be deplored) to make the individual more free to dispose of his means and efforts religiously according to his own will, and not according to the will of others. This movement, which has now continued for six hundred years, and extended over the whole area of Christendom in spite of the most persevering and zealous efforts of the most varied kinds to reverse and repress it- a movement which has been the very flower of human progress, seeing that all the highest races of the world have concurred in it, and that the advance of human society through all the ages which have preceded it was not to be compared with it-such a movement cannot surely, by any one but a Manichean, be deemed other than one specially ordained by God's providence for a wise and good end.

-The movement of human progress, judged by what we see up to our own days, may be compared with the development of the individual man. It is a process of intellectual differentiation and integration, a movement from direct and simple apprehensions to more and more reflex, self-conscious, and complex comprehensions. What has taken place for ages unconsciously and unintentionally now takes place with deliberation and full consciousness; and this characteristic of deliberate self-consciousness is daily spreading over a wider area of human action. All these facts being borne in mind, and the lesson they convey being seen to coincide with and reinforce the meaning of life before arrived at, we may venture without presumption to affirm that the course of human life has been arranged so to afford a constantly increasing field for more and more intelligent and deliberate and fully intentioned right volitions performed by individuals presenting the greatest variety of increasing individual excellences, and grouped in aggregations tending to manifest greater and greater degrees of utility, harmony, and beauty, and, above all, more and more favouring the full and free development of the individual conscience.

But the considerations here put forward as to the totality of human life may be yet further extended, and some speculation may be even hazarded as to the aim and meaning of all life which exists or has existed in the beautiful world we inhabit.

Geology and paleontology show us that a great process of evolution has taken place in the past, and is taking place now. In examining the creatures around us we see varying degrees of perfection expressed by the terms organic and inorganic, animal and vegetable existence. Science gives good grounds for believing that before this world was the theatre of organic life-processes, it had existed as an inorganic mass of highly complex materials, each with its special properties, and that animal life (at least in all but its lowest forms) was preceded in existence by the kingdom of plants. Certainly we may affirm that all these forms of life-the merely inorganic, the vital, and the sentient-coexisted for untold ages before the introduction into the world of the self-conscious life of man.

The inorganic world long existed alone, and could so have persisted indefinitely. It had, and has, no need of living organisms for its being. The vegetable world, which feeds upon inorganic matter, could not exist without what had preceded it, but might for untold ages, or for ever, have lived and flourished, nourished but by showers and breezes, fertilised but by the wind, with no hum of insects about its inconspicuous flowers, and with no songsters amidst its groves.

The animal world, which is necessitated ever directly or indirectly to feed upon the vegetable world, could not exist without the earth's green, vital, but insentient vesture, yet might for untold ages, or for ever, have lived, undominated by the hunter and with no experience of domestication and pampered servitude.

Man, though capable of sustaining life on vegetable food alone, could never have attained his high civilisation without the aid of his dogs and horses, his flocks and herds. The animal world has been necessary to him, as he is.

Thus an increase of service, and a consequently increased dependence, are manifest as we ascend through these degrees of existence. Cosmical entities and their laws serve organic being more than inorganic, sentient being more than insentient, rational being more than sentient. Now, if the purposes of God are revealed by history in which the free will of man intervenes, a fortiori they are revealed by the history of the irrational creation, which responds absolutely to his will. Every Theist, therefore, is logically compelled to affirm that God has evidently willed most service to man of all his earthly creatures. Thus a successively increasing purpose runs through the irrational creation up to him. All the lower creatures have ministered to him, and have as a fact prepared the way for his existence. Therefore, whatever ends they also serve, they exist especially for him. But, as the aim of the life of man is the exercise of right volition, the life of all lower creatures must have been ultimately directed to that same end. When, then, in the unfathomable abyss of past time, the first algoid film of vegetable substance coloured the water of some primeval pool, and when the most undifferentiated and nascent organisms first moved upon the surface of some dismal, silent morass, the true end of existence of such lowly forms of incipient life was the fulfilment of the moral law, a fulfilment to be brought about after what seems an eternity to the imagination, but which reason cannot doubt to have been but in its due time and season.

And now, before concluding, dare we ask yet one more question? Why was this so ordained? To ourselves, from the human standpoint, we see that the fulfilment of duty is a sufficient end for the life of each of us, and, convinced that happiness and virtue must ultimately coincide, it is a fully satisfying end. But our mind seems, in speculating on the ultimate cause, to demand some further answer. Those who feel satisfied that they have a right to conclude analogically

from the human to the Divine mind, may look within and see if it be possible to conceive some further motive underlying even the moral law. Falteringly we may answer that but one conception seems capable of satisfying our minds, utterly inadequate as any idea of ours must necessarily be to respond to the inconceivable reality. That one conception, the conception which seems to take us even deeper into God's essence than the conception of 'right,' is the correlated conception expressed by the sublimest and noblest of all words—'Love.'

The meaning of life, which it has thus been sought to extract from the combined interrogations of consciousness and consideration of natural phenomena, has given occasion to the enunciation of various maxims and principles, some of which have probably struck the reader as altogether abstract and devoid of practical utility. But the writer has purposely deferred the consideration of circumstances and limitations which must necessarily be entered upon in considering human life in the concrete. Here causes and principles have alone occupied us, but this is because the subject is to be completed in another paper devoted to the consideration, not of the aim and meaning of life, but of its ordering and government.

ST. GEORGE MIVART.

PHILOSOPHY OF THE PURE SCIENCES.1

III. THE UNIVERSAL STATEMENTS OF ARITHMETIC.

We have now to consider a series of alleged universal statements, the truth of which nobody has ever doubted. They are statements belonging to arithmetic, to the science of quantity, to pure logic, and to a branch of the science of space which is of quite recent origin, which applies to other objects besides space, and is called the analysis of position. I shall endeavour to show that the case of these statements is entirely different from that of the statements about space which I examined in my last lecture. There were four of those statements: that the space of three dimensions which we perceive is a continuous aggregate of points, that it is flat in its smallest parts, that figures may be moved in it without alteration of size or shape, and that similar figures of different sizes may be constructed in it. And the conclusion which I endeavoured to establish about these statements was, that, for all we know, any or all of them may be false. In regard to the statements we have now to examine, I shall not maintain a similar doctrine; I shall only maintain that, for all we know, there may be times and places where they are unmeaning and inapplicable. If I am asked what two and two make, I shall not reply that it depends upon circumstances, and that they make sometimes three and sometimes five; but I shall endeavour to show, that unless our experience had certain definite characters, there would be no such conception as two, or three, or four, and still less such a conception as the adding together of two numbers; and that we have no warrant for the absolute universality of these definite characters of experience.

In the first place it is clear that the moment we use language at all, we may make statements which are apparently universal, but which really only assign the meaning of words. Whenever we have called a thing by two names, so that every individual of a certain class bears the name A and also the name B, then we may affirm

This paper completes the publication of the substance of a series of three lectures delivered at the Royal Institution in 1873. The first two were published in the Contemporary Review while it was under the guidance of the present Editor of the Nineteenth Century. The third lecture contained-besides the substance of what is now here published—a statement of the doctrines afterwards set forth in a paper on the nature of things in themselves published in Mind, 1878.

the apparently universal proposition that every A is B. But it is really only the particular proposition that the name A has been conventionally settled to have the same meaning as the name B. I may, for example, enunciate the proposition that all depth is profundity, and all profundity is depth. This statement appears to be of universal generality; and nobody doubts that it is true. But for all that it is not a statement of some fact which is true of nature as a whole; it is only a statement about the use of certain words in the English language. In this case the meaning of the two words is coextensive; one means exactly as much as, and no more than, the other. But if we suppose the word crow to mean a black bird having certain peculiarities of structure, the statement, All crows are black,' is in a similar case. For the word black has part of the meaning of the word crow; and the proposition only states this connection between the two words. Are the propositions of arithmetic, then, mere statements about the meanings of words? No; but these examples will help us to understand them. Language is part of the apparatus of thought; it is that by which I am able to talk to myself. But it is not all of the apparatus of thought; and just as these apparently general propositions, 'All crows are black,' 'All depth is profundity,' are really statements about language, so I shall endeavour to show that the statements of arithmetic are really statements about certain other apparatus of thought.

We know that six and three are nine. Wherever we find six things, if we put three things to them, there are nine things altogether. The terms are so simple and so familiar, that it seems as if there were no more to be said, as if we could not examine into the nature of these statements any further.

No more there is, if we are obliged to take words as they stand, with the complex meanings which at present belong to them. But the real fact is that the meanings of six and three are already complex meanings, and are capable of being resolved into their elements. This resolution is due-I believe equally and independently-to two great living mathematicians, by whose other achievements this country has retained the scientific position which Newton won for her at a time of fierce competition when no ordinary genius could possibly have retained it. The conception of number, as represented by that word and also by the particular signs, three, six, and so on, has been shown to embody in itself a certain proposition, upon the repetition of which the whole science of arithmetic is based. By means of this remark of CAYLEY and SYLVESTER, we are able to assign the true nature of arithmetical propositions, and to pass from thence by an obvious analogy to those other cases that we have to consider.

What do I do to find out that a certain set of things are six in number? I count them; and all counting, like the names of