The science of crystallography is that perhaps in which the most precise and general laws have been detected, but it would be 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 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 generalisation, for it establishes a general resemblance between the forms of crystallisation 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 readily be shown that in whatever direction we extend our investigations and successfully harmonise 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 come to some recorded phenomenon of which no 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.
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 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 schoolboy are miraculously great. The schoolboy cannot comprehend the vastly greater powers of the student, who has acquired facility in algebraic processes. The student can but look with feelings of surprise and reverence at the powers of a Newton or a Laplace. But the question at once suggests itself, Do the powers of the highest human intellect bear a finite 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 methods 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, and omit to point out that what they do is but 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 as 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 chance out of the whole number which might be proposed, the probability is infinitely slight that a human mathematician could solve it. Just as the numbers we can count are nothing compared with the numbers which might exist, so the accomplishments of a Laplace or a Lagrange are, as it were, the little corner of the multiplication-table, which has really an infinite extent.
I have pointed out that the rude character of our observations prevents us from being aware of the greater number of effects and actions in nature. It must be added that, if we perceive 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