Commentary Aristotle Posterior Analytics
Lecture 25
How demonstration quia differs from demonstration propter quid when the former pertains to one science and the latter to another b34. But there is another way too in which the fact and the reasoned fact differ, and that is when they are investigated respectively by different sciences.
b35. This occurs in the case of problems related to one another as subordinate and superior, as when optical problems are subordinated to geometry, mechanical problems to stereometry, harmonic problems to arithmetic, the data of observation to astronomy.
a1. Some of these sciences bear almost the same name; e.g., mathematical and nautical astronomy, mathematical and acoustical harmonics.
a3. Here it is the business of the empirical observers to know the fact, of the mathematicians to know the reasoned fact; for the latter are in possession of the demonstrations giving the causes, and are often ignorant of the fact: just as we have often a clear insight into a universal, but through lack of observation are ignorant of some of its particular instances. These connexions have a perceptible existence though they are manifestations of forms. For the mathematical sciences concern forms: they do not demonstrate properties of a substratum, since, even though the geometrical subjects are predicable as properties of a perceptible substratum, it is not as thus predicable that the mathematician demonstrates properties of them.
a10. As optics is related to geometry, so another science is related to optics, namely the theory of the rainbow. Here knowledge of the fact is within the province of the natural philosopher, knowledge of the reasoned fact within that of the optician, either qua optician or qua mathematical optician.
a13. Many sciences not standing in this mutual relation enter into it at points; e.g., medicine and geometry: it is the physician’s business to know that circular wounds heal more slowly, the geometer’s to know the reason why.
But there is another way. After showing how demonstration 'quia' differs from demonstration 'propter quid' in the same science, the Philosopher shows how they differ in sciences that are diverse.
And he does two things.
First, he states his proposition, saying (78b34) that in a way other than the above the 'propter quid' differs from the 'quia' due to the fact that they are considered in diverse sciences, that is, that the 'propter quid' pertains to one science and the 'quia' to another.
Second, at: this occurs in the case (78b34) he elucidates his proposition.
First, he elucidates it in sciences one of which is under the other.
Second, in sciences one of which is not under the other, at: many sciences not standing (79a13; [212]).
Concerning the first he does two things.
First, he shows how those sciences are related, one of which is under the other and to one of which pertains the 'quia' and to the other the 'propter quid'.
Second, he shows how in these sciences the 'quia' pertains to one and the 'propter quid' to the other, at: here it is the business (79a3; [210]).
Concerning the first he does two things.
First, he shows how such sciences relate to one another as to order.
Second, how they relate to one another as to agreement, at: some of these sciences (79a1; [209]).
He says therefore first (78b35) that these sciences (that is, to one of which pertains 'quia' and to the other 'propter quid') are the ones so related that one is under the other.
But this occurs in two ways: in one way, when the subject of one science is a species of the subject of the higher science, as animal is a species of natural body—consequently, the science of animals is under natural science; in another way, when the subject of the lower science is not a species of the subject of the higher science but is compared to the latter as material to formal.
An example of this latter way of one science being under another is the way specular, that is, optics, is under geometry. For geometry is concerned with lines and other magnitudes, whereas optics is concerned with a line determined to matter, that is, the visual line. Now, the visual line is not, strictly speaking, a species of line any more than wooden triangle is a species of triangle: for wooden is not a difference in respect to triangle. In like manner mechanical engineering, that is, the science of making machines, is related to stereometry, that is, the science of measuring bodies. This science is said to be under the other as applying the formal to the material. For the measures of bodies absolutely are compared to the measures of wood and other material required for machines as the application of the formal to the material. In like manner, harmonics, that is, music, is related to arithmetic: for music applies formal number, which arithmetic considers, to matter, that is, to sounds. And the same is true of appearance, that is, nautical science (which considers the signs indicative of calm or storm), as compared to astronomy, which considers the motions and positions of the stars.
Then, at: some of these sciences (79a1) he shows how the aforesaid sciences relate to one another in point of agreement. And he says that these sciences are almost mutually univocal. And he says “almost” because they agree in the name of their genus and not in the name of their species. Thus, all the sciences mentioned above are called 'mathematical': some, indeed, because they are concerned with a subject which is abstracted from matter, as geometry and arithmetic, which are absolutely mathematical; but others are so through applying mathematical principles to material things, as astronomy is called mathematical and as nautical science is. Similarly, harmonics, that is, music, is called mathematical and so is acoustics, that is, practical music, which knows sounds through experience based on hearing. Or it might be said that they are univocal because they also agree in the name of their species, for we speak of nautical astronomy as astronomy and of practical music as music. But because this is not so in all but only in some, he says “almost”.
Then, at: here it is the business (79a3) he shows how in these sciences the 'quia' pertains to the one science and the 'propter quid' to the other.
In regard to this there are two points.
First, he shows how those sciences which contain others under them state the 'propter quid'.
Second, how sciences which are contained under another one state the 'propter quid' of some others, at: as optics is related to geometry (79a10; [211]).
One should note, therefore, with respect to the first point that in all the above-mentioned sciences, the ones which are contained under others apply mathematical principles to sensible things, while the ones which contain others under them are more mathematical. Accordingly, the Philosopher first of all says (79a3) that to know the 'quia' pertains to the sensible, that is, to the lower sciences, which make application to sensible things; but to know the 'propter quid' pertains to the mathematical, that is, to those sciences whose principles are not applied to sensible things. For these latter sciences are concerned with demonstrating matters which are accepted as causes in the lower sciences.
But because someone might suppose that whoever knows the 'propter quid' also necessarily knows the 'quia', he dismisses this, saying that very frequently those who know the 'propter quid' do not know the 'quia'. And he gives as an example of this those who, when considering the universal often take no account of the singulars precisely because their speculation does not consider them: thus, one who knows that every mule is sterile, does not know this of the one he does not consider. In like manner, a mathematician who demonstrates 'propter quid', now and then does not know the quia, because he does not apply the principles of the higher sciences to matters demonstrated in the lower science.
And because he had said that it belongs to mathematics to know the 'propter quid', he proposes to indicate which genus of cause is used by mathematics. Hence he says that those sciences which receive the 'propter quid' from mathematics are something else, that is, they differ from the mathematical according to subject, that is, insofar as they make application to matter. Hence these latter use forms, that is, formal principles, which they receive from mathematics: for the mathematical sciences concern forms. For their considerations do not bear on the subject, that is, on matter, because although the items which geometry considers exist in matter, for example, the line, plane and so on, nevertheless geometry does not consider them precisely as they are in matter, but as abstracted. For those things that are in matter according to existence, geometry abstracts from matter according to consideration. Conversely, the sciences subalternated to it accept those things which were considered in the abstract by geometry and apply them to matter. Hence it is plain that it is according to the formal cause that geometry states the 'propter quid' in those sciences.
Then, at: as optics is related to geometry (79a10), he shows that even the subalternated sciences state the 'propter quid', not of its subalternating science but of some other science. Thus, optics is subalternated to geometry, so that if we compare the one with the other, optics states the 'quia' and geometry the 'propter quid'. But just as optics is subalternated to geometry, so the science of the rainbow is subalternated to optics, for it applies to a determinate matter the principles which optics hands down absolutely. Hence it belongs to the naturalist who treats of the rainbow to know the 'quia', but to the expert in optics to know the 'propter quid'. For the naturalist says that the cause of the rainbow is the convergence of a visual line at a cloud arranged in some relation to the sun; but the 'propter quid' he takes from optics.
Then, at: many sciences not standing (79a13) he shows how 'quia' and 'propter quid' differ among sciences that are diverse but not subalternate. And he says that many sciences which are not subalternate are nevertheless related, that is, in such a way that one states the 'quia' and the other the 'propter quid'. This is true of medicine and geometry. For the subject of medicine is not subsumed under the subject of geometry as the subject of optics is. Nevertheless, the principles of geometry are applicable to certain conclusions reached in medicine: for example, it belongs to the man of medicine who observes it to know quia that circular wounds heal rather slowly; but to know the 'propter quid' belongs to the geometer, whose business it is to know that a circle is a figure without corners. Hence the edges of a circular wound are not close enough to each other to allow them to be easily joined.
It should also be noted that this difference of 'quia' and 'propter quid' between sciences that are diverse is contained under one of the modes previously discussed, namely, when the demonstration is made through a remote cause.