The human mind, in order to cover this immense field, the extension of which it was very late in apprehending, may pursue two different methods. Perfect geometry would, indeed, be the one which would demonstrate all the properties of all imaginable forms, and this can be obtained in two ways. Either we can successively conceive each of the forms, the triangles, the circle, the sphere, the ellipse, etc., and seek for the properties of each one of them. Or else we can group together the corresponding properties of various geometrical forms, in such a way as to study them together, and, so to speak, to know beforehand their application to such and such a form which we have not yet examined. “In a word,” says Comte, “the whole of geometry can be ordered, either in relation to bodies which are being studied, or in relation to phenomena which are to be considered.” The first plan is that of the geometry of the ancients, or special geometry; the second is that of the geometry since Descartes, or general geometry.[110]
At its origin geometry could only be special. The ancients, for instance, studied the circle, the ellipse, the parabola, etc., endeavouring, in the case of each geometrical form, to add to the number of known properties. But, if this line of advance had been the only one which could be followed, the progress of geometry would never have been a very rapid one. The method invented by Descartes has transformed this science, by enabling it to become general, and to abandon the individual study of geometrical forms for the common study of their properties. This revolution has not always been well understood. Often in teaching mathematics, its bearings are not sufficiently shown. From the manner in which it is usually presented, this “admirable method” would at first seem to have no other end than the simplification of the study of conic sections or of some other curves, always considered one by one according to the spirit of ancient geometry. This would not be of great importance. The distinctive character of our modern geometry consists in studying in a general way the various questions relating to any lines or surfaces whatever by transforming geometrical considerations and researches into analytical considerations and researches.[111]
All geometrical ideas necessarily relate to the three universal categories; magnitude, form, position. Magnitude already belongs to the domain of quantity. Form can be reduced to position, since every form can be considered as the result of the advance of a point, that is to say of its successive positions. The problem is therefore to bring all ideas of situation whatever back to ideas of magnitude. How did Descartes solve it? By generalising a process which we may say is natural to the human mind, since it comes spontaneously into being under the stress of necessity. Indeed, if we must indicate the situation of an object without showing it immediately, do we not refer it to others which are known, by stating the magnitude of geometrical elements by which we conceive the object to be connected with them? Geographers act in the same way in their science to determine the longitude and latitude of a place, and astronomers to determine the right ascension and the declination of a star. These geographical and astronomical co-ordinates fulfil the same office as the Cartesian co-ordinates. The only difference, but it is a capital one, consists in the fact that Descartes carried this method to the highest degree of abstract generality thus giving it its maximum of fertility and power.
Although general geometry is infinitely superior to special geometry it cannot, nevertheless, altogether dispense with the latter. As the ancients did, so it will always be necessary to begin with special geometry. For general geometry rests upon the use of calculation. But if, as Comte has said, geometry is truly a science of facts calculation will evidently never be able to supply us with the first knowledge of these facts. In order to lay the foundations of a natural science simple mathematical analysis would never suffice, nor could it give a fresh demonstration of it, when these foundations have already been laid. Before all things a direct study of the subject is necessary, until the precise relations are discovered. “The application of mathematical analysis can never begin any science whatever, since it could never take place except when the science has been sufficiently elaborated to establish, in relation to the phenomena under consideration, some equations which might serve as a starting-point for analytical work.”[112] In a word, the creation of analytical geometry does not prevent geometry from remaining a natural science. Even when it has become as purely rational as possible, it none the less remains rooted in experience.
IV.
The second part of concrete mathematics (mechanics) is also one of the natural sciences which owes its marvellous progress to analysis. Here again we must distinguish the data which are at the basis of science, and which are facts, from the abstract development undergone by this science because of the simplicity of these facts and the precision of the relations which exist between them. The distinction between what is “really physical” and what is “purely logical”[113] is not always an easy one. We must, however, separate facts furnished by experience, from artificial conceptions whose object is to facilitate the establishment of general laws of equilibrium and of motion.
Only to consider inertia in bodies is a fiction of this kind. Physically the force of inertia does not exist. Nature nowhere shows us bodies which are devoid of internal activity. We term those which are not alive inorganic, but not inert. Were gravitation alone common to all molecules, it would suffice to prevent the conception of matter as devoid of force. Nevertheless, mechanics only considers the inertia of bodies. Why? Because this abstraction presents many advantages for the study, “without, moreover, offering disadvantages in the application.” Indeed, if mechanics had to take into account the internal forces of bodies and the variations of these forces, the complications would immediately become such that the facts could never be submitted to calculation. Mechanics would run the risk of losing its character as a mathematical science. And, on the other hand, as it only considers the movements in themselves, regardless of their mode of production, it is always lawful for mechanics to replace, if necessary, the internal forces by an equivalent external force” applied to the body. The inertia of matter is therefore an abstraction, the end of which is to secure the perfect homogeneity of mechanical science, by allowing us to consider all moving bodies as identical in kind, and all forces as of the same nature.
The “physical” character of this science is again evident from the consideration of the three fundamental laws upon which it rests.[114]
The first, called Kepler’s law, is thus defined: “All movement is naturally rectilinear and uniform; that is to say, any body subject to the action of a single force which acts upon it instantaneously, moves constantly in a straight line with invariable speed.” It has been said that this law is derived from the principle of sufficient reason. The body must continue in a straight line because there is no reason why it should deviate from it more on one side than on the other. But, answers Comte, how do we know that there is no reason for the body to deviate, except precisely because we see that it does not deviate? The reasoning “reduces itself to the repetition in abstract terms of the fact itself, and to saying that bodies have a natural tendency to move in a straight line, which is precisely the proposition which we have to establish.” It is by similar arguments that the philosophers of antiquity, and especially Aristotle, had, on the contrary been led to regard circular motion as natural to the stars, in that it is the most perfect of all, a conception which is only the abstract enunciation of a imperfectly analysed phenomenon. The tendency of bodies to move in a straight line with constant speed is known to us by experience.
The second fundamental law of mechanics, called Newton’s law, expresses the constant equality of action and reaction. It is pretty generally agreed to-day to consider this law as resulting from the observation of facts. Newton himself understood it so.