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senses must soon stop, whereas the mental powers of deductive reasoning can proceed to an unlimited degree of approximation. Geometrical truths, then, are incapable of verification; and, if so, they cannot even be learnt by observation. How can I have learnt by observation a proposition of which I cannot even prove the truth by observation, when I am in possession of it? All that observation or empirical trial can do is to suggest propositions, of which the truth may afterwards be proved deductively.

If Viviani's story is to be believed, Galileo endeavoured to satisfy himself about the area of the cycloid by cutting out several large cycloids in pasteboard, and then comparing the areas of the curve and the generating circle by weighing them. In every trial the curve seemed to be rather less than three times the circle, so that Galileo, we are told, began to suspect that the ratio was not precisely 3 to I. It is quite clear, however, that no process of weighing or measuring could ever prove truths like these, and it remained for Torricelli to show what his master Galileo had only guessed at.1

Much has been said about the peculiar certainty of mathematical reasoning, but it is only certainty of deductive reasoning, and equal certainty attaches to all correct logical deduction. If a triangle be right-angled, the square on the hypothenuse will undoubtedly equal the sum of the two squares on the other sides; but I can never be sure that a triangle is right-angled: so I can be certain that nitric acid will not dissolve gold, provided I know that the substances employed really correspond to those on which I tried the experiment previously. Here is like certainty of inference, and like doubt as to the facts.

Discrimination of Certainty and Probability.

We can never recur too often to the truth that our knowledge of the laws and future events of the external world is only probable. The mind itself is quite capable of possessing certain knowledge, and it is well to discriminate carefully between what we can and cannot know

1 Life of Galileo, Society for the Diffusion of Useful Knowledge,

p. 102.

with certainty. In the first place, whatever feeling is actually present to the mind is certainly known to that mind. If I see blue sky, I may be quite sure that I do experience the sensation of blueness. Whatever I do feel, I do feel beyond all doubt. We are indeed very likely to confuse what we really feel with what we are inclined to associate with it, or infer inductively from it; but the whole of our consciousness, as far as it is the result of pure intuition and free from inference, is certain knowledge beyond all doubt.

In the second place, we may have certainty of inference; the fundamental laws of thought, and the rule of substitution. (p. 9), are certainly true; and if my senses could inform me that A was indistinguishable in colour from B, and B from C, then I should be equally certain that A was indistinguishable from C. In short, whatever truth there is in the premises, I can certainly embody in their correct logical result. But the certainty generally assumes a hypothetical character. I never can be quite sure that two colours are exactly alike, that two magnitudes are exactly equal, or that two bodies whatsoever are identical even in their apparent qualities. Almost all our judgments involve quantitative relations, and, as will be shown in succeeding. chapters, we can never attain exactness and certainty where continuous quantity enters. Judgments concerning discontinuous quantity or numbers, however, allow of certainty; 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 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.1

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 potassium.2 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 known 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

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

2 Roscoe's Spectrum Analysis, 1st edit., p. 98.

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.1 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 uncertainty enters, because we cannot be sure that the stones of the Magdeburg Church resemble other stones in all their properties.

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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. entertained no appreciable doubt that the acid existing in sea salt also contained oxygen; 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 ever will 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 mistake. We may be quite certain that a comet will go on moving in a similar path if all circumstances remain the same before; but if we leave out this extensive qualification, our predictions will always be subject to the chance of falsification by some unexpected event, such as the division of Biela's comet or the interference of an unknown gravitating body.

as

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

2 Lavoisier's Chemistry, translated by Kerr. 3rd ed., pp. 114,

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Inductive inference might attain to certainty if our 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 has been made.

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