discontinuous quantity or numbers, however, allow of certainty; for I may establish beyond doubt, for instance, that the difference of the squares of 17 and 13 is the product of 17+ 13 and 17-13, and is therefore 30 × 4, or 120.

Inferences which we draw concerning natural objects are never certain except in a hypothetical point of view. It might seem, indeed, to be certain that iron is magnetic, or that gold is incapable of solution in nitric acid; but, if we carefully investigate the meanings of these statements, they will be found to involve no certainty but that of consciousness and that of hypothetical inference. For what do I mean by iron or gold? If I choose a remarkable piece of yellow substance, call it gold, and then immerse it in a liquid which I call nitric acid, and find that there is no change called solution, then consciousness has certainly informed me that with my meaning of the terms, Gold is insoluble in nitric acid.' I may further be certain of something else; for if this gold and nitric acid remain what they were, I may be sure there will be no solution on again trying the experiment. If I take other portions of gold and nitric acid, and am sure that they really are identical in properties with the former portions, I can be certain that there will be no solution. But at this point my knowledge becomes purely hypothetical; for how can I be sure without trial that the gold and acid are really identical in nature with what I formerly called gold and nitric acid. How do I know gold when I see it? If I judge by the apparent qualities-colour, ductility, specific gravity, &c., I may be misled, because there may always exist a substance which to the colour, ductility, specific gravity, and other specified qualities, joins others which we do not expect. Similarly, if iron is magnetic, as shown by an experiment with objects answering to those names, then all iron is magnetic, meaning all pieces of matter identical

with my assumed piece. But in trying to identify iron, I am always open to mistake. Nor is this liability to mistake a matter of speculation only V.

The history of chemistry shows that the most confident inferences may have been falsified by the confusion of one substance with another. Thus strontia was never discriminated from baryta until Klaproth and Haüy detected differences between some of their properties. Accordingly chemists must often have inferred concerning strontia what was only true of baryta, and vice versa. There is now no doubt that the recently discovered substances, cæsium and rubidium were long mistaken for potassiumy. Other elements have often been confused together, for instance, tantalum and niobium; sulphur and selenium; cerium, lanthanum, and didymium; yttrium and erbium.

Even the best-established laws of physical science do not exclude false inference. No law of nature has been better established than that of universal gravitation, and we believe with the utmost confidence that any body capable of affecting the senses will attract other bodies, and fall to the earth if not prevented. Euler remarks that, although he had never made trial of the stones. which compose the church of Magdeburg, yet he had not the least doubt that all of them were heavy, and would fall if unsupported. But he adds, that it would be extremely difficult to give any satisfactory explanation of this confident belief. The fact is, that the belief ought not to amount to certainty until the experiment has been tried, and in the meantime a slight amount of uncer

▾ Professor Bowen has excellently stated this view. Treatise on Logic.' Cambridge, U.S.A., 1866. P. 354.

x Whewell's History of the Inductive Sciences,' vol. iii. p. 174.

y Roscoe's 'Spectrum Analysis,' 1st edit. p. 99.

z Euler's Letters to a German Princess,' translated by Hunter. 2nd ed. vol. ii. pp. 17-18.

T

tainty enters, because we cannot be sure that the stones of the Magdeburg church resemble other stones in all their properties.

In like manner, not one of the inductive truths which men have established, or think they have established, is really safe from exception or reversal. Lavoisier, when laying the foundations of chemistry, met with so many instances tending to show the existence of oxygen in all acids, that he adopted a general conclusion to that effect, and devised the name oxygen accordingly. He entertained no appreciable doubt that the acid existing in sea salt also contained oxygena; yet subsequent experience falsified his expectations.

This instance refers to a science in its infancy, speaking relatively to the possible achievements of men. But all sciences are and will ever remain in their infancy, relatively to the extent and complexity of the universe which they undertake to investigate. Euler expresses no more than the truth when he says that it would be impossible to fix on any one thing really existing, of which we could have so perfect a knowledge as to put us beyond the reach of mistakeb.

Like remarks may be made concerning all other inductive inferences. We may be quite certain that a comet will go on moving in a similar path if all circumstances remain the same as before; but if we leave out this extensive qualification, our predictions will always be subject to the chance of falsification by some wholly unexpected event, such as the division of Biela's comet, or the unforeseen interference of some planetary or other gravitating body.

Inductive inference might attain to certainty if our

a Lavoisier's Chemistry,' translated by Kerr. 3rd edit. pp. 114, 121, b Euler's Letters,' vol. ii. p. 21.

123.

knowledge of the agents existing throughout the universe were complete, and if we were at the same time certain that the same Power which created the universe would allow it to proceed without arbitrary change. There is always a possibility of causes being in existence without our knowledge, and these may at any moment produce an unexpected effect. Even when by the theory of probabilities we succeed in forming some notion of the comparative confidence with which we should receive inductive results, it yet appears to me that we must make an assumption. Events come out like balls from the vast ballot-box of nature, and close observation will enable us to form some notion, as we shall see in the next chapter, of the contents of that ballot-box. But we must still assume that between the time of an observation and that to which our inferences relate, no change in the ballot-box shall have been made.

CHAPTER XII.

THE INDUCTIVE OR INVERSE APPLICATION OF THE

THEORY OF PROBABILITIES.

We have hitherto considered the theory of probability only in its simple deductive employment, by which it enables us to determine from given conditions the probable character of events happening under those conditions. But as deductive reasoning when inversely applied constitutes the process of induction, so the calculation of probabilities may be inversely applied; from the known character of certain events we may argue backwards to the probability of a certain law or condition governing those events. Having satisfactorily accomplished this work, we may indeed calculate forwards to the probable character of future events happening under the same conditions; but this part of the process is a direct use of deductive reasoning (p. 260).

Now it is highly instructive to find that whether the theory of probabilities be deductively or inductively applied, the calculation is always performed according to the principles and rules of deduction. The probability that an event has a particular condition entirely depends upon the probability that if the condition existed the event would follow. If we take up a pack of common playing cards, and observe that they are arranged in perfect numerical order, we conclude beyond all reasonable doubt that they have been thus intentionally arranged