Proclus on pure and applied mathematics
Proclus Diadochus, in his Commentary on Euclid's Elements, describes the divisions of mathematics into pure mathematics and applied mathematics. The following extract is taken from his commentary on Book I:-
Such is the doctrine of the Pythagoreans and their fourfold division of mathematics [arithmetic, music, geometry, astronomy]. But others, among them Geminus, prefer to divide mathematics in another way. They consider mathematics on the one hand as concerned with things conceived by the mind, and on the other hand as concerned with and applied to things perceived by the senses. By things conceived by the mind they mean those that the psyche makes objects of contemplation by itself, when it completely divorces itself from forms connected with matter. In that division of mathematics which has to do with things conceived by the mind they place the two most basic and important branches, arithmetic and geometry. On the other hand, of that part of mathematics that devotes its attention to objects perceived by the senses they list six branches: mechanics, astronomy, optics, geodesy, canonics, and logistics.
They do not think, as some others do, that military science should be considered as one of the branches of mathematics, though, to be sure, it sometimes uses logistics, for example in the enumeration of companies, and at other times geodesy, for example in the division and measurement of areas. Similarly, there is even more reason why neither history nor medicine may be considered a part of mathematics, even though historians often use mathematical theorems, as when they give the location of climata [zones of latitude] or calculate the size and the diameter or circumference of cities. Physicians, too, by similar methods clarify many of the problems relevant to their field. For the importance of astronomy in medicine is made clear by Hippocrates and, in fact, by all who have discoursed on seasons and places. In the same way, then, the military tactician, without being a mathematician himself, uses the theorems of mathematics when, on occasion, he wishes to make the number of his troops appear as small as possible, and so arranges them in a circle, or, again, when he wishes to make the number appear as large as possible, and so arranges them in a square or pentagon or some other polygon.
While these are the branches of mathematics as a whole, geometry, in its turn, is divided into the theory of the plane and stereometry [solid geometry]. For the study of points and lines cannot constitute a separate branch since the figures they form must be either plane or solid. Everywhere it is the task of geometry, in the case both of planes and solids, either to construct, or to compare or divide what has been constructed.
Similarly, arithmetic is divided into the study of linear, of plane, and of solid numbers. For it treats of the kinds of numbers in and of themselves, as they proceed from unity, the generation of plane numbers both similar and dissimilar [square numbers aa, and rectangular numbers ab], and the progression to the third dimension.
Geodesy and logistics are analogous to geometry and arithmetic, respectively, but are concerned with things perceived by the senses, not with numbers or figures as conceived by the mind. For it is not the business of geodesy to measure the cylinder or the cone, but to measure mounds as if they were cones, or wells as if they were cylinders. Geodesy deals, then, with straight lines, not as conceived by the mind, but as perceived by the senses, sometimes more sharply defined, as in the case of the rays of the sun, sometimes less sharply defined, as in the case of cords or a plumb-line.
Similarly, one who employs logistics does not consider the properties of numbers by themselves, but always in connection with perceptible objects. Hence he gives to numbers a name after the objects that are being computed and thus speaks of melites [number of apples or sheep] and phialites [weight of liquid measures]. He does not admit an [absolute] minimum unit, as does the arithmetician, yet he does use a minimum unit in relation to some subject matter. Thus one man serves the logistician as the unit of a crowd of men.
Again, optics and canonics are derived from geometry and arithmetic, respectively. The science of optics makes use of lines as visual rays and makes use also of the angles formed by these lines. The divisions of optics are: (a) the study which is properly called optics and accounts for illusions in the perception of objects at a distance, for example, the apparent convergence of parallel lines or the appearance of square objects at a distance as circular; (b) catoptrics, a subject which deals, in its entirety, with every kind of reflection of light and embraces the theory of images; (c) scenography (scene-painting), as it is called, which shows how objects at various distances and of various heights may so be represented in drawings that they will not appear out of proportion and distorted in shape.
The science of canonics [musical intervals or harmonics] deals with the observed proportions of the notes of the musical scales and investigates the divisions of the monochord. It makes use, throughout, of sense perception and attributes greater importance to the ear than to the intellect, as Plato says.
In addition to these there is the science called mechanics that is a division of the study of material objects perceived by the senses. The science of mechanics embraces: (a) the manufacture of engines useful in war, for example, the engines of defence which Archimedes is said to have constructed against the besiegers of Syracuse; (b) the manufacture of wonderful devices, including those based on (1) air currents, e.g., devices such as Ctesibius and Hero describe, (2) weights (lack of equilibrium producing motion, and equilibrium producing rest, according to the definition in the Timaeus), (3) ropes and cables, by means of which the motion of living beings may be imitated; (c) the study of equilibrium, in general, and of so-called centres of gravity; (d) sphere construction, for depicting the revolutions of the heavenly bodies, a field in which Archimedes worked; (e) in general, the whole subject of the kinetics of material bodies.
There remains astronomy, which is concerned with cosmic motions, the sizes and forms of the heavenly bodies, their illumination, their distances from the earth, and all other subjects of this sort. It makes wide use of sense perception but also has close connection with physical theory. Its branches are: (a) gnomonics, which is concerned with the measuring of the hours by the proper placing of gnomons; (b) meteoroscopy, which investigates the different elevations and distances of stars, and sets forth numerous other theorems of various sorts in the field of astronomy; (c) the science of dioptrics, which, with the use of the proper instruments, investigates the positions of the sun, moon, and the other stars.
The above is the account of the branches of mathematics that we have found in the ancient writers.