and, added to this, the labours of mathematicians who studied the manner in which solid particles could be arranged so as to correspond to the geometrical and physical proportions thus determined by experiment.

But all these were experiments and reasoning upon matter which appears to be as nearly as possible inert; they entirely ignore the power of growth possessed by crystals; indeed, no such power is contemplated by the ordinary theories of crystal structure or could be predicted from them.

So far, then, we may regard the comparison of crystals with plants and all the fanciful ideas concerning their connexion with the origin of life which were suggested by that comparison as only one more example of the dangers of reasoning from analogy. The recollections that I retain of the allusions to that process—in text-books of logic and treatises on Science alike—give me the impression that examples of reasoning from analogy are generally quoted only as instances of its danger and futility. That, indeed, might seem to be the conclusion of my own inaugural lecture; for, when we come to examine the growth and structure of crystals, so far from finding that there is any real likeness to the life and structure of the plants which they resemble, we find nothing but a profound difference.

That, however, is not my real opinion, and was not my opinion fifteen years ago. On the contrary, I have the greatest belief in analogy as one of the most useful guides to discovery, and as the means by which in practice new lines of investigation are most frequently opened and new hypotheses suggested. In fact, I think that most of the advances which are made in science, and especially in scientific theory, have been made with the help of analogies.

If an explanation of any fact in Nature consists in correlating it with some apparently distinct fact, and showing that the two have a common cause, or are connected in a definite manner, how often has the explanation of a new occurrence been suggested by the analogy of some other known occurrence which is brought to mind by the memory; not by a conscious effort of the reason, but by the recollection of a resemblance. Do not some of the standard methods which are familiar in the descriptions or criticisms of scientific discovery, such as ‘reasoning from the known to the unknown’, ‘the adoption of a working hypothesis’, ‘the scientific use of the imagination’, often resolve themselves on analysis into the simple process of being struck by an analogy and being led by it to adopt an explanation or to try an experiment suggested by it?

I daresay that scientific discoverers are ashamed to confess that they may have been led to a theory by a superficial analogy just as they are ashamed to confess that they have hit upon a discovery or an invention by chance, and have found one thing when they were looking for another. But there is no need to be ashamed, for the discoveries only come to those who have the eyes to see, or the knowledge which enables them to remember a resemblance, and who have further the intellectual power to make use of it. Science grows on the acquisition of new knowledge and we must not hesitate to grasp it where we can. There is a danger lest the formulation of the methods of science may deter the inquisitive student from seeking knowledge wherever it is to be found, and make him believe that it is only by the orthodox processes of reasoning based upon a lifetime of training that he is to discover anything new. Rather let him be encouraged to seek any resemblance or analogy that may point a way in the gloom.

It is true that in the past the argument from analogy has often proved dangerous when, because the things possess certain attributes in common, it was inferred that they are alike in other respects. Yet even here it may prove useful. Newton only asserted that the diamond is inflammable because it resembles other inflammable substances in possessing a high refractive power; and yet he turned out to be right, although his analogy was wrong. Moreover, it was the analogy which prompted the experiment. To detect an analogy, to test it, and to find that it cannot be maintained, may be as useful an addition to knowledge as the establishment of a real causal relationship: it has had the effect of setting the worker on to new experiments or observations, and every such step is necessarily an advance. The conscientious search for one thing almost invariably leads to the discovery of another; and, even if it does not, who shall say that it may not be as important and useful, say, to establish a difference as to prove the resemblance which has been suspected?

If, however, reasoning from an analogy may be fallacious when inferences are drawn from the fact that two things possess the same attributes in common, it is, I think, a safer guide when it is used to suggest a new hypothesis; a corpuscular theory of light may be suggested by the analogy between the reflexion of light and the rebounding of elastic bodies; an undulatory theory by the analogy of a wave propagated along a string or on the surface of water. The one theory may be found to fit the facts and the other may be condemned; but without some analogy as a guide how are such theories to be devised? What we require in science are continual stimuli to start us on new experiments, new observations, and new ideas, and prevent us, especially those who are teachers, from constantly repeating the old ones.

I shall be satisfied if the present lecture is regarded as a plea for the use of reasoning from analogy, and as an illustration of its value.

Let me, then, after this exhortation, return to my problem of the growth of crystals, and show how I still think that we may be guided by analogy in seeking to understand it.

So far from regarding the growth of a crystal as of no further importance after it has been proved to be quite different from the growth of a plant, we must still think of it as one of the most interesting and mysterious events, and the one through which we may hope to get the clearest insight into the nature of the crystal itself.

For, after all, when after making physical experiments upon the solid crystal and studying its action upon heat and light we are led to speculate upon the manner in which it is constructed, we must not forget that it grew, and that the material was laid down under conditions which we can only understand by studying what happens on the surface as it grows.

Lord Curzon devoted his Romanes Lecture to the consideration of Frontiers, and explained their interest and importance in the growth of nations. In the study of Science also nothing is more interesting and important than frontier problems. In two senses is this true. The problems which lie on the borderland between two sciences are the most fruitful of all, because they throw light upon each science, and, by bringing into harmony things that were previously distinct and separate, lead to an immediate extension of knowledge.

And also in a more restricted sense. You will find, I think, that the scenes of the most interesting events in Nature are generally those places where different things come into contact and where there is consequently stir and action; where two substances meet to form a new chemical compound; where two bodies touch and react upon each other; at the surface of a solid or a liquid; these are the regions in which events are taking place that we can study and measure with the prospect of discovery.

Impressed by the analogy between the growth of crystals and of living things, I have always felt that, for a proper understanding of the things themselves, the study of their growth is as important for the one as for the other; that to obtain this understanding it is necessary to