The Divergent Scope for New Discovery.

In the writings of some recent philosophers, especially of Auguste Comte, and in some degree John Stuart Mill, there is an erroneous and hurtful tendency to represent our knowledge as assuming an approximately complete character. At least these and many other writers fail to impress upon their readers a truth which I think cannot be too constantly borne in mind, namely, that the utmost successes which our scientific method can accomplish will not enable us to comprehend more than an infinitesimal fraction of what there doubtless is to comprehend. Professor Tyndall seems to me open to the same charge in a less degree. He remarks1 that we can probably never bring natural phenomena completely under mathematical laws, because the approach of our sciences towards completeness may perhaps be asymptotic, so that however far we may go, there may still remain some facts not subject to scientific explanation. He thus likens the supply of novel phenomena to a convergent series, the earlier and larger terms of which have been successfully disposed of, so that only comparatively minor groups of phenomena remain for future investigators to occupy themselves upon. On the contrary, as it appears to me, the supply of new and unexplained facts is divergent in extent, so that the more we have explained, the more there is to explain. The further we advance in any generalization, the more numerous and intricate are the exceptional cases still demanding further treatment. The experiments of Boyle, Mariotte, Dalton, Gay-Lussac, and others, upon the physical properties of gases might seem to have exhausted that subject by showing that all gases obey the

VOL. II.

1 'Fragments of Science,' p. 362.

G g

same laws as regards temperature, pressure, and volume. But in reality these laws are only approximately true, and the divergences have afforded a wide and yet quite unexhausted field for further generalization. The more recent discoveries of Cagniard de la Tour and Professor Andrews might seem to have summed up many of these exceptional facts under a wider generalization, but in reality they have opened to us vast new regions of interesting inquiry, and they leave wholly untouched the question why one gas or one substance behaves differently from another.

The science of Crystallography is that perhaps in which the most precise and general laws have been detected, but it would be utterly untrue to assert that it has lessened the area of future discovery. We can show that each one of the seven or eight hundred forms of calcite is derivable by plain geometrical modifications from an hexagonal prism, but who has attempted to explain the molecular forces producing these modifications, or the chemical conditions in which they arise? The law of isomorphism is an important generalization, for it establishes a general resemblance between the forms of crystallization of natural classes of elements. But if we examine a little more closely we find that these forms are only approximately alike, and the divergence peculiar to each substance is an unexplained exception.

By many similar illustrations it might be readily shown that in whatever direction we extend our investigations and successfully harmonize a few facts, the result is only to raise up a host of other unexplained facts. Can any scientific man venture to state that there is less opening now for new discoveries than there was three centuries ago? Is it not rather true that we have but to open a scientific book and read a page or two, and we shall in all probability come to some recorded phenomenon of which

no precise explanation can yet be given? In every such fact there is a possible opening for new discoveries, and it can only be the fault of the investigator's mind if he can look around him and find no scope for the exercise of his faculties.

The Infinite Incompleteness of the Mathematical

Sciences.

There is one privilege which a certain amount of knowledge should confer; it is that of becoming aware of the indefinite weakness of our powers compared with the tasks which they might undertake if stronger. To the poor savage who cannot count twenty, the arithmetical accomplishments of the ordinary schoolboy are miraculously great in comparison. The schoolboy cannot comprehend the almost infinitely greater powers of the student, who has acquired facility with algebraic processes. The student can but look with feelings of surprise and reverence at the powers of a Newton or Laplace. But the question at once suggests itself, Do the powers of the highest human intellect bear any moderate ratio to the things which are to be understood and calculated? How many further steps must we take in the rise of mental ability and the extension of mathematical method before we begin to exhaust the knowable?

I am inclined to find fault with mathematical writers because they often exult in what they can accomplish, but omit to point out that what they do is but an indefinitely, nay an infinitely, small part of what might be done. They exhibit a general inclination, with few exceptions, not to do so much as mention the existence of problems of an impracticable character. This may be excusable so far as the immediate practical result of their researches is in question, but the custom has the effect of misleading the

general public into the fallacious notion that mathematics is a perfect science, which accomplishes what it undertakes in a complete manner. On the contrary, it may be said that if a mathematical problem were selected by pure chance out of the whole variety which might be proposed, the probability is infinitely slight that a human mathe matician could solve it. Just as the numbers we can count or frame to the mind are literally nothing compared with the numbers which might exist, so the whole accomplishments of a Laplace or a Lagrange are, as it were, the little corner of the multiplication table, which has really an indefinite extent.

I have sufficiently pointed out that the rude character of all our observations prevents us from being aware of the existence of the greater number of effects and actions of nature. It must be added that, if we perceived them, we should usually be incapable of including them in our theories from want of mathematical power. Some persons may be surprised that though nearly two centuries have elapsed since the time of Newton's discoveries, we have yet no general theory of molecular action. Some approximations have been made towards such a theory. Joule and Clausius have measured the velocity of gaseous atoms, or even determined the distance between the collision of atom and atom. Sir W. Thomson has approximated to the number of atoms in a given bulk of substance. Rankine has formed some reasonable hypotheses as to the actual constitution of atoms, but it would be a mistake to suppose that these ingenious results of theory and experiment form any appreciable approach to a complete solution of molecular motions. There is every reason to believe, judging from the spectra of the elements, and from other reasons, that even chemical atoms are very complicated structures. An atom of pure iron is probably a vastly more complicated system than that of the planets

and their satellites. A compound atom may perhaps be compared with a stellar system, each star a minor system in itself. The smallest particle of solid substance will consist of a vast number of such stellar systems united in regular order, each bounded by the other, communicating with it in some manner yet wholly imcomprehensible. Now what are our mathematical powers in comparison with this problem?

After two centuries of continuous labour, the most gifted men have succeeded in calculating the mutual effects of three bodies each upon the other, under the simple hypothesis of the law of gravity. Concerning these calculations we must further remember that they are purely approximate, and that the methods would not apply where four or more bodies are acting, and all produce considerable effects each upon the other. There is every reason to believe that each constituent of a chemical atom must go through an orbit in the millionth part of the twinkling of an eye, in which it successively or simultaneously is under the influence of many other constituents, or possibly comes into collision with them. It is, I apprehend, no exaggeration to say that mathematicians have scarcely a notion of the way in which they could successfully attack so difficult a problem of forces and motions. Each of these particles is for ever solving differential equations, which, if written out in full, might perhaps belt the earth, as Sir J. Herschel has beautifully remarked m.

Some of the most extensive calculations ever made, were those required for the reduction of the measurements executed in the course of the Trigonometrical Survey of Great Britain. The calculations arising out of the principal triangulation alone occupied twenty calculators during three or four years, in the course of which the m Familiar Lectures on Scientific Subjects,' p. 458.