logism; that the price of corn in England has never been so high since 1847 as it was in that year; that the price of the English funds has never been lower than it was on the 23rd of January, 1798, when it fell to 474.

It has been urged against this process of Perfect Induction that it gives no new information, and is merely a summing up in a brief form of a multitude of particulars. But mere abbreviation of mental labour is one of the most important aids we can enjoy in the acquisition of knowledge. The powers of the human mind are so limited that multiplicity of detail is alone sufficient to prevent its progress in many directions. Thought would be practically impossible if every separate fact had to be separately thought and treated. Economy of mental power may be considered one of the main conditions on which our elevated intellectual position depends. Most mathematical processes are but abbreviations of the simpler acts of addition and subtraction. The invention of logarithms was one of the most striking additions ever made to human power: yet it was a mere abbreviation of operations which could have been done before had a sufficient amount of labour been available. Similar additions to our power will, it is hoped, be made from time to time, for the number of mathematical problems hitherto solved is but an indefinitely small portion of those which await solution, because the labour they have hitherto demanded renders them impracticable. So it is really throughout all regions of thought. The amount of our knowledge depends upon our powers of bringing it within practicable compass. Unless we arrange and classify facts, and condense them into general truths, they soon surpass our powers of memory, and serve but to confuse. Hence Perfect Induction, even as a process of abbreviation, is absolutely essential to any high degree of mental achievement.

Transition from Perfect to Imperfect Induction.

It is a question of profound difficulty on what grounds we are warranted in inferring the future from the present, or the nature of undiscovered objects from those which we have examined with our senses. We pass from Perfect to Imperfect Induction when once we allow our conclusion to pass, at all events apparently, beyond the data on which it was founded. In making such a step we seem to gain a nett addition to our knowledge; for we learn the nature of what was unknown. We reap where we have never sown. We appear to possess the divine power of creating knowledge, and reaching with our mental arms far beyond the sphere of our own observation. I shall, indeed, have to point out certain methods of reasoning in which we do pass altogether beyond the sphere of the senses, and acquire accurate knowledge which observation could never have given; but it is not imperfect induction that accomplishes such a task. Of imperfect induction itself, I venture to assert that it never makes any real addition to our knowledge, in the meaning of the expression sometimes accepted. As in other cases of inference it merely unfolds the information contained in past observations or events; it merely renders explicit what was implicit in previous experience. It transmutes knowledge, but certainly does not create knowledge.

There is no fact which I shall more constantly keep before the reader's mind in the following pages than that the results of imperfect induction, however well authenticated and verified, are never more than probable. We never can be sure that the future will be as the present. We hang ever upon the Will of the Creator: and it is only so far as He has created two things alike, or maintains the framework of the world unchanged from moment to

moment, that our most careful inferences can be fulfilled. All predictions, all inferences which reach beyond their data, are purely hypothetical, and proceed on the assumption that new events will conform to the conditions detected in our observation of past events. No experience of finite duration can be expected to give an exhaustive knowledge of all the forces which are in operation. There is thus a double uncertainty; even supposing the Universe as a whole to proceed unchanged, we do not really know the Universe as a whole. Comparatively speaking we know only a point in its infinite extent, and a moment in its infinite duration. We cannot be sure, then, that our observations have not escaped some fact, which will cause the future to be apparently different from the past; nor can we be sure that the future really will be the outcome of the past. We proceed then in all our inferences to unexamined objects and times on the assumptions—

1. That our past observation gives us a complete knowledge of what exists.

2. That the conditions of things which did exist will continue to be the conditions of things which will exist.

We shall often need to illustrate the character of our knowledge of nature by the simile of a ballot-box, so often employed by mathematical writers in the theory of probability. Nature is to us like an infinite ballot-box, the contents of which are being continually drawn, ball after ball, and exhibited to us. Science is but the careful observation of the succession in which balls of various character usually present themselves; we register the combinations, notice those which seem to be excluded from occurrence, and from the proportional frequency of those which usually appear we infer the probable character of future drawings. But under such circumstances certainty of prediction depends on two conditions:

1. That we acquire a perfect knowledge of the comparative numbers of balls of each kind within the box. 2. That the contents of the ballot-box remain unchanged. Of the latter assumption, or rather that concerning the constitution of the world which it illustrates, the logician or physicist can have nothing to say. As the Creation of the Universe is necessarily an act passing all experience and all conception, so any change in that Creation, or, it may be, a termination of it, must likewise be infinitely beyond the bounds of our mental faculties. No science, no reasoning upon the subject, can have any validity; for without experience we are without the basis and materials of knowledge. It is the fundamental postulate accordingly of all inference concerning the future, that there shall be no arbitrary change in the subject of inference; of the probability or improbability of such a change I conceive that our faculties can give no estimate.

The other condition of inductive inference-that we aequire an approximately complete knowledge of the combinations in which events do occur, is at least in some degree within the bounds of our perceptive and mental powers. There are many branches of science in which phenomena seem to be governed by conditions of a most fixed and general character. We have much ground in such cases for believing that the future occurrence of such phenomena may be calculated and predicted. But the whole question now becomes one of probability and improbability. We leave the region of pure logic to enter one in which the number of events is the ground of inference. We do not leave the region of logic; we only leave that where certainty, affirmative or negative, is the result, and the agreement or disagreement of qualities the means of inference. For the future, number and quantity will enter into most of our processes of reasoning; but then I hold that number and quantity are but portions of the

great logical domain. I venture to assert that number is wholly logical, both in its fundamental nature and in all its complex developments. Quantity in all its forms is but a development of number. That which is mathematical is not the less logical; if anything it is the more logical, in the sense that it presents logical results in the highest degree of complexity and variety.

Before proceeding then from Perfect to Imperfect Induction I break off in some degree the course of the work, to treat of the logical conditions of number. I shall then employ number to estimate the variety of combinations in which natural phenomena may present themselves, and the probability or improbability of their occurrence under definite circumstances. It is in later parts of the work that I must endeavour to establish, in a complete manner, the notions which I have set forth upon the subject of Imperfect Induction, as applied in the investigation of Nature, which notions may be thus briefly stated:

1. Imperfect Induction entirely rests upon Perfect Induction for its materials.

2. The logical process by which we seem to pass directly from examined to unexamined cases consists in an inverse and complex application of deductive inference, so that all reasoning may be said to be either directly or inversely deductive.

3. The result is always of a hypothetical character, and is never more than probable.

4. No nett addition is ever made to our knowledge by

reasoning; what we know of future events or unexamined objects is only the unfolded contents of our previous knowledge, and it becomes less and less probable as it is more boldly extended to remote cases.