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been sufficiently realised. Cosmologists endeavoured for a long time, and have not ceased endeavouring, to reduce biology to a mere appendage to their own science. And though the best biologists have begun to resist the pressure thus put upon them, they have fallen into a retrograde course of thought in their wish to prosecute the study of life, irrespectively of its essential relations to the study of the world. The two schools-Materialist and Spiritualist -are alike inconsistent with the true spirit of natural philosophy, whether from the scientific or logical aspect. They both ignore the very root of the fundamental dogma of Positivism, the invariability of external relations....... The mere conception of such relations pre-supposes, as Kant so clearly saw, an object governed by them, a subject conscious of their existence........... In a word, every phenomenon implies a definite relation between an object and a subject ...... Ultimately the existence of life pre-supposes a passive environment, directly or indirectly sustaining the organism, and fixing its habitation."

The laws of that "passive environment," man's milieu, are summed up in Cosmology. The title of Cosmology stands for a whole group of inorganic sciences—mathematics, astronomy, physics, and chemistry. As it is obvious that the world was before man, it is fitting that the study of it should come first.

"Our world, properly so-called-that is to say, the total sum of the inorganic objects with which Humanity is concerned-exhibits, broadly speaking, two series of phenomena quite distinct in character. There are phenomena which are wholly inaccessible to human intervention; there are others more or less within its range. Hence cosmology, subjectively considered, falls into two great sciences, differing in their logical character, differing also in their objective features. Phenomena which cannot be modified have necessarily more independence and generality than all others, the existence of these others being conditional upon their own. Thus the classification indicated by the subjective point of view is identical with the objective division of science, which formed the guiding principle of my Positive Philosophy.

"In defining the scope of these two inorganic sciences, it may be remarked that they are both occupied with one common object-the earth. The first explains its general relations with the other heavenly bodies; the second deals with the earth itself specially. The distinction exists already

in the two positive meanings of the world. We shall, therefore, speak for the future of the two divisions of our subject under the terms of celestial and terrestrial cosmology. The latter must be regarded as subordinate to the former, on the same principle on which social philosophy is subordinate to natural philosophy, and on which biology is subordinate to cosmology."

Comte, as we said, arranges the sciences upon the principle of decreasing generality and increasing speciality, commencing with Mathematics and ending with Sociology. Mathematics reveal inductive principles, while training the mind in deductive principles. Wallis treated mathematics inductively (see the opening of his work).

But, the hearer may say, what does Comte mean by the term law? and, further, what does he mean by "cosmology subjectively considered "? By law Comte meant the perception of unity among variety in phenomena, many of which may be inconstant. This is a purely relative conception-that is, it relates to man, the subject, the percipient. The unity man perceives may not (most likely does not) exist outside himself-exist, that is, objectively. Law, therefore, is a convenient term for registering his subjective impressions and experiences of objective phenomena. This should constantly be borne in mind, for there is a tendency in most Scientists now-a-days, to treat law as if it were something objective, something external to man, having an existence of its own independent of man, than which a falser notion could not exist. It is one of the imperfections of thought, and must be allowed for as such.

Dr. Bridges observes, in his "Evolution and Positivism," that "law is no expression of an outside reality. It is an expression of an altogether human fact. It is simply the perception on man's part, more or less perfect, of regularity in the midst of irregularity, of unity amidst plurality. I see before me a thousand triangles, and I could easily construct a thousand or a million, none of which shall have the same shape. But in one thing they all agree-viz., that their angles are equal to two right angles. That perception of unity amidst divergence is what I call a law."

The tendency to treat Ideals as Reals, Abstractions as Concretes, and to reason from them as such, has been pointed out by Mr. George H. Lewes in his "Problems of Life and Mind." The diligent student of Comte is constantly on his guard in this matter, and is not allowed to fall into an error too common among mere Scientists.

By "Cosmology subjectively considered," Comte meant that knowledge of the laws of the External World which would most promote the well-being of Humanity-material, mental, and moral. Humanity is to be the central thought of the study of all phenomena whatsoever. "All that is necessary is that the Subjective Method be no longer directed to the useless search after Causes. It must be applied, as the Objective Method is applied, exclusively to the discovery of Laws: the end in view being to ameliorate human nature and human circumstances. In a word, the Subjective method must cease to be theological, and must become Sociological. And this transformation is now, for the first time, rendered possible by the recent application of positive science to the general evolution of Humanity." Humanity is, in Positivism, what God is in the old theology-the central idea and object of all thought, love, and action; and as the glory of God is the object of all Catholic (or Christian) worship, so the glory and well-being of Humanity-a real, and not a fictitious being-is the object of all Positivist devotion. To study Science, without reference to Humanity, would be as foolish as to study theology without reference to God. And as all knowledge in the old theology was subordinated to the knowledge of God, so in Positivism all knowledges (or sciences) are subordinated to that of Humanity. There is, it is obvious, a cardinal distinction between the two beings-Humanity being Relative, while God is non-Relative. The theologist might occasionally doubt whether he was promoting God's glory, and, if he were promoting it, how much the better God was for it. The Great Being Positivists worship needs their best services, and is, directly or indirectly, improved by them. There is no mysticism in this statement. This fact is demonstrably true to those who meditate on it. The thought of Humanity acts with Positivists as a controlling principle, regulating their thoughts, emotions, and actions. It prevents them studying the Sciences for their own sake, as if that study were the Be-all and End-all here. It does not comprehend what is meant by the "pursuit of knowledge for its own sake." It regards such an object as purely chimerical, not to say immoral. "Knowledge for its own sake" means, if it has any meaning at all, knowledge pursued for individualist ends and purposes-for the promotion of vanity and self-conceit. To the Positivist each Science is to be studied sufficiently for the requirements of that which stands next above it in the scale. The Cosmo

logy which is out of relation to Humanity is to him no Cosmology at all.

It is time that Comte spoke in his own person, and we shall venture to quote somewhat freely his profound and comprehensive views of the philosophy of cosmology.

"Mathematics study directly universal existence when reduced to its simplest-and consequently, lowest-phenomena, the phenomena on which necessarily rest all other real attributes of that existence. These fundamental properties of any being whatever are number, extension, movement. Whatever cannot be considered under these three points can have no existence except in our understanding. But nature shows us many beings of whom we can know nothing beyond the elementary attributes. Such are the stars. They are from their distance only accessible to us by the sight. They admit, therefore, only of mathematical study, quite sufficient, it must be allowed, to know our true relations towards them. Therefore it is that astronomy will always furnish the most direct and complete application of mathematical science. Still, if the general laws of number, extension, motion, could have been studied nowhere but in the heavenly bodies, they never would have been discovered, extremely simple though they be. But, as you find them everywhere, they were open to discovery in more accessible cases. We had but to put aside by a series of unconscious abstractions the other attributes of matter which complicated the question.

"And here let me point out how, under the auspices of our hierarchical principal, we may get the true internal distribution of each great science as naturally as the general co-ordination of the sciences. Mathematics cannot be reduced to less than three constituent parts-the calculus, geometry, mechanics. These three are a progressive scries from the historical no less than the dogmatical point of view-a series essentially analogous to that which is seen on a larger scale in the whole of the abstract system. The ideas of number are certainly more universal and simpler than those even of extension, and these on the same ground, in their turn, precede those of motion.

"In the case of most of the stars our real knowledge cannot extend beyond an accurate enumeration. We cannot go so far even as to say what their shape or size is, nor are we concerned with it. Phenomena as well as beings admit the numerical point of view. It leaves all things indistinct; but it is really the only one universally applicable, since

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none of our thoughts but come under it. It is, of course, but a rough view, but it may be put to a good use. may use number to perfect our ideas of harmony and stability. In fact the best types of these are found in number, and so you see, children of themselves begin their initiation in abstract science by simple speculations in number long before they begin to think on the attributes of space. Comte goes on to say that "the Greeks, forward as they were in geometry, never could master even the outline of mechanics, except for some cases of equilibrium, as they never had a glimpse of the elementary laws of motion, the highest part of the domain of mathematics."

"If we compare these three essential parts of mathematics, we see that the calculus of which algebra, rather than arithmetic, is the principal development, has a logical destination in particular, over and above its peculiar and direct use. Its real essential adaptation is to enlarge to the utmost our power of deduction. The study of extension and motion acquires, by the introduction of the calculus, a character of generality and coherence which they could not have had unless all their problems were transformed into mere questions of number. From the scientific point of view, on the other hand, it is geometry and mechanics which mainly constitute mathematics, for they, and they alone, enable us to form our theory of universal existence, viewed as passive in geometry, in action in mechanics. "Mechanics, then, take an important encyclopædic position as the necessary transition between mathematics and physics. The peculiar characteristics of both are found in close combination in mechanics. In them logic is seen to be no longer purely deductive, as it was supposed to be in geometry, owing to the extreme facility of making the required inductions. In mechanics we begin to feel distinctly the need of an inductive basis, nor is it easy to trace such a basis among our concrete observations. We require it to allow free scope for the abstract conceptions which are to connect with the general problem of the composition and communication of motion. It was the want of this external foundation which prevented mechanics as a science from being developed till the seventeenth century."

The three laws of motion which were inductively discovered by Kepler, Galileo, and Newton are these:

"All motion is naturally rectilinear and uniform;" or, as it is expressed in the tenth law of the "First Philosophy": "Every state, statical or dynamical, has an in

herent tendency to continue as it is without change, resisting all disturbance from without." (Kepler.)

2. "The independence of the movements of the bodies that form parts of a system-the independence, that is, in regard to their movement in common as a system." Thus expressed in the eleventh law of the "First Philosophy": "Every system maintains its constitution, whether in exercise or at rest, when its constituent parts are subjected to simultaneous changes, provided that the changes affect all the parts in equal degree." (Galileo.)

3. "In every case of mechanical collision action is always accompanied by equal and contrary reaction." Thus expressed in the twelfth law of the "First Philosophy": "Reaction and action are always equivalent if the degree of each is measured in accordance with the peculiar nature of each collision." (Newton.)

"These three laws together are sufficient to enable us to enter on a deductive solution of the general problem of mechanics by bringing the more complicated cases under the more simple."

These laws Comte regards as mainly Objective. Mathematics had before their discovery established Subjective laws, the only laws which geometry and the calculus reveal. Subjective implies objective laws. This truth, he says, men do not as yet see.

"These general laws will be of use to you as giving a direct explanation of numbers of phenomena of daily experience, in the midst of which you live without understanding or even perceiving them. They are eminently fitted to make you feel what it is which constitutes the true genius of science. Finally, you should remark how each one of them naturally ranges itself under a law common to all phenomena whatever-to social and moral quite as much as to simply material phenomena. The first law connects with the law of persistence which we trace everywhere. The second connects with the law which recognises the independence of the action of the part as regards the conditions common to the whole. By this law, in social questions we find it possible to reconcile order with progress. As for the third, it is at once universally applicable, the application varying only as the influences vary. This philosophical connection of the three laws of motion completes our estimate of the importance, in an encyclopædic point of view, of mechanics, the last portion of the domain of mathematics."

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