equation of the curve, they are as truly part of the curve as the more regular branches; and he will perceive that which the ocular observer does not suspect and cannot believe, namely, that these apparent anomalies and useless excrescences cannot be removed without absolutely destroying the system of which they form a part.

I remember reading many years ago (I think in an early number of the 'Cambridge Mathematical Journal') a paper on conjugate points, which supplies a thought capable of useful application either to Babbage's illustration of miracles, or to that adaptation of his illustration to nature which I have ventured to make. The writer of the paper showed that, under a more general treatment of the equation of a curve, conjugate points might be made to disappear altogether, or, at least, to take their place as points in continuous branches. According to this view the locus of the equation was to be sought not in a single plane, as is commonly done, but in the three dimensions of space; and the writer showed that, according to his method of interpretation, a conjugate point would be the point in which a branch of the curve not lying in the ordinary plane of reference crossed that plane. I think that Babbage might have much improved his illustration if he had happened to be familiar with this idea; he might have argued that a miracle would cease to be a miracle if you could regard it outside the plane of human experience;

but, leaving out of consideration what Babbage might have done, I should wish to remark, in connection with the subject of this paper, that possibly the anomalies of nature may, like conjugate points, only be so because we are compelled to move (as it were) in one plane, and that, if we were free from the trammels of experience and laws of human thought, we might possibly discover that even the anomalies of nature are part of a continuous and consistent law.

There is one other illustration of the problems of biology which may be drawn from the simpler problem of the dynamics of a particle, and which may be suitably introduced into this essay.

Every mathematician knows that when it is required to determine the orbit of a particle about a given centre of force, the mere assignment of the law of force is not sufficient for the solution of the problem. This assignment will enable him to integrate his equation, if it be integrable; but his integral will contain three arbitrary constants, for the determination of which he will require to know three elements -namely, the distance, the direction, and the velocity of projection. Almost as much will depend upon these conditions of projection as upon the law of force. I have already pointed out that, in the case of the natural law of the inverse square, the path of a particle may be either a parabola, an ellipse, or a hyperbola; and the question which of these the path will be depends upon the conditions of projection, or the initial

circumstances of motion. Now, in the case of the planets we cannot actually conceive of conditions of projection or initial circumstances of motion; but we positively know that there must have been something equivalent to these; there must have been something corresponding to the three data of an ordinary problem in central forces, which fixed the precise orbit of the earth, for example, and determined its eccentricity.

This being realised, let us pass from the known to the comparatively unknown, from force to life. Let it be granted that all living things have been developed according to some law, not necessarily known, or even capable of description in words, but still a real law of development; does this give us all the elements necessary for the solution of the life problrm? If we say yes, do we not run into the mistake of a beginner who fancies that he can solve a problem of motion round a centre when he has been told what is the law of force? Is it not necessary to know the conditions of projection, the initial circumstances of motion or development? and may not this portion of the data be quite as important as the knowledge of the law of force?

It seems to me, that they who are most anxious to establish the principle of evolution should be the most ready to perceive the necessity of taking into account the consideration of initial circumstances. It is not a complete account of the earth's motion to say

that it is the result of gravitation towards the sun. When a body is once in motion, the forces acting upon it may sufficiently account for all subsequent phenomena; but the distance of the earth from the sun, the small eccentricity of her orbit, and so forth, have nothing whatever to do with gravitation ; they depend upon quite different causes. You may speculate that the planets were originally rings thrown off from the sun, and thus get one step nearer to the beginning of things; but even then there is no cause which can be assigned why the planets should be situated as they are, and why the conditions of our own planet (to go no further than the body with which we are most familiar) should have been such as they are. Given a slowly revolving mass of cooling vapourous matter, and given the possibility of this mass being transformed into a system of bodies, with the sun in the centre, and the planets revolving round it in orbits nearly circular, they themselves also assuming forms nearly spherical, you still need an initial causation which shall determine the configuration of the system and shall make it to be what it is, and no other.

In like manner a quantity of protoplasm with an assumed power of development will not account for existing forms of life, without the additional hypothesis of some causative power to determine the initial circumstances. Given an original germ, and given some power which shall direct the particular original

cause of the development of that germ, and the whole subsequent development is conceivable; but the germ and the law of development left to themselves may be as insufficient as the particle and the law of

attraction.

This view seems undoubtedly to let in the idea of purpose which Haeckel is so anxious to exclude. I do not say that purpose does not come in at an earlier point; but, anyhow, when we come to the consideration of a number of results-all of which are possible under an original law-it would seem difficult to dispense with the supposition of some power, some will, some choice, which has caused one form rather than another to have been adopted in any given part of the kingdom of nature.

Let me add one more suggestion founded upon the supposed analogy between dynamics and biology.

We have seen that the parabola, the ellipse, and the hyperbola are all possible curves for a particle moving round a centre of force. Only one of these curves—namely, the ellipse, and only the ellipse under the condition of small eccentricity, or approximate circularity—can suffice for the orbit of a planet which shall be the home of the highest form of life; namely, that of man. A body moving round a centre of force acting according to the law of the inverse square will not, therefore, form a sufficient definition of a world like our own. The original conditions of motion, the initial circumstances as a mathematician