Here is Maurer's translation of Exposition of Boethius's De trinitate 5.1c, ad 6-10; 5.4c, ad 3-8. I've made one change at the end of 5.1c (important) and in ad 7.

Exposition of Boethius's De trinitate 5.1: Is Speculative Science Appropriately Divided into these Three Parts: Natural, Mathematical, and Divine?

We proceed as follows to the first article: It seems that speculative science is not appropriately divided into these three parts, for

1. The parts of speculative science are the habits that perfect the contemplative part of the soul. But the Philosopher says in the Ethics that the scientific part of the soul, which is its contemplative part, is perfected by three habits, namely, wisdom, science, and understanding. Therefore these are the three divisions of speculative science, not those proposed in the text.

2. Again, Augustine says that rational philosophy, or logic, is included under contemplative or speculative philosophy. Consequently, since no mention is made of it, it seems the division is inadequate.

3. Again, philosophy is commonly divided into seven liberal arts, which include neither natural nor divine science, but only rational and mathematical science. Hence natural and divine should not be called parts of speculative science.

4. Again, medicine seems to be the most practical science, and yet it is said to contain a speculative part and a practical part. By the same token, therefore, all the other practical sciences have a speculative part. Consequently, even though it is a practical science, ethics or moral science should be mentioned in this division because of its speculative part.

5. Again, the science of medicine is a branch of physics, and similarly certain other arts called "mechanical," like the science of agriculture, alchemy, and others of the same sort. Therefore, since these sciences are practical, it seems that natural science should not be included without qualification under speculative science.

6. Again, a whole should not be contradistinguished from its part. But divine science seems to be a whole in relation to physics and mathematics, since their subjects are parts of its subject. For the subject of divine science or first philosophy is being; and changeable substance, which the natural scientist considers, and also quantity, which the mathematician considers, are parts of being. This is clear in the Metaphysics. Therefore, divine science should not be contradistinguished from natural science and mathematics.

7. Again, as it is said in the De Anima, sciences are divided in the same manner as things. But philosophy concerns being, for it is knowledge of being, as Dionysius says. Now being is primarily divided into potency and act, one and many, substance and accident. So it seems that the parts of philosophy ought to be distinguished by such divisions of being.

8. Again, there are many other divisions of beings studied by sciences more essential than the divisions into mobile and immobile and into abstract and non-abstract; for example, the divisions into corporeal and incorporeal and into living and non-living, and the like. Therefore differences of this sort should be the basis for the division of the parts of philosophy rather than those mentioned here.

9. Again, that science on which others depend must be prior to them. Now all the other sciences depend on divine science because it is its business to prove their principles. Therefore Boethius should have placed divine science before the others.

10. Again, mathematics should be studied before natural science, for the young can easily learn mathematics, but only the more advanced natural science, as is said in the Ethics. This is why the ancients are said to have observed the following order in learning the sciences: first logic, then mathematics, then natural science, after that moral science, and finally men studied divine science. Therefore, Boethius should have placed mathematics before natural science. And so it seems that this division is unsuitable.

On the contrary, the Philosopher proves the appropriateness of this division in the Metaphysics, where he says, "There will be three philosophical and theoretical sciences: mathematics, physics, and theology."

Moreover, in the Physics three methods of the sciences are proposed which indeed seem to belong to these three.

Moreover, Ptolemy also uses this division in the beginning of his Almagest.

Reply: The theoretical or speculative intellect is properly distinguished from the operative or practical intellect by the fact that the speculative intellect has for its end the truth that it contemplates, while the practical intellect directs the truth under consideration to activity as to an end. So the Philosopher says in the De Anima that they differ from each other by their ends; and in the Metaphysics he states that "the end of speculative knowledge is truth, but the end of practical knowledge is action."

Now, since matter must be proportionate to the end, the subject-matter of the practical sciences must be things that can be made or done by us, so that we can direct the knowledge of them to activity as to an end. On the other hand, the subject-matter of the speculative sciences must be things that cannot be made or done by us, so that our knowledge of them cannot be directed to activity as to an end. And the speculative sciences must differ according to the distinctions among these things.

Now we must realize that when habits or powers are differentiated by their objects they do not differ according to just any distinction among these objects, but according to the distinctions that are essential to the objects as objects. For example, it is incidental to a sense object as such whether it be an animal or a plant. Accordingly, the distinction between the senses is not based upon this difference but rather upon the difference between color and sound. So the speculative sciences must be divided according to differences between objects of speculation, considered precisely as such. Now an object of this kind--namely, an object of a speculative power--derives one characteristic from the side of the power of intellect and another from the side of the habit of science that perfects the intellect. From the side of the intellect it has the fact that it is immaterial, because the intellect itself is immaterial. From the side of the habit of science it has the fact that it is necessary, for science treats of necessary matters, as is shown in the Posterior Analytics. Now everything that is necessary is, as such, immobile, because everything changeable is, as such, able to be or not to be, either absolutely or in a certain respect, as is said in the Metaphysics. Consequently, separation from matter and motion, or connection with them, essentially belongs to an object of speculation, which is the object of speculative science. As a result, the speculative sciences are differentiated according to their degree of separation from matter and motion.

Now there are some objects of speculation that depend on matter for their being, for they can exist only in matter. And these are subdivided. Some depend on matter both for their being and for their being understood, as do those things whose definition contains sensible matter and which, as a consequence, cannot be understood without sensible matter. For example, it is necessary to include flesh and bones in the definition of man. It is things of this sort that physics or natural science studies. On the other hand, there are some things that, although dependent upon matter for their being, do not depend upon it for their being understood, because sensible matter is not included in their definitions. This is the case with lines and numbers--the kind of objects with which mathematics deals. There are still other objects of speculative knowledge that do not depend upon matter for their being, because they can exist without matter; either they never exist in matter, as in the case of God and the angels, or they exist in matter in some instances and not in others, as in the case of substance, quality, being, potency, act, one and many, and the like. The science that treats of all these is theology or divine science, which is so called because the chief of the things known in it is God. By another name it is called metaphysics; that is to say, beyond physics, because it ought to be learned by us after physics; for we have to proceed from sensible things to those that are non-sensible. It is also called first philosophy, inasmuch as all the other sciences, receiving their principles from it, come after it. Now there can be nothing that depends upon matter for its being understood but not for its being, because by its very nature the intellect is immaterial. So there is no fourth kind of philosophy besides the ones mentioned.

Replies to the Opposing Arguments:

Reply to 1. In the Ethics, the Philosopher considers the intellectual habits insofar as they are intellectual virtues. Now they are called virtues because they perfect the intellect in its operation; for "virtue makes its possessor good and renders his work good." So he distinguishes between virtues of this sort inasmuch as speculative habits perfect the intellect in different ways. In one way the speculative part of the soul is perfected by understanding, which is the habit of principles, through which some things become known of themselves. In another way it is perfected by a habit through which conclusions demonstrated from these principles are known, whether the demonstration proceeds from inferior causes, as in science, or from the highest causes, as in wisdom. But when sciences are differentiated insofar as they are habits, they must be distinguished according to their objects, that is, according to the things of which the sciences treat. And it is in this way that both here and in the Metaphysics speculative philosophy is distinguished into three parts.

Reply to 2. As is evident in the beginning of the Metaphysics, the speculative sciences concern things the knowledge of which is sought for their own sake. However, we do not seek to know the things studied by logic for themselves, but as a help to the other sciences. So logic is not included under speculative philosophy as a principal part but as something brought under speculative philosophy as furnishing speculative thought with its instruments, namely, syllogisms, definitions, and the like, which we need in the speculative sciences. Thus, according to Boethius, logic is not so much a science as the instrument of science.

Reply to 3. The seven liberal arts do not adequately divide theoretical philosophy; but, as Hugh of St. Victor says, seven arts are grouped together (leaving out certain other ones), because those who wanted to learn philosophy were first instructed in them. And the reason why they are divided into the trivium and quadrivium is that "they are as it were paths (viae) introducing the quick mind to the secrets of philosophy." This is also in harmony with the Philosopher's statement in the Metaphysics, that we must investigate the method of scientific thinking before the sciences themselves. And the Commentator says in the same place that before all the other sciences a person should learn logic, which teaches the method of all the sciences; and the trivium concerns logic. The Philosopher also says in the Ethics that the young can know mathematics but not physics, because it requires experience. So we are given to understand that after logic we should learn mathematics, which the quadrivium concerns. These, then, are like paths leading the mind to the other philosophical disciplines.

We may add that among the other sciences these are called arts because they involve not only knowledge but also a work that is directly a product of reason itself; for example, producing a composition, syllogism or discourse, numbering, measuring, composing melodies, and reckoning the course of the stars. Other sciences (such as divine and natural science) either do not involve a work produced but only knowledge, and so we cannot call them arts, because, as the Metaphysics says, art is "productive reason"; or they involve some bodily activity, as in the case of medicine, alchemy, and other sciences of this kind. These latter, then, cannot be called liberal arts because such activity belongs to man on the side of his nature in which he is not free, namely, on the side of his body. And although moral science is directed to action, still that action is not the act of the science but rather of virtue, as is clear in the Ethics. So we cannot call moral science an art; but rather in these actions virtue takes the place of art. Thus, as Augustine says, the ancients defined virtue as the art of noble and well-ordered living.

Reply to 4. As Avicenna says, the distinction between theoretical and practical is not the same when philosophy is divided into theoretical and practical, when the arts are divided into theoretical and practical, and when medicine is so divided. For when we distinguish philosophy or the arts into theoretical and practical we must do so on the basis of their end, calling that theoretical which is directed solely to knowledge of the truth, and that practical which is directed to operation. However, there is this difference when we distinguish the whole of philosophy and the arts on this basis: We divide philosophy with respect to the final end or happiness, to which the whole of human life is directed. For, as Augustine says, following Varro, "There is no other reason for a man philosophizing except to be happy." And since the philosophers teach that there is a twofold happiness, one contemplative and the other active, as is clear in the Ethics, they have accordingly also distinguished between two parts of philosophy, calling moral philosophy practical and natural and rational philosophy theoretical. But when they call some arts speculative and some practical, this is on the basis of some special ends of those arts; as when we say that agriculture is a practical art but dialectic is theoretical.

However, when we divide medicine into theoretical and practical, the division is not on the basis of the end. For on that basis the whole of medicine is practical, since it is directed to practice. But the above division is made on the basis of whether what is studied in medicine is proximate to, or remote from practice. Thus we call that part of medicine practical which teaches the method of healing; for instance, that these particular medicines should be given for these abscesses. On the other hand, we call that part theoretical which teaches the principles directing a man in his practice, although not immediately; for instance, that there are three virtues, and that there are so many kinds of fever. Consequently, if we call some part of a practical science theoretical, we should not on that account place that part under speculative philosophy.

Reply to 5. One science is contained under another in two ways: in one way, as its part, because its subject is part of the subject of that other science, as plant is part of natural body. So the science of plants is also contained under natural science as one of its parts. In another way, one science is contained under another as subalternated to it. This occurs when in a higher science there is given the reason for what a lower science knows only as a fact. This is how music is contained under arithmetic.

Medicine, therefore, is not contained under physics as a part, for the subject of medicine is not part of the subject of natural science from the point of view from which it is the subject of medicine. For although the curable body is a natural body, it is not the subject of medicine insofar as it is curable by nature, but insofar as it is curable by art. But because art is nature's handmaid in healing (in which art too plays a part, for health is brought about through the power of nature with the assistance of art), it follows that the reason for the practices used in the art must be based on the properties of natural things. So medicine is subalternated to physics, and for the same reason so too are alchemy, the science of agriculture, and all sciences of this sort. We conclude, then, that physics in itself and in all its parts is speculative, although some practical sciences are subalternated to it.

Reply to 6. Although the subjects of the other sciences are parts of being, which is the subject of metaphysics, the other sciences are not necessarily parts of metaphysics. For each science treats of one part of being in a special way distinct from that in which metaphysics treats of being. So its subject is not properly speaking a part of the subject of metaphysics, for it is not a part of being from the point of view from which being is the subject of metaphysics; from this viewpoint it is a special science distinct from the others. However, the science treating of potency, or that treating of act or unity or anything of this sort, could be called a part of metaphysics, because these are considered in the same manner as being, which is the subject of metaphysics.

Reply to 7. These parts of being require the same manner of consideration as being-in-general (ens commune) because they too are not dependent on matter. For this reason the science dealing with them is not distinct from the science of being-in-general.

Reply to 8. The other diversities of things mentioned in the objection do not differentiate those things essentially as objects of knowledge. So the sciences are not distinguished according to them.

Reply to 9. Although divine science is by nature the first of all the sciences, with respect to us the other sciences come before it. For, as Avicenna says, the position of this science is that it be learned after the natural sciences, which explain many things used by metaphysics, such as generation, corruption, motion, and the like. It should also be learned after mathematics, because to know the separate substances metaphysics has to know the number and disposition of the heavenly spheres, and this is impossible without astronomy, which presupposes the whole of mathematics. Other sciences, such as music, ethics, and the like, contribute to its fullness of perfection.

Nor is there necessarily a vicious circle because metaphysics presupposes conclusions proved in the other sciences while it itself proves their principles. For the principles that another science (such as natural philosophy) takes from first philosophy do not prove the points which the first philosopher takes from the natural philosopher, but they are proved through other self-
evident principles. Similarly the first philosopher does not prove the principles he gives the natural philosopher by principles he receives from him, but by other self-evident principles. So there is no vicious circle in their definitions.

Moreover, the sensible effects on which the demonstrations of natural science are based are more evident to us in the beginning. But when we come to know the first causes through them, these causes will reveal to us the reason for the effects, from which they were proved by a demonstration quia. In this way natural science also contributes something to divine science, and nevertheless it is divine science that explains its principles. That is why Boethius places divine science last, because it is the last relative to us.

Reply to 10. Although we should learn natural science after mathematics because the general proofs of natural science require experience and time, still, since natural things fall under the senses, they are by nature better known than the mathematical entities abstracted from sensible matter.