In its essence, the positive method is one, as science is one. For it ever tends towards the same end: the establishment of the invariable relations which constitute the effective laws of all observable events, “thus capable of being rationally foreseen from one another.” The positive method proceeds to this by means of a threefold abstraction. It first separates the practical requirements from theoretical knowledge, to be only concerned with the latter, it seeks for the laws of phenomena without troubling itself, at least provisionally, with any possible applications. It also puts aside æsthetic considerations, which ought not to intervene in scientific investigation. Finally--and here is the condition for the very existence of science--the positive method always carefully distinguishes between the abstract and the concrete point of view. It studies not beings, but phenomena. Even in the simplest cases, in astronomy for instance, no general law can be established so long as bodies are considered in their concrete existence. The principal phenomenon has had to be detached, so to speak, so as to submit it alone to an abstract study, afterwards allowing us to return successfully to the consideration of more complex realities. This is what the ancients had known how to do in geometry; and this is what Comte himself has done in the most complex of all sciences, in sociology. Instead of stopping at the concrete reality of history, he determined, by a bold abstraction, the law of the essential movement in human society “leaving to subsequent labours the care of bringing apparent anomalies into line with it.”[90] In the main, these general characteristics of the positive method bring it singularly near to the Cartesian method. Comte’s “Threefold gradual abstraction” seems indeed to have for its end, like Descartes’ analysis, to go back to what is simplest and easiest to know, and then to come down, by a synthetic and progressive advance, towards the reality which is given to us in experience. The one and the other of these methods witness, here, to an effort towards generalising the spirit of the mathematical method. Let us never forget, writes Comte, that the general spirit of positive philosophy was first formed by the culture of mathematics, and that we must necessarily go back so far, in order to know this spirit in its elementary purity. The mathematical processes and formulæ are rarely capable of being applied to the effective study of natural phenomena, when we wish to go beyond the most extreme simplicity in the real conditions of the problems. But “the true mathematical spirit, so distinct from the algebraical spirit, with which it is too often confounded, is on the contrary, constantly of value.”[91]