanistic as well as in the aristotelian system of thought.]. It is worth the trouble to remember this. Also in other places we will meet with specifications that can be applied to two opposing systems. In these cases the specification -- which we experimentally find and theoretically, i.e. by the method of the explicative hypothesis, process -- provides the  nearest  cause of the phenomena. Thus, the properties of light are explained by undulations as nearest cause. The deeper causes we will find when we apply this specification to a deeper, a philosophical principle. So for the phenomenon of light :  when we learn to know the undulations as to be qualitative (electromagnetic) changes of a field [or, when we learn to know the undulations as changes of place (application to the system of mechanicism), but this application turned out to run into great trouble.]. In the same way we will find all this confirmed in other theories, especially in atomic theory.

The electromagnetic theory of light, one of the most elegant results of physics, thus is a theory in the end having turned out to be built on an aristotelian fundament. It is a specification of aristotelian principles. And nevertheless we hear a man even of the calibre of  E. Meyerson repeatedly reporting about the sterility and the definitive elimination of Aristotle's philosophy in the physical sciences. [In HOENEN's text a french quotation follows]. What is actually true of this? Partly this :  that in the Middle Ages [during which the philosophy of Aristotle was dominant, and that means dominant for a long time] one has not succeeded to construct a system of theories, as modern physics possesses one [such system], or even "classical physics" -- the science of the mechanical view of nature -- had one. Anyway, this is only partially true. We will see that the Middle Ages were already nicely on the way to provide the basics of the atomic theory. And we should also realize the fact that the Middle Ages did make all the relevant preparations demanded by the right mathematical treatment of intensities. But, taken globally, the absense of a system of such theories is a historical fact. But what is the explanation of that fact? Is it because, as Meyerson asserts, the principles of Aristotle were inappropriate to be a fundament of the necessary specifications, while the mechanical view of Nature was so appropriate? The development of the theory of light not only contradicts this, it proves the opposite, and thus demonstrates that Aristotle's philosophy is not sterile. And the same conclusion derives from all data given in the above  summary.  So we must find another explanation [of the absence of genuine physical theories in the Middle Ages], and, after what we have expounded, this other explanation is evident. The medievals did not have at their disposal the necessary instrument to introduce those specifications, neither in experimentation nor in theory, i.e. in searching, what we have called above, the  nearest  causes of the phenomena. This necessary tool is the mathematical method in experiment and theory. When [this mathematical method was] found and perfected, physics -- after much struggling with mechanistic preconceptions, and without intending and even knowing it -- has become :  a specification of Aristotle's philosophy. And with this, the reader in fact has to do with an application of our principle of elimination in the investigation of a historical problem [i.e. that things that in essence really don't matter when looking for the causes of the fact that the medievals did not develop such theories, are eliminated, that is, things such as the adherence to one or another philosphical system of thought.]
And the result is :  Aristotle's philosophy is far from sterile as to the development of physics. On the contrary, it is the only fruitful one.

No fields of ruins of physical theories.
From this consideration we may draw a valuable conclusion. The mechanistic elements in these classical theories were superfluous and turn out to be false. Therefore, one speaks of the demise of the mechanical view of Nature. Now, this fact is sometimes described as if scrutinizing the history of physical theories was a walk through a field of ruins. Our principle of elimination, and all what is connected with it, demonstrates that this is false. True, the philosophical preconception of classical physics, mechanicism, turned out to be false. But the principle also shows us that the proper physical theories, describing the  nearest  causes of the phenomena, are correct anyway. The only error then is, that one unjustifiably took them to be specifications of the general mechanical principles, while in fact they can and must be taken as specifications of the naturalism of Aristotle. How this is possible is evident from our consideration of the principle of elimination. The same we will later encounter in a most important theory :  the atomic theory [= theory of atoms, chemical elements, and molecules]. So it follows that the land of physical theories is not that field of ruins, although, of course, many false specific theories have been constructed and collapsed in the course of centuries.

Mathematical method.
These successes classical physics owes to the powerful instrument, the mathematical method. And we add to it :  Notwithstanding the fundamental falsity of the mechanistic principles, [classical physics owes] also [its success] to the circumstance, which by our analysis was placed into a clear light, that the same specifications hold good in the general principles of Aristotle and scholasticism demonstrating the very possibility of variable qualities. For us it remains to further work out the way of application of the mathematical methods to qualities and qualitative changes. In passing, we already said that the Middle Ages had already prepared all that was needed for the right mathematical treatment of intensities, and that the principles of Aristotle are suitable to be specified by modern, mathematically organized, experiment and theory, [this theory] as an explanation of experimental data in terms of nearest causes.
For, superficially seen, but only if so seen, a difficulty may arise and it had actually arisen. When we, namely, are going to apply mathematical methods, one may ask :  But aren't they, then, solely applicable to mathematics' own subject, quantity in the strict sense of the word, i.e. to numbers and extensa? And the very word itself, which one uses to name these methods -- a term, which we, until now, deliberately have avoided -- the word "quantitative method", seems to confirm the difficulty, formulated in this question. Based on this, one then reasons as follows :  If modern, mathematical physics applies quantitative methods to experimentally measure observed features and to set up deductive theories, it automatically transforms the objects of these measurements into quantities. And, because Nature is also qualitative, science, in so proceeding, loses its grip on reality, it then works merely with symbols, it doesn't  know  Nature herself anymore. From which further would follow that the results of such a mathematical natural science, and thus of modern physics, chemistry, and crystallography, are not useful anymore for the natural philosopher who  does  want to reveal reality. A different formulation of this same thought reads :  Mathematical physics performs measurements, and is satisfied with the results of these measurements, numbers, being brought into connection with other numbers having the same origin. These relationships between abstract numbers constitute physics, not desiring anything more, and not teaching anything more [than that], leaving Nature as unknown as she was before.
These assertions are still an echo of positivistic energetism of the beginning of the 20st century, especially in the form supported by Duhem. Possibly also influence of Bergson's philosophy is present [and the same ideas were spread in the 70's and 80's as a reaction against the forceful dominance of humanity and science over Nature and the continuing destruction of the latter as a result of thinking of sheer numbers (money, profit, etc.) only.]
Let us now see that we could rightly say that the medievals had prepared all for a mathematical, we now say quantitative treatment of qualities, like the new physics has in fact introduced. And let us further see that, in addition, as a result of that [quantitative] treatment, the qualities do not lose anything whatsoever of their qualitative nature, and that they, as a result, become uncomparably better known, and that thus modern physics doesn't lose contact with reality in applying this method, but, on the contrary, obtains sufficient contact with reality by this method only. From both facts then follows that this quantitative method is indeed a working-out from the philosophical principles of Aristotle.

The medievals distinguished (1) "quantity in the strict sense", as such coinciding with extension, and [also] that what originates as a result of division, i.e. categoric multiplicity,  and (2)  "quantity in a broad sense", which is nothing else than the intensity of a quality, for instance the degree of heat, the temperature.
The first quantity belongs to the second "category" in Aristotle's list of categories, while the other belongs to the third "category". The first quantity, quantity in the strict sense, St Thomas calls  quantitas extensiva  or  dimensiva, or  molis,  which we may translate as :  quantity of extension, quantity as to dimension, as to volume.
Quantity in the broad sense is, as contrasted with it (it has resemblance with the first, but also differs from it) :  quantitas virtualis  or  virtutis  or  perfectionis,  or  quantitas intensiva  or  intensio.  We translate :  virtual quantity or quantity of capacity or perfection, or intensive quantity or intensity.
This distinction-with-resemblance is strongly accentuated in an impressive series of texts of St Thomas and is classic to all medievals.
According to the terminology of St Thomas, intensity or intensive quantity  per se  belongs to a quality. But in addition he finds in bodily qualities yet another quantity which they have  per accidens.  In this sense :  If a body (wholly or partially) is white, then this quality has, as has the body itself, also extension. It may then also be measured as to this extension and be subjected to ordinary mathematical treatment, like every extension. We may appropriately call this  quantity per accidens  "extensive quality", like the  quantitas per se, i.e. the intensity, is called "intensive quantity" [for example, the intensive quantity of a given quality (= intensity of that quality.  Quality has quantity per se (intensity), and may have quantity per accidens (accidental extensity)].
But, according to Aristotle and St Thomas more can be measured. Quality also has, as to its intensity [thus not only as to its extension], a measure. According to both philosophers indeed the very good definition applies :  measure is that in virtue of which the quantity of something is known.  Both also apply the definition in the case of quantity in the broader sense, intensity. So they are of the opinion that also intensity can be  measured.  Not surprisingly. They knew that an intensive quality can get stronger or weaker -- for Aristotle a typical property of these qualities -- and did understand the correspondence, alongside the difference, of this change [of intensity] with the increase or decrease of an extensum [i.e. a given body or bodily part may become bigger or smaller]. Therefore, they speak of quantity in a broader sense. Therefore, the medievals also speak of growth (augmentum) of an intensity. About the metaphysical explanation of this growth there was a classic disagreement, but its existence was unanimously agreed upon. Not surprisingly that it also became classical that numbers were used, in order to indicate different degrees of intensity. Heat of four, of eight degrees (calor ut quattuor, ut octo) is an expression that is often encountered. These are, of course, not degrees of one of our scales of temperature experimentally established. These numbers are only the expression of an  idea, and of a correct idea, lying at the foundation of our measurement of temperature. The medievals had not yet at their disposal the elaborated experimental methods to also practically determine these different values. The way of expression merely presupposes that possibility.
We deliberately say :  elaborated methods. Because also the principle, lying at the foundation of the modern methods, was well known to them. St Thomas points to it where he defines intensive quantity [the intensity of a quality]. The "bigger" such a quantity is, the bigger (now in the strict sense) quantitative effects it can generate [Knowing that the quantity meant here is the intensivity of a quality, we can, equivalently, say :  The stronger the quality, the larger the quantitative effect that it can generate.]. And thus intensity is measured by these effects [Indeed, according to our theory of the Implicate/Explicate Orders, in the latter Order there is no intensity whatsoever -- intensity resides in the Implicate Order. So all what we can do as to measuring intensity is measuring its extensive effect. And the latter indeed resides in the Explicate Order because this is the domain of the extensive, the explicated.]. See here just a single short but clear text of St Thomas :  "intensive quantity (quantitas virtutis) is usually measured by the effects." He even sometimes points to yet another method also applied in modern physics :  Sometimes the  causes  of a given intensity contain something that is strictly quantitative, and this then provides a measure of the intensity of the effect.
So without exaggeration we could legitimately say that the medievals had all prepared what is in theory demanded by a right mathematical treatment of qualities. From this then also follows that the methods of modern, measuring, physics does not represent a break with the Middle Ages and with Aristotle. And we should, moreover, realize that this mathematical treatment [expressing degrees of intensity by numbers, to begin with.] makes us know the qualities uncomparably better, without spoiling the nature of these qualities as intensive magnitudes, and thus sharpening our knowledge of real Nature much and much more.

This case of measuring the  quantitas per accidens,  of extensive quality as we call it, is in fact without problems. Thus we may be succinct and only indicate where one does encounter this measurement. Recall what the term signifies, namely the extension shared by a given quality of a body with that body itself (the whole of it, or a part). Think of color, of density, of tension, of light-intensity, and the like [We may then have, in a single body, a pattern of color, of density, of tension, etc.].
It is evident that this extension can be mathematically taken and measured without transforming the quality itself into a quantity. Here three important examples.
A. - One example was well known already in the Middle Ages. It was dealt with in the science then called "perspectiva", and today called "geometric optics", the doctrine of reflection and that of refraction. The fact that in reflection the angle of entry is equal to the angle of back-bouncing was known. This is the result of an, albeit elementary, measurement. Also about refraction of light-rays something was known. There was even an explanation of the colors of the rainbow from a two-fold refraction. One may, without precise measurement, determine that a light-ray coming from air and entering [under an oblique angle] water or glass is broken, and that the angle of refraction is smaller than the angle of entrance. If now one measures this quantitatively and then arriving at the sinus-law of Snellius and Descartes, then one surely knows the refraction, a property of light, much more precisely. And yet, one does not, thereby, transform light, which is a quality, into a quantity.
B. - An extensive quality brings with it the possibility of a, as to this quality, heterogeneous  continuum, as we had discussed this already earlier [Here, i.e. in the expositions of HOENEN, we must clearly realize what HOENEN understands here by a "continuum". The latter is here not mathematically defined but metaphysically :  a continuum is a particular ens extensum, namely an ens, a being, that is an intrinsic whole, not a mere aggregate. It is a substance in the metaphysical sense. So a "continuum" is here always an "ens continuum".]. This heterogeneity may imply a very different distribution of different qualities or of different grades of a same quality in different parts of a continuum, of one body. The possibility of different distribution is investigated in the mathematical sciences, first of all the analysis situs [topology]. But then also metric considerations may be added. In classical physics and chemistry one has applied these methods to possible distribution of particles. It is clear that one may, without changing anything in the reasoning, apply them to the distribution of extensive qualities.
We will mention two domains where this method has led to rich results :  structural chemistry and stereo-chemistry, and the structure of crystals. Later we will see how these theories, without any loss of results, can be applied to the distribution, not of particles, but of qualities. Here it is sufficient to establish the fact that in applying this method, the quality remains what it is.
C. - Yet in a third case we can immediately establish the same. Light consists, as we saw, in a periodic change of qualitative states in an extensum, the aether. A period has, as to a determined simple color, a certain temporal duration. So per unit of time we find a determined number of changes, the frequency. Light propagates through the medium with a certain speed. For a given color, same phases of these changes lie at same distances, the wave-lengths. So we find in these qualitative changes two quantitative factors, frequency and wave-length [the amplitude of a light-wave is an intensive factor], which can both, as to their effects, be experimentally measured, and thus be expressed by numbers, increasing our knowledge of these qualities and changes of which light consists enormously, specifying it. And in that knowledge, light remains what it is in Nature :  A qualitative touch onto a spatial medium, and a periodic change of that touch. It doesn't lose anything of its qualitative nature.
[So, while many a quality, such as density, color, etc., of a given body is spatially distributed on or in that body, and so does contain a concurrent, but extrinsic quantitative aspect, there are other qualities that contain intrinsic quantitative aspects as well. Thus, light, as a qualitative determination of the aether and unevenly distributed over it (this is the extrinsic quantitative aspect) necessarily brings with it frequency and wavelength, i.e. certain intrinsic quantitative aspects.]

In all these measurements (A, B, C) it is immediately confirmed what we have said :  Mathematical methods, applied to extensive qualities, leave the quality what it is, and allow us to let it be known with great precision. One can only wonder that somebody, considering these measurements, may say that quantitative physical methods let go our contact with reality, or only consider numbers instead of things.

But physics not only wants to measure extensive qualities, as described above, but also intensive quantities, i.e. intensities. In this case our analysis -- leading to the same result -- must go a bit deeper. We will see that the numbers, resulting from such measurements, at first sight are ordinal numbers. And this is already sufficient for the use physics makes of them. A deeper analysis will have these numbers more approaching cardinal numbers, without, however, ever reaching the nature of cardinal numbers.

In the philosophy of arithmetic one diputes the question whether it is the cardinal number or the ordinal number that is primary. Helmholtz, Dedekind, and others, adhere to the second opinion [ordinal numbers primary], while Höldert, Husserl, and others, to the first opinion [cardinal numbers primary]. We [Hoenen] think we must make a distinction. The generic idea, i.e. multitude or number [of elements of a set or class] in general, seems to us primarily cardinal, while the specific idea, the determined, specified, number [for instance the number 138], primarily originates as an ordinal number. This, of course, asks for more explanation. In the case of intensities the ordinal nature will have primacy.

As to an intensive quality [in fact a quality having intensive quantity, intensivity] we have, already earlier, through simple insight, established the fact that its general properties may be expressed by the following propositions :
If, as to a same quality (for instance heat) a thing A is as intense (as hot) as is B  (symbolically :  A = B )  and, morover,  B = C  [ B is as hot as C ],  then we also have A = C  [A is as hot as C ].
A second proposition concerning intensities reads :
If A is less intense (less hot) than is B  (symbolically :  A < B)  and  if  B < C,  then A < C.
In the modern logic of relations, the relations, symbolized by the signs  " = "  and  " < ",  are called  transitive  because of this property. In the first relation,  " = ",  moreover does hold :  A = A,  and,  if A = B, then also B = A. Therefore this relation is also  reflexive  resp.  symmetric.
The relation  " < "  is also transitive. But because  " A < A " is necessarily false, the relation is called  irreflexive.  Also  " if A < B, then B < A "  is necessarily false. Therefore this relation is called  asymmetric.
These two relations, now, determine an order in the possible intensities of a given quality (for example degrees of heat, temperatures). That order cannot be  cyclical :  If one, starting from a given intensity, continues to heat a given body, one never will re-encounter the starting temperature. The fact that we immediately also see this, is a criterium of the fact that we have, as to the nature of intensive quality, a certain, albeit only generic, insight.
So these two relations,  " = "  and  " < ",  do determine a certain order in the intensities of a same quality. And if we add to this the continuity of change in intensity, then we have all the conditions that are demanded by a physical measurement.

From the two relations in intensive quality immediately follows the possibility of indexing, and thus of measurement, which consists in an attribution of purely ordinal numbers. Let us take as an example a series of points lying on an open line, for example on a straight line, a line which we here only consider as to its topological nature [concerning things like (spatially) before and after, inside, outside, closure, etc.]. If we go along this line in a specified direction we meet point A before point B, B before C, etc., which we may symbolize as :  A < B  and  B < C.  This relation evidently is transitive, irreflexive, and asymmetric  ( If we assume that differently labelled points may coincide, then also the relation "coincide" holds, which has with the equality relation in common that it is transitive, reflexive, and symmetric).
Suppose that ten points are given on the line. Then we may index them successively 1, 2, 3, ...10.  We now know that point 4 lies before point 5, and 5 before 6 (this by definition). See next Figure.

From the properties of the point series we know that then also 4 lies before 6. And if one places a point between 6 and 7, then this point may be indexed by a fraction between both, for instance 6.5. Which, of course, does not imply that it lies at the same distance from the points 6 and 7 [the same holds, for instance of the point with the numeral 3.5, lying between 3 and 4.]. In a purely topological consideration it doesn't even make sense to speak of distances [If we do, the consideration becomes metric]. In the considered concrete case, to be sure, distances do really exist. But equal [arithmetic] differences between attributed numerals do not need to correspond with equal distances on the line.
Something similar also holds for an intensive quality. Say, we have ten bodies, all differing as to the degree of heat (temperature). Then these bodies may be ordered into an ascending series. First the least hot body, then the least hot of the remaining nine bodies, and so on. Indexing these bodies as to this order, we have then attributed a  degree  to each intensity, which [degree] is a pure number, a pure ordinal number. We now can make the same inference as we did in the point series. Only as to the last remark we must make a restriction. In comparing  differences  in intensity it may make no sense, at least not the same sense as in line segments, to speak of "distances" between two degrees. Such a measurement, as described here, with a limited number of given degrees is still rather rough, but has applications in natural science. Let us consider an example.

The scale of hardness of Mohs,  Seger-cones.
In mineralogy one speaks of hardness of minerals, established as follows :  If A is scratched by B, B is harder than A.  If  B is scratched by C, then C is harder than B.  Now experience teaches us (perhaps one may expect this result, but a priori certainty one yet cannot have, because hardness is a property of which we know too little) that A is also scratched by C.  Were this not the case, then hardness could not be an intensive quality, and a sort of cyclic order might be present in it. Now, according to experience it turns out to have this transitive property. Thus one has chosen some ten minerals as standards of measurement (from talcum to diamond). If one has a mineral that can be scratched by mineral 7, and which itself scratches 6, then a number between 6 and 7 indicates its hardness, which is now roughly measured. The fact that this is a pure ordinal determination [determination of rank] is clear from the fact that one cannot reasonably speak of summation or subtraction of hardnesses [We don't know, in this case, how far 6 lies from 7].
A similar method to roughly determine very high temperatures (in glass-ovens, and the like) is, in industry, used by means of a series of so-called Seger-cones, of which one melts at a lower temperature than another does. When one melts and the other still not, it indicates a point in the temperature interval between both melting-points.

Perfect indexing (by numbers)  of intensities from the effects.

In the considered cases we had a discontinuous series of specimens of intensity of a same quality. But, as is motion, also the change of intensity is continuous [If we neglect for the moment all the contradictions involved in the notion of continuity] and one should look for a more detailed indexing of grades of intensity. Physics has found it by applying the principle of St Thomas :  Intensity is measured (is becoming known as to its value) by [measuring] its purely quantitative effects. Let us take heat as an example.
Heat is a quality, which has a purely quantitative effect :  As a result of heating the volume of the heated body changes (and also, of course, its linear dimensions), and it does so in a continuous fashion together with the intensity if no change in the state of aggregation takes place [If the body doesn't melt, or, if it is a liquid, it doesn't boil]. In what, now, does consist the construction of a "meter" [measuring device] of heat intensity? Well, one takes a thermometer-body, for example [a certain volume of] mercury [in a small glass container], puts it into melting ice and marks the volume of the mercury. This particular heat intensity then obtains the number 0.  Then one puts it into water which is, at "normal" pressures, boiling, and again marks the volume, and this intensity of heat obtains, for example, the number 100.  The increase of the volume of mercury between these two conditions is divided into 100 equal parts, and these in turn into equal parts, as far as one wants to go, and as far as the precision of the manipulations admit. Theoretically up into infinity. To the results of the first division of the scale one successively attributes the numbers 1 to 99, and those of the other divisions are indexed (numbered) according to the decimal system. And indeed in this way one has the possibility to index the intensities in theoretically a continuous fashion. Practical deviations from this ideal are due to the imprecision of our experiments, not from the very notion itself which is being discussed here. So one now has a method to precisely determine every heat-intensity whatsoever by a number. One obtains a continuous series of numbers which, corresponding to the real intensities in Nature itself, have the above described relations. [In fact we do not measure the very intensity itself. We assume that the intensive scale of intensities can be mapped onto some extensive scale. And while in the latter scale the points (possible values) spatially lie  outside  one another, the points of an intensive scale, points, indicating possible intensities, lie  inside  one another.]
So what does it mean to speak about a temperature of 35.427 degrees? This means that it indicates that particular heat-intensity that causes the thermometer-body (the mercury) to expand [as to its volume], reckoned from the tenperature of melting ice, precisely 0.35427 as many times as is its expansion which it undergoes from being at the temperature of melting ice to being at a temperature of boiling water. This heat-intensity is now, as a result, completely defined, and every physicist may reproduce the same intensity. Difficulties of a practical nature we do not, of course, discuss here. Of philosophical interest is the following :  In this way one has a means, to  know  every  intensity of a given intensive quality, which, according to the definition of Aristotle, means :  to measure.  This measurement leaves, as is clear, the qualitative nature of heat unaffected. We have a means to learn to know its intensity uncomparably more exactly than [learning it] by a mere so-called "qualitative observation" (which says, it is cold, it is lukewarm, it is hot), learning it to know, thay is, as to its qualitative nature.

The just described measurement of intensities was one, based on purely quantitative effects. But also the reverse method is known :  Measurement based on partially quantitative causes. Photometry provides a well known example. We may index (with numbers) the intensity of a given exposure by light as to the number of lit candles, one, or two, etc., of wholly the same structure (composition and size) exposing, from an equal distance, a given surface. If this would be all, then we would have the same poor method, poor that is, as a result of the discontinuity of this cause (the series of lit candles). But in this cause we have a genuine quantitative element, which is changeable in a continuous fashion :  the distance. At a large distance the exposure by the same light-source is weaker, than it is at a smaller distance.
If we, in the first case, index the intensity as to the number of lit candles, and in the second case index this intensity inversely proportional to the distance of the one candle, then different types of indexing will result. But nevertheless, this indexing is such that the order between two intensities in the two indexings always remains the same [i.e. the topological structure remains the same, while the metric structure differs.]. And because our measurement is of a purely ordinal nature, the one method is, for our knowledge of intensities, not better than the other. If we, in the second method, make the indexing by numbers such that it is inversely proportional with the  square  of the distance (and this one in fact does), the same indexing results as in the first method. One may ask which of the two definitions is the better one. Without digging deeper into this, we succinctly answer :  As long as we stick to the purely ordinal consideration of numbering, which is suffcient for the justification of the method of physics, one cannot hold one definition to be better than the other. This becomes different if one also considers the somewhat cardinal nature which the numbering may (and in the end must) have. Then, the second definition will be the better one. Here we will not dig deeper into this cardinal nature.
We like to note that the method of photometry is used also the other way around :  From the effect (a certain degree of exposure by light) one measures the strength (the number of lit candles) of the light-source.
So in most physical measurements of intensities, one or another quantitative factor connected with the intensity, is finally measured. Most measurements in the end boil down to the comparison of lengths or observing the position of a pointer, which is, in effect, also a comparison of lengths. Sometimes qualities, for example equal light-intensities, or nuances of color, are compared with each other. So in photometry, colorimetry, and polarimetry. But here, always the measurement of some quantitative factor is also involved, a quantitative factor like volume, length, rotation according to some angle.
But for all these measurements holds what we saw above :  Observation of the intensity of a given quality is rendered uncomparably more precise. As a result, a given intensity is exactly distinguished from another intensity, i.e. known, measured. The true nature of it we come to know much better by these mathematical methods.

Application to physical laws and theories.

The thus precisely known qualities may then be investigated as to their mutual influence, and this, quantitatively so. Let us take an example :  After having precisely defined "conductivity" of a metal as to electrical current, one may determine how this conductivity is influenced by temperature. One then finds the experimental not qualitative but quantitative law, saying that this conductivity is inversely proportional with temperature. The connection may of course much more complicated. In, generally, this way, experimental physics is constructed.
In the same way also theoretical physics. Also features appearing in explicative hypotheses, and their effects, are quantitatively formulated, and then mathematically conclusions may be deduced, which has been the great power of physical theories. After all, the result now must also be found  quantitatively  in observation and experiment, if it wants to be a confirmation of the theory. How much sharper has now become the verification! A most striking example we saw in the hypothesis of Lesage. " Qualitatively" it looked quite impressive and could account for the attraction by the sun of the Earth. But as soon as the elaboration of the hypothesis could become quantitative, it was at one stroke clear how far it was from the truth.

From our analysis this is clear :  These mathematically physical methods only suppose measurements, i.e. coming to know intensities, by purely ordinal numbers. All what is quality remains quality in these measurements, remains to belong to the third category of Aristotle, it does not become quantity in the strict sense. Physics, has, following natural reason, elaborated these methods. Above we saw that they simply extend the line [of thought] indicated by Aristotle and St Thomas. These methods do nothing else than further elaborate the fundament laid down in the Middle Ages with all clarity.
Earlier in this document we found out that the physics that based itself on mechanicism and initially had banned all qualities, was, as a result of the development of experimental investigation and theoretical detection of nearest causes, automatically led to re-introduce qualities. It became, from machanical, aristotelian. The later part of the present document teaches us that also physics' mathematical methods, the instrument of this grand development, purely lie in the line [of thought] of Aristotle. The first part of this line, determining the direction, was drawn in the Middle Ages.
One more remark. Our conclusion was :  For the physical methods, a purely ordinal nature of the numbers resulting from measurement is sufficient. In this way there is a certain, wider, analogy with quantities in the strict sense, namely with topologically considered extensions [So intensities are analogous with topologically-viewed extensities.]. The philosopher may go further and ask whether there doesn't exist a closer analogy without disappearance of the difference between both categories (also this may influence physical considerations). We think we must answer this question by a yes. However, we here cannot dig deeper into this problem.
[Our [JB] theory of the Implicate/Explicate Orders will make clearer the distinction between intensity and extension. For the time being we may say that the distinction between true (intensive) qualities and true (extensive) quantities is that they vary along different dimensions. A true quality varies within an intensive dimension, a dimension in which the possible points (representing values) are inside one another, while a true quantity varies within an extensive dimension, a dimension in which the possible points (representing values) are (spatially) extended from one another.].

To be continued . . .

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