perceives only transient, isolated, internal conditions; he makes use of these to affirm and name exterior states, positions, movements, changes, and avails himself of them for nothing else. He can only attain to facts, whether within or without, sometimes transient, when his impression is not repeated; sometimes permanent, when his impression many times repeated, makes him suppose that it will be repeated as often as he wishes to experience it. He only grasps colors, sounds, resistances, movements, sometimes momentary and variable, sometimes like one another, and renewed. To group these facts more advantageously, he supposes, by an artifice of language, qualities and properties. We go even further than you: we think that there are neither minds nor bodies, but simply groups of present or possible movements or thoughts. We believe that there are no substances, but only systems of facts. We regard the idea of substance as a psychological illusion. We consider substance, force, and all the modern metaphysical existences, as the remains of scholastic entities. We think that there exists nothing but facts and laws, that is, events and the relations between them; and we recognize, with you, that all knowledge consists first of all in connecting or adding fact to fact. But when this is done, a new operation begins, the most fertile of all, which consists in reducing these complex into simple facts. A splendid faculty appears, the source of language, the interpreter of nature, the parent of religions and philosophies, the only genuine distinction, which according to its degree, separates man from the brute, and great from little men. I mean Abstraction, which is the power of isolating the elements of facts, and of considering them one by one. My eyes follow the outline of a square, and abstraction isolates its two constituent properties, the equality of its sides and angles. My fingers touch the surface of a cylinder, and abstraction isolates its two generative elements, the idea of a rectangle, and of the revolution of this rectangle about one of its sides as an axis. A hundred thousand experiments develop for me, by an infinite number of details, the series of physiological operations which constitute life; and abstraction isolates the law of this series, which is a round of constant loss and continual reparation. Twelve hundred pages teach me Mill's opinion on the various facts of science, and abstraction isolates his fundamental idea, namely, that the only fertile propositions are those which connect a fact

with another not contained in the first. Everywhere the case is the same. A fact, or a series of facts, can always be resolved into its components. It is this resolution which forms our problem, when we ask what is the nature of an object. It is these components we look for when we wish to penetrate into the inner nature of a being. These we designate under the names of forces, causes, laws, essences, primitive properties. They are not new facts added to the first, but an essence or extract from them; they are contained in the first, they have no existence apart from the facts themselves. When we discover them, we do not pass from one fact to another, but from one to another aspect of the same fact; from the whole to a part, from the compound to the components. We only see the same thing under two forms; first, as a whole, then as divided: we only translate the same idea from one language into another, from the language of the senses into abstract language, just as we express a curve by an equation, or a cube as a function of its side. It signifies little whether this translation be difficult or not; or that we generally need the accumulation or comparison of a vast number of facts to arrive at it, and whether our mind may not often succumb before accomplishing it. However this may be, in this operation, which is evidently fertile, instead of proceeding from one fact to another, we go from the same to the same; instead of adding experiment to experiment, we set aside some portion of the first; instead of advancing, we pause to examine the ground we stand on. There are, thus, fruitful judgments, which, however, are not the results of experience: there are essential propositions, which, however, are not merely verbal : there is, thus, an operation, differing from experience, which acts by cutting down instead of by addition; which, instead of acquiring, devotes itself to acquired data; and which, going farther than observation, opening a new field to the sciences, defines their nature, determines their progress, completes their resources, and marks out their end.

This is the great omission of your system. Abstraction is left in the background, barely mentioned, concealed by the other operations of the mind, treated as an appendage of Experience; we have but to re-establish it in the general theory, in order to reform the particular theories in which it is absent.

III.

To begin with Definitions. Mill teaches that there is no definition of things, and that when you define a sphere as the solid generated by the revolution of a semicircle about its diameter, you only define a name. Doubtless you tell me by this the meaning of a name, but you also teach me a good deal more. You state that all the properties of every sphere are derived from this generating formula; you reduce an infinitely complex system of facts to two elements; you transform sensible into abstract data; you express the essence of the sphere, that is to say, the inner and primordial cause of all its properties. Such is the nature of every true definition; it is not content with explaining a name, it is not a mere description; it does not simply indicate a distinctive property; it does not limit itself to that ticketing of an object which will cause it to be distinguished from all others. There are, besides its definition, several other ways of causing the object to be recognized; there are other properties belonging to it exclusively: we might describe a sphere by saying that, of all bodies having an equal surface, it occupies the most space; or in many other ways. But such descriptions are not definitions ; they lay down a characteristic and derived property, not a generating and primitive one; they do not reduce the thing to its factors, and reconstruct it before our eyes; they do not show its inner nature and its irreducible elements. A definition is a proposition which marks in an object that quality from which its others are derived, but which is not derived from others. Such a proposition is not verbal, for it teaches the quality of a thing. It is not the affirmation of an ordinary quality, for it reveals to us the quality which is the source of the rest. It is an assertion of an extraordinary kind, the most fertile and valuable of all, which sums up a whole science, and in which it is the aim of every science to be summed up. There is a definition in every science, and one for each object. We do not in every case possess it, but we search for it everywhere. We have arrived at defining the planetary motion by the tangential force and attraction which compose it; we can already partially define a chemical body by the notion of equivalent, and a living body by the notion of type. We are striving to transform every group of phenomena into certain laws, forces, or abstract notions. We endeavor to attain in every object the generating elements, as we

do attain them in the sphere, the cylinder, the circle, the cone, and in all mathematical loci. We reduce natural bodies to twc or three kinds of movement-attraction, vibration, polarization —as we reduce geometrical bodies to two or three kinds of elements-the point, the movement, the line; and we consider our science partial or complete, provisional or definite, according as this reduction is approximate or absolute, imperfect or complete.

IV.

The same alteration is required in the Theory of Proof. According to Mill, we do not prove that Prince Albert will die by premising that all men are mortal, for that would be asserting the same thing twice over; but from the facts that John, Peter, and others, in short, all men of whom we have ever heard, have died. -I reply that the real source of our inference lies neither in the mortality of John, Peter, and company, nor in the mortality of all men, but elsewhere. We prove a fact, says Aristotle,1 by showing its cause. We shall therefore prove the mortality of Prince Albert by showing the cause which produces his death. And why will he die? Because the human body, being an unstable chemical compound, must in time be resolved; in other words, because mortality is added to the quality of man. Here is the cause and the proof. It is this abstract law which, present in nature, will cause the death of the prince, and which, being present to my mind, shows me that he will die. It is this abstract proposition which is demonstrative; it is neither the particular nor the general propositions. In fact, the abstract proposition proves the others. If John, Peter, and others are dead, it is because mortality is added to the quality of man. If all men are dead, or will die, it is still because mortality is added to the quality of man. Here, again, the part played by Abstraction has been overlooked. Mill has confounded it with Experience: he has not distinguished the proof from the materials of the proof, the abstract law from the finite or indefinite number of its applications. The applications contain the law and the proof, but are themselves neither law nor proof. The examples of Peter, John, and others, contain the cause, but they are not the cause. It is not sufficient to add up the cases, we must extract from them

1 See the Posterior Analytics, which are much superior to the Prior-di airiwv kai προτέρων.

the law. It is not enough to experimentalize, we must abstract. This is the great scientific operation. Syllogism does not proceed from the particular to the particular, as Mill says, nor from the general to the particular, as the ordinary logicians teach, but from the abstract to the concrete; that is to say, from cause to effect. It is on this ground that it forms part of science, the links of which it makes and marks out; it connects principles with effects; it brings together definitions and phenomena. It diffuses through the whole range of science that Abstraction which defi nition has carried to its summit.

V.

Abstraction explains also axioms. According to Mill, if we know that when equal magnitudes are added to equal magnitudes the wholes are equal, or that two straight lines cannot enclose a space, it is by external ocular experiment, or by an internal experiment by the aid of imagination. Doubtless we may thus arrive at the conclusion that two straight lines cannot enclose a space, but we might recognize it also in another manner. We might represent a straight line in imagination, and we may also form a conception of it by reason. We may either study its form or its definition. We can observe it in itself, or in its generating elements. I can represent to myself a line ready drawn, but I can also resolve it into its elements. I can go back to its formation, and discover the abstract elements which produce it, as I have watched the formation of the cylinder and discover the revolution of the rectangle which generated it. It will not do to say that a straight line is the shortest from one point to another, for that is a derived property; but I may say that it is the line described by a point, tending to approach towards another point, and towards that point only: which amounts to saying that two points suffice to determine a straight line; in other words, that two straight lines, having two points in common, coincide in their entire length; from which we see that if two straight lines approach to enclose a space, they would form but one straight line, and enclose nothing at all. Here is a second method of arriving at a knowledge of the axiom, and it is clear that it differs much from the first. In the first we verify; in the second we deduce it. In the first we find by experience that it is true; in the second we prove it to be true. In the first we admit the