Didascalicon, Book II: the sciences, their parts and their discipline
Chapter XVIII. What is proper to each art
Though all the arts tend to one end for philosophers, yet they do not all run by one road, but each has its own considerations, by which they differ from one another. Logical consideration is in things, attending to the understandings of things, either by intelligence: so that neither these things nor their likenesses exist; or by reason, so that these things do not exist, but their likenesses do. Logic considers, then, the species and genera of things. It is proper to mathematics to attend by reason, without confusion, to acts that are confused: for example, in the actual being of things a line is not found without surface and solidity (for no body is so long only as to lack breadth or height; but in every body these three are together); yet reason attends to the line purely by itself, without surface and thickness; which is the mathematical, not because in the thing it is so or can be so, but because reason often considers the acts of things not as they are but as they can be; not in themselves, but as to reason itself, that is, as reason would suffer them to be. According to this consideration it has been said that continuous quantity decreases to infinity, and discrete quantity increases to infinity. For such is the liveliness of reason that it divides every long into longs, broad into broads, and so on; and that to reason itself nothing lacking interval generates interval. It is proper to physics, on the other hand, to attend unmixedly, through things mixed, to the acts of things. For the acts of the bodies of the world are not pure, but composed of the acts of the pure ones, which physics, though they are not found pure by themselves, nevertheless considers purely and by themselves: namely the pure act of fire, or of earth, or of air, or of water; and from the nature of each considered by itself it judges of the concretion and efficiency of the whole. This also is not to be passed over, that physics alone properly treats of things, all the others of the understandings of things. Logic treats of the understandings themselves, according to the predicamental constitution. Mathematics treats of them according to integral composition; and therefore logic sometimes uses pure intelligence, but mathematics is never without imagination; and therefore nothing is truly simple. Since logic and mathematics are prior in the order of learning to physics, and in a way serve it as instruments, with which each one must first be informed before he gives his effort to physical speculation, it was necessary that they should place their consideration not in the acts of things, where experiment is deceptive, but in reason alone, where unshaken truth remains; and then, with reason itself going before, descend to the experience of things. Now that we have shown how the division of the theoretical which Boethius gives agrees with the preceding one, let us briefly repeat both, to compare each word of the two divisions with the other.